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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.01536v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S1" title="In Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S1.SS0.SSS0.Px1" title="In 1 Introduction ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Contributions.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S1.SS0.SSS0.Px2" title="In 1 Introduction ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Theoretical analysis.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S1.SS0.SSS0.Px3" title="In 1 Introduction ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Practical consequences.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S1.SS0.SSS0.Px4" title="In 1 Introduction ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Related Work.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S1.SS0.SSS0.Px5" title="In 1 Introduction ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Content.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2" title="In Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Background</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.SS0.SSS0.Px1" title="In 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Notation.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.SS0.SSS0.Px2" title="In 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Syntax.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.SS0.SSS0.Px3" title="In 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Semantics.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.SS0.SSS0.Px4" title="In 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Partial assignments.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.SS0.SSS0.Px5" title="In 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Existentially-quantified formulas.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.SS0.SSS0.Px6" title="In 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">CNF-ization.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3" title="In Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>A theoretical Analysis of verificationand entailment</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS1" title="In 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Verification and entailment of plain formulas.</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS1.SSS0.Px1" title="In 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Definitions.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS1.SSS0.Px2" title="In 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Verification vs. Entailment.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS2" title="In 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Other candidate forms of partial-assignment satisfaction.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS3" title="In 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Verification and entailment of existentially-quantified formulas.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS4" title="In 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>Verification and entailment of CNF-ized non-CNF formulas.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4" title="In Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Practical consequences of using verification or entailment</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS1" title="In 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Verification vs. entailment in solving, enumeration and compilation</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS1.SSS0.Px1" title="In 4.1 Verification vs. entailment in solving, enumeration and compilation ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Verification vs. entailment for native CNF formulas.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS1.SSS0.Px2" title="In 4.1 Verification vs. entailment in solving, enumeration and compilation ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Verification vs. entailment in satisfiability.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS1.SSS0.Px3" title="In 4.1 Verification vs. entailment in solving, enumeration and compilation ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Verification vs. entailment for enumeration on non-CNF formulas.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS1.SSS0.Px4" title="In 4.1 Verification vs. entailment in solving, enumeration and compilation ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Verification vs. entailment for projected enumeration.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS1.SSS0.Px5" title="In 4.1 Verification vs. entailment in solving, enumeration and compilation ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Verification vs. entailment for CNF-ized non-CNF formulas.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS1.SSS0.Px6" title="In 4.1 Verification vs. entailment in solving, enumeration and compilation ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Verification and entailment in formula compilation.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS1.SSS0.Px7" title="In 4.1 Verification vs. entailment in solving, enumeration and compilation ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title">Verification and entailment for #SMT and WMI.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS2" title="In 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Verification vs. entailment in search procedures and formula compilers.</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.SS3" title="In 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Implementing entailment within enumeration procedures</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S5" title="In Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Conclusions and Future Work</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"><span class="ltx_note ltx_role_institutetext" id="id1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_note_type">institutetext: </span>DISI, University of Trento, Italy</span></span></span> <h1 class="ltx_title ltx_title_document">Entailment vs. Verification <br class="ltx_break"/>for Partial-assignment Satisfiability and Enumeration </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Roberto Sebastiani </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">Many procedures for SAT-related problems, in particular for those requiring the complete enumeration of satisfying truth assignments, rely their efficiency and effectiveness on the detection of (possibly small) <span class="ltx_text ltx_font_italic" id="id1.id1.1">partial</span> assignments satisfying an input formula. Surprisingly, there seems to be no unique universally-agreed definition of formula satisfaction by a partial assignment in the literature.</p> <p class="ltx_p" id="id2.id2">In this paper we analyze in deep the issue of satisfaction by partial assignments, raising a flag about some ambiguities and subtleties of this concept, and investigating their practical consequences. We identify two alternative notions that are implicitly used in the literature, namely <span class="ltx_text ltx_font_italic" id="id2.id2.1">verification</span> and <span class="ltx_text ltx_font_italic" id="id2.id2.2">entailment</span>, which coincide if applied to CNF formulas but differ and present complementary properties if applied to non-CNF or to existentially-quantified formulas. We show that, although the former is easier to check and as such is implicitly used by most current search procedures, the latter has better theoretical properties, and can improve the efficiency and effectiveness of enumeration procedures.</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.5"><em class="ltx_emph ltx_font_italic" id="S1.p1.5.1">Motivations.</em> Many search procedures for SAT-related problems (e.g. Analytic Tableaux <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib38" title="">38</a>]</cite>, DPLL <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib14" title="">14</a>]</cite>, circuit AllSAT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>]</cite>) and many formula compilers (e.g., d-DNNFs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib13" title="">13</a>]</cite>, OBDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib4" title="">4</a>]</cite> and SDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib12" title="">12</a>]</cite>) rely their efficiency and effectiveness on the detection of <span class="ltx_text ltx_font_italic" id="S1.p1.5.2">partial</span> truth assignments <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_μ</annotation></semantics></math> satisfying an input propositional formula <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.p1.2.m2.1"><semantics id="S1.p1.2.m2.1a"><mi id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.2.m2.1b"><ci id="S1.p1.2.m2.1.1.cmml" xref="S1.p1.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_φ</annotation></semantics></math>, which allows to state that not only <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mi id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><ci id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_φ</annotation></semantics></math> is satisfiable, but also all total assignments extending <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p1.4.m4.1"><semantics id="S1.p1.4.m4.1a"><mi id="S1.p1.4.m4.1.1" xref="S1.p1.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.4.m4.1b"><ci id="S1.p1.4.m4.1.1.cmml" xref="S1.p1.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">italic_μ</annotation></semantics></math> satisfy <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.p1.5.m5.1"><semantics id="S1.p1.5.m5.1a"><mi id="S1.p1.5.m5.1.1" xref="S1.p1.5.m5.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.5.m5.1b"><ci id="S1.p1.5.m5.1.1.cmml" xref="S1.p1.5.m5.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m5.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.1d">italic_φ</annotation></semantics></math>. In particular, when it comes to SAT-based problems requiring the <span class="ltx_text ltx_font_italic" id="S1.p1.5.3">complete enumeration</span> of satisfying assignments (e.g. #SAT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib20" title="">20</a>]</cite>, Lazy SMT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib3" title="">3</a>]</cite>, OMT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib37" title="">37</a>]</cite>, AllSAT and AllSMT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>]</cite>, #SMT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib31" title="">31</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib7" title="">7</a>]</cite>, Projected #SAT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib2" title="">2</a>]</cite>, Projected AllSAT/AllSMT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib40" title="">40</a>]</cite>, knowledge compilation into d-DNNF <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib23" title="">23</a>]</cite>, satisfiability in modal and description logics <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib36" title="">36</a>]</cite>, Weighted Model Integration <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib39" title="">39</a>]</cite>), the ability of enumerating satisfying partial assignments which are as small as possible is essential, because each of them avoids the enumeration of the whole subtree of total assignments extending it, whose size is exponential in the number of unassigned atoms.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.2">We start by raising a very-basic question: <em class="ltx_emph ltx_font_italic" id="S1.p2.2.2">What should we mean by “a partial assignment <math alttext="\mu" class="ltx_Math" display="inline" id="S1.p2.1.1.m1.1"><semantics id="S1.p2.1.1.m1.1a"><mi id="S1.p2.1.1.m1.1.1" xref="S1.p2.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.1.m1.1b"><ci id="S1.p2.1.1.m1.1.1.cmml" xref="S1.p2.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.1.m1.1d">italic_μ</annotation></semantics></math> satisfies a formula <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.p2.2.2.m2.1"><semantics id="S1.p2.2.2.m2.1a"><mi id="S1.p2.2.2.m2.1.1" xref="S1.p2.2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.2.m2.1b"><ci id="S1.p2.2.2.m2.1.1.cmml" xref="S1.p2.2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.2.m2.1d">italic_φ</annotation></semantics></math>”?</em> We notice that, quite surprisingly and despite its widespread implicit usage in algorithms, there is no unique and universally-agreed definition for formula satisfaction by partial assignments in the literature: most authors do not define it explicitly; a few others define it only when dealing with CNF formulas (e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib22" title="">22</a>]</cite>); the very few who define it, adopt one of the following distinct definitions:</p> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">“<em class="ltx_emph ltx_font_italic" id="S1.I1.ix1.3.3.3">applying <math alttext="\mu" class="ltx_Math" display="inline" id="S1.I1.ix1.1.1.1.m1.1"><semantics id="S1.I1.ix1.1.1.1.m1.1b"><mi id="S1.I1.ix1.1.1.1.m1.1.1" xref="S1.I1.ix1.1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix1.1.1.1.m1.1c"><ci id="S1.I1.ix1.1.1.1.m1.1.1.cmml" xref="S1.I1.ix1.1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix1.1.1.1.m1.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix1.1.1.1.m1.1e">italic_μ</annotation></semantics></math> to <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.I1.ix1.2.2.2.m2.1"><semantics id="S1.I1.ix1.2.2.2.m2.1b"><mi id="S1.I1.ix1.2.2.2.m2.1.1" xref="S1.I1.ix1.2.2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix1.2.2.2.m2.1c"><ci id="S1.I1.ix1.2.2.2.m2.1.1.cmml" xref="S1.I1.ix1.2.2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix1.2.2.2.m2.1d">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix1.2.2.2.m2.1e">italic_φ</annotation></semantics></math> makes <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.I1.ix1.3.3.3.m3.1"><semantics id="S1.I1.ix1.3.3.3.m3.1b"><mi id="S1.I1.ix1.3.3.3.m3.1.1" xref="S1.I1.ix1.3.3.3.m3.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix1.3.3.3.m3.1c"><ci id="S1.I1.ix1.3.3.3.m3.1.1.cmml" xref="S1.I1.ix1.3.3.3.m3.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix1.3.3.3.m3.1d">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix1.3.3.3.m3.1e">italic_φ</annotation></semantics></math> true</em>”</span> <div class="ltx_para" id="S1.I1.ix1.p1"> <p class="ltx_p" id="S1.I1.ix1.p1.2">(e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>]</cite>). We say that <em class="ltx_emph ltx_font_italic" id="S1.I1.ix1.p1.2.2"><math alttext="\mu" class="ltx_Math" display="inline" id="S1.I1.ix1.p1.1.1.m1.1"><semantics id="S1.I1.ix1.p1.1.1.m1.1a"><mi id="S1.I1.ix1.p1.1.1.m1.1.1" xref="S1.I1.ix1.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix1.p1.1.1.m1.1b"><ci id="S1.I1.ix1.p1.1.1.m1.1.1.cmml" xref="S1.I1.ix1.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix1.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix1.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> verifies <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.I1.ix1.p1.2.2.m2.1"><semantics id="S1.I1.ix1.p1.2.2.m2.1a"><mi id="S1.I1.ix1.p1.2.2.m2.1.1" xref="S1.I1.ix1.p1.2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix1.p1.2.2.m2.1b"><ci id="S1.I1.ix1.p1.2.2.m2.1.1.cmml" xref="S1.I1.ix1.p1.2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix1.p1.2.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix1.p1.2.2.m2.1d">italic_φ</annotation></semantics></math></em>; <span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>The etymology of “to verify” derives from Latin “<span class="ltx_text ltx_font_italic" id="footnote1.1">verum facere</span>”, meaning “to make true”.</span></span></span> </p> </div> </li> <li class="ltx_item" id="S1.I1.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">“<em class="ltx_emph ltx_font_italic" id="S1.I1.ix2.2.2.2">all total assignments extending <math alttext="\mu" class="ltx_Math" display="inline" id="S1.I1.ix2.1.1.1.m1.1"><semantics id="S1.I1.ix2.1.1.1.m1.1b"><mi id="S1.I1.ix2.1.1.1.m1.1.1" xref="S1.I1.ix2.1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix2.1.1.1.m1.1c"><ci id="S1.I1.ix2.1.1.1.m1.1.1.cmml" xref="S1.I1.ix2.1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix2.1.1.1.m1.1d">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix2.1.1.1.m1.1e">italic_μ</annotation></semantics></math> satisfy <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.I1.ix2.2.2.2.m2.1"><semantics id="S1.I1.ix2.2.2.2.m2.1b"><mi id="S1.I1.ix2.2.2.2.m2.1.1" xref="S1.I1.ix2.2.2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix2.2.2.2.m2.1c"><ci id="S1.I1.ix2.2.2.2.m2.1.1.cmml" xref="S1.I1.ix2.2.2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix2.2.2.2.m2.1d">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix2.2.2.2.m2.1e">italic_φ</annotation></semantics></math></em>”</span> <div class="ltx_para" id="S1.I1.ix2.p1"> <p class="ltx_p" id="S1.I1.ix2.p1.2">(e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib34" title="">34</a>]</cite>). We say that <em class="ltx_emph ltx_font_italic" id="S1.I1.ix2.p1.2.2"><math alttext="\mu" class="ltx_Math" display="inline" id="S1.I1.ix2.p1.1.1.m1.1"><semantics id="S1.I1.ix2.p1.1.1.m1.1a"><mi id="S1.I1.ix2.p1.1.1.m1.1.1" xref="S1.I1.ix2.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix2.p1.1.1.m1.1b"><ci id="S1.I1.ix2.p1.1.1.m1.1.1.cmml" xref="S1.I1.ix2.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix2.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix2.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> entails <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.I1.ix2.p1.2.2.m2.1"><semantics id="S1.I1.ix2.p1.2.2.m2.1a"><mi id="S1.I1.ix2.p1.2.2.m2.1.1" xref="S1.I1.ix2.p1.2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.I1.ix2.p1.2.2.m2.1b"><ci id="S1.I1.ix2.p1.2.2.m2.1.1.cmml" xref="S1.I1.ix2.p1.2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.ix2.p1.2.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.I1.ix2.p1.2.2.m2.1d">italic_φ</annotation></semantics></math></em>.</p> </div> </li> </ul> <p class="ltx_p" id="S1.p2.3">Notice that this is not simply an issue of the meaning of the word “satisfy”: regardless which verb we may use for it (e.g. “satisfy”, “entail”, “verify”, “imply”,…), it should be desirable to have a unique and universally-agreed criterion to establish it. </p> </div> <section class="ltx_paragraph" id="S1.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Contributions.</h4> <div class="ltx_para" id="S1.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S1.SS0.SSS0.Px1.p1.1">In this paper we analyze in deep the notion of <span class="ltx_text ltx_font_italic" id="S1.SS0.SSS0.Px1.p1.1.1">partial-assignment satisfiability</span>, in particular when dealing with non-CNF and existentially-quantified formulas, raising a flag about the ambiguities and subtleties of this concept, and investigating their consequences. Our contributions are both theoretical and practical.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Theoretical analysis.</h4> <div class="ltx_para" id="S1.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S1.SS0.SSS0.Px2.p1.1">We show, analyze and discuss the following theoretical facts. </p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px2.p2"> <ul class="ltx_itemize" id="S1.SS0.SSS0.Px2.p2.4"> <li class="ltx_item" id="S1.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S1.I2.i1.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S1.I2.i1.1.1.1.1">a</span>)</span></span> <div class="ltx_para" id="S1.I2.i1.p1"> <p class="ltx_p" id="S1.I2.i1.p1.1">Whereas for (tautology-free<span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>A CNF formula is tautology-free if it contains no tautological clause in the form <math alttext="(l\vee\neg l\vee...)" class="ltx_Math" display="inline" id="footnote2.m1.1"><semantics id="footnote2.m1.1b"><mrow id="footnote2.m1.1.1.1" xref="footnote2.m1.1.1.1.1.cmml"><mo id="footnote2.m1.1.1.1.2" stretchy="false" xref="footnote2.m1.1.1.1.1.cmml">(</mo><mrow id="footnote2.m1.1.1.1.1" xref="footnote2.m1.1.1.1.1.cmml"><mi id="footnote2.m1.1.1.1.1.2" xref="footnote2.m1.1.1.1.1.2.cmml">l</mi><mo id="footnote2.m1.1.1.1.1.1" xref="footnote2.m1.1.1.1.1.1.cmml">∨</mo><mrow id="footnote2.m1.1.1.1.1.3" xref="footnote2.m1.1.1.1.1.3.cmml"><mo id="footnote2.m1.1.1.1.1.3.1" rspace="0.167em" xref="footnote2.m1.1.1.1.1.3.1.cmml">¬</mo><mi id="footnote2.m1.1.1.1.1.3.2" xref="footnote2.m1.1.1.1.1.3.2.cmml">l</mi></mrow><mo id="footnote2.m1.1.1.1.1.1b" xref="footnote2.m1.1.1.1.1.1.cmml">∨</mo><mi id="footnote2.m1.1.1.1.1.4" mathvariant="normal" xref="footnote2.m1.1.1.1.1.4.cmml">…</mi></mrow><mo id="footnote2.m1.1.1.1.3" stretchy="false" xref="footnote2.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><apply id="footnote2.m1.1.1.1.1.cmml" xref="footnote2.m1.1.1.1"><or id="footnote2.m1.1.1.1.1.1.cmml" xref="footnote2.m1.1.1.1.1.1"></or><ci id="footnote2.m1.1.1.1.1.2.cmml" xref="footnote2.m1.1.1.1.1.2">𝑙</ci><apply id="footnote2.m1.1.1.1.1.3.cmml" xref="footnote2.m1.1.1.1.1.3"><not id="footnote2.m1.1.1.1.1.3.1.cmml" xref="footnote2.m1.1.1.1.1.3.1"></not><ci id="footnote2.m1.1.1.1.1.3.2.cmml" xref="footnote2.m1.1.1.1.1.3.2">𝑙</ci></apply><ci id="footnote2.m1.1.1.1.1.4.cmml" xref="footnote2.m1.1.1.1.1.4">…</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">(l\vee\neg l\vee...)</annotation><annotation encoding="application/x-llamapun" id="footnote2.m1.1e">( italic_l ∨ ¬ italic_l ∨ … )</annotation></semantics></math>. Since tautological clause can be easily removed upfront by simple preprocessing, hereafter we often assume w.l.o.g. that CNF formulas are tautology-free.</span></span></span>) CNF formulas partial-assignment satisfiability can be indifferently be interpreted either as verification or as entailment because in this case the two concepts coincide, <span class="ltx_text ltx_font_italic" id="S1.I2.i1.p1.1.1">for non-CNF formulas verification is strictly stronger than entailment</span>, and they have complementary properties.</p> </div> </li> <li class="ltx_item" id="S1.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S1.I2.i2.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S1.I2.i2.1.1.1.1">b</span>)</span></span> <div class="ltx_para" id="S1.I2.i2.p1"> <p class="ltx_p" id="S1.I2.i2.p1.1">Whereas two equivalent formulas are always entailed by the same partial assignments, <span class="ltx_text ltx_font_italic" id="S1.I2.i2.p1.1.1">they are not always verified by the same partial assignments</span>.</p> </div> </li> <li class="ltx_item" id="S1.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S1.I2.i3.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S1.I2.i3.1.1.1.1">c</span>)</span></span> <div class="ltx_para" id="S1.I2.i3.p1"> <p class="ltx_p" id="S1.I2.i3.p1.1">Verification checks that the residual of a formula under a partial assignment is the true formula, whereas entailment checks it is a valid formula. Thus verification can be computationally much cheaper to check than entailment.</p> </div> </li> <li class="ltx_item" id="S1.I2.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S1.I2.i4.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S1.I2.i4.1.1.1.1">d</span>)</span></span> <div class="ltx_para" id="S1.I2.i4.p1"> <p class="ltx_p" id="S1.I2.i4.p1.2"><math alttext="(a)" class="ltx_Math" display="inline" id="S1.I2.i4.p1.1.m1.1"><semantics id="S1.I2.i4.p1.1.m1.1a"><mrow id="S1.I2.i4.p1.1.m1.1.2.2"><mo id="S1.I2.i4.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.I2.i4.p1.1.m1.1.1" xref="S1.I2.i4.p1.1.m1.1.1.cmml">a</mi><mo id="S1.I2.i4.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i4.p1.1.m1.1b"><ci id="S1.I2.i4.p1.1.m1.1.1.cmml" xref="S1.I2.i4.p1.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i4.p1.1.m1.1c">(a)</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i4.p1.1.m1.1d">( italic_a )</annotation></semantics></math>-<math alttext="(c)" class="ltx_Math" display="inline" id="S1.I2.i4.p1.2.m2.1"><semantics id="S1.I2.i4.p1.2.m2.1a"><mrow id="S1.I2.i4.p1.2.m2.1.2.2"><mo id="S1.I2.i4.p1.2.m2.1.2.2.1" stretchy="false">(</mo><mi id="S1.I2.i4.p1.2.m2.1.1" xref="S1.I2.i4.p1.2.m2.1.1.cmml">c</mi><mo id="S1.I2.i4.p1.2.m2.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i4.p1.2.m2.1b"><ci id="S1.I2.i4.p1.2.m2.1.1.cmml" xref="S1.I2.i4.p1.2.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i4.p1.2.m2.1c">(c)</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i4.p1.2.m2.1d">( italic_c )</annotation></semantics></math> apply also to existentially-quantified formulas, <em class="ltx_emph ltx_font_italic" id="S1.I2.i4.p1.2.1">even those in CNF</em>.</p> </div> </li> <li class="ltx_item" id="S1.I2.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S1.I2.i5.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S1.I2.i5.1.1.1.1">e</span>)</span></span> <div class="ltx_para" id="S1.I2.i5.p1"> <p class="ltx_p" id="S1.I2.i5.p1.2"><math alttext="(a)" class="ltx_Math" display="inline" id="S1.I2.i5.p1.1.m1.1"><semantics id="S1.I2.i5.p1.1.m1.1a"><mrow id="S1.I2.i5.p1.1.m1.1.2.2"><mo id="S1.I2.i5.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.I2.i5.p1.1.m1.1.1" xref="S1.I2.i5.p1.1.m1.1.1.cmml">a</mi><mo id="S1.I2.i5.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i5.p1.1.m1.1b"><ci id="S1.I2.i5.p1.1.m1.1.1.cmml" xref="S1.I2.i5.p1.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i5.p1.1.m1.1c">(a)</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i5.p1.1.m1.1d">( italic_a )</annotation></semantics></math>-<math alttext="(c)" class="ltx_Math" display="inline" id="S1.I2.i5.p1.2.m2.1"><semantics id="S1.I2.i5.p1.2.m2.1a"><mrow id="S1.I2.i5.p1.2.m2.1.2.2"><mo id="S1.I2.i5.p1.2.m2.1.2.2.1" stretchy="false">(</mo><mi id="S1.I2.i5.p1.2.m2.1.1" xref="S1.I2.i5.p1.2.m2.1.1.cmml">c</mi><mo id="S1.I2.i5.p1.2.m2.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i5.p1.2.m2.1b"><ci id="S1.I2.i5.p1.2.m2.1.1.cmml" xref="S1.I2.i5.p1.2.m2.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i5.p1.2.m2.1c">(c)</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i5.p1.2.m2.1d">( italic_c )</annotation></semantics></math> apply also to existentially-quantified formulas resulting from CNF-ization.</p> </div> </li> </ul> <p class="ltx_p" id="S1.SS0.SSS0.Px2.p2.3">We stress the fact that <math alttext="(b)" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px2.p2.1.m1.1"><semantics id="S1.SS0.SSS0.Px2.p2.1.m1.1a"><mrow id="S1.SS0.SSS0.Px2.p2.1.m1.1.2.2"><mo id="S1.SS0.SSS0.Px2.p2.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.SS0.SSS0.Px2.p2.1.m1.1.1" xref="S1.SS0.SSS0.Px2.p2.1.m1.1.1.cmml">b</mi><mo id="S1.SS0.SSS0.Px2.p2.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px2.p2.1.m1.1b"><ci id="S1.SS0.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S1.SS0.SSS0.Px2.p2.1.m1.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px2.p2.1.m1.1c">(b)</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px2.p2.1.m1.1d">( italic_b )</annotation></semantics></math> (and <math alttext="(d)" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px2.p2.2.m2.1"><semantics id="S1.SS0.SSS0.Px2.p2.2.m2.1a"><mrow id="S1.SS0.SSS0.Px2.p2.2.m2.1.2.2"><mo id="S1.SS0.SSS0.Px2.p2.2.m2.1.2.2.1" stretchy="false">(</mo><mi id="S1.SS0.SSS0.Px2.p2.2.m2.1.1" xref="S1.SS0.SSS0.Px2.p2.2.m2.1.1.cmml">d</mi><mo id="S1.SS0.SSS0.Px2.p2.2.m2.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px2.p2.2.m2.1b"><ci id="S1.SS0.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S1.SS0.SSS0.Px2.p2.2.m2.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px2.p2.2.m2.1c">(d)</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px2.p2.2.m2.1d">( italic_d )</annotation></semantics></math> and <math alttext="(e)" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px2.p2.3.m3.1"><semantics id="S1.SS0.SSS0.Px2.p2.3.m3.1a"><mrow id="S1.SS0.SSS0.Px2.p2.3.m3.1.2.2"><mo id="S1.SS0.SSS0.Px2.p2.3.m3.1.2.2.1" stretchy="false">(</mo><mi id="S1.SS0.SSS0.Px2.p2.3.m3.1.1" xref="S1.SS0.SSS0.Px2.p2.3.m3.1.1.cmml">e</mi><mo id="S1.SS0.SSS0.Px2.p2.3.m3.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px2.p2.3.m3.1b"><ci id="S1.SS0.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S1.SS0.SSS0.Px2.p2.3.m3.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px2.p2.3.m3.1c">(e)</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px2.p2.3.m3.1d">( italic_e )</annotation></semantics></math> consequently) would be an embarrassing fact if we adopted verification as the definition of partial-assignment satisfiability. As such, we champion the idea that the latter should be defined as entailment <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib36" title="">36</a>]</cite>, and that verification should be considered an easy-to-check sufficient condition for it.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS0.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Practical consequences.</h4> <div class="ltx_para" id="S1.SS0.SSS0.Px3.p1"> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p1.1">The above facts have the following practical consequences. </p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px3.p2"> <ol class="ltx_enumerate" id="S1.I3"> <li class="ltx_item" id="S1.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S1.I3.i1.p1"> <p class="ltx_p" id="S1.I3.i1.p1.4">In plain satisfiability we need finding only one total assignment <math alttext="\eta" class="ltx_Math" display="inline" id="S1.I3.i1.p1.1.m1.1"><semantics id="S1.I3.i1.p1.1.m1.1a"><mi id="S1.I3.i1.p1.1.m1.1.1" xref="S1.I3.i1.p1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S1.I3.i1.p1.1.m1.1b"><ci id="S1.I3.i1.p1.1.m1.1.1.cmml" xref="S1.I3.i1.p1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i1.p1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i1.p1.1.m1.1d">italic_η</annotation></semantics></math> extending <math alttext="\mu" class="ltx_Math" display="inline" id="S1.I3.i1.p1.2.m2.1"><semantics id="S1.I3.i1.p1.2.m2.1a"><mi id="S1.I3.i1.p1.2.m2.1.1" xref="S1.I3.i1.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.I3.i1.p1.2.m2.1b"><ci id="S1.I3.i1.p1.2.m2.1.1.cmml" xref="S1.I3.i1.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i1.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i1.p1.2.m2.1d">italic_μ</annotation></semantics></math>, so that entailment produces no benefits wrt. verification and is more expensive <math alttext="(c)" class="ltx_Math" display="inline" id="S1.I3.i1.p1.3.m3.1"><semantics id="S1.I3.i1.p1.3.m3.1a"><mrow id="S1.I3.i1.p1.3.m3.1.2.2"><mo id="S1.I3.i1.p1.3.m3.1.2.2.1" stretchy="false">(</mo><mi id="S1.I3.i1.p1.3.m3.1.1" xref="S1.I3.i1.p1.3.m3.1.1.cmml">c</mi><mo id="S1.I3.i1.p1.3.m3.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i1.p1.3.m3.1b"><ci id="S1.I3.i1.p1.3.m3.1.1.cmml" xref="S1.I3.i1.p1.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i1.p1.3.m3.1c">(c)</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i1.p1.3.m3.1d">( italic_c )</annotation></semantics></math>. Also, due to <math alttext="(a)" class="ltx_Math" display="inline" id="S1.I3.i1.p1.4.m4.1"><semantics id="S1.I3.i1.p1.4.m4.1a"><mrow id="S1.I3.i1.p1.4.m4.1.2.2"><mo id="S1.I3.i1.p1.4.m4.1.2.2.1" stretchy="false">(</mo><mi id="S1.I3.i1.p1.4.m4.1.1" xref="S1.I3.i1.p1.4.m4.1.1.cmml">a</mi><mo id="S1.I3.i1.p1.4.m4.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I3.i1.p1.4.m4.1b"><ci id="S1.I3.i1.p1.4.m4.1.1.cmml" xref="S1.I3.i1.p1.4.m4.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I3.i1.p1.4.m4.1c">(a)</annotation><annotation encoding="application/x-llamapun" id="S1.I3.i1.p1.4.m4.1d">( italic_a )</annotation></semantics></math>, when the input problem is natively in CNF the distinction between verification and entailment is not relevant, because these two concepts coincide.</p> </div> </li> </ol> </div> <div class="ltx_para ltx_noindent" id="S1.SS0.SSS0.Px3.p3"> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p3.1">Consequently, both the algorithms for satisfiability and these for enumeration with CNF formulas typically use verification as satisfiability criterion for the current partial assignments, because it is much cheaper and easier to implement than entailment. This is the case, e.g., of classical procedures like Analytic Tableaux <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib38" title="">38</a>]</cite> and DPLL <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib1" title="">1</a>]</cite>, or more-recent enumeration, counting or knowledge compilation procedures, e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib23" title="">23</a>]</cite>. A few notable exceptions are the <span class="ltx_text ltx_font_sansserif" id="S1.SS0.SSS0.Px3.p3.1.1">Dualiza</span> procedure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib27" title="">27</a>]</cite> and the procedures we described in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib30" title="">30</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib16" title="">16</a>]</cite>; also OBDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib4" title="">4</a>]</cite> and SDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib12" title="">12</a>]</cite> formula compilers implicitly use entailment to prune branches so that to guarantee canonicity (see below and §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4" title="4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">4</span></a>).</p> </div> <div class="ltx_para ltx_noindent" id="S1.SS0.SSS0.Px3.p4"> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p4.1">The scenario changes when we deal with enumeration-based algorithms applied to non-CNF formulas, or to existentially-quantified formulas, or to CNF-ized formulas. </p> <ol class="ltx_enumerate" id="S1.I4"> <li class="ltx_item" id="S1.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S1.I4.i2.p1"> <p class="ltx_p" id="S1.I4.i2.p1.1">Due to <math alttext="(a)" class="ltx_Math" display="inline" id="S1.I4.i2.p1.1.m1.1"><semantics id="S1.I4.i2.p1.1.m1.1a"><mrow id="S1.I4.i2.p1.1.m1.1.2.2"><mo id="S1.I4.i2.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.I4.i2.p1.1.m1.1.1" xref="S1.I4.i2.p1.1.m1.1.1.cmml">a</mi><mo id="S1.I4.i2.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I4.i2.p1.1.m1.1b"><ci id="S1.I4.i2.p1.1.m1.1.1.cmml" xref="S1.I4.i2.p1.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I4.i2.p1.1.m1.1c">(a)</annotation><annotation encoding="application/x-llamapun" id="S1.I4.i2.p1.1.m1.1d">( italic_a )</annotation></semantics></math>, a partial assignment may entail a formula without verifying it. Thus adopting entailment as partial-assignment satisfiability criterion during the search allows for detecting satisfiability earlier than with verification, and thus for producing smaller partial truth assignments.</p> </div> </li> </ol> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p4.2">Although this fact is not very interesting for plain satisfiability, it may become fundamental for enumeration, because the detection of a satisfying partial assignment avoids the enumeration of the whole subtree of total assignments extending it, whose size is exponential in the number of unassigned atoms: <em class="ltx_emph ltx_font_italic" id="S1.SS0.SSS0.Px3.p4.2.1">the earlier satisfiability is detected, the (up to exponentially) less assignments are enumerated</em>.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px3.p5"> <ol class="ltx_enumerate" id="S1.I5"> <li class="ltx_item" id="S1.I5.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S1.I5.i3.p1"> <p class="ltx_p" id="S1.I5.i3.p1.2">Due to <math alttext="(d)" class="ltx_Math" display="inline" id="S1.I5.i3.p1.1.m1.1"><semantics id="S1.I5.i3.p1.1.m1.1a"><mrow id="S1.I5.i3.p1.1.m1.1.2.2"><mo id="S1.I5.i3.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.I5.i3.p1.1.m1.1.1" xref="S1.I5.i3.p1.1.m1.1.1.cmml">d</mi><mo id="S1.I5.i3.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I5.i3.p1.1.m1.1b"><ci id="S1.I5.i3.p1.1.m1.1.1.cmml" xref="S1.I5.i3.p1.1.m1.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I5.i3.p1.1.m1.1c">(d)</annotation><annotation encoding="application/x-llamapun" id="S1.I5.i3.p1.1.m1.1d">( italic_d )</annotation></semantics></math>, a partial assignment may entail an existentially-quantified formula, even a CNF one, without verifying it. Thus, as with non-CNF formulas <math alttext="(b)" class="ltx_Math" display="inline" id="S1.I5.i3.p1.2.m2.1"><semantics id="S1.I5.i3.p1.2.m2.1a"><mrow id="S1.I5.i3.p1.2.m2.1.2.2"><mo id="S1.I5.i3.p1.2.m2.1.2.2.1" stretchy="false">(</mo><mi id="S1.I5.i3.p1.2.m2.1.1" xref="S1.I5.i3.p1.2.m2.1.1.cmml">b</mi><mo id="S1.I5.i3.p1.2.m2.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I5.i3.p1.2.m2.1b"><ci id="S1.I5.i3.p1.2.m2.1.1.cmml" xref="S1.I5.i3.p1.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I5.i3.p1.2.m2.1c">(b)</annotation><annotation encoding="application/x-llamapun" id="S1.I5.i3.p1.2.m2.1d">( italic_b )</annotation></semantics></math>, adopting entailment as partial-assignment satisfiability criterion during the search allows for detecting satisfiability earlier than with verification.</p> </div> </li> </ol> </div> <div class="ltx_para ltx_noindent" id="S1.SS0.SSS0.Px3.p6"> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p6.1">This is important because in many application domains fundamental operations —like <span class="ltx_text ltx_font_italic" id="S1.SS0.SSS0.Px3.p6.1.1">pre-image computation</span> in symbolic model checking (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib5" title="">5</a>]</cite>) or <span class="ltx_text ltx_font_italic" id="S1.SS0.SSS0.Px3.p6.1.2">predicate abstraction</span> in SW verification (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib24" title="">24</a>]</cite>) or <span class="ltx_text ltx_font_italic" id="S1.SS0.SSS0.Px3.p6.1.3">projected enumeration</span> (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib18" title="">18</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib40" title="">40</a>]</cite>) or <span class="ltx_text ltx_font_italic" id="S1.SS0.SSS0.Px3.p6.1.4">projected model counting</span> (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib2" title="">2</a>]</cite>) — require dealing with existentially-quantified formulas and with the enumeration of partial assignments satisfying them.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px3.p7"> <ol class="ltx_enumerate" id="S1.I6"> <li class="ltx_item" id="S1.I6.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">4.</span> <div class="ltx_para" id="S1.I6.i4.p1"> <p class="ltx_p" id="S1.I6.i4.p1.3">Due to <math alttext="(e)" class="ltx_Math" display="inline" id="S1.I6.i4.p1.1.m1.1"><semantics id="S1.I6.i4.p1.1.m1.1a"><mrow id="S1.I6.i4.p1.1.m1.1.2.2"><mo id="S1.I6.i4.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.I6.i4.p1.1.m1.1.1" xref="S1.I6.i4.p1.1.m1.1.1.cmml">e</mi><mo id="S1.I6.i4.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I6.i4.p1.1.m1.1b"><ci id="S1.I6.i4.p1.1.m1.1.1.cmml" xref="S1.I6.i4.p1.1.m1.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I6.i4.p1.1.m1.1c">(e)</annotation><annotation encoding="application/x-llamapun" id="S1.I6.i4.p1.1.m1.1d">( italic_e )</annotation></semantics></math>, CNF-izing upfront the non-CNF formula with the standard techniques <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib32" title="">32</a>]</cite> does not fix issues <math alttext="(a)" class="ltx_Math" display="inline" id="S1.I6.i4.p1.2.m2.1"><semantics id="S1.I6.i4.p1.2.m2.1a"><mrow id="S1.I6.i4.p1.2.m2.1.2.2"><mo id="S1.I6.i4.p1.2.m2.1.2.2.1" stretchy="false">(</mo><mi id="S1.I6.i4.p1.2.m2.1.1" xref="S1.I6.i4.p1.2.m2.1.1.cmml">a</mi><mo id="S1.I6.i4.p1.2.m2.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I6.i4.p1.2.m2.1b"><ci id="S1.I6.i4.p1.2.m2.1.1.cmml" xref="S1.I6.i4.p1.2.m2.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I6.i4.p1.2.m2.1c">(a)</annotation><annotation encoding="application/x-llamapun" id="S1.I6.i4.p1.2.m2.1d">( italic_a )</annotation></semantics></math>-<math alttext="(c)" class="ltx_Math" display="inline" id="S1.I6.i4.p1.3.m3.1"><semantics id="S1.I6.i4.p1.3.m3.1a"><mrow id="S1.I6.i4.p1.3.m3.1.2.2"><mo id="S1.I6.i4.p1.3.m3.1.2.2.1" stretchy="false">(</mo><mi id="S1.I6.i4.p1.3.m3.1.1" xref="S1.I6.i4.p1.3.m3.1.1.cmml">c</mi><mo id="S1.I6.i4.p1.3.m3.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I6.i4.p1.3.m3.1b"><ci id="S1.I6.i4.p1.3.m3.1.1.cmml" xref="S1.I6.i4.p1.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I6.i4.p1.3.m3.1c">(c)</annotation><annotation encoding="application/x-llamapun" id="S1.I6.i4.p1.3.m3.1d">( italic_c )</annotation></semantics></math>, since fresh atoms are introduced, and <em class="ltx_emph ltx_font_italic" id="S1.I6.i4.p1.3.1">a partial assignment over the original atoms may entail the existentially-quantified CNF-ized formula without verifying it.</em></p> </div> </li> </ol> </div> <div class="ltx_para ltx_noindent" id="S1.SS0.SSS0.Px3.p8"> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p8.1">This is important because it shows that, although verification and entailment coincide for CNF formulas, fact 2. above cannot be fixed by simply CNF-izing a formula upfront and running a CNF enumeration procedure based on partial-assignment verification, projecting out the fresh variables introduced by the CNF-ization.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px3.p9"> <ol class="ltx_enumerate" id="S1.I7"> <li class="ltx_item" id="S1.I7.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">5.</span> <div class="ltx_para" id="S1.I7.i5.p1"> <p class="ltx_p" id="S1.I7.i5.p1.1">Due to <math alttext="(c)" class="ltx_Math" display="inline" id="S1.I7.i5.p1.1.m1.1"><semantics id="S1.I7.i5.p1.1.m1.1a"><mrow id="S1.I7.i5.p1.1.m1.1.2.2"><mo id="S1.I7.i5.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.I7.i5.p1.1.m1.1.1" xref="S1.I7.i5.p1.1.m1.1.1.cmml">c</mi><mo id="S1.I7.i5.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.I7.i5.p1.1.m1.1b"><ci id="S1.I7.i5.p1.1.m1.1.1.cmml" xref="S1.I7.i5.p1.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I7.i5.p1.1.m1.1c">(c)</annotation><annotation encoding="application/x-llamapun" id="S1.I7.i5.p1.1.m1.1d">( italic_c )</annotation></semantics></math>, checking verification is polynomial, whereas checking entailment is co-NP-complete, since it consists in checking the validity of the residual formula.</p> </div> </li> </ol> </div> <div class="ltx_para ltx_noindent" id="S1.SS0.SSS0.Px3.p10"> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p10.6">This is the main argument in favour of verification vs. entailment. We notice, however, that if <math alttext="\mu" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px3.p10.1.m1.1"><semantics id="S1.SS0.SSS0.Px3.p10.1.m1.1a"><mi id="S1.SS0.SSS0.Px3.p10.1.m1.1.1" xref="S1.SS0.SSS0.Px3.p10.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px3.p10.1.m1.1b"><ci id="S1.SS0.SSS0.Px3.p10.1.m1.1.1.cmml" xref="S1.SS0.SSS0.Px3.p10.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px3.p10.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px3.p10.1.m1.1d">italic_μ</annotation></semantics></math> entails <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px3.p10.2.m2.1"><semantics id="S1.SS0.SSS0.Px3.p10.2.m2.1a"><mi id="S1.SS0.SSS0.Px3.p10.2.m2.1.1" xref="S1.SS0.SSS0.Px3.p10.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px3.p10.2.m2.1b"><ci id="S1.SS0.SSS0.Px3.p10.2.m2.1.1.cmml" xref="S1.SS0.SSS0.Px3.p10.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px3.p10.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px3.p10.2.m2.1d">italic_φ</annotation></semantics></math>, then the residual of <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px3.p10.3.m3.1"><semantics id="S1.SS0.SSS0.Px3.p10.3.m3.1a"><mi id="S1.SS0.SSS0.Px3.p10.3.m3.1.1" xref="S1.SS0.SSS0.Px3.p10.3.m3.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px3.p10.3.m3.1b"><ci id="S1.SS0.SSS0.Px3.p10.3.m3.1.1.cmml" xref="S1.SS0.SSS0.Px3.p10.3.m3.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px3.p10.3.m3.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px3.p10.3.m3.1d">italic_φ</annotation></semantics></math> under <math alttext="\mu" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px3.p10.4.m4.1"><semantics id="S1.SS0.SSS0.Px3.p10.4.m4.1a"><mi id="S1.SS0.SSS0.Px3.p10.4.m4.1.1" xref="S1.SS0.SSS0.Px3.p10.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px3.p10.4.m4.1b"><ci id="S1.SS0.SSS0.Px3.p10.4.m4.1.1.cmml" xref="S1.SS0.SSS0.Px3.p10.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px3.p10.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px3.p10.4.m4.1d">italic_μ</annotation></semantics></math> is typically much smaller than <math alttext="\varphi" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px3.p10.5.m5.1"><semantics id="S1.SS0.SSS0.Px3.p10.5.m5.1a"><mi id="S1.SS0.SSS0.Px3.p10.5.m5.1.1" xref="S1.SS0.SSS0.Px3.p10.5.m5.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px3.p10.5.m5.1b"><ci id="S1.SS0.SSS0.Px3.p10.5.m5.1.1.cmml" xref="S1.SS0.SSS0.Px3.p10.5.m5.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px3.p10.5.m5.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px3.p10.5.m5.1d">italic_φ</annotation></semantics></math> with a much smaller search space, since its atoms are only (a subset of) those which are not assigned by <math alttext="\mu" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px3.p10.6.m6.1"><semantics id="S1.SS0.SSS0.Px3.p10.6.m6.1a"><mi id="S1.SS0.SSS0.Px3.p10.6.m6.1.1" xref="S1.SS0.SSS0.Px3.p10.6.m6.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px3.p10.6.m6.1b"><ci id="S1.SS0.SSS0.Px3.p10.6.m6.1.1.cmml" xref="S1.SS0.SSS0.Px3.p10.6.m6.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px3.p10.6.m6.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px3.p10.6.m6.1d">italic_μ</annotation></semantics></math>. Notice also that, when this happens, entailment prevents from enumerating a number of assignments which is exponential in the number of the atoms in the residual. Therefore, unlike with plain satisfiability, for enumeration the tradeoff may be favourable to entailment checks.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px3.p11"> <p class="ltx_p" id="S1.SS0.SSS0.Px3.p11.1">Based on the considerations above, not only we suggest to adopt entailment rather than verification as unique definition of partial-assignment satisfiability, but also we champion its usage inside enumeration-based search procedures.</p> </div> </section> <section class="ltx_paragraph" id="S1.SS0.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Related Work.</h4> <div class="ltx_para" id="S1.SS0.SSS0.Px4.p1"> <p class="ltx_p" id="S1.SS0.SSS0.Px4.p1.1">This paper is a revised version of a 2020 unpublished paper <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib35" title="">35</a>]</cite>, which at that time was not published due to lack of algorithmic support and empirical evidence. These came afterwards. Based on the theoretical analysis in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib35" title="">35</a>]</cite> in combination with the idea of dualized search from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib27" title="">27</a>]</cite>, in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib30" title="">30</a>]</cite> we proposed novel enumeration and counting procedures based on entailment. Unfortunately we were not able to produce any implementation efficient enough to compete with s.o.a. enumeration procedures based on verification. Only very recently, in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib16" title="">16</a>]</cite> we have enhanced the <span class="ltx_text ltx_font_sansserif" id="S1.SS0.SSS0.Px4.p1.1.1">HALL</span> enumeration tool for circuits from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>]</cite> by substituting verification checks with entailment checks for partial-assignment reductions (“generalizations” in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib16" title="">16</a>]</cite>), boosting its performance in terms of both time efficiency and size of partial-assignment sets.</p> </div> <div class="ltx_para" id="S1.SS0.SSS0.Px4.p2"> <p class="ltx_p" id="S1.SS0.SSS0.Px4.p2.2">In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib25" title="">25</a>]</cite> we addressed a problem which is different although related to <math alttext="(e)" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px4.p2.1.m1.1"><semantics id="S1.SS0.SSS0.Px4.p2.1.m1.1a"><mrow id="S1.SS0.SSS0.Px4.p2.1.m1.1.2.2"><mo id="S1.SS0.SSS0.Px4.p2.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.SS0.SSS0.Px4.p2.1.m1.1.1" xref="S1.SS0.SSS0.Px4.p2.1.m1.1.1.cmml">e</mi><mo id="S1.SS0.SSS0.Px4.p2.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px4.p2.1.m1.1b"><ci id="S1.SS0.SSS0.Px4.p2.1.m1.1.1.cmml" xref="S1.SS0.SSS0.Px4.p2.1.m1.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px4.p2.1.m1.1c">(e)</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px4.p2.1.m1.1d">( italic_e )</annotation></semantics></math>, showing that adopting a form of CNF-ization different from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib32" title="">32</a>]</cite> improves the efficiency and effectiveness of enumeration. Such encoding, however, does not fix fact <math alttext="(e)" class="ltx_Math" display="inline" id="S1.SS0.SSS0.Px4.p2.2.m2.1"><semantics id="S1.SS0.SSS0.Px4.p2.2.m2.1a"><mrow id="S1.SS0.SSS0.Px4.p2.2.m2.1.2.2"><mo id="S1.SS0.SSS0.Px4.p2.2.m2.1.2.2.1" stretchy="false">(</mo><mi id="S1.SS0.SSS0.Px4.p2.2.m2.1.1" xref="S1.SS0.SSS0.Px4.p2.2.m2.1.1.cmml">e</mi><mo id="S1.SS0.SSS0.Px4.p2.2.m2.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS0.SSS0.Px4.p2.2.m2.1b"><ci id="S1.SS0.SSS0.Px4.p2.2.m2.1.1.cmml" xref="S1.SS0.SSS0.Px4.p2.2.m2.1.1">𝑒</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS0.SSS0.Px4.p2.2.m2.1c">(e)</annotation><annotation encoding="application/x-llamapun" id="S1.SS0.SSS0.Px4.p2.2.m2.1d">( italic_e )</annotation></semantics></math> (see §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS4" title="3.4 Verification and entailment of CNF-ized non-CNF formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.4</span></a>).</p> </div> </section> <section class="ltx_paragraph" id="S1.SS0.SSS0.Px5"> <h4 class="ltx_title ltx_title_paragraph">Content.</h4> <div class="ltx_para" id="S1.SS0.SSS0.Px5.p1"> <p class="ltx_p" id="S1.SS0.SSS0.Px5.p1.1">The rest of the paper is organized as follows. §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2" title="2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a> provides the necessary notation, terminology and concepts used in the paper. §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3" title="3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3</span></a> presents our theoretical analysis: §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS1" title="3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.1</span></a> introduces verification and entailment for generic propositional formulas and discusses their relative properties; §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS2" title="3.2 Other candidate forms of partial-assignment satisfaction. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.2</span></a> analyzes other candidate forms of partial-assignment satisfaction; §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS3" title="3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.3</span></a> lifts the discussion to existentially-quantified formulas; §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS4" title="3.4 Verification and entailment of CNF-ized non-CNF formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.4</span></a> discusses the role of CNF-ization in this context. §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4" title="4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">4</span></a> discusses the practical consequences of entailment vs. verification. §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S5" title="5 Conclusions and Future Work ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">5</span></a> provides some conclusions and suggestions for future research.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Background</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">In this section we introduce the notation and terminology adopted in this paper, and we recall the standard syntax, semantics and basic facts of propositional logics. </p> </div> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Notation.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p1.22">In what follows <span class="ltx_text ltx_markedasmath ltx_font_sansserif" id="S2.SS0.SSS0.Px1.p1.22.1">T</span>, <span class="ltx_text ltx_markedasmath ltx_font_sansserif" id="S2.SS0.SSS0.Px1.p1.22.2">F</span>, <span class="ltx_text ltx_markedasmath ltx_font_sansserif" id="S2.SS0.SSS0.Px1.p1.22.3">?</span> denote the truth values “true”, “false” and “unknown” respectively; <math alttext="\top" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.p1.4.m4.1a"><mo id="S2.SS0.SSS0.Px1.p1.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.4.m4.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.4.m4.1c">\top</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.4.m4.1d">⊤</annotation></semantics></math>, <math alttext="\bot" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.p1.5.m5.1a"><mo id="S2.SS0.SSS0.Px1.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.cmml">⊥</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.5.m5.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1">bottom</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.5.m5.1c">\bot</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.5.m5.1d">⊥</annotation></semantics></math> denote the logic constants “true” and “false” respectively; <math alttext="A" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.p1.6.m6.1a"><mi id="S2.SS0.SSS0.Px1.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.6.m6.1b"><ci id="S2.SS0.SSS0.Px1.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.6.m6.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.6.m6.1c">A</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.6.m6.1d">italic_A</annotation></semantics></math>, <math alttext="B" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.p1.7.m7.1a"><mi id="S2.SS0.SSS0.Px1.p1.7.m7.1.1" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.7.m7.1b"><ci id="S2.SS0.SSS0.Px1.p1.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.7.m7.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.7.m7.1c">B</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.7.m7.1d">italic_B</annotation></semantics></math> denote propositional atoms; <math alttext="\varphi,\phi,\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.8.m8.3"><semantics id="S2.SS0.SSS0.Px1.p1.8.m8.3a"><mrow id="S2.SS0.SSS0.Px1.p1.8.m8.3.4.2" xref="S2.SS0.SSS0.Px1.p1.8.m8.3.4.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.8.m8.1.1" xref="S2.SS0.SSS0.Px1.p1.8.m8.1.1.cmml">φ</mi><mo id="S2.SS0.SSS0.Px1.p1.8.m8.3.4.2.1" xref="S2.SS0.SSS0.Px1.p1.8.m8.3.4.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p1.8.m8.2.2" xref="S2.SS0.SSS0.Px1.p1.8.m8.2.2.cmml">ϕ</mi><mo id="S2.SS0.SSS0.Px1.p1.8.m8.3.4.2.2" xref="S2.SS0.SSS0.Px1.p1.8.m8.3.4.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p1.8.m8.3.3" xref="S2.SS0.SSS0.Px1.p1.8.m8.3.3.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.8.m8.3b"><list id="S2.SS0.SSS0.Px1.p1.8.m8.3.4.1.cmml" xref="S2.SS0.SSS0.Px1.p1.8.m8.3.4.2"><ci id="S2.SS0.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.8.m8.1.1">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p1.8.m8.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.8.m8.2.2">italic-ϕ</ci><ci id="S2.SS0.SSS0.Px1.p1.8.m8.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.8.m8.3.3">𝜓</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.8.m8.3c">\varphi,\phi,\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.8.m8.3d">italic_φ , italic_ϕ , italic_ψ</annotation></semantics></math> denote propositional formulas; <math alttext="\mu,\eta,\gamma" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.9.m9.3"><semantics id="S2.SS0.SSS0.Px1.p1.9.m9.3a"><mrow id="S2.SS0.SSS0.Px1.p1.9.m9.3.4.2" xref="S2.SS0.SSS0.Px1.p1.9.m9.3.4.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p1.9.m9.1.1.cmml">μ</mi><mo id="S2.SS0.SSS0.Px1.p1.9.m9.3.4.2.1" xref="S2.SS0.SSS0.Px1.p1.9.m9.3.4.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p1.9.m9.2.2" xref="S2.SS0.SSS0.Px1.p1.9.m9.2.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px1.p1.9.m9.3.4.2.2" xref="S2.SS0.SSS0.Px1.p1.9.m9.3.4.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p1.9.m9.3.3" xref="S2.SS0.SSS0.Px1.p1.9.m9.3.3.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.9.m9.3b"><list id="S2.SS0.SSS0.Px1.p1.9.m9.3.4.1.cmml" xref="S2.SS0.SSS0.Px1.p1.9.m9.3.4.2"><ci id="S2.SS0.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.9.m9.1.1">𝜇</ci><ci id="S2.SS0.SSS0.Px1.p1.9.m9.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.9.m9.2.2">𝜂</ci><ci id="S2.SS0.SSS0.Px1.p1.9.m9.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.9.m9.3.3">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.9.m9.3c">\mu,\eta,\gamma</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.9.m9.3d">italic_μ , italic_η , italic_γ</annotation></semantics></math> denote truth value assignments. 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id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2.2">𝐵</ci><ci id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2.3">𝐾</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.11.m11.3c">{\mathbf{B}}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{B_{1},...,B_{K}}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.11.m11.3d">bold_B start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_B start_POSTSUBSCRIPT italic_K end_POSTSUBSCRIPT }</annotation></semantics></math> denote disjoint sets of propositional atoms. More precisely, <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.12.m12.1"><semantics id="S2.SS0.SSS0.Px1.p1.12.m12.1a"><mi id="S2.SS0.SSS0.Px1.p1.12.m12.1.1" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.12.m12.1b"><ci id="S2.SS0.SSS0.Px1.p1.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.12.m12.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.12.m12.1d">italic_φ</annotation></semantics></math>, <math alttext="\phi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.13.m13.1"><semantics id="S2.SS0.SSS0.Px1.p1.13.m13.1a"><mi id="S2.SS0.SSS0.Px1.p1.13.m13.1.1" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.1.cmml">ϕ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.13.m13.1b"><ci id="S2.SS0.SSS0.Px1.p1.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.1">italic-ϕ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.13.m13.1c">\phi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.13.m13.1d">italic_ϕ</annotation></semantics></math> and <math alttext="\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.14.m14.1"><semantics id="S2.SS0.SSS0.Px1.p1.14.m14.1a"><mi id="S2.SS0.SSS0.Px1.p1.14.m14.1.1" xref="S2.SS0.SSS0.Px1.p1.14.m14.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.14.m14.1b"><ci id="S2.SS0.SSS0.Px1.p1.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.14.m14.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.14.m14.1d">italic_ψ</annotation></semantics></math> denote generic propositional formulas built on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.15.m15.1"><semantics id="S2.SS0.SSS0.Px1.p1.15.m15.1a"><mi id="S2.SS0.SSS0.Px1.p1.15.m15.1.1" xref="S2.SS0.SSS0.Px1.p1.15.m15.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.15.m15.1b"><ci id="S2.SS0.SSS0.Px1.p1.15.m15.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.15.m15.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.15.m15.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.15.m15.1d">bold_A</annotation></semantics></math>, <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.16.m16.1"><semantics id="S2.SS0.SSS0.Px1.p1.16.m16.1a"><mi id="S2.SS0.SSS0.Px1.p1.16.m16.1.1" xref="S2.SS0.SSS0.Px1.p1.16.m16.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.16.m16.1b"><ci id="S2.SS0.SSS0.Px1.p1.16.m16.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.16.m16.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.16.m16.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.16.m16.1d">bold_B</annotation></semantics></math> and <math alttext="{\mathbf{A}}\cup{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.17.m17.1"><semantics id="S2.SS0.SSS0.Px1.p1.17.m17.1a"><mrow id="S2.SS0.SSS0.Px1.p1.17.m17.1.1" xref="S2.SS0.SSS0.Px1.p1.17.m17.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.17.m17.1.1.2" xref="S2.SS0.SSS0.Px1.p1.17.m17.1.1.2.cmml">𝐀</mi><mo id="S2.SS0.SSS0.Px1.p1.17.m17.1.1.1" xref="S2.SS0.SSS0.Px1.p1.17.m17.1.1.1.cmml">∪</mo><mi id="S2.SS0.SSS0.Px1.p1.17.m17.1.1.3" xref="S2.SS0.SSS0.Px1.p1.17.m17.1.1.3.cmml">𝐁</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.17.m17.1b"><apply id="S2.SS0.SSS0.Px1.p1.17.m17.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.17.m17.1.1"><union id="S2.SS0.SSS0.Px1.p1.17.m17.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.17.m17.1.1.1"></union><ci id="S2.SS0.SSS0.Px1.p1.17.m17.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.17.m17.1.1.2">𝐀</ci><ci id="S2.SS0.SSS0.Px1.p1.17.m17.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.17.m17.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.17.m17.1c">{\mathbf{A}}\cup{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.17.m17.1d">bold_A ∪ bold_B</annotation></semantics></math> respectively; <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.18.m18.1"><semantics id="S2.SS0.SSS0.Px1.p1.18.m18.1a"><mi id="S2.SS0.SSS0.Px1.p1.18.m18.1.1" xref="S2.SS0.SSS0.Px1.p1.18.m18.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.18.m18.1b"><ci id="S2.SS0.SSS0.Px1.p1.18.m18.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.18.m18.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.18.m18.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.18.m18.1d">italic_η</annotation></semantics></math> and <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.19.m19.1"><semantics id="S2.SS0.SSS0.Px1.p1.19.m19.1a"><mi id="S2.SS0.SSS0.Px1.p1.19.m19.1.1" xref="S2.SS0.SSS0.Px1.p1.19.m19.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.19.m19.1b"><ci id="S2.SS0.SSS0.Px1.p1.19.m19.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.19.m19.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.19.m19.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.19.m19.1d">italic_μ</annotation></semantics></math> denote total and a partial assignments on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.20.m20.1"><semantics id="S2.SS0.SSS0.Px1.p1.20.m20.1a"><mi id="S2.SS0.SSS0.Px1.p1.20.m20.1.1" xref="S2.SS0.SSS0.Px1.p1.20.m20.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.20.m20.1b"><ci id="S2.SS0.SSS0.Px1.p1.20.m20.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.20.m20.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.20.m20.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.20.m20.1d">bold_A</annotation></semantics></math> respectively; <math alttext="\delta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.21.m21.1"><semantics id="S2.SS0.SSS0.Px1.p1.21.m21.1a"><mi id="S2.SS0.SSS0.Px1.p1.21.m21.1.1" xref="S2.SS0.SSS0.Px1.p1.21.m21.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.21.m21.1b"><ci id="S2.SS0.SSS0.Px1.p1.21.m21.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.21.m21.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.21.m21.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.21.m21.1d">italic_δ</annotation></semantics></math> denote total assignments on <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.22.m22.1"><semantics id="S2.SS0.SSS0.Px1.p1.22.m22.1a"><mi id="S2.SS0.SSS0.Px1.p1.22.m22.1.1" xref="S2.SS0.SSS0.Px1.p1.22.m22.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.22.m22.1b"><ci id="S2.SS0.SSS0.Px1.p1.22.m22.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.22.m22.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.22.m22.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.22.m22.1d">bold_B</annotation></semantics></math>. (All above symbols may possibly have subscripts.)</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Syntax.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px2.p1.22">A <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px2.p1.22.1">propositional formula</span> is defined inductively as follows: the constants <math alttext="\top" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.1.m1.1"><semantics id="S2.SS0.SSS0.Px2.p1.1.m1.1a"><mo id="S2.SS0.SSS0.Px2.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.1.m1.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.1.m1.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.1.m1.1c">\top</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.1.m1.1d">⊤</annotation></semantics></math> and <math alttext="\bot" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.2.m2.1"><semantics id="S2.SS0.SSS0.Px2.p1.2.m2.1a"><mo id="S2.SS0.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1.cmml">⊥</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.2.m2.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.2.m2.1.1">bottom</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.2.m2.1c">\bot</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.2.m2.1d">⊥</annotation></semantics></math> (denoting the truth values true and false) are formulas; a propositional atom <math alttext="A_{1},A_{2},A_{3},..." class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.3.m3.4"><semantics id="S2.SS0.SSS0.Px2.p1.3.m3.4a"><mrow id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.4.cmml"><msub id="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1" xref="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.cmml"><mi id="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.2" xref="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.2.cmml">A</mi><mn id="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.3" xref="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.4" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.4.cmml">,</mo><msub id="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2" xref="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.2" xref="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.2.cmml">A</mi><mn id="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.3" xref="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.5" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.4.cmml">,</mo><msub id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.2" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.2.cmml">A</mi><mn id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.3" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.3.cmml">3</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.6" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.4.cmml">,</mo><mi id="S2.SS0.SSS0.Px2.p1.3.m3.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px2.p1.3.m3.1.1.cmml">…</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.3.m3.4b"><list id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.4.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3"><apply id="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.2">𝐴</ci><cn id="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.3.m3.2.2.1.1.3">1</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.2">𝐴</ci><cn id="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.3.m3.3.3.2.2.3">2</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.2">𝐴</ci><cn id="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.3.m3.4.4.3.3.3">3</cn></apply><ci id="S2.SS0.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.3.m3.1.1">…</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.3.m3.4c">A_{1},A_{2},A_{3},...</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.3.m3.4d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , …</annotation></semantics></math> is a formula; if <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.4.m4.1"><semantics id="S2.SS0.SSS0.Px2.p1.4.m4.1a"><msub id="S2.SS0.SSS0.Px2.p1.4.m4.1.1" xref="S2.SS0.SSS0.Px2.p1.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px2.p1.4.m4.1.1.2" xref="S2.SS0.SSS0.Px2.p1.4.m4.1.1.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.4.m4.1.1.3" xref="S2.SS0.SSS0.Px2.p1.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.4.m4.1b"><apply id="S2.SS0.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.4.m4.1.1.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.4.m4.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.4.m4.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.4.m4.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.5.m5.1"><semantics id="S2.SS0.SSS0.Px2.p1.5.m5.1a"><msub id="S2.SS0.SSS0.Px2.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px2.p1.5.m5.1.1.cmml"><mi id="S2.SS0.SSS0.Px2.p1.5.m5.1.1.2" xref="S2.SS0.SSS0.Px2.p1.5.m5.1.1.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.5.m5.1.1.3" xref="S2.SS0.SSS0.Px2.p1.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.5.m5.1b"><apply id="S2.SS0.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.5.m5.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.5.m5.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.5.m5.1.1.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.5.m5.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.5.m5.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.5.m5.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are formulas, then <math alttext="\neg\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.6.m6.1"><semantics id="S2.SS0.SSS0.Px2.p1.6.m6.1a"><mrow id="S2.SS0.SSS0.Px2.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.6.m6.1b"><apply id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1"><not id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.1"></not><apply id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.6.m6.1.1.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.6.m6.1c">\neg\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.6.m6.1d">¬ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{1}\wedge\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.7.m7.1"><semantics id="S2.SS0.SSS0.Px2.p1.7.m7.1a"><mrow id="S2.SS0.SSS0.Px2.p1.7.m7.1.1" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.1" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.1.cmml">∧</mo><msub id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.3" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.7.m7.1b"><apply id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1"><and id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.1"></and><apply id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.7.m7.1c">\varphi_{1}\wedge\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.7.m7.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are formulas. We use the standard Boolean abbreviations: “<math alttext="\varphi_{1}\vee\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.8.m8.1"><semantics id="S2.SS0.SSS0.Px2.p1.8.m8.1a"><mrow id="S2.SS0.SSS0.Px2.p1.8.m8.1.1" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.1" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.1.cmml">∨</mo><msub id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.3" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.8.m8.1b"><apply id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1"><or id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.1"></or><apply id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.8.m8.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.8.m8.1c">\varphi_{1}\vee\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.8.m8.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>” for “<math alttext="\neg(\neg\varphi_{1}\wedge\neg\varphi_{2})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.9.m9.1"><semantics id="S2.SS0.SSS0.Px2.p1.9.m9.1a"><mrow id="S2.SS0.SSS0.Px2.p1.9.m9.1.1" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.2" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.2.cmml">¬</mo><mrow id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.cmml"><mrow id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.cmml"><mo id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.2" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.3" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.3.cmml">1</mn></msub></mrow><mo id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.1.cmml">∧</mo><mrow id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.cmml"><mo id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.2" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.3" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.9.m9.1b"><apply id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1"><not id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.2"></not><apply id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1"><and id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.1"></and><apply id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2"><not id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.1"></not><apply id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.2.2.3">1</cn></apply></apply><apply id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3"><not id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.1"></not><apply id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.9.m9.1.1.1.1.1.3.2.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.9.m9.1c">\neg(\neg\varphi_{1}\wedge\neg\varphi_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.9.m9.1d">¬ ( ¬ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>”, “<math alttext="\varphi_{1}\rightarrow\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.10.m10.1"><semantics id="S2.SS0.SSS0.Px2.p1.10.m10.1a"><mrow id="S2.SS0.SSS0.Px2.p1.10.m10.1.1" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.1.cmml">→</mo><msub id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.3" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.10.m10.1b"><apply id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1"><ci id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.1">→</ci><apply id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.10.m10.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.10.m10.1c">\varphi_{1}\rightarrow\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.10.m10.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>” for “<math alttext="\neg(\varphi_{1}\wedge\neg\varphi_{2})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.11.m11.1"><semantics id="S2.SS0.SSS0.Px2.p1.11.m11.1a"><mrow id="S2.SS0.SSS0.Px2.p1.11.m11.1.1" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.2" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.2.cmml">¬</mo><mrow id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.1.cmml">∧</mo><mrow id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.cmml"><mo id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.2" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.3" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.11.m11.1b"><apply id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1"><not id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.2"></not><apply id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1"><and id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.1"></and><apply id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3"><not id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.1"></not><apply id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.11.m11.1.1.1.1.1.3.2.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.11.m11.1c">\neg(\varphi_{1}\wedge\neg\varphi_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.11.m11.1d">¬ ( italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>”, “<math alttext="\varphi_{1}\leftrightarrow\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.12.m12.1"><semantics id="S2.SS0.SSS0.Px2.p1.12.m12.1a"><mrow id="S2.SS0.SSS0.Px2.p1.12.m12.1.1" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.1.cmml">↔</mo><msub id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.3" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.12.m12.1b"><apply id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1"><ci id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.1">↔</ci><apply id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.12.m12.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.12.m12.1c">\varphi_{1}\leftrightarrow\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.12.m12.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>” for “<math alttext="\neg(\varphi_{1}\wedge\neg\varphi_{2})\wedge\neg(\varphi_{2}\wedge\neg\varphi_% {1})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.13.m13.2"><semantics id="S2.SS0.SSS0.Px2.p1.13.m13.2a"><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.cmml"><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.2.cmml">¬</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.1.cmml">∧</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.cmml"><mo id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.3.cmml">∧</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.cmml"><mo id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.2.cmml">¬</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.1" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.1.cmml">∧</mo><mrow id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.cmml"><mo id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.2" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.3" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.3.cmml">1</mn></msub></mrow></mrow><mo id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.13.m13.2b"><apply id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2"><and id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.3"></and><apply id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1"><not id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.2"></not><apply id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1"><and id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.1"></and><apply id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3"><not id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.1"></not><apply id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.13.m13.1.1.1.1.1.1.3.2.3">2</cn></apply></apply></apply></apply><apply id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2"><not id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.2"></not><apply id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1"><and id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.1"></and><apply id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.2.3">2</cn></apply><apply id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3"><not id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.1"></not><apply id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px2.p1.13.m13.2.2.2.1.1.1.3.2.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.13.m13.2c">\neg(\varphi_{1}\wedge\neg\varphi_{2})\wedge\neg(\varphi_{2}\wedge\neg\varphi_% {1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.13.m13.2d">¬ ( italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∧ ¬ ( italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∧ ¬ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>”. A <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px2.p1.22.2">literal</span> is either an atom (a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px2.p1.22.3">positive literal</span>) or its negation (a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px2.p1.22.4">negative literal</span>). (If <math alttext="l" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.14.m14.1"><semantics id="S2.SS0.SSS0.Px2.p1.14.m14.1a"><mi id="S2.SS0.SSS0.Px2.p1.14.m14.1.1" xref="S2.SS0.SSS0.Px2.p1.14.m14.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.14.m14.1b"><ci id="S2.SS0.SSS0.Px2.p1.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.14.m14.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.14.m14.1c">l</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.14.m14.1d">italic_l</annotation></semantics></math> is a negative literal <math alttext="\neg A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.15.m15.1"><semantics id="S2.SS0.SSS0.Px2.p1.15.m15.1a"><mrow id="S2.SS0.SSS0.Px2.p1.15.m15.1.1" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.15.m15.1b"><apply id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1"><not id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.1"></not><apply id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.2">𝐴</ci><ci id="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.15.m15.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.15.m15.1c">\neg A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.15.m15.1d">¬ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, then by “<math alttext="\neg l" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.16.m16.1"><semantics id="S2.SS0.SSS0.Px2.p1.16.m16.1a"><mrow id="S2.SS0.SSS0.Px2.p1.16.m16.1.1" xref="S2.SS0.SSS0.Px2.p1.16.m16.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.16.m16.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.16.m16.1.1.1.cmml">¬</mo><mi id="S2.SS0.SSS0.Px2.p1.16.m16.1.1.2" xref="S2.SS0.SSS0.Px2.p1.16.m16.1.1.2.cmml">l</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.16.m16.1b"><apply id="S2.SS0.SSS0.Px2.p1.16.m16.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.16.m16.1.1"><not id="S2.SS0.SSS0.Px2.p1.16.m16.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.16.m16.1.1.1"></not><ci id="S2.SS0.SSS0.Px2.p1.16.m16.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.16.m16.1.1.2">𝑙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.16.m16.1c">\neg l</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.16.m16.1d">¬ italic_l</annotation></semantics></math>” we conventionally mean <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.17.m17.1"><semantics id="S2.SS0.SSS0.Px2.p1.17.m17.1a"><msub id="S2.SS0.SSS0.Px2.p1.17.m17.1.1" xref="S2.SS0.SSS0.Px2.p1.17.m17.1.1.cmml"><mi id="S2.SS0.SSS0.Px2.p1.17.m17.1.1.2" xref="S2.SS0.SSS0.Px2.p1.17.m17.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px2.p1.17.m17.1.1.3" xref="S2.SS0.SSS0.Px2.p1.17.m17.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.17.m17.1b"><apply id="S2.SS0.SSS0.Px2.p1.17.m17.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.17.m17.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.17.m17.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.17.m17.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.17.m17.1.1.2">𝐴</ci><ci id="S2.SS0.SSS0.Px2.p1.17.m17.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.17.m17.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.17.m17.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.17.m17.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> rather than <math alttext="\neg\neg A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.18.m18.1"><semantics id="S2.SS0.SSS0.Px2.p1.18.m18.1a"><mrow id="S2.SS0.SSS0.Px2.p1.18.m18.1.1" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.1.cmml">¬</mo><mrow id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.cmml"><mo id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.1" rspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.2" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.3" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.18.m18.1b"><apply id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1"><not id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.1"></not><apply id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2"><not id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.1"></not><apply id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.2">𝐴</ci><ci id="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.18.m18.1.1.2.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.18.m18.1c">\neg\neg A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.18.m18.1d">¬ ¬ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.) A <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px2.p1.22.5">clause</span> is a disjunction of literals <math alttext="\bigvee_{j}l_{j}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.19.m19.1"><semantics id="S2.SS0.SSS0.Px2.p1.19.m19.1a"><mrow id="S2.SS0.SSS0.Px2.p1.19.m19.1.1" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.2.cmml">⋁</mo><mi id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.3.cmml">j</mi></msub><msub id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.2.cmml">l</mi><mi id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.19.m19.1b"><apply id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1"><apply id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1">subscript</csymbol><or id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.2"></or><ci id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.1.3">𝑗</ci></apply><apply id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.2">𝑙</ci><ci id="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.19.m19.1.1.2.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.19.m19.1c">\bigvee_{j}l_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.19.m19.1d">⋁ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_l start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. A <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px2.p1.22.6">cube</span> is a conjunction of literals <math alttext="\bigwedge_{j}l_{j}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.20.m20.1"><semantics id="S2.SS0.SSS0.Px2.p1.20.m20.1a"><mrow id="S2.SS0.SSS0.Px2.p1.20.m20.1.1" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.cmml"><msub id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.2" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.2.cmml">⋀</mo><mi id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.3.cmml">j</mi></msub><msub id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.2.cmml">l</mi><mi id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.20.m20.1b"><apply id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1"><apply id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1">subscript</csymbol><and id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.2"></and><ci id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.1.3">𝑗</ci></apply><apply id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.2">𝑙</ci><ci id="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px2.p1.20.m20.1.1.2.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.20.m20.1c">\bigwedge_{j}l_{j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.20.m20.1d">⋀ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_l start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.21.m21.1"><semantics id="S2.SS0.SSS0.Px2.p1.21.m21.1a"><mi id="S2.SS0.SSS0.Px2.p1.21.m21.1.1" xref="S2.SS0.SSS0.Px2.p1.21.m21.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.21.m21.1b"><ci id="S2.SS0.SSS0.Px2.p1.21.m21.1.1.cmml" xref="S2.SS0.SSS0.Px2.p1.21.m21.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.21.m21.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.21.m21.1d">italic_φ</annotation></semantics></math> is in <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px2.p1.22.7">Conjunctive Normal Form (CNF)</span> iff it is a conjunction of clauses: <math alttext="\bigwedge_{i=1}^{L}\bigvee_{j_{i}=1}^{K_{i}}l_{j_{i}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px2.p1.22.m22.1"><semantics id="S2.SS0.SSS0.Px2.p1.22.m22.1a"><mrow id="S2.SS0.SSS0.Px2.p1.22.m22.1.1" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.cmml"><msubsup id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.2.cmml">⋀</mo><mrow id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.3.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.3.2.cmml">i</mi><mo id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.3.1" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.3.1.cmml">=</mo><mn id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.3.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.1.3.cmml">L</mi></msubsup><mrow id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.cmml"><msubsup id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.cmml"><mo id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.2" lspace="0.167em" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.2.cmml">⋁</mo><mrow id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.cmml"><msub id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.2.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.2.2.cmml">j</mi><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.2.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.1" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.1.cmml">=</mo><mn id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.2.3.3.cmml">1</mn></mrow><msub id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.3.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.3.2.cmml">K</mi><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.3.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.1.3.3.cmml">i</mi></msub></msubsup><msub id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.2.cmml">l</mi><msub id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.3.cmml"><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.3.2" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.3.2.cmml">j</mi><mi id="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.3.3" xref="S2.SS0.SSS0.Px2.p1.22.m22.1.1.2.2.3.3.cmml">i</mi></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px2.p1.22.m22.1b"><apply 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encoding="application/x-tex" id="S2.SS0.SSS0.Px2.p1.22.m22.1c">\bigwedge_{i=1}^{L}\bigvee_{j_{i}=1}^{K_{i}}l_{j_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px2.p1.22.m22.1d">⋀ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_L end_POSTSUPERSCRIPT ⋁ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_l start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Semantics.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px3.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px3.p1.12">Given <math alttext="{\mathbf{A}}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1},...,A_{N}}\}" class="ltx_Math" 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start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_A start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT }</annotation></semantics></math>, a map <math alttext="\eta:{\mathbf{A}}\longmapsto\{{\mbox{{\sf T}},\mbox{{\sf F}}}\}^{N}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.2.m2.2"><semantics id="S2.SS0.SSS0.Px3.p1.2.m2.2a"><mrow id="S2.SS0.SSS0.Px3.p1.2.m2.2.3" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.cmml"><mi id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.2" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.1.cmml">:</mo><mrow id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.cmml"><mi id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.2" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.2.cmml">𝐀</mi><mo id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.1" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.1.cmml">⟼</mo><msup id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3.cmml"><mrow 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xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.1">:</ci><ci id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.2.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3"><ci id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.1">⟼</ci><ci id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.2.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.2">𝐀</ci><apply id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3.1.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3">superscript</csymbol><set id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3.2.2"><ci id="S2.SS0.SSS0.Px3.p1.2.m2.1.1a.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.1.1"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.1.1">T</mtext></ci><ci id="S2.SS0.SSS0.Px3.p1.2.m2.2.2a.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.2.m2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.2">F</mtext></ci></set><ci id="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3.3.cmml" xref="S2.SS0.SSS0.Px3.p1.2.m2.2.3.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.2.m2.2c">\eta:{\mathbf{A}}\longmapsto\{{\mbox{{\sf T}},\mbox{{\sf F}}}\}^{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.2.m2.2d">italic_η : bold_A ⟼ { T , F } start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> is a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p1.12.1">total truth assignment</span> for <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.3.m3.1"><semantics id="S2.SS0.SSS0.Px3.p1.3.m3.1a"><mi id="S2.SS0.SSS0.Px3.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px3.p1.3.m3.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.3.m3.1b"><ci id="S2.SS0.SSS0.Px3.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.3.m3.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.3.m3.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.3.m3.1d">bold_A</annotation></semantics></math>. We assume <math alttext="\eta(\top)\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mbox{{\sf T}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.4.m4.1"><semantics id="S2.SS0.SSS0.Px3.p1.4.m4.1a"><mrow id="S2.SS0.SSS0.Px3.p1.4.m4.1.2" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.cmml"><mrow id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.2" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.1" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.3.2" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml"><mo id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml">(</mo><mo id="S2.SS0.SSS0.Px3.p1.4.m4.1.1" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.1.cmml">⊤</mo><mo id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mover id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.2" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.2.cmml">=</mo><mtext id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.3" mathsize="71%" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.3a.cmml">def</mtext></mover><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.3" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.3a.cmml">T</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.4.m4.1b"><apply id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2"><apply id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1">superscript</csymbol><eq id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.2"></eq><ci id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.3a.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.3"><mtext id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.3.cmml" mathsize="50%" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.1.3">def</mtext></ci></apply><apply id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2"><times id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.1"></times><ci id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.2.2">𝜂</ci><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.1">top</csymbol></apply><ci id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.3a.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.4.m4.1.2.3.cmml" xref="S2.SS0.SSS0.Px3.p1.4.m4.1.2.3">T</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.4.m4.1c">\eta(\top)\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mbox{{\sf T}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.4.m4.1d">italic_η ( ⊤ ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP T</annotation></semantics></math> and <math alttext="\eta(\bot)\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mbox{{\sf F}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.5.m5.1"><semantics id="S2.SS0.SSS0.Px3.p1.5.m5.1a"><mrow id="S2.SS0.SSS0.Px3.p1.5.m5.1.2" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.cmml"><mrow id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.2" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.1" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.3.2" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.cmml"><mo id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.cmml">(</mo><mo id="S2.SS0.SSS0.Px3.p1.5.m5.1.1" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.1.cmml">⊥</mo><mo id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mover id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.2" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.2.cmml">=</mo><mtext id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.3" mathsize="71%" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.3a.cmml">def</mtext></mover><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.3" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.3a.cmml">F</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.5.m5.1b"><apply id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2"><apply id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1">superscript</csymbol><eq id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.2"></eq><ci id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.3a.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.3"><mtext id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.3.cmml" mathsize="50%" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.1.3">def</mtext></ci></apply><apply id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2"><times id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.1"></times><ci id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.2.2">𝜂</ci><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.1">bottom</csymbol></apply><ci id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.3a.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.5.m5.1.2.3.cmml" xref="S2.SS0.SSS0.Px3.p1.5.m5.1.2.3">F</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.5.m5.1c">\eta(\bot)\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mbox{{\sf F}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.5.m5.1d">italic_η ( ⊥ ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP F</annotation></semantics></math>. We represent <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.6.m6.1"><semantics id="S2.SS0.SSS0.Px3.p1.6.m6.1a"><mi id="S2.SS0.SSS0.Px3.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px3.p1.6.m6.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.6.m6.1b"><ci id="S2.SS0.SSS0.Px3.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.6.m6.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.6.m6.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.6.m6.1d">italic_η</annotation></semantics></math> as a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p1.12.2">set of literals</span> <math alttext="\eta\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{i}\ |\ \eta(A_{i})=% \mbox{{\sf T}}}\}\cup\{{\neg A_{i}\ |\ \eta(A_{i})=\mbox{{\sf F}}}\}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.7.m7.4"><semantics id="S2.SS0.SSS0.Px3.p1.7.m7.4a"><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.4.4" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.cmml"><mi id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.6" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.6.cmml">η</mi><mover id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.cmml"><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.2.cmml">=</mo><mtext id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.3" mathsize="71%" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.3a.cmml">def</mtext></mover><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.cmml"><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.3.cmml"><mo id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.3.1.cmml">{</mo><msub id="S2.SS0.SSS0.Px3.p1.7.m7.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.7.m7.1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.7.m7.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p1.7.m7.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p1.7.m7.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.4" lspace="0em" rspace="0.500em" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.3.1.cmml">|</mo><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.cmml"><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.3" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.3.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.cmml">(</mo><msub id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.2.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.3" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.3a.cmml">T</mtext></mrow><mo id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.5" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.3.1.cmml">}</mo></mrow><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.5" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.5.cmml">∪</mo><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.3.cmml"><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.3" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.3.1.cmml">{</mo><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1" xref="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.3" xref="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.3.cmml">i</mi></msub></mrow><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.4" lspace="0em" rspace="0.500em" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.3.1.cmml">|</mo><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.cmml"><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.3" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.3.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.cmml">(</mo><msub id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.2" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.2.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.3" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.3a.cmml">F</mtext></mrow><mo id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.5" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.7.m7.4b"><apply id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4"><apply id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.1.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5">superscript</csymbol><eq id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.2"></eq><ci id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.3a.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.3"><mtext id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.5.3.cmml" mathsize="50%" 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xref="S2.SS0.SSS0.Px3.p1.7.m7.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2"><eq id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.2"></eq><apply id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1"><times id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.2"></times><ci id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.3.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.3">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.2.2.2.2.2.2.1.1.1.1.2">𝐴</ci><ci 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xref="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.3.3.3.3.1.1.2.3">𝑖</ci></apply></apply><apply id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2"><eq id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.2"></eq><apply id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1"><times id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.2"></times><ci id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.3.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.3">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.1.1.1.1.3">𝑖</ci></apply></apply><ci id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.3a.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.3.cmml" xref="S2.SS0.SSS0.Px3.p1.7.m7.4.4.4.4.2.2.3">F</mtext></ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.7.m7.4c">\eta\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{i}\ |\ \eta(A_{i})=% \mbox{{\sf T}}}\}\cup\{{\neg A_{i}\ |\ \eta(A_{i})=\mbox{{\sf F}}}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.7.m7.4d">italic_η start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_η ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = T } ∪ { ¬ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_η ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = F }</annotation></semantics></math>. We sometimes represent <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.8.m8.1"><semantics id="S2.SS0.SSS0.Px3.p1.8.m8.1a"><mi id="S2.SS0.SSS0.Px3.p1.8.m8.1.1" xref="S2.SS0.SSS0.Px3.p1.8.m8.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.8.m8.1b"><ci id="S2.SS0.SSS0.Px3.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.8.m8.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.8.m8.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.8.m8.1d">italic_η</annotation></semantics></math> also as a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p1.12.3">cube</span> <math alttext="\bigwedge_{\eta(A_{i})=\tiny{\mbox{{\sf T}}}}A_{i}\wedge\bigwedge_{\eta(A_{i})% =\tiny{\mbox{{\sf F}}}}\neg A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.9.m9.2"><semantics id="S2.SS0.SSS0.Px3.p1.9.m9.2a"><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.2.3" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.cmml"><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.cmml"><msub id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.1.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.1.2.cmml">⋀</mo><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.cmml"><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.3.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.2.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.3" mathsize="71%" xref="S2.SS0.SSS0.Px3.p1.9.m9.1.1.1.3a.cmml">T</mtext></mrow></msub><msub id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.2.cmml"><mi id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.2.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.2.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.2.3" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.2.2.3.cmml">i</mi></msub></mrow><mo id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.1" rspace="0.055em" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.1.cmml">∧</mo><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.cmml"><msub id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.1.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.1.2.cmml">⋀</mo><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.cmml"><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.3" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.3.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.2" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.cmml">(</mo><msub id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.cmml"><mi 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xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1"><times id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.2"></times><ci id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.3">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.1.1.1.1.3">𝑖</ci></apply></apply><ci id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.3a.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.3.cmml" mathsize="50%" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.2.1.3">F</mtext></ci></apply></apply><apply id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2"><not id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.1"></not><apply id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.2.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.2.3.cmml" xref="S2.SS0.SSS0.Px3.p1.9.m9.2.3.3.2.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.9.m9.2c">\bigwedge_{\eta(A_{i})=\tiny{\mbox{{\sf T}}}}A_{i}\wedge\bigwedge_{\eta(A_{i})% =\tiny{\mbox{{\sf F}}}}\neg A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.9.m9.2d">⋀ start_POSTSUBSCRIPT italic_η ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = T end_POSTSUBSCRIPT italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∧ ⋀ start_POSTSUBSCRIPT italic_η ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = F end_POSTSUBSCRIPT ¬ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> which we denote as “<math alttext="\bigwedge\!\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.10.m10.1"><semantics id="S2.SS0.SSS0.Px3.p1.10.m10.1a"><mrow id="S2.SS0.SSS0.Px3.p1.10.m10.1.1" xref="S2.SS0.SSS0.Px3.p1.10.m10.1.1.cmml"><mpadded width="0.747em"><mo id="S2.SS0.SSS0.Px3.p1.10.m10.1.1.1" xref="S2.SS0.SSS0.Px3.p1.10.m10.1.1.1.cmml">⋀</mo></mpadded><mi id="S2.SS0.SSS0.Px3.p1.10.m10.1.1.2" xref="S2.SS0.SSS0.Px3.p1.10.m10.1.1.2.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.10.m10.1b"><apply id="S2.SS0.SSS0.Px3.p1.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.10.m10.1.1"><and id="S2.SS0.SSS0.Px3.p1.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.10.m10.1.1.1"></and><ci id="S2.SS0.SSS0.Px3.p1.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p1.10.m10.1.1.2">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.10.m10.1c">\bigwedge\!\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.10.m10.1d">⋀ italic_η</annotation></semantics></math>” so that to distinguish the set and the cube representations. We denote by <math alttext="|\eta|" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.11.m11.1"><semantics id="S2.SS0.SSS0.Px3.p1.11.m11.1a"><mrow id="S2.SS0.SSS0.Px3.p1.11.m11.1.2.2" xref="S2.SS0.SSS0.Px3.p1.11.m11.1.2.1.cmml"><mo id="S2.SS0.SSS0.Px3.p1.11.m11.1.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.11.m11.1.2.1.1.cmml">|</mo><mi id="S2.SS0.SSS0.Px3.p1.11.m11.1.1" xref="S2.SS0.SSS0.Px3.p1.11.m11.1.1.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p1.11.m11.1.2.2.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p1.11.m11.1.2.1.1.cmml">|</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.11.m11.1b"><apply id="S2.SS0.SSS0.Px3.p1.11.m11.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p1.11.m11.1.2.2"><abs id="S2.SS0.SSS0.Px3.p1.11.m11.1.2.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.11.m11.1.2.2.1"></abs><ci id="S2.SS0.SSS0.Px3.p1.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.11.m11.1.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.11.m11.1c">|\eta|</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.11.m11.1d">| italic_η |</annotation></semantics></math> the number of literals in <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p1.12.m12.1"><semantics id="S2.SS0.SSS0.Px3.p1.12.m12.1a"><mi id="S2.SS0.SSS0.Px3.p1.12.m12.1.1" xref="S2.SS0.SSS0.Px3.p1.12.m12.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p1.12.m12.1b"><ci id="S2.SS0.SSS0.Px3.p1.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px3.p1.12.m12.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p1.12.m12.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p1.12.m12.1d">italic_η</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px3.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px3.p2.50">Given a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.50.4">total</span> truth assignment <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.1.m1.1"><semantics id="S2.SS0.SSS0.Px3.p2.1.m1.1a"><mi id="S2.SS0.SSS0.Px3.p2.1.m1.1.1" xref="S2.SS0.SSS0.Px3.p2.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.1.m1.1b"><ci id="S2.SS0.SSS0.Px3.p2.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.1.m1.1d">italic_η</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.2.m2.1"><semantics id="S2.SS0.SSS0.Px3.p2.2.m2.1a"><mi id="S2.SS0.SSS0.Px3.p2.2.m2.1.1" xref="S2.SS0.SSS0.Px3.p2.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.2.m2.1b"><ci id="S2.SS0.SSS0.Px3.p2.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.2.m2.1d">bold_A</annotation></semantics></math> and some formulas <math alttext="\varphi,\varphi_{1},\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.3.m3.3"><semantics id="S2.SS0.SSS0.Px3.p2.3.m3.3a"><mrow id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.3.m3.1.1" xref="S2.SS0.SSS0.Px3.p2.3.m3.1.1.cmml">φ</mi><mo id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.3" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1" xref="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.2" xref="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.3" xref="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.4" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.2" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.3" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.3.m3.3b"><list id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.3.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2"><ci id="S2.SS0.SSS0.Px3.p2.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.1.1">𝜑</ci><apply id="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.3.m3.2.2.1.1.3">1</cn></apply><apply id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.3.m3.3.3.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.3.m3.3c">\varphi,\varphi_{1},\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.3.m3.3d">italic_φ , italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.4.m4.1"><semantics id="S2.SS0.SSS0.Px3.p2.4.m4.1a"><mi id="S2.SS0.SSS0.Px3.p2.4.m4.1.1" xref="S2.SS0.SSS0.Px3.p2.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.4.m4.1b"><ci id="S2.SS0.SSS0.Px3.p2.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.4.m4.1d">bold_A</annotation></semantics></math>, the sentence “<math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.5.m5.1"><semantics id="S2.SS0.SSS0.Px3.p2.5.m5.1a"><mi id="S2.SS0.SSS0.Px3.p2.5.m5.1.1" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.5.m5.1b"><ci id="S2.SS0.SSS0.Px3.p2.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.5.m5.1d">italic_η</annotation></semantics></math> <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.50.5">satisfies</span> <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.6.m6.1"><semantics id="S2.SS0.SSS0.Px3.p2.6.m6.1a"><mi id="S2.SS0.SSS0.Px3.p2.6.m6.1.1" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.6.m6.1b"><ci id="S2.SS0.SSS0.Px3.p2.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.6.m6.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.6.m6.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.6.m6.1d">italic_φ</annotation></semantics></math>”, written “<math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.7.m7.1"><semantics id="S2.SS0.SSS0.Px3.p2.7.m7.1a"><mrow id="S2.SS0.SSS0.Px3.p2.7.m7.1.1" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.7.m7.1.1.2" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.7.m7.1.1.1" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1.1.cmml">⊧</mo><mi id="S2.SS0.SSS0.Px3.p2.7.m7.1.1.3" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.7.m7.1b"><apply id="S2.SS0.SSS0.Px3.p2.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1.2">𝜂</ci><ci id="S2.SS0.SSS0.Px3.p2.7.m7.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.7.m7.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.7.m7.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.7.m7.1d">italic_η ⊧ italic_φ</annotation></semantics></math>”, is defined recursively on the structure of <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.8.m8.1"><semantics id="S2.SS0.SSS0.Px3.p2.8.m8.1a"><mi id="S2.SS0.SSS0.Px3.p2.8.m8.1.1" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.8.m8.1b"><ci id="S2.SS0.SSS0.Px3.p2.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.8.m8.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.8.m8.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.8.m8.1d">italic_φ</annotation></semantics></math> as follows: <math alttext="\eta\models\top" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.9.m9.1"><semantics id="S2.SS0.SSS0.Px3.p2.9.m9.1a"><mrow id="S2.SS0.SSS0.Px3.p2.9.m9.1.1" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.9.m9.1.1.2" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.9.m9.1.1.1" rspace="0em" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1.1.cmml">⊧</mo><mo id="S2.SS0.SSS0.Px3.p2.9.m9.1.1.3" lspace="0em" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.9.m9.1b"><apply id="S2.SS0.SSS0.Px3.p2.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1.2">𝜂</ci><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.9.m9.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.9.m9.1.1.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.9.m9.1c">\eta\models\top</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.9.m9.1d">italic_η ⊧ ⊤</annotation></semantics></math>, <math alttext="\eta\not\models\bot" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.10.m10.1"><semantics id="S2.SS0.SSS0.Px3.p2.10.m10.1a"><mrow id="S2.SS0.SSS0.Px3.p2.10.m10.1.1" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.2" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.1" rspace="0em" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.1.cmml">⊧̸</mo><mo id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3" lspace="0em" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.cmml">⊥</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.10.m10.1b"><apply id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.1">not-models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.2">𝜂</ci><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.10.m10.1.1.3">bottom</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.10.m10.1c">\eta\not\models\bot</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.10.m10.1d">italic_η ⊧̸ ⊥</annotation></semantics></math>, <math alttext="\eta\models A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.11.m11.1"><semantics id="S2.SS0.SSS0.Px3.p2.11.m11.1a"><mrow id="S2.SS0.SSS0.Px3.p2.11.m11.1.1" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.2" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.1" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.11.m11.1b"><apply id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px3.p2.11.m11.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.11.m11.1c">\eta\models A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.11.m11.1d">italic_η ⊧ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\eta(A_{i})=\mbox{{\sf T}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.12.m12.1"><semantics id="S2.SS0.SSS0.Px3.p2.12.m12.1a"><mrow id="S2.SS0.SSS0.Px3.p2.12.m12.1.1" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.cmml"><mrow id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.3" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.3.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.2" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.2" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.2.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.3" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.3a.cmml">T</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.12.m12.1b"><apply id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1"><eq id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.2"></eq><apply id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1"><times id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.2"></times><ci id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.3">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.1.1.1.1.3">𝑖</ci></apply></apply><ci id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.3a.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px3.p2.12.m12.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.12.m12.1.1.3">T</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.12.m12.1c">\eta(A_{i})=\mbox{{\sf T}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.12.m12.1d">italic_η ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = T</annotation></semantics></math>, <math alttext="\eta\models\neg\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.13.m13.1"><semantics id="S2.SS0.SSS0.Px3.p2.13.m13.1a"><mrow id="S2.SS0.SSS0.Px3.p2.13.m13.1.1" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.2" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.1" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.1.cmml">⊧</mo><mrow id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.cmml"><mo id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.1" rspace="0.167em" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.2" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.3" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.13.m13.1b"><apply id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3"><not id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.1"></not><apply id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.13.m13.1.1.3.2.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.13.m13.1c">\eta\models\neg\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.13.m13.1d">italic_η ⊧ ¬ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\eta\not\models\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.14.m14.1"><semantics id="S2.SS0.SSS0.Px3.p2.14.m14.1a"><mrow id="S2.SS0.SSS0.Px3.p2.14.m14.1.1" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.2" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.1" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.1.cmml">⊧̸</mo><msub id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.14.m14.1b"><apply id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.1">not-models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.14.m14.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.14.m14.1c">\eta\not\models\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.14.m14.1d">italic_η ⊧̸ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\eta\models\varphi_{1}\wedge\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.15.m15.1"><semantics id="S2.SS0.SSS0.Px3.p2.15.m15.1a"><mrow id="S2.SS0.SSS0.Px3.p2.15.m15.1.1" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.2" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.1" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.1.cmml">⊧</mo><mrow id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.cmml"><msub id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.2" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.3" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.1" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.1.cmml">∧</mo><msub id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.2" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.3" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.15.m15.1b"><apply id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3"><and id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.1"></and><apply id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.15.m15.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.15.m15.1c">\eta\models\varphi_{1}\wedge\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.15.m15.1d">italic_η ⊧ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> if and only if <math alttext="\eta\models\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.16.m16.1"><semantics id="S2.SS0.SSS0.Px3.p2.16.m16.1a"><mrow id="S2.SS0.SSS0.Px3.p2.16.m16.1.1" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.2" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.1" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.16.m16.1b"><apply id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.16.m16.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.16.m16.1c">\eta\models\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.16.m16.1d">italic_η ⊧ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\eta\models\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.17.m17.1"><semantics id="S2.SS0.SSS0.Px3.p2.17.m17.1a"><mrow id="S2.SS0.SSS0.Px3.p2.17.m17.1.1" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.2" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.1" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.17.m17.1b"><apply id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.17.m17.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.17.m17.1c">\eta\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.17.m17.1d">italic_η ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. (The definition of <math alttext="\eta\models\varphi_{1}\bowtie\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.18.m18.1"><semantics id="S2.SS0.SSS0.Px3.p2.18.m18.1a"><mrow id="S2.SS0.SSS0.Px3.p2.18.m18.1.1" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.2" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.3" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.3.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.cmml"><mi id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.2" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.3" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.5" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.5.cmml">⋈</mo><msub id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.cmml"><mi id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.2" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.3" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.18.m18.1b"><apply id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1"><and id="S2.SS0.SSS0.Px3.p2.18.m18.1.1a.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1"></and><apply id="S2.SS0.SSS0.Px3.p2.18.m18.1.1b.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.3">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.1.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.2.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.3">1</cn></apply></apply><apply id="S2.SS0.SSS0.Px3.p2.18.m18.1.1c.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1"><ci id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.5.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.5">⋈</ci><share href="https://arxiv.org/html/2503.01536v1#S2.SS0.SSS0.Px3.p2.18.m18.1.1.4.cmml" id="S2.SS0.SSS0.Px3.p2.18.m18.1.1d.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1"></share><apply id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.1.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.2.cmml" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.18.m18.1.1.6.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.18.m18.1c">\eta\models\varphi_{1}\bowtie\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.18.m18.1d">italic_η ⊧ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋈ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> for the other connectives follows straightforwardly from their definition in terms of <math alttext="\neg,\wedge" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.19.m19.2"><semantics id="S2.SS0.SSS0.Px3.p2.19.m19.2a"><mrow id="S2.SS0.SSS0.Px3.p2.19.m19.2.3.2" xref="S2.SS0.SSS0.Px3.p2.19.m19.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px3.p2.19.m19.1.1" xref="S2.SS0.SSS0.Px3.p2.19.m19.1.1.cmml">¬</mo><mo id="S2.SS0.SSS0.Px3.p2.19.m19.2.3.2.1" rspace="0em" xref="S2.SS0.SSS0.Px3.p2.19.m19.2.3.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px3.p2.19.m19.2.2" lspace="0em" xref="S2.SS0.SSS0.Px3.p2.19.m19.2.2.cmml">∧</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.19.m19.2b"><list id="S2.SS0.SSS0.Px3.p2.19.m19.2.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.19.m19.2.3.2"><not id="S2.SS0.SSS0.Px3.p2.19.m19.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.19.m19.1.1"></not><and id="S2.SS0.SSS0.Px3.p2.19.m19.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.19.m19.2.2"></and></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.19.m19.2c">\neg,\wedge</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.19.m19.2d">¬ , ∧</annotation></semantics></math>.) <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.20.m20.1"><semantics id="S2.SS0.SSS0.Px3.p2.20.m20.1a"><mi id="S2.SS0.SSS0.Px3.p2.20.m20.1.1" xref="S2.SS0.SSS0.Px3.p2.20.m20.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.20.m20.1b"><ci id="S2.SS0.SSS0.Px3.p2.20.m20.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.20.m20.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.20.m20.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.20.m20.1d">italic_φ</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.50.6">satisfiable</span> iff <math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.21.m21.1"><semantics id="S2.SS0.SSS0.Px3.p2.21.m21.1a"><mrow id="S2.SS0.SSS0.Px3.p2.21.m21.1.1" xref="S2.SS0.SSS0.Px3.p2.21.m21.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.21.m21.1.1.2" xref="S2.SS0.SSS0.Px3.p2.21.m21.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.21.m21.1.1.1" xref="S2.SS0.SSS0.Px3.p2.21.m21.1.1.1.cmml">⊧</mo><mi id="S2.SS0.SSS0.Px3.p2.21.m21.1.1.3" xref="S2.SS0.SSS0.Px3.p2.21.m21.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.21.m21.1b"><apply id="S2.SS0.SSS0.Px3.p2.21.m21.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.21.m21.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.21.m21.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.21.m21.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.21.m21.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.21.m21.1.1.2">𝜂</ci><ci id="S2.SS0.SSS0.Px3.p2.21.m21.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.21.m21.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.21.m21.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.21.m21.1d">italic_η ⊧ italic_φ</annotation></semantics></math> for some total truth assignment <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.22.m22.1"><semantics id="S2.SS0.SSS0.Px3.p2.22.m22.1a"><mi id="S2.SS0.SSS0.Px3.p2.22.m22.1.1" xref="S2.SS0.SSS0.Px3.p2.22.m22.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.22.m22.1b"><ci id="S2.SS0.SSS0.Px3.p2.22.m22.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.22.m22.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.22.m22.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.22.m22.1d">italic_η</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.23.m23.1"><semantics id="S2.SS0.SSS0.Px3.p2.23.m23.1a"><mi id="S2.SS0.SSS0.Px3.p2.23.m23.1.1" xref="S2.SS0.SSS0.Px3.p2.23.m23.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.23.m23.1b"><ci id="S2.SS0.SSS0.Px3.p2.23.m23.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.23.m23.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.23.m23.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.23.m23.1d">bold_A</annotation></semantics></math>. <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.24.m24.1"><semantics id="S2.SS0.SSS0.Px3.p2.24.m24.1a"><mi id="S2.SS0.SSS0.Px3.p2.24.m24.1.1" xref="S2.SS0.SSS0.Px3.p2.24.m24.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.24.m24.1b"><ci id="S2.SS0.SSS0.Px3.p2.24.m24.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.24.m24.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.24.m24.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.24.m24.1d">italic_φ</annotation></semantics></math> is <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.50.7">valid</span> (written “<math alttext="\models\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.25.m25.1"><semantics id="S2.SS0.SSS0.Px3.p2.25.m25.1a"><mrow id="S2.SS0.SSS0.Px3.p2.25.m25.1.1" xref="S2.SS0.SSS0.Px3.p2.25.m25.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.25.m25.1.1.2" xref="S2.SS0.SSS0.Px3.p2.25.m25.1.1.2.cmml"></mi><mo id="S2.SS0.SSS0.Px3.p2.25.m25.1.1.1" xref="S2.SS0.SSS0.Px3.p2.25.m25.1.1.1.cmml">⊧</mo><mi id="S2.SS0.SSS0.Px3.p2.25.m25.1.1.3" xref="S2.SS0.SSS0.Px3.p2.25.m25.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.25.m25.1b"><apply id="S2.SS0.SSS0.Px3.p2.25.m25.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.25.m25.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.25.m25.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.25.m25.1.1.1">models</csymbol><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.25.m25.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.25.m25.1.1.2">absent</csymbol><ci id="S2.SS0.SSS0.Px3.p2.25.m25.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.25.m25.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.25.m25.1c">\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.25.m25.1d">⊧ italic_φ</annotation></semantics></math>”) iff <math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.26.m26.1"><semantics id="S2.SS0.SSS0.Px3.p2.26.m26.1a"><mrow id="S2.SS0.SSS0.Px3.p2.26.m26.1.1" xref="S2.SS0.SSS0.Px3.p2.26.m26.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.26.m26.1.1.2" xref="S2.SS0.SSS0.Px3.p2.26.m26.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.26.m26.1.1.1" xref="S2.SS0.SSS0.Px3.p2.26.m26.1.1.1.cmml">⊧</mo><mi id="S2.SS0.SSS0.Px3.p2.26.m26.1.1.3" xref="S2.SS0.SSS0.Px3.p2.26.m26.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.26.m26.1b"><apply id="S2.SS0.SSS0.Px3.p2.26.m26.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.26.m26.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.26.m26.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.26.m26.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.26.m26.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.26.m26.1.1.2">𝜂</ci><ci id="S2.SS0.SSS0.Px3.p2.26.m26.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.26.m26.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.26.m26.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.26.m26.1d">italic_η ⊧ italic_φ</annotation></semantics></math> for every total truth assignment <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.27.m27.1"><semantics id="S2.SS0.SSS0.Px3.p2.27.m27.1a"><mi id="S2.SS0.SSS0.Px3.p2.27.m27.1.1" xref="S2.SS0.SSS0.Px3.p2.27.m27.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.27.m27.1b"><ci id="S2.SS0.SSS0.Px3.p2.27.m27.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.27.m27.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.27.m27.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.27.m27.1d">italic_η</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.28.m28.1"><semantics id="S2.SS0.SSS0.Px3.p2.28.m28.1a"><mi id="S2.SS0.SSS0.Px3.p2.28.m28.1.1" xref="S2.SS0.SSS0.Px3.p2.28.m28.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.28.m28.1b"><ci id="S2.SS0.SSS0.Px3.p2.28.m28.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.28.m28.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.28.m28.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.28.m28.1d">bold_A</annotation></semantics></math>. <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.29.m29.1"><semantics id="S2.SS0.SSS0.Px3.p2.29.m29.1a"><msub id="S2.SS0.SSS0.Px3.p2.29.m29.1.1" xref="S2.SS0.SSS0.Px3.p2.29.m29.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.29.m29.1.1.2" xref="S2.SS0.SSS0.Px3.p2.29.m29.1.1.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.29.m29.1.1.3" xref="S2.SS0.SSS0.Px3.p2.29.m29.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.29.m29.1b"><apply id="S2.SS0.SSS0.Px3.p2.29.m29.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.29.m29.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.29.m29.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.29.m29.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.29.m29.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.29.m29.1.1.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.29.m29.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.29.m29.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.29.m29.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.29.m29.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.30.1"> entails <math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.30.1.m1.1"><semantics id="S2.SS0.SSS0.Px3.p2.30.1.m1.1a"><msub id="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1" xref="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.2" xref="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.3" xref="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.30.1.m1.1b"><apply id="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.30.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.30.1.m1.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.30.1.m1.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math></span> (written <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.31.2">“<math alttext="\varphi_{1}\models\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.31.2.m1.1"><semantics id="S2.SS0.SSS0.Px3.p2.31.2.m1.1a"><mrow id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.cmml"><msub id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.3" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.1" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.31.2.m1.1b"><apply id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.1">models</csymbol><apply id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.31.2.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.31.2.m1.1c">\varphi_{1}\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.31.2.m1.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>”</span>) iff, for every total assignment <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.32.m30.1"><semantics id="S2.SS0.SSS0.Px3.p2.32.m30.1a"><mi id="S2.SS0.SSS0.Px3.p2.32.m30.1.1" xref="S2.SS0.SSS0.Px3.p2.32.m30.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.32.m30.1b"><ci id="S2.SS0.SSS0.Px3.p2.32.m30.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.32.m30.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.32.m30.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.32.m30.1d">italic_η</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.33.m31.1"><semantics id="S2.SS0.SSS0.Px3.p2.33.m31.1a"><mi id="S2.SS0.SSS0.Px3.p2.33.m31.1.1" xref="S2.SS0.SSS0.Px3.p2.33.m31.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.33.m31.1b"><ci id="S2.SS0.SSS0.Px3.p2.33.m31.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.33.m31.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.33.m31.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.33.m31.1d">bold_A</annotation></semantics></math>, if <math alttext="\eta\models\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.34.m32.1"><semantics id="S2.SS0.SSS0.Px3.p2.34.m32.1a"><mrow id="S2.SS0.SSS0.Px3.p2.34.m32.1.1" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.2" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.1" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.34.m32.1b"><apply id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.34.m32.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.34.m32.1c">\eta\models\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.34.m32.1d">italic_η ⊧ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> then <math alttext="\eta\models\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.35.m33.1"><semantics id="S2.SS0.SSS0.Px3.p2.35.m33.1a"><mrow id="S2.SS0.SSS0.Px3.p2.35.m33.1.1" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.2" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.1" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.35.m33.1b"><apply id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.35.m33.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.35.m33.1c">\eta\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.35.m33.1d">italic_η ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.36.m34.1"><semantics id="S2.SS0.SSS0.Px3.p2.36.m34.1a"><msub id="S2.SS0.SSS0.Px3.p2.36.m34.1.1" xref="S2.SS0.SSS0.Px3.p2.36.m34.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.36.m34.1.1.2" xref="S2.SS0.SSS0.Px3.p2.36.m34.1.1.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.36.m34.1.1.3" xref="S2.SS0.SSS0.Px3.p2.36.m34.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.36.m34.1b"><apply id="S2.SS0.SSS0.Px3.p2.36.m34.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.36.m34.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.36.m34.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.36.m34.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.36.m34.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.36.m34.1.1.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.36.m34.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.36.m34.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.36.m34.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.36.m34.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.37.m35.1"><semantics id="S2.SS0.SSS0.Px3.p2.37.m35.1a"><msub id="S2.SS0.SSS0.Px3.p2.37.m35.1.1" xref="S2.SS0.SSS0.Px3.p2.37.m35.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.37.m35.1.1.2" xref="S2.SS0.SSS0.Px3.p2.37.m35.1.1.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.37.m35.1.1.3" xref="S2.SS0.SSS0.Px3.p2.37.m35.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.37.m35.1b"><apply id="S2.SS0.SSS0.Px3.p2.37.m35.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.37.m35.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.37.m35.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.37.m35.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.37.m35.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.37.m35.1.1.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.37.m35.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.37.m35.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.37.m35.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.37.m35.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.50.8"> equivalent</span> (written <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.38.3">“<math alttext="\varphi_{1}\equiv\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.38.3.m1.1"><semantics id="S2.SS0.SSS0.Px3.p2.38.3.m1.1a"><mrow id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.cmml"><msub id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.3" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.1" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.1.cmml">≡</mo><msub id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.38.3.m1.1b"><apply id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1"><equivalent id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.1"></equivalent><apply id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.38.3.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.38.3.m1.1c">\varphi_{1}\equiv\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.38.3.m1.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≡ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>”</span>) iff <math alttext="\varphi_{1}\models\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.39.m36.1"><semantics id="S2.SS0.SSS0.Px3.p2.39.m36.1a"><mrow id="S2.SS0.SSS0.Px3.p2.39.m36.1.1" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.cmml"><msub id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.3" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.1" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.39.m36.1b"><apply id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.1">models</csymbol><apply id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.39.m36.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.39.m36.1c">\varphi_{1}\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.39.m36.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{2}\models\varphi_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.40.m37.1"><semantics id="S2.SS0.SSS0.Px3.p2.40.m37.1a"><mrow id="S2.SS0.SSS0.Px3.p2.40.m37.1.1" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.cmml"><msub id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.3" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.1" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.40.m37.1b"><apply id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.1">models</csymbol><apply id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.2.3">2</cn></apply><apply id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.40.m37.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.40.m37.1c">\varphi_{2}\models\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.40.m37.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊧ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Consequently: <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.41.m38.1"><semantics id="S2.SS0.SSS0.Px3.p2.41.m38.1a"><mi id="S2.SS0.SSS0.Px3.p2.41.m38.1.1" xref="S2.SS0.SSS0.Px3.p2.41.m38.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.41.m38.1b"><ci id="S2.SS0.SSS0.Px3.p2.41.m38.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.41.m38.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.41.m38.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.41.m38.1d">italic_φ</annotation></semantics></math> is unsatisfiable iff <math alttext="\neg\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.42.m39.1"><semantics id="S2.SS0.SSS0.Px3.p2.42.m39.1a"><mrow id="S2.SS0.SSS0.Px3.p2.42.m39.1.1" xref="S2.SS0.SSS0.Px3.p2.42.m39.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p2.42.m39.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px3.p2.42.m39.1.1.1.cmml">¬</mo><mi id="S2.SS0.SSS0.Px3.p2.42.m39.1.1.2" xref="S2.SS0.SSS0.Px3.p2.42.m39.1.1.2.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.42.m39.1b"><apply id="S2.SS0.SSS0.Px3.p2.42.m39.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.42.m39.1.1"><not id="S2.SS0.SSS0.Px3.p2.42.m39.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.42.m39.1.1.1"></not><ci id="S2.SS0.SSS0.Px3.p2.42.m39.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.42.m39.1.1.2">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.42.m39.1c">\neg\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.42.m39.1d">¬ italic_φ</annotation></semantics></math> is valid; <math alttext="\varphi_{1}\models\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.43.m40.1"><semantics id="S2.SS0.SSS0.Px3.p2.43.m40.1a"><mrow id="S2.SS0.SSS0.Px3.p2.43.m40.1.1" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.cmml"><msub id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.3" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.1" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.1.cmml">⊧</mo><msub id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.43.m40.1b"><apply id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.1">models</csymbol><apply id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.43.m40.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.43.m40.1c">\varphi_{1}\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.43.m40.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> iff <math alttext="\varphi_{1}\rightarrow\varphi_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.44.m41.1"><semantics id="S2.SS0.SSS0.Px3.p2.44.m41.1a"><mrow id="S2.SS0.SSS0.Px3.p2.44.m41.1.1" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.cmml"><msub id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.3" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.1.cmml">→</mo><msub id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.2" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.3" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.44.m41.1b"><apply id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1"><ci id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.1">→</ci><apply id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.2.3">1</cn></apply><apply id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.2">𝜑</ci><cn id="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px3.p2.44.m41.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.44.m41.1c">\varphi_{1}\rightarrow\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.44.m41.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is valid; a clause <math alttext="\bigvee_{i}l_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.45.m42.1"><semantics id="S2.SS0.SSS0.Px3.p2.45.m42.1a"><mrow id="S2.SS0.SSS0.Px3.p2.45.m42.1.1" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.cmml"><msub id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.2" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.2.cmml">⋁</mo><mi id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.3" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.3.cmml">i</mi></msub><msub id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.2.cmml">l</mi><mi id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.3" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.45.m42.1b"><apply id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1"><apply id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1">subscript</csymbol><or id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.2"></or><ci id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.1.3">𝑖</ci></apply><apply id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.2">𝑙</ci><ci id="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px3.p2.45.m42.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.45.m42.1c">\bigvee_{i}l_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.45.m42.1d">⋁ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_l start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is valid (aka is a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.50.9">tautology</span>) iff both <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.46.m43.1"><semantics id="S2.SS0.SSS0.Px3.p2.46.m43.1a"><msub id="S2.SS0.SSS0.Px3.p2.46.m43.1.1" xref="S2.SS0.SSS0.Px3.p2.46.m43.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.46.m43.1.1.2" xref="S2.SS0.SSS0.Px3.p2.46.m43.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p2.46.m43.1.1.3" xref="S2.SS0.SSS0.Px3.p2.46.m43.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.46.m43.1b"><apply id="S2.SS0.SSS0.Px3.p2.46.m43.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.46.m43.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.46.m43.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.46.m43.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.46.m43.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.46.m43.1.1.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p2.46.m43.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.46.m43.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.46.m43.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.46.m43.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\neg A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.47.m44.1"><semantics id="S2.SS0.SSS0.Px3.p2.47.m44.1a"><mrow id="S2.SS0.SSS0.Px3.p2.47.m44.1.1" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.cmml"><mo id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.1.cmml">¬</mo><msub id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.2" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.3" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.47.m44.1b"><apply id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1"><not id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.1"></not><apply id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px3.p2.47.m44.1.1.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.47.m44.1c">\neg A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.47.m44.1d">¬ italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> occur in it for some <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.48.m45.1"><semantics id="S2.SS0.SSS0.Px3.p2.48.m45.1a"><msub id="S2.SS0.SSS0.Px3.p2.48.m45.1.1" xref="S2.SS0.SSS0.Px3.p2.48.m45.1.1.cmml"><mi id="S2.SS0.SSS0.Px3.p2.48.m45.1.1.2" xref="S2.SS0.SSS0.Px3.p2.48.m45.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px3.p2.48.m45.1.1.3" xref="S2.SS0.SSS0.Px3.p2.48.m45.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.48.m45.1b"><apply id="S2.SS0.SSS0.Px3.p2.48.m45.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.48.m45.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px3.p2.48.m45.1.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.48.m45.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px3.p2.48.m45.1.1.2.cmml" xref="S2.SS0.SSS0.Px3.p2.48.m45.1.1.2">𝐴</ci><ci id="S2.SS0.SSS0.Px3.p2.48.m45.1.1.3.cmml" xref="S2.SS0.SSS0.Px3.p2.48.m45.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.48.m45.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.48.m45.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; a CNF formula <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.49.m46.1"><semantics id="S2.SS0.SSS0.Px3.p2.49.m46.1a"><mi id="S2.SS0.SSS0.Px3.p2.49.m46.1.1" xref="S2.SS0.SSS0.Px3.p2.49.m46.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.49.m46.1b"><ci id="S2.SS0.SSS0.Px3.p2.49.m46.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.49.m46.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.49.m46.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.49.m46.1d">italic_φ</annotation></semantics></math> is valid iff either it is <math alttext="\top" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px3.p2.50.m47.1"><semantics id="S2.SS0.SSS0.Px3.p2.50.m47.1a"><mo id="S2.SS0.SSS0.Px3.p2.50.m47.1.1" xref="S2.SS0.SSS0.Px3.p2.50.m47.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px3.p2.50.m47.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px3.p2.50.m47.1.1.cmml" xref="S2.SS0.SSS0.Px3.p2.50.m47.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px3.p2.50.m47.1c">\top</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px3.p2.50.m47.1d">⊤</annotation></semantics></math> or all its clauses are tautologies. We say that a CNF formula is <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px3.p2.50.10">tautology-free</span> iff none of its clauses is a tautology.</p> </div> <figure class="ltx_figure" id="S2.F2"> <table class="ltx_tabular ltx_align_middle" id="S2.F2.6"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S2.F2.6.6"> <td class="ltx_td ltx_align_center" id="S2.F2.5.5.5"> <div class="ltx_block ltx_minipage ltx_align_top" id="S2.F2.5.5.5.5" style="width:195.1pt;"> <p class="ltx_p" id="S2.F2.1.1.1.1.1"><math alttext="\begin{array}[]{||c||l|l|l|l|l|l|l|l|l||}\hline\cr\mu(\varphi_{1})&\mbox{{\sf T% }}&\mbox{{\sf T}}&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf?}}&% \mbox{{\sf F}}&\mbox{{\sf F}}&\mbox{{\sf F}}\\ \mu(\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf F}}&\mbox{{\sf T}}&% \mbox{{\sf?}}&\mbox{{\sf F}}&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf F}}\\ \hline\cr\mu(\neg\varphi_{1})&\mbox{{\sf F}}&\mbox{{\sf F}}&\mbox{{\sf F}}&% \mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf T}}&\mbox{{\sf T}}&\mbox{% {\sf T}}\\ \mu(\varphi_{1}\wedge\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf F}}&% \mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf F}}&\mbox{{\sf F}}&\mbox{{\sf F}}&\mbox% {{\sf F}}\\ \mu(\varphi_{1}\vee\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf T}}&\mbox{{\sf T}}&% \mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{% {\sf F}}\\ \mu(\varphi_{1}\rightarrow\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf F% }}&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf T}}&\mbox{{\sf T}}&% \mbox{{\sf T}}\\ \mu(\varphi_{1}\leftrightarrow\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{% {\sf F}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf F}}&\mbox{{\sf?}% }&\mbox{{\sf T}}\\ \hline\cr\end{array}" class="ltx_math_unparsed" display="inline" id="S2.F2.1.1.1.1.1.m1.6"><semantics id="S2.F2.1.1.1.1.1.m1.6a"><mtable columnspacing="5pt" id="S2.F2.1.1.1.1.1.m1.6.6" rowspacing="0pt"><mtr id="S2.F2.1.1.1.1.1.m1.6.6a"><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6b"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6c"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6d"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6e"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6f"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6g"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6h"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6i"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6j"></mtd><mtd class="ltx_border_t" id="S2.F2.1.1.1.1.1.m1.6.6k"></mtd></mtr><mtr id="S2.F2.1.1.1.1.1.m1.6.6l"><mtd class="ltx_border_ll ltx_border_rr" id="S2.F2.1.1.1.1.1.m1.6.6m"><mrow id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1"><mi id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1.3" mathsize="90%">μ</mi><mo id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1.2">⁢</mo><mrow id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1.1.1"><mo id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1.1.1.2" maxsize="90%" minsize="90%">(</mo><msub id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1.1.1.1"><mi id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1.1.1.1.2" mathsize="90%">φ</mi><mn id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1.1.1.1.3" mathsize="90%">1</mn></msub><mo id="S2.F2.1.1.1.1.1.m1.1.1.1.1.1.1.1.3" maxsize="90%" minsize="90%">)</mo></mrow></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6n"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.2.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6o"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.3.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6p"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.4.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6q"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.5.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6r"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.6.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6s"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.7.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6t"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.8.1" mathsize="90%">F</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6u"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.9.1" mathsize="90%">F</mtext></mtd><mtd class="ltx_border_rr" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6v"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.1.1.1.10.1" mathsize="90%">F</mtext></mtd></mtr><mtr id="S2.F2.1.1.1.1.1.m1.6.6w"><mtd class="ltx_border_ll 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columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6bv"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.5.5.5.7.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6bw"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.5.5.5.8.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6bx"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.5.5.5.9.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_rr" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6by"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.5.5.5.10.1" mathsize="90%">F</mtext></mtd></mtr><mtr id="S2.F2.1.1.1.1.1.m1.6.6bz"><mtd class="ltx_border_ll ltx_border_rr" id="S2.F2.1.1.1.1.1.m1.6.6ca"><mrow id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1"><mi id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.3" mathsize="90%">μ</mi><mo id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.2">⁢</mo><mrow id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1"><mo id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.2" maxsize="90%" minsize="90%">(</mo><mrow id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.1"><msub id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.1.2"><mi id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.1.2.2" mathsize="90%">φ</mi><mn id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.1.2.3" mathsize="90%">1</mn></msub><mo id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.1.1" mathsize="90%" stretchy="false">→</mo><msub id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.1.3"><mi id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.1.3.2" mathsize="90%">φ</mi><mn id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.1.3.3" mathsize="90%">2</mn></msub></mrow><mo id="S2.F2.1.1.1.1.1.m1.6.6.6.1.1.1.1.3" maxsize="90%" minsize="90%">)</mo></mrow></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cb"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.2.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cc"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.3.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cd"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.4.1" mathsize="90%">F</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6ce"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.5.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cf"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.6.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cg"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.7.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6ch"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.8.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6ci"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.9.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_rr" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cj"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.6.10.1" mathsize="90%">T</mtext></mtd></mtr><mtr id="S2.F2.1.1.1.1.1.m1.6.6ck"><mtd class="ltx_border_b ltx_border_ll ltx_border_rr" id="S2.F2.1.1.1.1.1.m1.6.6cl"><mrow id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1"><mi id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.1" mathsize="90%">μ</mi><mrow id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2"><mo id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.1" maxsize="90%" minsize="90%">(</mo><msub id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.2"><mi id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.2.2" mathsize="90%">φ</mi><mn id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.2.3" mathsize="90%">1</mn></msub><mo id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.3" mathsize="90%" stretchy="false">↔</mo><msub id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.4"><mi id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.4.2" mathsize="90%">φ</mi><mn id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.4.3" mathsize="90%">2</mn></msub><mo id="S2.F2.1.1.1.1.1.m1.6.6.9.1.1.2.5" maxsize="90%" minsize="90%">)</mo></mrow></mrow></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cm"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.2.1" mathsize="90%">T</mtext></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cn"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.3.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6co"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.4.1" mathsize="90%">F</mtext></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cp"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.5.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cq"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.6.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cr"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.7.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cs"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.8.1" mathsize="90%">F</mtext></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6ct"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.9.1" mathsize="90%">?</mtext></mtd><mtd class="ltx_border_b ltx_border_rr" columnalign="left" id="S2.F2.1.1.1.1.1.m1.6.6cu"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.1.1.1.1.1.m1.6.6.9.10.1" mathsize="90%">T</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.F2.1.1.1.1.1.m1.6b">\begin{array}[]{||c||l|l|l|l|l|l|l|l|l||}\hline\cr\mu(\varphi_{1})&\mbox{{\sf T% }}&\mbox{{\sf T}}&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf?}}&% \mbox{{\sf F}}&\mbox{{\sf F}}&\mbox{{\sf F}}\\ \mu(\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf F}}&\mbox{{\sf T}}&% \mbox{{\sf?}}&\mbox{{\sf F}}&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf F}}\\ \hline\cr\mu(\neg\varphi_{1})&\mbox{{\sf F}}&\mbox{{\sf F}}&\mbox{{\sf F}}&% \mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf T}}&\mbox{{\sf T}}&\mbox{% {\sf T}}\\ \mu(\varphi_{1}\wedge\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf F}}&% \mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf F}}&\mbox{{\sf F}}&\mbox{{\sf F}}&\mbox% {{\sf F}}\\ \mu(\varphi_{1}\vee\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf T}}&\mbox{{\sf T}}&% \mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{% {\sf F}}\\ \mu(\varphi_{1}\rightarrow\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf F% }}&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf T}}&\mbox{{\sf T}}&% \mbox{{\sf T}}\\ \mu(\varphi_{1}\leftrightarrow\varphi_{2})&\mbox{{\sf T}}&\mbox{{\sf?}}&\mbox{% {\sf F}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf?}}&\mbox{{\sf F}}&\mbox{{\sf?}% }&\mbox{{\sf T}}\\ \hline\cr\end{array}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.1.1.1.1.1.m1.6c">start_ARRAY start_ROW start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_μ ( italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL T end_CELL start_CELL T end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL F end_CELL start_CELL F end_CELL start_CELL F end_CELL end_ROW start_ROW start_CELL italic_μ ( italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL F end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL F end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL F end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_μ ( ¬ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL F end_CELL start_CELL F end_CELL start_CELL F end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL T end_CELL start_CELL T end_CELL start_CELL T end_CELL end_ROW start_ROW start_CELL italic_μ ( italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL F end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL F end_CELL start_CELL F end_CELL start_CELL F end_CELL start_CELL F end_CELL end_ROW start_ROW start_CELL italic_μ ( italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL start_CELL T end_CELL start_CELL T end_CELL start_CELL T end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL F end_CELL end_ROW start_ROW start_CELL italic_μ ( italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL F end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL T end_CELL start_CELL T end_CELL start_CELL T end_CELL end_ROW start_ROW start_CELL italic_μ ( italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↔ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_CELL start_CELL T end_CELL start_CELL ? end_CELL start_CELL F end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL ? end_CELL start_CELL F end_CELL start_CELL ? end_CELL start_CELL T end_CELL end_ROW end_ARRAY</annotation></semantics></math><span class="ltx_text" id="S2.F2.1.1.1.1.1.1" style="font-size:90%;"></span></p> <figcaption class="ltx_caption" style="font-size:90%;"><span class="ltx_tag ltx_tag_block">Figure 1: </span> Three-value-semantics of <math alttext="\mu(\varphi)" class="ltx_Math" display="inline" id="S2.F2.4.4.4.4.4.m1.1"><semantics id="S2.F2.4.4.4.4.4.m1.1a"><mrow id="S2.F2.4.4.4.4.4.m1.1.2" xref="S2.F2.4.4.4.4.4.m1.1.2.cmml"><mi id="S2.F2.4.4.4.4.4.m1.1.2.2" xref="S2.F2.4.4.4.4.4.m1.1.2.2.cmml">μ</mi><mo id="S2.F2.4.4.4.4.4.m1.1.2.1" xref="S2.F2.4.4.4.4.4.m1.1.2.1.cmml">⁢</mo><mrow id="S2.F2.4.4.4.4.4.m1.1.2.3.2" xref="S2.F2.4.4.4.4.4.m1.1.2.cmml"><mo id="S2.F2.4.4.4.4.4.m1.1.2.3.2.1" stretchy="false" xref="S2.F2.4.4.4.4.4.m1.1.2.cmml">(</mo><mi id="S2.F2.4.4.4.4.4.m1.1.1" xref="S2.F2.4.4.4.4.4.m1.1.1.cmml">φ</mi><mo id="S2.F2.4.4.4.4.4.m1.1.2.3.2.2" stretchy="false" xref="S2.F2.4.4.4.4.4.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.4.4.4.4.4.m1.1b"><apply id="S2.F2.4.4.4.4.4.m1.1.2.cmml" xref="S2.F2.4.4.4.4.4.m1.1.2"><times id="S2.F2.4.4.4.4.4.m1.1.2.1.cmml" xref="S2.F2.4.4.4.4.4.m1.1.2.1"></times><ci id="S2.F2.4.4.4.4.4.m1.1.2.2.cmml" xref="S2.F2.4.4.4.4.4.m1.1.2.2">𝜇</ci><ci id="S2.F2.4.4.4.4.4.m1.1.1.cmml" xref="S2.F2.4.4.4.4.4.m1.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.4.4.4.4.4.m1.1c">\mu(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S2.F2.4.4.4.4.4.m1.1d">italic_μ ( italic_φ )</annotation></semantics></math> in terms of <math alttext="\{{\mbox{{\sf T}},\mbox{{\sf F}},\mbox{{\sf?}}}\}" class="ltx_Math" display="inline" id="S2.F2.5.5.5.5.5.m2.3"><semantics id="S2.F2.5.5.5.5.5.m2.3a"><mrow id="S2.F2.5.5.5.5.5.m2.3.4.2" xref="S2.F2.5.5.5.5.5.m2.3.4.1.cmml"><mo id="S2.F2.5.5.5.5.5.m2.3.4.2.1" stretchy="false" xref="S2.F2.5.5.5.5.5.m2.3.4.1.cmml">{</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.5.5.5.5.5.m2.1.1" xref="S2.F2.5.5.5.5.5.m2.1.1a.cmml">T</mtext><mo id="S2.F2.5.5.5.5.5.m2.3.4.2.2" xref="S2.F2.5.5.5.5.5.m2.3.4.1.cmml">,</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.5.5.5.5.5.m2.2.2" xref="S2.F2.5.5.5.5.5.m2.2.2a.cmml">F</mtext><mo id="S2.F2.5.5.5.5.5.m2.3.4.2.3" xref="S2.F2.5.5.5.5.5.m2.3.4.1.cmml">,</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.5.5.5.5.5.m2.3.3" xref="S2.F2.5.5.5.5.5.m2.3.3a.cmml">?</mtext><mo id="S2.F2.5.5.5.5.5.m2.3.4.2.4" stretchy="false" xref="S2.F2.5.5.5.5.5.m2.3.4.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.F2.5.5.5.5.5.m2.3b"><set id="S2.F2.5.5.5.5.5.m2.3.4.1.cmml" xref="S2.F2.5.5.5.5.5.m2.3.4.2"><ci id="S2.F2.5.5.5.5.5.m2.1.1a.cmml" xref="S2.F2.5.5.5.5.5.m2.1.1"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.5.5.5.5.5.m2.1.1.cmml" xref="S2.F2.5.5.5.5.5.m2.1.1">T</mtext></ci><ci id="S2.F2.5.5.5.5.5.m2.2.2a.cmml" xref="S2.F2.5.5.5.5.5.m2.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.5.5.5.5.5.m2.2.2.cmml" xref="S2.F2.5.5.5.5.5.m2.2.2">F</mtext></ci><ci id="S2.F2.5.5.5.5.5.m2.3.3a.cmml" xref="S2.F2.5.5.5.5.5.m2.3.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.F2.5.5.5.5.5.m2.3.3.cmml" xref="S2.F2.5.5.5.5.5.m2.3.3">?</mtext></ci></set></annotation-xml><annotation encoding="application/x-tex" id="S2.F2.5.5.5.5.5.m2.3c">\{{\mbox{{\sf T}},\mbox{{\sf F}},\mbox{{\sf?}}}\}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.5.5.5.5.5.m2.3d">{ T , F , ? }</annotation></semantics></math>. <br class="ltx_break"/></figcaption> </div> </td> <td class="ltx_td" id="S2.F2.6.6.7"></td> <td class="ltx_td ltx_align_center" id="S2.F2.6.6.6"> <div class="ltx_block ltx_minipage ltx_align_top" id="S2.F2.6.6.6.1" style="width:173.4pt;"> <p class="ltx_p" id="S2.F2.6.6.6.1.1"><math alttext="\begin{array}[]{|cl|cl|}\hline\cr\neg\top&\Rightarrow\bot&\neg\bot&\Rightarrow% \top\\ \top\wedge\varphi,\varphi\wedge\top&\Rightarrow\varphi&\bot\wedge\varphi,% \varphi\wedge\bot&\Rightarrow\bot\\ \top\vee\varphi,\varphi\vee\top&\Rightarrow\top&\bot\vee\varphi,\varphi\vee% \bot&\Rightarrow\varphi\\ \top\rightarrow\varphi&\Rightarrow\varphi&\bot\rightarrow\varphi&\Rightarrow% \top\\ \varphi\rightarrow\top&\Rightarrow\top&\varphi\rightarrow\bot&\Rightarrow\neg% \varphi\\ \top\leftrightarrow\varphi,\varphi\leftrightarrow\top&\Rightarrow\varphi&\bot% \leftrightarrow\varphi,\varphi\leftrightarrow\bot&\Rightarrow\neg\varphi\\ \hline\cr\end{array}" class="ltx_math_unparsed" display="inline" id="S2.F2.6.6.6.1.1.m1.16"><semantics id="S2.F2.6.6.6.1.1.m1.16a"><mtable columnspacing="5pt" id="S2.F2.6.6.6.1.1.m1.16.16" rowspacing="0pt"><mtr id="S2.F2.6.6.6.1.1.m1.16.16a"><mtd class="ltx_border_t" id="S2.F2.6.6.6.1.1.m1.16.16b"></mtd><mtd class="ltx_border_t" id="S2.F2.6.6.6.1.1.m1.16.16c"></mtd><mtd class="ltx_border_t" columnspan="-1" id="S2.F2.6.6.6.1.1.m1.16.16d"></mtd><mtd class="ltx_border_t" id="S2.F2.6.6.6.1.1.m1.16.16e"></mtd></mtr><mtr id="S2.F2.6.6.6.1.1.m1.16.16f"><mtd class="ltx_border_l" id="S2.F2.6.6.6.1.1.m1.16.16g"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.18.1.1"><mo id="S2.F2.6.6.6.1.1.m1.16.16.18.1.1.2" mathsize="90%">¬</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.18.1.1.3" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16h"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.18.2.1"><mi id="S2.F2.6.6.6.1.1.m1.16.16.18.2.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.18.2.1.1" mathsize="90%" rspace="0em" stretchy="false">⇒</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.18.2.1.3" lspace="0em" mathsize="90%">⊥</mo></mrow></mtd><mtd id="S2.F2.6.6.6.1.1.m1.16.16i"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.18.3.1"><mo id="S2.F2.6.6.6.1.1.m1.16.16.18.3.1.2" mathsize="90%">¬</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.18.3.1.3" lspace="0em" mathsize="90%">⊥</mo></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16j"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.18.4.1"><mi id="S2.F2.6.6.6.1.1.m1.16.16.18.4.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.18.4.1.1" mathsize="90%" rspace="0em" stretchy="false">⇒</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.18.4.1.3" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd></mtr><mtr id="S2.F2.6.6.6.1.1.m1.16.16k"><mtd class="ltx_border_l" id="S2.F2.6.6.6.1.1.m1.16.16l"><mrow id="S2.F2.6.6.6.1.1.m1.2.2.2.2.2"><mo id="S2.F2.6.6.6.1.1.m1.1.1.1.1.1.1" mathsize="90%" rspace="0em">⊤</mo><mo id="S2.F2.6.6.6.1.1.m1.2.2.2.2.2.2" lspace="0em" mathsize="90%">∧</mo><mi id="S2.F2.6.6.6.1.1.m1.2.2.2.2.2.3" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.2.2.2.2.2.4" mathsize="90%">,</mo><mi id="S2.F2.6.6.6.1.1.m1.2.2.2.2.2.5" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.2.2.2.2.2.6" mathsize="90%" rspace="0em">∧</mo><mo id="S2.F2.6.6.6.1.1.m1.2.2.2.2.2.7" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16m"><mrow id="S2.F2.6.6.6.1.1.m1.4.4.4.5.1"><mi id="S2.F2.6.6.6.1.1.m1.4.4.4.5.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.4.4.4.5.1.1" mathsize="90%" stretchy="false">⇒</mo><mi id="S2.F2.6.6.6.1.1.m1.4.4.4.5.1.3" mathsize="90%">φ</mi></mrow></mtd><mtd id="S2.F2.6.6.6.1.1.m1.16.16n"><mrow id="S2.F2.6.6.6.1.1.m1.4.4.4.4.2"><mo id="S2.F2.6.6.6.1.1.m1.3.3.3.3.1.1" mathsize="90%" rspace="0em">⊥</mo><mo id="S2.F2.6.6.6.1.1.m1.4.4.4.4.2.2" lspace="0em" mathsize="90%">∧</mo><mi id="S2.F2.6.6.6.1.1.m1.4.4.4.4.2.3" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.4.4.4.4.2.4" mathsize="90%">,</mo><mi id="S2.F2.6.6.6.1.1.m1.4.4.4.4.2.5" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.4.4.4.4.2.6" mathsize="90%" rspace="0em">∧</mo><mo id="S2.F2.6.6.6.1.1.m1.4.4.4.4.2.7" lspace="0em" mathsize="90%">⊥</mo></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16o"><mrow id="S2.F2.6.6.6.1.1.m1.4.4.4.6.1"><mi id="S2.F2.6.6.6.1.1.m1.4.4.4.6.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.4.4.4.6.1.1" mathsize="90%" rspace="0em" stretchy="false">⇒</mo><mo id="S2.F2.6.6.6.1.1.m1.4.4.4.6.1.3" lspace="0em" mathsize="90%">⊥</mo></mrow></mtd></mtr><mtr id="S2.F2.6.6.6.1.1.m1.16.16p"><mtd class="ltx_border_l" id="S2.F2.6.6.6.1.1.m1.16.16q"><mrow id="S2.F2.6.6.6.1.1.m1.6.6.6.2.2"><mo id="S2.F2.6.6.6.1.1.m1.5.5.5.1.1.1" mathsize="90%" rspace="0em">⊤</mo><mo id="S2.F2.6.6.6.1.1.m1.6.6.6.2.2.2" lspace="0em" mathsize="90%">∨</mo><mi id="S2.F2.6.6.6.1.1.m1.6.6.6.2.2.3" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.6.6.6.2.2.4" mathsize="90%">,</mo><mi id="S2.F2.6.6.6.1.1.m1.6.6.6.2.2.5" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.6.6.6.2.2.6" mathsize="90%" rspace="0em">∨</mo><mo id="S2.F2.6.6.6.1.1.m1.6.6.6.2.2.7" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16r"><mrow id="S2.F2.6.6.6.1.1.m1.8.8.8.5.1"><mi id="S2.F2.6.6.6.1.1.m1.8.8.8.5.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.8.8.8.5.1.1" mathsize="90%" rspace="0em" stretchy="false">⇒</mo><mo id="S2.F2.6.6.6.1.1.m1.8.8.8.5.1.3" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd><mtd id="S2.F2.6.6.6.1.1.m1.16.16s"><mrow id="S2.F2.6.6.6.1.1.m1.8.8.8.4.2"><mo id="S2.F2.6.6.6.1.1.m1.7.7.7.3.1.1" mathsize="90%" rspace="0em">⊥</mo><mo id="S2.F2.6.6.6.1.1.m1.8.8.8.4.2.2" lspace="0em" mathsize="90%">∨</mo><mi id="S2.F2.6.6.6.1.1.m1.8.8.8.4.2.3" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.8.8.8.4.2.4" mathsize="90%">,</mo><mi id="S2.F2.6.6.6.1.1.m1.8.8.8.4.2.5" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.8.8.8.4.2.6" mathsize="90%" rspace="0em">∨</mo><mo id="S2.F2.6.6.6.1.1.m1.8.8.8.4.2.7" lspace="0em" mathsize="90%">⊥</mo></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16t"><mrow id="S2.F2.6.6.6.1.1.m1.8.8.8.6.1"><mi id="S2.F2.6.6.6.1.1.m1.8.8.8.6.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.8.8.8.6.1.1" mathsize="90%" stretchy="false">⇒</mo><mi id="S2.F2.6.6.6.1.1.m1.8.8.8.6.1.3" mathsize="90%">φ</mi></mrow></mtd></mtr><mtr id="S2.F2.6.6.6.1.1.m1.16.16u"><mtd class="ltx_border_l" id="S2.F2.6.6.6.1.1.m1.16.16v"><mrow id="S2.F2.6.6.6.1.1.m1.10.10.10.2.2"><mo id="S2.F2.6.6.6.1.1.m1.9.9.9.1.1.1" mathsize="90%" rspace="0em">⊤</mo><mo id="S2.F2.6.6.6.1.1.m1.10.10.10.2.2.2" lspace="0em" mathsize="90%" stretchy="false">→</mo><mi id="S2.F2.6.6.6.1.1.m1.10.10.10.2.2.3" mathsize="90%">φ</mi></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16w"><mrow id="S2.F2.6.6.6.1.1.m1.12.12.12.5.1"><mi id="S2.F2.6.6.6.1.1.m1.12.12.12.5.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.12.12.12.5.1.1" mathsize="90%" stretchy="false">⇒</mo><mi id="S2.F2.6.6.6.1.1.m1.12.12.12.5.1.3" mathsize="90%">φ</mi></mrow></mtd><mtd id="S2.F2.6.6.6.1.1.m1.16.16x"><mrow id="S2.F2.6.6.6.1.1.m1.12.12.12.4.2"><mo id="S2.F2.6.6.6.1.1.m1.11.11.11.3.1.1" mathsize="90%" rspace="0em">⊥</mo><mo id="S2.F2.6.6.6.1.1.m1.12.12.12.4.2.2" lspace="0em" mathsize="90%" stretchy="false">→</mo><mi id="S2.F2.6.6.6.1.1.m1.12.12.12.4.2.3" mathsize="90%">φ</mi></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16y"><mrow id="S2.F2.6.6.6.1.1.m1.12.12.12.6.1"><mi id="S2.F2.6.6.6.1.1.m1.12.12.12.6.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.12.12.12.6.1.1" mathsize="90%" rspace="0em" stretchy="false">⇒</mo><mo id="S2.F2.6.6.6.1.1.m1.12.12.12.6.1.3" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd></mtr><mtr id="S2.F2.6.6.6.1.1.m1.16.16z"><mtd class="ltx_border_l" id="S2.F2.6.6.6.1.1.m1.16.16aa"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.19.1.1"><mi id="S2.F2.6.6.6.1.1.m1.16.16.19.1.1.2" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.19.1.1.1" mathsize="90%" rspace="0em" stretchy="false">→</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.19.1.1.3" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16ab"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.19.2.1"><mi id="S2.F2.6.6.6.1.1.m1.16.16.19.2.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.19.2.1.1" mathsize="90%" rspace="0em" stretchy="false">⇒</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.19.2.1.3" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd><mtd id="S2.F2.6.6.6.1.1.m1.16.16ac"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.19.3.1"><mi id="S2.F2.6.6.6.1.1.m1.16.16.19.3.1.2" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.19.3.1.1" mathsize="90%" rspace="0em" stretchy="false">→</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.19.3.1.3" lspace="0em" mathsize="90%">⊥</mo></mrow></mtd><mtd class="ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16ad"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.19.4.1"><mi id="S2.F2.6.6.6.1.1.m1.16.16.19.4.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.19.4.1.1" mathsize="90%" stretchy="false">⇒</mo><mrow id="S2.F2.6.6.6.1.1.m1.16.16.19.4.1.3"><mo id="S2.F2.6.6.6.1.1.m1.16.16.19.4.1.3.1" mathsize="90%" rspace="0.167em">¬</mo><mi id="S2.F2.6.6.6.1.1.m1.16.16.19.4.1.3.2" mathsize="90%">φ</mi></mrow></mrow></mtd></mtr><mtr id="S2.F2.6.6.6.1.1.m1.16.16ae"><mtd class="ltx_border_b ltx_border_l" id="S2.F2.6.6.6.1.1.m1.16.16af"><mrow id="S2.F2.6.6.6.1.1.m1.14.14.14.2.2"><mo id="S2.F2.6.6.6.1.1.m1.13.13.13.1.1.1" mathsize="90%" rspace="0em">⊤</mo><mo id="S2.F2.6.6.6.1.1.m1.14.14.14.2.2.2" lspace="0em" mathsize="90%" stretchy="false">↔</mo><mi id="S2.F2.6.6.6.1.1.m1.14.14.14.2.2.3" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.14.14.14.2.2.4" mathsize="90%">,</mo><mi id="S2.F2.6.6.6.1.1.m1.14.14.14.2.2.5" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.14.14.14.2.2.6" mathsize="90%" rspace="0em" stretchy="false">↔</mo><mo id="S2.F2.6.6.6.1.1.m1.14.14.14.2.2.7" lspace="0em" mathsize="90%">⊤</mo></mrow></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16ag"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.16.5.1"><mi id="S2.F2.6.6.6.1.1.m1.16.16.16.5.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.16.5.1.1" mathsize="90%" stretchy="false">⇒</mo><mi id="S2.F2.6.6.6.1.1.m1.16.16.16.5.1.3" mathsize="90%">φ</mi></mrow></mtd><mtd class="ltx_border_b" id="S2.F2.6.6.6.1.1.m1.16.16ah"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.16.4.2"><mo id="S2.F2.6.6.6.1.1.m1.15.15.15.3.1.1" mathsize="90%" rspace="0em">⊥</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.16.4.2.2" lspace="0em" mathsize="90%" stretchy="false">↔</mo><mi id="S2.F2.6.6.6.1.1.m1.16.16.16.4.2.3" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.16.4.2.4" mathsize="90%">,</mo><mi id="S2.F2.6.6.6.1.1.m1.16.16.16.4.2.5" mathsize="90%">φ</mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.16.4.2.6" mathsize="90%" rspace="0em" stretchy="false">↔</mo><mo id="S2.F2.6.6.6.1.1.m1.16.16.16.4.2.7" lspace="0em" mathsize="90%">⊥</mo></mrow></mtd><mtd class="ltx_border_b ltx_border_r" columnalign="left" id="S2.F2.6.6.6.1.1.m1.16.16ai"><mrow id="S2.F2.6.6.6.1.1.m1.16.16.16.6.1"><mi id="S2.F2.6.6.6.1.1.m1.16.16.16.6.1.2"></mi><mo id="S2.F2.6.6.6.1.1.m1.16.16.16.6.1.1" mathsize="90%" stretchy="false">⇒</mo><mrow id="S2.F2.6.6.6.1.1.m1.16.16.16.6.1.3"><mo id="S2.F2.6.6.6.1.1.m1.16.16.16.6.1.3.1" mathsize="90%" rspace="0.167em">¬</mo><mi id="S2.F2.6.6.6.1.1.m1.16.16.16.6.1.3.2" mathsize="90%">φ</mi></mrow></mrow></mtd></mtr></mtable><annotation encoding="application/x-tex" id="S2.F2.6.6.6.1.1.m1.16b">\begin{array}[]{|cl|cl|}\hline\cr\neg\top&\Rightarrow\bot&\neg\bot&\Rightarrow% \top\\ \top\wedge\varphi,\varphi\wedge\top&\Rightarrow\varphi&\bot\wedge\varphi,% \varphi\wedge\bot&\Rightarrow\bot\\ \top\vee\varphi,\varphi\vee\top&\Rightarrow\top&\bot\vee\varphi,\varphi\vee% \bot&\Rightarrow\varphi\\ \top\rightarrow\varphi&\Rightarrow\varphi&\bot\rightarrow\varphi&\Rightarrow% \top\\ \varphi\rightarrow\top&\Rightarrow\top&\varphi\rightarrow\bot&\Rightarrow\neg% \varphi\\ \top\leftrightarrow\varphi,\varphi\leftrightarrow\top&\Rightarrow\varphi&\bot% \leftrightarrow\varphi,\varphi\leftrightarrow\bot&\Rightarrow\neg\varphi\\ \hline\cr\end{array}</annotation><annotation encoding="application/x-llamapun" id="S2.F2.6.6.6.1.1.m1.16c">start_ARRAY start_ROW start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL ¬ ⊤ end_CELL start_CELL ⇒ ⊥ end_CELL start_CELL ¬ ⊥ end_CELL start_CELL ⇒ ⊤ end_CELL end_ROW start_ROW start_CELL ⊤ ∧ italic_φ , italic_φ ∧ ⊤ end_CELL start_CELL ⇒ italic_φ end_CELL start_CELL ⊥ ∧ italic_φ , italic_φ ∧ ⊥ end_CELL start_CELL ⇒ ⊥ end_CELL end_ROW start_ROW start_CELL ⊤ ∨ italic_φ , italic_φ ∨ ⊤ end_CELL start_CELL ⇒ ⊤ end_CELL start_CELL ⊥ ∨ italic_φ , italic_φ ∨ ⊥ end_CELL start_CELL ⇒ italic_φ end_CELL end_ROW start_ROW start_CELL ⊤ → italic_φ end_CELL start_CELL ⇒ italic_φ end_CELL start_CELL ⊥ → italic_φ end_CELL start_CELL ⇒ ⊤ end_CELL end_ROW start_ROW start_CELL italic_φ → ⊤ end_CELL start_CELL ⇒ ⊤ end_CELL start_CELL italic_φ → ⊥ end_CELL start_CELL ⇒ ¬ italic_φ end_CELL end_ROW start_ROW start_CELL ⊤ ↔ italic_φ , italic_φ ↔ ⊤ end_CELL start_CELL ⇒ italic_φ end_CELL start_CELL ⊥ ↔ italic_φ , italic_φ ↔ ⊥ end_CELL start_CELL ⇒ ¬ italic_φ end_CELL end_ROW end_ARRAY</annotation></semantics></math></p> <figcaption class="ltx_caption" style="font-size:90%;"><span class="ltx_tag ltx_tag_block">Figure 2: </span> Propagation of truth values through the Boolean connectives.</figcaption> </div> </td> </tr> </tbody> </table> </figure> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Partial assignments.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px4.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px4.p1.18">A map <math alttext="\mu:{\mathbf{A}}^{\prime}\longmapsto\{{\mbox{{\sf T}},\mbox{{\sf F}}}\}^{N^{% \prime}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.1.m1.2"><semantics 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xref="S2.SS0.SSS0.Px4.p1.2.m2.1.1.3">𝐀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.2.m2.1c">{\mathbf{A}}^{\prime}\subseteq{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.2.m2.1d">bold_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ bold_A</annotation></semantics></math> and <math alttext="N^{\prime}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}|{\mathbf{A}}^{\prime}|" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.3.m3.1"><semantics id="S2.SS0.SSS0.Px4.p1.3.m3.1a"><mrow id="S2.SS0.SSS0.Px4.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.cmml"><msup id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.2.cmml">N</mi><mo id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.3.cmml">′</mo></msup><mover id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2" 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xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1"><apply id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2">superscript</csymbol><eq id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.2"></eq><ci id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.3a.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.3"><mtext id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.3.cmml" mathsize="50%" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.2.3">def</mtext></ci></apply><apply id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3">superscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.2">𝑁</ci><ci id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.3.3">′</ci></apply><apply id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1"><abs id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.2"></abs><apply id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.1.2">𝐀</ci><ci id="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.3.m3.1.1.1.1.1.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.3.m3.1c">N^{\prime}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}|{\mathbf{A}}^{\prime}|</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.3.m3.1d">italic_N start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP | bold_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT |</annotation></semantics></math> is a <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px4.p1.18.1">partial truth assignment</span> for <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.4.m4.1"><semantics id="S2.SS0.SSS0.Px4.p1.4.m4.1a"><mi id="S2.SS0.SSS0.Px4.p1.4.m4.1.1" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.4.m4.1b"><ci id="S2.SS0.SSS0.Px4.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.4.m4.1d">bold_A</annotation></semantics></math>. As with total assignments, we can represent <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.5.m5.1"><semantics id="S2.SS0.SSS0.Px4.p1.5.m5.1a"><mi id="S2.SS0.SSS0.Px4.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.5.m5.1b"><ci id="S2.SS0.SSS0.Px4.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.5.m5.1d">italic_μ</annotation></semantics></math> as a set of literals or as a cube, denoted with “<math alttext="\bigwedge\!\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.6.m6.1"><semantics id="S2.SS0.SSS0.Px4.p1.6.m6.1a"><mrow id="S2.SS0.SSS0.Px4.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.cmml"><mpadded width="0.747em"><mo id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.cmml">⋀</mo></mpadded><mi id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.2" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.2.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.6.m6.1b"><apply id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1"><and id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.1"></and><ci id="S2.SS0.SSS0.Px4.p1.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.6.m6.1.1.2">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.6.m6.1c">\bigwedge\!\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.6.m6.1d">⋀ italic_μ</annotation></semantics></math>”. Using a three-value logic we extend <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.7.m7.1"><semantics id="S2.SS0.SSS0.Px4.p1.7.m7.1a"><mi id="S2.SS0.SSS0.Px4.p1.7.m7.1.1" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.7.m7.1b"><ci id="S2.SS0.SSS0.Px4.p1.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.7.m7.1d">italic_μ</annotation></semantics></math> to <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.8.m8.1"><semantics id="S2.SS0.SSS0.Px4.p1.8.m8.1a"><mi id="S2.SS0.SSS0.Px4.p1.8.m8.1.1" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.8.m8.1b"><ci id="S2.SS0.SSS0.Px4.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.8.m8.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.8.m8.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.8.m8.1d">bold_A</annotation></semantics></math> as <math alttext="\mu:{\mathbf{A}}\longmapsto\{{\mbox{{\sf T}},\mbox{{\sf F}},\mbox{{\sf?}}}\}^{N}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.9.m9.3"><semantics id="S2.SS0.SSS0.Px4.p1.9.m9.3a"><mrow id="S2.SS0.SSS0.Px4.p1.9.m9.3.4" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.cmml"><mi id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.2" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.2.cmml">μ</mi><mo id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.1" lspace="0.278em" rspace="0.278em" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.1.cmml">:</mo><mrow id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.cmml"><mi id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.2" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.2.cmml">𝐀</mi><mo id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.1" stretchy="false" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.1.cmml">⟼</mo><msup id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.cmml"><mrow id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.2" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.1.cmml"><mo id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.1.cmml">{</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.9.m9.1.1" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1a.cmml">T</mtext><mo id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.2.2" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.1.cmml">,</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.9.m9.2.2" xref="S2.SS0.SSS0.Px4.p1.9.m9.2.2a.cmml">F</mtext><mo id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.2.3" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.1.cmml">,</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.9.m9.3.3" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.3a.cmml">?</mtext><mo id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.2.4" stretchy="false" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.1.cmml">}</mo></mrow><mi id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.3" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.3.cmml">N</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.9.m9.3b"><apply id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4"><ci id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.1.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.1">:</ci><ci id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.2.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.2">𝜇</ci><apply id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3"><ci id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.1">⟼</ci><ci id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.2">𝐀</ci><apply id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3">superscript</csymbol><set id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.2.2"><ci id="S2.SS0.SSS0.Px4.p1.9.m9.1.1a.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.1.1">T</mtext></ci><ci id="S2.SS0.SSS0.Px4.p1.9.m9.2.2a.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.9.m9.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.2.2">F</mtext></ci><ci id="S2.SS0.SSS0.Px4.p1.9.m9.3.3a.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.9.m9.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.3">?</mtext></ci></set><ci id="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.9.m9.3.4.3.3.3">𝑁</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.9.m9.3c">\mu:{\mathbf{A}}\longmapsto\{{\mbox{{\sf T}},\mbox{{\sf F}},\mbox{{\sf?}}}\}^{N}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.9.m9.3d">italic_μ : bold_A ⟼ { T , F , ? } start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> by assigning to <span class="ltx_text ltx_markedasmath ltx_font_sansserif" id="S2.SS0.SSS0.Px4.p1.18.2">?</span> (unknown) the unassigned atoms in <math alttext="{\mathbf{A}}\setminus{\mathbf{A}}^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.11.m11.1"><semantics id="S2.SS0.SSS0.Px4.p1.11.m11.1a"><mrow id="S2.SS0.SSS0.Px4.p1.11.m11.1.1" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.2" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.2.cmml">𝐀</mi><mo id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.1" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.1.cmml">∖</mo><msup id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.2" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.2.cmml">𝐀</mi><mo id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.3" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.11.m11.1b"><apply id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1"><setdiff id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.1"></setdiff><ci id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.2">𝐀</ci><apply id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3">superscript</csymbol><ci id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.2">𝐀</ci><ci id="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px4.p1.11.m11.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.11.m11.1c">{\mathbf{A}}\setminus{\mathbf{A}}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.11.m11.1d">bold_A ∖ bold_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Then we extend the semantics of <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.12.m12.1"><semantics id="S2.SS0.SSS0.Px4.p1.12.m12.1a"><mi id="S2.SS0.SSS0.Px4.p1.12.m12.1.1" xref="S2.SS0.SSS0.Px4.p1.12.m12.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.12.m12.1b"><ci id="S2.SS0.SSS0.Px4.p1.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.12.m12.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.12.m12.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.12.m12.1d">italic_μ</annotation></semantics></math> to any formula <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.13.m13.1"><semantics id="S2.SS0.SSS0.Px4.p1.13.m13.1a"><mi id="S2.SS0.SSS0.Px4.p1.13.m13.1.1" xref="S2.SS0.SSS0.Px4.p1.13.m13.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.13.m13.1b"><ci id="S2.SS0.SSS0.Px4.p1.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.13.m13.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.13.m13.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.13.m13.1d">italic_φ</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.14.m14.1"><semantics id="S2.SS0.SSS0.Px4.p1.14.m14.1a"><mi id="S2.SS0.SSS0.Px4.p1.14.m14.1.1" xref="S2.SS0.SSS0.Px4.p1.14.m14.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.14.m14.1b"><ci id="S2.SS0.SSS0.Px4.p1.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.14.m14.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.14.m14.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.14.m14.1d">bold_A</annotation></semantics></math> as described in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.F2" title="Figure 2 ‣ Semantics. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>. We say that <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.15.m15.1"><semantics id="S2.SS0.SSS0.Px4.p1.15.m15.1a"><mi id="S2.SS0.SSS0.Px4.p1.15.m15.1.1" xref="S2.SS0.SSS0.Px4.p1.15.m15.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.15.m15.1b"><ci id="S2.SS0.SSS0.Px4.p1.15.m15.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.15.m15.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.15.m15.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.15.m15.1d">italic_μ</annotation></semantics></math> <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px4.p1.18.3">verifies [resp. falsifies]</span> <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.16.m16.1"><semantics id="S2.SS0.SSS0.Px4.p1.16.m16.1a"><mi id="S2.SS0.SSS0.Px4.p1.16.m16.1.1" xref="S2.SS0.SSS0.Px4.p1.16.m16.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.16.m16.1b"><ci id="S2.SS0.SSS0.Px4.p1.16.m16.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.16.m16.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.16.m16.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.16.m16.1d">italic_φ</annotation></semantics></math> if <math alttext="\mu(\varphi)=\mbox{{\sf T}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.17.m17.1"><semantics id="S2.SS0.SSS0.Px4.p1.17.m17.1a"><mrow id="S2.SS0.SSS0.Px4.p1.17.m17.1.2" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.cmml"><mrow id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.2" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.2.cmml">μ</mi><mo id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.1" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.3.2" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.cmml"><mo id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px4.p1.17.m17.1.1" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.1.cmml">φ</mi><mo id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.1" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.1.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.3" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.3a.cmml">T</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.17.m17.1b"><apply id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2"><eq id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.1"></eq><apply id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2"><times id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.1"></times><ci id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.2.2">𝜇</ci><ci id="S2.SS0.SSS0.Px4.p1.17.m17.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.1">𝜑</ci></apply><ci id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.3a.cmml" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.17.m17.1.2.3.cmml" xref="S2.SS0.SSS0.Px4.p1.17.m17.1.2.3">T</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.17.m17.1c">\mu(\varphi)=\mbox{{\sf T}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.17.m17.1d">italic_μ ( italic_φ ) = T</annotation></semantics></math> [resp. <math alttext="\mu(\varphi)=\mbox{{\sf F}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p1.18.m18.1"><semantics id="S2.SS0.SSS0.Px4.p1.18.m18.1a"><mrow id="S2.SS0.SSS0.Px4.p1.18.m18.1.2" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.cmml"><mrow id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.2" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.2.cmml">μ</mi><mo id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.1" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.3.2" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.cmml"><mo id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px4.p1.18.m18.1.1" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.1.cmml">φ</mi><mo id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.1" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.1.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.3" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.3a.cmml">F</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.18.m18.1b"><apply id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.cmml" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2"><eq id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.1"></eq><apply id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2"><times id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.1"></times><ci id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.2.2">𝜇</ci><ci id="S2.SS0.SSS0.Px4.p1.18.m18.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.1">𝜑</ci></apply><ci id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.3a.cmml" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px4.p1.18.m18.1.2.3.cmml" xref="S2.SS0.SSS0.Px4.p1.18.m18.1.2.3">F</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.18.m18.1c">\mu(\varphi)=\mbox{{\sf F}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p1.18.m18.1d">italic_μ ( italic_φ ) = F</annotation></semantics></math>].</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px4.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px4.p2.11">By “<span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px4.p2.2.2">apply a partial assignment <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.1.1.m1.1"><semantics id="S2.SS0.SSS0.Px4.p2.1.1.m1.1a"><mi id="S2.SS0.SSS0.Px4.p2.1.1.m1.1.1" xref="S2.SS0.SSS0.Px4.p2.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.1.1.m1.1b"><ci id="S2.SS0.SSS0.Px4.p2.1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.1.1.m1.1d">italic_μ</annotation></semantics></math> to <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.2.2.m2.1"><semantics id="S2.SS0.SSS0.Px4.p2.2.2.m2.1a"><mi id="S2.SS0.SSS0.Px4.p2.2.2.m2.1.1" xref="S2.SS0.SSS0.Px4.p2.2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.2.2.m2.1b"><ci id="S2.SS0.SSS0.Px4.p2.2.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.2.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.2.2.m2.1d">italic_φ</annotation></semantics></math></span>” we mean “substitute all instances of each assigned <math alttext="A_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.3.m1.1"><semantics id="S2.SS0.SSS0.Px4.p2.3.m1.1a"><msub id="S2.SS0.SSS0.Px4.p2.3.m1.1.1" xref="S2.SS0.SSS0.Px4.p2.3.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px4.p2.3.m1.1.1.2" xref="S2.SS0.SSS0.Px4.p2.3.m1.1.1.2.cmml">A</mi><mi id="S2.SS0.SSS0.Px4.p2.3.m1.1.1.3" xref="S2.SS0.SSS0.Px4.p2.3.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.3.m1.1b"><apply id="S2.SS0.SSS0.Px4.p2.3.m1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.3.m1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px4.p2.3.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.3.m1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px4.p2.3.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px4.p2.3.m1.1.1.2">𝐴</ci><ci id="S2.SS0.SSS0.Px4.p2.3.m1.1.1.3.cmml" xref="S2.SS0.SSS0.Px4.p2.3.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.3.m1.1c">A_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.3.m1.1d">italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.4.m2.1"><semantics id="S2.SS0.SSS0.Px4.p2.4.m2.1a"><mi id="S2.SS0.SSS0.Px4.p2.4.m2.1.1" xref="S2.SS0.SSS0.Px4.p2.4.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.4.m2.1b"><ci id="S2.SS0.SSS0.Px4.p2.4.m2.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.4.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.4.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.4.m2.1d">italic_φ</annotation></semantics></math> with the truth constants in <math alttext="\{{\top,\bot}\}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.5.m3.2"><semantics id="S2.SS0.SSS0.Px4.p2.5.m3.2a"><mrow id="S2.SS0.SSS0.Px4.p2.5.m3.2.3.2" xref="S2.SS0.SSS0.Px4.p2.5.m3.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px4.p2.5.m3.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px4.p2.5.m3.2.3.1.cmml">{</mo><mo id="S2.SS0.SSS0.Px4.p2.5.m3.1.1" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px4.p2.5.m3.1.1.cmml">⊤</mo><mo id="S2.SS0.SSS0.Px4.p2.5.m3.2.3.2.2" rspace="0em" xref="S2.SS0.SSS0.Px4.p2.5.m3.2.3.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px4.p2.5.m3.2.2" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px4.p2.5.m3.2.2.cmml">⊥</mo><mo id="S2.SS0.SSS0.Px4.p2.5.m3.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px4.p2.5.m3.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.5.m3.2b"><set id="S2.SS0.SSS0.Px4.p2.5.m3.2.3.1.cmml" xref="S2.SS0.SSS0.Px4.p2.5.m3.2.3.2"><csymbol cd="latexml" id="S2.SS0.SSS0.Px4.p2.5.m3.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.5.m3.1.1">top</csymbol><csymbol cd="latexml" id="S2.SS0.SSS0.Px4.p2.5.m3.2.2.cmml" xref="S2.SS0.SSS0.Px4.p2.5.m3.2.2">bottom</csymbol></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.5.m3.2c">\{{\top,\bot}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.5.m3.2d">{ ⊤ , ⊥ }</annotation></semantics></math> corresponding to the truth value assigned by <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.6.m4.1"><semantics id="S2.SS0.SSS0.Px4.p2.6.m4.1a"><mi id="S2.SS0.SSS0.Px4.p2.6.m4.1.1" xref="S2.SS0.SSS0.Px4.p2.6.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.6.m4.1b"><ci id="S2.SS0.SSS0.Px4.p2.6.m4.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.6.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.6.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.6.m4.1d">italic_μ</annotation></semantics></math>, and then apply recursively the standard propagation of truth constants through the Boolean connectives described in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.F2" title="Figure 2 ‣ Semantics. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>. We denote by “<math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.7.m5.2"><semantics id="S2.SS0.SSS0.Px4.p2.7.m5.2a"><msub id="S2.SS0.SSS0.Px4.p2.7.m5.2.3.2" xref="S2.SS0.SSS0.Px4.p2.7.m5.2.3.1.cmml"><mrow id="S2.SS0.SSS0.Px4.p2.7.m5.2.3.2.2" xref="S2.SS0.SSS0.Px4.p2.7.m5.2.3.1.cmml"><mi id="S2.SS0.SSS0.Px4.p2.7.m5.1.1" xref="S2.SS0.SSS0.Px4.p2.7.m5.1.1.cmml">φ</mi><mo id="S2.SS0.SSS0.Px4.p2.7.m5.2.3.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px4.p2.7.m5.2.3.1.1.cmml">|</mo></mrow><mi id="S2.SS0.SSS0.Px4.p2.7.m5.2.2.1" xref="S2.SS0.SSS0.Px4.p2.7.m5.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.7.m5.2b"><apply id="S2.SS0.SSS0.Px4.p2.7.m5.2.3.1.cmml" xref="S2.SS0.SSS0.Px4.p2.7.m5.2.3.2"><csymbol cd="latexml" id="S2.SS0.SSS0.Px4.p2.7.m5.2.3.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.7.m5.2.3.2.2.1">evaluated-at</csymbol><ci id="S2.SS0.SSS0.Px4.p2.7.m5.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.7.m5.1.1">𝜑</ci><ci id="S2.SS0.SSS0.Px4.p2.7.m5.2.2.1.cmml" xref="S2.SS0.SSS0.Px4.p2.7.m5.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.7.m5.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.7.m5.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math>” (“<span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px4.p2.9.4">residual of <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.8.3.m1.1"><semantics id="S2.SS0.SSS0.Px4.p2.8.3.m1.1a"><mi id="S2.SS0.SSS0.Px4.p2.8.3.m1.1.1" xref="S2.SS0.SSS0.Px4.p2.8.3.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.8.3.m1.1b"><ci id="S2.SS0.SSS0.Px4.p2.8.3.m1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.8.3.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.8.3.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.8.3.m1.1d">italic_φ</annotation></semantics></math> under <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.9.4.m2.1"><semantics id="S2.SS0.SSS0.Px4.p2.9.4.m2.1a"><mi id="S2.SS0.SSS0.Px4.p2.9.4.m2.1.1" xref="S2.SS0.SSS0.Px4.p2.9.4.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.9.4.m2.1b"><ci id="S2.SS0.SSS0.Px4.p2.9.4.m2.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.9.4.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.9.4.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.9.4.m2.1d">italic_μ</annotation></semantics></math>”</span>) the formula resulting from applying <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.10.m6.1"><semantics id="S2.SS0.SSS0.Px4.p2.10.m6.1a"><mi id="S2.SS0.SSS0.Px4.p2.10.m6.1.1" xref="S2.SS0.SSS0.Px4.p2.10.m6.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.10.m6.1b"><ci id="S2.SS0.SSS0.Px4.p2.10.m6.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.10.m6.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.10.m6.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.10.m6.1d">italic_μ</annotation></semantics></math> to <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p2.11.m7.1"><semantics id="S2.SS0.SSS0.Px4.p2.11.m7.1a"><mi id="S2.SS0.SSS0.Px4.p2.11.m7.1.1" xref="S2.SS0.SSS0.Px4.p2.11.m7.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p2.11.m7.1b"><ci id="S2.SS0.SSS0.Px4.p2.11.m7.1.1.cmml" xref="S2.SS0.SSS0.Px4.p2.11.m7.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p2.11.m7.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p2.11.m7.1d">italic_φ</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px4.p3"> <p class="ltx_p" id="S2.SS0.SSS0.Px4.p3.1">The following facts follow straightforwardly and are of interest for our discussion. </p> </div> <div class="ltx_theorem ltx_theorem_property" id="Thmproperty1"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Property 1</span></h6> <div class="ltx_para" id="Thmproperty1.p1"> <p class="ltx_p" id="Thmproperty1.p1.4">Let <math alttext="\eta" class="ltx_Math" display="inline" id="Thmproperty1.p1.1.m1.1"><semantics id="Thmproperty1.p1.1.m1.1a"><mi id="Thmproperty1.p1.1.m1.1.1" xref="Thmproperty1.p1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmproperty1.p1.1.m1.1b"><ci id="Thmproperty1.p1.1.m1.1.1.cmml" xref="Thmproperty1.p1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty1.p1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmproperty1.p1.1.m1.1d">italic_η</annotation></semantics></math> be a total truth assignment on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty1.p1.2.m2.1"><semantics id="Thmproperty1.p1.2.m2.1a"><mi id="Thmproperty1.p1.2.m2.1.1" xref="Thmproperty1.p1.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty1.p1.2.m2.1b"><ci id="Thmproperty1.p1.2.m2.1.1.cmml" xref="Thmproperty1.p1.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty1.p1.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty1.p1.2.m2.1d">bold_A</annotation></semantics></math> and <math alttext="\varphi,\varphi_{1},\varphi_{2}" class="ltx_Math" display="inline" id="Thmproperty1.p1.3.m3.3"><semantics id="Thmproperty1.p1.3.m3.3a"><mrow id="Thmproperty1.p1.3.m3.3.3.2" xref="Thmproperty1.p1.3.m3.3.3.3.cmml"><mi id="Thmproperty1.p1.3.m3.1.1" xref="Thmproperty1.p1.3.m3.1.1.cmml">φ</mi><mo id="Thmproperty1.p1.3.m3.3.3.2.3" xref="Thmproperty1.p1.3.m3.3.3.3.cmml">,</mo><msub id="Thmproperty1.p1.3.m3.2.2.1.1" xref="Thmproperty1.p1.3.m3.2.2.1.1.cmml"><mi id="Thmproperty1.p1.3.m3.2.2.1.1.2" xref="Thmproperty1.p1.3.m3.2.2.1.1.2.cmml">φ</mi><mn id="Thmproperty1.p1.3.m3.2.2.1.1.3" xref="Thmproperty1.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmproperty1.p1.3.m3.3.3.2.4" xref="Thmproperty1.p1.3.m3.3.3.3.cmml">,</mo><msub id="Thmproperty1.p1.3.m3.3.3.2.2" xref="Thmproperty1.p1.3.m3.3.3.2.2.cmml"><mi id="Thmproperty1.p1.3.m3.3.3.2.2.2" xref="Thmproperty1.p1.3.m3.3.3.2.2.2.cmml">φ</mi><mn id="Thmproperty1.p1.3.m3.3.3.2.2.3" xref="Thmproperty1.p1.3.m3.3.3.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty1.p1.3.m3.3b"><list id="Thmproperty1.p1.3.m3.3.3.3.cmml" xref="Thmproperty1.p1.3.m3.3.3.2"><ci id="Thmproperty1.p1.3.m3.1.1.cmml" xref="Thmproperty1.p1.3.m3.1.1">𝜑</ci><apply id="Thmproperty1.p1.3.m3.2.2.1.1.cmml" xref="Thmproperty1.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="Thmproperty1.p1.3.m3.2.2.1.1.1.cmml" xref="Thmproperty1.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="Thmproperty1.p1.3.m3.2.2.1.1.2.cmml" xref="Thmproperty1.p1.3.m3.2.2.1.1.2">𝜑</ci><cn id="Thmproperty1.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="Thmproperty1.p1.3.m3.2.2.1.1.3">1</cn></apply><apply id="Thmproperty1.p1.3.m3.3.3.2.2.cmml" xref="Thmproperty1.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="Thmproperty1.p1.3.m3.3.3.2.2.1.cmml" xref="Thmproperty1.p1.3.m3.3.3.2.2">subscript</csymbol><ci id="Thmproperty1.p1.3.m3.3.3.2.2.2.cmml" xref="Thmproperty1.p1.3.m3.3.3.2.2.2">𝜑</ci><cn id="Thmproperty1.p1.3.m3.3.3.2.2.3.cmml" type="integer" xref="Thmproperty1.p1.3.m3.3.3.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty1.p1.3.m3.3c">\varphi,\varphi_{1},\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty1.p1.3.m3.3d">italic_φ , italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be formulas on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty1.p1.4.m4.1"><semantics id="Thmproperty1.p1.4.m4.1a"><mi id="Thmproperty1.p1.4.m4.1.1" xref="Thmproperty1.p1.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty1.p1.4.m4.1b"><ci id="Thmproperty1.p1.4.m4.1.1.cmml" xref="Thmproperty1.p1.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty1.p1.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty1.p1.4.m4.1d">bold_A</annotation></semantics></math>.</p> <ul class="ltx_itemize" id="Thmproperty1.p1.5"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S2.I1.i1.1.1.1" style="width:0.0pt;">(i)</span></span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.2"><math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="S2.I1.i1.p1.1.m1.1"><semantics id="S2.I1.i1.p1.1.m1.1a"><mrow id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml"><mi id="S2.I1.i1.p1.1.m1.1.1.2" xref="S2.I1.i1.p1.1.m1.1.1.2.cmml">η</mi><mo id="S2.I1.i1.p1.1.m1.1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.1.cmml">⊧</mo><mi id="S2.I1.i1.p1.1.m1.1.1.3" xref="S2.I1.i1.p1.1.m1.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><apply id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1"><csymbol cd="latexml" id="S2.I1.i1.p1.1.m1.1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1.1">models</csymbol><ci id="S2.I1.i1.p1.1.m1.1.1.2.cmml" xref="S2.I1.i1.p1.1.m1.1.1.2">𝜂</ci><ci id="S2.I1.i1.p1.1.m1.1.1.3.cmml" xref="S2.I1.i1.p1.1.m1.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.1d">italic_η ⊧ italic_φ</annotation></semantics></math> iff <math alttext="\bigwedge\!\eta\models\varphi" class="ltx_Math" display="inline" id="S2.I1.i1.p1.2.m2.1"><semantics id="S2.I1.i1.p1.2.m2.1a"><mrow id="S2.I1.i1.p1.2.m2.1.1" xref="S2.I1.i1.p1.2.m2.1.1.cmml"><mrow id="S2.I1.i1.p1.2.m2.1.1.2" xref="S2.I1.i1.p1.2.m2.1.1.2.cmml"><mpadded width="0.747em"><mo id="S2.I1.i1.p1.2.m2.1.1.2.1" xref="S2.I1.i1.p1.2.m2.1.1.2.1.cmml">⋀</mo></mpadded><mi id="S2.I1.i1.p1.2.m2.1.1.2.2" xref="S2.I1.i1.p1.2.m2.1.1.2.2.cmml">η</mi></mrow><mo id="S2.I1.i1.p1.2.m2.1.1.1" xref="S2.I1.i1.p1.2.m2.1.1.1.cmml">⊧</mo><mi id="S2.I1.i1.p1.2.m2.1.1.3" xref="S2.I1.i1.p1.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.2.m2.1b"><apply id="S2.I1.i1.p1.2.m2.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1"><csymbol cd="latexml" id="S2.I1.i1.p1.2.m2.1.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1.1">models</csymbol><apply id="S2.I1.i1.p1.2.m2.1.1.2.cmml" xref="S2.I1.i1.p1.2.m2.1.1.2"><and id="S2.I1.i1.p1.2.m2.1.1.2.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1.2.1"></and><ci id="S2.I1.i1.p1.2.m2.1.1.2.2.cmml" xref="S2.I1.i1.p1.2.m2.1.1.2.2">𝜂</ci></apply><ci id="S2.I1.i1.p1.2.m2.1.1.3.cmml" xref="S2.I1.i1.p1.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.2.m2.1c">\bigwedge\!\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.2.m2.1d">⋀ italic_η ⊧ italic_φ</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S2.I1.i2.1.1.1" style="width:0.0pt;">(ii)</span></span> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.4">If <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.1.m1.1"><semantics id="S2.I1.i2.p1.1.m1.1a"><msub id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml"><mi id="S2.I1.i2.p1.1.m1.1.1.2" xref="S2.I1.i2.p1.1.m1.1.1.2.cmml">φ</mi><mn id="S2.I1.i2.p1.1.m1.1.1.3" xref="S2.I1.i2.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.1b"><apply id="S2.I1.i2.p1.1.m1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.1.m1.1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S2.I1.i2.p1.1.m1.1.1.2.cmml" xref="S2.I1.i2.p1.1.m1.1.1.2">𝜑</ci><cn id="S2.I1.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.I1.i2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.1.m1.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.2.m2.1"><semantics id="S2.I1.i2.p1.2.m2.1a"><msub id="S2.I1.i2.p1.2.m2.1.1" xref="S2.I1.i2.p1.2.m2.1.1.cmml"><mi id="S2.I1.i2.p1.2.m2.1.1.2" xref="S2.I1.i2.p1.2.m2.1.1.2.cmml">φ</mi><mn id="S2.I1.i2.p1.2.m2.1.1.3" xref="S2.I1.i2.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.1b"><apply id="S2.I1.i2.p1.2.m2.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.I1.i2.p1.2.m2.1.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S2.I1.i2.p1.2.m2.1.1.2.cmml" xref="S2.I1.i2.p1.2.m2.1.1.2">𝜑</ci><cn id="S2.I1.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.I1.i2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.2.m2.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are equivalent, then <math alttext="\eta\models\varphi_{1}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.3.m3.1"><semantics id="S2.I1.i2.p1.3.m3.1a"><mrow id="S2.I1.i2.p1.3.m3.1.1" xref="S2.I1.i2.p1.3.m3.1.1.cmml"><mi id="S2.I1.i2.p1.3.m3.1.1.2" xref="S2.I1.i2.p1.3.m3.1.1.2.cmml">η</mi><mo id="S2.I1.i2.p1.3.m3.1.1.1" xref="S2.I1.i2.p1.3.m3.1.1.1.cmml">⊧</mo><msub id="S2.I1.i2.p1.3.m3.1.1.3" xref="S2.I1.i2.p1.3.m3.1.1.3.cmml"><mi id="S2.I1.i2.p1.3.m3.1.1.3.2" xref="S2.I1.i2.p1.3.m3.1.1.3.2.cmml">φ</mi><mn id="S2.I1.i2.p1.3.m3.1.1.3.3" xref="S2.I1.i2.p1.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.3.m3.1b"><apply id="S2.I1.i2.p1.3.m3.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1"><csymbol cd="latexml" id="S2.I1.i2.p1.3.m3.1.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1.1">models</csymbol><ci id="S2.I1.i2.p1.3.m3.1.1.2.cmml" xref="S2.I1.i2.p1.3.m3.1.1.2">𝜂</ci><apply id="S2.I1.i2.p1.3.m3.1.1.3.cmml" xref="S2.I1.i2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.I1.i2.p1.3.m3.1.1.3.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S2.I1.i2.p1.3.m3.1.1.3.2.cmml" xref="S2.I1.i2.p1.3.m3.1.1.3.2">𝜑</ci><cn id="S2.I1.i2.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.I1.i2.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.3.m3.1c">\eta\models\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.3.m3.1d">italic_η ⊧ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> iff <math alttext="\eta\models\varphi_{2}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.4.m4.1"><semantics id="S2.I1.i2.p1.4.m4.1a"><mrow id="S2.I1.i2.p1.4.m4.1.1" xref="S2.I1.i2.p1.4.m4.1.1.cmml"><mi id="S2.I1.i2.p1.4.m4.1.1.2" xref="S2.I1.i2.p1.4.m4.1.1.2.cmml">η</mi><mo id="S2.I1.i2.p1.4.m4.1.1.1" xref="S2.I1.i2.p1.4.m4.1.1.1.cmml">⊧</mo><msub id="S2.I1.i2.p1.4.m4.1.1.3" xref="S2.I1.i2.p1.4.m4.1.1.3.cmml"><mi id="S2.I1.i2.p1.4.m4.1.1.3.2" xref="S2.I1.i2.p1.4.m4.1.1.3.2.cmml">φ</mi><mn id="S2.I1.i2.p1.4.m4.1.1.3.3" xref="S2.I1.i2.p1.4.m4.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.4.m4.1b"><apply id="S2.I1.i2.p1.4.m4.1.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1"><csymbol cd="latexml" id="S2.I1.i2.p1.4.m4.1.1.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1.1">models</csymbol><ci id="S2.I1.i2.p1.4.m4.1.1.2.cmml" xref="S2.I1.i2.p1.4.m4.1.1.2">𝜂</ci><apply id="S2.I1.i2.p1.4.m4.1.1.3.cmml" xref="S2.I1.i2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.I1.i2.p1.4.m4.1.1.3.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1.3">subscript</csymbol><ci id="S2.I1.i2.p1.4.m4.1.1.3.2.cmml" xref="S2.I1.i2.p1.4.m4.1.1.3.2">𝜑</ci><cn id="S2.I1.i2.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S2.I1.i2.p1.4.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.4.m4.1c">\eta\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.4.m4.1d">italic_η ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S2.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S2.I1.i3.1.1.1" style="width:0.0pt;">(iii)</span></span> <div class="ltx_para" id="S2.I1.i3.p1"> <p class="ltx_p" id="S2.I1.i3.p1.4"><math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="S2.I1.i3.p1.1.m1.1"><semantics id="S2.I1.i3.p1.1.m1.1a"><mrow id="S2.I1.i3.p1.1.m1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.cmml"><mi id="S2.I1.i3.p1.1.m1.1.1.2" xref="S2.I1.i3.p1.1.m1.1.1.2.cmml">η</mi><mo id="S2.I1.i3.p1.1.m1.1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.1.cmml">⊧</mo><mi id="S2.I1.i3.p1.1.m1.1.1.3" xref="S2.I1.i3.p1.1.m1.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.1.m1.1b"><apply id="S2.I1.i3.p1.1.m1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1"><csymbol cd="latexml" id="S2.I1.i3.p1.1.m1.1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1.1">models</csymbol><ci id="S2.I1.i3.p1.1.m1.1.1.2.cmml" xref="S2.I1.i3.p1.1.m1.1.1.2">𝜂</ci><ci id="S2.I1.i3.p1.1.m1.1.1.3.cmml" xref="S2.I1.i3.p1.1.m1.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.1.m1.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.1.m1.1d">italic_η ⊧ italic_φ</annotation></semantics></math> iff <math alttext="\varphi|_{\eta}" class="ltx_Math" display="inline" id="S2.I1.i3.p1.2.m2.2"><semantics id="S2.I1.i3.p1.2.m2.2a"><msub id="S2.I1.i3.p1.2.m2.2.3.2" xref="S2.I1.i3.p1.2.m2.2.3.1.cmml"><mrow id="S2.I1.i3.p1.2.m2.2.3.2.2" xref="S2.I1.i3.p1.2.m2.2.3.1.cmml"><mi id="S2.I1.i3.p1.2.m2.1.1" xref="S2.I1.i3.p1.2.m2.1.1.cmml">φ</mi><mo id="S2.I1.i3.p1.2.m2.2.3.2.2.1" stretchy="false" xref="S2.I1.i3.p1.2.m2.2.3.1.1.cmml">|</mo></mrow><mi id="S2.I1.i3.p1.2.m2.2.2.1" xref="S2.I1.i3.p1.2.m2.2.2.1.cmml">η</mi></msub><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.2.m2.2b"><apply id="S2.I1.i3.p1.2.m2.2.3.1.cmml" xref="S2.I1.i3.p1.2.m2.2.3.2"><csymbol cd="latexml" id="S2.I1.i3.p1.2.m2.2.3.1.1.cmml" xref="S2.I1.i3.p1.2.m2.2.3.2.2.1">evaluated-at</csymbol><ci id="S2.I1.i3.p1.2.m2.1.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1">𝜑</ci><ci id="S2.I1.i3.p1.2.m2.2.2.1.cmml" xref="S2.I1.i3.p1.2.m2.2.2.1">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.2.m2.2c">\varphi|_{\eta}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.2.m2.2d">italic_φ | start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\top" class="ltx_Math" display="inline" id="S2.I1.i3.p1.3.m3.1"><semantics id="S2.I1.i3.p1.3.m3.1a"><mo id="S2.I1.i3.p1.3.m3.1.1" xref="S2.I1.i3.p1.3.m3.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.3.m3.1b"><csymbol cd="latexml" id="S2.I1.i3.p1.3.m3.1.1.cmml" xref="S2.I1.i3.p1.3.m3.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.3.m3.1c">\top</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.3.m3.1d">⊤</annotation></semantics></math> (also, by property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty2" title="Property 2 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>, iff <math alttext="\eta(\varphi)=\mbox{{\sf T}}" class="ltx_Math" display="inline" id="S2.I1.i3.p1.4.m4.1"><semantics id="S2.I1.i3.p1.4.m4.1a"><mrow id="S2.I1.i3.p1.4.m4.1.2" xref="S2.I1.i3.p1.4.m4.1.2.cmml"><mrow id="S2.I1.i3.p1.4.m4.1.2.2" xref="S2.I1.i3.p1.4.m4.1.2.2.cmml"><mi id="S2.I1.i3.p1.4.m4.1.2.2.2" xref="S2.I1.i3.p1.4.m4.1.2.2.2.cmml">η</mi><mo id="S2.I1.i3.p1.4.m4.1.2.2.1" xref="S2.I1.i3.p1.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.I1.i3.p1.4.m4.1.2.2.3.2" xref="S2.I1.i3.p1.4.m4.1.2.2.cmml"><mo id="S2.I1.i3.p1.4.m4.1.2.2.3.2.1" stretchy="false" xref="S2.I1.i3.p1.4.m4.1.2.2.cmml">(</mo><mi id="S2.I1.i3.p1.4.m4.1.1" xref="S2.I1.i3.p1.4.m4.1.1.cmml">φ</mi><mo id="S2.I1.i3.p1.4.m4.1.2.2.3.2.2" stretchy="false" xref="S2.I1.i3.p1.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.I1.i3.p1.4.m4.1.2.1" xref="S2.I1.i3.p1.4.m4.1.2.1.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.I1.i3.p1.4.m4.1.2.3" xref="S2.I1.i3.p1.4.m4.1.2.3a.cmml">T</mtext></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.4.m4.1b"><apply id="S2.I1.i3.p1.4.m4.1.2.cmml" xref="S2.I1.i3.p1.4.m4.1.2"><eq id="S2.I1.i3.p1.4.m4.1.2.1.cmml" xref="S2.I1.i3.p1.4.m4.1.2.1"></eq><apply id="S2.I1.i3.p1.4.m4.1.2.2.cmml" xref="S2.I1.i3.p1.4.m4.1.2.2"><times id="S2.I1.i3.p1.4.m4.1.2.2.1.cmml" xref="S2.I1.i3.p1.4.m4.1.2.2.1"></times><ci id="S2.I1.i3.p1.4.m4.1.2.2.2.cmml" xref="S2.I1.i3.p1.4.m4.1.2.2.2">𝜂</ci><ci id="S2.I1.i3.p1.4.m4.1.1.cmml" xref="S2.I1.i3.p1.4.m4.1.1">𝜑</ci></apply><ci id="S2.I1.i3.p1.4.m4.1.2.3a.cmml" xref="S2.I1.i3.p1.4.m4.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.I1.i3.p1.4.m4.1.2.3.cmml" xref="S2.I1.i3.p1.4.m4.1.2.3">T</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.4.m4.1c">\eta(\varphi)=\mbox{{\sf T}}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.4.m4.1d">italic_η ( italic_φ ) = T</annotation></semantics></math>).</p> </div> </li> <li class="ltx_item" id="S2.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S2.I1.i4.1.1.1" style="width:0.0pt;">(iv)</span></span> <div class="ltx_para" id="S2.I1.i4.p1"> <p class="ltx_p" id="S2.I1.i4.p1.1">Checking <math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="S2.I1.i4.p1.1.m1.1"><semantics id="S2.I1.i4.p1.1.m1.1a"><mrow id="S2.I1.i4.p1.1.m1.1.1" xref="S2.I1.i4.p1.1.m1.1.1.cmml"><mi id="S2.I1.i4.p1.1.m1.1.1.2" xref="S2.I1.i4.p1.1.m1.1.1.2.cmml">η</mi><mo id="S2.I1.i4.p1.1.m1.1.1.1" xref="S2.I1.i4.p1.1.m1.1.1.1.cmml">⊧</mo><mi id="S2.I1.i4.p1.1.m1.1.1.3" xref="S2.I1.i4.p1.1.m1.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i4.p1.1.m1.1b"><apply id="S2.I1.i4.p1.1.m1.1.1.cmml" xref="S2.I1.i4.p1.1.m1.1.1"><csymbol cd="latexml" id="S2.I1.i4.p1.1.m1.1.1.1.cmml" xref="S2.I1.i4.p1.1.m1.1.1.1">models</csymbol><ci id="S2.I1.i4.p1.1.m1.1.1.2.cmml" xref="S2.I1.i4.p1.1.m1.1.1.2">𝜂</ci><ci id="S2.I1.i4.p1.1.m1.1.1.3.cmml" xref="S2.I1.i4.p1.1.m1.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i4.p1.1.m1.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i4.p1.1.m1.1d">italic_η ⊧ italic_φ</annotation></semantics></math> requires at most a polynomial amount of steps.</p> </div> </li> </ul> </div> </div> <div class="ltx_para ltx_noindent" id="S2.SS0.SSS0.Px4.p4"> <p class="ltx_p" id="S2.SS0.SSS0.Px4.p4.1">Notice that property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty1" title="Property 1 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a>(i) justifies the usage of “<math alttext="\models" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p4.1.m1.1"><semantics id="S2.SS0.SSS0.Px4.p4.1.m1.1a"><mo id="S2.SS0.SSS0.Px4.p4.1.m1.1.1" xref="S2.SS0.SSS0.Px4.p4.1.m1.1.1.cmml">⊧</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p4.1.m1.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px4.p4.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p4.1.m1.1.1">models</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p4.1.m1.1c">\models</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p4.1.m1.1d">⊧</annotation></semantics></math>” for both satisfiability and entailment.</p> </div> <div class="ltx_theorem ltx_theorem_property" id="Thmproperty2"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Property 2</span></h6> <div class="ltx_para" id="Thmproperty2.p1"> <p class="ltx_p" id="Thmproperty2.p1.10">Let <math alttext="\mu" class="ltx_Math" display="inline" id="Thmproperty2.p1.1.m1.1"><semantics id="Thmproperty2.p1.1.m1.1a"><mi id="Thmproperty2.p1.1.m1.1.1" xref="Thmproperty2.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.1.m1.1b"><ci id="Thmproperty2.p1.1.m1.1.1.cmml" xref="Thmproperty2.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.1.m1.1d">italic_μ</annotation></semantics></math> be a partial assignment on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty2.p1.2.m2.1"><semantics id="Thmproperty2.p1.2.m2.1a"><mi id="Thmproperty2.p1.2.m2.1.1" xref="Thmproperty2.p1.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.2.m2.1b"><ci id="Thmproperty2.p1.2.m2.1.1.cmml" xref="Thmproperty2.p1.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.2.m2.1d">bold_A</annotation></semantics></math> and <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmproperty2.p1.3.m3.1"><semantics id="Thmproperty2.p1.3.m3.1a"><mi id="Thmproperty2.p1.3.m3.1.1" xref="Thmproperty2.p1.3.m3.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.3.m3.1b"><ci id="Thmproperty2.p1.3.m3.1.1.cmml" xref="Thmproperty2.p1.3.m3.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.3.m3.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.3.m3.1d">italic_φ</annotation></semantics></math> be a formula on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty2.p1.4.m4.1"><semantics id="Thmproperty2.p1.4.m4.1a"><mi id="Thmproperty2.p1.4.m4.1.1" xref="Thmproperty2.p1.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.4.m4.1b"><ci id="Thmproperty2.p1.4.m4.1.1.cmml" xref="Thmproperty2.p1.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.4.m4.1d">bold_A</annotation></semantics></math>. <br class="ltx_break"/>Then <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="Thmproperty2.p1.5.m5.2"><semantics id="Thmproperty2.p1.5.m5.2a"><msub id="Thmproperty2.p1.5.m5.2.3.2" xref="Thmproperty2.p1.5.m5.2.3.1.cmml"><mrow id="Thmproperty2.p1.5.m5.2.3.2.2" xref="Thmproperty2.p1.5.m5.2.3.1.cmml"><mi id="Thmproperty2.p1.5.m5.1.1" xref="Thmproperty2.p1.5.m5.1.1.cmml">φ</mi><mo id="Thmproperty2.p1.5.m5.2.3.2.2.1" stretchy="false" xref="Thmproperty2.p1.5.m5.2.3.1.1.cmml">|</mo></mrow><mi id="Thmproperty2.p1.5.m5.2.2.1" xref="Thmproperty2.p1.5.m5.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.5.m5.2b"><apply id="Thmproperty2.p1.5.m5.2.3.1.cmml" xref="Thmproperty2.p1.5.m5.2.3.2"><csymbol cd="latexml" id="Thmproperty2.p1.5.m5.2.3.1.1.cmml" xref="Thmproperty2.p1.5.m5.2.3.2.2.1">evaluated-at</csymbol><ci id="Thmproperty2.p1.5.m5.1.1.cmml" xref="Thmproperty2.p1.5.m5.1.1">𝜑</ci><ci id="Thmproperty2.p1.5.m5.2.2.1.cmml" xref="Thmproperty2.p1.5.m5.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.5.m5.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.5.m5.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\top" class="ltx_Math" display="inline" id="Thmproperty2.p1.6.m6.1"><semantics id="Thmproperty2.p1.6.m6.1a"><mo id="Thmproperty2.p1.6.m6.1.1" xref="Thmproperty2.p1.6.m6.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.6.m6.1b"><csymbol cd="latexml" id="Thmproperty2.p1.6.m6.1.1.cmml" xref="Thmproperty2.p1.6.m6.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.6.m6.1c">\top</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.6.m6.1d">⊤</annotation></semantics></math> iff <math alttext="\mu(\varphi)=\mbox{{\sf T}}" class="ltx_Math" display="inline" id="Thmproperty2.p1.7.m7.1"><semantics id="Thmproperty2.p1.7.m7.1a"><mrow id="Thmproperty2.p1.7.m7.1.2" xref="Thmproperty2.p1.7.m7.1.2.cmml"><mrow id="Thmproperty2.p1.7.m7.1.2.2" xref="Thmproperty2.p1.7.m7.1.2.2.cmml"><mi id="Thmproperty2.p1.7.m7.1.2.2.2" xref="Thmproperty2.p1.7.m7.1.2.2.2.cmml">μ</mi><mo id="Thmproperty2.p1.7.m7.1.2.2.1" xref="Thmproperty2.p1.7.m7.1.2.2.1.cmml">⁢</mo><mrow id="Thmproperty2.p1.7.m7.1.2.2.3.2" xref="Thmproperty2.p1.7.m7.1.2.2.cmml"><mo id="Thmproperty2.p1.7.m7.1.2.2.3.2.1" stretchy="false" xref="Thmproperty2.p1.7.m7.1.2.2.cmml">(</mo><mi id="Thmproperty2.p1.7.m7.1.1" xref="Thmproperty2.p1.7.m7.1.1.cmml">φ</mi><mo id="Thmproperty2.p1.7.m7.1.2.2.3.2.2" stretchy="false" xref="Thmproperty2.p1.7.m7.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmproperty2.p1.7.m7.1.2.1" xref="Thmproperty2.p1.7.m7.1.2.1.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="Thmproperty2.p1.7.m7.1.2.3" xref="Thmproperty2.p1.7.m7.1.2.3a.cmml">T</mtext></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.7.m7.1b"><apply id="Thmproperty2.p1.7.m7.1.2.cmml" xref="Thmproperty2.p1.7.m7.1.2"><eq id="Thmproperty2.p1.7.m7.1.2.1.cmml" xref="Thmproperty2.p1.7.m7.1.2.1"></eq><apply id="Thmproperty2.p1.7.m7.1.2.2.cmml" xref="Thmproperty2.p1.7.m7.1.2.2"><times id="Thmproperty2.p1.7.m7.1.2.2.1.cmml" xref="Thmproperty2.p1.7.m7.1.2.2.1"></times><ci id="Thmproperty2.p1.7.m7.1.2.2.2.cmml" xref="Thmproperty2.p1.7.m7.1.2.2.2">𝜇</ci><ci id="Thmproperty2.p1.7.m7.1.1.cmml" xref="Thmproperty2.p1.7.m7.1.1">𝜑</ci></apply><ci id="Thmproperty2.p1.7.m7.1.2.3a.cmml" xref="Thmproperty2.p1.7.m7.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="Thmproperty2.p1.7.m7.1.2.3.cmml" xref="Thmproperty2.p1.7.m7.1.2.3">T</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.7.m7.1c">\mu(\varphi)=\mbox{{\sf T}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.7.m7.1d">italic_μ ( italic_φ ) = T</annotation></semantics></math> and <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="Thmproperty2.p1.8.m8.2"><semantics id="Thmproperty2.p1.8.m8.2a"><msub id="Thmproperty2.p1.8.m8.2.3.2" xref="Thmproperty2.p1.8.m8.2.3.1.cmml"><mrow id="Thmproperty2.p1.8.m8.2.3.2.2" xref="Thmproperty2.p1.8.m8.2.3.1.cmml"><mi id="Thmproperty2.p1.8.m8.1.1" xref="Thmproperty2.p1.8.m8.1.1.cmml">φ</mi><mo id="Thmproperty2.p1.8.m8.2.3.2.2.1" stretchy="false" xref="Thmproperty2.p1.8.m8.2.3.1.1.cmml">|</mo></mrow><mi id="Thmproperty2.p1.8.m8.2.2.1" xref="Thmproperty2.p1.8.m8.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.8.m8.2b"><apply id="Thmproperty2.p1.8.m8.2.3.1.cmml" xref="Thmproperty2.p1.8.m8.2.3.2"><csymbol cd="latexml" id="Thmproperty2.p1.8.m8.2.3.1.1.cmml" xref="Thmproperty2.p1.8.m8.2.3.2.2.1">evaluated-at</csymbol><ci id="Thmproperty2.p1.8.m8.1.1.cmml" xref="Thmproperty2.p1.8.m8.1.1">𝜑</ci><ci id="Thmproperty2.p1.8.m8.2.2.1.cmml" xref="Thmproperty2.p1.8.m8.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.8.m8.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.8.m8.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\bot" class="ltx_Math" display="inline" id="Thmproperty2.p1.9.m9.1"><semantics id="Thmproperty2.p1.9.m9.1a"><mo id="Thmproperty2.p1.9.m9.1.1" xref="Thmproperty2.p1.9.m9.1.1.cmml">⊥</mo><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.9.m9.1b"><csymbol cd="latexml" id="Thmproperty2.p1.9.m9.1.1.cmml" xref="Thmproperty2.p1.9.m9.1.1">bottom</csymbol></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.9.m9.1c">\bot</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.9.m9.1d">⊥</annotation></semantics></math> iff <math alttext="\mu(\varphi)=\mbox{{\sf F}}" class="ltx_Math" display="inline" id="Thmproperty2.p1.10.m10.1"><semantics id="Thmproperty2.p1.10.m10.1a"><mrow id="Thmproperty2.p1.10.m10.1.2" xref="Thmproperty2.p1.10.m10.1.2.cmml"><mrow id="Thmproperty2.p1.10.m10.1.2.2" xref="Thmproperty2.p1.10.m10.1.2.2.cmml"><mi id="Thmproperty2.p1.10.m10.1.2.2.2" xref="Thmproperty2.p1.10.m10.1.2.2.2.cmml">μ</mi><mo id="Thmproperty2.p1.10.m10.1.2.2.1" xref="Thmproperty2.p1.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="Thmproperty2.p1.10.m10.1.2.2.3.2" xref="Thmproperty2.p1.10.m10.1.2.2.cmml"><mo id="Thmproperty2.p1.10.m10.1.2.2.3.2.1" stretchy="false" xref="Thmproperty2.p1.10.m10.1.2.2.cmml">(</mo><mi id="Thmproperty2.p1.10.m10.1.1" xref="Thmproperty2.p1.10.m10.1.1.cmml">φ</mi><mo id="Thmproperty2.p1.10.m10.1.2.2.3.2.2" stretchy="false" xref="Thmproperty2.p1.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmproperty2.p1.10.m10.1.2.1" xref="Thmproperty2.p1.10.m10.1.2.1.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="Thmproperty2.p1.10.m10.1.2.3" xref="Thmproperty2.p1.10.m10.1.2.3a.cmml">F</mtext></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty2.p1.10.m10.1b"><apply id="Thmproperty2.p1.10.m10.1.2.cmml" xref="Thmproperty2.p1.10.m10.1.2"><eq id="Thmproperty2.p1.10.m10.1.2.1.cmml" xref="Thmproperty2.p1.10.m10.1.2.1"></eq><apply id="Thmproperty2.p1.10.m10.1.2.2.cmml" xref="Thmproperty2.p1.10.m10.1.2.2"><times id="Thmproperty2.p1.10.m10.1.2.2.1.cmml" xref="Thmproperty2.p1.10.m10.1.2.2.1"></times><ci id="Thmproperty2.p1.10.m10.1.2.2.2.cmml" xref="Thmproperty2.p1.10.m10.1.2.2.2">𝜇</ci><ci id="Thmproperty2.p1.10.m10.1.1.cmml" xref="Thmproperty2.p1.10.m10.1.1">𝜑</ci></apply><ci id="Thmproperty2.p1.10.m10.1.2.3a.cmml" xref="Thmproperty2.p1.10.m10.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="Thmproperty2.p1.10.m10.1.2.3.cmml" xref="Thmproperty2.p1.10.m10.1.2.3">F</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty2.p1.10.m10.1c">\mu(\varphi)=\mbox{{\sf F}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty2.p1.10.m10.1d">italic_μ ( italic_φ ) = F</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para ltx_noindent" id="S2.SS0.SSS0.Px4.p5"> <p class="ltx_p" id="S2.SS0.SSS0.Px4.p5.1">Notice that total assignments are a subcase of partial ones, so that the definition of residual and property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty2" title="Property 2 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a> apply also to total assignments <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px4.p5.1.m1.1"><semantics id="S2.SS0.SSS0.Px4.p5.1.m1.1a"><mi id="S2.SS0.SSS0.Px4.p5.1.m1.1.1" xref="S2.SS0.SSS0.Px4.p5.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p5.1.m1.1b"><ci id="S2.SS0.SSS0.Px4.p5.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p5.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p5.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px4.p5.1.m1.1d">italic_η</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px5"> <h4 class="ltx_title ltx_title_paragraph">Existentially-quantified formulas.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px5.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px5.p1.9">A total truth assignment <math alttext="\eta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.1.m1.1"><semantics id="S2.SS0.SSS0.Px5.p1.1.m1.1a"><mi id="S2.SS0.SSS0.Px5.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px5.p1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.1.m1.1b"><ci id="S2.SS0.SSS0.Px5.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.1.m1.1d">italic_η</annotation></semantics></math> satisfies <math alttext="\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.2.m2.2"><semantics id="S2.SS0.SSS0.Px5.p1.2.m2.2a"><mrow id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.2.cmml"><mrow id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.cmml"><mo id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.1.cmml">∃</mo><mi id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.2" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.2.cmml">𝐁</mi></mrow><mo id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.2" lspace="0em" rspace="0.167em" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.2a.cmml">.</mo><mi id="S2.SS0.SSS0.Px5.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px5.p1.2.m2.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.2.m2.2b"><apply id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.2.cmml" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.2a.cmml" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.2">formulae-sequence</csymbol><apply id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1"><exists id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.1"></exists><ci id="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px5.p1.2.m2.2.2.1.1.2">𝐁</ci></apply><ci id="S2.SS0.SSS0.Px5.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.2.m2.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.2.m2.2c">\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.2.m2.2d">∃ bold_B . italic_ψ</annotation></semantics></math>, written “<math alttext="\eta\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.3.m3.2"><semantics id="S2.SS0.SSS0.Px5.p1.3.m3.2a"><mrow id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.2.cmml"><mrow id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.cmml"><mi id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.2" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.1" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.1.cmml">⊧</mo><mrow id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.cmml"><mo id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.1" rspace="0.167em" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.1.cmml">∃</mo><mi id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.2" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.2" lspace="0em" rspace="0.167em" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.2a.cmml">.</mo><mi id="S2.SS0.SSS0.Px5.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px5.p1.3.m3.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.3.m3.2b"><apply id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.2.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.2a.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.2">formulae-sequence</csymbol><apply id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.1">models</csymbol><ci id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.2">𝜂</ci><apply id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3"><exists id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.1"></exists><ci id="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="S2.SS0.SSS0.Px5.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.3.m3.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.3.m3.2c">\eta\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.3.m3.2d">italic_η ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math>”, iff exists a total truth assignment <math alttext="\delta" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.4.m4.1"><semantics id="S2.SS0.SSS0.Px5.p1.4.m4.1a"><mi id="S2.SS0.SSS0.Px5.p1.4.m4.1.1" xref="S2.SS0.SSS0.Px5.p1.4.m4.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.4.m4.1b"><ci id="S2.SS0.SSS0.Px5.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.4.m4.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.4.m4.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.4.m4.1d">italic_δ</annotation></semantics></math> on <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.5.m5.1"><semantics id="S2.SS0.SSS0.Px5.p1.5.m5.1a"><mi id="S2.SS0.SSS0.Px5.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px5.p1.5.m5.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.5.m5.1b"><ci id="S2.SS0.SSS0.Px5.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.5.m5.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.5.m5.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.5.m5.1d">bold_B</annotation></semantics></math> s.t. <math alttext="\eta\cup\delta\models\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.6.m6.1"><semantics id="S2.SS0.SSS0.Px5.p1.6.m6.1a"><mrow id="S2.SS0.SSS0.Px5.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.cmml"><mrow id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.2" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.2.cmml">η</mi><mo id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.1" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.1.cmml">∪</mo><mi id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.3" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.3.cmml">δ</mi></mrow><mo id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.1" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.1.cmml">⊧</mo><mi id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.3" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.3.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.6.m6.1b"><apply id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.1">models</csymbol><apply id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2"><union id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.1"></union><ci id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.2">𝜂</ci><ci id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.2.3">𝛿</ci></apply><ci id="S2.SS0.SSS0.Px5.p1.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px5.p1.6.m6.1.1.3">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.6.m6.1c">\eta\cup\delta\models\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.6.m6.1d">italic_η ∪ italic_δ ⊧ italic_ψ</annotation></semantics></math>. We call the <span class="ltx_text ltx_font_italic" id="S2.SS0.SSS0.Px5.p1.9.1">Shannon expansion</span> <math alttext="{\sf SE}[{\exists{\mathbf{B}}}.\psi" class="ltx_math_unparsed" display="inline" id="S2.SS0.SSS0.Px5.p1.7.m7.1"><semantics id="S2.SS0.SSS0.Px5.p1.7.m7.1a"><mrow id="S2.SS0.SSS0.Px5.p1.7.m7.1b"><mi id="S2.SS0.SSS0.Px5.p1.7.m7.1.1">𝖲𝖤</mi><mrow id="S2.SS0.SSS0.Px5.p1.7.m7.1.2"><mo id="S2.SS0.SSS0.Px5.p1.7.m7.1.2.1" stretchy="false">[</mo><mo id="S2.SS0.SSS0.Px5.p1.7.m7.1.2.2" rspace="0.167em">∃</mo><mi id="S2.SS0.SSS0.Px5.p1.7.m7.1.2.3">𝐁</mi><mo id="S2.SS0.SSS0.Px5.p1.7.m7.1.2.4" lspace="0em" rspace="0.167em">.</mo><mi id="S2.SS0.SSS0.Px5.p1.7.m7.1.2.5">ψ</mi></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.7.m7.1c">{\sf SE}[{\exists{\mathbf{B}}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.7.m7.1d">sansserif_SE [ ∃ bold_B . italic_ψ</annotation></semantics></math>] of the existentially-quantified formula <math alttext="\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.8.m8.2"><semantics id="S2.SS0.SSS0.Px5.p1.8.m8.2a"><mrow id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.2.cmml"><mrow id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.cmml"><mo id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.1" rspace="0.167em" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.1.cmml">∃</mo><mi id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.2" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.2.cmml">𝐁</mi></mrow><mo id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.2" lspace="0em" rspace="0.167em" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.2a.cmml">.</mo><mi id="S2.SS0.SSS0.Px5.p1.8.m8.1.1" xref="S2.SS0.SSS0.Px5.p1.8.m8.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.8.m8.2b"><apply id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.2.cmml" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.2a.cmml" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.2">formulae-sequence</csymbol><apply id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1"><exists id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.1"></exists><ci id="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px5.p1.8.m8.2.2.1.1.2">𝐁</ci></apply><ci id="S2.SS0.SSS0.Px5.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.8.m8.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.8.m8.2c">\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.8.m8.2d">∃ bold_B . italic_ψ</annotation></semantics></math> the propositional formula on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.9.m9.1"><semantics id="S2.SS0.SSS0.Px5.p1.9.m9.1a"><mi id="S2.SS0.SSS0.Px5.p1.9.m9.1.1" xref="S2.SS0.SSS0.Px5.p1.9.m9.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.9.m9.1b"><ci id="S2.SS0.SSS0.Px5.p1.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.9.m9.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.9.m9.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.9.m9.1d">bold_A</annotation></semantics></math> defined as</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S5.EGx1"> <tbody id="S2.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\sf SE}[{\exists{\mathbf{B}}}.\psi]\stackrel{{\scriptstyle\text{% \tiny def}}}{{=}}\textstyle\bigvee_{\delta_{i}\in\{{{\scriptsize\mbox{{\sf T}}% ,\mbox{{\sf F}}}}\}^{|{\mathbf{B}}|}}\psi|_{\delta_{i}}" class="ltx_math_unparsed" display="inline" id="S2.E1.m1.3"><semantics id="S2.E1.m1.3a"><mrow id="S2.E1.m1.3b"><mi id="S2.E1.m1.3.4">𝖲𝖤</mi><mrow id="S2.E1.m1.3.5"><mo id="S2.E1.m1.3.5.1" stretchy="false">[</mo><mo id="S2.E1.m1.3.5.2" rspace="0.167em">∃</mo><mi id="S2.E1.m1.3.5.3">𝐁</mi><mo id="S2.E1.m1.3.5.4" lspace="0em" rspace="0.167em">.</mo><mi id="S2.E1.m1.3.5.5">ψ</mi><mo id="S2.E1.m1.3.5.6" stretchy="false">]</mo></mrow><mover id="S2.E1.m1.3.6"><mo id="S2.E1.m1.3.6.2" rspace="0.111em">=</mo><mtext id="S2.E1.m1.3.6.3" mathsize="71%">def</mtext></mover><msub id="S2.E1.m1.3.7"><mo id="S2.E1.m1.3.7.2">⋁</mo><mrow id="S2.E1.m1.3.3.3"><msub id="S2.E1.m1.3.3.3.5"><mi id="S2.E1.m1.3.3.3.5.2">δ</mi><mi id="S2.E1.m1.3.3.3.5.3">i</mi></msub><mo id="S2.E1.m1.3.3.3.4">∈</mo><msup id="S2.E1.m1.3.3.3.6"><mrow id="S2.E1.m1.3.3.3.6.2.2"><mo id="S2.E1.m1.3.3.3.6.2.2.1" stretchy="false">{</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.E1.m1.2.2.2.2">T</mtext><mo id="S2.E1.m1.3.3.3.6.2.2.2">,</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.E1.m1.3.3.3.3">F</mtext><mo id="S2.E1.m1.3.3.3.6.2.2.3" stretchy="false">}</mo></mrow><mrow id="S2.E1.m1.1.1.1.1.1.3"><mo id="S2.E1.m1.1.1.1.1.1.3.1" stretchy="false">|</mo><mi id="S2.E1.m1.1.1.1.1.1.1">𝐁</mi><mo id="S2.E1.m1.1.1.1.1.1.3.2" stretchy="false">|</mo></mrow></msup></mrow></msub><mi id="S2.E1.m1.3.8">ψ</mi><msub id="S2.E1.m1.3.9"><mo fence="false" id="S2.E1.m1.3.9.2" stretchy="false">|</mo><msub id="S2.E1.m1.3.9.3"><mi id="S2.E1.m1.3.9.3.2">δ</mi><mi id="S2.E1.m1.3.9.3.3">i</mi></msub></msub></mrow><annotation encoding="application/x-tex" id="S2.E1.m1.3c">\displaystyle{\sf SE}[{\exists{\mathbf{B}}}.\psi]\stackrel{{\scriptstyle\text{% \tiny def}}}{{=}}\textstyle\bigvee_{\delta_{i}\in\{{{\scriptsize\mbox{{\sf T}}% ,\mbox{{\sf F}}}}\}^{|{\mathbf{B}}|}}\psi|_{\delta_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.3d">sansserif_SE [ ∃ bold_B . italic_ψ ] start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ⋁ start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { T , F } start_POSTSUPERSCRIPT | bold_B | end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ψ | start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS0.SSS0.Px5.p1.13">where <math alttext="\{{{\mbox{{\sf T}},\mbox{{\sf F}}}}\}^{|{\mathbf{B}}|}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.10.m1.3"><semantics id="S2.SS0.SSS0.Px5.p1.10.m1.3a"><msup id="S2.SS0.SSS0.Px5.p1.10.m1.3.4" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.4.cmml"><mrow id="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.2" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.1.cmml"><mo id="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.1.cmml">{</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px5.p1.10.m1.2.2" xref="S2.SS0.SSS0.Px5.p1.10.m1.2.2a.cmml">T</mtext><mo id="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.2.2" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.1.cmml">,</mo><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px5.p1.10.m1.3.3" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.3a.cmml">F</mtext><mo id="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.1.cmml">}</mo></mrow><mrow id="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.3" xref="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.2.cmml"><mo id="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.3.1" stretchy="false" xref="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.2.1.cmml">|</mo><mi id="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.1" xref="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.1.cmml">𝐁</mi><mo id="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.3.2" stretchy="false" xref="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.2.1.cmml">|</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.10.m1.3b"><apply id="S2.SS0.SSS0.Px5.p1.10.m1.3.4.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.4"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px5.p1.10.m1.3.4.1.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.4">superscript</csymbol><set id="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.1.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.4.2.2"><ci id="S2.SS0.SSS0.Px5.p1.10.m1.2.2a.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.2.2"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px5.p1.10.m1.2.2.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.2.2">T</mtext></ci><ci id="S2.SS0.SSS0.Px5.p1.10.m1.3.3a.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.3"><mtext class="ltx_mathvariant_sans-serif" id="S2.SS0.SSS0.Px5.p1.10.m1.3.3.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.3.3">F</mtext></ci></set><apply id="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.3"><abs id="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.3.1"></abs><ci id="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.10.m1.1.1.1.1">𝐁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.10.m1.3c">\{{{\mbox{{\sf T}},\mbox{{\sf F}}}}\}^{|{\mathbf{B}}|}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.10.m1.3d">{ T , F } start_POSTSUPERSCRIPT | bold_B | end_POSTSUPERSCRIPT</annotation></semantics></math> is the set of all total assignments on <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.11.m2.1"><semantics id="S2.SS0.SSS0.Px5.p1.11.m2.1a"><mi id="S2.SS0.SSS0.Px5.p1.11.m2.1.1" xref="S2.SS0.SSS0.Px5.p1.11.m2.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.11.m2.1b"><ci id="S2.SS0.SSS0.Px5.p1.11.m2.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.11.m2.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.11.m2.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.11.m2.1d">bold_B</annotation></semantics></math>. Notice that some <math alttext="\psi|_{\delta_{i}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.12.m3.2"><semantics id="S2.SS0.SSS0.Px5.p1.12.m3.2a"><msub id="S2.SS0.SSS0.Px5.p1.12.m3.2.3.2" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.3.1.cmml"><mrow id="S2.SS0.SSS0.Px5.p1.12.m3.2.3.2.2" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.3.1.cmml"><mi id="S2.SS0.SSS0.Px5.p1.12.m3.1.1" xref="S2.SS0.SSS0.Px5.p1.12.m3.1.1.cmml">ψ</mi><mo id="S2.SS0.SSS0.Px5.p1.12.m3.2.3.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.3.1.1.cmml">|</mo></mrow><msub id="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.cmml"><mi id="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.2" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.2.cmml">δ</mi><mi id="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.3" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.3.cmml">i</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.12.m3.2b"><apply id="S2.SS0.SSS0.Px5.p1.12.m3.2.3.1.cmml" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.3.2"><csymbol cd="latexml" id="S2.SS0.SSS0.Px5.p1.12.m3.2.3.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.3.2.2.1">evaluated-at</csymbol><ci id="S2.SS0.SSS0.Px5.p1.12.m3.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.12.m3.1.1">𝜓</ci><apply id="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.cmml" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.2.cmml" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.2">𝛿</ci><ci id="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.3.cmml" xref="S2.SS0.SSS0.Px5.p1.12.m3.2.2.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.12.m3.2c">\psi|_{\delta_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.12.m3.2d">italic_ψ | start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> may be inconsistent or <math alttext="\bot" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px5.p1.13.m4.1"><semantics id="S2.SS0.SSS0.Px5.p1.13.m4.1a"><mo id="S2.SS0.SSS0.Px5.p1.13.m4.1.1" xref="S2.SS0.SSS0.Px5.p1.13.m4.1.1.cmml">⊥</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px5.p1.13.m4.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px5.p1.13.m4.1.1.cmml" xref="S2.SS0.SSS0.Px5.p1.13.m4.1.1">bottom</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px5.p1.13.m4.1c">\bot</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px5.p1.13.m4.1d">⊥</annotation></semantics></math>. The following property derives directly from the above definitions.</p> </div> <div class="ltx_theorem ltx_theorem_property" id="Thmproperty3"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Property 3</span></h6> <div class="ltx_para" id="Thmproperty3.p1"> <p class="ltx_p" id="Thmproperty3.p1.7">Let <math alttext="\psi" class="ltx_Math" display="inline" id="Thmproperty3.p1.1.m1.1"><semantics id="Thmproperty3.p1.1.m1.1a"><mi id="Thmproperty3.p1.1.m1.1.1" xref="Thmproperty3.p1.1.m1.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="Thmproperty3.p1.1.m1.1b"><ci id="Thmproperty3.p1.1.m1.1.1.cmml" xref="Thmproperty3.p1.1.m1.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty3.p1.1.m1.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproperty3.p1.1.m1.1d">italic_ψ</annotation></semantics></math> be a formula on <math alttext="{\mathbf{A}}\cup{\mathbf{B}}" class="ltx_Math" display="inline" id="Thmproperty3.p1.2.m2.1"><semantics id="Thmproperty3.p1.2.m2.1a"><mrow id="Thmproperty3.p1.2.m2.1.1" xref="Thmproperty3.p1.2.m2.1.1.cmml"><mi id="Thmproperty3.p1.2.m2.1.1.2" xref="Thmproperty3.p1.2.m2.1.1.2.cmml">𝐀</mi><mo id="Thmproperty3.p1.2.m2.1.1.1" xref="Thmproperty3.p1.2.m2.1.1.1.cmml">∪</mo><mi id="Thmproperty3.p1.2.m2.1.1.3" xref="Thmproperty3.p1.2.m2.1.1.3.cmml">𝐁</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty3.p1.2.m2.1b"><apply id="Thmproperty3.p1.2.m2.1.1.cmml" xref="Thmproperty3.p1.2.m2.1.1"><union id="Thmproperty3.p1.2.m2.1.1.1.cmml" xref="Thmproperty3.p1.2.m2.1.1.1"></union><ci id="Thmproperty3.p1.2.m2.1.1.2.cmml" xref="Thmproperty3.p1.2.m2.1.1.2">𝐀</ci><ci id="Thmproperty3.p1.2.m2.1.1.3.cmml" xref="Thmproperty3.p1.2.m2.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty3.p1.2.m2.1c">{\mathbf{A}}\cup{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty3.p1.2.m2.1d">bold_A ∪ bold_B</annotation></semantics></math> and <math alttext="\eta" class="ltx_Math" display="inline" id="Thmproperty3.p1.3.m3.1"><semantics id="Thmproperty3.p1.3.m3.1a"><mi id="Thmproperty3.p1.3.m3.1.1" xref="Thmproperty3.p1.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmproperty3.p1.3.m3.1b"><ci id="Thmproperty3.p1.3.m3.1.1.cmml" xref="Thmproperty3.p1.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty3.p1.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmproperty3.p1.3.m3.1d">italic_η</annotation></semantics></math> be a total truth assignment on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty3.p1.4.m4.1"><semantics id="Thmproperty3.p1.4.m4.1a"><mi id="Thmproperty3.p1.4.m4.1.1" xref="Thmproperty3.p1.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty3.p1.4.m4.1b"><ci id="Thmproperty3.p1.4.m4.1.1.cmml" xref="Thmproperty3.p1.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty3.p1.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty3.p1.4.m4.1d">bold_A</annotation></semantics></math>. <br class="ltx_break"/>Then <math alttext="\eta\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmproperty3.p1.5.m5.2"><semantics id="Thmproperty3.p1.5.m5.2a"><mrow id="Thmproperty3.p1.5.m5.2.2.1" xref="Thmproperty3.p1.5.m5.2.2.2.cmml"><mrow id="Thmproperty3.p1.5.m5.2.2.1.1" xref="Thmproperty3.p1.5.m5.2.2.1.1.cmml"><mi id="Thmproperty3.p1.5.m5.2.2.1.1.2" xref="Thmproperty3.p1.5.m5.2.2.1.1.2.cmml">η</mi><mo id="Thmproperty3.p1.5.m5.2.2.1.1.1" xref="Thmproperty3.p1.5.m5.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmproperty3.p1.5.m5.2.2.1.1.3" xref="Thmproperty3.p1.5.m5.2.2.1.1.3.cmml"><mo id="Thmproperty3.p1.5.m5.2.2.1.1.3.1" rspace="0.167em" xref="Thmproperty3.p1.5.m5.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmproperty3.p1.5.m5.2.2.1.1.3.2" xref="Thmproperty3.p1.5.m5.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmproperty3.p1.5.m5.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmproperty3.p1.5.m5.2.2.2a.cmml">.</mo><mi id="Thmproperty3.p1.5.m5.1.1" xref="Thmproperty3.p1.5.m5.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty3.p1.5.m5.2b"><apply id="Thmproperty3.p1.5.m5.2.2.2.cmml" xref="Thmproperty3.p1.5.m5.2.2.1"><csymbol cd="ambiguous" id="Thmproperty3.p1.5.m5.2.2.2a.cmml" xref="Thmproperty3.p1.5.m5.2.2.1.2">formulae-sequence</csymbol><apply id="Thmproperty3.p1.5.m5.2.2.1.1.cmml" xref="Thmproperty3.p1.5.m5.2.2.1.1"><csymbol cd="latexml" id="Thmproperty3.p1.5.m5.2.2.1.1.1.cmml" xref="Thmproperty3.p1.5.m5.2.2.1.1.1">models</csymbol><ci id="Thmproperty3.p1.5.m5.2.2.1.1.2.cmml" xref="Thmproperty3.p1.5.m5.2.2.1.1.2">𝜂</ci><apply id="Thmproperty3.p1.5.m5.2.2.1.1.3.cmml" xref="Thmproperty3.p1.5.m5.2.2.1.1.3"><exists id="Thmproperty3.p1.5.m5.2.2.1.1.3.1.cmml" xref="Thmproperty3.p1.5.m5.2.2.1.1.3.1"></exists><ci id="Thmproperty3.p1.5.m5.2.2.1.1.3.2.cmml" xref="Thmproperty3.p1.5.m5.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="Thmproperty3.p1.5.m5.1.1.cmml" xref="Thmproperty3.p1.5.m5.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty3.p1.5.m5.2c">\eta\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproperty3.p1.5.m5.2d">italic_η ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math> iff <math alttext="\eta\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="Thmproperty3.p1.6.m6.1"><semantics id="Thmproperty3.p1.6.m6.1a"><mrow id="Thmproperty3.p1.6.m6.1b"><mi id="Thmproperty3.p1.6.m6.1.1">η</mi><mo id="Thmproperty3.p1.6.m6.1.2">⊧</mo><mi id="Thmproperty3.p1.6.m6.1.3">𝖲𝖤</mi><mrow id="Thmproperty3.p1.6.m6.1.4"><mo id="Thmproperty3.p1.6.m6.1.4.1" stretchy="false">[</mo><mo id="Thmproperty3.p1.6.m6.1.4.2" rspace="0.167em">∃</mo><mi id="Thmproperty3.p1.6.m6.1.4.3">𝐁</mi><mo id="Thmproperty3.p1.6.m6.1.4.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproperty3.p1.6.m6.1.4.5">ψ</mi><mo id="Thmproperty3.p1.6.m6.1.4.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmproperty3.p1.6.m6.1c">\eta\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="Thmproperty3.p1.6.m6.1d">italic_η ⊧ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math>, that is, <math alttext="\exists{\mathbf{B}}.\psi\equiv{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="Thmproperty3.p1.7.m7.1"><semantics id="Thmproperty3.p1.7.m7.1a"><mrow id="Thmproperty3.p1.7.m7.1b"><mo id="Thmproperty3.p1.7.m7.1.1" rspace="0.167em">∃</mo><mi id="Thmproperty3.p1.7.m7.1.2">𝐁</mi><mo id="Thmproperty3.p1.7.m7.1.3" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproperty3.p1.7.m7.1.4">ψ</mi><mo id="Thmproperty3.p1.7.m7.1.5">≡</mo><mi id="Thmproperty3.p1.7.m7.1.6">𝖲𝖤</mi><mrow id="Thmproperty3.p1.7.m7.1.7"><mo id="Thmproperty3.p1.7.m7.1.7.1" stretchy="false">[</mo><mo id="Thmproperty3.p1.7.m7.1.7.2" rspace="0.167em">∃</mo><mi id="Thmproperty3.p1.7.m7.1.7.3">𝐁</mi><mo id="Thmproperty3.p1.7.m7.1.7.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproperty3.p1.7.m7.1.7.5">ψ</mi><mo id="Thmproperty3.p1.7.m7.1.7.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmproperty3.p1.7.m7.1c">\exists{\mathbf{B}}.\psi\equiv{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="Thmproperty3.p1.7.m7.1d">∃ bold_B . italic_ψ ≡ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math>.</p> </div> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px6"> <h4 class="ltx_title ltx_title_paragraph">CNF-ization.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px6.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px6.p1.7">Every generic formula <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.1.m1.1"><semantics id="S2.SS0.SSS0.Px6.p1.1.m1.1a"><mi id="S2.SS0.SSS0.Px6.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px6.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.1.m1.1b"><ci id="S2.SS0.SSS0.Px6.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.1.m1.1d">italic_φ</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.2.m2.1"><semantics id="S2.SS0.SSS0.Px6.p1.2.m2.1a"><mi id="S2.SS0.SSS0.Px6.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px6.p1.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.2.m2.1b"><ci id="S2.SS0.SSS0.Px6.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.2.m2.1d">bold_A</annotation></semantics></math> can be encoded into a CNF formula <math alttext="\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.3.m3.1"><semantics id="S2.SS0.SSS0.Px6.p1.3.m3.1a"><mi id="S2.SS0.SSS0.Px6.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px6.p1.3.m3.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.3.m3.1b"><ci id="S2.SS0.SSS0.Px6.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.3.m3.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.3.m3.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.3.m3.1d">italic_ψ</annotation></semantics></math> on <math alttext="{\mathbf{A}}\cup{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.4.m4.1"><semantics id="S2.SS0.SSS0.Px6.p1.4.m4.1a"><mrow id="S2.SS0.SSS0.Px6.p1.4.m4.1.1" xref="S2.SS0.SSS0.Px6.p1.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px6.p1.4.m4.1.1.2" xref="S2.SS0.SSS0.Px6.p1.4.m4.1.1.2.cmml">𝐀</mi><mo id="S2.SS0.SSS0.Px6.p1.4.m4.1.1.1" xref="S2.SS0.SSS0.Px6.p1.4.m4.1.1.1.cmml">∪</mo><mi id="S2.SS0.SSS0.Px6.p1.4.m4.1.1.3" xref="S2.SS0.SSS0.Px6.p1.4.m4.1.1.3.cmml">𝐁</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.4.m4.1b"><apply id="S2.SS0.SSS0.Px6.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.4.m4.1.1"><union id="S2.SS0.SSS0.Px6.p1.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.4.m4.1.1.1"></union><ci id="S2.SS0.SSS0.Px6.p1.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.4.m4.1.1.2">𝐀</ci><ci id="S2.SS0.SSS0.Px6.p1.4.m4.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.4.m4.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.4.m4.1c">{\mathbf{A}}\cup{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.4.m4.1d">bold_A ∪ bold_B</annotation></semantics></math> for some <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.5.m5.1"><semantics id="S2.SS0.SSS0.Px6.p1.5.m5.1a"><mi id="S2.SS0.SSS0.Px6.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px6.p1.5.m5.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.5.m5.1b"><ci id="S2.SS0.SSS0.Px6.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.5.m5.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.5.m5.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.5.m5.1d">bold_B</annotation></semantics></math> by applying (variants of) Tseitin CNF-ization <math alttext="\mathsf{CNF_{Ts}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.6.m6.1"><semantics id="S2.SS0.SSS0.Px6.p1.6.m6.1a"><msub id="S2.SS0.SSS0.Px6.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px6.p1.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px6.p1.6.m6.1.1.2" xref="S2.SS0.SSS0.Px6.p1.6.m6.1.1.2.cmml">𝖢𝖭𝖥</mi><mi id="S2.SS0.SSS0.Px6.p1.6.m6.1.1.3" xref="S2.SS0.SSS0.Px6.p1.6.m6.1.1.3.cmml">𝖳𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.6.m6.1b"><apply id="S2.SS0.SSS0.Px6.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.6.m6.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.6.m6.1.1.2">𝖢𝖭𝖥</ci><ci id="S2.SS0.SSS0.Px6.p1.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.6.m6.1.1.3">𝖳𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.6.m6.1c">\mathsf{CNF_{Ts}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.6.m6.1d">sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT</annotation></semantics></math>(<math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.7.m7.1"><semantics id="S2.SS0.SSS0.Px6.p1.7.m7.1a"><mi id="S2.SS0.SSS0.Px6.p1.7.m7.1.1" xref="S2.SS0.SSS0.Px6.p1.7.m7.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.7.m7.1b"><ci id="S2.SS0.SSS0.Px6.p1.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.7.m7.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.7.m7.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.7.m7.1d">italic_φ</annotation></semantics></math>) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib41" title="">41</a>]</cite>, consisting in applying recursively bottom-up the rewriting rule:</p> <table class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table" id="S5.EGx2"> <tbody id="S2.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\varphi" class="ltx_Math" display="inline" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.E2.m1.1b"><ci id="S2.E2.m1.1.1.cmml" xref="S2.E2.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.1c">\displaystyle\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.1d">italic_φ</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_eqn_cell"><math alttext="\displaystyle\Rightarrow" class="ltx_Math" display="inline" id="S2.E2.m2.1"><semantics id="S2.E2.m2.1a"><mo id="S2.E2.m2.1.1" stretchy="false" xref="S2.E2.m2.1.1.cmml">⇒</mo><annotation-xml encoding="MathML-Content" id="S2.E2.m2.1b"><ci id="S2.E2.m2.1.1.cmml" xref="S2.E2.m2.1.1">⇒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m2.1c">\displaystyle\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m2.1d">⇒</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\varphi[(l_{j1}\bowtie l_{j2})\mapsto B_{j}]\wedge\mathsf{CNF_{DM% }}(B_{j}\leftrightarrow(l_{j1}\bowtie l_{j2}))" class="ltx_math_unparsed" display="inline" id="S2.E2.m3.1"><semantics id="S2.E2.m3.1a"><mrow id="S2.E2.m3.1b"><mi id="S2.E2.m3.1.1">φ</mi><mrow id="S2.E2.m3.1.2"><mo id="S2.E2.m3.1.2.1" stretchy="false">[</mo><mrow id="S2.E2.m3.1.2.2"><mo id="S2.E2.m3.1.2.2.1" stretchy="false">(</mo><msub id="S2.E2.m3.1.2.2.2"><mi id="S2.E2.m3.1.2.2.2.2">l</mi><mrow id="S2.E2.m3.1.2.2.2.3"><mi id="S2.E2.m3.1.2.2.2.3.2">j</mi><mo id="S2.E2.m3.1.2.2.2.3.1">⁢</mo><mn id="S2.E2.m3.1.2.2.2.3.3">1</mn></mrow></msub><mo id="S2.E2.m3.1.2.2.3">⋈</mo><msub id="S2.E2.m3.1.2.2.4"><mi id="S2.E2.m3.1.2.2.4.2">l</mi><mrow id="S2.E2.m3.1.2.2.4.3"><mi id="S2.E2.m3.1.2.2.4.3.2">j</mi><mo id="S2.E2.m3.1.2.2.4.3.1">⁢</mo><mn id="S2.E2.m3.1.2.2.4.3.3">2</mn></mrow></msub><mo id="S2.E2.m3.1.2.2.5" stretchy="false">)</mo></mrow><mo id="S2.E2.m3.1.2.3" stretchy="false">↦</mo><msub id="S2.E2.m3.1.2.4"><mi id="S2.E2.m3.1.2.4.2">B</mi><mi id="S2.E2.m3.1.2.4.3">j</mi></msub><mo id="S2.E2.m3.1.2.5" stretchy="false">]</mo></mrow><mo id="S2.E2.m3.1.3">∧</mo><msub id="S2.E2.m3.1.4"><mi id="S2.E2.m3.1.4.2">𝖢𝖭𝖥</mi><mi id="S2.E2.m3.1.4.3">𝖣𝖬</mi></msub><mrow id="S2.E2.m3.1.5"><mo id="S2.E2.m3.1.5.1" stretchy="false">(</mo><msub id="S2.E2.m3.1.5.2"><mi id="S2.E2.m3.1.5.2.2">B</mi><mi id="S2.E2.m3.1.5.2.3">j</mi></msub><mo id="S2.E2.m3.1.5.3" stretchy="false">↔</mo><mrow id="S2.E2.m3.1.5.4"><mo id="S2.E2.m3.1.5.4.1" stretchy="false">(</mo><msub id="S2.E2.m3.1.5.4.2"><mi id="S2.E2.m3.1.5.4.2.2">l</mi><mrow id="S2.E2.m3.1.5.4.2.3"><mi id="S2.E2.m3.1.5.4.2.3.2">j</mi><mo id="S2.E2.m3.1.5.4.2.3.1">⁢</mo><mn id="S2.E2.m3.1.5.4.2.3.3">1</mn></mrow></msub><mo id="S2.E2.m3.1.5.4.3">⋈</mo><msub id="S2.E2.m3.1.5.4.4"><mi id="S2.E2.m3.1.5.4.4.2">l</mi><mrow id="S2.E2.m3.1.5.4.4.3"><mi id="S2.E2.m3.1.5.4.4.3.2">j</mi><mo id="S2.E2.m3.1.5.4.4.3.1">⁢</mo><mn id="S2.E2.m3.1.5.4.4.3.3">2</mn></mrow></msub><mo id="S2.E2.m3.1.5.4.5" stretchy="false">)</mo></mrow><mo id="S2.E2.m3.1.5.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.E2.m3.1c">\displaystyle\varphi[(l_{j1}\bowtie l_{j2})\mapsto B_{j}]\wedge\mathsf{CNF_{DM% }}(B_{j}\leftrightarrow(l_{j1}\bowtie l_{j2}))</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m3.1d">italic_φ [ ( italic_l start_POSTSUBSCRIPT italic_j 1 end_POSTSUBSCRIPT ⋈ italic_l start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT ) ↦ italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ] ∧ sansserif_CNF start_POSTSUBSCRIPT sansserif_DM end_POSTSUBSCRIPT ( italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ↔ ( italic_l start_POSTSUBSCRIPT italic_j 1 end_POSTSUBSCRIPT ⋈ italic_l start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS0.SSS0.Px6.p1.21">until the resulting formula <math alttext="\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.8.m1.1"><semantics id="S2.SS0.SSS0.Px6.p1.8.m1.1a"><mi id="S2.SS0.SSS0.Px6.p1.8.m1.1.1" xref="S2.SS0.SSS0.Px6.p1.8.m1.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.8.m1.1b"><ci id="S2.SS0.SSS0.Px6.p1.8.m1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.8.m1.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.8.m1.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.8.m1.1d">italic_ψ</annotation></semantics></math> is in CNF, where <math alttext="l_{j1},l_{j2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.9.m2.2"><semantics id="S2.SS0.SSS0.Px6.p1.9.m2.2a"><mrow id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.3.cmml"><msub id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.2" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.1" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.3" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.3" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.2" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.2" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.1" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.3" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.3.cmml">2</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.9.m2.2b"><list id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.3.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2"><apply id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3"><times id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.9.m2.1.1.1.1.3.3">1</cn></apply></apply><apply id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3"><times id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.9.m2.2.2.2.2.3.3">2</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.9.m2.2c">l_{j1},l_{j2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.9.m2.2d">italic_l start_POSTSUBSCRIPT italic_j 1 end_POSTSUBSCRIPT , italic_l start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT</annotation></semantics></math> are literals, <math alttext="\bowtie\ \in\ \{{\wedge,\vee,\rightarrow,\leftarrow,\leftrightarrow}\}" class="ltx_math_unparsed" display="inline" id="S2.SS0.SSS0.Px6.p1.10.m3.2"><semantics id="S2.SS0.SSS0.Px6.p1.10.m3.2a"><mrow id="S2.SS0.SSS0.Px6.p1.10.m3.2b"><mo id="S2.SS0.SSS0.Px6.p1.10.m3.1.1" rspace="0.222em">⋈</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.2" rspace="0.778em">∈</mo><mrow id="S2.SS0.SSS0.Px6.p1.10.m3.2.3"><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.1" stretchy="false">{</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.2" lspace="0em" rspace="0em">∧</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.3" rspace="0em">,</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.4" lspace="0em" rspace="0em">∨</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.5">,</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.6" lspace="0em" rspace="0em" stretchy="false">→</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.7">,</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.8" lspace="0em" rspace="0em" stretchy="false">←</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.9">,</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.10" lspace="0em" rspace="0em" stretchy="false">↔</mo><mo id="S2.SS0.SSS0.Px6.p1.10.m3.2.3.11" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.10.m3.2c">\bowtie\ \in\ \{{\wedge,\vee,\rightarrow,\leftarrow,\leftrightarrow}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.10.m3.2d">⋈ ∈ { ∧ , ∨ , → , ← , ↔ }</annotation></semantics></math> and <math alttext="\mathsf{CNF_{DM}}()" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.11.m4.1"><semantics id="S2.SS0.SSS0.Px6.p1.11.m4.1a"><mrow id="S2.SS0.SSS0.Px6.p1.11.m4.1.1" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.cmml"><msub id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.2" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.2.cmml">𝖢𝖭𝖥</mi><mi id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.3" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.3.cmml">𝖣𝖬</mi></msub><mo id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.1" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.3.2" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.cmml"><mo id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.3.1.cmml">(</mo><mo id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.11.m4.1b"><apply id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1"><times id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.1"></times><apply id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.2">𝖢𝖭𝖥</ci><ci id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.2.3">𝖣𝖬</ci></apply><list id="S2.SS0.SSS0.Px6.p1.11.m4.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.11.m4.1.1.3.2.1"></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.11.m4.1c">\mathsf{CNF_{DM}}()</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.11.m4.1d">sansserif_CNF start_POSTSUBSCRIPT sansserif_DM end_POSTSUBSCRIPT ( )</annotation></semantics></math> is the validity-preserving CNF conversion based on DeMorgan rules (e.g., <math alttext="\mathsf{CNF_{DM}}(B\leftrightarrow(l_{1}\wedge l_{2}))\stackrel{{\scriptstyle% \text{\tiny def}}}{{=}}(\neg B\vee l_{1})\wedge(\neg B\vee l_{2})\wedge(B\vee% \neg l_{1}\vee\neg l_{2})" class="ltx_math_unparsed" display="inline" id="S2.SS0.SSS0.Px6.p1.12.m5.1"><semantics id="S2.SS0.SSS0.Px6.p1.12.m5.1a"><mrow id="S2.SS0.SSS0.Px6.p1.12.m5.1b"><msub id="S2.SS0.SSS0.Px6.p1.12.m5.1.1"><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.1.2">𝖢𝖭𝖥</mi><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.1.3">𝖣𝖬</mi></msub><mrow id="S2.SS0.SSS0.Px6.p1.12.m5.1.2"><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.1" stretchy="false">(</mo><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.2">B</mi><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.3" stretchy="false">↔</mo><mrow id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4"><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.1" stretchy="false">(</mo><msub id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.2"><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.2.2">l</mi><mn id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.2.3">1</mn></msub><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.3">∧</mo><msub id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.4"><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.4.2">l</mi><mn id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.4.3">2</mn></msub><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.4.5" stretchy="false">)</mo></mrow><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.2.5" stretchy="false">)</mo></mrow><mover id="S2.SS0.SSS0.Px6.p1.12.m5.1.3"><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.3.2">=</mo><mtext id="S2.SS0.SSS0.Px6.p1.12.m5.1.3.3" mathsize="71%">def</mtext></mover><mrow id="S2.SS0.SSS0.Px6.p1.12.m5.1.4"><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.4.1" stretchy="false">(</mo><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.4.2" rspace="0.167em">¬</mo><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.4.3">B</mi><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.4.4">∨</mo><msub id="S2.SS0.SSS0.Px6.p1.12.m5.1.4.5"><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.4.5.2">l</mi><mn id="S2.SS0.SSS0.Px6.p1.12.m5.1.4.5.3">1</mn></msub><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.4.6" stretchy="false">)</mo></mrow><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.5">∧</mo><mrow id="S2.SS0.SSS0.Px6.p1.12.m5.1.6"><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.6.1" stretchy="false">(</mo><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.6.2" rspace="0.167em">¬</mo><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.6.3">B</mi><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.6.4">∨</mo><msub id="S2.SS0.SSS0.Px6.p1.12.m5.1.6.5"><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.6.5.2">l</mi><mn id="S2.SS0.SSS0.Px6.p1.12.m5.1.6.5.3">2</mn></msub><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.6.6" stretchy="false">)</mo></mrow><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.7">∧</mo><mrow id="S2.SS0.SSS0.Px6.p1.12.m5.1.8"><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.1" stretchy="false">(</mo><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.2">B</mi><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.3">∨</mo><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.4" rspace="0.167em">¬</mo><msub id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.5"><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.5.2">l</mi><mn id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.5.3">1</mn></msub><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.6">∨</mo><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.7" rspace="0.167em">¬</mo><msub id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.8"><mi id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.8.2">l</mi><mn id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.8.3">2</mn></msub><mo id="S2.SS0.SSS0.Px6.p1.12.m5.1.8.9" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.12.m5.1c">\mathsf{CNF_{DM}}(B\leftrightarrow(l_{1}\wedge l_{2}))\stackrel{{\scriptstyle% \text{\tiny def}}}{{=}}(\neg B\vee l_{1})\wedge(\neg B\vee l_{2})\wedge(B\vee% \neg l_{1}\vee\neg l_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.12.m5.1d">sansserif_CNF start_POSTSUBSCRIPT sansserif_DM end_POSTSUBSCRIPT ( italic_B ↔ ( italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ( ¬ italic_B ∨ italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∧ ( ¬ italic_B ∨ italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∧ ( italic_B ∨ ¬ italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ ¬ italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>). The size of <math alttext="\psi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.13.m6.1"><semantics id="S2.SS0.SSS0.Px6.p1.13.m6.1a"><mi id="S2.SS0.SSS0.Px6.p1.13.m6.1.1" xref="S2.SS0.SSS0.Px6.p1.13.m6.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.13.m6.1b"><ci id="S2.SS0.SSS0.Px6.p1.13.m6.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.13.m6.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.13.m6.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.13.m6.1d">italic_ψ</annotation></semantics></math> is linear wrt. that of <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.14.m7.1"><semantics id="S2.SS0.SSS0.Px6.p1.14.m7.1a"><mi id="S2.SS0.SSS0.Px6.p1.14.m7.1.1" xref="S2.SS0.SSS0.Px6.p1.14.m7.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.14.m7.1b"><ci id="S2.SS0.SSS0.Px6.p1.14.m7.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.14.m7.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.14.m7.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.14.m7.1d">italic_φ</annotation></semantics></math>. With Plaisted&Greenbaum CNF-ization <math alttext="\mathsf{CNF_{PG}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.15.m8.1"><semantics id="S2.SS0.SSS0.Px6.p1.15.m8.1a"><msub id="S2.SS0.SSS0.Px6.p1.15.m8.1.1" xref="S2.SS0.SSS0.Px6.p1.15.m8.1.1.cmml"><mi id="S2.SS0.SSS0.Px6.p1.15.m8.1.1.2" xref="S2.SS0.SSS0.Px6.p1.15.m8.1.1.2.cmml">𝖢𝖭𝖥</mi><mi id="S2.SS0.SSS0.Px6.p1.15.m8.1.1.3" xref="S2.SS0.SSS0.Px6.p1.15.m8.1.1.3.cmml">𝖯𝖦</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.15.m8.1b"><apply id="S2.SS0.SSS0.Px6.p1.15.m8.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.15.m8.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.15.m8.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.15.m8.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.15.m8.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.15.m8.1.1.2">𝖢𝖭𝖥</ci><ci id="S2.SS0.SSS0.Px6.p1.15.m8.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.15.m8.1.1.3">𝖯𝖦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.15.m8.1c">\mathsf{CNF_{PG}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.15.m8.1d">sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT</annotation></semantics></math>(<math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.16.m9.1"><semantics id="S2.SS0.SSS0.Px6.p1.16.m9.1a"><mi id="S2.SS0.SSS0.Px6.p1.16.m9.1.1" xref="S2.SS0.SSS0.Px6.p1.16.m9.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.16.m9.1b"><ci id="S2.SS0.SSS0.Px6.p1.16.m9.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.16.m9.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.16.m9.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.16.m9.1d">italic_φ</annotation></semantics></math>) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib32" title="">32</a>]</cite> the term <math alttext="B_{j}\leftrightarrow(l_{j1}\bowtie l_{j2})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.17.m10.1"><semantics id="S2.SS0.SSS0.Px6.p1.17.m10.1a"><mrow id="S2.SS0.SSS0.Px6.p1.17.m10.1.1" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.cmml"><msub id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.2" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.2.cmml">B</mi><mi id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.3" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.3.cmml">j</mi></msub><mo id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.2.cmml">↔</mo><mrow id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.2" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.1" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.3" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.1.cmml">⋈</mo><msub id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.2" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.1" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.3" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.3.cmml">2</mn></mrow></msub></mrow><mo id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.17.m10.1b"><apply id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1"><ci id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.2">↔</ci><apply id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.2">𝐵</ci><ci id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.3.3">𝑗</ci></apply><apply id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1"><ci id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.1">⋈</ci><apply id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3"><times id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3"><times id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.17.m10.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.17.m10.1c">B_{j}\leftrightarrow(l_{j1}\bowtie l_{j2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.17.m10.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ↔ ( italic_l start_POSTSUBSCRIPT italic_j 1 end_POSTSUBSCRIPT ⋈ italic_l start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.E2" title="Equation 2 ‣ CNF-ization. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>) is substituted with <math alttext="B_{j}\rightarrow(l_{j1}\bowtie l_{j2})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.18.m11.1"><semantics id="S2.SS0.SSS0.Px6.p1.18.m11.1a"><mrow id="S2.SS0.SSS0.Px6.p1.18.m11.1.1" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.cmml"><msub id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.2" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.2.cmml">B</mi><mi id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.3" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.3.cmml">j</mi></msub><mo id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.2.cmml">→</mo><mrow id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.2" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.1" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.3" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.1.cmml">⋈</mo><msub id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.2" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.1" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.3" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.3.cmml">2</mn></mrow></msub></mrow><mo id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.18.m11.1b"><apply id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1"><ci id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.2">→</ci><apply id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.2">𝐵</ci><ci id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.3.3">𝑗</ci></apply><apply id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1"><ci id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.1">⋈</ci><apply id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3"><times id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3"><times id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.18.m11.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.18.m11.1c">B_{j}\rightarrow(l_{j1}\bowtie l_{j2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.18.m11.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT → ( italic_l start_POSTSUBSCRIPT italic_j 1 end_POSTSUBSCRIPT ⋈ italic_l start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT )</annotation></semantics></math> [resp. <math alttext="B_{j}\leftarrow(l_{j1}\bowtie l_{j2})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.19.m12.1"><semantics id="S2.SS0.SSS0.Px6.p1.19.m12.1a"><mrow id="S2.SS0.SSS0.Px6.p1.19.m12.1.1" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.cmml"><msub id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.3" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.3.2" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.3.2.cmml">B</mi><mi id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.3.3" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.3.3.cmml">j</mi></msub><mo id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.2.cmml">←</mo><mrow id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.2" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.1" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.3" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.1.cmml">⋈</mo><msub id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.2" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.2" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.1" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.3" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.3.cmml">2</mn></mrow></msub></mrow><mo id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml 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id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3"><times id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3"><times id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.19.m12.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.19.m12.1c">B_{j}\leftarrow(l_{j1}\bowtie l_{j2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.19.m12.1d">italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ← ( italic_l start_POSTSUBSCRIPT italic_j 1 end_POSTSUBSCRIPT ⋈ italic_l start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT )</annotation></semantics></math>] if <math alttext="(l_{j1}\bowtie l_{j2})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.20.m13.1"><semantics id="S2.SS0.SSS0.Px6.p1.20.m13.1a"><mrow id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.2" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.2.cmml">l</mi><mrow id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.2" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.2.cmml">j</mi><mo id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.1" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.3" 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id="S2.SS0.SSS0.Px6.p1.20.m13.1b"><apply id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1"><ci id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.1">⋈</ci><apply id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3"><times id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.2.3.3">1</cn></apply></apply><apply id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.2">𝑙</ci><apply id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.3"><times id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.3.1.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.3.1"></times><ci id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.3.2.cmml" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.3.2">𝑗</ci><cn id="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px6.p1.20.m13.1.1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.20.m13.1c">(l_{j1}\bowtie l_{j2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.20.m13.1d">( italic_l start_POSTSUBSCRIPT italic_j 1 end_POSTSUBSCRIPT ⋈ italic_l start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT )</annotation></semantics></math> occurs only positively [resp. negatively] in <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px6.p1.21.m14.1"><semantics id="S2.SS0.SSS0.Px6.p1.21.m14.1a"><mi id="S2.SS0.SSS0.Px6.p1.21.m14.1.1" xref="S2.SS0.SSS0.Px6.p1.21.m14.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px6.p1.21.m14.1b"><ci id="S2.SS0.SSS0.Px6.p1.21.m14.1.1.cmml" xref="S2.SS0.SSS0.Px6.p1.21.m14.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px6.p1.21.m14.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px6.p1.21.m14.1d">italic_φ</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_property" id="Thmproperty4"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Property 4</span></h6> <div class="ltx_para" id="Thmproperty4.p1"> <p class="ltx_p" id="Thmproperty4.p1.7">If <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{Ts}}(\varphi)" class="ltx_Math" display="inline" id="Thmproperty4.p1.1.m1.1"><semantics id="Thmproperty4.p1.1.m1.1a"><mrow id="Thmproperty4.p1.1.m1.1.2" xref="Thmproperty4.p1.1.m1.1.2.cmml"><mi id="Thmproperty4.p1.1.m1.1.2.2" xref="Thmproperty4.p1.1.m1.1.2.2.cmml">ψ</mi><mover id="Thmproperty4.p1.1.m1.1.2.1" xref="Thmproperty4.p1.1.m1.1.2.1.cmml"><mo id="Thmproperty4.p1.1.m1.1.2.1.2" xref="Thmproperty4.p1.1.m1.1.2.1.2.cmml">=</mo><mtext id="Thmproperty4.p1.1.m1.1.2.1.3" mathsize="71%" xref="Thmproperty4.p1.1.m1.1.2.1.3a.cmml">def</mtext></mover><mrow id="Thmproperty4.p1.1.m1.1.2.3" xref="Thmproperty4.p1.1.m1.1.2.3.cmml"><msub id="Thmproperty4.p1.1.m1.1.2.3.2" xref="Thmproperty4.p1.1.m1.1.2.3.2.cmml"><mi id="Thmproperty4.p1.1.m1.1.2.3.2.2" xref="Thmproperty4.p1.1.m1.1.2.3.2.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmproperty4.p1.1.m1.1.2.3.2.3" xref="Thmproperty4.p1.1.m1.1.2.3.2.3.cmml">𝖳𝗌</mi></msub><mo id="Thmproperty4.p1.1.m1.1.2.3.1" xref="Thmproperty4.p1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="Thmproperty4.p1.1.m1.1.2.3.3.2" xref="Thmproperty4.p1.1.m1.1.2.3.cmml"><mo id="Thmproperty4.p1.1.m1.1.2.3.3.2.1" stretchy="false" xref="Thmproperty4.p1.1.m1.1.2.3.cmml">(</mo><mi id="Thmproperty4.p1.1.m1.1.1" xref="Thmproperty4.p1.1.m1.1.1.cmml">φ</mi><mo id="Thmproperty4.p1.1.m1.1.2.3.3.2.2" stretchy="false" xref="Thmproperty4.p1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty4.p1.1.m1.1b"><apply id="Thmproperty4.p1.1.m1.1.2.cmml" xref="Thmproperty4.p1.1.m1.1.2"><apply id="Thmproperty4.p1.1.m1.1.2.1.cmml" xref="Thmproperty4.p1.1.m1.1.2.1"><csymbol cd="ambiguous" id="Thmproperty4.p1.1.m1.1.2.1.1.cmml" xref="Thmproperty4.p1.1.m1.1.2.1">superscript</csymbol><eq id="Thmproperty4.p1.1.m1.1.2.1.2.cmml" xref="Thmproperty4.p1.1.m1.1.2.1.2"></eq><ci id="Thmproperty4.p1.1.m1.1.2.1.3a.cmml" xref="Thmproperty4.p1.1.m1.1.2.1.3"><mtext id="Thmproperty4.p1.1.m1.1.2.1.3.cmml" mathsize="50%" xref="Thmproperty4.p1.1.m1.1.2.1.3">def</mtext></ci></apply><ci id="Thmproperty4.p1.1.m1.1.2.2.cmml" xref="Thmproperty4.p1.1.m1.1.2.2">𝜓</ci><apply id="Thmproperty4.p1.1.m1.1.2.3.cmml" xref="Thmproperty4.p1.1.m1.1.2.3"><times id="Thmproperty4.p1.1.m1.1.2.3.1.cmml" xref="Thmproperty4.p1.1.m1.1.2.3.1"></times><apply id="Thmproperty4.p1.1.m1.1.2.3.2.cmml" xref="Thmproperty4.p1.1.m1.1.2.3.2"><csymbol cd="ambiguous" id="Thmproperty4.p1.1.m1.1.2.3.2.1.cmml" xref="Thmproperty4.p1.1.m1.1.2.3.2">subscript</csymbol><ci id="Thmproperty4.p1.1.m1.1.2.3.2.2.cmml" xref="Thmproperty4.p1.1.m1.1.2.3.2.2">𝖢𝖭𝖥</ci><ci id="Thmproperty4.p1.1.m1.1.2.3.2.3.cmml" xref="Thmproperty4.p1.1.m1.1.2.3.2.3">𝖳𝗌</ci></apply><ci id="Thmproperty4.p1.1.m1.1.1.cmml" xref="Thmproperty4.p1.1.m1.1.1">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty4.p1.1.m1.1c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{Ts}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmproperty4.p1.1.m1.1d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math> or <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\varphi)" class="ltx_Math" display="inline" id="Thmproperty4.p1.2.m2.1"><semantics id="Thmproperty4.p1.2.m2.1a"><mrow id="Thmproperty4.p1.2.m2.1.2" xref="Thmproperty4.p1.2.m2.1.2.cmml"><mi id="Thmproperty4.p1.2.m2.1.2.2" xref="Thmproperty4.p1.2.m2.1.2.2.cmml">ψ</mi><mover id="Thmproperty4.p1.2.m2.1.2.1" xref="Thmproperty4.p1.2.m2.1.2.1.cmml"><mo id="Thmproperty4.p1.2.m2.1.2.1.2" xref="Thmproperty4.p1.2.m2.1.2.1.2.cmml">=</mo><mtext id="Thmproperty4.p1.2.m2.1.2.1.3" mathsize="71%" xref="Thmproperty4.p1.2.m2.1.2.1.3a.cmml">def</mtext></mover><mrow id="Thmproperty4.p1.2.m2.1.2.3" xref="Thmproperty4.p1.2.m2.1.2.3.cmml"><msub id="Thmproperty4.p1.2.m2.1.2.3.2" xref="Thmproperty4.p1.2.m2.1.2.3.2.cmml"><mi id="Thmproperty4.p1.2.m2.1.2.3.2.2" xref="Thmproperty4.p1.2.m2.1.2.3.2.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmproperty4.p1.2.m2.1.2.3.2.3" xref="Thmproperty4.p1.2.m2.1.2.3.2.3.cmml">𝖯𝖦</mi></msub><mo id="Thmproperty4.p1.2.m2.1.2.3.1" xref="Thmproperty4.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="Thmproperty4.p1.2.m2.1.2.3.3.2" xref="Thmproperty4.p1.2.m2.1.2.3.cmml"><mo id="Thmproperty4.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="Thmproperty4.p1.2.m2.1.2.3.cmml">(</mo><mi id="Thmproperty4.p1.2.m2.1.1" xref="Thmproperty4.p1.2.m2.1.1.cmml">φ</mi><mo id="Thmproperty4.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="Thmproperty4.p1.2.m2.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty4.p1.2.m2.1b"><apply id="Thmproperty4.p1.2.m2.1.2.cmml" xref="Thmproperty4.p1.2.m2.1.2"><apply id="Thmproperty4.p1.2.m2.1.2.1.cmml" xref="Thmproperty4.p1.2.m2.1.2.1"><csymbol cd="ambiguous" id="Thmproperty4.p1.2.m2.1.2.1.1.cmml" xref="Thmproperty4.p1.2.m2.1.2.1">superscript</csymbol><eq id="Thmproperty4.p1.2.m2.1.2.1.2.cmml" xref="Thmproperty4.p1.2.m2.1.2.1.2"></eq><ci id="Thmproperty4.p1.2.m2.1.2.1.3a.cmml" xref="Thmproperty4.p1.2.m2.1.2.1.3"><mtext id="Thmproperty4.p1.2.m2.1.2.1.3.cmml" mathsize="50%" xref="Thmproperty4.p1.2.m2.1.2.1.3">def</mtext></ci></apply><ci id="Thmproperty4.p1.2.m2.1.2.2.cmml" xref="Thmproperty4.p1.2.m2.1.2.2">𝜓</ci><apply id="Thmproperty4.p1.2.m2.1.2.3.cmml" xref="Thmproperty4.p1.2.m2.1.2.3"><times id="Thmproperty4.p1.2.m2.1.2.3.1.cmml" xref="Thmproperty4.p1.2.m2.1.2.3.1"></times><apply id="Thmproperty4.p1.2.m2.1.2.3.2.cmml" xref="Thmproperty4.p1.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="Thmproperty4.p1.2.m2.1.2.3.2.1.cmml" xref="Thmproperty4.p1.2.m2.1.2.3.2">subscript</csymbol><ci id="Thmproperty4.p1.2.m2.1.2.3.2.2.cmml" xref="Thmproperty4.p1.2.m2.1.2.3.2.2">𝖢𝖭𝖥</ci><ci id="Thmproperty4.p1.2.m2.1.2.3.2.3.cmml" xref="Thmproperty4.p1.2.m2.1.2.3.2.3">𝖯𝖦</ci></apply><ci id="Thmproperty4.p1.2.m2.1.1.cmml" xref="Thmproperty4.p1.2.m2.1.1">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty4.p1.2.m2.1c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmproperty4.p1.2.m2.1d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math>, then <math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="Thmproperty4.p1.3.m3.1"><semantics id="Thmproperty4.p1.3.m3.1a"><mrow id="Thmproperty4.p1.3.m3.1.1" xref="Thmproperty4.p1.3.m3.1.1.cmml"><mi id="Thmproperty4.p1.3.m3.1.1.2" xref="Thmproperty4.p1.3.m3.1.1.2.cmml">η</mi><mo id="Thmproperty4.p1.3.m3.1.1.1" xref="Thmproperty4.p1.3.m3.1.1.1.cmml">⊧</mo><mi id="Thmproperty4.p1.3.m3.1.1.3" xref="Thmproperty4.p1.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty4.p1.3.m3.1b"><apply id="Thmproperty4.p1.3.m3.1.1.cmml" xref="Thmproperty4.p1.3.m3.1.1"><csymbol cd="latexml" id="Thmproperty4.p1.3.m3.1.1.1.cmml" xref="Thmproperty4.p1.3.m3.1.1.1">models</csymbol><ci id="Thmproperty4.p1.3.m3.1.1.2.cmml" xref="Thmproperty4.p1.3.m3.1.1.2">𝜂</ci><ci id="Thmproperty4.p1.3.m3.1.1.3.cmml" xref="Thmproperty4.p1.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty4.p1.3.m3.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproperty4.p1.3.m3.1d">italic_η ⊧ italic_φ</annotation></semantics></math> iff exists a total assignment <math alttext="\delta" class="ltx_Math" display="inline" id="Thmproperty4.p1.4.m4.1"><semantics id="Thmproperty4.p1.4.m4.1a"><mi id="Thmproperty4.p1.4.m4.1.1" xref="Thmproperty4.p1.4.m4.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="Thmproperty4.p1.4.m4.1b"><ci id="Thmproperty4.p1.4.m4.1.1.cmml" xref="Thmproperty4.p1.4.m4.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty4.p1.4.m4.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="Thmproperty4.p1.4.m4.1d">italic_δ</annotation></semantics></math> on <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="Thmproperty4.p1.5.m5.1"><semantics id="Thmproperty4.p1.5.m5.1a"><mi id="Thmproperty4.p1.5.m5.1.1" xref="Thmproperty4.p1.5.m5.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="Thmproperty4.p1.5.m5.1b"><ci id="Thmproperty4.p1.5.m5.1.1.cmml" xref="Thmproperty4.p1.5.m5.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty4.p1.5.m5.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty4.p1.5.m5.1d">bold_B</annotation></semantics></math> s.t. <math alttext="\eta\cup\delta\models\psi" class="ltx_Math" display="inline" id="Thmproperty4.p1.6.m6.1"><semantics id="Thmproperty4.p1.6.m6.1a"><mrow id="Thmproperty4.p1.6.m6.1.1" xref="Thmproperty4.p1.6.m6.1.1.cmml"><mrow id="Thmproperty4.p1.6.m6.1.1.2" xref="Thmproperty4.p1.6.m6.1.1.2.cmml"><mi id="Thmproperty4.p1.6.m6.1.1.2.2" xref="Thmproperty4.p1.6.m6.1.1.2.2.cmml">η</mi><mo id="Thmproperty4.p1.6.m6.1.1.2.1" xref="Thmproperty4.p1.6.m6.1.1.2.1.cmml">∪</mo><mi id="Thmproperty4.p1.6.m6.1.1.2.3" xref="Thmproperty4.p1.6.m6.1.1.2.3.cmml">δ</mi></mrow><mo id="Thmproperty4.p1.6.m6.1.1.1" xref="Thmproperty4.p1.6.m6.1.1.1.cmml">⊧</mo><mi id="Thmproperty4.p1.6.m6.1.1.3" xref="Thmproperty4.p1.6.m6.1.1.3.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty4.p1.6.m6.1b"><apply id="Thmproperty4.p1.6.m6.1.1.cmml" xref="Thmproperty4.p1.6.m6.1.1"><csymbol cd="latexml" id="Thmproperty4.p1.6.m6.1.1.1.cmml" xref="Thmproperty4.p1.6.m6.1.1.1">models</csymbol><apply id="Thmproperty4.p1.6.m6.1.1.2.cmml" xref="Thmproperty4.p1.6.m6.1.1.2"><union id="Thmproperty4.p1.6.m6.1.1.2.1.cmml" xref="Thmproperty4.p1.6.m6.1.1.2.1"></union><ci id="Thmproperty4.p1.6.m6.1.1.2.2.cmml" xref="Thmproperty4.p1.6.m6.1.1.2.2">𝜂</ci><ci id="Thmproperty4.p1.6.m6.1.1.2.3.cmml" xref="Thmproperty4.p1.6.m6.1.1.2.3">𝛿</ci></apply><ci id="Thmproperty4.p1.6.m6.1.1.3.cmml" xref="Thmproperty4.p1.6.m6.1.1.3">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty4.p1.6.m6.1c">\eta\cup\delta\models\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproperty4.p1.6.m6.1d">italic_η ∪ italic_δ ⊧ italic_ψ</annotation></semantics></math>, that is, <math alttext="\varphi\equiv\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmproperty4.p1.7.m7.2"><semantics id="Thmproperty4.p1.7.m7.2a"><mrow id="Thmproperty4.p1.7.m7.2.2.1" xref="Thmproperty4.p1.7.m7.2.2.2.cmml"><mrow id="Thmproperty4.p1.7.m7.2.2.1.1" xref="Thmproperty4.p1.7.m7.2.2.1.1.cmml"><mi id="Thmproperty4.p1.7.m7.2.2.1.1.2" xref="Thmproperty4.p1.7.m7.2.2.1.1.2.cmml">φ</mi><mo id="Thmproperty4.p1.7.m7.2.2.1.1.1" xref="Thmproperty4.p1.7.m7.2.2.1.1.1.cmml">≡</mo><mrow id="Thmproperty4.p1.7.m7.2.2.1.1.3" xref="Thmproperty4.p1.7.m7.2.2.1.1.3.cmml"><mo id="Thmproperty4.p1.7.m7.2.2.1.1.3.1" rspace="0.167em" xref="Thmproperty4.p1.7.m7.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmproperty4.p1.7.m7.2.2.1.1.3.2" xref="Thmproperty4.p1.7.m7.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmproperty4.p1.7.m7.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmproperty4.p1.7.m7.2.2.2a.cmml">.</mo><mi id="Thmproperty4.p1.7.m7.1.1" xref="Thmproperty4.p1.7.m7.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty4.p1.7.m7.2b"><apply id="Thmproperty4.p1.7.m7.2.2.2.cmml" xref="Thmproperty4.p1.7.m7.2.2.1"><csymbol cd="ambiguous" id="Thmproperty4.p1.7.m7.2.2.2a.cmml" xref="Thmproperty4.p1.7.m7.2.2.1.2">formulae-sequence</csymbol><apply id="Thmproperty4.p1.7.m7.2.2.1.1.cmml" xref="Thmproperty4.p1.7.m7.2.2.1.1"><equivalent id="Thmproperty4.p1.7.m7.2.2.1.1.1.cmml" xref="Thmproperty4.p1.7.m7.2.2.1.1.1"></equivalent><ci id="Thmproperty4.p1.7.m7.2.2.1.1.2.cmml" xref="Thmproperty4.p1.7.m7.2.2.1.1.2">𝜑</ci><apply id="Thmproperty4.p1.7.m7.2.2.1.1.3.cmml" xref="Thmproperty4.p1.7.m7.2.2.1.1.3"><exists id="Thmproperty4.p1.7.m7.2.2.1.1.3.1.cmml" xref="Thmproperty4.p1.7.m7.2.2.1.1.3.1"></exists><ci id="Thmproperty4.p1.7.m7.2.2.1.1.3.2.cmml" xref="Thmproperty4.p1.7.m7.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="Thmproperty4.p1.7.m7.1.1.cmml" xref="Thmproperty4.p1.7.m7.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty4.p1.7.m7.2c">\varphi\equiv\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproperty4.p1.7.m7.2d">italic_φ ≡ ∃ bold_B . italic_ψ</annotation></semantics></math>.</p> </div> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>A theoretical Analysis of verificationand entailment</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.2">We address the following basic question: <span class="ltx_text ltx_font_italic" id="S3.p1.2.2">What should we mean by “a partial assignment <math alttext="\mu" class="ltx_Math" display="inline" id="S3.p1.1.1.m1.1"><semantics id="S3.p1.1.1.m1.1a"><mi id="S3.p1.1.1.m1.1.1" xref="S3.p1.1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.1.m1.1b"><ci id="S3.p1.1.1.m1.1.1.cmml" xref="S3.p1.1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.1.m1.1d">italic_μ</annotation></semantics></math> satisfies <math alttext="\varphi" class="ltx_Math" display="inline" id="S3.p1.2.2.m2.1"><semantics id="S3.p1.2.2.m2.1a"><mi id="S3.p1.2.2.m2.1.1" xref="S3.p1.2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.p1.2.2.m2.1b"><ci id="S3.p1.2.2.m2.1.1.cmml" xref="S3.p1.2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.2.m2.1d">italic_φ</annotation></semantics></math>"?</span> We wish to provide a satisfactory definition of partial-assignment satisfiability for a generic propositional formula —i.e., non only for (tautology-free) CNF. Ideally, a suitable definition of partial-assignment satisfiability should verify all statements in property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty1" title="Property 1 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a>, in particular (ii) and (iv). In practice, unfortunately, at least for generic (non-CNF) formulas, we show this is not the case.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Verification and entailment of plain formulas.</h3> <section class="ltx_paragraph" id="S3.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Definitions.</h4> <div class="ltx_para" id="S3.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p1.1">The first candidate definition of partial-assignment satisfaction, which extends to partial assignments property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty1" title="Property 1 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a>(iii), is <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px1.p1.1.1">verification</span>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition1"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Definition 1</span></h6> <div class="ltx_para" id="Thmdefinition1.p1"> <p class="ltx_p" id="Thmdefinition1.p1.5">We say that a <span class="ltx_text ltx_font_italic" id="Thmdefinition1.p1.5.1">partial</span> truth assignment <math alttext="\mu" class="ltx_Math" display="inline" id="Thmdefinition1.p1.1.m1.1"><semantics id="Thmdefinition1.p1.1.m1.1a"><mi id="Thmdefinition1.p1.1.m1.1.1" xref="Thmdefinition1.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.1.m1.1b"><ci id="Thmdefinition1.p1.1.m1.1.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.1.m1.1d">italic_μ</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="Thmdefinition1.p1.5.2">verifies</span> <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmdefinition1.p1.2.m2.1"><semantics id="Thmdefinition1.p1.2.m2.1a"><mi id="Thmdefinition1.p1.2.m2.1.1" xref="Thmdefinition1.p1.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.2.m2.1b"><ci id="Thmdefinition1.p1.2.m2.1.1.cmml" xref="Thmdefinition1.p1.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.2.m2.1d">italic_φ</annotation></semantics></math> iff <math alttext="\mu(\varphi)=\mbox{{\sf T}}" class="ltx_Math" display="inline" id="Thmdefinition1.p1.3.m3.1"><semantics id="Thmdefinition1.p1.3.m3.1a"><mrow id="Thmdefinition1.p1.3.m3.1.2" xref="Thmdefinition1.p1.3.m3.1.2.cmml"><mrow id="Thmdefinition1.p1.3.m3.1.2.2" xref="Thmdefinition1.p1.3.m3.1.2.2.cmml"><mi id="Thmdefinition1.p1.3.m3.1.2.2.2" xref="Thmdefinition1.p1.3.m3.1.2.2.2.cmml">μ</mi><mo id="Thmdefinition1.p1.3.m3.1.2.2.1" xref="Thmdefinition1.p1.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="Thmdefinition1.p1.3.m3.1.2.2.3.2" xref="Thmdefinition1.p1.3.m3.1.2.2.cmml"><mo id="Thmdefinition1.p1.3.m3.1.2.2.3.2.1" stretchy="false" xref="Thmdefinition1.p1.3.m3.1.2.2.cmml">(</mo><mi id="Thmdefinition1.p1.3.m3.1.1" xref="Thmdefinition1.p1.3.m3.1.1.cmml">φ</mi><mo id="Thmdefinition1.p1.3.m3.1.2.2.3.2.2" stretchy="false" xref="Thmdefinition1.p1.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="Thmdefinition1.p1.3.m3.1.2.1" xref="Thmdefinition1.p1.3.m3.1.2.1.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="Thmdefinition1.p1.3.m3.1.2.3" xref="Thmdefinition1.p1.3.m3.1.2.3a.cmml">T</mtext></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.3.m3.1b"><apply id="Thmdefinition1.p1.3.m3.1.2.cmml" xref="Thmdefinition1.p1.3.m3.1.2"><eq id="Thmdefinition1.p1.3.m3.1.2.1.cmml" xref="Thmdefinition1.p1.3.m3.1.2.1"></eq><apply id="Thmdefinition1.p1.3.m3.1.2.2.cmml" xref="Thmdefinition1.p1.3.m3.1.2.2"><times id="Thmdefinition1.p1.3.m3.1.2.2.1.cmml" xref="Thmdefinition1.p1.3.m3.1.2.2.1"></times><ci id="Thmdefinition1.p1.3.m3.1.2.2.2.cmml" xref="Thmdefinition1.p1.3.m3.1.2.2.2">𝜇</ci><ci id="Thmdefinition1.p1.3.m3.1.1.cmml" xref="Thmdefinition1.p1.3.m3.1.1">𝜑</ci></apply><ci id="Thmdefinition1.p1.3.m3.1.2.3a.cmml" xref="Thmdefinition1.p1.3.m3.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="Thmdefinition1.p1.3.m3.1.2.3.cmml" xref="Thmdefinition1.p1.3.m3.1.2.3">T</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.3.m3.1c">\mu(\varphi)=\mbox{{\sf T}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.3.m3.1d">italic_μ ( italic_φ ) = T</annotation></semantics></math> (or, equivalently by property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty2" title="Property 2 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>, iff <math alttext="\varphi|_{\mu}=\top" class="ltx_Math" display="inline" id="Thmdefinition1.p1.4.m4.2"><semantics id="Thmdefinition1.p1.4.m4.2a"><mrow id="Thmdefinition1.p1.4.m4.2.3" xref="Thmdefinition1.p1.4.m4.2.3.cmml"><msub id="Thmdefinition1.p1.4.m4.2.3.2.2" xref="Thmdefinition1.p1.4.m4.2.3.2.1.cmml"><mrow id="Thmdefinition1.p1.4.m4.2.3.2.2.2" xref="Thmdefinition1.p1.4.m4.2.3.2.1.cmml"><mi id="Thmdefinition1.p1.4.m4.1.1" xref="Thmdefinition1.p1.4.m4.1.1.cmml">φ</mi><mo id="Thmdefinition1.p1.4.m4.2.3.2.2.2.1" stretchy="false" xref="Thmdefinition1.p1.4.m4.2.3.2.1.1.cmml">|</mo></mrow><mi id="Thmdefinition1.p1.4.m4.2.2.1" xref="Thmdefinition1.p1.4.m4.2.2.1.cmml">μ</mi></msub><mo id="Thmdefinition1.p1.4.m4.2.3.1" rspace="0em" xref="Thmdefinition1.p1.4.m4.2.3.1.cmml">=</mo><mo id="Thmdefinition1.p1.4.m4.2.3.3" lspace="0em" xref="Thmdefinition1.p1.4.m4.2.3.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.4.m4.2b"><apply id="Thmdefinition1.p1.4.m4.2.3.cmml" xref="Thmdefinition1.p1.4.m4.2.3"><eq id="Thmdefinition1.p1.4.m4.2.3.1.cmml" xref="Thmdefinition1.p1.4.m4.2.3.1"></eq><apply id="Thmdefinition1.p1.4.m4.2.3.2.1.cmml" xref="Thmdefinition1.p1.4.m4.2.3.2.2"><csymbol cd="latexml" id="Thmdefinition1.p1.4.m4.2.3.2.1.1.cmml" xref="Thmdefinition1.p1.4.m4.2.3.2.2.2.1">evaluated-at</csymbol><ci id="Thmdefinition1.p1.4.m4.1.1.cmml" xref="Thmdefinition1.p1.4.m4.1.1">𝜑</ci><ci id="Thmdefinition1.p1.4.m4.2.2.1.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1">𝜇</ci></apply><csymbol cd="latexml" id="Thmdefinition1.p1.4.m4.2.3.3.cmml" xref="Thmdefinition1.p1.4.m4.2.3.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.4.m4.2c">\varphi|_{\mu}=\top</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.4.m4.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math>). We denote this fact with “<math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="Thmdefinition1.p1.5.m5.1"><semantics id="Thmdefinition1.p1.5.m5.1a"><mrow id="Thmdefinition1.p1.5.m5.1b"><mi id="Thmdefinition1.p1.5.m5.1.1">μ</mi><mpadded id="Thmdefinition1.p1.5.m5.1c" width="0.219em"><mo id="Thmdefinition1.p1.5.m5.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmdefinition1.p1.5.m5.1.3">≈</mo><mi id="Thmdefinition1.p1.5.m5.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmdefinition1.p1.5.m5.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition1.p1.5.m5.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math>”.</p> </div> </div> <div class="ltx_para ltx_noindent" id="S3.SS1.SSS0.Px1.p2"> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p2.2">We stress the fact that verification is a <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px1.p2.2.1">semantic</span> definition. Nevertheless, due to property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty2" title="Property 2 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>, it has also an easy-to-check syntactic definition: <math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S3.SS1.SSS0.Px1.p2.1.m1.1"><semantics id="S3.SS1.SSS0.Px1.p2.1.m1.1a"><mrow id="S3.SS1.SSS0.Px1.p2.1.m1.1b"><mi id="S3.SS1.SSS0.Px1.p2.1.m1.1.1">μ</mi><mpadded id="S3.SS1.SSS0.Px1.p2.1.m1.1c" width="0.219em"><mo id="S3.SS1.SSS0.Px1.p2.1.m1.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.SS1.SSS0.Px1.p2.1.m1.1.3">≈</mo><mi id="S3.SS1.SSS0.Px1.p2.1.m1.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p2.1.m1.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p2.1.m1.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math> iff <math alttext="\varphi|_{\mu}=\top" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px1.p2.2.m2.2"><semantics id="S3.SS1.SSS0.Px1.p2.2.m2.2a"><mrow id="S3.SS1.SSS0.Px1.p2.2.m2.2.3" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.cmml"><msub id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.2" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.1.cmml"><mrow id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.2.2" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.1.cmml"><mi id="S3.SS1.SSS0.Px1.p2.2.m2.1.1" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1.cmml">φ</mi><mo id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.2.2.1" stretchy="false" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.1.1.cmml">|</mo></mrow><mi id="S3.SS1.SSS0.Px1.p2.2.m2.2.2.1" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.2.1.cmml">μ</mi></msub><mo id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.1" rspace="0em" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.1.cmml">=</mo><mo id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.3" lspace="0em" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px1.p2.2.m2.2b"><apply id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3"><eq id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.1.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.1"></eq><apply id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.1.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.2"><csymbol cd="latexml" id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.2.2.2.1">evaluated-at</csymbol><ci id="S3.SS1.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.1.1">𝜑</ci><ci id="S3.SS1.SSS0.Px1.p2.2.m2.2.2.1.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.2.1">𝜇</ci></apply><csymbol cd="latexml" id="S3.SS1.SSS0.Px1.p2.2.m2.2.3.3.cmml" xref="S3.SS1.SSS0.Px1.p2.2.m2.2.3.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px1.p2.2.m2.2c">\varphi|_{\mu}=\top</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px1.p2.2.m2.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.SSS0.Px1.p3"> <p class="ltx_p" id="S3.SS1.SSS0.Px1.p3.1">The second candidate definition of partial-assignment satisfaction, which extends to partial assignments property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty1" title="Property 1 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a>(i), is <span class="ltx_text ltx_font_italic" id="S3.SS1.SSS0.Px1.p3.1.1">entailment</span>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition2"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Definition 2</span></h6> <div class="ltx_para" id="Thmdefinition2.p1"> <p class="ltx_p" id="Thmdefinition2.p1.8">We say that a <span class="ltx_text ltx_font_italic" id="Thmdefinition2.p1.8.1">partial</span> truth assignment <math alttext="\mu" class="ltx_Math" display="inline" id="Thmdefinition2.p1.1.m1.1"><semantics id="Thmdefinition2.p1.1.m1.1a"><mi id="Thmdefinition2.p1.1.m1.1.1" xref="Thmdefinition2.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.1.m1.1b"><ci id="Thmdefinition2.p1.1.m1.1.1.cmml" xref="Thmdefinition2.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.1.m1.1d">italic_μ</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="Thmdefinition2.p1.8.2">entails</span> <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmdefinition2.p1.2.m2.1"><semantics id="Thmdefinition2.p1.2.m2.1a"><mi id="Thmdefinition2.p1.2.m2.1.1" xref="Thmdefinition2.p1.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.2.m2.1b"><ci id="Thmdefinition2.p1.2.m2.1.1.cmml" xref="Thmdefinition2.p1.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.2.m2.1d">italic_φ</annotation></semantics></math> if and only if, for every total truth assignments <math alttext="\eta" class="ltx_Math" display="inline" id="Thmdefinition2.p1.3.m3.1"><semantics id="Thmdefinition2.p1.3.m3.1a"><mi id="Thmdefinition2.p1.3.m3.1.1" xref="Thmdefinition2.p1.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.3.m3.1b"><ci id="Thmdefinition2.p1.3.m3.1.1.cmml" xref="Thmdefinition2.p1.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.3.m3.1d">italic_η</annotation></semantics></math> s.t.<math alttext="\mu\subseteq\eta" class="ltx_Math" display="inline" id="Thmdefinition2.p1.4.m4.1"><semantics id="Thmdefinition2.p1.4.m4.1a"><mrow id="Thmdefinition2.p1.4.m4.1.1" xref="Thmdefinition2.p1.4.m4.1.1.cmml"><mi id="Thmdefinition2.p1.4.m4.1.1.2" xref="Thmdefinition2.p1.4.m4.1.1.2.cmml">μ</mi><mo id="Thmdefinition2.p1.4.m4.1.1.1" xref="Thmdefinition2.p1.4.m4.1.1.1.cmml">⊆</mo><mi id="Thmdefinition2.p1.4.m4.1.1.3" xref="Thmdefinition2.p1.4.m4.1.1.3.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.4.m4.1b"><apply id="Thmdefinition2.p1.4.m4.1.1.cmml" xref="Thmdefinition2.p1.4.m4.1.1"><subset id="Thmdefinition2.p1.4.m4.1.1.1.cmml" xref="Thmdefinition2.p1.4.m4.1.1.1"></subset><ci id="Thmdefinition2.p1.4.m4.1.1.2.cmml" xref="Thmdefinition2.p1.4.m4.1.1.2">𝜇</ci><ci id="Thmdefinition2.p1.4.m4.1.1.3.cmml" xref="Thmdefinition2.p1.4.m4.1.1.3">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.4.m4.1c">\mu\subseteq\eta</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.4.m4.1d">italic_μ ⊆ italic_η</annotation></semantics></math>, <math alttext="\eta" class="ltx_Math" display="inline" id="Thmdefinition2.p1.5.m5.1"><semantics id="Thmdefinition2.p1.5.m5.1a"><mi id="Thmdefinition2.p1.5.m5.1.1" xref="Thmdefinition2.p1.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.5.m5.1b"><ci id="Thmdefinition2.p1.5.m5.1.1.cmml" xref="Thmdefinition2.p1.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.5.m5.1d">italic_η</annotation></semantics></math> satisfies <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmdefinition2.p1.6.m6.1"><semantics id="Thmdefinition2.p1.6.m6.1a"><mi id="Thmdefinition2.p1.6.m6.1.1" xref="Thmdefinition2.p1.6.m6.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.6.m6.1b"><ci id="Thmdefinition2.p1.6.m6.1.1.cmml" xref="Thmdefinition2.p1.6.m6.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.6.m6.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.6.m6.1d">italic_φ</annotation></semantics></math> (or equivalently, iff <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="Thmdefinition2.p1.7.m7.2"><semantics id="Thmdefinition2.p1.7.m7.2a"><msub id="Thmdefinition2.p1.7.m7.2.3.2" xref="Thmdefinition2.p1.7.m7.2.3.1.cmml"><mrow id="Thmdefinition2.p1.7.m7.2.3.2.2" xref="Thmdefinition2.p1.7.m7.2.3.1.cmml"><mi id="Thmdefinition2.p1.7.m7.1.1" xref="Thmdefinition2.p1.7.m7.1.1.cmml">φ</mi><mo id="Thmdefinition2.p1.7.m7.2.3.2.2.1" stretchy="false" xref="Thmdefinition2.p1.7.m7.2.3.1.1.cmml">|</mo></mrow><mi id="Thmdefinition2.p1.7.m7.2.2.1" xref="Thmdefinition2.p1.7.m7.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.7.m7.2b"><apply id="Thmdefinition2.p1.7.m7.2.3.1.cmml" xref="Thmdefinition2.p1.7.m7.2.3.2"><csymbol cd="latexml" id="Thmdefinition2.p1.7.m7.2.3.1.1.cmml" xref="Thmdefinition2.p1.7.m7.2.3.2.2.1">evaluated-at</csymbol><ci id="Thmdefinition2.p1.7.m7.1.1.cmml" xref="Thmdefinition2.p1.7.m7.1.1">𝜑</ci><ci id="Thmdefinition2.p1.7.m7.2.2.1.cmml" xref="Thmdefinition2.p1.7.m7.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.7.m7.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.7.m7.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is valid). We denote this fact with “<math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmdefinition2.p1.8.m8.1"><semantics id="Thmdefinition2.p1.8.m8.1a"><mrow id="Thmdefinition2.p1.8.m8.1.1" xref="Thmdefinition2.p1.8.m8.1.1.cmml"><mi id="Thmdefinition2.p1.8.m8.1.1.2" xref="Thmdefinition2.p1.8.m8.1.1.2.cmml">μ</mi><mo id="Thmdefinition2.p1.8.m8.1.1.1" xref="Thmdefinition2.p1.8.m8.1.1.1.cmml">⊧</mo><mi id="Thmdefinition2.p1.8.m8.1.1.3" xref="Thmdefinition2.p1.8.m8.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.8.m8.1b"><apply id="Thmdefinition2.p1.8.m8.1.1.cmml" xref="Thmdefinition2.p1.8.m8.1.1"><csymbol cd="latexml" id="Thmdefinition2.p1.8.m8.1.1.1.cmml" xref="Thmdefinition2.p1.8.m8.1.1.1">models</csymbol><ci id="Thmdefinition2.p1.8.m8.1.1.2.cmml" xref="Thmdefinition2.p1.8.m8.1.1.2">𝜇</ci><ci id="Thmdefinition2.p1.8.m8.1.1.3.cmml" xref="Thmdefinition2.p1.8.m8.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.8.m8.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition2.p1.8.m8.1d">italic_μ ⊧ italic_φ</annotation></semantics></math>”.</p> </div> </div> </section> <section class="ltx_paragraph" id="S3.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Verification vs. Entailment.</h4> <div class="ltx_para" id="S3.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S3.SS1.SSS0.Px2.p1.3">We show the following facts. When the formula <math alttext="\varphi" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="S3.SS1.SSS0.Px2.p1.1.m1.1a"><mi id="S3.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p1.1.m1.1b"><ci id="S3.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p1.1.m1.1d">italic_φ</annotation></semantics></math> is a tautology-free CNF then verification and entailment coincide: <math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S3.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="S3.SS1.SSS0.Px2.p1.2.m2.1a"><mrow id="S3.SS1.SSS0.Px2.p1.2.m2.1b"><mi id="S3.SS1.SSS0.Px2.p1.2.m2.1.1">μ</mi><mpadded id="S3.SS1.SSS0.Px2.p1.2.m2.1c" width="0.219em"><mo id="S3.SS1.SSS0.Px2.p1.2.m2.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.SS1.SSS0.Px2.p1.2.m2.1.3">≈</mo><mi id="S3.SS1.SSS0.Px2.p1.2.m2.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.2.m2.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p1.2.m2.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math> iff <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p1.3.m3.1"><semantics id="S3.SS1.SSS0.Px2.p1.3.m3.1a"><mrow id="S3.SS1.SSS0.Px2.p1.3.m3.1.1" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1.cmml"><mi id="S3.SS1.SSS0.Px2.p1.3.m3.1.1.2" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml">μ</mi><mo id="S3.SS1.SSS0.Px2.p1.3.m3.1.1.1" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1.1.cmml">⊧</mo><mi id="S3.SS1.SSS0.Px2.p1.3.m3.1.1.3" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p1.3.m3.1b"><apply id="S3.SS1.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="latexml" id="S3.SS1.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1.1">models</csymbol><ci id="S3.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1.2">𝜇</ci><ci id="S3.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="S3.SS1.SSS0.Px2.p1.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p1.3.m3.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p1.3.m3.1d">italic_μ ⊧ italic_φ</annotation></semantics></math>. Unfortunately, <em class="ltx_emph ltx_font_italic" id="S3.SS1.SSS0.Px2.p1.3.1">with generic formulas verification and entailment do not coincide, the former being strictly stronger than the latter</em>. These facts are illustrated in the following result. </p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 3.1</span></h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p1.2.2">Let <math alttext="\varphi" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.1.m1.1"><semantics id="S3.Thmtheorem1.p1.1.1.m1.1a"><mi id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.1.m1.1b"><ci id="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.1d">italic_φ</annotation></semantics></math> be a formula and <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.2.m2.1"><semantics id="S3.Thmtheorem1.p1.2.2.m2.1a"><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.2.2.m2.1b"><ci id="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.2.m2.1d">italic_μ</annotation></semantics></math> be a partial truth assignment over its atoms.</span></p> <ul class="ltx_itemize" id="S3.I1"> <li class="ltx_item" id="S3.I1.ix1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(a)</span> <div class="ltx_para" id="S3.I1.ix1.p1"> <p class="ltx_p" id="S3.I1.ix1.p1.3"><span class="ltx_text ltx_font_italic" id="S3.I1.ix1.p1.3.1">If </span><math alttext="\varphi" class="ltx_Math" display="inline" id="S3.I1.ix1.p1.1.m1.1"><semantics id="S3.I1.ix1.p1.1.m1.1a"><mi id="S3.I1.ix1.p1.1.m1.1.1" xref="S3.I1.ix1.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.I1.ix1.p1.1.m1.1b"><ci id="S3.I1.ix1.p1.1.m1.1.1.cmml" xref="S3.I1.ix1.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix1.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix1.p1.1.m1.1d">italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.ix1.p1.3.2"> a tautology-free CNF, then </span><math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S3.I1.ix1.p1.2.m2.1"><semantics id="S3.I1.ix1.p1.2.m2.1a"><mrow id="S3.I1.ix1.p1.2.m2.1.1" xref="S3.I1.ix1.p1.2.m2.1.1.cmml"><mi id="S3.I1.ix1.p1.2.m2.1.1.2" xref="S3.I1.ix1.p1.2.m2.1.1.2.cmml">μ</mi><mo id="S3.I1.ix1.p1.2.m2.1.1.1" xref="S3.I1.ix1.p1.2.m2.1.1.1.cmml">⊧</mo><mi id="S3.I1.ix1.p1.2.m2.1.1.3" xref="S3.I1.ix1.p1.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix1.p1.2.m2.1b"><apply id="S3.I1.ix1.p1.2.m2.1.1.cmml" xref="S3.I1.ix1.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.I1.ix1.p1.2.m2.1.1.1.cmml" xref="S3.I1.ix1.p1.2.m2.1.1.1">models</csymbol><ci id="S3.I1.ix1.p1.2.m2.1.1.2.cmml" xref="S3.I1.ix1.p1.2.m2.1.1.2">𝜇</ci><ci id="S3.I1.ix1.p1.2.m2.1.1.3.cmml" xref="S3.I1.ix1.p1.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix1.p1.2.m2.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix1.p1.2.m2.1d">italic_μ ⊧ italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.ix1.p1.3.3"> iff </span><math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S3.I1.ix1.p1.3.m3.1"><semantics id="S3.I1.ix1.p1.3.m3.1a"><mrow id="S3.I1.ix1.p1.3.m3.1b"><mi id="S3.I1.ix1.p1.3.m3.1.1">μ</mi><mpadded id="S3.I1.ix1.p1.3.m3.1c" width="0.219em"><mo id="S3.I1.ix1.p1.3.m3.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I1.ix1.p1.3.m3.1.3">≈</mo><mi id="S3.I1.ix1.p1.3.m3.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.I1.ix1.p1.3.m3.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix1.p1.3.m3.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.ix1.p1.3.4">.</span></p> </div> </li> <li class="ltx_item" id="S3.I1.ix2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(b)</span> <div class="ltx_para" id="S3.I1.ix2.p1"> <p class="ltx_p" id="S3.I1.ix2.p1.3"><span class="ltx_text ltx_font_italic" id="S3.I1.ix2.p1.3.1">If </span><math alttext="\varphi" class="ltx_Math" display="inline" id="S3.I1.ix2.p1.1.m1.1"><semantics id="S3.I1.ix2.p1.1.m1.1a"><mi id="S3.I1.ix2.p1.1.m1.1.1" xref="S3.I1.ix2.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.p1.1.m1.1b"><ci id="S3.I1.ix2.p1.1.m1.1.1.cmml" xref="S3.I1.ix2.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.p1.1.m1.1d">italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.ix2.p1.3.2"> is a generic formula, then </span><math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S3.I1.ix2.p1.2.m2.1"><semantics id="S3.I1.ix2.p1.2.m2.1a"><mrow id="S3.I1.ix2.p1.2.m2.1.1" xref="S3.I1.ix2.p1.2.m2.1.1.cmml"><mi id="S3.I1.ix2.p1.2.m2.1.1.2" xref="S3.I1.ix2.p1.2.m2.1.1.2.cmml">μ</mi><mo id="S3.I1.ix2.p1.2.m2.1.1.1" xref="S3.I1.ix2.p1.2.m2.1.1.1.cmml">⊧</mo><mi id="S3.I1.ix2.p1.2.m2.1.1.3" xref="S3.I1.ix2.p1.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.ix2.p1.2.m2.1b"><apply id="S3.I1.ix2.p1.2.m2.1.1.cmml" xref="S3.I1.ix2.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.I1.ix2.p1.2.m2.1.1.1.cmml" xref="S3.I1.ix2.p1.2.m2.1.1.1">models</csymbol><ci id="S3.I1.ix2.p1.2.m2.1.1.2.cmml" xref="S3.I1.ix2.p1.2.m2.1.1.2">𝜇</ci><ci id="S3.I1.ix2.p1.2.m2.1.1.3.cmml" xref="S3.I1.ix2.p1.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.ix2.p1.2.m2.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.p1.2.m2.1d">italic_μ ⊧ italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.ix2.p1.3.3"> if </span><math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S3.I1.ix2.p1.3.m3.1"><semantics id="S3.I1.ix2.p1.3.m3.1a"><mrow id="S3.I1.ix2.p1.3.m3.1b"><mi id="S3.I1.ix2.p1.3.m3.1.1">μ</mi><mpadded id="S3.I1.ix2.p1.3.m3.1c" width="0.219em"><mo id="S3.I1.ix2.p1.3.m3.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I1.ix2.p1.3.m3.1.3">≈</mo><mi id="S3.I1.ix2.p1.3.m3.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.I1.ix2.p1.3.m3.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I1.ix2.p1.3.m3.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I1.ix2.p1.3.4">, but the converse does not hold.</span></p> </div> </li> </ul> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx1"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx1.p1"> <p class="ltx_p" id="Thmproofx1.p1.16"><math alttext="(a)" class="ltx_Math" display="inline" id="Thmproofx1.p1.1.m1.1"><semantics id="Thmproofx1.p1.1.m1.1a"><mrow id="Thmproofx1.p1.1.m1.1.2.2"><mo id="Thmproofx1.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="Thmproofx1.p1.1.m1.1.1" xref="Thmproofx1.p1.1.m1.1.1.cmml">a</mi><mo id="Thmproofx1.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.1.m1.1b"><ci id="Thmproofx1.p1.1.m1.1.1.cmml" xref="Thmproofx1.p1.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.1.m1.1c">(a)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.1.m1.1d">( italic_a )</annotation></semantics></math>: Let <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmproofx1.p1.2.m2.1"><semantics id="Thmproofx1.p1.2.m2.1a"><mi id="Thmproofx1.p1.2.m2.1.1" xref="Thmproofx1.p1.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.2.m2.1b"><ci id="Thmproofx1.p1.2.m2.1.1.cmml" xref="Thmproofx1.p1.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.2.m2.1d">italic_φ</annotation></semantics></math> be a tautology-free CNF. If <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmproofx1.p1.3.m3.1"><semantics id="Thmproofx1.p1.3.m3.1a"><mrow id="Thmproofx1.p1.3.m3.1.1" xref="Thmproofx1.p1.3.m3.1.1.cmml"><mi id="Thmproofx1.p1.3.m3.1.1.2" xref="Thmproofx1.p1.3.m3.1.1.2.cmml">μ</mi><mo id="Thmproofx1.p1.3.m3.1.1.1" xref="Thmproofx1.p1.3.m3.1.1.1.cmml">⊧</mo><mi id="Thmproofx1.p1.3.m3.1.1.3" xref="Thmproofx1.p1.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.3.m3.1b"><apply id="Thmproofx1.p1.3.m3.1.1.cmml" xref="Thmproofx1.p1.3.m3.1.1"><csymbol cd="latexml" id="Thmproofx1.p1.3.m3.1.1.1.cmml" xref="Thmproofx1.p1.3.m3.1.1.1">models</csymbol><ci id="Thmproofx1.p1.3.m3.1.1.2.cmml" xref="Thmproofx1.p1.3.m3.1.1.2">𝜇</ci><ci id="Thmproofx1.p1.3.m3.1.1.3.cmml" xref="Thmproofx1.p1.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.3.m3.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.3.m3.1d">italic_μ ⊧ italic_φ</annotation></semantics></math>, then <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="Thmproofx1.p1.4.m4.2"><semantics id="Thmproofx1.p1.4.m4.2a"><msub id="Thmproofx1.p1.4.m4.2.3.2" xref="Thmproofx1.p1.4.m4.2.3.1.cmml"><mrow id="Thmproofx1.p1.4.m4.2.3.2.2" xref="Thmproofx1.p1.4.m4.2.3.1.cmml"><mi id="Thmproofx1.p1.4.m4.1.1" xref="Thmproofx1.p1.4.m4.1.1.cmml">φ</mi><mo id="Thmproofx1.p1.4.m4.2.3.2.2.1" stretchy="false" xref="Thmproofx1.p1.4.m4.2.3.1.1.cmml">|</mo></mrow><mi id="Thmproofx1.p1.4.m4.2.2.1" xref="Thmproofx1.p1.4.m4.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.4.m4.2b"><apply id="Thmproofx1.p1.4.m4.2.3.1.cmml" xref="Thmproofx1.p1.4.m4.2.3.2"><csymbol cd="latexml" id="Thmproofx1.p1.4.m4.2.3.1.1.cmml" xref="Thmproofx1.p1.4.m4.2.3.2.2.1">evaluated-at</csymbol><ci id="Thmproofx1.p1.4.m4.1.1.cmml" xref="Thmproofx1.p1.4.m4.1.1">𝜑</ci><ci id="Thmproofx1.p1.4.m4.2.2.1.cmml" xref="Thmproofx1.p1.4.m4.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.4.m4.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.4.m4.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is a valid CNF formula which does not contain valid clauses, so that <math alttext="\varphi|_{\mu}=\top" class="ltx_Math" display="inline" id="Thmproofx1.p1.5.m5.2"><semantics id="Thmproofx1.p1.5.m5.2a"><mrow id="Thmproofx1.p1.5.m5.2.3" xref="Thmproofx1.p1.5.m5.2.3.cmml"><msub id="Thmproofx1.p1.5.m5.2.3.2.2" xref="Thmproofx1.p1.5.m5.2.3.2.1.cmml"><mrow id="Thmproofx1.p1.5.m5.2.3.2.2.2" xref="Thmproofx1.p1.5.m5.2.3.2.1.cmml"><mi id="Thmproofx1.p1.5.m5.1.1" xref="Thmproofx1.p1.5.m5.1.1.cmml">φ</mi><mo id="Thmproofx1.p1.5.m5.2.3.2.2.2.1" stretchy="false" xref="Thmproofx1.p1.5.m5.2.3.2.1.1.cmml">|</mo></mrow><mi id="Thmproofx1.p1.5.m5.2.2.1" xref="Thmproofx1.p1.5.m5.2.2.1.cmml">μ</mi></msub><mo id="Thmproofx1.p1.5.m5.2.3.1" rspace="0em" xref="Thmproofx1.p1.5.m5.2.3.1.cmml">=</mo><mo id="Thmproofx1.p1.5.m5.2.3.3" lspace="0em" xref="Thmproofx1.p1.5.m5.2.3.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.5.m5.2b"><apply id="Thmproofx1.p1.5.m5.2.3.cmml" xref="Thmproofx1.p1.5.m5.2.3"><eq id="Thmproofx1.p1.5.m5.2.3.1.cmml" xref="Thmproofx1.p1.5.m5.2.3.1"></eq><apply id="Thmproofx1.p1.5.m5.2.3.2.1.cmml" xref="Thmproofx1.p1.5.m5.2.3.2.2"><csymbol cd="latexml" id="Thmproofx1.p1.5.m5.2.3.2.1.1.cmml" xref="Thmproofx1.p1.5.m5.2.3.2.2.2.1">evaluated-at</csymbol><ci id="Thmproofx1.p1.5.m5.1.1.cmml" xref="Thmproofx1.p1.5.m5.1.1">𝜑</ci><ci id="Thmproofx1.p1.5.m5.2.2.1.cmml" xref="Thmproofx1.p1.5.m5.2.2.1">𝜇</ci></apply><csymbol cd="latexml" id="Thmproofx1.p1.5.m5.2.3.3.cmml" xref="Thmproofx1.p1.5.m5.2.3.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.5.m5.2c">\varphi|_{\mu}=\top</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.5.m5.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math>, hence <math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="Thmproofx1.p1.6.m6.1"><semantics id="Thmproofx1.p1.6.m6.1a"><mrow id="Thmproofx1.p1.6.m6.1b"><mi id="Thmproofx1.p1.6.m6.1.1">μ</mi><mpadded id="Thmproofx1.p1.6.m6.1c" width="0.219em"><mo id="Thmproofx1.p1.6.m6.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx1.p1.6.m6.1.3">≈</mo><mi id="Thmproofx1.p1.6.m6.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmproofx1.p1.6.m6.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.6.m6.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math> by property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty2" title="Property 2 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>. <br class="ltx_break"/><math alttext="(a)" class="ltx_Math" display="inline" id="Thmproofx1.p1.7.m7.1"><semantics id="Thmproofx1.p1.7.m7.1a"><mrow id="Thmproofx1.p1.7.m7.1.2.2"><mo id="Thmproofx1.p1.7.m7.1.2.2.1" stretchy="false">(</mo><mi id="Thmproofx1.p1.7.m7.1.1" xref="Thmproofx1.p1.7.m7.1.1.cmml">a</mi><mo id="Thmproofx1.p1.7.m7.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.7.m7.1b"><ci id="Thmproofx1.p1.7.m7.1.1.cmml" xref="Thmproofx1.p1.7.m7.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.7.m7.1c">(a)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.7.m7.1d">( italic_a )</annotation></semantics></math> and <math alttext="(b)" class="ltx_Math" display="inline" id="Thmproofx1.p1.8.m8.1"><semantics id="Thmproofx1.p1.8.m8.1a"><mrow id="Thmproofx1.p1.8.m8.1.2.2"><mo id="Thmproofx1.p1.8.m8.1.2.2.1" stretchy="false">(</mo><mi id="Thmproofx1.p1.8.m8.1.1" xref="Thmproofx1.p1.8.m8.1.1.cmml">b</mi><mo id="Thmproofx1.p1.8.m8.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.8.m8.1b"><ci id="Thmproofx1.p1.8.m8.1.1.cmml" xref="Thmproofx1.p1.8.m8.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.8.m8.1c">(b)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.8.m8.1d">( italic_b )</annotation></semantics></math>: If <math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="Thmproofx1.p1.9.m9.1"><semantics id="Thmproofx1.p1.9.m9.1a"><mrow id="Thmproofx1.p1.9.m9.1b"><mi id="Thmproofx1.p1.9.m9.1.1">μ</mi><mpadded id="Thmproofx1.p1.9.m9.1c" width="0.219em"><mo id="Thmproofx1.p1.9.m9.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx1.p1.9.m9.1.3">≈</mo><mi id="Thmproofx1.p1.9.m9.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmproofx1.p1.9.m9.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.9.m9.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math> then, for every <math alttext="\eta" class="ltx_Math" display="inline" id="Thmproofx1.p1.10.m10.1"><semantics id="Thmproofx1.p1.10.m10.1a"><mi id="Thmproofx1.p1.10.m10.1.1" xref="Thmproofx1.p1.10.m10.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.10.m10.1b"><ci id="Thmproofx1.p1.10.m10.1.1.cmml" xref="Thmproofx1.p1.10.m10.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.10.m10.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.10.m10.1d">italic_η</annotation></semantics></math> s.t. <math alttext="\eta\supseteq\mu" class="ltx_Math" display="inline" id="Thmproofx1.p1.11.m11.1"><semantics id="Thmproofx1.p1.11.m11.1a"><mrow id="Thmproofx1.p1.11.m11.1.1" xref="Thmproofx1.p1.11.m11.1.1.cmml"><mi id="Thmproofx1.p1.11.m11.1.1.2" xref="Thmproofx1.p1.11.m11.1.1.2.cmml">η</mi><mo id="Thmproofx1.p1.11.m11.1.1.1" xref="Thmproofx1.p1.11.m11.1.1.cmml">⊇</mo><mi id="Thmproofx1.p1.11.m11.1.1.3" xref="Thmproofx1.p1.11.m11.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.11.m11.1b"><apply id="Thmproofx1.p1.11.m11.1.1.cmml" xref="Thmproofx1.p1.11.m11.1.1"><subset id="Thmproofx1.p1.11.m11.1.1a.cmml" xref="Thmproofx1.p1.11.m11.1.1"></subset><ci id="Thmproofx1.p1.11.m11.1.1.3.cmml" xref="Thmproofx1.p1.11.m11.1.1.3">𝜇</ci><ci id="Thmproofx1.p1.11.m11.1.1.2.cmml" xref="Thmproofx1.p1.11.m11.1.1.2">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.11.m11.1c">\eta\supseteq\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.11.m11.1d">italic_η ⊇ italic_μ</annotation></semantics></math>, <math alttext="\eta\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="Thmproofx1.p1.12.m12.1"><semantics id="Thmproofx1.p1.12.m12.1a"><mrow id="Thmproofx1.p1.12.m12.1b"><mi id="Thmproofx1.p1.12.m12.1.1">η</mi><mpadded id="Thmproofx1.p1.12.m12.1c" width="0.219em"><mo id="Thmproofx1.p1.12.m12.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx1.p1.12.m12.1.3">≈</mo><mi id="Thmproofx1.p1.12.m12.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmproofx1.p1.12.m12.1d">\eta\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.12.m12.1e">italic_η ∣ ≈ italic_φ</annotation></semantics></math> and thus <math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="Thmproofx1.p1.13.m13.1"><semantics id="Thmproofx1.p1.13.m13.1a"><mrow id="Thmproofx1.p1.13.m13.1.1" xref="Thmproofx1.p1.13.m13.1.1.cmml"><mi id="Thmproofx1.p1.13.m13.1.1.2" xref="Thmproofx1.p1.13.m13.1.1.2.cmml">η</mi><mo id="Thmproofx1.p1.13.m13.1.1.1" xref="Thmproofx1.p1.13.m13.1.1.1.cmml">⊧</mo><mi id="Thmproofx1.p1.13.m13.1.1.3" xref="Thmproofx1.p1.13.m13.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.13.m13.1b"><apply id="Thmproofx1.p1.13.m13.1.1.cmml" xref="Thmproofx1.p1.13.m13.1.1"><csymbol cd="latexml" id="Thmproofx1.p1.13.m13.1.1.1.cmml" xref="Thmproofx1.p1.13.m13.1.1.1">models</csymbol><ci id="Thmproofx1.p1.13.m13.1.1.2.cmml" xref="Thmproofx1.p1.13.m13.1.1.2">𝜂</ci><ci id="Thmproofx1.p1.13.m13.1.1.3.cmml" xref="Thmproofx1.p1.13.m13.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.13.m13.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.13.m13.1d">italic_η ⊧ italic_φ</annotation></semantics></math>, hence <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmproofx1.p1.14.m14.1"><semantics id="Thmproofx1.p1.14.m14.1a"><mrow id="Thmproofx1.p1.14.m14.1.1" xref="Thmproofx1.p1.14.m14.1.1.cmml"><mi id="Thmproofx1.p1.14.m14.1.1.2" xref="Thmproofx1.p1.14.m14.1.1.2.cmml">μ</mi><mo id="Thmproofx1.p1.14.m14.1.1.1" xref="Thmproofx1.p1.14.m14.1.1.1.cmml">⊧</mo><mi id="Thmproofx1.p1.14.m14.1.1.3" xref="Thmproofx1.p1.14.m14.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.14.m14.1b"><apply id="Thmproofx1.p1.14.m14.1.1.cmml" xref="Thmproofx1.p1.14.m14.1.1"><csymbol cd="latexml" id="Thmproofx1.p1.14.m14.1.1.1.cmml" xref="Thmproofx1.p1.14.m14.1.1.1">models</csymbol><ci id="Thmproofx1.p1.14.m14.1.1.2.cmml" xref="Thmproofx1.p1.14.m14.1.1.2">𝜇</ci><ci id="Thmproofx1.p1.14.m14.1.1.3.cmml" xref="Thmproofx1.p1.14.m14.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.14.m14.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.14.m14.1d">italic_μ ⊧ italic_φ</annotation></semantics></math>. <br class="ltx_break"/><math alttext="(b)" class="ltx_Math" display="inline" id="Thmproofx1.p1.15.m15.1"><semantics id="Thmproofx1.p1.15.m15.1a"><mrow id="Thmproofx1.p1.15.m15.1.2.2"><mo id="Thmproofx1.p1.15.m15.1.2.2.1" stretchy="false">(</mo><mi id="Thmproofx1.p1.15.m15.1.1" xref="Thmproofx1.p1.15.m15.1.1.cmml">b</mi><mo id="Thmproofx1.p1.15.m15.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx1.p1.15.m15.1b"><ci id="Thmproofx1.p1.15.m15.1.1.cmml" xref="Thmproofx1.p1.15.m15.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx1.p1.15.m15.1c">(b)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx1.p1.15.m15.1d">( italic_b )</annotation></semantics></math>: The fact that the converse does not hold is shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample1" title="Example 1 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">1</span></a>. <span class="ltx_text ltx_markedasmath" id="Thmproofx1.p1.16.1">∎</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample1"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 1</span></h6> <div class="ltx_para" id="Thmexample1.p1"> <p class="ltx_p" id="Thmexample1.p1.4">Let <math alttext="\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}(A_{1}\wedge A_{2})\vee(A% _{1}\wedge\neg A_{2})" class="ltx_Math" display="inline" id="Thmexample1.p1.1.m1.2"><semantics id="Thmexample1.p1.1.m1.2a"><mrow id="Thmexample1.p1.1.m1.2.2" xref="Thmexample1.p1.1.m1.2.2.cmml"><mi id="Thmexample1.p1.1.m1.2.2.4" xref="Thmexample1.p1.1.m1.2.2.4.cmml">φ</mi><mover id="Thmexample1.p1.1.m1.2.2.3" xref="Thmexample1.p1.1.m1.2.2.3.cmml"><mo id="Thmexample1.p1.1.m1.2.2.3.2" xref="Thmexample1.p1.1.m1.2.2.3.2.cmml">=</mo><mtext id="Thmexample1.p1.1.m1.2.2.3.3" mathsize="71%" xref="Thmexample1.p1.1.m1.2.2.3.3a.cmml">def</mtext></mover><mrow id="Thmexample1.p1.1.m1.2.2.2" xref="Thmexample1.p1.1.m1.2.2.2.cmml"><mrow id="Thmexample1.p1.1.m1.1.1.1.1.1" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmexample1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample1.p1.1.m1.1.1.1.1.1.1" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.cmml"><msub id="Thmexample1.p1.1.m1.1.1.1.1.1.1.2" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.cmml"><mi id="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.2" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.2.cmml">A</mi><mn id="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.3" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample1.p1.1.m1.1.1.1.1.1.1.1" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.1.cmml">∧</mo><msub id="Thmexample1.p1.1.m1.1.1.1.1.1.1.3" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.2" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.2.cmml">A</mi><mn id="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.3" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="Thmexample1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample1.p1.1.m1.2.2.2.3" xref="Thmexample1.p1.1.m1.2.2.2.3.cmml">∨</mo><mrow id="Thmexample1.p1.1.m1.2.2.2.2.1" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.cmml"><mo id="Thmexample1.p1.1.m1.2.2.2.2.1.2" stretchy="false" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="Thmexample1.p1.1.m1.2.2.2.2.1.1" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.cmml"><msub id="Thmexample1.p1.1.m1.2.2.2.2.1.1.2" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.cmml"><mi id="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.2" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.2.cmml">A</mi><mn id="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.3" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample1.p1.1.m1.2.2.2.2.1.1.1" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.1.cmml">∧</mo><mrow id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.cmml"><mo id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.1" rspace="0.167em" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.1.cmml">¬</mo><msub id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.cmml"><mi id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.2" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.2.cmml">A</mi><mn id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.3" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="Thmexample1.p1.1.m1.2.2.2.2.1.3" stretchy="false" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample1.p1.1.m1.2b"><apply id="Thmexample1.p1.1.m1.2.2.cmml" xref="Thmexample1.p1.1.m1.2.2"><apply id="Thmexample1.p1.1.m1.2.2.3.cmml" xref="Thmexample1.p1.1.m1.2.2.3"><csymbol cd="ambiguous" id="Thmexample1.p1.1.m1.2.2.3.1.cmml" xref="Thmexample1.p1.1.m1.2.2.3">superscript</csymbol><eq id="Thmexample1.p1.1.m1.2.2.3.2.cmml" xref="Thmexample1.p1.1.m1.2.2.3.2"></eq><ci id="Thmexample1.p1.1.m1.2.2.3.3a.cmml" xref="Thmexample1.p1.1.m1.2.2.3.3"><mtext id="Thmexample1.p1.1.m1.2.2.3.3.cmml" mathsize="50%" xref="Thmexample1.p1.1.m1.2.2.3.3">def</mtext></ci></apply><ci id="Thmexample1.p1.1.m1.2.2.4.cmml" xref="Thmexample1.p1.1.m1.2.2.4">𝜑</ci><apply id="Thmexample1.p1.1.m1.2.2.2.cmml" xref="Thmexample1.p1.1.m1.2.2.2"><or id="Thmexample1.p1.1.m1.2.2.2.3.cmml" xref="Thmexample1.p1.1.m1.2.2.2.3"></or><apply id="Thmexample1.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmexample1.p1.1.m1.1.1.1.1.1"><and id="Thmexample1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.1"></and><apply id="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.2">𝐴</ci><cn id="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.2.3">1</cn></apply><apply id="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.2">𝐴</ci><cn id="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="Thmexample1.p1.1.m1.1.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="Thmexample1.p1.1.m1.2.2.2.2.1.1.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1"><and id="Thmexample1.p1.1.m1.2.2.2.2.1.1.1.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.1"></and><apply id="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.2"><csymbol cd="ambiguous" id="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.1.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.2">subscript</csymbol><ci id="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.2.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.2">𝐴</ci><cn id="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.3.cmml" type="integer" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.2.3">1</cn></apply><apply id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3"><not id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.1.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.1"></not><apply id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2"><csymbol cd="ambiguous" id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.1.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2">subscript</csymbol><ci id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.2.cmml" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.2">𝐴</ci><cn id="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.3.cmml" type="integer" xref="Thmexample1.p1.1.m1.2.2.2.2.1.1.3.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p1.1.m1.2c">\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}(A_{1}\wedge A_{2})\vee(A% _{1}\wedge\neg A_{2})</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p1.1.m1.2d">italic_φ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1}}\}" class="ltx_Math" display="inline" id="Thmexample1.p1.2.m2.1"><semantics id="Thmexample1.p1.2.m2.1a"><mrow id="Thmexample1.p1.2.m2.1.1" xref="Thmexample1.p1.2.m2.1.1.cmml"><mi id="Thmexample1.p1.2.m2.1.1.3" xref="Thmexample1.p1.2.m2.1.1.3.cmml">μ</mi><mover id="Thmexample1.p1.2.m2.1.1.2" xref="Thmexample1.p1.2.m2.1.1.2.cmml"><mo id="Thmexample1.p1.2.m2.1.1.2.2" xref="Thmexample1.p1.2.m2.1.1.2.2.cmml">=</mo><mtext id="Thmexample1.p1.2.m2.1.1.2.3" mathsize="71%" xref="Thmexample1.p1.2.m2.1.1.2.3a.cmml">def</mtext></mover><mrow id="Thmexample1.p1.2.m2.1.1.1.1" xref="Thmexample1.p1.2.m2.1.1.1.2.cmml"><mo id="Thmexample1.p1.2.m2.1.1.1.1.2" stretchy="false" xref="Thmexample1.p1.2.m2.1.1.1.2.cmml">{</mo><msub id="Thmexample1.p1.2.m2.1.1.1.1.1" xref="Thmexample1.p1.2.m2.1.1.1.1.1.cmml"><mi id="Thmexample1.p1.2.m2.1.1.1.1.1.2" xref="Thmexample1.p1.2.m2.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample1.p1.2.m2.1.1.1.1.1.3" xref="Thmexample1.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample1.p1.2.m2.1.1.1.1.3" stretchy="false" xref="Thmexample1.p1.2.m2.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample1.p1.2.m2.1b"><apply id="Thmexample1.p1.2.m2.1.1.cmml" xref="Thmexample1.p1.2.m2.1.1"><apply id="Thmexample1.p1.2.m2.1.1.2.cmml" xref="Thmexample1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="Thmexample1.p1.2.m2.1.1.2.1.cmml" xref="Thmexample1.p1.2.m2.1.1.2">superscript</csymbol><eq id="Thmexample1.p1.2.m2.1.1.2.2.cmml" xref="Thmexample1.p1.2.m2.1.1.2.2"></eq><ci id="Thmexample1.p1.2.m2.1.1.2.3a.cmml" xref="Thmexample1.p1.2.m2.1.1.2.3"><mtext id="Thmexample1.p1.2.m2.1.1.2.3.cmml" mathsize="50%" xref="Thmexample1.p1.2.m2.1.1.2.3">def</mtext></ci></apply><ci id="Thmexample1.p1.2.m2.1.1.3.cmml" xref="Thmexample1.p1.2.m2.1.1.3">𝜇</ci><set id="Thmexample1.p1.2.m2.1.1.1.2.cmml" xref="Thmexample1.p1.2.m2.1.1.1.1"><apply id="Thmexample1.p1.2.m2.1.1.1.1.1.cmml" xref="Thmexample1.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample1.p1.2.m2.1.1.1.1.1.1.cmml" xref="Thmexample1.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="Thmexample1.p1.2.m2.1.1.1.1.1.2.cmml" xref="Thmexample1.p1.2.m2.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample1.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample1.p1.2.m2.1.1.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p1.2.m2.1c">\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p1.2.m2.1d">italic_μ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>. Then <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmexample1.p1.3.m3.1"><semantics id="Thmexample1.p1.3.m3.1a"><mrow id="Thmexample1.p1.3.m3.1.1" xref="Thmexample1.p1.3.m3.1.1.cmml"><mi id="Thmexample1.p1.3.m3.1.1.2" xref="Thmexample1.p1.3.m3.1.1.2.cmml">μ</mi><mo id="Thmexample1.p1.3.m3.1.1.1" xref="Thmexample1.p1.3.m3.1.1.1.cmml">⊧</mo><mi id="Thmexample1.p1.3.m3.1.1.3" xref="Thmexample1.p1.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample1.p1.3.m3.1b"><apply id="Thmexample1.p1.3.m3.1.1.cmml" xref="Thmexample1.p1.3.m3.1.1"><csymbol cd="latexml" id="Thmexample1.p1.3.m3.1.1.1.cmml" xref="Thmexample1.p1.3.m3.1.1.1">models</csymbol><ci id="Thmexample1.p1.3.m3.1.1.2.cmml" xref="Thmexample1.p1.3.m3.1.1.2">𝜇</ci><ci id="Thmexample1.p1.3.m3.1.1.3.cmml" xref="Thmexample1.p1.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample1.p1.3.m3.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p1.3.m3.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> but <math alttext="\mu\not\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="Thmexample1.p1.4.m4.1"><semantics id="Thmexample1.p1.4.m4.1a"><mrow id="Thmexample1.p1.4.m4.1b"><mi id="Thmexample1.p1.4.m4.1.1">μ</mi><mpadded id="Thmexample1.p1.4.m4.1c" width="0.969em"><mo id="Thmexample1.p1.4.m4.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample1.p1.4.m4.1.3">≈</mo><mi id="Thmexample1.p1.4.m4.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmexample1.p1.4.m4.1d">\mu\not\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample1.p1.4.m4.1e">italic_μ ∤ ≈ italic_φ</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para ltx_noindent" id="S3.SS1.SSS0.Px2.p2"> <p class="ltx_p" id="S3.SS1.SSS0.Px2.p2.1">Hereafter we implicitly assume w.l.o.g. that all CNF formulas are tautology free.</p> </div> <div class="ltx_para" id="S3.SS1.SSS0.Px2.p3"> <p class="ltx_p" id="S3.SS1.SSS0.Px2.p3.2">We try to build a counterpart of property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty1" title="Property 1 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a> for Definitions <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition1" title="Definition 1 ‣ Definitions. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition2" title="Definition 2 ‣ Definitions. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a> respectively, but in both cases we fail to achieve all points (i)-(iv), resulting into complementary situations. The following properties follow straightforward from Definitions <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition1" title="Definition 1 ‣ Definitions. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition2" title="Definition 2 ‣ Definitions. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>. (Here “<math alttext="{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p3.1.m1.1"><semantics id="S3.SS1.SSS0.Px2.p3.1.m1.1a"><mi id="S3.SS1.SSS0.Px2.p3.1.m1.1.1" mathcolor="#006300" mathsize="144%" mathvariant="normal" xref="S3.SS1.SSS0.Px2.p3.1.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p3.1.m1.1b"><ci id="S3.SS1.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S3.SS1.SSS0.Px2.p3.1.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p3.1.m1.1c">{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p3.1.m1.1d">✓</annotation></semantics></math>” [resp. “<math alttext="{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}" class="ltx_Math" display="inline" id="S3.SS1.SSS0.Px2.p3.2.m2.1"><semantics id="S3.SS1.SSS0.Px2.p3.2.m2.1a"><mo id="S3.SS1.SSS0.Px2.p3.2.m2.1.1" mathcolor="#FF0000" mathsize="173%" xref="S3.SS1.SSS0.Px2.p3.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S3.SS1.SSS0.Px2.p3.2.m2.1b"><times id="S3.SS1.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S3.SS1.SSS0.Px2.p3.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.SSS0.Px2.p3.2.m2.1c">{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.SSS0.Px2.p3.2.m2.1d">×</annotation></semantics></math>”] denotes facts from property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty1" title="Property 1 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a> which are [resp. are not] preserved.)</p> </div> <div class="ltx_theorem ltx_theorem_property" id="Thmproperty5"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Property 5</span></h6> <div class="ltx_para" id="Thmproperty5.p1"> <p class="ltx_p" id="Thmproperty5.p1.4">Let <math alttext="\mu" class="ltx_Math" display="inline" id="Thmproperty5.p1.1.m1.1"><semantics id="Thmproperty5.p1.1.m1.1a"><mi id="Thmproperty5.p1.1.m1.1.1" xref="Thmproperty5.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmproperty5.p1.1.m1.1b"><ci id="Thmproperty5.p1.1.m1.1.1.cmml" xref="Thmproperty5.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty5.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproperty5.p1.1.m1.1d">italic_μ</annotation></semantics></math> be a partial truth assignment on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty5.p1.2.m2.1"><semantics id="Thmproperty5.p1.2.m2.1a"><mi id="Thmproperty5.p1.2.m2.1.1" xref="Thmproperty5.p1.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty5.p1.2.m2.1b"><ci id="Thmproperty5.p1.2.m2.1.1.cmml" xref="Thmproperty5.p1.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty5.p1.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty5.p1.2.m2.1d">bold_A</annotation></semantics></math> and <math alttext="\varphi,\varphi_{1},\varphi_{2}" class="ltx_Math" display="inline" id="Thmproperty5.p1.3.m3.3"><semantics id="Thmproperty5.p1.3.m3.3a"><mrow id="Thmproperty5.p1.3.m3.3.3.2" xref="Thmproperty5.p1.3.m3.3.3.3.cmml"><mi id="Thmproperty5.p1.3.m3.1.1" xref="Thmproperty5.p1.3.m3.1.1.cmml">φ</mi><mo id="Thmproperty5.p1.3.m3.3.3.2.3" xref="Thmproperty5.p1.3.m3.3.3.3.cmml">,</mo><msub id="Thmproperty5.p1.3.m3.2.2.1.1" xref="Thmproperty5.p1.3.m3.2.2.1.1.cmml"><mi id="Thmproperty5.p1.3.m3.2.2.1.1.2" xref="Thmproperty5.p1.3.m3.2.2.1.1.2.cmml">φ</mi><mn id="Thmproperty5.p1.3.m3.2.2.1.1.3" xref="Thmproperty5.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmproperty5.p1.3.m3.3.3.2.4" xref="Thmproperty5.p1.3.m3.3.3.3.cmml">,</mo><msub id="Thmproperty5.p1.3.m3.3.3.2.2" xref="Thmproperty5.p1.3.m3.3.3.2.2.cmml"><mi id="Thmproperty5.p1.3.m3.3.3.2.2.2" xref="Thmproperty5.p1.3.m3.3.3.2.2.2.cmml">φ</mi><mn id="Thmproperty5.p1.3.m3.3.3.2.2.3" xref="Thmproperty5.p1.3.m3.3.3.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty5.p1.3.m3.3b"><list id="Thmproperty5.p1.3.m3.3.3.3.cmml" xref="Thmproperty5.p1.3.m3.3.3.2"><ci id="Thmproperty5.p1.3.m3.1.1.cmml" xref="Thmproperty5.p1.3.m3.1.1">𝜑</ci><apply id="Thmproperty5.p1.3.m3.2.2.1.1.cmml" xref="Thmproperty5.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="Thmproperty5.p1.3.m3.2.2.1.1.1.cmml" xref="Thmproperty5.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="Thmproperty5.p1.3.m3.2.2.1.1.2.cmml" xref="Thmproperty5.p1.3.m3.2.2.1.1.2">𝜑</ci><cn id="Thmproperty5.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="Thmproperty5.p1.3.m3.2.2.1.1.3">1</cn></apply><apply id="Thmproperty5.p1.3.m3.3.3.2.2.cmml" xref="Thmproperty5.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="Thmproperty5.p1.3.m3.3.3.2.2.1.cmml" xref="Thmproperty5.p1.3.m3.3.3.2.2">subscript</csymbol><ci id="Thmproperty5.p1.3.m3.3.3.2.2.2.cmml" xref="Thmproperty5.p1.3.m3.3.3.2.2.2">𝜑</ci><cn id="Thmproperty5.p1.3.m3.3.3.2.2.3.cmml" type="integer" xref="Thmproperty5.p1.3.m3.3.3.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty5.p1.3.m3.3c">\varphi,\varphi_{1},\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty5.p1.3.m3.3d">italic_φ , italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be formulas on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty5.p1.4.m4.1"><semantics id="Thmproperty5.p1.4.m4.1a"><mi id="Thmproperty5.p1.4.m4.1.1" xref="Thmproperty5.p1.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty5.p1.4.m4.1b"><ci id="Thmproperty5.p1.4.m4.1.1.cmml" xref="Thmproperty5.p1.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty5.p1.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty5.p1.4.m4.1d">bold_A</annotation></semantics></math>.</p> <ul class="ltx_itemize" id="Thmproperty5.p1.5"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I2.i1.1.1.1" style="width:0.0pt;">(i)</span></span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.3"><math alttext="{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.1"><semantics id="S3.I2.i1.p1.1.m1.1a"><mo id="S3.I2.i1.p1.1.m1.1.1" mathcolor="#FF0000" mathsize="173%" xref="S3.I2.i1.p1.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.1b"><times id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.1c">{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.1d">×</annotation></semantics></math> If <math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S3.I2.i1.p1.2.m2.1"><semantics id="S3.I2.i1.p1.2.m2.1a"><mrow id="S3.I2.i1.p1.2.m2.1b"><mi id="S3.I2.i1.p1.2.m2.1.1">μ</mi><mpadded id="S3.I2.i1.p1.2.m2.1c" width="0.219em"><mo id="S3.I2.i1.p1.2.m2.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I2.i1.p1.2.m2.1.3">≈</mo><mi id="S3.I2.i1.p1.2.m2.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math> then <math alttext="\bigwedge\!\mu\models\varphi" class="ltx_Math" display="inline" id="S3.I2.i1.p1.3.m3.1"><semantics id="S3.I2.i1.p1.3.m3.1a"><mrow id="S3.I2.i1.p1.3.m3.1.1" xref="S3.I2.i1.p1.3.m3.1.1.cmml"><mrow id="S3.I2.i1.p1.3.m3.1.1.2" xref="S3.I2.i1.p1.3.m3.1.1.2.cmml"><mpadded width="0.747em"><mo id="S3.I2.i1.p1.3.m3.1.1.2.1" xref="S3.I2.i1.p1.3.m3.1.1.2.1.cmml">⋀</mo></mpadded><mi id="S3.I2.i1.p1.3.m3.1.1.2.2" xref="S3.I2.i1.p1.3.m3.1.1.2.2.cmml">μ</mi></mrow><mo id="S3.I2.i1.p1.3.m3.1.1.1" xref="S3.I2.i1.p1.3.m3.1.1.1.cmml">⊧</mo><mi id="S3.I2.i1.p1.3.m3.1.1.3" xref="S3.I2.i1.p1.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.3.m3.1b"><apply id="S3.I2.i1.p1.3.m3.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1"><csymbol cd="latexml" id="S3.I2.i1.p1.3.m3.1.1.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.1">models</csymbol><apply id="S3.I2.i1.p1.3.m3.1.1.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2"><and id="S3.I2.i1.p1.3.m3.1.1.2.1.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2.1"></and><ci id="S3.I2.i1.p1.3.m3.1.1.2.2.cmml" xref="S3.I2.i1.p1.3.m3.1.1.2.2">𝜇</ci></apply><ci id="S3.I2.i1.p1.3.m3.1.1.3.cmml" xref="S3.I2.i1.p1.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.3.m3.1c">\bigwedge\!\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.3.m3.1d">⋀ italic_μ ⊧ italic_φ</annotation></semantics></math>, but not vice versa.</p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I2.i2.1.1.1" style="width:0.0pt;">(ii)</span></span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.5"><math alttext="{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><mo id="S3.I2.i2.p1.1.m1.1.1" mathcolor="#FF0000" mathsize="173%" xref="S3.I2.i2.p1.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><times id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">×</annotation></semantics></math> If <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><msub id="S3.I2.i2.p1.2.m2.1.1" xref="S3.I2.i2.p1.2.m2.1.1.cmml"><mi id="S3.I2.i2.p1.2.m2.1.1.2" xref="S3.I2.i2.p1.2.m2.1.1.2.cmml">φ</mi><mn id="S3.I2.i2.p1.2.m2.1.1.3" xref="S3.I2.i2.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.2.m2.1.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.2">𝜑</ci><cn id="S3.I2.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.I2.i2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.1"><semantics id="S3.I2.i2.p1.3.m3.1a"><msub id="S3.I2.i2.p1.3.m3.1.1" xref="S3.I2.i2.p1.3.m3.1.1.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.2.cmml">φ</mi><mn id="S3.I2.i2.p1.3.m3.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.1b"><apply id="S3.I2.i2.p1.3.m3.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.3.m3.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.3.m3.1.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2">𝜑</ci><cn id="S3.I2.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.I2.i2.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are equivalent, this does not imply that <math alttext="\mu\mid\!\approx\varphi_{1}" class="ltx_math_unparsed" display="inline" id="S3.I2.i2.p1.4.m4.1"><semantics id="S3.I2.i2.p1.4.m4.1a"><mrow id="S3.I2.i2.p1.4.m4.1b"><mi id="S3.I2.i2.p1.4.m4.1.1">μ</mi><mpadded id="S3.I2.i2.p1.4.m4.1c" width="0.219em"><mo id="S3.I2.i2.p1.4.m4.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I2.i2.p1.4.m4.1.3">≈</mo><msub id="S3.I2.i2.p1.4.m4.1.4"><mi id="S3.I2.i2.p1.4.m4.1.4.2">φ</mi><mn id="S3.I2.i2.p1.4.m4.1.4.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="S3.I2.i2.p1.4.m4.1d">\mu\mid\!\approx\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.4.m4.1e">italic_μ ∣ ≈ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> iff <math alttext="\mu\mid\!\approx\varphi_{2}" class="ltx_math_unparsed" display="inline" id="S3.I2.i2.p1.5.m5.1"><semantics id="S3.I2.i2.p1.5.m5.1a"><mrow id="S3.I2.i2.p1.5.m5.1b"><mi id="S3.I2.i2.p1.5.m5.1.1">μ</mi><mpadded id="S3.I2.i2.p1.5.m5.1c" width="0.219em"><mo id="S3.I2.i2.p1.5.m5.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I2.i2.p1.5.m5.1.3">≈</mo><msub id="S3.I2.i2.p1.5.m5.1.4"><mi id="S3.I2.i2.p1.5.m5.1.4.2">φ</mi><mn id="S3.I2.i2.p1.5.m5.1.4.3">2</mn></msub></mrow><annotation encoding="application/x-tex" id="S3.I2.i2.p1.5.m5.1d">\mu\mid\!\approx\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.5.m5.1e">italic_μ ∣ ≈ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I2.i3.1.1.1" style="width:0.0pt;">(iii)</span></span> <div class="ltx_para" id="S3.I2.i3.p1"> <p class="ltx_p" id="S3.I2.i3.p1.5"><math alttext="{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}" class="ltx_Math" display="inline" id="S3.I2.i3.p1.1.m1.1"><semantics id="S3.I2.i3.p1.1.m1.1a"><mi id="S3.I2.i3.p1.1.m1.1.1" mathcolor="#006300" mathsize="144%" mathvariant="normal" xref="S3.I2.i3.p1.1.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.1.m1.1b"><ci id="S3.I2.i3.p1.1.m1.1.1.cmml" xref="S3.I2.i3.p1.1.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.1.m1.1c">{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.1.m1.1d">✓</annotation></semantics></math> <math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S3.I2.i3.p1.2.m2.1"><semantics id="S3.I2.i3.p1.2.m2.1a"><mrow id="S3.I2.i3.p1.2.m2.1b"><mi id="S3.I2.i3.p1.2.m2.1.1">μ</mi><mpadded id="S3.I2.i3.p1.2.m2.1c" width="0.219em"><mo id="S3.I2.i3.p1.2.m2.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I2.i3.p1.2.m2.1.3">≈</mo><mi id="S3.I2.i3.p1.2.m2.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.I2.i3.p1.2.m2.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.2.m2.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math> iff <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S3.I2.i3.p1.3.m3.2"><semantics id="S3.I2.i3.p1.3.m3.2a"><msub id="S3.I2.i3.p1.3.m3.2.3.2" xref="S3.I2.i3.p1.3.m3.2.3.1.cmml"><mrow id="S3.I2.i3.p1.3.m3.2.3.2.2" xref="S3.I2.i3.p1.3.m3.2.3.1.cmml"><mi id="S3.I2.i3.p1.3.m3.1.1" xref="S3.I2.i3.p1.3.m3.1.1.cmml">φ</mi><mo id="S3.I2.i3.p1.3.m3.2.3.2.2.1" stretchy="false" xref="S3.I2.i3.p1.3.m3.2.3.1.1.cmml">|</mo></mrow><mi id="S3.I2.i3.p1.3.m3.2.2.1" xref="S3.I2.i3.p1.3.m3.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.3.m3.2b"><apply id="S3.I2.i3.p1.3.m3.2.3.1.cmml" xref="S3.I2.i3.p1.3.m3.2.3.2"><csymbol cd="latexml" id="S3.I2.i3.p1.3.m3.2.3.1.1.cmml" xref="S3.I2.i3.p1.3.m3.2.3.2.2.1">evaluated-at</csymbol><ci id="S3.I2.i3.p1.3.m3.1.1.cmml" xref="S3.I2.i3.p1.3.m3.1.1">𝜑</ci><ci id="S3.I2.i3.p1.3.m3.2.2.1.cmml" xref="S3.I2.i3.p1.3.m3.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.3.m3.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.3.m3.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\top" class="ltx_Math" display="inline" id="S3.I2.i3.p1.4.m4.1"><semantics id="S3.I2.i3.p1.4.m4.1a"><mo id="S3.I2.i3.p1.4.m4.1.1" xref="S3.I2.i3.p1.4.m4.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.4.m4.1b"><csymbol cd="latexml" id="S3.I2.i3.p1.4.m4.1.1.cmml" xref="S3.I2.i3.p1.4.m4.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.4.m4.1c">\top</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.4.m4.1d">⊤</annotation></semantics></math> (also, iff <math alttext="\mu(\varphi)=\mbox{{\sf T}}" class="ltx_Math" display="inline" id="S3.I2.i3.p1.5.m5.1"><semantics id="S3.I2.i3.p1.5.m5.1a"><mrow id="S3.I2.i3.p1.5.m5.1.2" xref="S3.I2.i3.p1.5.m5.1.2.cmml"><mrow id="S3.I2.i3.p1.5.m5.1.2.2" xref="S3.I2.i3.p1.5.m5.1.2.2.cmml"><mi id="S3.I2.i3.p1.5.m5.1.2.2.2" xref="S3.I2.i3.p1.5.m5.1.2.2.2.cmml">μ</mi><mo id="S3.I2.i3.p1.5.m5.1.2.2.1" xref="S3.I2.i3.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S3.I2.i3.p1.5.m5.1.2.2.3.2" xref="S3.I2.i3.p1.5.m5.1.2.2.cmml"><mo id="S3.I2.i3.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S3.I2.i3.p1.5.m5.1.2.2.cmml">(</mo><mi id="S3.I2.i3.p1.5.m5.1.1" xref="S3.I2.i3.p1.5.m5.1.1.cmml">φ</mi><mo id="S3.I2.i3.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S3.I2.i3.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I2.i3.p1.5.m5.1.2.1" xref="S3.I2.i3.p1.5.m5.1.2.1.cmml">=</mo><mtext class="ltx_mathvariant_sans-serif" id="S3.I2.i3.p1.5.m5.1.2.3" xref="S3.I2.i3.p1.5.m5.1.2.3a.cmml">T</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i3.p1.5.m5.1b"><apply id="S3.I2.i3.p1.5.m5.1.2.cmml" xref="S3.I2.i3.p1.5.m5.1.2"><eq id="S3.I2.i3.p1.5.m5.1.2.1.cmml" xref="S3.I2.i3.p1.5.m5.1.2.1"></eq><apply id="S3.I2.i3.p1.5.m5.1.2.2.cmml" xref="S3.I2.i3.p1.5.m5.1.2.2"><times id="S3.I2.i3.p1.5.m5.1.2.2.1.cmml" xref="S3.I2.i3.p1.5.m5.1.2.2.1"></times><ci id="S3.I2.i3.p1.5.m5.1.2.2.2.cmml" xref="S3.I2.i3.p1.5.m5.1.2.2.2">𝜇</ci><ci id="S3.I2.i3.p1.5.m5.1.1.cmml" xref="S3.I2.i3.p1.5.m5.1.1">𝜑</ci></apply><ci id="S3.I2.i3.p1.5.m5.1.2.3a.cmml" xref="S3.I2.i3.p1.5.m5.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.I2.i3.p1.5.m5.1.2.3.cmml" xref="S3.I2.i3.p1.5.m5.1.2.3">T</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i3.p1.5.m5.1c">\mu(\varphi)=\mbox{{\sf T}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i3.p1.5.m5.1d">italic_μ ( italic_φ ) = T</annotation></semantics></math> by property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty2" title="Property 2 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a>).</p> </div> </li> <li class="ltx_item" id="S3.I2.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I2.i4.1.1.1" style="width:0.0pt;">(iv)</span></span> <div class="ltx_para" id="S3.I2.i4.p1"> <p class="ltx_p" id="S3.I2.i4.p1.2"><math alttext="{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}" class="ltx_Math" display="inline" id="S3.I2.i4.p1.1.m1.1"><semantics id="S3.I2.i4.p1.1.m1.1a"><mi id="S3.I2.i4.p1.1.m1.1.1" mathcolor="#006300" mathsize="144%" mathvariant="normal" xref="S3.I2.i4.p1.1.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.I2.i4.p1.1.m1.1b"><ci id="S3.I2.i4.p1.1.m1.1.1.cmml" xref="S3.I2.i4.p1.1.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i4.p1.1.m1.1c">{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i4.p1.1.m1.1d">✓</annotation></semantics></math> Checking <math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S3.I2.i4.p1.2.m2.1"><semantics id="S3.I2.i4.p1.2.m2.1a"><mrow id="S3.I2.i4.p1.2.m2.1b"><mi id="S3.I2.i4.p1.2.m2.1.1">μ</mi><mpadded id="S3.I2.i4.p1.2.m2.1c" width="0.219em"><mo id="S3.I2.i4.p1.2.m2.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I2.i4.p1.2.m2.1.3">≈</mo><mi id="S3.I2.i4.p1.2.m2.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.I2.i4.p1.2.m2.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i4.p1.2.m2.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math> requires at most a polynomial amount of steps.</p> </div> </li> </ul> </div> </div> <div class="ltx_theorem ltx_theorem_property" id="Thmproperty6"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Property 6</span></h6> <div class="ltx_para" id="Thmproperty6.p1"> <p class="ltx_p" id="Thmproperty6.p1.4">Let <math alttext="\mu" class="ltx_Math" display="inline" id="Thmproperty6.p1.1.m1.1"><semantics id="Thmproperty6.p1.1.m1.1a"><mi id="Thmproperty6.p1.1.m1.1.1" xref="Thmproperty6.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmproperty6.p1.1.m1.1b"><ci id="Thmproperty6.p1.1.m1.1.1.cmml" xref="Thmproperty6.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty6.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproperty6.p1.1.m1.1d">italic_μ</annotation></semantics></math> be a partial truth assignment on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty6.p1.2.m2.1"><semantics id="Thmproperty6.p1.2.m2.1a"><mi id="Thmproperty6.p1.2.m2.1.1" xref="Thmproperty6.p1.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty6.p1.2.m2.1b"><ci id="Thmproperty6.p1.2.m2.1.1.cmml" xref="Thmproperty6.p1.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty6.p1.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty6.p1.2.m2.1d">bold_A</annotation></semantics></math> and <math alttext="\varphi,\varphi_{1},\varphi_{2}" class="ltx_Math" display="inline" id="Thmproperty6.p1.3.m3.3"><semantics id="Thmproperty6.p1.3.m3.3a"><mrow id="Thmproperty6.p1.3.m3.3.3.2" xref="Thmproperty6.p1.3.m3.3.3.3.cmml"><mi id="Thmproperty6.p1.3.m3.1.1" xref="Thmproperty6.p1.3.m3.1.1.cmml">φ</mi><mo id="Thmproperty6.p1.3.m3.3.3.2.3" xref="Thmproperty6.p1.3.m3.3.3.3.cmml">,</mo><msub id="Thmproperty6.p1.3.m3.2.2.1.1" xref="Thmproperty6.p1.3.m3.2.2.1.1.cmml"><mi id="Thmproperty6.p1.3.m3.2.2.1.1.2" xref="Thmproperty6.p1.3.m3.2.2.1.1.2.cmml">φ</mi><mn id="Thmproperty6.p1.3.m3.2.2.1.1.3" xref="Thmproperty6.p1.3.m3.2.2.1.1.3.cmml">1</mn></msub><mo id="Thmproperty6.p1.3.m3.3.3.2.4" xref="Thmproperty6.p1.3.m3.3.3.3.cmml">,</mo><msub id="Thmproperty6.p1.3.m3.3.3.2.2" xref="Thmproperty6.p1.3.m3.3.3.2.2.cmml"><mi id="Thmproperty6.p1.3.m3.3.3.2.2.2" xref="Thmproperty6.p1.3.m3.3.3.2.2.2.cmml">φ</mi><mn id="Thmproperty6.p1.3.m3.3.3.2.2.3" xref="Thmproperty6.p1.3.m3.3.3.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproperty6.p1.3.m3.3b"><list id="Thmproperty6.p1.3.m3.3.3.3.cmml" xref="Thmproperty6.p1.3.m3.3.3.2"><ci id="Thmproperty6.p1.3.m3.1.1.cmml" xref="Thmproperty6.p1.3.m3.1.1">𝜑</ci><apply id="Thmproperty6.p1.3.m3.2.2.1.1.cmml" xref="Thmproperty6.p1.3.m3.2.2.1.1"><csymbol cd="ambiguous" id="Thmproperty6.p1.3.m3.2.2.1.1.1.cmml" xref="Thmproperty6.p1.3.m3.2.2.1.1">subscript</csymbol><ci id="Thmproperty6.p1.3.m3.2.2.1.1.2.cmml" xref="Thmproperty6.p1.3.m3.2.2.1.1.2">𝜑</ci><cn id="Thmproperty6.p1.3.m3.2.2.1.1.3.cmml" type="integer" xref="Thmproperty6.p1.3.m3.2.2.1.1.3">1</cn></apply><apply id="Thmproperty6.p1.3.m3.3.3.2.2.cmml" xref="Thmproperty6.p1.3.m3.3.3.2.2"><csymbol cd="ambiguous" id="Thmproperty6.p1.3.m3.3.3.2.2.1.cmml" xref="Thmproperty6.p1.3.m3.3.3.2.2">subscript</csymbol><ci id="Thmproperty6.p1.3.m3.3.3.2.2.2.cmml" xref="Thmproperty6.p1.3.m3.3.3.2.2.2">𝜑</ci><cn id="Thmproperty6.p1.3.m3.3.3.2.2.3.cmml" type="integer" xref="Thmproperty6.p1.3.m3.3.3.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty6.p1.3.m3.3c">\varphi,\varphi_{1},\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty6.p1.3.m3.3d">italic_φ , italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be formulas on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmproperty6.p1.4.m4.1"><semantics id="Thmproperty6.p1.4.m4.1a"><mi id="Thmproperty6.p1.4.m4.1.1" xref="Thmproperty6.p1.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmproperty6.p1.4.m4.1b"><ci id="Thmproperty6.p1.4.m4.1.1.cmml" xref="Thmproperty6.p1.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproperty6.p1.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmproperty6.p1.4.m4.1d">bold_A</annotation></semantics></math>.</p> <ul class="ltx_itemize" id="Thmproperty6.p1.5"> <li class="ltx_item" id="S3.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I3.i1.1.1.1" style="width:0.0pt;">(i)</span></span> <div class="ltx_para" id="S3.I3.i1.p1"> <p class="ltx_p" id="S3.I3.i1.p1.3"><math alttext="{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}" class="ltx_Math" display="inline" id="S3.I3.i1.p1.1.m1.1"><semantics id="S3.I3.i1.p1.1.m1.1a"><mi id="S3.I3.i1.p1.1.m1.1.1" mathcolor="#006300" mathsize="144%" mathvariant="normal" xref="S3.I3.i1.p1.1.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.1.m1.1b"><ci id="S3.I3.i1.p1.1.m1.1.1.cmml" xref="S3.I3.i1.p1.1.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.1.m1.1c">{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.1.m1.1d">✓</annotation></semantics></math> <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S3.I3.i1.p1.2.m2.1"><semantics id="S3.I3.i1.p1.2.m2.1a"><mrow id="S3.I3.i1.p1.2.m2.1.1" xref="S3.I3.i1.p1.2.m2.1.1.cmml"><mi id="S3.I3.i1.p1.2.m2.1.1.2" xref="S3.I3.i1.p1.2.m2.1.1.2.cmml">μ</mi><mo id="S3.I3.i1.p1.2.m2.1.1.1" xref="S3.I3.i1.p1.2.m2.1.1.1.cmml">⊧</mo><mi id="S3.I3.i1.p1.2.m2.1.1.3" xref="S3.I3.i1.p1.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.2.m2.1b"><apply id="S3.I3.i1.p1.2.m2.1.1.cmml" xref="S3.I3.i1.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.I3.i1.p1.2.m2.1.1.1.cmml" xref="S3.I3.i1.p1.2.m2.1.1.1">models</csymbol><ci id="S3.I3.i1.p1.2.m2.1.1.2.cmml" xref="S3.I3.i1.p1.2.m2.1.1.2">𝜇</ci><ci id="S3.I3.i1.p1.2.m2.1.1.3.cmml" xref="S3.I3.i1.p1.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.2.m2.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.2.m2.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> iff <math alttext="\bigwedge\!\mu\models\varphi" class="ltx_Math" display="inline" id="S3.I3.i1.p1.3.m3.1"><semantics id="S3.I3.i1.p1.3.m3.1a"><mrow id="S3.I3.i1.p1.3.m3.1.1" xref="S3.I3.i1.p1.3.m3.1.1.cmml"><mrow id="S3.I3.i1.p1.3.m3.1.1.2" xref="S3.I3.i1.p1.3.m3.1.1.2.cmml"><mpadded width="0.747em"><mo id="S3.I3.i1.p1.3.m3.1.1.2.1" xref="S3.I3.i1.p1.3.m3.1.1.2.1.cmml">⋀</mo></mpadded><mi id="S3.I3.i1.p1.3.m3.1.1.2.2" xref="S3.I3.i1.p1.3.m3.1.1.2.2.cmml">μ</mi></mrow><mo id="S3.I3.i1.p1.3.m3.1.1.1" xref="S3.I3.i1.p1.3.m3.1.1.1.cmml">⊧</mo><mi id="S3.I3.i1.p1.3.m3.1.1.3" xref="S3.I3.i1.p1.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.3.m3.1b"><apply id="S3.I3.i1.p1.3.m3.1.1.cmml" xref="S3.I3.i1.p1.3.m3.1.1"><csymbol cd="latexml" id="S3.I3.i1.p1.3.m3.1.1.1.cmml" xref="S3.I3.i1.p1.3.m3.1.1.1">models</csymbol><apply id="S3.I3.i1.p1.3.m3.1.1.2.cmml" xref="S3.I3.i1.p1.3.m3.1.1.2"><and id="S3.I3.i1.p1.3.m3.1.1.2.1.cmml" xref="S3.I3.i1.p1.3.m3.1.1.2.1"></and><ci id="S3.I3.i1.p1.3.m3.1.1.2.2.cmml" xref="S3.I3.i1.p1.3.m3.1.1.2.2">𝜇</ci></apply><ci id="S3.I3.i1.p1.3.m3.1.1.3.cmml" xref="S3.I3.i1.p1.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.3.m3.1c">\bigwedge\!\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.3.m3.1d">⋀ italic_μ ⊧ italic_φ</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I3.i2.1.1.1" style="width:0.0pt;">(ii)</span></span> <div class="ltx_para" id="S3.I3.i2.p1"> <p class="ltx_p" id="S3.I3.i2.p1.5"><math alttext="{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.1.m1.1"><semantics id="S3.I3.i2.p1.1.m1.1a"><mi id="S3.I3.i2.p1.1.m1.1.1" mathcolor="#006300" mathsize="144%" mathvariant="normal" xref="S3.I3.i2.p1.1.m1.1.1.cmml">✓</mi><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.1.m1.1b"><ci id="S3.I3.i2.p1.1.m1.1.1.cmml" xref="S3.I3.i2.p1.1.m1.1.1">✓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.1.m1.1c">{\color[rgb]{0.00,0.39,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 0.00,0.39,0.00}\checkmark}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.1.m1.1d">✓</annotation></semantics></math> If <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.2.m2.1"><semantics id="S3.I3.i2.p1.2.m2.1a"><msub id="S3.I3.i2.p1.2.m2.1.1" xref="S3.I3.i2.p1.2.m2.1.1.cmml"><mi id="S3.I3.i2.p1.2.m2.1.1.2" xref="S3.I3.i2.p1.2.m2.1.1.2.cmml">φ</mi><mn id="S3.I3.i2.p1.2.m2.1.1.3" xref="S3.I3.i2.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.2.m2.1b"><apply id="S3.I3.i2.p1.2.m2.1.1.cmml" xref="S3.I3.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.2.m2.1.1.1.cmml" xref="S3.I3.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I3.i2.p1.2.m2.1.1.2.cmml" xref="S3.I3.i2.p1.2.m2.1.1.2">𝜑</ci><cn id="S3.I3.i2.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.I3.i2.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.2.m2.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.2.m2.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.3.m3.1"><semantics id="S3.I3.i2.p1.3.m3.1a"><msub id="S3.I3.i2.p1.3.m3.1.1" xref="S3.I3.i2.p1.3.m3.1.1.cmml"><mi id="S3.I3.i2.p1.3.m3.1.1.2" xref="S3.I3.i2.p1.3.m3.1.1.2.cmml">φ</mi><mn id="S3.I3.i2.p1.3.m3.1.1.3" xref="S3.I3.i2.p1.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.3.m3.1b"><apply id="S3.I3.i2.p1.3.m3.1.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I3.i2.p1.3.m3.1.1.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I3.i2.p1.3.m3.1.1.2.cmml" xref="S3.I3.i2.p1.3.m3.1.1.2">𝜑</ci><cn id="S3.I3.i2.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.I3.i2.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.3.m3.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.3.m3.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are equivalent, then <math alttext="\mu\models\varphi_{1}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.4.m4.1"><semantics id="S3.I3.i2.p1.4.m4.1a"><mrow id="S3.I3.i2.p1.4.m4.1.1" xref="S3.I3.i2.p1.4.m4.1.1.cmml"><mi id="S3.I3.i2.p1.4.m4.1.1.2" xref="S3.I3.i2.p1.4.m4.1.1.2.cmml">μ</mi><mo id="S3.I3.i2.p1.4.m4.1.1.1" xref="S3.I3.i2.p1.4.m4.1.1.1.cmml">⊧</mo><msub id="S3.I3.i2.p1.4.m4.1.1.3" xref="S3.I3.i2.p1.4.m4.1.1.3.cmml"><mi id="S3.I3.i2.p1.4.m4.1.1.3.2" xref="S3.I3.i2.p1.4.m4.1.1.3.2.cmml">φ</mi><mn id="S3.I3.i2.p1.4.m4.1.1.3.3" xref="S3.I3.i2.p1.4.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.4.m4.1b"><apply id="S3.I3.i2.p1.4.m4.1.1.cmml" xref="S3.I3.i2.p1.4.m4.1.1"><csymbol cd="latexml" id="S3.I3.i2.p1.4.m4.1.1.1.cmml" xref="S3.I3.i2.p1.4.m4.1.1.1">models</csymbol><ci id="S3.I3.i2.p1.4.m4.1.1.2.cmml" xref="S3.I3.i2.p1.4.m4.1.1.2">𝜇</ci><apply id="S3.I3.i2.p1.4.m4.1.1.3.cmml" xref="S3.I3.i2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.4.m4.1.1.3.1.cmml" xref="S3.I3.i2.p1.4.m4.1.1.3">subscript</csymbol><ci id="S3.I3.i2.p1.4.m4.1.1.3.2.cmml" xref="S3.I3.i2.p1.4.m4.1.1.3.2">𝜑</ci><cn id="S3.I3.i2.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S3.I3.i2.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.4.m4.1c">\mu\models\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.4.m4.1d">italic_μ ⊧ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> iff <math alttext="\mu\models\varphi_{2}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.5.m5.1"><semantics id="S3.I3.i2.p1.5.m5.1a"><mrow id="S3.I3.i2.p1.5.m5.1.1" xref="S3.I3.i2.p1.5.m5.1.1.cmml"><mi id="S3.I3.i2.p1.5.m5.1.1.2" xref="S3.I3.i2.p1.5.m5.1.1.2.cmml">μ</mi><mo id="S3.I3.i2.p1.5.m5.1.1.1" xref="S3.I3.i2.p1.5.m5.1.1.1.cmml">⊧</mo><msub id="S3.I3.i2.p1.5.m5.1.1.3" xref="S3.I3.i2.p1.5.m5.1.1.3.cmml"><mi id="S3.I3.i2.p1.5.m5.1.1.3.2" xref="S3.I3.i2.p1.5.m5.1.1.3.2.cmml">φ</mi><mn id="S3.I3.i2.p1.5.m5.1.1.3.3" xref="S3.I3.i2.p1.5.m5.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.5.m5.1b"><apply id="S3.I3.i2.p1.5.m5.1.1.cmml" xref="S3.I3.i2.p1.5.m5.1.1"><csymbol cd="latexml" id="S3.I3.i2.p1.5.m5.1.1.1.cmml" xref="S3.I3.i2.p1.5.m5.1.1.1">models</csymbol><ci id="S3.I3.i2.p1.5.m5.1.1.2.cmml" xref="S3.I3.i2.p1.5.m5.1.1.2">𝜇</ci><apply id="S3.I3.i2.p1.5.m5.1.1.3.cmml" xref="S3.I3.i2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.I3.i2.p1.5.m5.1.1.3.1.cmml" xref="S3.I3.i2.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.I3.i2.p1.5.m5.1.1.3.2.cmml" xref="S3.I3.i2.p1.5.m5.1.1.3.2">𝜑</ci><cn id="S3.I3.i2.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S3.I3.i2.p1.5.m5.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.5.m5.1c">\mu\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.5.m5.1d">italic_μ ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I3.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I3.i3.1.1.1" style="width:0.0pt;">(iii)</span></span> <div class="ltx_para" id="S3.I3.i3.p1"> <p class="ltx_p" id="S3.I3.i3.p1.5"><math alttext="{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}" class="ltx_Math" display="inline" id="S3.I3.i3.p1.1.m1.1"><semantics id="S3.I3.i3.p1.1.m1.1a"><mo id="S3.I3.i3.p1.1.m1.1.1" mathcolor="#FF0000" mathsize="173%" xref="S3.I3.i3.p1.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S3.I3.i3.p1.1.m1.1b"><times id="S3.I3.i3.p1.1.m1.1.1.cmml" xref="S3.I3.i3.p1.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i3.p1.1.m1.1c">{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i3.p1.1.m1.1d">×</annotation></semantics></math> <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S3.I3.i3.p1.2.m2.1"><semantics id="S3.I3.i3.p1.2.m2.1a"><mrow id="S3.I3.i3.p1.2.m2.1.1" xref="S3.I3.i3.p1.2.m2.1.1.cmml"><mi id="S3.I3.i3.p1.2.m2.1.1.2" xref="S3.I3.i3.p1.2.m2.1.1.2.cmml">μ</mi><mo id="S3.I3.i3.p1.2.m2.1.1.1" xref="S3.I3.i3.p1.2.m2.1.1.1.cmml">⊧</mo><mi id="S3.I3.i3.p1.2.m2.1.1.3" xref="S3.I3.i3.p1.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i3.p1.2.m2.1b"><apply id="S3.I3.i3.p1.2.m2.1.1.cmml" xref="S3.I3.i3.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.I3.i3.p1.2.m2.1.1.1.cmml" xref="S3.I3.i3.p1.2.m2.1.1.1">models</csymbol><ci id="S3.I3.i3.p1.2.m2.1.1.2.cmml" xref="S3.I3.i3.p1.2.m2.1.1.2">𝜇</ci><ci id="S3.I3.i3.p1.2.m2.1.1.3.cmml" xref="S3.I3.i3.p1.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i3.p1.2.m2.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i3.p1.2.m2.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> iff <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S3.I3.i3.p1.3.m3.2"><semantics id="S3.I3.i3.p1.3.m3.2a"><msub id="S3.I3.i3.p1.3.m3.2.3.2" xref="S3.I3.i3.p1.3.m3.2.3.1.cmml"><mrow id="S3.I3.i3.p1.3.m3.2.3.2.2" xref="S3.I3.i3.p1.3.m3.2.3.1.cmml"><mi id="S3.I3.i3.p1.3.m3.1.1" xref="S3.I3.i3.p1.3.m3.1.1.cmml">φ</mi><mo id="S3.I3.i3.p1.3.m3.2.3.2.2.1" stretchy="false" xref="S3.I3.i3.p1.3.m3.2.3.1.1.cmml">|</mo></mrow><mi id="S3.I3.i3.p1.3.m3.2.2.1" xref="S3.I3.i3.p1.3.m3.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I3.i3.p1.3.m3.2b"><apply id="S3.I3.i3.p1.3.m3.2.3.1.cmml" xref="S3.I3.i3.p1.3.m3.2.3.2"><csymbol cd="latexml" id="S3.I3.i3.p1.3.m3.2.3.1.1.cmml" xref="S3.I3.i3.p1.3.m3.2.3.2.2.1">evaluated-at</csymbol><ci id="S3.I3.i3.p1.3.m3.1.1.cmml" xref="S3.I3.i3.p1.3.m3.1.1">𝜑</ci><ci id="S3.I3.i3.p1.3.m3.2.2.1.cmml" xref="S3.I3.i3.p1.3.m3.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i3.p1.3.m3.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i3.p1.3.m3.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is a valid formula, not necessarily <math alttext="\top" class="ltx_Math" display="inline" id="S3.I3.i3.p1.4.m4.1"><semantics id="S3.I3.i3.p1.4.m4.1a"><mo id="S3.I3.i3.p1.4.m4.1.1" xref="S3.I3.i3.p1.4.m4.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="S3.I3.i3.p1.4.m4.1b"><csymbol cd="latexml" id="S3.I3.i3.p1.4.m4.1.1.cmml" xref="S3.I3.i3.p1.4.m4.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i3.p1.4.m4.1c">\top</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i3.p1.4.m4.1d">⊤</annotation></semantics></math> (also, in general <math alttext="\mu(\varphi)\neq\mbox{{\sf T}}" class="ltx_Math" display="inline" id="S3.I3.i3.p1.5.m5.1"><semantics id="S3.I3.i3.p1.5.m5.1a"><mrow id="S3.I3.i3.p1.5.m5.1.2" xref="S3.I3.i3.p1.5.m5.1.2.cmml"><mrow id="S3.I3.i3.p1.5.m5.1.2.2" xref="S3.I3.i3.p1.5.m5.1.2.2.cmml"><mi id="S3.I3.i3.p1.5.m5.1.2.2.2" xref="S3.I3.i3.p1.5.m5.1.2.2.2.cmml">μ</mi><mo id="S3.I3.i3.p1.5.m5.1.2.2.1" xref="S3.I3.i3.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S3.I3.i3.p1.5.m5.1.2.2.3.2" xref="S3.I3.i3.p1.5.m5.1.2.2.cmml"><mo id="S3.I3.i3.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S3.I3.i3.p1.5.m5.1.2.2.cmml">(</mo><mi id="S3.I3.i3.p1.5.m5.1.1" xref="S3.I3.i3.p1.5.m5.1.1.cmml">φ</mi><mo id="S3.I3.i3.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S3.I3.i3.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.I3.i3.p1.5.m5.1.2.1" xref="S3.I3.i3.p1.5.m5.1.2.1.cmml">≠</mo><mtext class="ltx_mathvariant_sans-serif" id="S3.I3.i3.p1.5.m5.1.2.3" xref="S3.I3.i3.p1.5.m5.1.2.3a.cmml">T</mtext></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i3.p1.5.m5.1b"><apply id="S3.I3.i3.p1.5.m5.1.2.cmml" xref="S3.I3.i3.p1.5.m5.1.2"><neq id="S3.I3.i3.p1.5.m5.1.2.1.cmml" xref="S3.I3.i3.p1.5.m5.1.2.1"></neq><apply id="S3.I3.i3.p1.5.m5.1.2.2.cmml" xref="S3.I3.i3.p1.5.m5.1.2.2"><times id="S3.I3.i3.p1.5.m5.1.2.2.1.cmml" xref="S3.I3.i3.p1.5.m5.1.2.2.1"></times><ci id="S3.I3.i3.p1.5.m5.1.2.2.2.cmml" xref="S3.I3.i3.p1.5.m5.1.2.2.2">𝜇</ci><ci id="S3.I3.i3.p1.5.m5.1.1.cmml" xref="S3.I3.i3.p1.5.m5.1.1">𝜑</ci></apply><ci id="S3.I3.i3.p1.5.m5.1.2.3a.cmml" xref="S3.I3.i3.p1.5.m5.1.2.3"><mtext class="ltx_mathvariant_sans-serif" id="S3.I3.i3.p1.5.m5.1.2.3.cmml" xref="S3.I3.i3.p1.5.m5.1.2.3">T</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i3.p1.5.m5.1c">\mu(\varphi)\neq\mbox{{\sf T}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i3.p1.5.m5.1d">italic_μ ( italic_φ ) ≠ T</annotation></semantics></math>).</p> </div> </li> <li class="ltx_item" id="S3.I3.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I3.i4.1.1.1" style="width:0.0pt;">(iv)</span></span> <div class="ltx_para" id="S3.I3.i4.p1"> <p class="ltx_p" id="S3.I3.i4.p1.2"><math alttext="{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}" class="ltx_Math" display="inline" id="S3.I3.i4.p1.1.m1.1"><semantics id="S3.I3.i4.p1.1.m1.1a"><mo id="S3.I3.i4.p1.1.m1.1.1" mathcolor="#FF0000" mathsize="173%" xref="S3.I3.i4.p1.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S3.I3.i4.p1.1.m1.1b"><times id="S3.I3.i4.p1.1.m1.1.1.cmml" xref="S3.I3.i4.p1.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i4.p1.1.m1.1c">{\color[rgb]{1.00,0.00,0.00}\definecolor[named]{pgfstrokecolor}{rgb}{% 1.00,0.00,0.00}\mathbf{\times}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i4.p1.1.m1.1d">×</annotation></semantics></math> Checking <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S3.I3.i4.p1.2.m2.1"><semantics id="S3.I3.i4.p1.2.m2.1a"><mrow id="S3.I3.i4.p1.2.m2.1.1" xref="S3.I3.i4.p1.2.m2.1.1.cmml"><mi id="S3.I3.i4.p1.2.m2.1.1.2" xref="S3.I3.i4.p1.2.m2.1.1.2.cmml">μ</mi><mo id="S3.I3.i4.p1.2.m2.1.1.1" xref="S3.I3.i4.p1.2.m2.1.1.1.cmml">⊧</mo><mi id="S3.I3.i4.p1.2.m2.1.1.3" xref="S3.I3.i4.p1.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i4.p1.2.m2.1b"><apply id="S3.I3.i4.p1.2.m2.1.1.cmml" xref="S3.I3.i4.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.I3.i4.p1.2.m2.1.1.1.cmml" xref="S3.I3.i4.p1.2.m2.1.1.1">models</csymbol><ci id="S3.I3.i4.p1.2.m2.1.1.2.cmml" xref="S3.I3.i4.p1.2.m2.1.1.2">𝜇</ci><ci id="S3.I3.i4.p1.2.m2.1.1.3.cmml" xref="S3.I3.i4.p1.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i4.p1.2.m2.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i4.p1.2.m2.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S3.I3.i4.p1.2.1">co-NP-complete</em>.</p> </div> </li> </ul> </div> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample2"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 2</span></h6> <div class="ltx_para" id="Thmexample2.p1"> <p class="ltx_p" id="Thmexample2.p1.11">Let <math alttext="\varphi_{1}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}(A_{1}\wedge A_{2})% \vee(A_{1}\wedge\neg A_{2})" class="ltx_Math" display="inline" id="Thmexample2.p1.1.m1.2"><semantics id="Thmexample2.p1.1.m1.2a"><mrow id="Thmexample2.p1.1.m1.2.2" xref="Thmexample2.p1.1.m1.2.2.cmml"><msub id="Thmexample2.p1.1.m1.2.2.4" xref="Thmexample2.p1.1.m1.2.2.4.cmml"><mi id="Thmexample2.p1.1.m1.2.2.4.2" xref="Thmexample2.p1.1.m1.2.2.4.2.cmml">φ</mi><mn id="Thmexample2.p1.1.m1.2.2.4.3" xref="Thmexample2.p1.1.m1.2.2.4.3.cmml">1</mn></msub><mover id="Thmexample2.p1.1.m1.2.2.3" xref="Thmexample2.p1.1.m1.2.2.3.cmml"><mo id="Thmexample2.p1.1.m1.2.2.3.2" xref="Thmexample2.p1.1.m1.2.2.3.2.cmml">=</mo><mtext id="Thmexample2.p1.1.m1.2.2.3.3" mathsize="71%" xref="Thmexample2.p1.1.m1.2.2.3.3a.cmml">def</mtext></mover><mrow id="Thmexample2.p1.1.m1.2.2.2" xref="Thmexample2.p1.1.m1.2.2.2.cmml"><mrow id="Thmexample2.p1.1.m1.1.1.1.1.1" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmexample2.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample2.p1.1.m1.1.1.1.1.1.1" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.cmml"><msub id="Thmexample2.p1.1.m1.1.1.1.1.1.1.2" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.cmml"><mi id="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.2" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.2.cmml">A</mi><mn id="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.3" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample2.p1.1.m1.1.1.1.1.1.1.1" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.1.cmml">∧</mo><msub id="Thmexample2.p1.1.m1.1.1.1.1.1.1.3" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.2" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.2.cmml">A</mi><mn id="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.3" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="Thmexample2.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample2.p1.1.m1.2.2.2.3" xref="Thmexample2.p1.1.m1.2.2.2.3.cmml">∨</mo><mrow id="Thmexample2.p1.1.m1.2.2.2.2.1" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.cmml"><mo id="Thmexample2.p1.1.m1.2.2.2.2.1.2" stretchy="false" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="Thmexample2.p1.1.m1.2.2.2.2.1.1" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.cmml"><msub id="Thmexample2.p1.1.m1.2.2.2.2.1.1.2" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.cmml"><mi id="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.2" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.2.cmml">A</mi><mn id="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.3" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample2.p1.1.m1.2.2.2.2.1.1.1" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.1.cmml">∧</mo><mrow id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.cmml"><mo id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.1" rspace="0.167em" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.1.cmml">¬</mo><msub id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.cmml"><mi id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.2" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.2.cmml">A</mi><mn id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.3" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="Thmexample2.p1.1.m1.2.2.2.2.1.3" stretchy="false" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.1.m1.2b"><apply id="Thmexample2.p1.1.m1.2.2.cmml" xref="Thmexample2.p1.1.m1.2.2"><apply id="Thmexample2.p1.1.m1.2.2.3.cmml" xref="Thmexample2.p1.1.m1.2.2.3"><csymbol cd="ambiguous" id="Thmexample2.p1.1.m1.2.2.3.1.cmml" xref="Thmexample2.p1.1.m1.2.2.3">superscript</csymbol><eq id="Thmexample2.p1.1.m1.2.2.3.2.cmml" xref="Thmexample2.p1.1.m1.2.2.3.2"></eq><ci id="Thmexample2.p1.1.m1.2.2.3.3a.cmml" xref="Thmexample2.p1.1.m1.2.2.3.3"><mtext id="Thmexample2.p1.1.m1.2.2.3.3.cmml" mathsize="50%" xref="Thmexample2.p1.1.m1.2.2.3.3">def</mtext></ci></apply><apply id="Thmexample2.p1.1.m1.2.2.4.cmml" xref="Thmexample2.p1.1.m1.2.2.4"><csymbol cd="ambiguous" id="Thmexample2.p1.1.m1.2.2.4.1.cmml" xref="Thmexample2.p1.1.m1.2.2.4">subscript</csymbol><ci id="Thmexample2.p1.1.m1.2.2.4.2.cmml" xref="Thmexample2.p1.1.m1.2.2.4.2">𝜑</ci><cn id="Thmexample2.p1.1.m1.2.2.4.3.cmml" type="integer" xref="Thmexample2.p1.1.m1.2.2.4.3">1</cn></apply><apply id="Thmexample2.p1.1.m1.2.2.2.cmml" xref="Thmexample2.p1.1.m1.2.2.2"><or id="Thmexample2.p1.1.m1.2.2.2.3.cmml" xref="Thmexample2.p1.1.m1.2.2.2.3"></or><apply id="Thmexample2.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmexample2.p1.1.m1.1.1.1.1.1"><and id="Thmexample2.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.1"></and><apply id="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.2">𝐴</ci><cn id="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.2.3">1</cn></apply><apply id="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.2">𝐴</ci><cn id="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="Thmexample2.p1.1.m1.1.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="Thmexample2.p1.1.m1.2.2.2.2.1.1.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1"><and id="Thmexample2.p1.1.m1.2.2.2.2.1.1.1.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.1"></and><apply id="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.2"><csymbol cd="ambiguous" id="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.1.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.2">subscript</csymbol><ci id="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.2.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.2">𝐴</ci><cn id="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.3.cmml" type="integer" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.2.3">1</cn></apply><apply id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3"><not id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.1.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.1"></not><apply id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2"><csymbol cd="ambiguous" id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.1.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2">subscript</csymbol><ci id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.2.cmml" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.2">𝐴</ci><cn id="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.3.cmml" type="integer" xref="Thmexample2.p1.1.m1.2.2.2.2.1.1.3.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.1.m1.2c">\varphi_{1}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}(A_{1}\wedge A_{2})% \vee(A_{1}\wedge\neg A_{2})</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.1.m1.2d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\varphi_{2}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}A_{1}" class="ltx_Math" display="inline" id="Thmexample2.p1.2.m2.1"><semantics id="Thmexample2.p1.2.m2.1a"><mrow id="Thmexample2.p1.2.m2.1.1" xref="Thmexample2.p1.2.m2.1.1.cmml"><msub id="Thmexample2.p1.2.m2.1.1.2" xref="Thmexample2.p1.2.m2.1.1.2.cmml"><mi id="Thmexample2.p1.2.m2.1.1.2.2" xref="Thmexample2.p1.2.m2.1.1.2.2.cmml">φ</mi><mn id="Thmexample2.p1.2.m2.1.1.2.3" xref="Thmexample2.p1.2.m2.1.1.2.3.cmml">2</mn></msub><mover id="Thmexample2.p1.2.m2.1.1.1" xref="Thmexample2.p1.2.m2.1.1.1.cmml"><mo id="Thmexample2.p1.2.m2.1.1.1.2" xref="Thmexample2.p1.2.m2.1.1.1.2.cmml">=</mo><mtext id="Thmexample2.p1.2.m2.1.1.1.3" mathsize="71%" xref="Thmexample2.p1.2.m2.1.1.1.3a.cmml">def</mtext></mover><msub id="Thmexample2.p1.2.m2.1.1.3" xref="Thmexample2.p1.2.m2.1.1.3.cmml"><mi id="Thmexample2.p1.2.m2.1.1.3.2" xref="Thmexample2.p1.2.m2.1.1.3.2.cmml">A</mi><mn id="Thmexample2.p1.2.m2.1.1.3.3" xref="Thmexample2.p1.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.2.m2.1b"><apply id="Thmexample2.p1.2.m2.1.1.cmml" xref="Thmexample2.p1.2.m2.1.1"><apply id="Thmexample2.p1.2.m2.1.1.1.cmml" xref="Thmexample2.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="Thmexample2.p1.2.m2.1.1.1.1.cmml" xref="Thmexample2.p1.2.m2.1.1.1">superscript</csymbol><eq id="Thmexample2.p1.2.m2.1.1.1.2.cmml" xref="Thmexample2.p1.2.m2.1.1.1.2"></eq><ci id="Thmexample2.p1.2.m2.1.1.1.3a.cmml" xref="Thmexample2.p1.2.m2.1.1.1.3"><mtext id="Thmexample2.p1.2.m2.1.1.1.3.cmml" mathsize="50%" xref="Thmexample2.p1.2.m2.1.1.1.3">def</mtext></ci></apply><apply id="Thmexample2.p1.2.m2.1.1.2.cmml" xref="Thmexample2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="Thmexample2.p1.2.m2.1.1.2.1.cmml" xref="Thmexample2.p1.2.m2.1.1.2">subscript</csymbol><ci id="Thmexample2.p1.2.m2.1.1.2.2.cmml" xref="Thmexample2.p1.2.m2.1.1.2.2">𝜑</ci><cn id="Thmexample2.p1.2.m2.1.1.2.3.cmml" type="integer" xref="Thmexample2.p1.2.m2.1.1.2.3">2</cn></apply><apply id="Thmexample2.p1.2.m2.1.1.3.cmml" xref="Thmexample2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmexample2.p1.2.m2.1.1.3.1.cmml" xref="Thmexample2.p1.2.m2.1.1.3">subscript</csymbol><ci id="Thmexample2.p1.2.m2.1.1.3.2.cmml" xref="Thmexample2.p1.2.m2.1.1.3.2">𝐴</ci><cn id="Thmexample2.p1.2.m2.1.1.3.3.cmml" type="integer" xref="Thmexample2.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.2.m2.1c">\varphi_{2}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}A_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.2.m2.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1}}\}" class="ltx_Math" display="inline" id="Thmexample2.p1.3.m3.1"><semantics id="Thmexample2.p1.3.m3.1a"><mrow id="Thmexample2.p1.3.m3.1.1" xref="Thmexample2.p1.3.m3.1.1.cmml"><mi id="Thmexample2.p1.3.m3.1.1.3" xref="Thmexample2.p1.3.m3.1.1.3.cmml">μ</mi><mover id="Thmexample2.p1.3.m3.1.1.2" xref="Thmexample2.p1.3.m3.1.1.2.cmml"><mo id="Thmexample2.p1.3.m3.1.1.2.2" xref="Thmexample2.p1.3.m3.1.1.2.2.cmml">=</mo><mtext id="Thmexample2.p1.3.m3.1.1.2.3" mathsize="71%" xref="Thmexample2.p1.3.m3.1.1.2.3a.cmml">def</mtext></mover><mrow id="Thmexample2.p1.3.m3.1.1.1.1" xref="Thmexample2.p1.3.m3.1.1.1.2.cmml"><mo id="Thmexample2.p1.3.m3.1.1.1.1.2" stretchy="false" xref="Thmexample2.p1.3.m3.1.1.1.2.cmml">{</mo><msub id="Thmexample2.p1.3.m3.1.1.1.1.1" xref="Thmexample2.p1.3.m3.1.1.1.1.1.cmml"><mi id="Thmexample2.p1.3.m3.1.1.1.1.1.2" xref="Thmexample2.p1.3.m3.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample2.p1.3.m3.1.1.1.1.1.3" xref="Thmexample2.p1.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample2.p1.3.m3.1.1.1.1.3" stretchy="false" xref="Thmexample2.p1.3.m3.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.3.m3.1b"><apply id="Thmexample2.p1.3.m3.1.1.cmml" xref="Thmexample2.p1.3.m3.1.1"><apply id="Thmexample2.p1.3.m3.1.1.2.cmml" xref="Thmexample2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="Thmexample2.p1.3.m3.1.1.2.1.cmml" xref="Thmexample2.p1.3.m3.1.1.2">superscript</csymbol><eq id="Thmexample2.p1.3.m3.1.1.2.2.cmml" xref="Thmexample2.p1.3.m3.1.1.2.2"></eq><ci id="Thmexample2.p1.3.m3.1.1.2.3a.cmml" xref="Thmexample2.p1.3.m3.1.1.2.3"><mtext id="Thmexample2.p1.3.m3.1.1.2.3.cmml" mathsize="50%" xref="Thmexample2.p1.3.m3.1.1.2.3">def</mtext></ci></apply><ci id="Thmexample2.p1.3.m3.1.1.3.cmml" xref="Thmexample2.p1.3.m3.1.1.3">𝜇</ci><set id="Thmexample2.p1.3.m3.1.1.1.2.cmml" xref="Thmexample2.p1.3.m3.1.1.1.1"><apply id="Thmexample2.p1.3.m3.1.1.1.1.1.cmml" xref="Thmexample2.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample2.p1.3.m3.1.1.1.1.1.1.cmml" xref="Thmexample2.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="Thmexample2.p1.3.m3.1.1.1.1.1.2.cmml" xref="Thmexample2.p1.3.m3.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample2.p1.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample2.p1.3.m3.1.1.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.3.m3.1c">\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.3.m3.1d">italic_μ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>. Then <math alttext="\varphi_{1}\equiv\varphi_{2}" class="ltx_Math" display="inline" id="Thmexample2.p1.4.m4.1"><semantics id="Thmexample2.p1.4.m4.1a"><mrow id="Thmexample2.p1.4.m4.1.1" xref="Thmexample2.p1.4.m4.1.1.cmml"><msub id="Thmexample2.p1.4.m4.1.1.2" xref="Thmexample2.p1.4.m4.1.1.2.cmml"><mi id="Thmexample2.p1.4.m4.1.1.2.2" xref="Thmexample2.p1.4.m4.1.1.2.2.cmml">φ</mi><mn id="Thmexample2.p1.4.m4.1.1.2.3" xref="Thmexample2.p1.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample2.p1.4.m4.1.1.1" xref="Thmexample2.p1.4.m4.1.1.1.cmml">≡</mo><msub id="Thmexample2.p1.4.m4.1.1.3" xref="Thmexample2.p1.4.m4.1.1.3.cmml"><mi id="Thmexample2.p1.4.m4.1.1.3.2" xref="Thmexample2.p1.4.m4.1.1.3.2.cmml">φ</mi><mn id="Thmexample2.p1.4.m4.1.1.3.3" xref="Thmexample2.p1.4.m4.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.4.m4.1b"><apply id="Thmexample2.p1.4.m4.1.1.cmml" xref="Thmexample2.p1.4.m4.1.1"><equivalent id="Thmexample2.p1.4.m4.1.1.1.cmml" xref="Thmexample2.p1.4.m4.1.1.1"></equivalent><apply id="Thmexample2.p1.4.m4.1.1.2.cmml" xref="Thmexample2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="Thmexample2.p1.4.m4.1.1.2.1.cmml" xref="Thmexample2.p1.4.m4.1.1.2">subscript</csymbol><ci id="Thmexample2.p1.4.m4.1.1.2.2.cmml" xref="Thmexample2.p1.4.m4.1.1.2.2">𝜑</ci><cn id="Thmexample2.p1.4.m4.1.1.2.3.cmml" type="integer" xref="Thmexample2.p1.4.m4.1.1.2.3">1</cn></apply><apply id="Thmexample2.p1.4.m4.1.1.3.cmml" xref="Thmexample2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="Thmexample2.p1.4.m4.1.1.3.1.cmml" xref="Thmexample2.p1.4.m4.1.1.3">subscript</csymbol><ci id="Thmexample2.p1.4.m4.1.1.3.2.cmml" xref="Thmexample2.p1.4.m4.1.1.3.2">𝜑</ci><cn id="Thmexample2.p1.4.m4.1.1.3.3.cmml" type="integer" xref="Thmexample2.p1.4.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.4.m4.1c">\varphi_{1}\equiv\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.4.m4.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≡ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\varphi_{1}|_{\mu}=A_{2}\vee\neg A_{2}\neq\top" class="ltx_Math" display="inline" id="Thmexample2.p1.5.m5.2"><semantics id="Thmexample2.p1.5.m5.2a"><mrow id="Thmexample2.p1.5.m5.2.2" xref="Thmexample2.p1.5.m5.2.2.cmml"><msub id="Thmexample2.p1.5.m5.2.2.1.1" xref="Thmexample2.p1.5.m5.2.2.1.2.cmml"><mrow id="Thmexample2.p1.5.m5.2.2.1.1.1" xref="Thmexample2.p1.5.m5.2.2.1.2.cmml"><msub id="Thmexample2.p1.5.m5.2.2.1.1.1.1" xref="Thmexample2.p1.5.m5.2.2.1.1.1.1.cmml"><mi id="Thmexample2.p1.5.m5.2.2.1.1.1.1.2" xref="Thmexample2.p1.5.m5.2.2.1.1.1.1.2.cmml">φ</mi><mn id="Thmexample2.p1.5.m5.2.2.1.1.1.1.3" xref="Thmexample2.p1.5.m5.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample2.p1.5.m5.2.2.1.1.1.2" stretchy="false" xref="Thmexample2.p1.5.m5.2.2.1.2.1.cmml">|</mo></mrow><mi id="Thmexample2.p1.5.m5.1.1.1" xref="Thmexample2.p1.5.m5.1.1.1.cmml">μ</mi></msub><mo id="Thmexample2.p1.5.m5.2.2.3" xref="Thmexample2.p1.5.m5.2.2.3.cmml">=</mo><mrow id="Thmexample2.p1.5.m5.2.2.4" xref="Thmexample2.p1.5.m5.2.2.4.cmml"><msub id="Thmexample2.p1.5.m5.2.2.4.2" xref="Thmexample2.p1.5.m5.2.2.4.2.cmml"><mi id="Thmexample2.p1.5.m5.2.2.4.2.2" xref="Thmexample2.p1.5.m5.2.2.4.2.2.cmml">A</mi><mn id="Thmexample2.p1.5.m5.2.2.4.2.3" xref="Thmexample2.p1.5.m5.2.2.4.2.3.cmml">2</mn></msub><mo id="Thmexample2.p1.5.m5.2.2.4.1" xref="Thmexample2.p1.5.m5.2.2.4.1.cmml">∨</mo><mrow id="Thmexample2.p1.5.m5.2.2.4.3" xref="Thmexample2.p1.5.m5.2.2.4.3.cmml"><mo id="Thmexample2.p1.5.m5.2.2.4.3.1" rspace="0.167em" xref="Thmexample2.p1.5.m5.2.2.4.3.1.cmml">¬</mo><msub id="Thmexample2.p1.5.m5.2.2.4.3.2" xref="Thmexample2.p1.5.m5.2.2.4.3.2.cmml"><mi id="Thmexample2.p1.5.m5.2.2.4.3.2.2" xref="Thmexample2.p1.5.m5.2.2.4.3.2.2.cmml">A</mi><mn id="Thmexample2.p1.5.m5.2.2.4.3.2.3" xref="Thmexample2.p1.5.m5.2.2.4.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="Thmexample2.p1.5.m5.2.2.5" rspace="0em" xref="Thmexample2.p1.5.m5.2.2.5.cmml">≠</mo><mo id="Thmexample2.p1.5.m5.2.2.6" lspace="0em" xref="Thmexample2.p1.5.m5.2.2.6.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.5.m5.2b"><apply id="Thmexample2.p1.5.m5.2.2.cmml" xref="Thmexample2.p1.5.m5.2.2"><and id="Thmexample2.p1.5.m5.2.2a.cmml" xref="Thmexample2.p1.5.m5.2.2"></and><apply id="Thmexample2.p1.5.m5.2.2b.cmml" xref="Thmexample2.p1.5.m5.2.2"><eq id="Thmexample2.p1.5.m5.2.2.3.cmml" xref="Thmexample2.p1.5.m5.2.2.3"></eq><apply id="Thmexample2.p1.5.m5.2.2.1.2.cmml" xref="Thmexample2.p1.5.m5.2.2.1.1"><csymbol cd="latexml" id="Thmexample2.p1.5.m5.2.2.1.2.1.cmml" xref="Thmexample2.p1.5.m5.2.2.1.1.1.2">evaluated-at</csymbol><apply id="Thmexample2.p1.5.m5.2.2.1.1.1.1.cmml" xref="Thmexample2.p1.5.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample2.p1.5.m5.2.2.1.1.1.1.1.cmml" xref="Thmexample2.p1.5.m5.2.2.1.1.1.1">subscript</csymbol><ci id="Thmexample2.p1.5.m5.2.2.1.1.1.1.2.cmml" xref="Thmexample2.p1.5.m5.2.2.1.1.1.1.2">𝜑</ci><cn id="Thmexample2.p1.5.m5.2.2.1.1.1.1.3.cmml" type="integer" xref="Thmexample2.p1.5.m5.2.2.1.1.1.1.3">1</cn></apply><ci id="Thmexample2.p1.5.m5.1.1.1.cmml" xref="Thmexample2.p1.5.m5.1.1.1">𝜇</ci></apply><apply id="Thmexample2.p1.5.m5.2.2.4.cmml" xref="Thmexample2.p1.5.m5.2.2.4"><or id="Thmexample2.p1.5.m5.2.2.4.1.cmml" xref="Thmexample2.p1.5.m5.2.2.4.1"></or><apply id="Thmexample2.p1.5.m5.2.2.4.2.cmml" xref="Thmexample2.p1.5.m5.2.2.4.2"><csymbol cd="ambiguous" id="Thmexample2.p1.5.m5.2.2.4.2.1.cmml" xref="Thmexample2.p1.5.m5.2.2.4.2">subscript</csymbol><ci id="Thmexample2.p1.5.m5.2.2.4.2.2.cmml" xref="Thmexample2.p1.5.m5.2.2.4.2.2">𝐴</ci><cn id="Thmexample2.p1.5.m5.2.2.4.2.3.cmml" type="integer" xref="Thmexample2.p1.5.m5.2.2.4.2.3">2</cn></apply><apply id="Thmexample2.p1.5.m5.2.2.4.3.cmml" xref="Thmexample2.p1.5.m5.2.2.4.3"><not id="Thmexample2.p1.5.m5.2.2.4.3.1.cmml" xref="Thmexample2.p1.5.m5.2.2.4.3.1"></not><apply id="Thmexample2.p1.5.m5.2.2.4.3.2.cmml" xref="Thmexample2.p1.5.m5.2.2.4.3.2"><csymbol cd="ambiguous" id="Thmexample2.p1.5.m5.2.2.4.3.2.1.cmml" xref="Thmexample2.p1.5.m5.2.2.4.3.2">subscript</csymbol><ci id="Thmexample2.p1.5.m5.2.2.4.3.2.2.cmml" xref="Thmexample2.p1.5.m5.2.2.4.3.2.2">𝐴</ci><cn id="Thmexample2.p1.5.m5.2.2.4.3.2.3.cmml" type="integer" xref="Thmexample2.p1.5.m5.2.2.4.3.2.3">2</cn></apply></apply></apply></apply><apply id="Thmexample2.p1.5.m5.2.2c.cmml" xref="Thmexample2.p1.5.m5.2.2"><neq id="Thmexample2.p1.5.m5.2.2.5.cmml" xref="Thmexample2.p1.5.m5.2.2.5"></neq><share href="https://arxiv.org/html/2503.01536v1#Thmexample2.p1.5.m5.2.2.4.cmml" id="Thmexample2.p1.5.m5.2.2d.cmml" xref="Thmexample2.p1.5.m5.2.2"></share><csymbol cd="latexml" id="Thmexample2.p1.5.m5.2.2.6.cmml" xref="Thmexample2.p1.5.m5.2.2.6">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.5.m5.2c">\varphi_{1}|_{\mu}=A_{2}\vee\neg A_{2}\neq\top</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.5.m5.2d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∨ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≠ ⊤</annotation></semantics></math> and <math alttext="\varphi_{2}|_{\mu}=\top" class="ltx_Math" display="inline" id="Thmexample2.p1.6.m6.2"><semantics id="Thmexample2.p1.6.m6.2a"><mrow id="Thmexample2.p1.6.m6.2.2" xref="Thmexample2.p1.6.m6.2.2.cmml"><msub id="Thmexample2.p1.6.m6.2.2.1.1" xref="Thmexample2.p1.6.m6.2.2.1.2.cmml"><mrow id="Thmexample2.p1.6.m6.2.2.1.1.1" xref="Thmexample2.p1.6.m6.2.2.1.2.cmml"><msub id="Thmexample2.p1.6.m6.2.2.1.1.1.1" xref="Thmexample2.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="Thmexample2.p1.6.m6.2.2.1.1.1.1.2" xref="Thmexample2.p1.6.m6.2.2.1.1.1.1.2.cmml">φ</mi><mn id="Thmexample2.p1.6.m6.2.2.1.1.1.1.3" xref="Thmexample2.p1.6.m6.2.2.1.1.1.1.3.cmml">2</mn></msub><mo id="Thmexample2.p1.6.m6.2.2.1.1.1.2" stretchy="false" xref="Thmexample2.p1.6.m6.2.2.1.2.1.cmml">|</mo></mrow><mi id="Thmexample2.p1.6.m6.1.1.1" xref="Thmexample2.p1.6.m6.1.1.1.cmml">μ</mi></msub><mo id="Thmexample2.p1.6.m6.2.2.2" rspace="0em" xref="Thmexample2.p1.6.m6.2.2.2.cmml">=</mo><mo id="Thmexample2.p1.6.m6.2.2.3" lspace="0em" xref="Thmexample2.p1.6.m6.2.2.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.6.m6.2b"><apply id="Thmexample2.p1.6.m6.2.2.cmml" xref="Thmexample2.p1.6.m6.2.2"><eq id="Thmexample2.p1.6.m6.2.2.2.cmml" xref="Thmexample2.p1.6.m6.2.2.2"></eq><apply id="Thmexample2.p1.6.m6.2.2.1.2.cmml" xref="Thmexample2.p1.6.m6.2.2.1.1"><csymbol cd="latexml" id="Thmexample2.p1.6.m6.2.2.1.2.1.cmml" xref="Thmexample2.p1.6.m6.2.2.1.1.1.2">evaluated-at</csymbol><apply id="Thmexample2.p1.6.m6.2.2.1.1.1.1.cmml" xref="Thmexample2.p1.6.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample2.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="Thmexample2.p1.6.m6.2.2.1.1.1.1">subscript</csymbol><ci id="Thmexample2.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="Thmexample2.p1.6.m6.2.2.1.1.1.1.2">𝜑</ci><cn id="Thmexample2.p1.6.m6.2.2.1.1.1.1.3.cmml" type="integer" xref="Thmexample2.p1.6.m6.2.2.1.1.1.1.3">2</cn></apply><ci id="Thmexample2.p1.6.m6.1.1.1.cmml" xref="Thmexample2.p1.6.m6.1.1.1">𝜇</ci></apply><csymbol cd="latexml" id="Thmexample2.p1.6.m6.2.2.3.cmml" xref="Thmexample2.p1.6.m6.2.2.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.6.m6.2c">\varphi_{2}|_{\mu}=\top</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.6.m6.2d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math>. Thus <math alttext="\mu\models\varphi_{1}" class="ltx_Math" display="inline" id="Thmexample2.p1.7.m7.1"><semantics id="Thmexample2.p1.7.m7.1a"><mrow id="Thmexample2.p1.7.m7.1.1" xref="Thmexample2.p1.7.m7.1.1.cmml"><mi id="Thmexample2.p1.7.m7.1.1.2" xref="Thmexample2.p1.7.m7.1.1.2.cmml">μ</mi><mo id="Thmexample2.p1.7.m7.1.1.1" xref="Thmexample2.p1.7.m7.1.1.1.cmml">⊧</mo><msub id="Thmexample2.p1.7.m7.1.1.3" xref="Thmexample2.p1.7.m7.1.1.3.cmml"><mi id="Thmexample2.p1.7.m7.1.1.3.2" xref="Thmexample2.p1.7.m7.1.1.3.2.cmml">φ</mi><mn id="Thmexample2.p1.7.m7.1.1.3.3" xref="Thmexample2.p1.7.m7.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.7.m7.1b"><apply id="Thmexample2.p1.7.m7.1.1.cmml" xref="Thmexample2.p1.7.m7.1.1"><csymbol cd="latexml" id="Thmexample2.p1.7.m7.1.1.1.cmml" xref="Thmexample2.p1.7.m7.1.1.1">models</csymbol><ci id="Thmexample2.p1.7.m7.1.1.2.cmml" xref="Thmexample2.p1.7.m7.1.1.2">𝜇</ci><apply id="Thmexample2.p1.7.m7.1.1.3.cmml" xref="Thmexample2.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="Thmexample2.p1.7.m7.1.1.3.1.cmml" xref="Thmexample2.p1.7.m7.1.1.3">subscript</csymbol><ci id="Thmexample2.p1.7.m7.1.1.3.2.cmml" xref="Thmexample2.p1.7.m7.1.1.3.2">𝜑</ci><cn id="Thmexample2.p1.7.m7.1.1.3.3.cmml" type="integer" xref="Thmexample2.p1.7.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.7.m7.1c">\mu\models\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.7.m7.1d">italic_μ ⊧ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mu\models\varphi_{2}" class="ltx_Math" display="inline" id="Thmexample2.p1.8.m8.1"><semantics id="Thmexample2.p1.8.m8.1a"><mrow id="Thmexample2.p1.8.m8.1.1" xref="Thmexample2.p1.8.m8.1.1.cmml"><mi id="Thmexample2.p1.8.m8.1.1.2" xref="Thmexample2.p1.8.m8.1.1.2.cmml">μ</mi><mo id="Thmexample2.p1.8.m8.1.1.1" xref="Thmexample2.p1.8.m8.1.1.1.cmml">⊧</mo><msub id="Thmexample2.p1.8.m8.1.1.3" xref="Thmexample2.p1.8.m8.1.1.3.cmml"><mi id="Thmexample2.p1.8.m8.1.1.3.2" xref="Thmexample2.p1.8.m8.1.1.3.2.cmml">φ</mi><mn id="Thmexample2.p1.8.m8.1.1.3.3" xref="Thmexample2.p1.8.m8.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.8.m8.1b"><apply id="Thmexample2.p1.8.m8.1.1.cmml" xref="Thmexample2.p1.8.m8.1.1"><csymbol cd="latexml" id="Thmexample2.p1.8.m8.1.1.1.cmml" xref="Thmexample2.p1.8.m8.1.1.1">models</csymbol><ci id="Thmexample2.p1.8.m8.1.1.2.cmml" xref="Thmexample2.p1.8.m8.1.1.2">𝜇</ci><apply id="Thmexample2.p1.8.m8.1.1.3.cmml" xref="Thmexample2.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="Thmexample2.p1.8.m8.1.1.3.1.cmml" xref="Thmexample2.p1.8.m8.1.1.3">subscript</csymbol><ci id="Thmexample2.p1.8.m8.1.1.3.2.cmml" xref="Thmexample2.p1.8.m8.1.1.3.2">𝜑</ci><cn id="Thmexample2.p1.8.m8.1.1.3.3.cmml" type="integer" xref="Thmexample2.p1.8.m8.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.8.m8.1c">\mu\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.8.m8.1d">italic_μ ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, whereas <math alttext="\mu\not\mid\!\approx\varphi_{1}" class="ltx_math_unparsed" display="inline" id="Thmexample2.p1.9.m9.1"><semantics id="Thmexample2.p1.9.m9.1a"><mrow id="Thmexample2.p1.9.m9.1b"><mi id="Thmexample2.p1.9.m9.1.1">μ</mi><mpadded id="Thmexample2.p1.9.m9.1c" width="0.969em"><mo id="Thmexample2.p1.9.m9.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample2.p1.9.m9.1.3">≈</mo><msub id="Thmexample2.p1.9.m9.1.4"><mi id="Thmexample2.p1.9.m9.1.4.2">φ</mi><mn id="Thmexample2.p1.9.m9.1.4.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="Thmexample2.p1.9.m9.1d">\mu\not\mid\!\approx\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.9.m9.1e">italic_μ ∤ ≈ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> but <math alttext="\mu\mid\!\approx\varphi_{2}" class="ltx_math_unparsed" display="inline" id="Thmexample2.p1.10.m10.1"><semantics id="Thmexample2.p1.10.m10.1a"><mrow id="Thmexample2.p1.10.m10.1b"><mi id="Thmexample2.p1.10.m10.1.1">μ</mi><mpadded id="Thmexample2.p1.10.m10.1c" width="0.219em"><mo id="Thmexample2.p1.10.m10.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmexample2.p1.10.m10.1.3">≈</mo><msub id="Thmexample2.p1.10.m10.1.4"><mi id="Thmexample2.p1.10.m10.1.4.2">φ</mi><mn id="Thmexample2.p1.10.m10.1.4.3">2</mn></msub></mrow><annotation encoding="application/x-tex" id="Thmexample2.p1.10.m10.1d">\mu\mid\!\approx\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.10.m10.1e">italic_μ ∣ ≈ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. <math alttext="\diamond" class="ltx_Math" display="inline" id="Thmexample2.p1.11.m11.1"><semantics id="Thmexample2.p1.11.m11.1a"><mo id="Thmexample2.p1.11.m11.1.1" xref="Thmexample2.p1.11.m11.1.1.cmml">⋄</mo><annotation-xml encoding="MathML-Content" id="Thmexample2.p1.11.m11.1b"><ci id="Thmexample2.p1.11.m11.1.1.cmml" xref="Thmexample2.p1.11.m11.1.1">⋄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample2.p1.11.m11.1c">\diamond</annotation><annotation encoding="application/x-llamapun" id="Thmexample2.p1.11.m11.1d">⋄</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S3.SS1.SSS0.Px2.p4"> <p class="ltx_p" id="S3.SS1.SSS0.Px2.p4.1">From a theoretical viewpoint, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample2" title="Example 2 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">2</span></a> spotlights the difference between property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty5" title="Property 5 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">5</span></a>(ii) and property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty6" title="Property 6 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">6</span></a>(ii) (see also Property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty1" title="Property 1 ‣ Partial assignments. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a>(ii)): <em class="ltx_emph ltx_font_italic" id="S3.SS1.SSS0.Px2.p4.1.1">whereas entailment matches the intuition that equivalent formulas should be satisfied by the same (partial) assignments, verification does not</em>. The latter fact looks theoretically awkward. In particular, if we adopted verification as the general definition of partial-assignment satisfiability, then we believe that it would be embarrassing to have that equivalent formulas were not “satisfied” by the same assignments. </p> </div> </section> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Other candidate forms of partial-assignment satisfaction.</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">Given the above facts, we wonder if we could use other notions of partial-assignment satisfiability. (E.g., we could enrich <math alttext="\mid\!\approx" class="ltx_math_unparsed" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mrow id="S3.SS2.p1.1.m1.1b"><mpadded id="S3.SS2.p1.1.m1.1c" width="0.219em"><mo id="S3.SS2.p1.1.m1.1.1">∣</mo></mpadded><mo id="S3.SS2.p1.1.m1.1.2">≈</mo></mrow><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1d">\mid\!\approx</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1e">∣ ≈</annotation></semantics></math> with some form of formula simplification.) In particular, we wonder if there exists some suitable notion of partial-assignment satisfaction which is strictly stronger then entailment and fixes the “embarrassing fact” for verification of Property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty5" title="Property 5 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">5</span></a>(ii). The following theorem states this is not the case.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 3.2</span></h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.1"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.1.1">Let <math alttext="\mid\!\simeq" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p1.1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem2.p1.1.1.m1.1b"><mpadded id="S3.Thmtheorem2.p1.1.1.m1.1c" width="0.219em"><mo id="S3.Thmtheorem2.p1.1.1.m1.1.1">∣</mo></mpadded><mo id="S3.Thmtheorem2.p1.1.1.m1.1.2">≃</mo></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.1.m1.1d">\mid\!\simeq</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.1.m1.1e">∣ ≃</annotation></semantics></math> denote some relation such that </span></p> <ul class="ltx_itemize" id="S3.Thmtheorem2.p1.6"> <li class="ltx_item" id="S3.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I4.i1.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S3.I4.i1.1.1.1.1">a</span>)</span></span> <div class="ltx_para" id="S3.I4.i1.p1"> <p class="ltx_p" id="S3.I4.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.4.1">for every </span><math alttext="\mu" class="ltx_Math" display="inline" id="S3.I4.i1.p1.1.m1.1"><semantics id="S3.I4.i1.p1.1.m1.1a"><mi id="S3.I4.i1.p1.1.m1.1.1" xref="S3.I4.i1.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.1.m1.1b"><ci id="S3.I4.i1.p1.1.m1.1.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.1.m1.1d">italic_μ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.4.2"> and </span><math alttext="\varphi" class="ltx_Math" display="inline" id="S3.I4.i1.p1.2.m2.1"><semantics id="S3.I4.i1.p1.2.m2.1a"><mi id="S3.I4.i1.p1.2.m2.1.1" xref="S3.I4.i1.p1.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.2.m2.1b"><ci id="S3.I4.i1.p1.2.m2.1.1.cmml" xref="S3.I4.i1.p1.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.2.m2.1d">italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.4.3">, </span><math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S3.I4.i1.p1.3.m3.1"><semantics id="S3.I4.i1.p1.3.m3.1a"><mrow id="S3.I4.i1.p1.3.m3.1.1" xref="S3.I4.i1.p1.3.m3.1.1.cmml"><mi id="S3.I4.i1.p1.3.m3.1.1.2" xref="S3.I4.i1.p1.3.m3.1.1.2.cmml">μ</mi><mo id="S3.I4.i1.p1.3.m3.1.1.1" xref="S3.I4.i1.p1.3.m3.1.1.1.cmml">⊧</mo><mi id="S3.I4.i1.p1.3.m3.1.1.3" xref="S3.I4.i1.p1.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.3.m3.1b"><apply id="S3.I4.i1.p1.3.m3.1.1.cmml" xref="S3.I4.i1.p1.3.m3.1.1"><csymbol cd="latexml" id="S3.I4.i1.p1.3.m3.1.1.1.cmml" xref="S3.I4.i1.p1.3.m3.1.1.1">models</csymbol><ci id="S3.I4.i1.p1.3.m3.1.1.2.cmml" xref="S3.I4.i1.p1.3.m3.1.1.2">𝜇</ci><ci id="S3.I4.i1.p1.3.m3.1.1.3.cmml" xref="S3.I4.i1.p1.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.3.m3.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.3.m3.1d">italic_μ ⊧ italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.4.4"> if </span><math alttext="\mu\mid\!\simeq\varphi" class="ltx_math_unparsed" display="inline" id="S3.I4.i1.p1.4.m4.1"><semantics id="S3.I4.i1.p1.4.m4.1a"><mrow id="S3.I4.i1.p1.4.m4.1b"><mi id="S3.I4.i1.p1.4.m4.1.1">μ</mi><mpadded id="S3.I4.i1.p1.4.m4.1c" width="0.219em"><mo id="S3.I4.i1.p1.4.m4.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I4.i1.p1.4.m4.1.3">≃</mo><mi id="S3.I4.i1.p1.4.m4.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.I4.i1.p1.4.m4.1d">\mu\mid\!\simeq\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.4.m4.1e">italic_μ ∣ ≃ italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.4.5">;</span></p> </div> </li> <li class="ltx_item" id="S3.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I4.i2.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S3.I4.i2.1.1.1.1">b</span>)</span></span> <div class="ltx_para" id="S3.I4.i2.p1"> <p class="ltx_p" id="S3.I4.i2.p1.6"><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.6.1">for every </span><math alttext="\eta" class="ltx_Math" display="inline" id="S3.I4.i2.p1.1.m1.1"><semantics id="S3.I4.i2.p1.1.m1.1a"><mi id="S3.I4.i2.p1.1.m1.1.1" xref="S3.I4.i2.p1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.1.m1.1b"><ci id="S3.I4.i2.p1.1.m1.1.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.1.m1.1d">italic_η</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.6.2"> and </span><math alttext="\varphi" class="ltx_Math" display="inline" id="S3.I4.i2.p1.2.m2.1"><semantics id="S3.I4.i2.p1.2.m2.1a"><mi id="S3.I4.i2.p1.2.m2.1.1" xref="S3.I4.i2.p1.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.2.m2.1b"><ci id="S3.I4.i2.p1.2.m2.1.1.cmml" xref="S3.I4.i2.p1.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.2.m2.1d">italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.6.3">, if </span><math alttext="\eta" class="ltx_Math" display="inline" id="S3.I4.i2.p1.3.m3.1"><semantics id="S3.I4.i2.p1.3.m3.1a"><mi id="S3.I4.i2.p1.3.m3.1.1" xref="S3.I4.i2.p1.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.3.m3.1b"><ci id="S3.I4.i2.p1.3.m3.1.1.cmml" xref="S3.I4.i2.p1.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.3.m3.1d">italic_η</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.6.4"> is total for the set of atoms in </span><math alttext="\varphi" class="ltx_Math" display="inline" id="S3.I4.i2.p1.4.m4.1"><semantics id="S3.I4.i2.p1.4.m4.1a"><mi id="S3.I4.i2.p1.4.m4.1.1" xref="S3.I4.i2.p1.4.m4.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.4.m4.1b"><ci id="S3.I4.i2.p1.4.m4.1.1.cmml" xref="S3.I4.i2.p1.4.m4.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.4.m4.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.4.m4.1d">italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.6.5">, then </span><math alttext="\eta\mid\!\simeq\varphi" class="ltx_math_unparsed" display="inline" id="S3.I4.i2.p1.5.m5.1"><semantics id="S3.I4.i2.p1.5.m5.1a"><mrow id="S3.I4.i2.p1.5.m5.1b"><mi id="S3.I4.i2.p1.5.m5.1.1">η</mi><mpadded id="S3.I4.i2.p1.5.m5.1c" width="0.219em"><mo id="S3.I4.i2.p1.5.m5.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I4.i2.p1.5.m5.1.3">≃</mo><mi id="S3.I4.i2.p1.5.m5.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.I4.i2.p1.5.m5.1d">\eta\mid\!\simeq\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.5.m5.1e">italic_η ∣ ≃ italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.6.6"> iff </span><math alttext="\eta\models\varphi" class="ltx_Math" display="inline" id="S3.I4.i2.p1.6.m6.1"><semantics id="S3.I4.i2.p1.6.m6.1a"><mrow id="S3.I4.i2.p1.6.m6.1.1" xref="S3.I4.i2.p1.6.m6.1.1.cmml"><mi id="S3.I4.i2.p1.6.m6.1.1.2" xref="S3.I4.i2.p1.6.m6.1.1.2.cmml">η</mi><mo id="S3.I4.i2.p1.6.m6.1.1.1" xref="S3.I4.i2.p1.6.m6.1.1.1.cmml">⊧</mo><mi id="S3.I4.i2.p1.6.m6.1.1.3" xref="S3.I4.i2.p1.6.m6.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.6.m6.1b"><apply id="S3.I4.i2.p1.6.m6.1.1.cmml" xref="S3.I4.i2.p1.6.m6.1.1"><csymbol cd="latexml" id="S3.I4.i2.p1.6.m6.1.1.1.cmml" xref="S3.I4.i2.p1.6.m6.1.1.1">models</csymbol><ci id="S3.I4.i2.p1.6.m6.1.1.2.cmml" xref="S3.I4.i2.p1.6.m6.1.1.2">𝜂</ci><ci id="S3.I4.i2.p1.6.m6.1.1.3.cmml" xref="S3.I4.i2.p1.6.m6.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.6.m6.1c">\eta\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.6.m6.1d">italic_η ⊧ italic_φ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.6.7">;</span></p> </div> </li> <li class="ltx_item" id="S3.I4.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I4.i3.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S3.I4.i3.1.1.1.1">c</span>)</span></span> <div class="ltx_para" id="S3.I4.i3.p1"> <p class="ltx_p" id="S3.I4.i3.p1.6"><span class="ltx_text ltx_font_italic" id="S3.I4.i3.p1.6.1">for every </span><math alttext="\mu" class="ltx_Math" display="inline" id="S3.I4.i3.p1.1.m1.1"><semantics id="S3.I4.i3.p1.1.m1.1a"><mi id="S3.I4.i3.p1.1.m1.1.1" xref="S3.I4.i3.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.1.m1.1b"><ci id="S3.I4.i3.p1.1.m1.1.1.cmml" xref="S3.I4.i3.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.1.m1.1d">italic_μ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i3.p1.6.2">, </span><math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S3.I4.i3.p1.2.m2.1"><semantics id="S3.I4.i3.p1.2.m2.1a"><msub id="S3.I4.i3.p1.2.m2.1.1" xref="S3.I4.i3.p1.2.m2.1.1.cmml"><mi id="S3.I4.i3.p1.2.m2.1.1.2" xref="S3.I4.i3.p1.2.m2.1.1.2.cmml">φ</mi><mn id="S3.I4.i3.p1.2.m2.1.1.3" xref="S3.I4.i3.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.2.m2.1b"><apply id="S3.I4.i3.p1.2.m2.1.1.cmml" xref="S3.I4.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I4.i3.p1.2.m2.1.1.1.cmml" xref="S3.I4.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I4.i3.p1.2.m2.1.1.2.cmml" xref="S3.I4.i3.p1.2.m2.1.1.2">𝜑</ci><cn id="S3.I4.i3.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.I4.i3.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.2.m2.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.2.m2.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i3.p1.6.3"> and </span><math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S3.I4.i3.p1.3.m3.1"><semantics id="S3.I4.i3.p1.3.m3.1a"><msub id="S3.I4.i3.p1.3.m3.1.1" xref="S3.I4.i3.p1.3.m3.1.1.cmml"><mi id="S3.I4.i3.p1.3.m3.1.1.2" xref="S3.I4.i3.p1.3.m3.1.1.2.cmml">φ</mi><mn id="S3.I4.i3.p1.3.m3.1.1.3" xref="S3.I4.i3.p1.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.I4.i3.p1.3.m3.1b"><apply id="S3.I4.i3.p1.3.m3.1.1.cmml" xref="S3.I4.i3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I4.i3.p1.3.m3.1.1.1.cmml" xref="S3.I4.i3.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I4.i3.p1.3.m3.1.1.2.cmml" xref="S3.I4.i3.p1.3.m3.1.1.2">𝜑</ci><cn id="S3.I4.i3.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.I4.i3.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i3.p1.3.m3.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.3.m3.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i3.p1.6.4">, if </span><math alttext="\mu\mid\!\simeq\varphi_{1}\wedge\varphi_{2}" class="ltx_math_unparsed" display="inline" id="S3.I4.i3.p1.4.m4.1"><semantics id="S3.I4.i3.p1.4.m4.1a"><mrow id="S3.I4.i3.p1.4.m4.1b"><mi id="S3.I4.i3.p1.4.m4.1.1">μ</mi><mpadded id="S3.I4.i3.p1.4.m4.1c" width="0.219em"><mo id="S3.I4.i3.p1.4.m4.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I4.i3.p1.4.m4.1.3">≃</mo><msub id="S3.I4.i3.p1.4.m4.1.4"><mi id="S3.I4.i3.p1.4.m4.1.4.2">φ</mi><mn id="S3.I4.i3.p1.4.m4.1.4.3">1</mn></msub><mo id="S3.I4.i3.p1.4.m4.1.5">∧</mo><msub id="S3.I4.i3.p1.4.m4.1.6"><mi id="S3.I4.i3.p1.4.m4.1.6.2">φ</mi><mn id="S3.I4.i3.p1.4.m4.1.6.3">2</mn></msub></mrow><annotation encoding="application/x-tex" id="S3.I4.i3.p1.4.m4.1d">\mu\mid\!\simeq\varphi_{1}\wedge\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.4.m4.1e">italic_μ ∣ ≃ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i3.p1.6.5">, then </span><math alttext="\mu\mid\!\simeq\varphi_{1}" class="ltx_math_unparsed" display="inline" id="S3.I4.i3.p1.5.m5.1"><semantics id="S3.I4.i3.p1.5.m5.1a"><mrow id="S3.I4.i3.p1.5.m5.1b"><mi id="S3.I4.i3.p1.5.m5.1.1">μ</mi><mpadded id="S3.I4.i3.p1.5.m5.1c" width="0.219em"><mo id="S3.I4.i3.p1.5.m5.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I4.i3.p1.5.m5.1.3">≃</mo><msub id="S3.I4.i3.p1.5.m5.1.4"><mi id="S3.I4.i3.p1.5.m5.1.4.2">φ</mi><mn id="S3.I4.i3.p1.5.m5.1.4.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="S3.I4.i3.p1.5.m5.1d">\mu\mid\!\simeq\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.5.m5.1e">italic_μ ∣ ≃ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i3.p1.6.6"> and </span><math alttext="\mu\mid\!\simeq\varphi_{2}" class="ltx_math_unparsed" display="inline" id="S3.I4.i3.p1.6.m6.1"><semantics id="S3.I4.i3.p1.6.m6.1a"><mrow id="S3.I4.i3.p1.6.m6.1b"><mi id="S3.I4.i3.p1.6.m6.1.1">μ</mi><mpadded id="S3.I4.i3.p1.6.m6.1c" width="0.219em"><mo id="S3.I4.i3.p1.6.m6.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I4.i3.p1.6.m6.1.3">≃</mo><msub id="S3.I4.i3.p1.6.m6.1.4"><mi id="S3.I4.i3.p1.6.m6.1.4.2">φ</mi><mn id="S3.I4.i3.p1.6.m6.1.4.3">2</mn></msub></mrow><annotation encoding="application/x-tex" id="S3.I4.i3.p1.6.m6.1d">\mu\mid\!\simeq\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i3.p1.6.m6.1e">italic_μ ∣ ≃ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i3.p1.6.7">;</span></p> </div> </li> <li class="ltx_item" id="S3.I4.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item"><span class="ltx_text ltx_align_right ltx_inline-block" id="S3.I4.i4.1.1.1" style="width:0.0pt;">(<span class="ltx_text ltx_font_italic" id="S3.I4.i4.1.1.1.1">d</span>)</span></span> <div class="ltx_para" id="S3.I4.i4.p1"> <p class="ltx_p" id="S3.I4.i4.p1.6"><span class="ltx_text ltx_font_italic" id="S3.I4.i4.p1.6.1">for every </span><math alttext="\mu" class="ltx_Math" display="inline" id="S3.I4.i4.p1.1.m1.1"><semantics id="S3.I4.i4.p1.1.m1.1a"><mi id="S3.I4.i4.p1.1.m1.1.1" xref="S3.I4.i4.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i4.p1.1.m1.1b"><ci id="S3.I4.i4.p1.1.m1.1.1.cmml" xref="S3.I4.i4.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i4.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i4.p1.1.m1.1d">italic_μ</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i4.p1.6.2">, </span><math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S3.I4.i4.p1.2.m2.1"><semantics id="S3.I4.i4.p1.2.m2.1a"><msub id="S3.I4.i4.p1.2.m2.1.1" xref="S3.I4.i4.p1.2.m2.1.1.cmml"><mi id="S3.I4.i4.p1.2.m2.1.1.2" xref="S3.I4.i4.p1.2.m2.1.1.2.cmml">φ</mi><mn id="S3.I4.i4.p1.2.m2.1.1.3" xref="S3.I4.i4.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.I4.i4.p1.2.m2.1b"><apply id="S3.I4.i4.p1.2.m2.1.1.cmml" xref="S3.I4.i4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I4.i4.p1.2.m2.1.1.1.cmml" xref="S3.I4.i4.p1.2.m2.1.1">subscript</csymbol><ci id="S3.I4.i4.p1.2.m2.1.1.2.cmml" xref="S3.I4.i4.p1.2.m2.1.1.2">𝜑</ci><cn id="S3.I4.i4.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.I4.i4.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i4.p1.2.m2.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i4.p1.2.m2.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i4.p1.6.3"> and </span><math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S3.I4.i4.p1.3.m3.1"><semantics id="S3.I4.i4.p1.3.m3.1a"><msub id="S3.I4.i4.p1.3.m3.1.1" xref="S3.I4.i4.p1.3.m3.1.1.cmml"><mi id="S3.I4.i4.p1.3.m3.1.1.2" xref="S3.I4.i4.p1.3.m3.1.1.2.cmml">φ</mi><mn id="S3.I4.i4.p1.3.m3.1.1.3" xref="S3.I4.i4.p1.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.I4.i4.p1.3.m3.1b"><apply id="S3.I4.i4.p1.3.m3.1.1.cmml" xref="S3.I4.i4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I4.i4.p1.3.m3.1.1.1.cmml" xref="S3.I4.i4.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I4.i4.p1.3.m3.1.1.2.cmml" xref="S3.I4.i4.p1.3.m3.1.1.2">𝜑</ci><cn id="S3.I4.i4.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.I4.i4.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i4.p1.3.m3.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i4.p1.3.m3.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i4.p1.6.4">, if </span><math alttext="\varphi_{1}\equiv\varphi_{2}" class="ltx_Math" display="inline" id="S3.I4.i4.p1.4.m4.1"><semantics id="S3.I4.i4.p1.4.m4.1a"><mrow id="S3.I4.i4.p1.4.m4.1.1" xref="S3.I4.i4.p1.4.m4.1.1.cmml"><msub id="S3.I4.i4.p1.4.m4.1.1.2" xref="S3.I4.i4.p1.4.m4.1.1.2.cmml"><mi id="S3.I4.i4.p1.4.m4.1.1.2.2" xref="S3.I4.i4.p1.4.m4.1.1.2.2.cmml">φ</mi><mn id="S3.I4.i4.p1.4.m4.1.1.2.3" xref="S3.I4.i4.p1.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="S3.I4.i4.p1.4.m4.1.1.1" xref="S3.I4.i4.p1.4.m4.1.1.1.cmml">≡</mo><msub id="S3.I4.i4.p1.4.m4.1.1.3" xref="S3.I4.i4.p1.4.m4.1.1.3.cmml"><mi id="S3.I4.i4.p1.4.m4.1.1.3.2" xref="S3.I4.i4.p1.4.m4.1.1.3.2.cmml">φ</mi><mn id="S3.I4.i4.p1.4.m4.1.1.3.3" xref="S3.I4.i4.p1.4.m4.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i4.p1.4.m4.1b"><apply id="S3.I4.i4.p1.4.m4.1.1.cmml" xref="S3.I4.i4.p1.4.m4.1.1"><equivalent id="S3.I4.i4.p1.4.m4.1.1.1.cmml" xref="S3.I4.i4.p1.4.m4.1.1.1"></equivalent><apply id="S3.I4.i4.p1.4.m4.1.1.2.cmml" xref="S3.I4.i4.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.I4.i4.p1.4.m4.1.1.2.1.cmml" xref="S3.I4.i4.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.I4.i4.p1.4.m4.1.1.2.2.cmml" xref="S3.I4.i4.p1.4.m4.1.1.2.2">𝜑</ci><cn id="S3.I4.i4.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S3.I4.i4.p1.4.m4.1.1.2.3">1</cn></apply><apply id="S3.I4.i4.p1.4.m4.1.1.3.cmml" xref="S3.I4.i4.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S3.I4.i4.p1.4.m4.1.1.3.1.cmml" xref="S3.I4.i4.p1.4.m4.1.1.3">subscript</csymbol><ci id="S3.I4.i4.p1.4.m4.1.1.3.2.cmml" xref="S3.I4.i4.p1.4.m4.1.1.3.2">𝜑</ci><cn id="S3.I4.i4.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S3.I4.i4.p1.4.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i4.p1.4.m4.1c">\varphi_{1}\equiv\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i4.p1.4.m4.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≡ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i4.p1.6.5">, then </span><math alttext="\mu\mid\!\simeq\varphi_{1}" class="ltx_math_unparsed" display="inline" id="S3.I4.i4.p1.5.m5.1"><semantics id="S3.I4.i4.p1.5.m5.1a"><mrow id="S3.I4.i4.p1.5.m5.1b"><mi id="S3.I4.i4.p1.5.m5.1.1">μ</mi><mpadded id="S3.I4.i4.p1.5.m5.1c" width="0.219em"><mo id="S3.I4.i4.p1.5.m5.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I4.i4.p1.5.m5.1.3">≃</mo><msub id="S3.I4.i4.p1.5.m5.1.4"><mi id="S3.I4.i4.p1.5.m5.1.4.2">φ</mi><mn id="S3.I4.i4.p1.5.m5.1.4.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="S3.I4.i4.p1.5.m5.1d">\mu\mid\!\simeq\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i4.p1.5.m5.1e">italic_μ ∣ ≃ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i4.p1.6.6"> iff </span><math alttext="\mu\mid\!\simeq\varphi_{2}" class="ltx_math_unparsed" display="inline" id="S3.I4.i4.p1.6.m6.1"><semantics id="S3.I4.i4.p1.6.m6.1a"><mrow id="S3.I4.i4.p1.6.m6.1b"><mi id="S3.I4.i4.p1.6.m6.1.1">μ</mi><mpadded id="S3.I4.i4.p1.6.m6.1c" width="0.219em"><mo id="S3.I4.i4.p1.6.m6.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.I4.i4.p1.6.m6.1.3">≃</mo><msub id="S3.I4.i4.p1.6.m6.1.4"><mi id="S3.I4.i4.p1.6.m6.1.4.2">φ</mi><mn id="S3.I4.i4.p1.6.m6.1.4.3">2</mn></msub></mrow><annotation encoding="application/x-tex" id="S3.I4.i4.p1.6.m6.1d">\mu\mid\!\simeq\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i4.p1.6.m6.1e">italic_μ ∣ ≃ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i4.p1.6.7"></span></p> </div> </li> </ul> <p class="ltx_p" id="S3.Thmtheorem2.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.5.4">Then, for every <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.1.m1.1"><semantics id="S3.Thmtheorem2.p1.2.1.m1.1a"><mi id="S3.Thmtheorem2.p1.2.1.m1.1.1" xref="S3.Thmtheorem2.p1.2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.1.m1.1b"><ci id="S3.Thmtheorem2.p1.2.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.1.m1.1d">italic_μ</annotation></semantics></math>, <math alttext="\varphi" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.2.m2.1"><semantics id="S3.Thmtheorem2.p1.3.2.m2.1a"><mi id="S3.Thmtheorem2.p1.3.2.m2.1.1" xref="S3.Thmtheorem2.p1.3.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.2.m2.1b"><ci id="S3.Thmtheorem2.p1.3.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.3.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.2.m2.1d">italic_φ</annotation></semantics></math>, we have that <math alttext="\mu\mid\!\simeq\varphi" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p1.4.3.m3.1"><semantics id="S3.Thmtheorem2.p1.4.3.m3.1a"><mrow id="S3.Thmtheorem2.p1.4.3.m3.1b"><mi id="S3.Thmtheorem2.p1.4.3.m3.1.1">μ</mi><mpadded id="S3.Thmtheorem2.p1.4.3.m3.1c" width="0.219em"><mo id="S3.Thmtheorem2.p1.4.3.m3.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.Thmtheorem2.p1.4.3.m3.1.3">≃</mo><mi id="S3.Thmtheorem2.p1.4.3.m3.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.3.m3.1d">\mu\mid\!\simeq\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.3.m3.1e">italic_μ ∣ ≃ italic_φ</annotation></semantics></math> if and only if <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.5.4.m4.1"><semantics id="S3.Thmtheorem2.p1.5.4.m4.1a"><mrow id="S3.Thmtheorem2.p1.5.4.m4.1.1" xref="S3.Thmtheorem2.p1.5.4.m4.1.1.cmml"><mi id="S3.Thmtheorem2.p1.5.4.m4.1.1.2" xref="S3.Thmtheorem2.p1.5.4.m4.1.1.2.cmml">μ</mi><mo id="S3.Thmtheorem2.p1.5.4.m4.1.1.1" xref="S3.Thmtheorem2.p1.5.4.m4.1.1.1.cmml">⊧</mo><mi id="S3.Thmtheorem2.p1.5.4.m4.1.1.3" xref="S3.Thmtheorem2.p1.5.4.m4.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.5.4.m4.1b"><apply id="S3.Thmtheorem2.p1.5.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p1.5.4.m4.1.1"><csymbol cd="latexml" id="S3.Thmtheorem2.p1.5.4.m4.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.4.m4.1.1.1">models</csymbol><ci id="S3.Thmtheorem2.p1.5.4.m4.1.1.2.cmml" xref="S3.Thmtheorem2.p1.5.4.m4.1.1.2">𝜇</ci><ci id="S3.Thmtheorem2.p1.5.4.m4.1.1.3.cmml" xref="S3.Thmtheorem2.p1.5.4.m4.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.4.m4.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.4.m4.1d">italic_μ ⊧ italic_φ</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx2"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx2.p1"> <p class="ltx_p" id="Thmproofx2.p1.12">The “only if” part is hypothesis <span class="ltx_text ltx_font_italic" id="Thmproofx2.p1.12.1">(a)</span>. <br class="ltx_break"/>The “if” part: By absurd, suppose there exist <math alttext="\mu" class="ltx_Math" display="inline" id="Thmproofx2.p1.1.m1.1"><semantics id="Thmproofx2.p1.1.m1.1a"><mi id="Thmproofx2.p1.1.m1.1.1" xref="Thmproofx2.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.1.m1.1b"><ci id="Thmproofx2.p1.1.m1.1.1.cmml" xref="Thmproofx2.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.1.m1.1d">italic_μ</annotation></semantics></math> and <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmproofx2.p1.2.m2.1"><semantics id="Thmproofx2.p1.2.m2.1a"><mi id="Thmproofx2.p1.2.m2.1.1" xref="Thmproofx2.p1.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.2.m2.1b"><ci id="Thmproofx2.p1.2.m2.1.1.cmml" xref="Thmproofx2.p1.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.2.m2.1d">italic_φ</annotation></semantics></math> s.t. <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmproofx2.p1.3.m3.1"><semantics id="Thmproofx2.p1.3.m3.1a"><mrow id="Thmproofx2.p1.3.m3.1.1" xref="Thmproofx2.p1.3.m3.1.1.cmml"><mi id="Thmproofx2.p1.3.m3.1.1.2" xref="Thmproofx2.p1.3.m3.1.1.2.cmml">μ</mi><mo id="Thmproofx2.p1.3.m3.1.1.1" xref="Thmproofx2.p1.3.m3.1.1.1.cmml">⊧</mo><mi id="Thmproofx2.p1.3.m3.1.1.3" xref="Thmproofx2.p1.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.3.m3.1b"><apply id="Thmproofx2.p1.3.m3.1.1.cmml" xref="Thmproofx2.p1.3.m3.1.1"><csymbol cd="latexml" id="Thmproofx2.p1.3.m3.1.1.1.cmml" xref="Thmproofx2.p1.3.m3.1.1.1">models</csymbol><ci id="Thmproofx2.p1.3.m3.1.1.2.cmml" xref="Thmproofx2.p1.3.m3.1.1.2">𝜇</ci><ci id="Thmproofx2.p1.3.m3.1.1.3.cmml" xref="Thmproofx2.p1.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.3.m3.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.3.m3.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> and <math alttext="\mu\not\mid\!\simeq\varphi" class="ltx_math_unparsed" display="inline" id="Thmproofx2.p1.4.m4.1"><semantics id="Thmproofx2.p1.4.m4.1a"><mrow id="Thmproofx2.p1.4.m4.1b"><mi id="Thmproofx2.p1.4.m4.1.1">μ</mi><mpadded id="Thmproofx2.p1.4.m4.1c" width="0.969em"><mo id="Thmproofx2.p1.4.m4.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmproofx2.p1.4.m4.1.3">≃</mo><mi id="Thmproofx2.p1.4.m4.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmproofx2.p1.4.m4.1d">\mu\not\mid\!\simeq\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.4.m4.1e">italic_μ ∤ ≃ italic_φ</annotation></semantics></math> . <br class="ltx_break"/>Let <math alttext="\varphi_{1}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\bigwedge\!\mu" class="ltx_Math" display="inline" id="Thmproofx2.p1.5.m5.1"><semantics id="Thmproofx2.p1.5.m5.1a"><mrow id="Thmproofx2.p1.5.m5.1.1" xref="Thmproofx2.p1.5.m5.1.1.cmml"><msub id="Thmproofx2.p1.5.m5.1.1.2" xref="Thmproofx2.p1.5.m5.1.1.2.cmml"><mi id="Thmproofx2.p1.5.m5.1.1.2.2" xref="Thmproofx2.p1.5.m5.1.1.2.2.cmml">φ</mi><mn id="Thmproofx2.p1.5.m5.1.1.2.3" xref="Thmproofx2.p1.5.m5.1.1.2.3.cmml">1</mn></msub><mover id="Thmproofx2.p1.5.m5.1.1.1" xref="Thmproofx2.p1.5.m5.1.1.1.cmml"><mo id="Thmproofx2.p1.5.m5.1.1.1.2" rspace="0.111em" xref="Thmproofx2.p1.5.m5.1.1.1.2.cmml">=</mo><mtext id="Thmproofx2.p1.5.m5.1.1.1.3" mathsize="71%" xref="Thmproofx2.p1.5.m5.1.1.1.3a.cmml">def</mtext></mover><mrow id="Thmproofx2.p1.5.m5.1.1.3" xref="Thmproofx2.p1.5.m5.1.1.3.cmml"><mpadded width="0.747em"><mo id="Thmproofx2.p1.5.m5.1.1.3.1" xref="Thmproofx2.p1.5.m5.1.1.3.1.cmml">⋀</mo></mpadded><mi id="Thmproofx2.p1.5.m5.1.1.3.2" xref="Thmproofx2.p1.5.m5.1.1.3.2.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.5.m5.1b"><apply id="Thmproofx2.p1.5.m5.1.1.cmml" xref="Thmproofx2.p1.5.m5.1.1"><apply id="Thmproofx2.p1.5.m5.1.1.1.cmml" xref="Thmproofx2.p1.5.m5.1.1.1"><csymbol cd="ambiguous" id="Thmproofx2.p1.5.m5.1.1.1.1.cmml" xref="Thmproofx2.p1.5.m5.1.1.1">superscript</csymbol><eq id="Thmproofx2.p1.5.m5.1.1.1.2.cmml" xref="Thmproofx2.p1.5.m5.1.1.1.2"></eq><ci id="Thmproofx2.p1.5.m5.1.1.1.3a.cmml" xref="Thmproofx2.p1.5.m5.1.1.1.3"><mtext id="Thmproofx2.p1.5.m5.1.1.1.3.cmml" mathsize="50%" xref="Thmproofx2.p1.5.m5.1.1.1.3">def</mtext></ci></apply><apply id="Thmproofx2.p1.5.m5.1.1.2.cmml" xref="Thmproofx2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmproofx2.p1.5.m5.1.1.2.1.cmml" xref="Thmproofx2.p1.5.m5.1.1.2">subscript</csymbol><ci id="Thmproofx2.p1.5.m5.1.1.2.2.cmml" xref="Thmproofx2.p1.5.m5.1.1.2.2">𝜑</ci><cn id="Thmproofx2.p1.5.m5.1.1.2.3.cmml" type="integer" xref="Thmproofx2.p1.5.m5.1.1.2.3">1</cn></apply><apply id="Thmproofx2.p1.5.m5.1.1.3.cmml" xref="Thmproofx2.p1.5.m5.1.1.3"><and id="Thmproofx2.p1.5.m5.1.1.3.1.cmml" xref="Thmproofx2.p1.5.m5.1.1.3.1"></and><ci id="Thmproofx2.p1.5.m5.1.1.3.2.cmml" xref="Thmproofx2.p1.5.m5.1.1.3.2">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.5.m5.1c">\varphi_{1}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\bigwedge\!\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.5.m5.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ⋀ italic_μ</annotation></semantics></math> and <math alttext="\varphi_{2}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\bigwedge\!\mu\wedge\varphi" class="ltx_Math" display="inline" id="Thmproofx2.p1.6.m6.1"><semantics id="Thmproofx2.p1.6.m6.1a"><mrow id="Thmproofx2.p1.6.m6.1.1" xref="Thmproofx2.p1.6.m6.1.1.cmml"><msub id="Thmproofx2.p1.6.m6.1.1.2" xref="Thmproofx2.p1.6.m6.1.1.2.cmml"><mi id="Thmproofx2.p1.6.m6.1.1.2.2" xref="Thmproofx2.p1.6.m6.1.1.2.2.cmml">φ</mi><mn id="Thmproofx2.p1.6.m6.1.1.2.3" xref="Thmproofx2.p1.6.m6.1.1.2.3.cmml">2</mn></msub><mover id="Thmproofx2.p1.6.m6.1.1.1" xref="Thmproofx2.p1.6.m6.1.1.1.cmml"><mo id="Thmproofx2.p1.6.m6.1.1.1.2" rspace="0.111em" xref="Thmproofx2.p1.6.m6.1.1.1.2.cmml">=</mo><mtext id="Thmproofx2.p1.6.m6.1.1.1.3" mathsize="71%" xref="Thmproofx2.p1.6.m6.1.1.1.3a.cmml">def</mtext></mover><mrow id="Thmproofx2.p1.6.m6.1.1.3" xref="Thmproofx2.p1.6.m6.1.1.3.cmml"><mrow id="Thmproofx2.p1.6.m6.1.1.3.2" xref="Thmproofx2.p1.6.m6.1.1.3.2.cmml"><mpadded width="0.747em"><mo id="Thmproofx2.p1.6.m6.1.1.3.2.1" xref="Thmproofx2.p1.6.m6.1.1.3.2.1.cmml">⋀</mo></mpadded><mi id="Thmproofx2.p1.6.m6.1.1.3.2.2" xref="Thmproofx2.p1.6.m6.1.1.3.2.2.cmml">μ</mi></mrow><mo id="Thmproofx2.p1.6.m6.1.1.3.1" xref="Thmproofx2.p1.6.m6.1.1.3.1.cmml">∧</mo><mi id="Thmproofx2.p1.6.m6.1.1.3.3" xref="Thmproofx2.p1.6.m6.1.1.3.3.cmml">φ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.6.m6.1b"><apply id="Thmproofx2.p1.6.m6.1.1.cmml" xref="Thmproofx2.p1.6.m6.1.1"><apply id="Thmproofx2.p1.6.m6.1.1.1.cmml" xref="Thmproofx2.p1.6.m6.1.1.1"><csymbol cd="ambiguous" id="Thmproofx2.p1.6.m6.1.1.1.1.cmml" xref="Thmproofx2.p1.6.m6.1.1.1">superscript</csymbol><eq id="Thmproofx2.p1.6.m6.1.1.1.2.cmml" xref="Thmproofx2.p1.6.m6.1.1.1.2"></eq><ci id="Thmproofx2.p1.6.m6.1.1.1.3a.cmml" xref="Thmproofx2.p1.6.m6.1.1.1.3"><mtext id="Thmproofx2.p1.6.m6.1.1.1.3.cmml" mathsize="50%" xref="Thmproofx2.p1.6.m6.1.1.1.3">def</mtext></ci></apply><apply id="Thmproofx2.p1.6.m6.1.1.2.cmml" xref="Thmproofx2.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="Thmproofx2.p1.6.m6.1.1.2.1.cmml" xref="Thmproofx2.p1.6.m6.1.1.2">subscript</csymbol><ci id="Thmproofx2.p1.6.m6.1.1.2.2.cmml" xref="Thmproofx2.p1.6.m6.1.1.2.2">𝜑</ci><cn id="Thmproofx2.p1.6.m6.1.1.2.3.cmml" type="integer" xref="Thmproofx2.p1.6.m6.1.1.2.3">2</cn></apply><apply id="Thmproofx2.p1.6.m6.1.1.3.cmml" xref="Thmproofx2.p1.6.m6.1.1.3"><and id="Thmproofx2.p1.6.m6.1.1.3.1.cmml" xref="Thmproofx2.p1.6.m6.1.1.3.1"></and><apply id="Thmproofx2.p1.6.m6.1.1.3.2.cmml" xref="Thmproofx2.p1.6.m6.1.1.3.2"><and id="Thmproofx2.p1.6.m6.1.1.3.2.1.cmml" xref="Thmproofx2.p1.6.m6.1.1.3.2.1"></and><ci id="Thmproofx2.p1.6.m6.1.1.3.2.2.cmml" xref="Thmproofx2.p1.6.m6.1.1.3.2.2">𝜇</ci></apply><ci id="Thmproofx2.p1.6.m6.1.1.3.3.cmml" xref="Thmproofx2.p1.6.m6.1.1.3.3">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.6.m6.1c">\varphi_{2}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\bigwedge\!\mu\wedge\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.6.m6.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ⋀ italic_μ ∧ italic_φ</annotation></semantics></math>. Then we have that: <br class="ltx_break"/><math alttext="\varphi_{1}\equiv\varphi_{2}" class="ltx_Math" display="inline" id="Thmproofx2.p1.7.m7.1"><semantics id="Thmproofx2.p1.7.m7.1a"><mrow id="Thmproofx2.p1.7.m7.1.1" xref="Thmproofx2.p1.7.m7.1.1.cmml"><msub id="Thmproofx2.p1.7.m7.1.1.2" xref="Thmproofx2.p1.7.m7.1.1.2.cmml"><mi id="Thmproofx2.p1.7.m7.1.1.2.2" xref="Thmproofx2.p1.7.m7.1.1.2.2.cmml">φ</mi><mn id="Thmproofx2.p1.7.m7.1.1.2.3" xref="Thmproofx2.p1.7.m7.1.1.2.3.cmml">1</mn></msub><mo id="Thmproofx2.p1.7.m7.1.1.1" xref="Thmproofx2.p1.7.m7.1.1.1.cmml">≡</mo><msub id="Thmproofx2.p1.7.m7.1.1.3" xref="Thmproofx2.p1.7.m7.1.1.3.cmml"><mi id="Thmproofx2.p1.7.m7.1.1.3.2" xref="Thmproofx2.p1.7.m7.1.1.3.2.cmml">φ</mi><mn id="Thmproofx2.p1.7.m7.1.1.3.3" xref="Thmproofx2.p1.7.m7.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.7.m7.1b"><apply id="Thmproofx2.p1.7.m7.1.1.cmml" xref="Thmproofx2.p1.7.m7.1.1"><equivalent id="Thmproofx2.p1.7.m7.1.1.1.cmml" xref="Thmproofx2.p1.7.m7.1.1.1"></equivalent><apply id="Thmproofx2.p1.7.m7.1.1.2.cmml" xref="Thmproofx2.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="Thmproofx2.p1.7.m7.1.1.2.1.cmml" xref="Thmproofx2.p1.7.m7.1.1.2">subscript</csymbol><ci id="Thmproofx2.p1.7.m7.1.1.2.2.cmml" xref="Thmproofx2.p1.7.m7.1.1.2.2">𝜑</ci><cn id="Thmproofx2.p1.7.m7.1.1.2.3.cmml" type="integer" xref="Thmproofx2.p1.7.m7.1.1.2.3">1</cn></apply><apply id="Thmproofx2.p1.7.m7.1.1.3.cmml" xref="Thmproofx2.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="Thmproofx2.p1.7.m7.1.1.3.1.cmml" xref="Thmproofx2.p1.7.m7.1.1.3">subscript</csymbol><ci id="Thmproofx2.p1.7.m7.1.1.3.2.cmml" xref="Thmproofx2.p1.7.m7.1.1.3.2">𝜑</ci><cn id="Thmproofx2.p1.7.m7.1.1.3.3.cmml" type="integer" xref="Thmproofx2.p1.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.7.m7.1c">\varphi_{1}\equiv\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.7.m7.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≡ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, because <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmproofx2.p1.8.m8.1"><semantics id="Thmproofx2.p1.8.m8.1a"><mrow id="Thmproofx2.p1.8.m8.1.1" xref="Thmproofx2.p1.8.m8.1.1.cmml"><mi id="Thmproofx2.p1.8.m8.1.1.2" xref="Thmproofx2.p1.8.m8.1.1.2.cmml">μ</mi><mo id="Thmproofx2.p1.8.m8.1.1.1" xref="Thmproofx2.p1.8.m8.1.1.1.cmml">⊧</mo><mi id="Thmproofx2.p1.8.m8.1.1.3" xref="Thmproofx2.p1.8.m8.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.8.m8.1b"><apply id="Thmproofx2.p1.8.m8.1.1.cmml" xref="Thmproofx2.p1.8.m8.1.1"><csymbol cd="latexml" id="Thmproofx2.p1.8.m8.1.1.1.cmml" xref="Thmproofx2.p1.8.m8.1.1.1">models</csymbol><ci id="Thmproofx2.p1.8.m8.1.1.2.cmml" xref="Thmproofx2.p1.8.m8.1.1.2">𝜇</ci><ci id="Thmproofx2.p1.8.m8.1.1.3.cmml" xref="Thmproofx2.p1.8.m8.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.8.m8.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.8.m8.1d">italic_μ ⊧ italic_φ</annotation></semantics></math>, and hence <math alttext="\bigwedge\!\mu\models\varphi" class="ltx_Math" display="inline" id="Thmproofx2.p1.9.m9.1"><semantics id="Thmproofx2.p1.9.m9.1a"><mrow id="Thmproofx2.p1.9.m9.1.1" xref="Thmproofx2.p1.9.m9.1.1.cmml"><mrow id="Thmproofx2.p1.9.m9.1.1.2" xref="Thmproofx2.p1.9.m9.1.1.2.cmml"><mpadded width="0.747em"><mo id="Thmproofx2.p1.9.m9.1.1.2.1" xref="Thmproofx2.p1.9.m9.1.1.2.1.cmml">⋀</mo></mpadded><mi id="Thmproofx2.p1.9.m9.1.1.2.2" xref="Thmproofx2.p1.9.m9.1.1.2.2.cmml">μ</mi></mrow><mo id="Thmproofx2.p1.9.m9.1.1.1" xref="Thmproofx2.p1.9.m9.1.1.1.cmml">⊧</mo><mi id="Thmproofx2.p1.9.m9.1.1.3" xref="Thmproofx2.p1.9.m9.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx2.p1.9.m9.1b"><apply id="Thmproofx2.p1.9.m9.1.1.cmml" xref="Thmproofx2.p1.9.m9.1.1"><csymbol cd="latexml" id="Thmproofx2.p1.9.m9.1.1.1.cmml" xref="Thmproofx2.p1.9.m9.1.1.1">models</csymbol><apply id="Thmproofx2.p1.9.m9.1.1.2.cmml" xref="Thmproofx2.p1.9.m9.1.1.2"><and id="Thmproofx2.p1.9.m9.1.1.2.1.cmml" xref="Thmproofx2.p1.9.m9.1.1.2.1"></and><ci id="Thmproofx2.p1.9.m9.1.1.2.2.cmml" xref="Thmproofx2.p1.9.m9.1.1.2.2">𝜇</ci></apply><ci id="Thmproofx2.p1.9.m9.1.1.3.cmml" xref="Thmproofx2.p1.9.m9.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx2.p1.9.m9.1c">\bigwedge\!\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.9.m9.1d">⋀ italic_μ ⊧ italic_φ</annotation></semantics></math> because of Property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty6" title="Property 6 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">6</span></a>(i); <br class="ltx_break"/><math alttext="\mu\mid\!\simeq\varphi_{1}" class="ltx_math_unparsed" display="inline" id="Thmproofx2.p1.10.m10.1"><semantics id="Thmproofx2.p1.10.m10.1a"><mrow id="Thmproofx2.p1.10.m10.1b"><mi id="Thmproofx2.p1.10.m10.1.1">μ</mi><mpadded id="Thmproofx2.p1.10.m10.1c" width="0.219em"><mo id="Thmproofx2.p1.10.m10.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx2.p1.10.m10.1.3">≃</mo><msub id="Thmproofx2.p1.10.m10.1.4"><mi id="Thmproofx2.p1.10.m10.1.4.2">φ</mi><mn id="Thmproofx2.p1.10.m10.1.4.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="Thmproofx2.p1.10.m10.1d">\mu\mid\!\simeq\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.10.m10.1e">italic_μ ∣ ≃ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, because of hypothesis <span class="ltx_text ltx_font_italic" id="Thmproofx2.p1.12.2">(b)</span>; <br class="ltx_break"/><math alttext="\mu\not\mid\!\simeq\varphi_{2}" class="ltx_math_unparsed" display="inline" id="Thmproofx2.p1.11.m11.1"><semantics id="Thmproofx2.p1.11.m11.1a"><mrow id="Thmproofx2.p1.11.m11.1b"><mi id="Thmproofx2.p1.11.m11.1.1">μ</mi><mpadded id="Thmproofx2.p1.11.m11.1c" width="0.969em"><mo id="Thmproofx2.p1.11.m11.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmproofx2.p1.11.m11.1.3">≃</mo><msub id="Thmproofx2.p1.11.m11.1.4"><mi id="Thmproofx2.p1.11.m11.1.4.2">φ</mi><mn id="Thmproofx2.p1.11.m11.1.4.3">2</mn></msub></mrow><annotation encoding="application/x-tex" id="Thmproofx2.p1.11.m11.1d">\mu\not\mid\!\simeq\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx2.p1.11.m11.1e">italic_μ ∤ ≃ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, because of hypothesis <span class="ltx_text ltx_font_italic" id="Thmproofx2.p1.12.3">(c)</span>. <br class="ltx_break"/>The latter three facts violate hypothesis <span class="ltx_text ltx_font_italic" id="Thmproofx2.p1.12.4">(d)</span>. <span class="ltx_text ltx_markedasmath" id="Thmproofx2.p1.12.5">∎</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.7">Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.Thmtheorem2" title="Theorem 3.2 ‣ 3.2 Other candidate forms of partial-assignment satisfaction. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.2</span></a> says that entailment is the only relation which <math alttext="(a)" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mrow id="S3.SS2.p2.1.m1.1.2.2"><mo id="S3.SS2.p2.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml">a</mi><mo id="S3.SS2.p2.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><ci id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">𝑎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">(a)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">( italic_a )</annotation></semantics></math> is stronger or equal than entailment, <math alttext="(b)" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mrow id="S3.SS2.p2.2.m2.1.2.2"><mo id="S3.SS2.p2.2.m2.1.2.2.1" stretchy="false">(</mo><mi id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml">b</mi><mo id="S3.SS2.p2.2.m2.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><ci id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">(b)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">( italic_b )</annotation></semantics></math> extends to partial assignments the standard notion of total-assignment satisfaction, <math alttext="(c)" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.1"><semantics id="S3.SS2.p2.3.m3.1a"><mrow id="S3.SS2.p2.3.m3.1.2.2"><mo id="S3.SS2.p2.3.m3.1.2.2.1" stretchy="false">(</mo><mi id="S3.SS2.p2.3.m3.1.1" xref="S3.SS2.p2.3.m3.1.1.cmml">c</mi><mo id="S3.SS2.p2.3.m3.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.1b"><ci id="S3.SS2.p2.3.m3.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m3.1c">(c)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m3.1d">( italic_c )</annotation></semantics></math> maintains the standard semantics of <math alttext="\wedge" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m4.1"><semantics id="S3.SS2.p2.4.m4.1a"><mo id="S3.SS2.p2.4.m4.1.1" xref="S3.SS2.p2.4.m4.1.1.cmml">∧</mo><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.1b"><and id="S3.SS2.p2.4.m4.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1"></and></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.1c">\wedge</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.1d">∧</annotation></semantics></math>, and <math alttext="(d)" class="ltx_Math" display="inline" id="S3.SS2.p2.5.m5.1"><semantics id="S3.SS2.p2.5.m5.1a"><mrow id="S3.SS2.p2.5.m5.1.2.2"><mo id="S3.SS2.p2.5.m5.1.2.2.1" stretchy="false">(</mo><mi id="S3.SS2.p2.5.m5.1.1" xref="S3.SS2.p2.5.m5.1.1.cmml">d</mi><mo id="S3.SS2.p2.5.m5.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.5.m5.1b"><ci id="S3.SS2.p2.5.m5.1.1.cmml" xref="S3.SS2.p2.5.m5.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.5.m5.1c">(d)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.5.m5.1d">( italic_d )</annotation></semantics></math> verifies Property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty6" title="Property 6 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">6</span></a>(ii). Thus, any such relation <math alttext="\mid\!\simeq" class="ltx_math_unparsed" display="inline" id="S3.SS2.p2.6.m6.1"><semantics id="S3.SS2.p2.6.m6.1a"><mrow id="S3.SS2.p2.6.m6.1b"><mpadded id="S3.SS2.p2.6.m6.1c" width="0.219em"><mo id="S3.SS2.p2.6.m6.1.1">∣</mo></mpadded><mo id="S3.SS2.p2.6.m6.1.2">≃</mo></mrow><annotation encoding="application/x-tex" id="S3.SS2.p2.6.m6.1d">\mid\!\simeq</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.6.m6.1e">∣ ≃</annotation></semantics></math> which is strictly stronger that <math alttext="\models" class="ltx_Math" display="inline" id="S3.SS2.p2.7.m7.1"><semantics id="S3.SS2.p2.7.m7.1a"><mo id="S3.SS2.p2.7.m7.1.1" xref="S3.SS2.p2.7.m7.1.1.cmml">⊧</mo><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.7.m7.1b"><csymbol cd="latexml" id="S3.SS2.p2.7.m7.1.1.cmml" xref="S3.SS2.p2.7.m7.1.1">models</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.7.m7.1c">\models</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.7.m7.1d">⊧</annotation></semantics></math> suffers for the same “embarrassing fact” of Property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty5" title="Property 5 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">5</span></a>(ii).</p> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample3"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 3</span></h6> <div class="ltx_para" id="Thmexample3.p1"> <p class="ltx_p" id="Thmexample3.p1.8">Let “<math alttext="\mid\!\simeq" class="ltx_math_unparsed" display="inline" id="Thmexample3.p1.1.m1.1"><semantics id="Thmexample3.p1.1.m1.1a"><mrow id="Thmexample3.p1.1.m1.1b"><mpadded id="Thmexample3.p1.1.m1.1c" width="0.219em"><mo id="Thmexample3.p1.1.m1.1.1">∣</mo></mpadded><mo id="Thmexample3.p1.1.m1.1.2">≃</mo></mrow><annotation encoding="application/x-tex" id="Thmexample3.p1.1.m1.1d">\mid\!\simeq</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p1.1.m1.1e">∣ ≃</annotation></semantics></math>” denote a variant of <math alttext="\mid\!\approx" class="ltx_math_unparsed" display="inline" id="Thmexample3.p1.2.m2.1"><semantics id="Thmexample3.p1.2.m2.1a"><mrow id="Thmexample3.p1.2.m2.1b"><mpadded id="Thmexample3.p1.2.m2.1c" width="0.219em"><mo id="Thmexample3.p1.2.m2.1.1">∣</mo></mpadded><mo id="Thmexample3.p1.2.m2.1.2">≈</mo></mrow><annotation encoding="application/x-tex" id="Thmexample3.p1.2.m2.1d">\mid\!\approx</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p1.2.m2.1e">∣ ≈</annotation></semantics></math> such that <math alttext="\mu\mid\!\simeq\varphi" class="ltx_math_unparsed" display="inline" id="Thmexample3.p1.3.m3.1"><semantics id="Thmexample3.p1.3.m3.1a"><mrow id="Thmexample3.p1.3.m3.1b"><mi id="Thmexample3.p1.3.m3.1.1">μ</mi><mpadded id="Thmexample3.p1.3.m3.1c" width="0.219em"><mo id="Thmexample3.p1.3.m3.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmexample3.p1.3.m3.1.3">≃</mo><mi id="Thmexample3.p1.3.m3.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmexample3.p1.3.m3.1d">\mu\mid\!\simeq\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p1.3.m3.1e">italic_μ ∣ ≃ italic_φ</annotation></semantics></math> implements a syntactic check which adds “<math alttext="l\vee\neg l\Rightarrow\top" class="ltx_Math" display="inline" id="Thmexample3.p1.4.m4.1"><semantics id="Thmexample3.p1.4.m4.1a"><mrow id="Thmexample3.p1.4.m4.1.1" xref="Thmexample3.p1.4.m4.1.1.cmml"><mrow id="Thmexample3.p1.4.m4.1.1.2" xref="Thmexample3.p1.4.m4.1.1.2.cmml"><mi id="Thmexample3.p1.4.m4.1.1.2.2" xref="Thmexample3.p1.4.m4.1.1.2.2.cmml">l</mi><mo id="Thmexample3.p1.4.m4.1.1.2.1" xref="Thmexample3.p1.4.m4.1.1.2.1.cmml">∨</mo><mrow id="Thmexample3.p1.4.m4.1.1.2.3" xref="Thmexample3.p1.4.m4.1.1.2.3.cmml"><mo id="Thmexample3.p1.4.m4.1.1.2.3.1" rspace="0.167em" xref="Thmexample3.p1.4.m4.1.1.2.3.1.cmml">¬</mo><mi id="Thmexample3.p1.4.m4.1.1.2.3.2" xref="Thmexample3.p1.4.m4.1.1.2.3.2.cmml">l</mi></mrow></mrow><mo id="Thmexample3.p1.4.m4.1.1.1" rspace="0em" stretchy="false" xref="Thmexample3.p1.4.m4.1.1.1.cmml">⇒</mo><mo id="Thmexample3.p1.4.m4.1.1.3" lspace="0em" xref="Thmexample3.p1.4.m4.1.1.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p1.4.m4.1b"><apply id="Thmexample3.p1.4.m4.1.1.cmml" xref="Thmexample3.p1.4.m4.1.1"><ci id="Thmexample3.p1.4.m4.1.1.1.cmml" xref="Thmexample3.p1.4.m4.1.1.1">⇒</ci><apply id="Thmexample3.p1.4.m4.1.1.2.cmml" xref="Thmexample3.p1.4.m4.1.1.2"><or id="Thmexample3.p1.4.m4.1.1.2.1.cmml" xref="Thmexample3.p1.4.m4.1.1.2.1"></or><ci id="Thmexample3.p1.4.m4.1.1.2.2.cmml" xref="Thmexample3.p1.4.m4.1.1.2.2">𝑙</ci><apply id="Thmexample3.p1.4.m4.1.1.2.3.cmml" xref="Thmexample3.p1.4.m4.1.1.2.3"><not id="Thmexample3.p1.4.m4.1.1.2.3.1.cmml" xref="Thmexample3.p1.4.m4.1.1.2.3.1"></not><ci id="Thmexample3.p1.4.m4.1.1.2.3.2.cmml" xref="Thmexample3.p1.4.m4.1.1.2.3.2">𝑙</ci></apply></apply><csymbol cd="latexml" id="Thmexample3.p1.4.m4.1.1.3.cmml" xref="Thmexample3.p1.4.m4.1.1.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p1.4.m4.1c">l\vee\neg l\Rightarrow\top</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p1.4.m4.1d">italic_l ∨ ¬ italic_l ⇒ ⊤</annotation></semantics></math>”, <math alttext="l" class="ltx_Math" display="inline" id="Thmexample3.p1.5.m5.1"><semantics id="Thmexample3.p1.5.m5.1a"><mi id="Thmexample3.p1.5.m5.1.1" xref="Thmexample3.p1.5.m5.1.1.cmml">l</mi><annotation-xml encoding="MathML-Content" id="Thmexample3.p1.5.m5.1b"><ci id="Thmexample3.p1.5.m5.1.1.cmml" xref="Thmexample3.p1.5.m5.1.1">𝑙</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p1.5.m5.1c">l</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p1.5.m5.1d">italic_l</annotation></semantics></math> being a literal, to the rules used in <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.F2" title="In Semantics. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">fig.</span> <span class="ltx_text ltx_ref_tag">2</span></a> for computing the residual <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="Thmexample3.p1.6.m6.2"><semantics id="Thmexample3.p1.6.m6.2a"><msub id="Thmexample3.p1.6.m6.2.3.2" xref="Thmexample3.p1.6.m6.2.3.1.cmml"><mrow id="Thmexample3.p1.6.m6.2.3.2.2" xref="Thmexample3.p1.6.m6.2.3.1.cmml"><mi id="Thmexample3.p1.6.m6.1.1" xref="Thmexample3.p1.6.m6.1.1.cmml">φ</mi><mo id="Thmexample3.p1.6.m6.2.3.2.2.1" stretchy="false" xref="Thmexample3.p1.6.m6.2.3.1.1.cmml">|</mo></mrow><mi id="Thmexample3.p1.6.m6.2.2.1" xref="Thmexample3.p1.6.m6.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmexample3.p1.6.m6.2b"><apply id="Thmexample3.p1.6.m6.2.3.1.cmml" xref="Thmexample3.p1.6.m6.2.3.2"><csymbol cd="latexml" id="Thmexample3.p1.6.m6.2.3.1.1.cmml" xref="Thmexample3.p1.6.m6.2.3.2.2.1">evaluated-at</csymbol><ci id="Thmexample3.p1.6.m6.1.1.cmml" xref="Thmexample3.p1.6.m6.1.1">𝜑</ci><ci id="Thmexample3.p1.6.m6.2.2.1.cmml" xref="Thmexample3.p1.6.m6.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p1.6.m6.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p1.6.m6.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math>. Thus, considering the formula <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="Thmexample3.p1.7.m7.1"><semantics id="Thmexample3.p1.7.m7.1a"><msub id="Thmexample3.p1.7.m7.1.1" xref="Thmexample3.p1.7.m7.1.1.cmml"><mi id="Thmexample3.p1.7.m7.1.1.2" xref="Thmexample3.p1.7.m7.1.1.2.cmml">φ</mi><mn id="Thmexample3.p1.7.m7.1.1.3" xref="Thmexample3.p1.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="Thmexample3.p1.7.m7.1b"><apply id="Thmexample3.p1.7.m7.1.1.cmml" xref="Thmexample3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="Thmexample3.p1.7.m7.1.1.1.cmml" xref="Thmexample3.p1.7.m7.1.1">subscript</csymbol><ci id="Thmexample3.p1.7.m7.1.1.2.cmml" xref="Thmexample3.p1.7.m7.1.1.2">𝜑</ci><cn id="Thmexample3.p1.7.m7.1.1.3.cmml" type="integer" xref="Thmexample3.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p1.7.m7.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p1.7.m7.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> of <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample2" title="Example 2 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">2</span></a> we have <math alttext="\mu\mid\!\simeq\varphi_{1}" class="ltx_math_unparsed" display="inline" id="Thmexample3.p1.8.m8.1"><semantics id="Thmexample3.p1.8.m8.1a"><mrow id="Thmexample3.p1.8.m8.1b"><mi id="Thmexample3.p1.8.m8.1.1">μ</mi><mpadded id="Thmexample3.p1.8.m8.1c" width="0.219em"><mo id="Thmexample3.p1.8.m8.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmexample3.p1.8.m8.1.3">≃</mo><msub id="Thmexample3.p1.8.m8.1.4"><mi id="Thmexample3.p1.8.m8.1.4.2">φ</mi><mn id="Thmexample3.p1.8.m8.1.4.3">1</mn></msub></mrow><annotation encoding="application/x-tex" id="Thmexample3.p1.8.m8.1d">\mu\mid\!\simeq\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p1.8.m8.1e">italic_μ ∣ ≃ italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmexample3.p2"> <p class="ltx_p" id="Thmexample3.p2.17">Nevertheless, let <math alttext="\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1},..,A_{M}}\}" class="ltx_math_unparsed" display="inline" id="Thmexample3.p2.1.m1.1"><semantics id="Thmexample3.p2.1.m1.1a"><mrow id="Thmexample3.p2.1.m1.1b"><mi id="Thmexample3.p2.1.m1.1.1">μ</mi><mover id="Thmexample3.p2.1.m1.1.2"><mo id="Thmexample3.p2.1.m1.1.2.2">=</mo><mtext id="Thmexample3.p2.1.m1.1.2.3" mathsize="71%">def</mtext></mover><mrow id="Thmexample3.p2.1.m1.1.3"><mo id="Thmexample3.p2.1.m1.1.3.1" stretchy="false">{</mo><msub id="Thmexample3.p2.1.m1.1.3.2"><mi id="Thmexample3.p2.1.m1.1.3.2.2">A</mi><mn id="Thmexample3.p2.1.m1.1.3.2.3">1</mn></msub><mo id="Thmexample3.p2.1.m1.1.3.3">,</mo><mo id="Thmexample3.p2.1.m1.1.3.4" lspace="0em" rspace="0.0835em">.</mo><mo id="Thmexample3.p2.1.m1.1.3.5" lspace="0.0835em" rspace="0.167em">.</mo><mo id="Thmexample3.p2.1.m1.1.3.6">,</mo><msub id="Thmexample3.p2.1.m1.1.3.7"><mi id="Thmexample3.p2.1.m1.1.3.7.2">A</mi><mi id="Thmexample3.p2.1.m1.1.3.7.3">M</mi></msub><mo id="Thmexample3.p2.1.m1.1.3.8" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmexample3.p2.1.m1.1c">\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1},..,A_{M}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.1.m1.1d">italic_μ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , . . , italic_A start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT }</annotation></semantics></math> s.t. <math alttext="\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\bigvee_{i=1}^{M}(A_{i}% \wedge cube_{i})" class="ltx_Math" display="inline" id="Thmexample3.p2.2.m2.1"><semantics id="Thmexample3.p2.2.m2.1a"><mrow id="Thmexample3.p2.2.m2.1.1" xref="Thmexample3.p2.2.m2.1.1.cmml"><mi id="Thmexample3.p2.2.m2.1.1.3" xref="Thmexample3.p2.2.m2.1.1.3.cmml">φ</mi><mover id="Thmexample3.p2.2.m2.1.1.2" xref="Thmexample3.p2.2.m2.1.1.2.cmml"><mo id="Thmexample3.p2.2.m2.1.1.2.2" rspace="0.111em" xref="Thmexample3.p2.2.m2.1.1.2.2.cmml">=</mo><mtext id="Thmexample3.p2.2.m2.1.1.2.3" mathsize="71%" xref="Thmexample3.p2.2.m2.1.1.2.3a.cmml">def</mtext></mover><mrow id="Thmexample3.p2.2.m2.1.1.1" xref="Thmexample3.p2.2.m2.1.1.1.cmml"><msubsup id="Thmexample3.p2.2.m2.1.1.1.2" xref="Thmexample3.p2.2.m2.1.1.1.2.cmml"><mo id="Thmexample3.p2.2.m2.1.1.1.2.2.2" rspace="0em" xref="Thmexample3.p2.2.m2.1.1.1.2.2.2.cmml">⋁</mo><mrow id="Thmexample3.p2.2.m2.1.1.1.2.2.3" xref="Thmexample3.p2.2.m2.1.1.1.2.2.3.cmml"><mi id="Thmexample3.p2.2.m2.1.1.1.2.2.3.2" xref="Thmexample3.p2.2.m2.1.1.1.2.2.3.2.cmml">i</mi><mo id="Thmexample3.p2.2.m2.1.1.1.2.2.3.1" xref="Thmexample3.p2.2.m2.1.1.1.2.2.3.1.cmml">=</mo><mn id="Thmexample3.p2.2.m2.1.1.1.2.2.3.3" xref="Thmexample3.p2.2.m2.1.1.1.2.2.3.3.cmml">1</mn></mrow><mi id="Thmexample3.p2.2.m2.1.1.1.2.3" xref="Thmexample3.p2.2.m2.1.1.1.2.3.cmml">M</mi></msubsup><mrow id="Thmexample3.p2.2.m2.1.1.1.1.1" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.cmml"><mo id="Thmexample3.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample3.p2.2.m2.1.1.1.1.1.1" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.cmml"><msub id="Thmexample3.p2.2.m2.1.1.1.1.1.1.2" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.cmml"><mi id="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.2" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.2.cmml">A</mi><mi id="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.3" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="Thmexample3.p2.2.m2.1.1.1.1.1.1.1" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.1.cmml">∧</mo><mrow id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.cmml"><mi id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.2" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.2.cmml">c</mi><mo id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.1" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.3" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.3.cmml">u</mi><mo id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.1a" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.4" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.4.cmml">b</mi><mo id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.1b" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.cmml"><mi id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.2" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.2.cmml">e</mi><mi id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.3" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.3.cmml">i</mi></msub></mrow></mrow><mo id="Thmexample3.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.2.m2.1b"><apply id="Thmexample3.p2.2.m2.1.1.cmml" xref="Thmexample3.p2.2.m2.1.1"><apply id="Thmexample3.p2.2.m2.1.1.2.cmml" xref="Thmexample3.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="Thmexample3.p2.2.m2.1.1.2.1.cmml" xref="Thmexample3.p2.2.m2.1.1.2">superscript</csymbol><eq id="Thmexample3.p2.2.m2.1.1.2.2.cmml" xref="Thmexample3.p2.2.m2.1.1.2.2"></eq><ci id="Thmexample3.p2.2.m2.1.1.2.3a.cmml" xref="Thmexample3.p2.2.m2.1.1.2.3"><mtext id="Thmexample3.p2.2.m2.1.1.2.3.cmml" mathsize="50%" xref="Thmexample3.p2.2.m2.1.1.2.3">def</mtext></ci></apply><ci id="Thmexample3.p2.2.m2.1.1.3.cmml" xref="Thmexample3.p2.2.m2.1.1.3">𝜑</ci><apply id="Thmexample3.p2.2.m2.1.1.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1"><apply id="Thmexample3.p2.2.m2.1.1.1.2.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample3.p2.2.m2.1.1.1.2.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2">superscript</csymbol><apply id="Thmexample3.p2.2.m2.1.1.1.2.2.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample3.p2.2.m2.1.1.1.2.2.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2">subscript</csymbol><or id="Thmexample3.p2.2.m2.1.1.1.2.2.2.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2.2.2"></or><apply id="Thmexample3.p2.2.m2.1.1.1.2.2.3.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2.2.3"><eq id="Thmexample3.p2.2.m2.1.1.1.2.2.3.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2.2.3.1"></eq><ci id="Thmexample3.p2.2.m2.1.1.1.2.2.3.2.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2.2.3.2">𝑖</ci><cn id="Thmexample3.p2.2.m2.1.1.1.2.2.3.3.cmml" type="integer" xref="Thmexample3.p2.2.m2.1.1.1.2.2.3.3">1</cn></apply></apply><ci id="Thmexample3.p2.2.m2.1.1.1.2.3.cmml" xref="Thmexample3.p2.2.m2.1.1.1.2.3">𝑀</ci></apply><apply id="Thmexample3.p2.2.m2.1.1.1.1.1.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1"><and id="Thmexample3.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.1"></and><apply id="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.2">subscript</csymbol><ci id="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.2.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.2">𝐴</ci><ci id="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.3.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3"><times id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.1"></times><ci id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.2.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.2">𝑐</ci><ci id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.3.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.3">𝑢</ci><ci id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.4.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.4">𝑏</ci><apply id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5"><csymbol cd="ambiguous" id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.1.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5">subscript</csymbol><ci id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.2.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.2">𝑒</ci><ci id="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.3.cmml" xref="Thmexample3.p2.2.m2.1.1.1.1.1.1.3.5.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.2.m2.1c">\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\bigvee_{i=1}^{M}(A_{i}% \wedge cube_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.2.m2.1d">italic_φ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ⋁ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT ( italic_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∧ italic_c italic_u italic_b italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> s.t. each <math alttext="cube_{i}" class="ltx_Math" display="inline" id="Thmexample3.p2.3.m3.1"><semantics id="Thmexample3.p2.3.m3.1a"><mrow id="Thmexample3.p2.3.m3.1.1" xref="Thmexample3.p2.3.m3.1.1.cmml"><mi id="Thmexample3.p2.3.m3.1.1.2" xref="Thmexample3.p2.3.m3.1.1.2.cmml">c</mi><mo id="Thmexample3.p2.3.m3.1.1.1" xref="Thmexample3.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="Thmexample3.p2.3.m3.1.1.3" xref="Thmexample3.p2.3.m3.1.1.3.cmml">u</mi><mo id="Thmexample3.p2.3.m3.1.1.1a" xref="Thmexample3.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="Thmexample3.p2.3.m3.1.1.4" xref="Thmexample3.p2.3.m3.1.1.4.cmml">b</mi><mo id="Thmexample3.p2.3.m3.1.1.1b" xref="Thmexample3.p2.3.m3.1.1.1.cmml">⁢</mo><msub id="Thmexample3.p2.3.m3.1.1.5" xref="Thmexample3.p2.3.m3.1.1.5.cmml"><mi id="Thmexample3.p2.3.m3.1.1.5.2" xref="Thmexample3.p2.3.m3.1.1.5.2.cmml">e</mi><mi id="Thmexample3.p2.3.m3.1.1.5.3" xref="Thmexample3.p2.3.m3.1.1.5.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.3.m3.1b"><apply id="Thmexample3.p2.3.m3.1.1.cmml" xref="Thmexample3.p2.3.m3.1.1"><times id="Thmexample3.p2.3.m3.1.1.1.cmml" xref="Thmexample3.p2.3.m3.1.1.1"></times><ci id="Thmexample3.p2.3.m3.1.1.2.cmml" xref="Thmexample3.p2.3.m3.1.1.2">𝑐</ci><ci id="Thmexample3.p2.3.m3.1.1.3.cmml" xref="Thmexample3.p2.3.m3.1.1.3">𝑢</ci><ci id="Thmexample3.p2.3.m3.1.1.4.cmml" xref="Thmexample3.p2.3.m3.1.1.4">𝑏</ci><apply id="Thmexample3.p2.3.m3.1.1.5.cmml" xref="Thmexample3.p2.3.m3.1.1.5"><csymbol cd="ambiguous" id="Thmexample3.p2.3.m3.1.1.5.1.cmml" xref="Thmexample3.p2.3.m3.1.1.5">subscript</csymbol><ci id="Thmexample3.p2.3.m3.1.1.5.2.cmml" xref="Thmexample3.p2.3.m3.1.1.5.2">𝑒</ci><ci id="Thmexample3.p2.3.m3.1.1.5.3.cmml" xref="Thmexample3.p2.3.m3.1.1.5.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.3.m3.1c">cube_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.3.m3.1d">italic_c italic_u italic_b italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a non-unary cube and <math alttext="\bigvee_{i=1}^{M}cube_{i}" class="ltx_Math" display="inline" id="Thmexample3.p2.4.m4.1"><semantics id="Thmexample3.p2.4.m4.1a"><mrow id="Thmexample3.p2.4.m4.1.1" xref="Thmexample3.p2.4.m4.1.1.cmml"><msubsup id="Thmexample3.p2.4.m4.1.1.1" xref="Thmexample3.p2.4.m4.1.1.1.cmml"><mo id="Thmexample3.p2.4.m4.1.1.1.2.2" xref="Thmexample3.p2.4.m4.1.1.1.2.2.cmml">⋁</mo><mrow id="Thmexample3.p2.4.m4.1.1.1.2.3" xref="Thmexample3.p2.4.m4.1.1.1.2.3.cmml"><mi id="Thmexample3.p2.4.m4.1.1.1.2.3.2" xref="Thmexample3.p2.4.m4.1.1.1.2.3.2.cmml">i</mi><mo id="Thmexample3.p2.4.m4.1.1.1.2.3.1" xref="Thmexample3.p2.4.m4.1.1.1.2.3.1.cmml">=</mo><mn id="Thmexample3.p2.4.m4.1.1.1.2.3.3" xref="Thmexample3.p2.4.m4.1.1.1.2.3.3.cmml">1</mn></mrow><mi id="Thmexample3.p2.4.m4.1.1.1.3" xref="Thmexample3.p2.4.m4.1.1.1.3.cmml">M</mi></msubsup><mrow id="Thmexample3.p2.4.m4.1.1.2" xref="Thmexample3.p2.4.m4.1.1.2.cmml"><mi id="Thmexample3.p2.4.m4.1.1.2.2" xref="Thmexample3.p2.4.m4.1.1.2.2.cmml">c</mi><mo id="Thmexample3.p2.4.m4.1.1.2.1" xref="Thmexample3.p2.4.m4.1.1.2.1.cmml">⁢</mo><mi id="Thmexample3.p2.4.m4.1.1.2.3" xref="Thmexample3.p2.4.m4.1.1.2.3.cmml">u</mi><mo id="Thmexample3.p2.4.m4.1.1.2.1a" xref="Thmexample3.p2.4.m4.1.1.2.1.cmml">⁢</mo><mi id="Thmexample3.p2.4.m4.1.1.2.4" xref="Thmexample3.p2.4.m4.1.1.2.4.cmml">b</mi><mo id="Thmexample3.p2.4.m4.1.1.2.1b" xref="Thmexample3.p2.4.m4.1.1.2.1.cmml">⁢</mo><msub id="Thmexample3.p2.4.m4.1.1.2.5" xref="Thmexample3.p2.4.m4.1.1.2.5.cmml"><mi id="Thmexample3.p2.4.m4.1.1.2.5.2" xref="Thmexample3.p2.4.m4.1.1.2.5.2.cmml">e</mi><mi id="Thmexample3.p2.4.m4.1.1.2.5.3" xref="Thmexample3.p2.4.m4.1.1.2.5.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.4.m4.1b"><apply id="Thmexample3.p2.4.m4.1.1.cmml" xref="Thmexample3.p2.4.m4.1.1"><apply id="Thmexample3.p2.4.m4.1.1.1.cmml" xref="Thmexample3.p2.4.m4.1.1.1"><csymbol cd="ambiguous" id="Thmexample3.p2.4.m4.1.1.1.1.cmml" xref="Thmexample3.p2.4.m4.1.1.1">superscript</csymbol><apply id="Thmexample3.p2.4.m4.1.1.1.2.cmml" xref="Thmexample3.p2.4.m4.1.1.1"><csymbol cd="ambiguous" id="Thmexample3.p2.4.m4.1.1.1.2.1.cmml" xref="Thmexample3.p2.4.m4.1.1.1">subscript</csymbol><or id="Thmexample3.p2.4.m4.1.1.1.2.2.cmml" xref="Thmexample3.p2.4.m4.1.1.1.2.2"></or><apply id="Thmexample3.p2.4.m4.1.1.1.2.3.cmml" xref="Thmexample3.p2.4.m4.1.1.1.2.3"><eq id="Thmexample3.p2.4.m4.1.1.1.2.3.1.cmml" xref="Thmexample3.p2.4.m4.1.1.1.2.3.1"></eq><ci id="Thmexample3.p2.4.m4.1.1.1.2.3.2.cmml" xref="Thmexample3.p2.4.m4.1.1.1.2.3.2">𝑖</ci><cn id="Thmexample3.p2.4.m4.1.1.1.2.3.3.cmml" type="integer" xref="Thmexample3.p2.4.m4.1.1.1.2.3.3">1</cn></apply></apply><ci id="Thmexample3.p2.4.m4.1.1.1.3.cmml" xref="Thmexample3.p2.4.m4.1.1.1.3">𝑀</ci></apply><apply id="Thmexample3.p2.4.m4.1.1.2.cmml" xref="Thmexample3.p2.4.m4.1.1.2"><times id="Thmexample3.p2.4.m4.1.1.2.1.cmml" xref="Thmexample3.p2.4.m4.1.1.2.1"></times><ci id="Thmexample3.p2.4.m4.1.1.2.2.cmml" xref="Thmexample3.p2.4.m4.1.1.2.2">𝑐</ci><ci id="Thmexample3.p2.4.m4.1.1.2.3.cmml" xref="Thmexample3.p2.4.m4.1.1.2.3">𝑢</ci><ci id="Thmexample3.p2.4.m4.1.1.2.4.cmml" xref="Thmexample3.p2.4.m4.1.1.2.4">𝑏</ci><apply id="Thmexample3.p2.4.m4.1.1.2.5.cmml" xref="Thmexample3.p2.4.m4.1.1.2.5"><csymbol cd="ambiguous" id="Thmexample3.p2.4.m4.1.1.2.5.1.cmml" xref="Thmexample3.p2.4.m4.1.1.2.5">subscript</csymbol><ci id="Thmexample3.p2.4.m4.1.1.2.5.2.cmml" xref="Thmexample3.p2.4.m4.1.1.2.5.2">𝑒</ci><ci id="Thmexample3.p2.4.m4.1.1.2.5.3.cmml" xref="Thmexample3.p2.4.m4.1.1.2.5.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.4.m4.1c">\bigvee_{i=1}^{M}cube_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.4.m4.1d">⋁ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_M end_POSTSUPERSCRIPT italic_c italic_u italic_b italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is valid and does not contain occurrences of the atoms <math alttext="A_{1},..,A_{M}" class="ltx_math_unparsed" display="inline" id="Thmexample3.p2.5.m5.1"><semantics id="Thmexample3.p2.5.m5.1a"><mrow id="Thmexample3.p2.5.m5.1b"><msub id="Thmexample3.p2.5.m5.1.1"><mi id="Thmexample3.p2.5.m5.1.1.2">A</mi><mn id="Thmexample3.p2.5.m5.1.1.3">1</mn></msub><mo id="Thmexample3.p2.5.m5.1.2">,</mo><mo id="Thmexample3.p2.5.m5.1.3" lspace="0em" rspace="0.0835em">.</mo><mo id="Thmexample3.p2.5.m5.1.4" lspace="0.0835em" rspace="0.167em">.</mo><mo id="Thmexample3.p2.5.m5.1.5">,</mo><msub id="Thmexample3.p2.5.m5.1.6"><mi id="Thmexample3.p2.5.m5.1.6.2">A</mi><mi id="Thmexample3.p2.5.m5.1.6.3">M</mi></msub></mrow><annotation encoding="application/x-tex" id="Thmexample3.p2.5.m5.1c">A_{1},..,A_{M}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.5.m5.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , . . , italic_A start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT</annotation></semantics></math>. Then <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmexample3.p2.6.m6.1"><semantics id="Thmexample3.p2.6.m6.1a"><mrow id="Thmexample3.p2.6.m6.1.1" xref="Thmexample3.p2.6.m6.1.1.cmml"><mi id="Thmexample3.p2.6.m6.1.1.2" xref="Thmexample3.p2.6.m6.1.1.2.cmml">μ</mi><mo id="Thmexample3.p2.6.m6.1.1.1" xref="Thmexample3.p2.6.m6.1.1.1.cmml">⊧</mo><mi id="Thmexample3.p2.6.m6.1.1.3" xref="Thmexample3.p2.6.m6.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.6.m6.1b"><apply id="Thmexample3.p2.6.m6.1.1.cmml" xref="Thmexample3.p2.6.m6.1.1"><csymbol cd="latexml" id="Thmexample3.p2.6.m6.1.1.1.cmml" xref="Thmexample3.p2.6.m6.1.1.1">models</csymbol><ci id="Thmexample3.p2.6.m6.1.1.2.cmml" xref="Thmexample3.p2.6.m6.1.1.2">𝜇</ci><ci id="Thmexample3.p2.6.m6.1.1.3.cmml" xref="Thmexample3.p2.6.m6.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.6.m6.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.6.m6.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> but <math alttext="\mu\not\mid\!\simeq\varphi" class="ltx_math_unparsed" display="inline" id="Thmexample3.p2.7.m7.1"><semantics id="Thmexample3.p2.7.m7.1a"><mrow id="Thmexample3.p2.7.m7.1b"><mi id="Thmexample3.p2.7.m7.1.1">μ</mi><mpadded id="Thmexample3.p2.7.m7.1c" width="0.969em"><mo id="Thmexample3.p2.7.m7.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample3.p2.7.m7.1.3">≃</mo><mi id="Thmexample3.p2.7.m7.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmexample3.p2.7.m7.1d">\mu\not\mid\!\simeq\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.7.m7.1e">italic_μ ∤ ≃ italic_φ</annotation></semantics></math>, because <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="Thmexample3.p2.8.m8.2"><semantics id="Thmexample3.p2.8.m8.2a"><msub id="Thmexample3.p2.8.m8.2.3.2" xref="Thmexample3.p2.8.m8.2.3.1.cmml"><mrow id="Thmexample3.p2.8.m8.2.3.2.2" xref="Thmexample3.p2.8.m8.2.3.1.cmml"><mi id="Thmexample3.p2.8.m8.1.1" xref="Thmexample3.p2.8.m8.1.1.cmml">φ</mi><mo id="Thmexample3.p2.8.m8.2.3.2.2.1" stretchy="false" xref="Thmexample3.p2.8.m8.2.3.1.1.cmml">|</mo></mrow><mi id="Thmexample3.p2.8.m8.2.2.1" xref="Thmexample3.p2.8.m8.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.8.m8.2b"><apply id="Thmexample3.p2.8.m8.2.3.1.cmml" xref="Thmexample3.p2.8.m8.2.3.2"><csymbol cd="latexml" id="Thmexample3.p2.8.m8.2.3.1.1.cmml" xref="Thmexample3.p2.8.m8.2.3.2.2.1">evaluated-at</csymbol><ci id="Thmexample3.p2.8.m8.1.1.cmml" xref="Thmexample3.p2.8.m8.1.1">𝜑</ci><ci id="Thmexample3.p2.8.m8.2.2.1.cmml" xref="Thmexample3.p2.8.m8.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.8.m8.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.8.m8.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is the valid formula <math alttext="\bigvee_{i}cube_{i}" class="ltx_Math" display="inline" id="Thmexample3.p2.9.m9.1"><semantics id="Thmexample3.p2.9.m9.1a"><mrow id="Thmexample3.p2.9.m9.1.1" xref="Thmexample3.p2.9.m9.1.1.cmml"><msub id="Thmexample3.p2.9.m9.1.1.1" xref="Thmexample3.p2.9.m9.1.1.1.cmml"><mo id="Thmexample3.p2.9.m9.1.1.1.2" xref="Thmexample3.p2.9.m9.1.1.1.2.cmml">⋁</mo><mi id="Thmexample3.p2.9.m9.1.1.1.3" xref="Thmexample3.p2.9.m9.1.1.1.3.cmml">i</mi></msub><mrow id="Thmexample3.p2.9.m9.1.1.2" xref="Thmexample3.p2.9.m9.1.1.2.cmml"><mi id="Thmexample3.p2.9.m9.1.1.2.2" xref="Thmexample3.p2.9.m9.1.1.2.2.cmml">c</mi><mo id="Thmexample3.p2.9.m9.1.1.2.1" xref="Thmexample3.p2.9.m9.1.1.2.1.cmml">⁢</mo><mi id="Thmexample3.p2.9.m9.1.1.2.3" xref="Thmexample3.p2.9.m9.1.1.2.3.cmml">u</mi><mo id="Thmexample3.p2.9.m9.1.1.2.1a" xref="Thmexample3.p2.9.m9.1.1.2.1.cmml">⁢</mo><mi id="Thmexample3.p2.9.m9.1.1.2.4" xref="Thmexample3.p2.9.m9.1.1.2.4.cmml">b</mi><mo id="Thmexample3.p2.9.m9.1.1.2.1b" xref="Thmexample3.p2.9.m9.1.1.2.1.cmml">⁢</mo><msub id="Thmexample3.p2.9.m9.1.1.2.5" xref="Thmexample3.p2.9.m9.1.1.2.5.cmml"><mi id="Thmexample3.p2.9.m9.1.1.2.5.2" xref="Thmexample3.p2.9.m9.1.1.2.5.2.cmml">e</mi><mi id="Thmexample3.p2.9.m9.1.1.2.5.3" xref="Thmexample3.p2.9.m9.1.1.2.5.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.9.m9.1b"><apply id="Thmexample3.p2.9.m9.1.1.cmml" xref="Thmexample3.p2.9.m9.1.1"><apply id="Thmexample3.p2.9.m9.1.1.1.cmml" xref="Thmexample3.p2.9.m9.1.1.1"><csymbol cd="ambiguous" id="Thmexample3.p2.9.m9.1.1.1.1.cmml" xref="Thmexample3.p2.9.m9.1.1.1">subscript</csymbol><or id="Thmexample3.p2.9.m9.1.1.1.2.cmml" xref="Thmexample3.p2.9.m9.1.1.1.2"></or><ci id="Thmexample3.p2.9.m9.1.1.1.3.cmml" xref="Thmexample3.p2.9.m9.1.1.1.3">𝑖</ci></apply><apply id="Thmexample3.p2.9.m9.1.1.2.cmml" xref="Thmexample3.p2.9.m9.1.1.2"><times id="Thmexample3.p2.9.m9.1.1.2.1.cmml" xref="Thmexample3.p2.9.m9.1.1.2.1"></times><ci id="Thmexample3.p2.9.m9.1.1.2.2.cmml" xref="Thmexample3.p2.9.m9.1.1.2.2">𝑐</ci><ci id="Thmexample3.p2.9.m9.1.1.2.3.cmml" xref="Thmexample3.p2.9.m9.1.1.2.3">𝑢</ci><ci id="Thmexample3.p2.9.m9.1.1.2.4.cmml" xref="Thmexample3.p2.9.m9.1.1.2.4">𝑏</ci><apply id="Thmexample3.p2.9.m9.1.1.2.5.cmml" xref="Thmexample3.p2.9.m9.1.1.2.5"><csymbol cd="ambiguous" id="Thmexample3.p2.9.m9.1.1.2.5.1.cmml" xref="Thmexample3.p2.9.m9.1.1.2.5">subscript</csymbol><ci id="Thmexample3.p2.9.m9.1.1.2.5.2.cmml" xref="Thmexample3.p2.9.m9.1.1.2.5.2">𝑒</ci><ci id="Thmexample3.p2.9.m9.1.1.2.5.3.cmml" xref="Thmexample3.p2.9.m9.1.1.2.5.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.9.m9.1c">\bigvee_{i}cube_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.9.m9.1d">⋁ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_c italic_u italic_b italic_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> which is not simplified by the above reduction. If <math alttext="\varphi_{2}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\bigwedge\mu" class="ltx_Math" display="inline" id="Thmexample3.p2.10.m10.1"><semantics id="Thmexample3.p2.10.m10.1a"><mrow id="Thmexample3.p2.10.m10.1.1" xref="Thmexample3.p2.10.m10.1.1.cmml"><msub id="Thmexample3.p2.10.m10.1.1.2" xref="Thmexample3.p2.10.m10.1.1.2.cmml"><mi id="Thmexample3.p2.10.m10.1.1.2.2" xref="Thmexample3.p2.10.m10.1.1.2.2.cmml">φ</mi><mn id="Thmexample3.p2.10.m10.1.1.2.3" xref="Thmexample3.p2.10.m10.1.1.2.3.cmml">2</mn></msub><mover id="Thmexample3.p2.10.m10.1.1.1" xref="Thmexample3.p2.10.m10.1.1.1.cmml"><mo id="Thmexample3.p2.10.m10.1.1.1.2" rspace="0.111em" xref="Thmexample3.p2.10.m10.1.1.1.2.cmml">=</mo><mtext id="Thmexample3.p2.10.m10.1.1.1.3" mathsize="71%" xref="Thmexample3.p2.10.m10.1.1.1.3a.cmml">def</mtext></mover><mrow id="Thmexample3.p2.10.m10.1.1.3" xref="Thmexample3.p2.10.m10.1.1.3.cmml"><mo id="Thmexample3.p2.10.m10.1.1.3.1" xref="Thmexample3.p2.10.m10.1.1.3.1.cmml">⋀</mo><mi id="Thmexample3.p2.10.m10.1.1.3.2" xref="Thmexample3.p2.10.m10.1.1.3.2.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.10.m10.1b"><apply id="Thmexample3.p2.10.m10.1.1.cmml" xref="Thmexample3.p2.10.m10.1.1"><apply id="Thmexample3.p2.10.m10.1.1.1.cmml" xref="Thmexample3.p2.10.m10.1.1.1"><csymbol cd="ambiguous" id="Thmexample3.p2.10.m10.1.1.1.1.cmml" xref="Thmexample3.p2.10.m10.1.1.1">superscript</csymbol><eq id="Thmexample3.p2.10.m10.1.1.1.2.cmml" xref="Thmexample3.p2.10.m10.1.1.1.2"></eq><ci id="Thmexample3.p2.10.m10.1.1.1.3a.cmml" xref="Thmexample3.p2.10.m10.1.1.1.3"><mtext id="Thmexample3.p2.10.m10.1.1.1.3.cmml" mathsize="50%" xref="Thmexample3.p2.10.m10.1.1.1.3">def</mtext></ci></apply><apply id="Thmexample3.p2.10.m10.1.1.2.cmml" xref="Thmexample3.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="Thmexample3.p2.10.m10.1.1.2.1.cmml" xref="Thmexample3.p2.10.m10.1.1.2">subscript</csymbol><ci id="Thmexample3.p2.10.m10.1.1.2.2.cmml" xref="Thmexample3.p2.10.m10.1.1.2.2">𝜑</ci><cn id="Thmexample3.p2.10.m10.1.1.2.3.cmml" type="integer" xref="Thmexample3.p2.10.m10.1.1.2.3">2</cn></apply><apply id="Thmexample3.p2.10.m10.1.1.3.cmml" xref="Thmexample3.p2.10.m10.1.1.3"><and id="Thmexample3.p2.10.m10.1.1.3.1.cmml" xref="Thmexample3.p2.10.m10.1.1.3.1"></and><ci id="Thmexample3.p2.10.m10.1.1.3.2.cmml" xref="Thmexample3.p2.10.m10.1.1.3.2">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.10.m10.1c">\varphi_{2}\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\bigwedge\mu</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.10.m10.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ⋀ italic_μ</annotation></semantics></math>, then <math alttext="\varphi\equiv\varphi_{2}" class="ltx_Math" display="inline" id="Thmexample3.p2.11.m11.1"><semantics id="Thmexample3.p2.11.m11.1a"><mrow id="Thmexample3.p2.11.m11.1.1" xref="Thmexample3.p2.11.m11.1.1.cmml"><mi id="Thmexample3.p2.11.m11.1.1.2" xref="Thmexample3.p2.11.m11.1.1.2.cmml">φ</mi><mo id="Thmexample3.p2.11.m11.1.1.1" xref="Thmexample3.p2.11.m11.1.1.1.cmml">≡</mo><msub id="Thmexample3.p2.11.m11.1.1.3" xref="Thmexample3.p2.11.m11.1.1.3.cmml"><mi id="Thmexample3.p2.11.m11.1.1.3.2" xref="Thmexample3.p2.11.m11.1.1.3.2.cmml">φ</mi><mn id="Thmexample3.p2.11.m11.1.1.3.3" xref="Thmexample3.p2.11.m11.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.11.m11.1b"><apply id="Thmexample3.p2.11.m11.1.1.cmml" xref="Thmexample3.p2.11.m11.1.1"><equivalent id="Thmexample3.p2.11.m11.1.1.1.cmml" xref="Thmexample3.p2.11.m11.1.1.1"></equivalent><ci id="Thmexample3.p2.11.m11.1.1.2.cmml" xref="Thmexample3.p2.11.m11.1.1.2">𝜑</ci><apply id="Thmexample3.p2.11.m11.1.1.3.cmml" xref="Thmexample3.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="Thmexample3.p2.11.m11.1.1.3.1.cmml" xref="Thmexample3.p2.11.m11.1.1.3">subscript</csymbol><ci id="Thmexample3.p2.11.m11.1.1.3.2.cmml" xref="Thmexample3.p2.11.m11.1.1.3.2">𝜑</ci><cn id="Thmexample3.p2.11.m11.1.1.3.3.cmml" type="integer" xref="Thmexample3.p2.11.m11.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.11.m11.1c">\varphi\equiv\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.11.m11.1d">italic_φ ≡ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\mu\not\mid\!\simeq\varphi" class="ltx_math_unparsed" display="inline" id="Thmexample3.p2.12.m12.1"><semantics id="Thmexample3.p2.12.m12.1a"><mrow id="Thmexample3.p2.12.m12.1b"><mi id="Thmexample3.p2.12.m12.1.1">μ</mi><mpadded id="Thmexample3.p2.12.m12.1c" width="0.969em"><mo id="Thmexample3.p2.12.m12.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample3.p2.12.m12.1.3">≃</mo><mi id="Thmexample3.p2.12.m12.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmexample3.p2.12.m12.1d">\mu\not\mid\!\simeq\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.12.m12.1e">italic_μ ∤ ≃ italic_φ</annotation></semantics></math> but <math alttext="\mu\mid\!\simeq\varphi_{2}" class="ltx_math_unparsed" display="inline" id="Thmexample3.p2.13.m13.1"><semantics id="Thmexample3.p2.13.m13.1a"><mrow id="Thmexample3.p2.13.m13.1b"><mi id="Thmexample3.p2.13.m13.1.1">μ</mi><mpadded id="Thmexample3.p2.13.m13.1c" width="0.219em"><mo id="Thmexample3.p2.13.m13.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmexample3.p2.13.m13.1.3">≃</mo><msub id="Thmexample3.p2.13.m13.1.4"><mi id="Thmexample3.p2.13.m13.1.4.2">φ</mi><mn id="Thmexample3.p2.13.m13.1.4.3">2</mn></msub></mrow><annotation encoding="application/x-tex" id="Thmexample3.p2.13.m13.1d">\mu\mid\!\simeq\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.13.m13.1e">italic_μ ∣ ≃ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, whereas <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmexample3.p2.14.m14.1"><semantics id="Thmexample3.p2.14.m14.1a"><mrow id="Thmexample3.p2.14.m14.1.1" xref="Thmexample3.p2.14.m14.1.1.cmml"><mi id="Thmexample3.p2.14.m14.1.1.2" xref="Thmexample3.p2.14.m14.1.1.2.cmml">μ</mi><mo id="Thmexample3.p2.14.m14.1.1.1" xref="Thmexample3.p2.14.m14.1.1.1.cmml">⊧</mo><mi id="Thmexample3.p2.14.m14.1.1.3" xref="Thmexample3.p2.14.m14.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.14.m14.1b"><apply id="Thmexample3.p2.14.m14.1.1.cmml" xref="Thmexample3.p2.14.m14.1.1"><csymbol cd="latexml" id="Thmexample3.p2.14.m14.1.1.1.cmml" xref="Thmexample3.p2.14.m14.1.1.1">models</csymbol><ci id="Thmexample3.p2.14.m14.1.1.2.cmml" xref="Thmexample3.p2.14.m14.1.1.2">𝜇</ci><ci id="Thmexample3.p2.14.m14.1.1.3.cmml" xref="Thmexample3.p2.14.m14.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.14.m14.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.14.m14.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> and <math alttext="\mu\models\varphi_{2}" class="ltx_Math" display="inline" id="Thmexample3.p2.15.m15.1"><semantics id="Thmexample3.p2.15.m15.1a"><mrow id="Thmexample3.p2.15.m15.1.1" xref="Thmexample3.p2.15.m15.1.1.cmml"><mi id="Thmexample3.p2.15.m15.1.1.2" xref="Thmexample3.p2.15.m15.1.1.2.cmml">μ</mi><mo id="Thmexample3.p2.15.m15.1.1.1" xref="Thmexample3.p2.15.m15.1.1.1.cmml">⊧</mo><msub id="Thmexample3.p2.15.m15.1.1.3" xref="Thmexample3.p2.15.m15.1.1.3.cmml"><mi id="Thmexample3.p2.15.m15.1.1.3.2" xref="Thmexample3.p2.15.m15.1.1.3.2.cmml">φ</mi><mn id="Thmexample3.p2.15.m15.1.1.3.3" xref="Thmexample3.p2.15.m15.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.15.m15.1b"><apply id="Thmexample3.p2.15.m15.1.1.cmml" xref="Thmexample3.p2.15.m15.1.1"><csymbol cd="latexml" id="Thmexample3.p2.15.m15.1.1.1.cmml" xref="Thmexample3.p2.15.m15.1.1.1">models</csymbol><ci id="Thmexample3.p2.15.m15.1.1.2.cmml" xref="Thmexample3.p2.15.m15.1.1.2">𝜇</ci><apply id="Thmexample3.p2.15.m15.1.1.3.cmml" xref="Thmexample3.p2.15.m15.1.1.3"><csymbol cd="ambiguous" id="Thmexample3.p2.15.m15.1.1.3.1.cmml" xref="Thmexample3.p2.15.m15.1.1.3">subscript</csymbol><ci id="Thmexample3.p2.15.m15.1.1.3.2.cmml" xref="Thmexample3.p2.15.m15.1.1.3.2">𝜑</ci><cn id="Thmexample3.p2.15.m15.1.1.3.3.cmml" type="integer" xref="Thmexample3.p2.15.m15.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.15.m15.1c">\mu\models\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.15.m15.1d">italic_μ ⊧ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Thus the “embarrassing fact” above applies also to <math alttext="\mid\!\simeq" class="ltx_math_unparsed" display="inline" id="Thmexample3.p2.16.m16.1"><semantics id="Thmexample3.p2.16.m16.1a"><mrow id="Thmexample3.p2.16.m16.1b"><mpadded id="Thmexample3.p2.16.m16.1c" width="0.219em"><mo id="Thmexample3.p2.16.m16.1.1">∣</mo></mpadded><mo id="Thmexample3.p2.16.m16.1.2">≃</mo></mrow><annotation encoding="application/x-tex" id="Thmexample3.p2.16.m16.1d">\mid\!\simeq</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.16.m16.1e">∣ ≃</annotation></semantics></math>. <math alttext="\diamond" class="ltx_Math" display="inline" id="Thmexample3.p2.17.m17.1"><semantics id="Thmexample3.p2.17.m17.1a"><mo id="Thmexample3.p2.17.m17.1.1" xref="Thmexample3.p2.17.m17.1.1.cmml">⋄</mo><annotation-xml encoding="MathML-Content" id="Thmexample3.p2.17.m17.1b"><ci id="Thmexample3.p2.17.m17.1.1.cmml" xref="Thmexample3.p2.17.m17.1.1">⋄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample3.p2.17.m17.1c">\diamond</annotation><annotation encoding="application/x-llamapun" id="Thmexample3.p2.17.m17.1d">⋄</annotation></semantics></math></p> </div> </div> </section> <section class="ltx_subsection ltx_pruned_first" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>Verification and entailment of existentially-quantified formulas.</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">We extend the analysis of §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS1" title="3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.1</span></a> to existentially-quantified propositional formulas, wishing to provide a satisfactory definition of partial-assignment satisfiability for this case as well. Property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty3" title="Property 3 ‣ Existentially-quantified formulas. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3</span></a> paves our way, telling how to extend the definitions of verification and entailment to existentially-quantified formulas.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.1">The first possibility is to see partial-assignment satisfiability as <span class="ltx_text ltx_font_italic" id="S3.SS3.p2.1.1">verification</span>, leveraging Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition1" title="Definition 1 ‣ Definitions. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a> to the existentially-quantified case by exploiting Shannon expansion. </p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition3"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Definition 3</span></h6> <div class="ltx_para" id="Thmdefinition3.p1"> <p class="ltx_p" id="Thmdefinition3.p1.7">We say that a <span class="ltx_text ltx_font_italic" id="Thmdefinition3.p1.7.1">partial</span> truth assignment <math alttext="\mu" class="ltx_Math" display="inline" id="Thmdefinition3.p1.1.m1.1"><semantics id="Thmdefinition3.p1.1.m1.1a"><mi id="Thmdefinition3.p1.1.m1.1.1" xref="Thmdefinition3.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.1.m1.1b"><ci id="Thmdefinition3.p1.1.m1.1.1.cmml" xref="Thmdefinition3.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.1.m1.1d">italic_μ</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmdefinition3.p1.2.m2.1"><semantics id="Thmdefinition3.p1.2.m2.1a"><mi id="Thmdefinition3.p1.2.m2.1.1" xref="Thmdefinition3.p1.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.2.m2.1b"><ci id="Thmdefinition3.p1.2.m2.1.1.cmml" xref="Thmdefinition3.p1.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.2.m2.1d">bold_A</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="Thmdefinition3.p1.7.2">verifies</span> <math alttext="\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmdefinition3.p1.3.m3.2"><semantics id="Thmdefinition3.p1.3.m3.2a"><mrow id="Thmdefinition3.p1.3.m3.2.2.1" xref="Thmdefinition3.p1.3.m3.2.2.2.cmml"><mrow id="Thmdefinition3.p1.3.m3.2.2.1.1" xref="Thmdefinition3.p1.3.m3.2.2.1.1.cmml"><mo id="Thmdefinition3.p1.3.m3.2.2.1.1.1" rspace="0.167em" xref="Thmdefinition3.p1.3.m3.2.2.1.1.1.cmml">∃</mo><mi id="Thmdefinition3.p1.3.m3.2.2.1.1.2" xref="Thmdefinition3.p1.3.m3.2.2.1.1.2.cmml">𝐁</mi></mrow><mo id="Thmdefinition3.p1.3.m3.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmdefinition3.p1.3.m3.2.2.2a.cmml">.</mo><mi id="Thmdefinition3.p1.3.m3.1.1" xref="Thmdefinition3.p1.3.m3.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.3.m3.2b"><apply id="Thmdefinition3.p1.3.m3.2.2.2.cmml" xref="Thmdefinition3.p1.3.m3.2.2.1"><csymbol cd="ambiguous" id="Thmdefinition3.p1.3.m3.2.2.2a.cmml" xref="Thmdefinition3.p1.3.m3.2.2.1.2">formulae-sequence</csymbol><apply id="Thmdefinition3.p1.3.m3.2.2.1.1.cmml" xref="Thmdefinition3.p1.3.m3.2.2.1.1"><exists id="Thmdefinition3.p1.3.m3.2.2.1.1.1.cmml" xref="Thmdefinition3.p1.3.m3.2.2.1.1.1"></exists><ci id="Thmdefinition3.p1.3.m3.2.2.1.1.2.cmml" xref="Thmdefinition3.p1.3.m3.2.2.1.1.2">𝐁</ci></apply><ci id="Thmdefinition3.p1.3.m3.1.1.cmml" xref="Thmdefinition3.p1.3.m3.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.3.m3.2c">\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.3.m3.2d">∃ bold_B . italic_ψ</annotation></semantics></math> if and only if, there exists a total truth assignment <math alttext="\delta" class="ltx_Math" display="inline" id="Thmdefinition3.p1.4.m4.1"><semantics id="Thmdefinition3.p1.4.m4.1a"><mi id="Thmdefinition3.p1.4.m4.1.1" xref="Thmdefinition3.p1.4.m4.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.4.m4.1b"><ci id="Thmdefinition3.p1.4.m4.1.1.cmml" xref="Thmdefinition3.p1.4.m4.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.4.m4.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.4.m4.1d">italic_δ</annotation></semantics></math> on <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="Thmdefinition3.p1.5.m5.1"><semantics id="Thmdefinition3.p1.5.m5.1a"><mi id="Thmdefinition3.p1.5.m5.1.1" xref="Thmdefinition3.p1.5.m5.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.5.m5.1b"><ci id="Thmdefinition3.p1.5.m5.1.1.cmml" xref="Thmdefinition3.p1.5.m5.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.5.m5.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.5.m5.1d">bold_B</annotation></semantics></math> s.t. <math alttext="\mu\cup\delta\mid\!\approx\psi" class="ltx_math_unparsed" display="inline" id="Thmdefinition3.p1.6.m6.1"><semantics id="Thmdefinition3.p1.6.m6.1a"><mrow id="Thmdefinition3.p1.6.m6.1b"><mi id="Thmdefinition3.p1.6.m6.1.1">μ</mi><mo id="Thmdefinition3.p1.6.m6.1.2">∪</mo><mi id="Thmdefinition3.p1.6.m6.1.3">δ</mi><mpadded id="Thmdefinition3.p1.6.m6.1c" width="0.219em"><mo id="Thmdefinition3.p1.6.m6.1.4" lspace="0em">∣</mo></mpadded><mo id="Thmdefinition3.p1.6.m6.1.5">≈</mo><mi id="Thmdefinition3.p1.6.m6.1.6">ψ</mi></mrow><annotation encoding="application/x-tex" id="Thmdefinition3.p1.6.m6.1d">\mu\cup\delta\mid\!\approx\psi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.6.m6.1e">italic_μ ∪ italic_δ ∣ ≈ italic_ψ</annotation></semantics></math>, that is, <math alttext="\psi|_{\mu\cup\delta}=\top" class="ltx_Math" display="inline" id="Thmdefinition3.p1.7.m7.2"><semantics id="Thmdefinition3.p1.7.m7.2a"><mrow id="Thmdefinition3.p1.7.m7.2.3" xref="Thmdefinition3.p1.7.m7.2.3.cmml"><msub id="Thmdefinition3.p1.7.m7.2.3.2.2" xref="Thmdefinition3.p1.7.m7.2.3.2.1.cmml"><mrow id="Thmdefinition3.p1.7.m7.2.3.2.2.2" xref="Thmdefinition3.p1.7.m7.2.3.2.1.cmml"><mi id="Thmdefinition3.p1.7.m7.1.1" xref="Thmdefinition3.p1.7.m7.1.1.cmml">ψ</mi><mo id="Thmdefinition3.p1.7.m7.2.3.2.2.2.1" stretchy="false" xref="Thmdefinition3.p1.7.m7.2.3.2.1.1.cmml">|</mo></mrow><mrow id="Thmdefinition3.p1.7.m7.2.2.1" xref="Thmdefinition3.p1.7.m7.2.2.1.cmml"><mi id="Thmdefinition3.p1.7.m7.2.2.1.2" xref="Thmdefinition3.p1.7.m7.2.2.1.2.cmml">μ</mi><mo id="Thmdefinition3.p1.7.m7.2.2.1.1" xref="Thmdefinition3.p1.7.m7.2.2.1.1.cmml">∪</mo><mi id="Thmdefinition3.p1.7.m7.2.2.1.3" xref="Thmdefinition3.p1.7.m7.2.2.1.3.cmml">δ</mi></mrow></msub><mo id="Thmdefinition3.p1.7.m7.2.3.1" rspace="0em" xref="Thmdefinition3.p1.7.m7.2.3.1.cmml">=</mo><mo id="Thmdefinition3.p1.7.m7.2.3.3" lspace="0em" xref="Thmdefinition3.p1.7.m7.2.3.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.7.m7.2b"><apply id="Thmdefinition3.p1.7.m7.2.3.cmml" xref="Thmdefinition3.p1.7.m7.2.3"><eq id="Thmdefinition3.p1.7.m7.2.3.1.cmml" xref="Thmdefinition3.p1.7.m7.2.3.1"></eq><apply id="Thmdefinition3.p1.7.m7.2.3.2.1.cmml" xref="Thmdefinition3.p1.7.m7.2.3.2.2"><csymbol cd="latexml" id="Thmdefinition3.p1.7.m7.2.3.2.1.1.cmml" xref="Thmdefinition3.p1.7.m7.2.3.2.2.2.1">evaluated-at</csymbol><ci id="Thmdefinition3.p1.7.m7.1.1.cmml" xref="Thmdefinition3.p1.7.m7.1.1">𝜓</ci><apply id="Thmdefinition3.p1.7.m7.2.2.1.cmml" xref="Thmdefinition3.p1.7.m7.2.2.1"><union id="Thmdefinition3.p1.7.m7.2.2.1.1.cmml" xref="Thmdefinition3.p1.7.m7.2.2.1.1"></union><ci id="Thmdefinition3.p1.7.m7.2.2.1.2.cmml" xref="Thmdefinition3.p1.7.m7.2.2.1.2">𝜇</ci><ci id="Thmdefinition3.p1.7.m7.2.2.1.3.cmml" xref="Thmdefinition3.p1.7.m7.2.2.1.3">𝛿</ci></apply></apply><csymbol cd="latexml" id="Thmdefinition3.p1.7.m7.2.3.3.cmml" xref="Thmdefinition3.p1.7.m7.2.3.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.7.m7.2c">\psi|_{\mu\cup\delta}=\top</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition3.p1.7.m7.2d">italic_ψ | start_POSTSUBSCRIPT italic_μ ∪ italic_δ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 3.3</span></h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.6.6">Let <math alttext="\psi" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.1.m1.1"><semantics id="S3.Thmtheorem3.p1.1.1.m1.1a"><mi id="S3.Thmtheorem3.p1.1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.1.1.m1.1b"><ci id="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.1.1.m1.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.1.1.m1.1d">italic_ψ</annotation></semantics></math> be a formula on <math alttext="{\mathbf{A}}\cup{\mathbf{B}}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.2.2.m2.1"><semantics id="S3.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem3.p1.2.2.m2.1.1" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.cmml">𝐀</mi><mo id="S3.Thmtheorem3.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.1.cmml">∪</mo><mi id="S3.Thmtheorem3.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.3.cmml">𝐁</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.2.2.m2.1b"><apply id="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1"><union id="S3.Thmtheorem3.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.1"></union><ci id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2">𝐀</ci><ci id="S3.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.2.2.m2.1c">{\mathbf{A}}\cup{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.2.2.m2.1d">bold_A ∪ bold_B</annotation></semantics></math> and <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.3.3.m3.1"><semantics id="S3.Thmtheorem3.p1.3.3.m3.1a"><mi id="S3.Thmtheorem3.p1.3.3.m3.1.1" xref="S3.Thmtheorem3.p1.3.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.3.3.m3.1b"><ci id="S3.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem3.p1.3.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.3.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.3.3.m3.1d">italic_μ</annotation></semantics></math> be a partial assignment on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.4.4.m4.1"><semantics id="S3.Thmtheorem3.p1.4.4.m4.1a"><mi id="S3.Thmtheorem3.p1.4.4.m4.1.1" xref="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.4.4.m4.1b"><ci id="S3.Thmtheorem3.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem3.p1.4.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.4.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.4.4.m4.1d">bold_A</annotation></semantics></math>. Then <br class="ltx_break"/><math alttext="\mu\mid\!\approx\exists{\mathbf{B}}.\psi" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem3.p1.5.5.m5.1"><semantics id="S3.Thmtheorem3.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem3.p1.5.5.m5.1b"><mi id="S3.Thmtheorem3.p1.5.5.m5.1.1">μ</mi><mpadded id="S3.Thmtheorem3.p1.5.5.m5.1c" width="0.219em"><mo id="S3.Thmtheorem3.p1.5.5.m5.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.Thmtheorem3.p1.5.5.m5.1.3">≈</mo><mo id="S3.Thmtheorem3.p1.5.5.m5.1.4" rspace="0.167em">∃</mo><mi id="S3.Thmtheorem3.p1.5.5.m5.1.5">𝐁</mi><mo id="S3.Thmtheorem3.p1.5.5.m5.1.6" lspace="0em" rspace="0.167em">.</mo><mi id="S3.Thmtheorem3.p1.5.5.m5.1.7">ψ</mi></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.5.5.m5.1d">\mu\mid\!\approx\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.5.5.m5.1e">italic_μ ∣ ≈ ∃ bold_B . italic_ψ</annotation></semantics></math> iff <math alttext="\mu\mid\!\approx{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem3.p1.6.6.m6.1"><semantics id="S3.Thmtheorem3.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem3.p1.6.6.m6.1b"><mi id="S3.Thmtheorem3.p1.6.6.m6.1.1">μ</mi><mpadded id="S3.Thmtheorem3.p1.6.6.m6.1c" width="0.219em"><mo id="S3.Thmtheorem3.p1.6.6.m6.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.Thmtheorem3.p1.6.6.m6.1.3">≈</mo><mi id="S3.Thmtheorem3.p1.6.6.m6.1.4">𝖲𝖤</mi><mrow id="S3.Thmtheorem3.p1.6.6.m6.1.5"><mo id="S3.Thmtheorem3.p1.6.6.m6.1.5.1" stretchy="false">[</mo><mo id="S3.Thmtheorem3.p1.6.6.m6.1.5.2" rspace="0.167em">∃</mo><mi id="S3.Thmtheorem3.p1.6.6.m6.1.5.3">𝐁</mi><mo id="S3.Thmtheorem3.p1.6.6.m6.1.5.4" lspace="0em" rspace="0.167em">.</mo><mi id="S3.Thmtheorem3.p1.6.6.m6.1.5.5">ψ</mi><mo id="S3.Thmtheorem3.p1.6.6.m6.1.5.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.6.6.m6.1d">\mu\mid\!\approx{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.6.6.m6.1e">italic_μ ∣ ≈ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx3"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx3.p1"> <p class="ltx_p" id="Thmproofx3.p1.9">By Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition1" title="Definition 1 ‣ Definitions. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a>, <math alttext="\mu\mid\!\approx{}{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="Thmproofx3.p1.1.m1.1"><semantics id="Thmproofx3.p1.1.m1.1a"><mrow id="Thmproofx3.p1.1.m1.1b"><mi id="Thmproofx3.p1.1.m1.1.1">μ</mi><mpadded id="Thmproofx3.p1.1.m1.1c" width="0.219em"><mo id="Thmproofx3.p1.1.m1.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx3.p1.1.m1.1.3">≈</mo><mi id="Thmproofx3.p1.1.m1.1.4">𝖲𝖤</mi><mrow id="Thmproofx3.p1.1.m1.1.5"><mo id="Thmproofx3.p1.1.m1.1.5.1" stretchy="false">[</mo><mo id="Thmproofx3.p1.1.m1.1.5.2" rspace="0.167em">∃</mo><mi id="Thmproofx3.p1.1.m1.1.5.3">𝐁</mi><mo id="Thmproofx3.p1.1.m1.1.5.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx3.p1.1.m1.1.5.5">ψ</mi><mo id="Thmproofx3.p1.1.m1.1.5.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmproofx3.p1.1.m1.1d">\mu\mid\!\approx{}{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.1.m1.1e">italic_μ ∣ ≈ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math> iff <math alttext="({\sf SE}[{\exists{\mathbf{B}}}.\psi])|_{\mu}=\top" class="ltx_math_unparsed" display="inline" id="Thmproofx3.p1.2.m2.1"><semantics id="Thmproofx3.p1.2.m2.1a"><mrow id="Thmproofx3.p1.2.m2.1b"><mrow id="Thmproofx3.p1.2.m2.1.1"><mo id="Thmproofx3.p1.2.m2.1.1.1" stretchy="false">(</mo><mi id="Thmproofx3.p1.2.m2.1.1.2">𝖲𝖤</mi><mrow id="Thmproofx3.p1.2.m2.1.1.3"><mo id="Thmproofx3.p1.2.m2.1.1.3.1" stretchy="false">[</mo><mo id="Thmproofx3.p1.2.m2.1.1.3.2" rspace="0.167em">∃</mo><mi id="Thmproofx3.p1.2.m2.1.1.3.3">𝐁</mi><mo id="Thmproofx3.p1.2.m2.1.1.3.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx3.p1.2.m2.1.1.3.5">ψ</mi><mo id="Thmproofx3.p1.2.m2.1.1.3.6" stretchy="false">]</mo></mrow><mo id="Thmproofx3.p1.2.m2.1.1.4" stretchy="false">)</mo></mrow><msub id="Thmproofx3.p1.2.m2.1.2"><mo fence="false" id="Thmproofx3.p1.2.m2.1.2.2" stretchy="false">|</mo><mi id="Thmproofx3.p1.2.m2.1.2.3">μ</mi></msub><mo id="Thmproofx3.p1.2.m2.1.3" lspace="0.167em" rspace="0em">=</mo><mo id="Thmproofx3.p1.2.m2.1.4" lspace="0em">⊤</mo></mrow><annotation encoding="application/x-tex" id="Thmproofx3.p1.2.m2.1c">({\sf SE}[{\exists{\mathbf{B}}}.\psi])|_{\mu}=\top</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.2.m2.1d">( sansserif_SE [ ∃ bold_B . italic_ψ ] ) | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math>, <br class="ltx_break"/>that is, by (<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S2.E1" title="Equation 1 ‣ Existentially-quantified formulas. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a>), iff there exists some <math alttext="\delta_{i}" class="ltx_Math" display="inline" id="Thmproofx3.p1.3.m3.1"><semantics id="Thmproofx3.p1.3.m3.1a"><msub id="Thmproofx3.p1.3.m3.1.1" xref="Thmproofx3.p1.3.m3.1.1.cmml"><mi id="Thmproofx3.p1.3.m3.1.1.2" xref="Thmproofx3.p1.3.m3.1.1.2.cmml">δ</mi><mi id="Thmproofx3.p1.3.m3.1.1.3" xref="Thmproofx3.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.3.m3.1b"><apply id="Thmproofx3.p1.3.m3.1.1.cmml" xref="Thmproofx3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.3.m3.1.1.1.cmml" xref="Thmproofx3.p1.3.m3.1.1">subscript</csymbol><ci id="Thmproofx3.p1.3.m3.1.1.2.cmml" xref="Thmproofx3.p1.3.m3.1.1.2">𝛿</ci><ci id="Thmproofx3.p1.3.m3.1.1.3.cmml" xref="Thmproofx3.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.3.m3.1c">\delta_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.3.m3.1d">italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> s.t. <math alttext="\psi|_{\delta_{i}}|_{\mu}=\top" class="ltx_Math" display="inline" id="Thmproofx3.p1.4.m4.4"><semantics id="Thmproofx3.p1.4.m4.4a"><mrow id="Thmproofx3.p1.4.m4.4.4" xref="Thmproofx3.p1.4.m4.4.4.cmml"><msub id="Thmproofx3.p1.4.m4.4.4.1.1" xref="Thmproofx3.p1.4.m4.4.4.1.2.cmml"><mrow id="Thmproofx3.p1.4.m4.4.4.1.1.1" xref="Thmproofx3.p1.4.m4.4.4.1.2.cmml"><msub id="Thmproofx3.p1.4.m4.4.4.1.1.1.1.2" xref="Thmproofx3.p1.4.m4.4.4.1.1.1.1.1.cmml"><mrow id="Thmproofx3.p1.4.m4.4.4.1.1.1.1.2.2" xref="Thmproofx3.p1.4.m4.4.4.1.1.1.1.1.cmml"><mi id="Thmproofx3.p1.4.m4.1.1" xref="Thmproofx3.p1.4.m4.1.1.cmml">ψ</mi><mo id="Thmproofx3.p1.4.m4.4.4.1.1.1.1.2.2.1" stretchy="false" xref="Thmproofx3.p1.4.m4.4.4.1.1.1.1.1.1.cmml">|</mo></mrow><msub id="Thmproofx3.p1.4.m4.2.2.1" xref="Thmproofx3.p1.4.m4.2.2.1.cmml"><mi id="Thmproofx3.p1.4.m4.2.2.1.2" xref="Thmproofx3.p1.4.m4.2.2.1.2.cmml">δ</mi><mi id="Thmproofx3.p1.4.m4.2.2.1.3" xref="Thmproofx3.p1.4.m4.2.2.1.3.cmml">i</mi></msub></msub><mo id="Thmproofx3.p1.4.m4.4.4.1.1.1.2" stretchy="false" xref="Thmproofx3.p1.4.m4.4.4.1.2.1.cmml">|</mo></mrow><mi id="Thmproofx3.p1.4.m4.3.3.1" xref="Thmproofx3.p1.4.m4.3.3.1.cmml">μ</mi></msub><mo id="Thmproofx3.p1.4.m4.4.4.2" rspace="0em" xref="Thmproofx3.p1.4.m4.4.4.2.cmml">=</mo><mo id="Thmproofx3.p1.4.m4.4.4.3" lspace="0em" xref="Thmproofx3.p1.4.m4.4.4.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.4.m4.4b"><apply id="Thmproofx3.p1.4.m4.4.4.cmml" xref="Thmproofx3.p1.4.m4.4.4"><eq id="Thmproofx3.p1.4.m4.4.4.2.cmml" xref="Thmproofx3.p1.4.m4.4.4.2"></eq><apply id="Thmproofx3.p1.4.m4.4.4.1.2.cmml" xref="Thmproofx3.p1.4.m4.4.4.1.1"><csymbol cd="latexml" id="Thmproofx3.p1.4.m4.4.4.1.2.1.cmml" xref="Thmproofx3.p1.4.m4.4.4.1.1.1.2">evaluated-at</csymbol><apply id="Thmproofx3.p1.4.m4.4.4.1.1.1.1.1.cmml" xref="Thmproofx3.p1.4.m4.4.4.1.1.1.1.2"><csymbol cd="latexml" id="Thmproofx3.p1.4.m4.4.4.1.1.1.1.1.1.cmml" xref="Thmproofx3.p1.4.m4.4.4.1.1.1.1.2.2.1">evaluated-at</csymbol><ci id="Thmproofx3.p1.4.m4.1.1.cmml" xref="Thmproofx3.p1.4.m4.1.1">𝜓</ci><apply id="Thmproofx3.p1.4.m4.2.2.1.cmml" xref="Thmproofx3.p1.4.m4.2.2.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.4.m4.2.2.1.1.cmml" xref="Thmproofx3.p1.4.m4.2.2.1">subscript</csymbol><ci id="Thmproofx3.p1.4.m4.2.2.1.2.cmml" xref="Thmproofx3.p1.4.m4.2.2.1.2">𝛿</ci><ci id="Thmproofx3.p1.4.m4.2.2.1.3.cmml" xref="Thmproofx3.p1.4.m4.2.2.1.3">𝑖</ci></apply></apply><ci id="Thmproofx3.p1.4.m4.3.3.1.cmml" xref="Thmproofx3.p1.4.m4.3.3.1">𝜇</ci></apply><csymbol cd="latexml" id="Thmproofx3.p1.4.m4.4.4.3.cmml" xref="Thmproofx3.p1.4.m4.4.4.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.4.m4.4c">\psi|_{\delta_{i}}|_{\mu}=\top</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.4.m4.4d">italic_ψ | start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math> (i.e., <math alttext="\psi|_{\delta_{i}\cup\mu}=\top" class="ltx_Math" display="inline" id="Thmproofx3.p1.5.m5.2"><semantics id="Thmproofx3.p1.5.m5.2a"><mrow id="Thmproofx3.p1.5.m5.2.3" xref="Thmproofx3.p1.5.m5.2.3.cmml"><msub id="Thmproofx3.p1.5.m5.2.3.2.2" xref="Thmproofx3.p1.5.m5.2.3.2.1.cmml"><mrow id="Thmproofx3.p1.5.m5.2.3.2.2.2" xref="Thmproofx3.p1.5.m5.2.3.2.1.cmml"><mi id="Thmproofx3.p1.5.m5.1.1" xref="Thmproofx3.p1.5.m5.1.1.cmml">ψ</mi><mo id="Thmproofx3.p1.5.m5.2.3.2.2.2.1" stretchy="false" xref="Thmproofx3.p1.5.m5.2.3.2.1.1.cmml">|</mo></mrow><mrow id="Thmproofx3.p1.5.m5.2.2.1" xref="Thmproofx3.p1.5.m5.2.2.1.cmml"><msub id="Thmproofx3.p1.5.m5.2.2.1.2" xref="Thmproofx3.p1.5.m5.2.2.1.2.cmml"><mi id="Thmproofx3.p1.5.m5.2.2.1.2.2" xref="Thmproofx3.p1.5.m5.2.2.1.2.2.cmml">δ</mi><mi id="Thmproofx3.p1.5.m5.2.2.1.2.3" xref="Thmproofx3.p1.5.m5.2.2.1.2.3.cmml">i</mi></msub><mo id="Thmproofx3.p1.5.m5.2.2.1.1" xref="Thmproofx3.p1.5.m5.2.2.1.1.cmml">∪</mo><mi id="Thmproofx3.p1.5.m5.2.2.1.3" xref="Thmproofx3.p1.5.m5.2.2.1.3.cmml">μ</mi></mrow></msub><mo id="Thmproofx3.p1.5.m5.2.3.1" rspace="0em" xref="Thmproofx3.p1.5.m5.2.3.1.cmml">=</mo><mo id="Thmproofx3.p1.5.m5.2.3.3" lspace="0em" xref="Thmproofx3.p1.5.m5.2.3.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.5.m5.2b"><apply id="Thmproofx3.p1.5.m5.2.3.cmml" xref="Thmproofx3.p1.5.m5.2.3"><eq id="Thmproofx3.p1.5.m5.2.3.1.cmml" xref="Thmproofx3.p1.5.m5.2.3.1"></eq><apply id="Thmproofx3.p1.5.m5.2.3.2.1.cmml" xref="Thmproofx3.p1.5.m5.2.3.2.2"><csymbol cd="latexml" id="Thmproofx3.p1.5.m5.2.3.2.1.1.cmml" xref="Thmproofx3.p1.5.m5.2.3.2.2.2.1">evaluated-at</csymbol><ci id="Thmproofx3.p1.5.m5.1.1.cmml" xref="Thmproofx3.p1.5.m5.1.1">𝜓</ci><apply id="Thmproofx3.p1.5.m5.2.2.1.cmml" xref="Thmproofx3.p1.5.m5.2.2.1"><union id="Thmproofx3.p1.5.m5.2.2.1.1.cmml" xref="Thmproofx3.p1.5.m5.2.2.1.1"></union><apply id="Thmproofx3.p1.5.m5.2.2.1.2.cmml" xref="Thmproofx3.p1.5.m5.2.2.1.2"><csymbol cd="ambiguous" id="Thmproofx3.p1.5.m5.2.2.1.2.1.cmml" xref="Thmproofx3.p1.5.m5.2.2.1.2">subscript</csymbol><ci id="Thmproofx3.p1.5.m5.2.2.1.2.2.cmml" xref="Thmproofx3.p1.5.m5.2.2.1.2.2">𝛿</ci><ci id="Thmproofx3.p1.5.m5.2.2.1.2.3.cmml" xref="Thmproofx3.p1.5.m5.2.2.1.2.3">𝑖</ci></apply><ci id="Thmproofx3.p1.5.m5.2.2.1.3.cmml" xref="Thmproofx3.p1.5.m5.2.2.1.3">𝜇</ci></apply></apply><csymbol cd="latexml" id="Thmproofx3.p1.5.m5.2.3.3.cmml" xref="Thmproofx3.p1.5.m5.2.3.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.5.m5.2c">\psi|_{\delta_{i}\cup\mu}=\top</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.5.m5.2d">italic_ψ | start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∪ italic_μ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math>), <br class="ltx_break"/>that is, by Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition1" title="Definition 1 ‣ Definitions. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">1</span></a>, iff there exists some <math alttext="\delta_{i}" class="ltx_Math" display="inline" id="Thmproofx3.p1.6.m6.1"><semantics id="Thmproofx3.p1.6.m6.1a"><msub id="Thmproofx3.p1.6.m6.1.1" xref="Thmproofx3.p1.6.m6.1.1.cmml"><mi id="Thmproofx3.p1.6.m6.1.1.2" xref="Thmproofx3.p1.6.m6.1.1.2.cmml">δ</mi><mi id="Thmproofx3.p1.6.m6.1.1.3" xref="Thmproofx3.p1.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="Thmproofx3.p1.6.m6.1b"><apply id="Thmproofx3.p1.6.m6.1.1.cmml" xref="Thmproofx3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="Thmproofx3.p1.6.m6.1.1.1.cmml" xref="Thmproofx3.p1.6.m6.1.1">subscript</csymbol><ci id="Thmproofx3.p1.6.m6.1.1.2.cmml" xref="Thmproofx3.p1.6.m6.1.1.2">𝛿</ci><ci id="Thmproofx3.p1.6.m6.1.1.3.cmml" xref="Thmproofx3.p1.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx3.p1.6.m6.1c">\delta_{i}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.6.m6.1d">italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> s.t. <math alttext="\mu\cup\delta_{i}\mid\!\approx\psi" class="ltx_math_unparsed" display="inline" id="Thmproofx3.p1.7.m7.1"><semantics id="Thmproofx3.p1.7.m7.1a"><mrow id="Thmproofx3.p1.7.m7.1b"><mi id="Thmproofx3.p1.7.m7.1.1">μ</mi><mo id="Thmproofx3.p1.7.m7.1.2">∪</mo><msub id="Thmproofx3.p1.7.m7.1.3"><mi id="Thmproofx3.p1.7.m7.1.3.2">δ</mi><mi id="Thmproofx3.p1.7.m7.1.3.3">i</mi></msub><mpadded id="Thmproofx3.p1.7.m7.1c" width="0.219em"><mo id="Thmproofx3.p1.7.m7.1.4" lspace="0em">∣</mo></mpadded><mo id="Thmproofx3.p1.7.m7.1.5">≈</mo><mi id="Thmproofx3.p1.7.m7.1.6">ψ</mi></mrow><annotation encoding="application/x-tex" id="Thmproofx3.p1.7.m7.1d">\mu\cup\delta_{i}\mid\!\approx\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.7.m7.1e">italic_μ ∪ italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∣ ≈ italic_ψ</annotation></semantics></math>, <br class="ltx_break"/>that is, by Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition3" title="Definition 3 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3</span></a>, iff <math alttext="\mu\mid\!\approx\exists{\mathbf{B}}.\psi" class="ltx_math_unparsed" display="inline" id="Thmproofx3.p1.8.m8.1"><semantics id="Thmproofx3.p1.8.m8.1a"><mrow id="Thmproofx3.p1.8.m8.1b"><mi id="Thmproofx3.p1.8.m8.1.1">μ</mi><mpadded id="Thmproofx3.p1.8.m8.1c" width="0.219em"><mo id="Thmproofx3.p1.8.m8.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx3.p1.8.m8.1.3">≈</mo><mo id="Thmproofx3.p1.8.m8.1.4" rspace="0.167em">∃</mo><mi id="Thmproofx3.p1.8.m8.1.5">𝐁</mi><mo id="Thmproofx3.p1.8.m8.1.6" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx3.p1.8.m8.1.7">ψ</mi></mrow><annotation encoding="application/x-tex" id="Thmproofx3.p1.8.m8.1d">\mu\mid\!\approx\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx3.p1.8.m8.1e">italic_μ ∣ ≈ ∃ bold_B . italic_ψ</annotation></semantics></math>. <span class="ltx_text ltx_markedasmath" id="Thmproofx3.p1.9.1">∎</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.1">Notice that <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.Thmtheorem3" title="Theorem 3.3 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">theorem</span> <span class="ltx_text ltx_ref_tag">3.3</span></a> is <span class="ltx_text ltx_font_italic" id="S3.SS3.p3.1.1">not</span> a straighforward consequence of property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty3" title="Property 3 ‣ Existentially-quantified formulas. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3</span></a>, because equivalent formulas are not necessarily verified by the same assignments (property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty5" title="Property 5 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">5</span></a>(ii)).</p> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.1">One second possibility is to see partial-assignment satisfiability as <span class="ltx_text ltx_font_italic" id="S3.SS3.p4.1.1">entailment</span>, leveraging Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition2" title="Definition 2 ‣ Definitions. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">2</span></a> and property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty3" title="Property 3 ‣ Existentially-quantified formulas. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3</span></a> to the existentially-quantified case. This leads to the following definition and relative property.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinition4"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Definition 4</span></h6> <div class="ltx_para" id="Thmdefinition4.p1"> <p class="ltx_p" id="Thmdefinition4.p1.11">We say that a <span class="ltx_text ltx_font_italic" id="Thmdefinition4.p1.11.1">partial</span> truth assignment <math alttext="\mu" class="ltx_Math" display="inline" id="Thmdefinition4.p1.1.m1.1"><semantics id="Thmdefinition4.p1.1.m1.1a"><mi id="Thmdefinition4.p1.1.m1.1.1" xref="Thmdefinition4.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.1.m1.1b"><ci id="Thmdefinition4.p1.1.m1.1.1.cmml" xref="Thmdefinition4.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.1.m1.1d">italic_μ</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmdefinition4.p1.2.m2.1"><semantics id="Thmdefinition4.p1.2.m2.1a"><mi id="Thmdefinition4.p1.2.m2.1.1" xref="Thmdefinition4.p1.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.2.m2.1b"><ci id="Thmdefinition4.p1.2.m2.1.1.cmml" xref="Thmdefinition4.p1.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.2.m2.1d">bold_A</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="Thmdefinition4.p1.11.2">entails</span> <math alttext="\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmdefinition4.p1.3.m3.2"><semantics id="Thmdefinition4.p1.3.m3.2a"><mrow id="Thmdefinition4.p1.3.m3.2.2.1" xref="Thmdefinition4.p1.3.m3.2.2.2.cmml"><mrow id="Thmdefinition4.p1.3.m3.2.2.1.1" xref="Thmdefinition4.p1.3.m3.2.2.1.1.cmml"><mo id="Thmdefinition4.p1.3.m3.2.2.1.1.1" rspace="0.167em" xref="Thmdefinition4.p1.3.m3.2.2.1.1.1.cmml">∃</mo><mi id="Thmdefinition4.p1.3.m3.2.2.1.1.2" xref="Thmdefinition4.p1.3.m3.2.2.1.1.2.cmml">𝐁</mi></mrow><mo id="Thmdefinition4.p1.3.m3.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmdefinition4.p1.3.m3.2.2.2a.cmml">.</mo><mi id="Thmdefinition4.p1.3.m3.1.1" xref="Thmdefinition4.p1.3.m3.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.3.m3.2b"><apply id="Thmdefinition4.p1.3.m3.2.2.2.cmml" xref="Thmdefinition4.p1.3.m3.2.2.1"><csymbol cd="ambiguous" id="Thmdefinition4.p1.3.m3.2.2.2a.cmml" xref="Thmdefinition4.p1.3.m3.2.2.1.2">formulae-sequence</csymbol><apply id="Thmdefinition4.p1.3.m3.2.2.1.1.cmml" xref="Thmdefinition4.p1.3.m3.2.2.1.1"><exists id="Thmdefinition4.p1.3.m3.2.2.1.1.1.cmml" xref="Thmdefinition4.p1.3.m3.2.2.1.1.1"></exists><ci id="Thmdefinition4.p1.3.m3.2.2.1.1.2.cmml" xref="Thmdefinition4.p1.3.m3.2.2.1.1.2">𝐁</ci></apply><ci id="Thmdefinition4.p1.3.m3.1.1.cmml" xref="Thmdefinition4.p1.3.m3.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.3.m3.2c">\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.3.m3.2d">∃ bold_B . italic_ψ</annotation></semantics></math>, written <math alttext="\mu\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmdefinition4.p1.4.m4.2"><semantics id="Thmdefinition4.p1.4.m4.2a"><mrow id="Thmdefinition4.p1.4.m4.2.2.1" xref="Thmdefinition4.p1.4.m4.2.2.2.cmml"><mrow id="Thmdefinition4.p1.4.m4.2.2.1.1" xref="Thmdefinition4.p1.4.m4.2.2.1.1.cmml"><mi id="Thmdefinition4.p1.4.m4.2.2.1.1.2" xref="Thmdefinition4.p1.4.m4.2.2.1.1.2.cmml">μ</mi><mo id="Thmdefinition4.p1.4.m4.2.2.1.1.1" xref="Thmdefinition4.p1.4.m4.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmdefinition4.p1.4.m4.2.2.1.1.3" xref="Thmdefinition4.p1.4.m4.2.2.1.1.3.cmml"><mo id="Thmdefinition4.p1.4.m4.2.2.1.1.3.1" rspace="0.167em" xref="Thmdefinition4.p1.4.m4.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmdefinition4.p1.4.m4.2.2.1.1.3.2" xref="Thmdefinition4.p1.4.m4.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmdefinition4.p1.4.m4.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmdefinition4.p1.4.m4.2.2.2a.cmml">.</mo><mi id="Thmdefinition4.p1.4.m4.1.1" xref="Thmdefinition4.p1.4.m4.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.4.m4.2b"><apply id="Thmdefinition4.p1.4.m4.2.2.2.cmml" xref="Thmdefinition4.p1.4.m4.2.2.1"><csymbol cd="ambiguous" id="Thmdefinition4.p1.4.m4.2.2.2a.cmml" xref="Thmdefinition4.p1.4.m4.2.2.1.2">formulae-sequence</csymbol><apply id="Thmdefinition4.p1.4.m4.2.2.1.1.cmml" xref="Thmdefinition4.p1.4.m4.2.2.1.1"><csymbol cd="latexml" id="Thmdefinition4.p1.4.m4.2.2.1.1.1.cmml" xref="Thmdefinition4.p1.4.m4.2.2.1.1.1">models</csymbol><ci id="Thmdefinition4.p1.4.m4.2.2.1.1.2.cmml" xref="Thmdefinition4.p1.4.m4.2.2.1.1.2">𝜇</ci><apply id="Thmdefinition4.p1.4.m4.2.2.1.1.3.cmml" xref="Thmdefinition4.p1.4.m4.2.2.1.1.3"><exists id="Thmdefinition4.p1.4.m4.2.2.1.1.3.1.cmml" xref="Thmdefinition4.p1.4.m4.2.2.1.1.3.1"></exists><ci id="Thmdefinition4.p1.4.m4.2.2.1.1.3.2.cmml" xref="Thmdefinition4.p1.4.m4.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="Thmdefinition4.p1.4.m4.1.1.cmml" xref="Thmdefinition4.p1.4.m4.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.4.m4.2c">\mu\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.4.m4.2d">italic_μ ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math>, if and only if , for every total truth assignment <math alttext="\eta" class="ltx_Math" display="inline" id="Thmdefinition4.p1.5.m5.1"><semantics id="Thmdefinition4.p1.5.m5.1a"><mi id="Thmdefinition4.p1.5.m5.1.1" xref="Thmdefinition4.p1.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.5.m5.1b"><ci id="Thmdefinition4.p1.5.m5.1.1.cmml" xref="Thmdefinition4.p1.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.5.m5.1d">italic_η</annotation></semantics></math> on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="Thmdefinition4.p1.6.m6.1"><semantics id="Thmdefinition4.p1.6.m6.1a"><mi id="Thmdefinition4.p1.6.m6.1.1" xref="Thmdefinition4.p1.6.m6.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.6.m6.1b"><ci id="Thmdefinition4.p1.6.m6.1.1.cmml" xref="Thmdefinition4.p1.6.m6.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.6.m6.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.6.m6.1d">bold_A</annotation></semantics></math> extending <math alttext="\mu" class="ltx_Math" display="inline" id="Thmdefinition4.p1.7.m7.1"><semantics id="Thmdefinition4.p1.7.m7.1a"><mi id="Thmdefinition4.p1.7.m7.1.1" xref="Thmdefinition4.p1.7.m7.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.7.m7.1b"><ci id="Thmdefinition4.p1.7.m7.1.1.cmml" xref="Thmdefinition4.p1.7.m7.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.7.m7.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.7.m7.1d">italic_μ</annotation></semantics></math>, there exists a total truth assignment <math alttext="\delta" class="ltx_Math" display="inline" id="Thmdefinition4.p1.8.m8.1"><semantics id="Thmdefinition4.p1.8.m8.1a"><mi id="Thmdefinition4.p1.8.m8.1.1" xref="Thmdefinition4.p1.8.m8.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.8.m8.1b"><ci id="Thmdefinition4.p1.8.m8.1.1.cmml" xref="Thmdefinition4.p1.8.m8.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.8.m8.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.8.m8.1d">italic_δ</annotation></semantics></math> on <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="Thmdefinition4.p1.9.m9.1"><semantics id="Thmdefinition4.p1.9.m9.1a"><mi id="Thmdefinition4.p1.9.m9.1.1" xref="Thmdefinition4.p1.9.m9.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.9.m9.1b"><ci id="Thmdefinition4.p1.9.m9.1.1.cmml" xref="Thmdefinition4.p1.9.m9.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.9.m9.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.9.m9.1d">bold_B</annotation></semantics></math> s.t. <math alttext="\eta\cup\delta" class="ltx_Math" display="inline" id="Thmdefinition4.p1.10.m10.1"><semantics id="Thmdefinition4.p1.10.m10.1a"><mrow id="Thmdefinition4.p1.10.m10.1.1" xref="Thmdefinition4.p1.10.m10.1.1.cmml"><mi id="Thmdefinition4.p1.10.m10.1.1.2" xref="Thmdefinition4.p1.10.m10.1.1.2.cmml">η</mi><mo id="Thmdefinition4.p1.10.m10.1.1.1" xref="Thmdefinition4.p1.10.m10.1.1.1.cmml">∪</mo><mi id="Thmdefinition4.p1.10.m10.1.1.3" xref="Thmdefinition4.p1.10.m10.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.10.m10.1b"><apply id="Thmdefinition4.p1.10.m10.1.1.cmml" xref="Thmdefinition4.p1.10.m10.1.1"><union id="Thmdefinition4.p1.10.m10.1.1.1.cmml" xref="Thmdefinition4.p1.10.m10.1.1.1"></union><ci id="Thmdefinition4.p1.10.m10.1.1.2.cmml" xref="Thmdefinition4.p1.10.m10.1.1.2">𝜂</ci><ci id="Thmdefinition4.p1.10.m10.1.1.3.cmml" xref="Thmdefinition4.p1.10.m10.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.10.m10.1c">\eta\cup\delta</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.10.m10.1d">italic_η ∪ italic_δ</annotation></semantics></math> satisfies <math alttext="\psi" class="ltx_Math" display="inline" id="Thmdefinition4.p1.11.m11.1"><semantics id="Thmdefinition4.p1.11.m11.1a"><mi id="Thmdefinition4.p1.11.m11.1.1" xref="Thmdefinition4.p1.11.m11.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition4.p1.11.m11.1b"><ci id="Thmdefinition4.p1.11.m11.1.1.cmml" xref="Thmdefinition4.p1.11.m11.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition4.p1.11.m11.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="Thmdefinition4.p1.11.m11.1d">italic_ψ</annotation></semantics></math>.</p> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 3.4</span></h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.6.6">Let <math alttext="\psi" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.1"><semantics id="S3.Thmtheorem4.p1.1.1.m1.1a"><mi id="S3.Thmtheorem4.p1.1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.1.m1.1b"><ci id="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.1.m1.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.1.m1.1d">italic_ψ</annotation></semantics></math> be a formula on <math alttext="{\mathbf{A}}\cup{\mathbf{B}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">𝐀</mi><mo id="S3.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">∪</mo><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">𝐁</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.1b"><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1"><union id="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.1"></union><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2">𝐀</ci><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.2.m2.1c">{\mathbf{A}}\cup{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.2.m2.1d">bold_A ∪ bold_B</annotation></semantics></math> and <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.3.3.m3.1"><semantics id="S3.Thmtheorem4.p1.3.3.m3.1a"><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.3.3.m3.1b"><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.3.m3.1d">italic_μ</annotation></semantics></math> be a partial assignment on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.4.m4.1"><semantics id="S3.Thmtheorem4.p1.4.4.m4.1a"><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.4.m4.1b"><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.4.m4.1d">bold_A</annotation></semantics></math>. Then <br class="ltx_break"/><math alttext="\mu\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.5.5.m5.2"><semantics id="S3.Thmtheorem4.p1.5.5.m5.2a"><mrow id="S3.Thmtheorem4.p1.5.5.m5.2.2.1" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.2.cmml"><mrow id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.cmml"><mi id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.2" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.2.cmml">μ</mi><mo id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.1" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.1.cmml">⊧</mo><mrow id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.cmml"><mo id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.1" rspace="0.167em" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.1.cmml">∃</mo><mi id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.2" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.2" lspace="0em" rspace="0.167em" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.2a.cmml">.</mo><mi id="S3.Thmtheorem4.p1.5.5.m5.1.1" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.5.5.m5.2b"><apply id="S3.Thmtheorem4.p1.5.5.m5.2.2.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.5.5.m5.2.2.2a.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.2">formulae-sequence</csymbol><apply id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1"><csymbol cd="latexml" id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.1">models</csymbol><ci id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.2">𝜇</ci><apply id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3"><exists id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.1"></exists><ci id="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.2.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.5.5.m5.2c">\mu\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.5.5.m5.2d">italic_μ ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math> iff <math alttext="\mu\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem4.p1.6.6.m6.1"><semantics id="S3.Thmtheorem4.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem4.p1.6.6.m6.1b"><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1">μ</mi><mo id="S3.Thmtheorem4.p1.6.6.m6.1.2">⊧</mo><mi id="S3.Thmtheorem4.p1.6.6.m6.1.3">𝖲𝖤</mi><mrow id="S3.Thmtheorem4.p1.6.6.m6.1.4"><mo id="S3.Thmtheorem4.p1.6.6.m6.1.4.1" stretchy="false">[</mo><mo id="S3.Thmtheorem4.p1.6.6.m6.1.4.2" rspace="0.167em">∃</mo><mi id="S3.Thmtheorem4.p1.6.6.m6.1.4.3">𝐁</mi><mo id="S3.Thmtheorem4.p1.6.6.m6.1.4.4" lspace="0em" rspace="0.167em">.</mo><mi id="S3.Thmtheorem4.p1.6.6.m6.1.4.5">ψ</mi><mo id="S3.Thmtheorem4.p1.6.6.m6.1.4.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.6.6.m6.1c">\mu\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.6.6.m6.1d">italic_μ ⊧ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx4"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx4.p1"> <p class="ltx_p" id="Thmproofx4.p1.11"><math alttext="\mu\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="Thmproofx4.p1.1.m1.1"><semantics id="Thmproofx4.p1.1.m1.1a"><mrow id="Thmproofx4.p1.1.m1.1b"><mi id="Thmproofx4.p1.1.m1.1.1">μ</mi><mo id="Thmproofx4.p1.1.m1.1.2">⊧</mo><mi id="Thmproofx4.p1.1.m1.1.3">𝖲𝖤</mi><mrow id="Thmproofx4.p1.1.m1.1.4"><mo id="Thmproofx4.p1.1.m1.1.4.1" stretchy="false">[</mo><mo id="Thmproofx4.p1.1.m1.1.4.2" rspace="0.167em">∃</mo><mi id="Thmproofx4.p1.1.m1.1.4.3">𝐁</mi><mo id="Thmproofx4.p1.1.m1.1.4.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx4.p1.1.m1.1.4.5">ψ</mi><mo id="Thmproofx4.p1.1.m1.1.4.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmproofx4.p1.1.m1.1c">\mu\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.1.m1.1d">italic_μ ⊧ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math> iff <math alttext="\eta\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="Thmproofx4.p1.2.m2.1"><semantics id="Thmproofx4.p1.2.m2.1a"><mrow id="Thmproofx4.p1.2.m2.1b"><mi id="Thmproofx4.p1.2.m2.1.1">η</mi><mo id="Thmproofx4.p1.2.m2.1.2">⊧</mo><mi id="Thmproofx4.p1.2.m2.1.3">𝖲𝖤</mi><mrow id="Thmproofx4.p1.2.m2.1.4"><mo id="Thmproofx4.p1.2.m2.1.4.1" stretchy="false">[</mo><mo id="Thmproofx4.p1.2.m2.1.4.2" rspace="0.167em">∃</mo><mi id="Thmproofx4.p1.2.m2.1.4.3">𝐁</mi><mo id="Thmproofx4.p1.2.m2.1.4.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx4.p1.2.m2.1.4.5">ψ</mi><mo id="Thmproofx4.p1.2.m2.1.4.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmproofx4.p1.2.m2.1c">\eta\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.2.m2.1d">italic_η ⊧ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math> for every total assignment <math alttext="\eta" class="ltx_Math" display="inline" id="Thmproofx4.p1.3.m3.1"><semantics id="Thmproofx4.p1.3.m3.1a"><mi id="Thmproofx4.p1.3.m3.1.1" xref="Thmproofx4.p1.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.3.m3.1b"><ci id="Thmproofx4.p1.3.m3.1.1.cmml" xref="Thmproofx4.p1.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.3.m3.1d">italic_η</annotation></semantics></math> s.t. <math alttext="\eta\supseteq\mu" class="ltx_Math" display="inline" id="Thmproofx4.p1.4.m4.1"><semantics id="Thmproofx4.p1.4.m4.1a"><mrow id="Thmproofx4.p1.4.m4.1.1" xref="Thmproofx4.p1.4.m4.1.1.cmml"><mi id="Thmproofx4.p1.4.m4.1.1.2" xref="Thmproofx4.p1.4.m4.1.1.2.cmml">η</mi><mo id="Thmproofx4.p1.4.m4.1.1.1" xref="Thmproofx4.p1.4.m4.1.1.cmml">⊇</mo><mi id="Thmproofx4.p1.4.m4.1.1.3" xref="Thmproofx4.p1.4.m4.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.4.m4.1b"><apply id="Thmproofx4.p1.4.m4.1.1.cmml" xref="Thmproofx4.p1.4.m4.1.1"><subset id="Thmproofx4.p1.4.m4.1.1a.cmml" xref="Thmproofx4.p1.4.m4.1.1"></subset><ci id="Thmproofx4.p1.4.m4.1.1.3.cmml" xref="Thmproofx4.p1.4.m4.1.1.3">𝜇</ci><ci id="Thmproofx4.p1.4.m4.1.1.2.cmml" xref="Thmproofx4.p1.4.m4.1.1.2">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.4.m4.1c">\eta\supseteq\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.4.m4.1d">italic_η ⊇ italic_μ</annotation></semantics></math>, that is, by property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty3" title="Property 3 ‣ Existentially-quantified formulas. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3</span></a>, iff for every total assignments <math alttext="\eta" class="ltx_Math" display="inline" id="Thmproofx4.p1.5.m5.1"><semantics id="Thmproofx4.p1.5.m5.1a"><mi id="Thmproofx4.p1.5.m5.1.1" xref="Thmproofx4.p1.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.5.m5.1b"><ci id="Thmproofx4.p1.5.m5.1.1.cmml" xref="Thmproofx4.p1.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.5.m5.1d">italic_η</annotation></semantics></math> s.t. <math alttext="\eta\supseteq\mu" class="ltx_Math" display="inline" id="Thmproofx4.p1.6.m6.1"><semantics id="Thmproofx4.p1.6.m6.1a"><mrow id="Thmproofx4.p1.6.m6.1.1" xref="Thmproofx4.p1.6.m6.1.1.cmml"><mi id="Thmproofx4.p1.6.m6.1.1.2" xref="Thmproofx4.p1.6.m6.1.1.2.cmml">η</mi><mo id="Thmproofx4.p1.6.m6.1.1.1" xref="Thmproofx4.p1.6.m6.1.1.cmml">⊇</mo><mi id="Thmproofx4.p1.6.m6.1.1.3" xref="Thmproofx4.p1.6.m6.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.6.m6.1b"><apply id="Thmproofx4.p1.6.m6.1.1.cmml" xref="Thmproofx4.p1.6.m6.1.1"><subset id="Thmproofx4.p1.6.m6.1.1a.cmml" xref="Thmproofx4.p1.6.m6.1.1"></subset><ci id="Thmproofx4.p1.6.m6.1.1.3.cmml" xref="Thmproofx4.p1.6.m6.1.1.3">𝜇</ci><ci id="Thmproofx4.p1.6.m6.1.1.2.cmml" xref="Thmproofx4.p1.6.m6.1.1.2">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.6.m6.1c">\eta\supseteq\mu</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.6.m6.1d">italic_η ⊇ italic_μ</annotation></semantics></math> exists a total assignment <math alttext="\delta" class="ltx_Math" display="inline" id="Thmproofx4.p1.7.m7.1"><semantics id="Thmproofx4.p1.7.m7.1a"><mi id="Thmproofx4.p1.7.m7.1.1" xref="Thmproofx4.p1.7.m7.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.7.m7.1b"><ci id="Thmproofx4.p1.7.m7.1.1.cmml" xref="Thmproofx4.p1.7.m7.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.7.m7.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.7.m7.1d">italic_δ</annotation></semantics></math> on <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="Thmproofx4.p1.8.m8.1"><semantics id="Thmproofx4.p1.8.m8.1a"><mi id="Thmproofx4.p1.8.m8.1.1" xref="Thmproofx4.p1.8.m8.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.8.m8.1b"><ci id="Thmproofx4.p1.8.m8.1.1.cmml" xref="Thmproofx4.p1.8.m8.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.8.m8.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.8.m8.1d">bold_B</annotation></semantics></math> s.t. <math alttext="\eta\cup\delta\models\psi" class="ltx_Math" display="inline" id="Thmproofx4.p1.9.m9.1"><semantics id="Thmproofx4.p1.9.m9.1a"><mrow id="Thmproofx4.p1.9.m9.1.1" xref="Thmproofx4.p1.9.m9.1.1.cmml"><mrow id="Thmproofx4.p1.9.m9.1.1.2" xref="Thmproofx4.p1.9.m9.1.1.2.cmml"><mi id="Thmproofx4.p1.9.m9.1.1.2.2" xref="Thmproofx4.p1.9.m9.1.1.2.2.cmml">η</mi><mo id="Thmproofx4.p1.9.m9.1.1.2.1" xref="Thmproofx4.p1.9.m9.1.1.2.1.cmml">∪</mo><mi id="Thmproofx4.p1.9.m9.1.1.2.3" xref="Thmproofx4.p1.9.m9.1.1.2.3.cmml">δ</mi></mrow><mo id="Thmproofx4.p1.9.m9.1.1.1" xref="Thmproofx4.p1.9.m9.1.1.1.cmml">⊧</mo><mi id="Thmproofx4.p1.9.m9.1.1.3" xref="Thmproofx4.p1.9.m9.1.1.3.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.9.m9.1b"><apply id="Thmproofx4.p1.9.m9.1.1.cmml" xref="Thmproofx4.p1.9.m9.1.1"><csymbol cd="latexml" id="Thmproofx4.p1.9.m9.1.1.1.cmml" xref="Thmproofx4.p1.9.m9.1.1.1">models</csymbol><apply id="Thmproofx4.p1.9.m9.1.1.2.cmml" xref="Thmproofx4.p1.9.m9.1.1.2"><union id="Thmproofx4.p1.9.m9.1.1.2.1.cmml" xref="Thmproofx4.p1.9.m9.1.1.2.1"></union><ci id="Thmproofx4.p1.9.m9.1.1.2.2.cmml" xref="Thmproofx4.p1.9.m9.1.1.2.2">𝜂</ci><ci id="Thmproofx4.p1.9.m9.1.1.2.3.cmml" xref="Thmproofx4.p1.9.m9.1.1.2.3">𝛿</ci></apply><ci id="Thmproofx4.p1.9.m9.1.1.3.cmml" xref="Thmproofx4.p1.9.m9.1.1.3">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.9.m9.1c">\eta\cup\delta\models\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.9.m9.1d">italic_η ∪ italic_δ ⊧ italic_ψ</annotation></semantics></math>, that is, iff <math alttext="\mu\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmproofx4.p1.10.m10.2"><semantics id="Thmproofx4.p1.10.m10.2a"><mrow id="Thmproofx4.p1.10.m10.2.2.1" xref="Thmproofx4.p1.10.m10.2.2.2.cmml"><mrow id="Thmproofx4.p1.10.m10.2.2.1.1" xref="Thmproofx4.p1.10.m10.2.2.1.1.cmml"><mi id="Thmproofx4.p1.10.m10.2.2.1.1.2" xref="Thmproofx4.p1.10.m10.2.2.1.1.2.cmml">μ</mi><mo id="Thmproofx4.p1.10.m10.2.2.1.1.1" xref="Thmproofx4.p1.10.m10.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmproofx4.p1.10.m10.2.2.1.1.3" xref="Thmproofx4.p1.10.m10.2.2.1.1.3.cmml"><mo id="Thmproofx4.p1.10.m10.2.2.1.1.3.1" rspace="0.167em" xref="Thmproofx4.p1.10.m10.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmproofx4.p1.10.m10.2.2.1.1.3.2" xref="Thmproofx4.p1.10.m10.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmproofx4.p1.10.m10.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmproofx4.p1.10.m10.2.2.2a.cmml">.</mo><mi id="Thmproofx4.p1.10.m10.1.1" xref="Thmproofx4.p1.10.m10.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx4.p1.10.m10.2b"><apply id="Thmproofx4.p1.10.m10.2.2.2.cmml" xref="Thmproofx4.p1.10.m10.2.2.1"><csymbol cd="ambiguous" id="Thmproofx4.p1.10.m10.2.2.2a.cmml" xref="Thmproofx4.p1.10.m10.2.2.1.2">formulae-sequence</csymbol><apply id="Thmproofx4.p1.10.m10.2.2.1.1.cmml" xref="Thmproofx4.p1.10.m10.2.2.1.1"><csymbol cd="latexml" id="Thmproofx4.p1.10.m10.2.2.1.1.1.cmml" xref="Thmproofx4.p1.10.m10.2.2.1.1.1">models</csymbol><ci id="Thmproofx4.p1.10.m10.2.2.1.1.2.cmml" xref="Thmproofx4.p1.10.m10.2.2.1.1.2">𝜇</ci><apply id="Thmproofx4.p1.10.m10.2.2.1.1.3.cmml" xref="Thmproofx4.p1.10.m10.2.2.1.1.3"><exists id="Thmproofx4.p1.10.m10.2.2.1.1.3.1.cmml" xref="Thmproofx4.p1.10.m10.2.2.1.1.3.1"></exists><ci id="Thmproofx4.p1.10.m10.2.2.1.1.3.2.cmml" xref="Thmproofx4.p1.10.m10.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="Thmproofx4.p1.10.m10.1.1.cmml" xref="Thmproofx4.p1.10.m10.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx4.p1.10.m10.2c">\mu\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx4.p1.10.m10.2d">italic_μ ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math>. <span class="ltx_text ltx_markedasmath" id="Thmproofx4.p1.11.1">∎</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S3.SS3.p5"> <p class="ltx_p" id="S3.SS3.p5.6">Notice the nesting order of forall/exists in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmdefinition4" title="Definition 4 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">4</span></a>: “for every <math alttext="\eta" class="ltx_Math" display="inline" id="S3.SS3.p5.1.m1.1"><semantics id="S3.SS3.p5.1.m1.1a"><mi id="S3.SS3.p5.1.m1.1.1" xref="S3.SS3.p5.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.1.m1.1b"><ci id="S3.SS3.p5.1.m1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.1.m1.1d">italic_η</annotation></semantics></math> exists <math alttext="\delta" class="ltx_Math" display="inline" id="S3.SS3.p5.2.m2.1"><semantics id="S3.SS3.p5.2.m2.1a"><mi id="S3.SS3.p5.2.m2.1.1" xref="S3.SS3.p5.2.m2.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.2.m2.1b"><ci id="S3.SS3.p5.2.m2.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.2.m2.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.2.m2.1d">italic_δ</annotation></semantics></math> s.t. …”. In fact, distinct <math alttext="\eta" class="ltx_Math" display="inline" id="S3.SS3.p5.3.m3.1"><semantics id="S3.SS3.p5.3.m3.1a"><mi id="S3.SS3.p5.3.m3.1.1" xref="S3.SS3.p5.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.3.m3.1b"><ci id="S3.SS3.p5.3.m3.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.3.m3.1d">italic_η</annotation></semantics></math>’s may satisfy distinct disjuncts <math alttext="\psi|_{\delta_{i}}" class="ltx_Math" display="inline" id="S3.SS3.p5.4.m4.2"><semantics id="S3.SS3.p5.4.m4.2a"><msub id="S3.SS3.p5.4.m4.2.3.2" xref="S3.SS3.p5.4.m4.2.3.1.cmml"><mrow id="S3.SS3.p5.4.m4.2.3.2.2" xref="S3.SS3.p5.4.m4.2.3.1.cmml"><mi id="S3.SS3.p5.4.m4.1.1" xref="S3.SS3.p5.4.m4.1.1.cmml">ψ</mi><mo id="S3.SS3.p5.4.m4.2.3.2.2.1" stretchy="false" xref="S3.SS3.p5.4.m4.2.3.1.1.cmml">|</mo></mrow><msub id="S3.SS3.p5.4.m4.2.2.1" xref="S3.SS3.p5.4.m4.2.2.1.cmml"><mi id="S3.SS3.p5.4.m4.2.2.1.2" xref="S3.SS3.p5.4.m4.2.2.1.2.cmml">δ</mi><mi id="S3.SS3.p5.4.m4.2.2.1.3" xref="S3.SS3.p5.4.m4.2.2.1.3.cmml">i</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.4.m4.2b"><apply id="S3.SS3.p5.4.m4.2.3.1.cmml" xref="S3.SS3.p5.4.m4.2.3.2"><csymbol cd="latexml" id="S3.SS3.p5.4.m4.2.3.1.1.cmml" xref="S3.SS3.p5.4.m4.2.3.2.2.1">evaluated-at</csymbol><ci id="S3.SS3.p5.4.m4.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1">𝜓</ci><apply id="S3.SS3.p5.4.m4.2.2.1.cmml" xref="S3.SS3.p5.4.m4.2.2.1"><csymbol cd="ambiguous" id="S3.SS3.p5.4.m4.2.2.1.1.cmml" xref="S3.SS3.p5.4.m4.2.2.1">subscript</csymbol><ci id="S3.SS3.p5.4.m4.2.2.1.2.cmml" xref="S3.SS3.p5.4.m4.2.2.1.2">𝛿</ci><ci id="S3.SS3.p5.4.m4.2.2.1.3.cmml" xref="S3.SS3.p5.4.m4.2.2.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.4.m4.2c">\psi|_{\delta_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.4.m4.2d">italic_ψ | start_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="{\sf SE}[{\exists{\mathbf{B}}}.\psi" class="ltx_math_unparsed" display="inline" id="S3.SS3.p5.5.m5.1"><semantics id="S3.SS3.p5.5.m5.1a"><mrow id="S3.SS3.p5.5.m5.1b"><mi id="S3.SS3.p5.5.m5.1.1">𝖲𝖤</mi><mrow id="S3.SS3.p5.5.m5.1.2"><mo id="S3.SS3.p5.5.m5.1.2.1" stretchy="false">[</mo><mo id="S3.SS3.p5.5.m5.1.2.2" rspace="0.167em">∃</mo><mi id="S3.SS3.p5.5.m5.1.2.3">𝐁</mi><mo id="S3.SS3.p5.5.m5.1.2.4" lspace="0em" rspace="0.167em">.</mo><mi id="S3.SS3.p5.5.m5.1.2.5">ψ</mi></mrow></mrow><annotation encoding="application/x-tex" id="S3.SS3.p5.5.m5.1c">{\sf SE}[{\exists{\mathbf{B}}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.5.m5.1d">sansserif_SE [ ∃ bold_B . italic_ψ</annotation></semantics></math>], requiring distinct <math alttext="\delta_{i}" class="ltx_Math" display="inline" id="S3.SS3.p5.6.m6.1"><semantics id="S3.SS3.p5.6.m6.1a"><msub id="S3.SS3.p5.6.m6.1.1" xref="S3.SS3.p5.6.m6.1.1.cmml"><mi id="S3.SS3.p5.6.m6.1.1.2" xref="S3.SS3.p5.6.m6.1.1.2.cmml">δ</mi><mi id="S3.SS3.p5.6.m6.1.1.3" xref="S3.SS3.p5.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.6.m6.1b"><apply id="S3.SS3.p5.6.m6.1.1.cmml" xref="S3.SS3.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS3.p5.6.m6.1.1.1.cmml" xref="S3.SS3.p5.6.m6.1.1">subscript</csymbol><ci id="S3.SS3.p5.6.m6.1.1.2.cmml" xref="S3.SS3.p5.6.m6.1.1.2">𝛿</ci><ci id="S3.SS3.p5.6.m6.1.1.3.cmml" xref="S3.SS3.p5.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.6.m6.1c">\delta_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.6.m6.1d">italic_δ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>’s.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem5"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 3.5</span></h6> <div class="ltx_para" id="S3.Thmtheorem5.p1"> <p class="ltx_p" id="S3.Thmtheorem5.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem5.p1.6.6">Let <math alttext="\psi" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.1.m1.1"><semantics id="S3.Thmtheorem5.p1.1.1.m1.1a"><mi id="S3.Thmtheorem5.p1.1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.1.m1.1b"><ci id="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.1.m1.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.1.m1.1d">italic_ψ</annotation></semantics></math> be a formula on <math alttext="{\mathbf{A}}\cup{\mathbf{B}}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem5.p1.2.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">𝐀</mi><mo id="S3.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">∪</mo><mi id="S3.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3.cmml">𝐁</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.2.m2.1b"><apply id="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1"><union id="S3.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.1"></union><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.2">𝐀</ci><ci id="S3.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.2.m2.1c">{\mathbf{A}}\cup{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.2.m2.1d">bold_A ∪ bold_B</annotation></semantics></math> and <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.3.m3.1"><semantics id="S3.Thmtheorem5.p1.3.3.m3.1a"><mi id="S3.Thmtheorem5.p1.3.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.3.m3.1b"><ci id="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.3.m3.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.3.m3.1d">italic_μ</annotation></semantics></math> be a partial truth assignment over <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.4.m4.1"><semantics id="S3.Thmtheorem5.p1.4.4.m4.1a"><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.4.m4.1b"><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.4.m4.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.4.m4.1d">bold_A</annotation></semantics></math>. Then <math alttext="\mu\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.5.5.m5.2"><semantics id="S3.Thmtheorem5.p1.5.5.m5.2a"><mrow id="S3.Thmtheorem5.p1.5.5.m5.2.2.1" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.2.cmml"><mrow id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.cmml"><mi id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.2" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.2.cmml">μ</mi><mo id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.1" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.cmml">⊧</mo><mrow id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.cmml"><mo id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.1" rspace="0.167em" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.1.cmml">∃</mo><mi id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.2" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.2" lspace="0em" rspace="0.167em" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.2a.cmml">.</mo><mi id="S3.Thmtheorem5.p1.5.5.m5.1.1" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.5.5.m5.2b"><apply id="S3.Thmtheorem5.p1.5.5.m5.2.2.2.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem5.p1.5.5.m5.2.2.2a.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.2">formulae-sequence</csymbol><apply id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1"><csymbol cd="latexml" id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.1">models</csymbol><ci id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.2.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.2">𝜇</ci><apply id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3"><exists id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.1"></exists><ci id="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="S3.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.5.5.m5.2c">\mu\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.5.5.m5.2d">italic_μ ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math> if <math alttext="\mu\mid\!\approx\exists{\mathbf{B}}.\psi" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem5.p1.6.6.m6.1"><semantics id="S3.Thmtheorem5.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem5.p1.6.6.m6.1b"><mi id="S3.Thmtheorem5.p1.6.6.m6.1.1">μ</mi><mpadded id="S3.Thmtheorem5.p1.6.6.m6.1c" width="0.219em"><mo id="S3.Thmtheorem5.p1.6.6.m6.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.Thmtheorem5.p1.6.6.m6.1.3">≈</mo><mo id="S3.Thmtheorem5.p1.6.6.m6.1.4" rspace="0.167em">∃</mo><mi id="S3.Thmtheorem5.p1.6.6.m6.1.5">𝐁</mi><mo id="S3.Thmtheorem5.p1.6.6.m6.1.6" lspace="0em" rspace="0.167em">.</mo><mi id="S3.Thmtheorem5.p1.6.6.m6.1.7">ψ</mi></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.6.6.m6.1d">\mu\mid\!\approx\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.6.6.m6.1e">italic_μ ∣ ≈ ∃ bold_B . italic_ψ</annotation></semantics></math>, but the converse does not hold.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx5"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx5.p1"> <p class="ltx_p" id="Thmproofx5.p1.7">From <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.Thmtheorem1" title="Theorem 3.1 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">theorem</span> <span class="ltx_text ltx_ref_tag">3.1</span></a><math alttext="(b)" class="ltx_Math" display="inline" id="Thmproofx5.p1.1.m1.1"><semantics id="Thmproofx5.p1.1.m1.1a"><mrow id="Thmproofx5.p1.1.m1.1.2.2"><mo id="Thmproofx5.p1.1.m1.1.2.2.1" stretchy="false">(</mo><mi id="Thmproofx5.p1.1.m1.1.1" xref="Thmproofx5.p1.1.m1.1.1.cmml">b</mi><mo id="Thmproofx5.p1.1.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx5.p1.1.m1.1b"><ci id="Thmproofx5.p1.1.m1.1.1.cmml" xref="Thmproofx5.p1.1.m1.1.1">𝑏</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx5.p1.1.m1.1c">(b)</annotation><annotation encoding="application/x-llamapun" id="Thmproofx5.p1.1.m1.1d">( italic_b )</annotation></semantics></math> with <math alttext="\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}{\sf SE}[{\exists{\mathbf% {B}}}.\psi]{}" class="ltx_math_unparsed" display="inline" id="Thmproofx5.p1.2.m2.1"><semantics id="Thmproofx5.p1.2.m2.1a"><mrow id="Thmproofx5.p1.2.m2.1b"><mi id="Thmproofx5.p1.2.m2.1.1">φ</mi><mover id="Thmproofx5.p1.2.m2.1.2"><mo id="Thmproofx5.p1.2.m2.1.2.2">=</mo><mtext id="Thmproofx5.p1.2.m2.1.2.3" mathsize="71%">def</mtext></mover><mi id="Thmproofx5.p1.2.m2.1.3">𝖲𝖤</mi><mrow id="Thmproofx5.p1.2.m2.1.4"><mo id="Thmproofx5.p1.2.m2.1.4.1" stretchy="false">[</mo><mo id="Thmproofx5.p1.2.m2.1.4.2" rspace="0.167em">∃</mo><mi id="Thmproofx5.p1.2.m2.1.4.3">𝐁</mi><mo id="Thmproofx5.p1.2.m2.1.4.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx5.p1.2.m2.1.4.5">ψ</mi><mo id="Thmproofx5.p1.2.m2.1.4.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmproofx5.p1.2.m2.1c">\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}{\sf SE}[{\exists{\mathbf% {B}}}.\psi]{}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx5.p1.2.m2.1d">italic_φ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math> we can deduce that <math alttext="\mu\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]{}" class="ltx_math_unparsed" display="inline" id="Thmproofx5.p1.3.m3.1"><semantics id="Thmproofx5.p1.3.m3.1a"><mrow id="Thmproofx5.p1.3.m3.1b"><mi id="Thmproofx5.p1.3.m3.1.1">μ</mi><mo id="Thmproofx5.p1.3.m3.1.2">⊧</mo><mi id="Thmproofx5.p1.3.m3.1.3">𝖲𝖤</mi><mrow id="Thmproofx5.p1.3.m3.1.4"><mo id="Thmproofx5.p1.3.m3.1.4.1" stretchy="false">[</mo><mo id="Thmproofx5.p1.3.m3.1.4.2" rspace="0.167em">∃</mo><mi id="Thmproofx5.p1.3.m3.1.4.3">𝐁</mi><mo id="Thmproofx5.p1.3.m3.1.4.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx5.p1.3.m3.1.4.5">ψ</mi><mo id="Thmproofx5.p1.3.m3.1.4.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmproofx5.p1.3.m3.1c">\mu\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]{}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx5.p1.3.m3.1d">italic_μ ⊧ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math> if <math alttext="\mu\mid\!\approx{\sf SE}[{\exists{\mathbf{B}}}.\psi]{}" class="ltx_math_unparsed" display="inline" id="Thmproofx5.p1.4.m4.1"><semantics id="Thmproofx5.p1.4.m4.1a"><mrow id="Thmproofx5.p1.4.m4.1b"><mi id="Thmproofx5.p1.4.m4.1.1">μ</mi><mpadded id="Thmproofx5.p1.4.m4.1c" width="0.219em"><mo id="Thmproofx5.p1.4.m4.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx5.p1.4.m4.1.3">≈</mo><mi id="Thmproofx5.p1.4.m4.1.4">𝖲𝖤</mi><mrow id="Thmproofx5.p1.4.m4.1.5"><mo id="Thmproofx5.p1.4.m4.1.5.1" stretchy="false">[</mo><mo id="Thmproofx5.p1.4.m4.1.5.2" rspace="0.167em">∃</mo><mi id="Thmproofx5.p1.4.m4.1.5.3">𝐁</mi><mo id="Thmproofx5.p1.4.m4.1.5.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx5.p1.4.m4.1.5.5">ψ</mi><mo id="Thmproofx5.p1.4.m4.1.5.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmproofx5.p1.4.m4.1d">\mu\mid\!\approx{\sf SE}[{\exists{\mathbf{B}}}.\psi]{}</annotation><annotation encoding="application/x-llamapun" id="Thmproofx5.p1.4.m4.1e">italic_μ ∣ ≈ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math>. By <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.Thmtheorem3" title="Theorem 3.3 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">theorem</span> <span class="ltx_text ltx_ref_tag">3.3</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.Thmtheorem4" title="Theorem 3.4 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">theorem</span> <span class="ltx_text ltx_ref_tag">3.4</span></a> we have that <math alttext="\mu\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmproofx5.p1.5.m5.2"><semantics id="Thmproofx5.p1.5.m5.2a"><mrow id="Thmproofx5.p1.5.m5.2.2.1" xref="Thmproofx5.p1.5.m5.2.2.2.cmml"><mrow id="Thmproofx5.p1.5.m5.2.2.1.1" xref="Thmproofx5.p1.5.m5.2.2.1.1.cmml"><mi id="Thmproofx5.p1.5.m5.2.2.1.1.2" xref="Thmproofx5.p1.5.m5.2.2.1.1.2.cmml">μ</mi><mo id="Thmproofx5.p1.5.m5.2.2.1.1.1" xref="Thmproofx5.p1.5.m5.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmproofx5.p1.5.m5.2.2.1.1.3" xref="Thmproofx5.p1.5.m5.2.2.1.1.3.cmml"><mo id="Thmproofx5.p1.5.m5.2.2.1.1.3.1" rspace="0.167em" xref="Thmproofx5.p1.5.m5.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmproofx5.p1.5.m5.2.2.1.1.3.2" xref="Thmproofx5.p1.5.m5.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmproofx5.p1.5.m5.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmproofx5.p1.5.m5.2.2.2a.cmml">.</mo><mi id="Thmproofx5.p1.5.m5.1.1" xref="Thmproofx5.p1.5.m5.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx5.p1.5.m5.2b"><apply id="Thmproofx5.p1.5.m5.2.2.2.cmml" xref="Thmproofx5.p1.5.m5.2.2.1"><csymbol cd="ambiguous" id="Thmproofx5.p1.5.m5.2.2.2a.cmml" xref="Thmproofx5.p1.5.m5.2.2.1.2">formulae-sequence</csymbol><apply id="Thmproofx5.p1.5.m5.2.2.1.1.cmml" xref="Thmproofx5.p1.5.m5.2.2.1.1"><csymbol cd="latexml" id="Thmproofx5.p1.5.m5.2.2.1.1.1.cmml" xref="Thmproofx5.p1.5.m5.2.2.1.1.1">models</csymbol><ci id="Thmproofx5.p1.5.m5.2.2.1.1.2.cmml" xref="Thmproofx5.p1.5.m5.2.2.1.1.2">𝜇</ci><apply id="Thmproofx5.p1.5.m5.2.2.1.1.3.cmml" xref="Thmproofx5.p1.5.m5.2.2.1.1.3"><exists id="Thmproofx5.p1.5.m5.2.2.1.1.3.1.cmml" xref="Thmproofx5.p1.5.m5.2.2.1.1.3.1"></exists><ci id="Thmproofx5.p1.5.m5.2.2.1.1.3.2.cmml" xref="Thmproofx5.p1.5.m5.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="Thmproofx5.p1.5.m5.1.1.cmml" xref="Thmproofx5.p1.5.m5.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx5.p1.5.m5.2c">\mu\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx5.p1.5.m5.2d">italic_μ ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math> if <math alttext="\mu\mid\!\approx\exists{\mathbf{B}}.\psi" class="ltx_math_unparsed" display="inline" id="Thmproofx5.p1.6.m6.1"><semantics id="Thmproofx5.p1.6.m6.1a"><mrow id="Thmproofx5.p1.6.m6.1b"><mi id="Thmproofx5.p1.6.m6.1.1">μ</mi><mpadded id="Thmproofx5.p1.6.m6.1c" width="0.219em"><mo id="Thmproofx5.p1.6.m6.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx5.p1.6.m6.1.3">≈</mo><mo id="Thmproofx5.p1.6.m6.1.4" rspace="0.167em">∃</mo><mi id="Thmproofx5.p1.6.m6.1.5">𝐁</mi><mo id="Thmproofx5.p1.6.m6.1.6" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx5.p1.6.m6.1.7">ψ</mi></mrow><annotation encoding="application/x-tex" id="Thmproofx5.p1.6.m6.1d">\mu\mid\!\approx\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx5.p1.6.m6.1e">italic_μ ∣ ≈ ∃ bold_B . italic_ψ</annotation></semantics></math>. <br class="ltx_break"/>The fact that the converse does not hold is shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample4" title="Example 4 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">4</span></a>. <span class="ltx_text ltx_markedasmath" id="Thmproofx5.p1.7.1">∎</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample4"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 4</span></h6> <div class="ltx_para" id="Thmexample4.p1"> <p class="ltx_p" id="Thmexample4.p1.8">Consider <math alttext="\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1}}\}" class="ltx_Math" display="inline" id="Thmexample4.p1.1.m1.1"><semantics id="Thmexample4.p1.1.m1.1a"><mrow id="Thmexample4.p1.1.m1.1.1" xref="Thmexample4.p1.1.m1.1.1.cmml"><mi id="Thmexample4.p1.1.m1.1.1.3" xref="Thmexample4.p1.1.m1.1.1.3.cmml">μ</mi><mover id="Thmexample4.p1.1.m1.1.1.2" xref="Thmexample4.p1.1.m1.1.1.2.cmml"><mo id="Thmexample4.p1.1.m1.1.1.2.2" xref="Thmexample4.p1.1.m1.1.1.2.2.cmml">=</mo><mtext id="Thmexample4.p1.1.m1.1.1.2.3" mathsize="71%" xref="Thmexample4.p1.1.m1.1.1.2.3a.cmml">def</mtext></mover><mrow id="Thmexample4.p1.1.m1.1.1.1.1" xref="Thmexample4.p1.1.m1.1.1.1.2.cmml"><mo id="Thmexample4.p1.1.m1.1.1.1.1.2" stretchy="false" xref="Thmexample4.p1.1.m1.1.1.1.2.cmml">{</mo><msub id="Thmexample4.p1.1.m1.1.1.1.1.1" xref="Thmexample4.p1.1.m1.1.1.1.1.1.cmml"><mi id="Thmexample4.p1.1.m1.1.1.1.1.1.2" xref="Thmexample4.p1.1.m1.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample4.p1.1.m1.1.1.1.1.1.3" xref="Thmexample4.p1.1.m1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample4.p1.1.m1.1.1.1.1.3" stretchy="false" xref="Thmexample4.p1.1.m1.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample4.p1.1.m1.1b"><apply id="Thmexample4.p1.1.m1.1.1.cmml" xref="Thmexample4.p1.1.m1.1.1"><apply id="Thmexample4.p1.1.m1.1.1.2.cmml" xref="Thmexample4.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="Thmexample4.p1.1.m1.1.1.2.1.cmml" xref="Thmexample4.p1.1.m1.1.1.2">superscript</csymbol><eq id="Thmexample4.p1.1.m1.1.1.2.2.cmml" xref="Thmexample4.p1.1.m1.1.1.2.2"></eq><ci id="Thmexample4.p1.1.m1.1.1.2.3a.cmml" xref="Thmexample4.p1.1.m1.1.1.2.3"><mtext id="Thmexample4.p1.1.m1.1.1.2.3.cmml" mathsize="50%" xref="Thmexample4.p1.1.m1.1.1.2.3">def</mtext></ci></apply><ci id="Thmexample4.p1.1.m1.1.1.3.cmml" xref="Thmexample4.p1.1.m1.1.1.3">𝜇</ci><set id="Thmexample4.p1.1.m1.1.1.1.2.cmml" xref="Thmexample4.p1.1.m1.1.1.1.1"><apply id="Thmexample4.p1.1.m1.1.1.1.1.1.cmml" xref="Thmexample4.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample4.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmexample4.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmexample4.p1.1.m1.1.1.1.1.1.2.cmml" xref="Thmexample4.p1.1.m1.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample4.p1.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample4.p1.1.m1.1.1.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample4.p1.1.m1.1c">\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample4.p1.1.m1.1d">italic_μ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> and the tautology-free CNF formula on <math alttext="{\mathbf{A}}\cup{\mathbf{B}}" class="ltx_Math" display="inline" id="Thmexample4.p1.2.m2.1"><semantics id="Thmexample4.p1.2.m2.1a"><mrow id="Thmexample4.p1.2.m2.1.1" xref="Thmexample4.p1.2.m2.1.1.cmml"><mi id="Thmexample4.p1.2.m2.1.1.2" xref="Thmexample4.p1.2.m2.1.1.2.cmml">𝐀</mi><mo id="Thmexample4.p1.2.m2.1.1.1" xref="Thmexample4.p1.2.m2.1.1.1.cmml">∪</mo><mi id="Thmexample4.p1.2.m2.1.1.3" xref="Thmexample4.p1.2.m2.1.1.3.cmml">𝐁</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample4.p1.2.m2.1b"><apply id="Thmexample4.p1.2.m2.1.1.cmml" xref="Thmexample4.p1.2.m2.1.1"><union id="Thmexample4.p1.2.m2.1.1.1.cmml" xref="Thmexample4.p1.2.m2.1.1.1"></union><ci id="Thmexample4.p1.2.m2.1.1.2.cmml" xref="Thmexample4.p1.2.m2.1.1.2">𝐀</ci><ci id="Thmexample4.p1.2.m2.1.1.3.cmml" xref="Thmexample4.p1.2.m2.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample4.p1.2.m2.1c">{\mathbf{A}}\cup{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="Thmexample4.p1.2.m2.1d">bold_A ∪ bold_B</annotation></semantics></math>: <br class="ltx_break"/><math alttext="\begin{array}[]{lll}\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}&(B_{1}% \vee B_{2})\ \wedge&\\ &(\neg B_{1}\vee\phantom{\neg}A_{1})\wedge(\neg B_{1}\vee\phantom{\neg}A_{2})% \wedge(B_{1}\vee\neg A_{1}\vee\neg A_{2})\ \wedge&\\ &(\neg B_{2}\vee\phantom{\neg}A_{1})\wedge(\neg B_{2}\vee\neg A_{2})\wedge(B_{% 2}\vee\neg A_{1}\vee\phantom{\neg}A_{2}).&\\ {\sf SE}[{\exists{\mathbf{B}}}.\psi]=&\xcancel{(A_{1}\wedge A_{2}\wedge\neg A_% {2})}\vee(A_{1}\wedge A_{2})\vee(A_{1}\wedge\neg A_{2})\vee\xcancel{\bot}.\\ \end{array}" class="ltx_math_unparsed" display="inline" id="Thmexample4.p1.3.m3.7"><semantics id="Thmexample4.p1.3.m3.7a"><mtable columnspacing="5pt" id="Thmexample4.p1.3.m3.7.7" rowspacing="0pt"><mtr id="Thmexample4.p1.3.m3.7.7a"><mtd class="ltx_align_left" columnalign="left" id="Thmexample4.p1.3.m3.7.7b"><mrow id="Thmexample4.p1.3.m3.1.1.1.2.1"><mi id="Thmexample4.p1.3.m3.1.1.1.2.1.2">ψ</mi><mover id="Thmexample4.p1.3.m3.1.1.1.2.1.1"><mo id="Thmexample4.p1.3.m3.1.1.1.2.1.1.2">=</mo><mtext id="Thmexample4.p1.3.m3.1.1.1.2.1.1.3" mathsize="71%">def</mtext></mover><mi id="Thmexample4.p1.3.m3.1.1.1.2.1.3"></mi></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="Thmexample4.p1.3.m3.7.7c"><mrow id="Thmexample4.p1.3.m3.1.1.1.1.1"><mrow id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1"><mo id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.1"><msub id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.1.2"><mi id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.1.2.2">B</mi><mn id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.1.2.3">1</mn></msub><mo id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.1.1">∨</mo><msub id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.1.3"><mi id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.1.3.2">B</mi><mn id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.1.3.3">2</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.1.1.1.1.1.1.1.3" rspace="0.500em" stretchy="false">)</mo></mrow><mo id="Thmexample4.p1.3.m3.1.1.1.1.1.3">∧</mo></mrow></mtd><mtd id="Thmexample4.p1.3.m3.7.7d"></mtd></mtr><mtr id="Thmexample4.p1.3.m3.7.7e"><mtd id="Thmexample4.p1.3.m3.7.7f"></mtd><mtd class="ltx_align_left" columnalign="left" id="Thmexample4.p1.3.m3.7.7g"><mrow id="Thmexample4.p1.3.m3.4.4.4.3.3"><mrow id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1"><mo id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1"><mrow id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.2"><mo id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.2.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.2.2"><mi id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.2.2.2">B</mi><mn id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.2.2.3">1</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.1" rspace="0.892em">∨</mo><msub id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.3"><mi id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.3.2">A</mi><mn id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.1.3.3">1</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.2.2.2.1.1.1.1.3" stretchy="false">)</mo></mrow><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.4">∧</mo><mrow id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1"><mo id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1"><mrow id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.2"><mo id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.2.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.2.2"><mi id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.2.2.2">B</mi><mn id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.2.2.3">1</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.1" rspace="0.892em">∨</mo><msub id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.3"><mi id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.3.2">A</mi><mn id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.1.3.3">2</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.3.3.3.2.2.2.1.3" stretchy="false">)</mo></mrow><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.4a">∧</mo><mrow id="Thmexample4.p1.3.m3.4.4.4.3.3.3"><mrow id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1"><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1"><msub id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.2"><mi id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.2.2">B</mi><mn id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.2.3">1</mn></msub><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.1">∨</mo><mrow id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.3"><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.3.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.3.2"><mi id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.3.2.2">A</mi><mn id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.3.2.3">1</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.1a">∨</mo><mrow id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.4"><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.4.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.4.2"><mi id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.4.2.2">A</mi><mn id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.1.4.2.3">2</mn></msub></mrow></mrow><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.3.1.1.3" rspace="0.500em" stretchy="false">)</mo></mrow><mo id="Thmexample4.p1.3.m3.4.4.4.3.3.3.3">∧</mo></mrow></mrow></mtd><mtd id="Thmexample4.p1.3.m3.7.7h"></mtd></mtr><mtr id="Thmexample4.p1.3.m3.7.7i"><mtd id="Thmexample4.p1.3.m3.7.7j"></mtd><mtd class="ltx_align_left" columnalign="left" id="Thmexample4.p1.3.m3.7.7k"><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1"><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1"><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1"><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1"><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.2"><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.2.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.2.2"><mi id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.2.2.2">B</mi><mn id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.2.2.3">2</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.1" rspace="0.892em">∨</mo><msub id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.3"><mi id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.3.2">A</mi><mn id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.1.3.3">1</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.1.1.3" stretchy="false">)</mo></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.4">∧</mo><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1"><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1"><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.2"><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.2.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.2.2"><mi id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.2.2.2">B</mi><mn id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.2.2.3">2</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.1">∨</mo><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.3"><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.3.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.3.2"><mi id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.3.2.2">A</mi><mn id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.1.3.2.3">2</mn></msub></mrow></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.2.1.3" stretchy="false">)</mo></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.4a">∧</mo><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1"><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1"><msub id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.2"><mi id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.2.2">B</mi><mn id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.2.3">2</mn></msub><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.1">∨</mo><mrow id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.3"><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.3.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.3.2"><mi id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.3.2.2">A</mi><mn id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.3.2.3">1</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.1a">∨</mo><msub id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.4"><mi id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.4.2">A</mi><mn id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.1.4.3">2</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.1.3.1.3" stretchy="false">)</mo></mrow></mrow><mo id="Thmexample4.p1.3.m3.5.5.5.1.1.1.2" lspace="0em">.</mo></mrow></mtd><mtd id="Thmexample4.p1.3.m3.7.7l"></mtd></mtr><mtr id="Thmexample4.p1.3.m3.7.7m"><mtd class="ltx_align_left" columnalign="left" id="Thmexample4.p1.3.m3.7.7n"><mrow id="Thmexample4.p1.3.m3.7.7.7.3.1"><mi id="Thmexample4.p1.3.m3.7.7.7.3.1.1">𝖲𝖤</mi><mrow id="Thmexample4.p1.3.m3.7.7.7.3.1.2"><mo id="Thmexample4.p1.3.m3.7.7.7.3.1.2.1" stretchy="false">[</mo><mo id="Thmexample4.p1.3.m3.7.7.7.3.1.2.2" rspace="0.167em">∃</mo><mi id="Thmexample4.p1.3.m3.7.7.7.3.1.2.3">𝐁</mi><mo id="Thmexample4.p1.3.m3.7.7.7.3.1.2.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmexample4.p1.3.m3.7.7.7.3.1.2.5">ψ</mi><mo id="Thmexample4.p1.3.m3.7.7.7.3.1.2.6" stretchy="false">]</mo></mrow><mo id="Thmexample4.p1.3.m3.7.7.7.3.1.3">=</mo></mrow></mtd><mtd class="ltx_align_left" columnalign="left" id="Thmexample4.p1.3.m3.7.7o"><mrow id="Thmexample4.p1.3.m3.7.7.7.2.2.2"><mrow id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1"><menclose id="Thmexample4.p1.3.m3.6.6.6.1.1.1" notation="updiagonalstrike downdiagonalstrike"><mrow id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1"><mo id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1"><msub id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.2"><mi id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.2.2">A</mi><mn id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.2.3">1</mn></msub><mo id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.1">∧</mo><msub id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.3"><mi id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.3.2">A</mi><mn id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.3.3">2</mn></msub><mo id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.1a">∧</mo><mrow id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.4"><mo id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.4.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.4.2"><mi id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.4.2.2">A</mi><mn id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.1.4.2.3">2</mn></msub></mrow></mrow><mo id="Thmexample4.p1.3.m3.6.6.6.1.1.1.1.1.3" stretchy="false">)</mo></mrow></menclose><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.3">∨</mo><mrow id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1"><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.1"><msub id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.1.2"><mi id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.1.2.2">A</mi><mn id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.1.2.3">1</mn></msub><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.1.1">∧</mo><msub id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.1.3"><mi id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.1.3.2">A</mi><mn id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.1.3.3">2</mn></msub></mrow><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.1.1.3" stretchy="false">)</mo></mrow><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.3a">∨</mo><mrow id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1"><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.2" stretchy="false">(</mo><mrow id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1"><msub id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.2"><mi id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.2.2">A</mi><mn id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.2.3">1</mn></msub><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.1">∧</mo><mrow id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.3"><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.3.1" rspace="0.167em">¬</mo><msub id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.3.2"><mi id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.3.2.2">A</mi><mn id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.1.3.2.3">2</mn></msub></mrow></mrow><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.2.1.3" stretchy="false">)</mo></mrow><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.3b">∨</mo><menclose id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.4" notation="updiagonalstrike downdiagonalstrike"><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.1.4.2">⊥</mo></menclose></mrow><mo id="Thmexample4.p1.3.m3.7.7.7.2.2.2.2" lspace="0em">.</mo></mrow></mtd><mtd id="Thmexample4.p1.3.m3.7.7p"></mtd></mtr></mtable><annotation encoding="application/x-tex" id="Thmexample4.p1.3.m3.7b">\begin{array}[]{lll}\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}&(B_{1}% \vee B_{2})\ \wedge&\\ &(\neg B_{1}\vee\phantom{\neg}A_{1})\wedge(\neg B_{1}\vee\phantom{\neg}A_{2})% \wedge(B_{1}\vee\neg A_{1}\vee\neg A_{2})\ \wedge&\\ &(\neg B_{2}\vee\phantom{\neg}A_{1})\wedge(\neg B_{2}\vee\neg A_{2})\wedge(B_{% 2}\vee\neg A_{1}\vee\phantom{\neg}A_{2}).&\\ {\sf SE}[{\exists{\mathbf{B}}}.\psi]=&\xcancel{(A_{1}\wedge A_{2}\wedge\neg A_% {2})}\vee(A_{1}\wedge A_{2})\vee(A_{1}\wedge\neg A_{2})\vee\xcancel{\bot}.\\ \end{array}</annotation><annotation encoding="application/x-llamapun" id="Thmexample4.p1.3.m3.7c">start_ARRAY start_ROW start_CELL italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP end_CELL start_CELL ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∧ end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( ¬ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∧ ( ¬ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∧ ( italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ ¬ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∧ end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( ¬ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∨ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∧ ( ¬ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∨ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∧ ( italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∨ ¬ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) . end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL sansserif_SE [ ∃ bold_B . italic_ψ ] = end_CELL start_CELL cancel ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ cancel ⊥ . end_CELL start_CELL end_CELL end_ROW end_ARRAY</annotation></semantics></math> <br class="ltx_break"/>Thus <math alttext="\mu\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="Thmexample4.p1.4.m4.1"><semantics id="Thmexample4.p1.4.m4.1a"><mrow id="Thmexample4.p1.4.m4.1b"><mi id="Thmexample4.p1.4.m4.1.1">μ</mi><mo id="Thmexample4.p1.4.m4.1.2">⊧</mo><mi id="Thmexample4.p1.4.m4.1.3">𝖲𝖤</mi><mrow id="Thmexample4.p1.4.m4.1.4"><mo id="Thmexample4.p1.4.m4.1.4.1" stretchy="false">[</mo><mo id="Thmexample4.p1.4.m4.1.4.2" rspace="0.167em">∃</mo><mi id="Thmexample4.p1.4.m4.1.4.3">𝐁</mi><mo id="Thmexample4.p1.4.m4.1.4.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmexample4.p1.4.m4.1.4.5">ψ</mi><mo id="Thmexample4.p1.4.m4.1.4.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmexample4.p1.4.m4.1c">\mu\models{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="Thmexample4.p1.4.m4.1d">italic_μ ⊧ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math> but <math alttext="\mu\not\mid\!\approx{\sf SE}[{\exists{\mathbf{B}}}.\psi]" class="ltx_math_unparsed" display="inline" id="Thmexample4.p1.5.m5.1"><semantics id="Thmexample4.p1.5.m5.1a"><mrow id="Thmexample4.p1.5.m5.1b"><mi id="Thmexample4.p1.5.m5.1.1">μ</mi><mpadded id="Thmexample4.p1.5.m5.1c" width="0.969em"><mo id="Thmexample4.p1.5.m5.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample4.p1.5.m5.1.3">≈</mo><mi id="Thmexample4.p1.5.m5.1.4">𝖲𝖤</mi><mrow id="Thmexample4.p1.5.m5.1.5"><mo id="Thmexample4.p1.5.m5.1.5.1" stretchy="false">[</mo><mo id="Thmexample4.p1.5.m5.1.5.2" rspace="0.167em">∃</mo><mi id="Thmexample4.p1.5.m5.1.5.3">𝐁</mi><mo id="Thmexample4.p1.5.m5.1.5.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmexample4.p1.5.m5.1.5.5">ψ</mi><mo id="Thmexample4.p1.5.m5.1.5.6" stretchy="false">]</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmexample4.p1.5.m5.1d">\mu\not\mid\!\approx{\sf SE}[{\exists{\mathbf{B}}}.\psi]</annotation><annotation encoding="application/x-llamapun" id="Thmexample4.p1.5.m5.1e">italic_μ ∤ ≈ sansserif_SE [ ∃ bold_B . italic_ψ ]</annotation></semantics></math>, so that <math alttext="\mu\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmexample4.p1.6.m6.2"><semantics id="Thmexample4.p1.6.m6.2a"><mrow id="Thmexample4.p1.6.m6.2.2.1" xref="Thmexample4.p1.6.m6.2.2.2.cmml"><mrow id="Thmexample4.p1.6.m6.2.2.1.1" xref="Thmexample4.p1.6.m6.2.2.1.1.cmml"><mi id="Thmexample4.p1.6.m6.2.2.1.1.2" xref="Thmexample4.p1.6.m6.2.2.1.1.2.cmml">μ</mi><mo id="Thmexample4.p1.6.m6.2.2.1.1.1" xref="Thmexample4.p1.6.m6.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmexample4.p1.6.m6.2.2.1.1.3" xref="Thmexample4.p1.6.m6.2.2.1.1.3.cmml"><mo id="Thmexample4.p1.6.m6.2.2.1.1.3.1" rspace="0.167em" xref="Thmexample4.p1.6.m6.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmexample4.p1.6.m6.2.2.1.1.3.2" xref="Thmexample4.p1.6.m6.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmexample4.p1.6.m6.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmexample4.p1.6.m6.2.2.2a.cmml">.</mo><mi id="Thmexample4.p1.6.m6.1.1" xref="Thmexample4.p1.6.m6.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample4.p1.6.m6.2b"><apply id="Thmexample4.p1.6.m6.2.2.2.cmml" xref="Thmexample4.p1.6.m6.2.2.1"><csymbol cd="ambiguous" id="Thmexample4.p1.6.m6.2.2.2a.cmml" xref="Thmexample4.p1.6.m6.2.2.1.2">formulae-sequence</csymbol><apply id="Thmexample4.p1.6.m6.2.2.1.1.cmml" xref="Thmexample4.p1.6.m6.2.2.1.1"><csymbol cd="latexml" id="Thmexample4.p1.6.m6.2.2.1.1.1.cmml" xref="Thmexample4.p1.6.m6.2.2.1.1.1">models</csymbol><ci id="Thmexample4.p1.6.m6.2.2.1.1.2.cmml" xref="Thmexample4.p1.6.m6.2.2.1.1.2">𝜇</ci><apply id="Thmexample4.p1.6.m6.2.2.1.1.3.cmml" xref="Thmexample4.p1.6.m6.2.2.1.1.3"><exists id="Thmexample4.p1.6.m6.2.2.1.1.3.1.cmml" xref="Thmexample4.p1.6.m6.2.2.1.1.3.1"></exists><ci id="Thmexample4.p1.6.m6.2.2.1.1.3.2.cmml" xref="Thmexample4.p1.6.m6.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="Thmexample4.p1.6.m6.1.1.cmml" xref="Thmexample4.p1.6.m6.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample4.p1.6.m6.2c">\mu\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmexample4.p1.6.m6.2d">italic_μ ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math> but <math alttext="\mu\not\mid\!\approx\exists{\mathbf{B}}.\psi" class="ltx_math_unparsed" display="inline" id="Thmexample4.p1.7.m7.1"><semantics id="Thmexample4.p1.7.m7.1a"><mrow id="Thmexample4.p1.7.m7.1b"><mi id="Thmexample4.p1.7.m7.1.1">μ</mi><mpadded id="Thmexample4.p1.7.m7.1c" width="0.969em"><mo id="Thmexample4.p1.7.m7.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample4.p1.7.m7.1.3">≈</mo><mo id="Thmexample4.p1.7.m7.1.4" rspace="0.167em">∃</mo><mi id="Thmexample4.p1.7.m7.1.5">𝐁</mi><mo id="Thmexample4.p1.7.m7.1.6" lspace="0em" rspace="0.167em">.</mo><mi id="Thmexample4.p1.7.m7.1.7">ψ</mi></mrow><annotation encoding="application/x-tex" id="Thmexample4.p1.7.m7.1d">\mu\not\mid\!\approx\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmexample4.p1.7.m7.1e">italic_μ ∤ ≈ ∃ bold_B . italic_ψ</annotation></semantics></math>. <math alttext="\diamond" class="ltx_Math" display="inline" id="Thmexample4.p1.8.m8.1"><semantics id="Thmexample4.p1.8.m8.1a"><mo id="Thmexample4.p1.8.m8.1.1" xref="Thmexample4.p1.8.m8.1.1.cmml">⋄</mo><annotation-xml encoding="MathML-Content" id="Thmexample4.p1.8.m8.1b"><ci id="Thmexample4.p1.8.m8.1.1.cmml" xref="Thmexample4.p1.8.m8.1.1">⋄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample4.p1.8.m8.1c">\diamond</annotation><annotation encoding="application/x-llamapun" id="Thmexample4.p1.8.m8.1d">⋄</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S3.SS3.p7"> <p class="ltx_p" id="S3.SS3.p7.3">Thus verification is strictly stronger than entailment also with existentially-quantified formulas. Remarkably, and unlike with the un-quantified case, this is the case <em class="ltx_emph ltx_font_italic" id="S3.SS3.p7.1.1">even if <math alttext="\psi" class="ltx_Math" display="inline" id="S3.SS3.p7.1.1.m1.1"><semantics id="S3.SS3.p7.1.1.m1.1a"><mi id="S3.SS3.p7.1.1.m1.1.1" xref="S3.SS3.p7.1.1.m1.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.1.1.m1.1b"><ci id="S3.SS3.p7.1.1.m1.1.1.cmml" xref="S3.SS3.p7.1.1.m1.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.1.1.m1.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.1.1.m1.1d">italic_ψ</annotation></semantics></math> is a tautology-free CNF formula</em>! (See <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample4" title="Example 4 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">4</span></a>.) Intuitively, this can be seen as a consequence of the fact that <math alttext="{\sf SE}[{\exists{\mathbf{B}}}.\psi" class="ltx_math_unparsed" display="inline" id="S3.SS3.p7.2.m1.1"><semantics id="S3.SS3.p7.2.m1.1a"><mrow id="S3.SS3.p7.2.m1.1b"><mi id="S3.SS3.p7.2.m1.1.1">𝖲𝖤</mi><mrow id="S3.SS3.p7.2.m1.1.2"><mo id="S3.SS3.p7.2.m1.1.2.1" stretchy="false">[</mo><mo id="S3.SS3.p7.2.m1.1.2.2" rspace="0.167em">∃</mo><mi id="S3.SS3.p7.2.m1.1.2.3">𝐁</mi><mo id="S3.SS3.p7.2.m1.1.2.4" lspace="0em" rspace="0.167em">.</mo><mi id="S3.SS3.p7.2.m1.1.2.5">ψ</mi></mrow></mrow><annotation encoding="application/x-tex" id="S3.SS3.p7.2.m1.1c">{\sf SE}[{\exists{\mathbf{B}}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.2.m1.1d">sansserif_SE [ ∃ bold_B . italic_ψ</annotation></semantics></math>] is not in CNF even if <math alttext="\psi" class="ltx_Math" display="inline" id="S3.SS3.p7.3.m2.1"><semantics id="S3.SS3.p7.3.m2.1a"><mi id="S3.SS3.p7.3.m2.1.1" xref="S3.SS3.p7.3.m2.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p7.3.m2.1b"><ci id="S3.SS3.p7.3.m2.1.1.cmml" xref="S3.SS3.p7.3.m2.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p7.3.m2.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p7.3.m2.1d">italic_ψ</annotation></semantics></math> is in CNF.</p> </div> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.4 </span>Verification and entailment of CNF-ized non-CNF formulas.</h3> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.1">One might argue that in SAT-related problems the distinction between verification and entailment in §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS1" title="3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.1</span></a> is not relevant in practice, because we could CNF-ize upfront the input non-CNF formulas and then remove tautological clauses, exploiting property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty4" title="Property 4 ‣ CNF-ization. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">4</span></a> and the fact that the above distinction does not hold for (tautology-free) CNF formulas.</p> </div> <div class="ltx_para" id="S3.SS4.p2"> <p class="ltx_p" id="S3.SS4.p2.1">Unfortunately, the following result shows this is not the case.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem6"> <h6 class="ltx_title ltx_font_bold ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Theorem 3.6</span></h6> <div class="ltx_para" id="S3.Thmtheorem6.p1"> <p class="ltx_p" id="S3.Thmtheorem6.p1.8"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem6.p1.8.8">Let <math alttext="\varphi" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.1.m1.1"><semantics id="S3.Thmtheorem6.p1.1.1.m1.1a"><mi id="S3.Thmtheorem6.p1.1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.1.m1.1b"><ci id="S3.Thmtheorem6.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.1.m1.1d">italic_φ</annotation></semantics></math> be a non-cnf formula on <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.2.m2.1"><semantics id="S3.Thmtheorem6.p1.2.2.m2.1a"><mi id="S3.Thmtheorem6.p1.2.2.m2.1.1" xref="S3.Thmtheorem6.p1.2.2.m2.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.2.m2.1b"><ci id="S3.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.2.2.m2.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.2.m2.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.2.m2.1d">bold_A</annotation></semantics></math> and <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{Ts}}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.3.3.m3.1"><semantics id="S3.Thmtheorem6.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem6.p1.3.3.m3.1.2" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.cmml"><mi id="S3.Thmtheorem6.p1.3.3.m3.1.2.2" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.2.cmml">ψ</mi><mover id="S3.Thmtheorem6.p1.3.3.m3.1.2.1" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.1.cmml"><mo id="S3.Thmtheorem6.p1.3.3.m3.1.2.1.2" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.1.2.cmml">=</mo><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem6.p1.3.3.m3.1.2.1.3" mathsize="71%" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.1.3a.cmml">def</mtext></mover><mrow id="S3.Thmtheorem6.p1.3.3.m3.1.2.3" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.cmml"><msub id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.cmml"><mi id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.2" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.2.cmml">𝖢𝖭𝖥</mi><mi id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.3" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.3.cmml">𝖳𝗌</mi></msub><mo id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.1" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.3.2" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.cmml"><mo id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.cmml">(</mo><mi id="S3.Thmtheorem6.p1.3.3.m3.1.1" xref="S3.Thmtheorem6.p1.3.3.m3.1.1.cmml">φ</mi><mo id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.3.3.m3.1b"><apply id="S3.Thmtheorem6.p1.3.3.m3.1.2.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2"><apply id="S3.Thmtheorem6.p1.3.3.m3.1.2.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.3.3.m3.1.2.1.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.1">superscript</csymbol><eq id="S3.Thmtheorem6.p1.3.3.m3.1.2.1.2.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.1.2"></eq><ci id="S3.Thmtheorem6.p1.3.3.m3.1.2.1.3a.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.1.3"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem6.p1.3.3.m3.1.2.1.3.cmml" mathsize="50%" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.1.3">def</mtext></ci></apply><ci id="S3.Thmtheorem6.p1.3.3.m3.1.2.2.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.2">𝜓</ci><apply id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3"><times id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.1"></times><apply id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.2.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.2">𝖢𝖭𝖥</ci><ci id="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.3.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.2.3.2.3">𝖳𝗌</ci></apply><ci id="S3.Thmtheorem6.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem6.p1.3.3.m3.1.1">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.3.m3.1c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{Ts}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.3.m3.1d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math> or <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.4.m4.1"><semantics id="S3.Thmtheorem6.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem6.p1.4.4.m4.1.2" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.cmml"><mi id="S3.Thmtheorem6.p1.4.4.m4.1.2.2" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.2.cmml">ψ</mi><mover id="S3.Thmtheorem6.p1.4.4.m4.1.2.1" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.1.cmml"><mo id="S3.Thmtheorem6.p1.4.4.m4.1.2.1.2" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.1.2.cmml">=</mo><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem6.p1.4.4.m4.1.2.1.3" mathsize="71%" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.1.3a.cmml">def</mtext></mover><mrow id="S3.Thmtheorem6.p1.4.4.m4.1.2.3" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.cmml"><msub id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.cmml"><mi id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.2" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.2.cmml">𝖢𝖭𝖥</mi><mi id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.3" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.3.cmml">𝖯𝖦</mi></msub><mo id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.1" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.3.2" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.cmml"><mo id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.cmml">(</mo><mi id="S3.Thmtheorem6.p1.4.4.m4.1.1" xref="S3.Thmtheorem6.p1.4.4.m4.1.1.cmml">φ</mi><mo id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.3.2.2" stretchy="false" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.4.m4.1b"><apply id="S3.Thmtheorem6.p1.4.4.m4.1.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2"><apply id="S3.Thmtheorem6.p1.4.4.m4.1.2.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.4.4.m4.1.2.1.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.1">superscript</csymbol><eq id="S3.Thmtheorem6.p1.4.4.m4.1.2.1.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.1.2"></eq><ci id="S3.Thmtheorem6.p1.4.4.m4.1.2.1.3a.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.1.3"><mtext class="ltx_mathvariant_italic" id="S3.Thmtheorem6.p1.4.4.m4.1.2.1.3.cmml" mathsize="50%" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.1.3">def</mtext></ci></apply><ci id="S3.Thmtheorem6.p1.4.4.m4.1.2.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.2">𝜓</ci><apply id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3"><times id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.1"></times><apply id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.2.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.2">𝖢𝖭𝖥</ci><ci id="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.3.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.2.3.2.3">𝖯𝖦</ci></apply><ci id="S3.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem6.p1.4.4.m4.1.1">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.4.m4.1c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.4.m4.1d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math>, and let <math alttext="\mu" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.5.5.m5.1"><semantics id="S3.Thmtheorem6.p1.5.5.m5.1a"><mi id="S3.Thmtheorem6.p1.5.5.m5.1.1" xref="S3.Thmtheorem6.p1.5.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.5.5.m5.1b"><ci id="S3.Thmtheorem6.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem6.p1.5.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.5.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.5.5.m5.1d">italic_μ</annotation></semantics></math> be a partial truth assignment over <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.6.6.m6.1"><semantics id="S3.Thmtheorem6.p1.6.6.m6.1a"><mi id="S3.Thmtheorem6.p1.6.6.m6.1.1" xref="S3.Thmtheorem6.p1.6.6.m6.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.6.6.m6.1b"><ci id="S3.Thmtheorem6.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem6.p1.6.6.m6.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.6.6.m6.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.6.6.m6.1d">bold_A</annotation></semantics></math>. Then <math alttext="\mu\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.7.7.m7.2"><semantics id="S3.Thmtheorem6.p1.7.7.m7.2a"><mrow id="S3.Thmtheorem6.p1.7.7.m7.2.2.1" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.2.cmml"><mrow id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.cmml"><mi id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.2" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.2.cmml">μ</mi><mo id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.1" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.1.cmml">⊧</mo><mrow id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.cmml"><mo id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.1" rspace="0.167em" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.1.cmml">∃</mo><mi id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.2" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.2" lspace="0em" rspace="0.167em" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.2a.cmml">.</mo><mi id="S3.Thmtheorem6.p1.7.7.m7.1.1" xref="S3.Thmtheorem6.p1.7.7.m7.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.7.7.m7.2b"><apply id="S3.Thmtheorem6.p1.7.7.m7.2.2.2.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.7.7.m7.2.2.2a.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.2">formulae-sequence</csymbol><apply id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1"><csymbol cd="latexml" id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.1">models</csymbol><ci id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.2">𝜇</ci><apply id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3"><exists id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.1.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.1"></exists><ci id="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.2.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="S3.Thmtheorem6.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem6.p1.7.7.m7.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.7.7.m7.2c">\mu\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.7.7.m7.2d">italic_μ ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math> if <math alttext="\mu\mid\!\approx\exists{\mathbf{B}}.\psi" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem6.p1.8.8.m8.1"><semantics id="S3.Thmtheorem6.p1.8.8.m8.1a"><mrow id="S3.Thmtheorem6.p1.8.8.m8.1b"><mi id="S3.Thmtheorem6.p1.8.8.m8.1.1">μ</mi><mpadded id="S3.Thmtheorem6.p1.8.8.m8.1c" width="0.219em"><mo id="S3.Thmtheorem6.p1.8.8.m8.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.Thmtheorem6.p1.8.8.m8.1.3">≈</mo><mo id="S3.Thmtheorem6.p1.8.8.m8.1.4" rspace="0.167em">∃</mo><mi id="S3.Thmtheorem6.p1.8.8.m8.1.5">𝐁</mi><mo id="S3.Thmtheorem6.p1.8.8.m8.1.6" lspace="0em" rspace="0.167em">.</mo><mi id="S3.Thmtheorem6.p1.8.8.m8.1.7">ψ</mi></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.8.8.m8.1d">\mu\mid\!\approx\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.8.8.m8.1e">italic_μ ∣ ≈ ∃ bold_B . italic_ψ</annotation></semantics></math>, but the converse does not hold.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="Thmproofx6"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Proof</span></h6> <div class="ltx_para" id="Thmproofx6.p1"> <p class="ltx_p" id="Thmproofx6.p1.3">As an instantiation of <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.Thmtheorem5" title="Theorem 3.5 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">theorem</span> <span class="ltx_text ltx_ref_tag">3.5</span></a> we have <math alttext="\mu\models\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="Thmproofx6.p1.1.m1.2"><semantics id="Thmproofx6.p1.1.m1.2a"><mrow id="Thmproofx6.p1.1.m1.2.2.1" xref="Thmproofx6.p1.1.m1.2.2.2.cmml"><mrow id="Thmproofx6.p1.1.m1.2.2.1.1" xref="Thmproofx6.p1.1.m1.2.2.1.1.cmml"><mi id="Thmproofx6.p1.1.m1.2.2.1.1.2" xref="Thmproofx6.p1.1.m1.2.2.1.1.2.cmml">μ</mi><mo id="Thmproofx6.p1.1.m1.2.2.1.1.1" xref="Thmproofx6.p1.1.m1.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmproofx6.p1.1.m1.2.2.1.1.3" xref="Thmproofx6.p1.1.m1.2.2.1.1.3.cmml"><mo id="Thmproofx6.p1.1.m1.2.2.1.1.3.1" rspace="0.167em" xref="Thmproofx6.p1.1.m1.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmproofx6.p1.1.m1.2.2.1.1.3.2" xref="Thmproofx6.p1.1.m1.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmproofx6.p1.1.m1.2.2.1.2" lspace="0em" rspace="0.167em" xref="Thmproofx6.p1.1.m1.2.2.2a.cmml">.</mo><mi id="Thmproofx6.p1.1.m1.1.1" xref="Thmproofx6.p1.1.m1.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmproofx6.p1.1.m1.2b"><apply id="Thmproofx6.p1.1.m1.2.2.2.cmml" xref="Thmproofx6.p1.1.m1.2.2.1"><csymbol cd="ambiguous" id="Thmproofx6.p1.1.m1.2.2.2a.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.2">formulae-sequence</csymbol><apply id="Thmproofx6.p1.1.m1.2.2.1.1.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1"><csymbol cd="latexml" id="Thmproofx6.p1.1.m1.2.2.1.1.1.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1.1">models</csymbol><ci id="Thmproofx6.p1.1.m1.2.2.1.1.2.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1.2">𝜇</ci><apply id="Thmproofx6.p1.1.m1.2.2.1.1.3.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1.3"><exists id="Thmproofx6.p1.1.m1.2.2.1.1.3.1.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1.3.1"></exists><ci id="Thmproofx6.p1.1.m1.2.2.1.1.3.2.cmml" xref="Thmproofx6.p1.1.m1.2.2.1.1.3.2">𝐁</ci></apply></apply><ci id="Thmproofx6.p1.1.m1.1.1.cmml" xref="Thmproofx6.p1.1.m1.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproofx6.p1.1.m1.2c">\mu\models\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.1.m1.2d">italic_μ ⊧ ∃ bold_B . italic_ψ</annotation></semantics></math> if <math alttext="\mu\mid\!\approx\exists{\mathbf{B}}.\psi" class="ltx_math_unparsed" display="inline" id="Thmproofx6.p1.2.m2.1"><semantics id="Thmproofx6.p1.2.m2.1a"><mrow id="Thmproofx6.p1.2.m2.1b"><mi id="Thmproofx6.p1.2.m2.1.1">μ</mi><mpadded id="Thmproofx6.p1.2.m2.1c" width="0.219em"><mo id="Thmproofx6.p1.2.m2.1.2" lspace="0em">∣</mo></mpadded><mo id="Thmproofx6.p1.2.m2.1.3">≈</mo><mo id="Thmproofx6.p1.2.m2.1.4" rspace="0.167em">∃</mo><mi id="Thmproofx6.p1.2.m2.1.5">𝐁</mi><mo id="Thmproofx6.p1.2.m2.1.6" lspace="0em" rspace="0.167em">.</mo><mi id="Thmproofx6.p1.2.m2.1.7">ψ</mi></mrow><annotation encoding="application/x-tex" id="Thmproofx6.p1.2.m2.1d">\mu\mid\!\approx\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="Thmproofx6.p1.2.m2.1e">italic_μ ∣ ≈ ∃ bold_B . italic_ψ</annotation></semantics></math>. <br class="ltx_break"/>The fact that the converse does not hold is shown in <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample5" title="Example 5 ‣ 3.4 Verification and entailment of CNF-ized non-CNF formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">5</span></a>. <span class="ltx_text ltx_markedasmath" id="Thmproofx6.p1.3.1">∎</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample5"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 5</span></h6> <div class="ltx_para" id="Thmexample5.p1"> <p class="ltx_p" id="Thmexample5.p1.13">Consider <math alttext="\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}(A_{1}\wedge A_{2})\vee(A% _{1}\wedge\neg A_{2})" class="ltx_Math" display="inline" id="Thmexample5.p1.1.m1.2"><semantics id="Thmexample5.p1.1.m1.2a"><mrow id="Thmexample5.p1.1.m1.2.2" xref="Thmexample5.p1.1.m1.2.2.cmml"><mi id="Thmexample5.p1.1.m1.2.2.4" xref="Thmexample5.p1.1.m1.2.2.4.cmml">φ</mi><mover id="Thmexample5.p1.1.m1.2.2.3" xref="Thmexample5.p1.1.m1.2.2.3.cmml"><mo id="Thmexample5.p1.1.m1.2.2.3.2" xref="Thmexample5.p1.1.m1.2.2.3.2.cmml">=</mo><mtext id="Thmexample5.p1.1.m1.2.2.3.3" mathsize="71%" xref="Thmexample5.p1.1.m1.2.2.3.3a.cmml">def</mtext></mover><mrow id="Thmexample5.p1.1.m1.2.2.2" xref="Thmexample5.p1.1.m1.2.2.2.cmml"><mrow id="Thmexample5.p1.1.m1.1.1.1.1.1" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmexample5.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample5.p1.1.m1.1.1.1.1.1.1" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.cmml"><msub id="Thmexample5.p1.1.m1.1.1.1.1.1.1.2" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.cmml"><mi id="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.2" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.2.cmml">A</mi><mn id="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.3" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample5.p1.1.m1.1.1.1.1.1.1.1" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.1.cmml">∧</mo><msub id="Thmexample5.p1.1.m1.1.1.1.1.1.1.3" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.cmml"><mi id="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.2" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.2.cmml">A</mi><mn id="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.3" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="Thmexample5.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample5.p1.1.m1.2.2.2.3" xref="Thmexample5.p1.1.m1.2.2.2.3.cmml">∨</mo><mrow id="Thmexample5.p1.1.m1.2.2.2.2.1" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.cmml"><mo id="Thmexample5.p1.1.m1.2.2.2.2.1.2" stretchy="false" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="Thmexample5.p1.1.m1.2.2.2.2.1.1" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.cmml"><msub id="Thmexample5.p1.1.m1.2.2.2.2.1.1.2" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.cmml"><mi id="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.2" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.2.cmml">A</mi><mn id="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.3" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample5.p1.1.m1.2.2.2.2.1.1.1" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.1.cmml">∧</mo><mrow id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.cmml"><mo id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.1" rspace="0.167em" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.1.cmml">¬</mo><msub id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.cmml"><mi id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.2" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.2.cmml">A</mi><mn id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.3" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="Thmexample5.p1.1.m1.2.2.2.2.1.3" stretchy="false" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.1.m1.2b"><apply id="Thmexample5.p1.1.m1.2.2.cmml" xref="Thmexample5.p1.1.m1.2.2"><apply id="Thmexample5.p1.1.m1.2.2.3.cmml" xref="Thmexample5.p1.1.m1.2.2.3"><csymbol cd="ambiguous" id="Thmexample5.p1.1.m1.2.2.3.1.cmml" xref="Thmexample5.p1.1.m1.2.2.3">superscript</csymbol><eq id="Thmexample5.p1.1.m1.2.2.3.2.cmml" xref="Thmexample5.p1.1.m1.2.2.3.2"></eq><ci id="Thmexample5.p1.1.m1.2.2.3.3a.cmml" xref="Thmexample5.p1.1.m1.2.2.3.3"><mtext id="Thmexample5.p1.1.m1.2.2.3.3.cmml" mathsize="50%" xref="Thmexample5.p1.1.m1.2.2.3.3">def</mtext></ci></apply><ci id="Thmexample5.p1.1.m1.2.2.4.cmml" xref="Thmexample5.p1.1.m1.2.2.4">𝜑</ci><apply id="Thmexample5.p1.1.m1.2.2.2.cmml" xref="Thmexample5.p1.1.m1.2.2.2"><or id="Thmexample5.p1.1.m1.2.2.2.3.cmml" xref="Thmexample5.p1.1.m1.2.2.2.3"></or><apply id="Thmexample5.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmexample5.p1.1.m1.1.1.1.1.1"><and id="Thmexample5.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.1"></and><apply id="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.1.cmml" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.2">subscript</csymbol><ci id="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.2">𝐴</ci><cn id="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.2.3">1</cn></apply><apply id="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.1.cmml" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.2.cmml" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.2">𝐴</ci><cn id="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="Thmexample5.p1.1.m1.1.1.1.1.1.1.3.3">2</cn></apply></apply><apply id="Thmexample5.p1.1.m1.2.2.2.2.1.1.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1"><and id="Thmexample5.p1.1.m1.2.2.2.2.1.1.1.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.1"></and><apply id="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.2"><csymbol cd="ambiguous" id="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.1.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.2">subscript</csymbol><ci id="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.2.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.2">𝐴</ci><cn id="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.3.cmml" type="integer" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.2.3">1</cn></apply><apply id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3"><not id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.1.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.1"></not><apply id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2"><csymbol cd="ambiguous" id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.1.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2">subscript</csymbol><ci id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.2.cmml" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.2">𝐴</ci><cn id="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.3.cmml" type="integer" xref="Thmexample5.p1.1.m1.2.2.2.2.1.1.3.2.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.1.m1.2c">\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}(A_{1}\wedge A_{2})\vee(A% _{1}\wedge\neg A_{2})</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.1.m1.2d">italic_φ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1}}\}" class="ltx_Math" display="inline" id="Thmexample5.p1.2.m2.1"><semantics id="Thmexample5.p1.2.m2.1a"><mrow id="Thmexample5.p1.2.m2.1.1" xref="Thmexample5.p1.2.m2.1.1.cmml"><mi id="Thmexample5.p1.2.m2.1.1.3" xref="Thmexample5.p1.2.m2.1.1.3.cmml">μ</mi><mover id="Thmexample5.p1.2.m2.1.1.2" xref="Thmexample5.p1.2.m2.1.1.2.cmml"><mo id="Thmexample5.p1.2.m2.1.1.2.2" xref="Thmexample5.p1.2.m2.1.1.2.2.cmml">=</mo><mtext id="Thmexample5.p1.2.m2.1.1.2.3" mathsize="71%" xref="Thmexample5.p1.2.m2.1.1.2.3a.cmml">def</mtext></mover><mrow id="Thmexample5.p1.2.m2.1.1.1.1" xref="Thmexample5.p1.2.m2.1.1.1.2.cmml"><mo id="Thmexample5.p1.2.m2.1.1.1.1.2" stretchy="false" xref="Thmexample5.p1.2.m2.1.1.1.2.cmml">{</mo><msub id="Thmexample5.p1.2.m2.1.1.1.1.1" xref="Thmexample5.p1.2.m2.1.1.1.1.1.cmml"><mi id="Thmexample5.p1.2.m2.1.1.1.1.1.2" xref="Thmexample5.p1.2.m2.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample5.p1.2.m2.1.1.1.1.1.3" xref="Thmexample5.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample5.p1.2.m2.1.1.1.1.3" stretchy="false" xref="Thmexample5.p1.2.m2.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.2.m2.1b"><apply id="Thmexample5.p1.2.m2.1.1.cmml" xref="Thmexample5.p1.2.m2.1.1"><apply id="Thmexample5.p1.2.m2.1.1.2.cmml" xref="Thmexample5.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="Thmexample5.p1.2.m2.1.1.2.1.cmml" xref="Thmexample5.p1.2.m2.1.1.2">superscript</csymbol><eq id="Thmexample5.p1.2.m2.1.1.2.2.cmml" xref="Thmexample5.p1.2.m2.1.1.2.2"></eq><ci id="Thmexample5.p1.2.m2.1.1.2.3a.cmml" xref="Thmexample5.p1.2.m2.1.1.2.3"><mtext id="Thmexample5.p1.2.m2.1.1.2.3.cmml" mathsize="50%" xref="Thmexample5.p1.2.m2.1.1.2.3">def</mtext></ci></apply><ci id="Thmexample5.p1.2.m2.1.1.3.cmml" xref="Thmexample5.p1.2.m2.1.1.3">𝜇</ci><set id="Thmexample5.p1.2.m2.1.1.1.2.cmml" xref="Thmexample5.p1.2.m2.1.1.1.1"><apply id="Thmexample5.p1.2.m2.1.1.1.1.1.cmml" xref="Thmexample5.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample5.p1.2.m2.1.1.1.1.1.1.cmml" xref="Thmexample5.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="Thmexample5.p1.2.m2.1.1.1.1.1.2.cmml" xref="Thmexample5.p1.2.m2.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample5.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample5.p1.2.m2.1.1.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.2.m2.1c">\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.2.m2.1d">italic_μ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>. Then <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{Ts}}(\varphi)" class="ltx_Math" display="inline" id="Thmexample5.p1.3.m3.1"><semantics id="Thmexample5.p1.3.m3.1a"><mrow id="Thmexample5.p1.3.m3.1.2" xref="Thmexample5.p1.3.m3.1.2.cmml"><mi id="Thmexample5.p1.3.m3.1.2.2" xref="Thmexample5.p1.3.m3.1.2.2.cmml">ψ</mi><mover id="Thmexample5.p1.3.m3.1.2.1" xref="Thmexample5.p1.3.m3.1.2.1.cmml"><mo id="Thmexample5.p1.3.m3.1.2.1.2" xref="Thmexample5.p1.3.m3.1.2.1.2.cmml">=</mo><mtext id="Thmexample5.p1.3.m3.1.2.1.3" mathsize="71%" xref="Thmexample5.p1.3.m3.1.2.1.3a.cmml">def</mtext></mover><mrow id="Thmexample5.p1.3.m3.1.2.3" xref="Thmexample5.p1.3.m3.1.2.3.cmml"><msub id="Thmexample5.p1.3.m3.1.2.3.2" xref="Thmexample5.p1.3.m3.1.2.3.2.cmml"><mi id="Thmexample5.p1.3.m3.1.2.3.2.2" xref="Thmexample5.p1.3.m3.1.2.3.2.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmexample5.p1.3.m3.1.2.3.2.3" xref="Thmexample5.p1.3.m3.1.2.3.2.3.cmml">𝖳𝗌</mi></msub><mo id="Thmexample5.p1.3.m3.1.2.3.1" xref="Thmexample5.p1.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="Thmexample5.p1.3.m3.1.2.3.3.2" xref="Thmexample5.p1.3.m3.1.2.3.cmml"><mo id="Thmexample5.p1.3.m3.1.2.3.3.2.1" stretchy="false" xref="Thmexample5.p1.3.m3.1.2.3.cmml">(</mo><mi id="Thmexample5.p1.3.m3.1.1" xref="Thmexample5.p1.3.m3.1.1.cmml">φ</mi><mo id="Thmexample5.p1.3.m3.1.2.3.3.2.2" stretchy="false" xref="Thmexample5.p1.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.3.m3.1b"><apply id="Thmexample5.p1.3.m3.1.2.cmml" xref="Thmexample5.p1.3.m3.1.2"><apply id="Thmexample5.p1.3.m3.1.2.1.cmml" xref="Thmexample5.p1.3.m3.1.2.1"><csymbol cd="ambiguous" id="Thmexample5.p1.3.m3.1.2.1.1.cmml" xref="Thmexample5.p1.3.m3.1.2.1">superscript</csymbol><eq id="Thmexample5.p1.3.m3.1.2.1.2.cmml" xref="Thmexample5.p1.3.m3.1.2.1.2"></eq><ci id="Thmexample5.p1.3.m3.1.2.1.3a.cmml" xref="Thmexample5.p1.3.m3.1.2.1.3"><mtext id="Thmexample5.p1.3.m3.1.2.1.3.cmml" mathsize="50%" xref="Thmexample5.p1.3.m3.1.2.1.3">def</mtext></ci></apply><ci id="Thmexample5.p1.3.m3.1.2.2.cmml" xref="Thmexample5.p1.3.m3.1.2.2">𝜓</ci><apply id="Thmexample5.p1.3.m3.1.2.3.cmml" xref="Thmexample5.p1.3.m3.1.2.3"><times id="Thmexample5.p1.3.m3.1.2.3.1.cmml" xref="Thmexample5.p1.3.m3.1.2.3.1"></times><apply id="Thmexample5.p1.3.m3.1.2.3.2.cmml" xref="Thmexample5.p1.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="Thmexample5.p1.3.m3.1.2.3.2.1.cmml" xref="Thmexample5.p1.3.m3.1.2.3.2">subscript</csymbol><ci id="Thmexample5.p1.3.m3.1.2.3.2.2.cmml" xref="Thmexample5.p1.3.m3.1.2.3.2.2">𝖢𝖭𝖥</ci><ci id="Thmexample5.p1.3.m3.1.2.3.2.3.cmml" xref="Thmexample5.p1.3.m3.1.2.3.2.3">𝖳𝗌</ci></apply><ci id="Thmexample5.p1.3.m3.1.1.cmml" xref="Thmexample5.p1.3.m3.1.1">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.3.m3.1c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{Ts}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.3.m3.1d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math> is the formula in <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample4" title="Example 4 ‣ 3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">4</span></a>. Then <math alttext="\mu\models\exists{\mathbf{B}}.\mathsf{CNF_{Ts}}(\varphi)" class="ltx_Math" display="inline" id="Thmexample5.p1.4.m4.3"><semantics id="Thmexample5.p1.4.m4.3a"><mrow id="Thmexample5.p1.4.m4.3.3.2" xref="Thmexample5.p1.4.m4.3.3.3.cmml"><mrow id="Thmexample5.p1.4.m4.2.2.1.1" xref="Thmexample5.p1.4.m4.2.2.1.1.cmml"><mi id="Thmexample5.p1.4.m4.2.2.1.1.2" xref="Thmexample5.p1.4.m4.2.2.1.1.2.cmml">μ</mi><mo id="Thmexample5.p1.4.m4.2.2.1.1.1" xref="Thmexample5.p1.4.m4.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmexample5.p1.4.m4.2.2.1.1.3" xref="Thmexample5.p1.4.m4.2.2.1.1.3.cmml"><mo id="Thmexample5.p1.4.m4.2.2.1.1.3.1" rspace="0.167em" xref="Thmexample5.p1.4.m4.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmexample5.p1.4.m4.2.2.1.1.3.2" xref="Thmexample5.p1.4.m4.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmexample5.p1.4.m4.3.3.2.3" lspace="0em" rspace="0.167em" xref="Thmexample5.p1.4.m4.3.3.3a.cmml">.</mo><mrow id="Thmexample5.p1.4.m4.3.3.2.2" xref="Thmexample5.p1.4.m4.3.3.2.2.cmml"><msub id="Thmexample5.p1.4.m4.3.3.2.2.2" xref="Thmexample5.p1.4.m4.3.3.2.2.2.cmml"><mi id="Thmexample5.p1.4.m4.3.3.2.2.2.2" xref="Thmexample5.p1.4.m4.3.3.2.2.2.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmexample5.p1.4.m4.3.3.2.2.2.3" xref="Thmexample5.p1.4.m4.3.3.2.2.2.3.cmml">𝖳𝗌</mi></msub><mo id="Thmexample5.p1.4.m4.3.3.2.2.1" xref="Thmexample5.p1.4.m4.3.3.2.2.1.cmml">⁢</mo><mrow id="Thmexample5.p1.4.m4.3.3.2.2.3.2" xref="Thmexample5.p1.4.m4.3.3.2.2.cmml"><mo id="Thmexample5.p1.4.m4.3.3.2.2.3.2.1" stretchy="false" xref="Thmexample5.p1.4.m4.3.3.2.2.cmml">(</mo><mi id="Thmexample5.p1.4.m4.1.1" xref="Thmexample5.p1.4.m4.1.1.cmml">φ</mi><mo id="Thmexample5.p1.4.m4.3.3.2.2.3.2.2" stretchy="false" xref="Thmexample5.p1.4.m4.3.3.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.4.m4.3b"><apply id="Thmexample5.p1.4.m4.3.3.3.cmml" xref="Thmexample5.p1.4.m4.3.3.2"><csymbol cd="ambiguous" id="Thmexample5.p1.4.m4.3.3.3a.cmml" xref="Thmexample5.p1.4.m4.3.3.2.3">formulae-sequence</csymbol><apply id="Thmexample5.p1.4.m4.2.2.1.1.cmml" xref="Thmexample5.p1.4.m4.2.2.1.1"><csymbol cd="latexml" id="Thmexample5.p1.4.m4.2.2.1.1.1.cmml" xref="Thmexample5.p1.4.m4.2.2.1.1.1">models</csymbol><ci id="Thmexample5.p1.4.m4.2.2.1.1.2.cmml" xref="Thmexample5.p1.4.m4.2.2.1.1.2">𝜇</ci><apply id="Thmexample5.p1.4.m4.2.2.1.1.3.cmml" xref="Thmexample5.p1.4.m4.2.2.1.1.3"><exists id="Thmexample5.p1.4.m4.2.2.1.1.3.1.cmml" xref="Thmexample5.p1.4.m4.2.2.1.1.3.1"></exists><ci id="Thmexample5.p1.4.m4.2.2.1.1.3.2.cmml" xref="Thmexample5.p1.4.m4.2.2.1.1.3.2">𝐁</ci></apply></apply><apply id="Thmexample5.p1.4.m4.3.3.2.2.cmml" xref="Thmexample5.p1.4.m4.3.3.2.2"><times id="Thmexample5.p1.4.m4.3.3.2.2.1.cmml" xref="Thmexample5.p1.4.m4.3.3.2.2.1"></times><apply id="Thmexample5.p1.4.m4.3.3.2.2.2.cmml" xref="Thmexample5.p1.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmexample5.p1.4.m4.3.3.2.2.2.1.cmml" xref="Thmexample5.p1.4.m4.3.3.2.2.2">subscript</csymbol><ci id="Thmexample5.p1.4.m4.3.3.2.2.2.2.cmml" xref="Thmexample5.p1.4.m4.3.3.2.2.2.2">𝖢𝖭𝖥</ci><ci id="Thmexample5.p1.4.m4.3.3.2.2.2.3.cmml" xref="Thmexample5.p1.4.m4.3.3.2.2.2.3">𝖳𝗌</ci></apply><ci id="Thmexample5.p1.4.m4.1.1.cmml" xref="Thmexample5.p1.4.m4.1.1">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.4.m4.3c">\mu\models\exists{\mathbf{B}}.\mathsf{CNF_{Ts}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.4.m4.3d">italic_μ ⊧ ∃ bold_B . sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math> but <math alttext="\mu\not\mid\!\approx\exists{\mathbf{B}}.\mathsf{CNF_{Ts}}(\varphi)" class="ltx_math_unparsed" display="inline" id="Thmexample5.p1.5.m5.1"><semantics id="Thmexample5.p1.5.m5.1a"><mrow id="Thmexample5.p1.5.m5.1b"><mi id="Thmexample5.p1.5.m5.1.1">μ</mi><mpadded id="Thmexample5.p1.5.m5.1c" width="0.969em"><mo id="Thmexample5.p1.5.m5.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample5.p1.5.m5.1.3">≈</mo><mo id="Thmexample5.p1.5.m5.1.4" rspace="0.167em">∃</mo><mi id="Thmexample5.p1.5.m5.1.5">𝐁</mi><mo id="Thmexample5.p1.5.m5.1.6" lspace="0em" rspace="0.167em">.</mo><msub id="Thmexample5.p1.5.m5.1.7"><mi id="Thmexample5.p1.5.m5.1.7.2">𝖢𝖭𝖥</mi><mi id="Thmexample5.p1.5.m5.1.7.3">𝖳𝗌</mi></msub><mrow id="Thmexample5.p1.5.m5.1.8"><mo id="Thmexample5.p1.5.m5.1.8.1" stretchy="false">(</mo><mi id="Thmexample5.p1.5.m5.1.8.2">φ</mi><mo id="Thmexample5.p1.5.m5.1.8.3" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmexample5.p1.5.m5.1d">\mu\not\mid\!\approx\exists{\mathbf{B}}.\mathsf{CNF_{Ts}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.5.m5.1e">italic_μ ∤ ≈ ∃ bold_B . sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math>. <br class="ltx_break"/>Let <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\varphi)" class="ltx_Math" display="inline" id="Thmexample5.p1.6.m6.1"><semantics id="Thmexample5.p1.6.m6.1a"><mrow id="Thmexample5.p1.6.m6.1.2" xref="Thmexample5.p1.6.m6.1.2.cmml"><mi id="Thmexample5.p1.6.m6.1.2.2" xref="Thmexample5.p1.6.m6.1.2.2.cmml">ψ</mi><mover id="Thmexample5.p1.6.m6.1.2.1" xref="Thmexample5.p1.6.m6.1.2.1.cmml"><mo id="Thmexample5.p1.6.m6.1.2.1.2" xref="Thmexample5.p1.6.m6.1.2.1.2.cmml">=</mo><mtext id="Thmexample5.p1.6.m6.1.2.1.3" mathsize="71%" xref="Thmexample5.p1.6.m6.1.2.1.3a.cmml">def</mtext></mover><mrow id="Thmexample5.p1.6.m6.1.2.3" xref="Thmexample5.p1.6.m6.1.2.3.cmml"><msub id="Thmexample5.p1.6.m6.1.2.3.2" xref="Thmexample5.p1.6.m6.1.2.3.2.cmml"><mi id="Thmexample5.p1.6.m6.1.2.3.2.2" xref="Thmexample5.p1.6.m6.1.2.3.2.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmexample5.p1.6.m6.1.2.3.2.3" xref="Thmexample5.p1.6.m6.1.2.3.2.3.cmml">𝖯𝖦</mi></msub><mo id="Thmexample5.p1.6.m6.1.2.3.1" xref="Thmexample5.p1.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="Thmexample5.p1.6.m6.1.2.3.3.2" xref="Thmexample5.p1.6.m6.1.2.3.cmml"><mo id="Thmexample5.p1.6.m6.1.2.3.3.2.1" stretchy="false" xref="Thmexample5.p1.6.m6.1.2.3.cmml">(</mo><mi id="Thmexample5.p1.6.m6.1.1" xref="Thmexample5.p1.6.m6.1.1.cmml">φ</mi><mo id="Thmexample5.p1.6.m6.1.2.3.3.2.2" stretchy="false" xref="Thmexample5.p1.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.6.m6.1b"><apply id="Thmexample5.p1.6.m6.1.2.cmml" xref="Thmexample5.p1.6.m6.1.2"><apply id="Thmexample5.p1.6.m6.1.2.1.cmml" xref="Thmexample5.p1.6.m6.1.2.1"><csymbol cd="ambiguous" id="Thmexample5.p1.6.m6.1.2.1.1.cmml" xref="Thmexample5.p1.6.m6.1.2.1">superscript</csymbol><eq id="Thmexample5.p1.6.m6.1.2.1.2.cmml" xref="Thmexample5.p1.6.m6.1.2.1.2"></eq><ci id="Thmexample5.p1.6.m6.1.2.1.3a.cmml" xref="Thmexample5.p1.6.m6.1.2.1.3"><mtext id="Thmexample5.p1.6.m6.1.2.1.3.cmml" mathsize="50%" xref="Thmexample5.p1.6.m6.1.2.1.3">def</mtext></ci></apply><ci id="Thmexample5.p1.6.m6.1.2.2.cmml" xref="Thmexample5.p1.6.m6.1.2.2">𝜓</ci><apply id="Thmexample5.p1.6.m6.1.2.3.cmml" xref="Thmexample5.p1.6.m6.1.2.3"><times id="Thmexample5.p1.6.m6.1.2.3.1.cmml" xref="Thmexample5.p1.6.m6.1.2.3.1"></times><apply id="Thmexample5.p1.6.m6.1.2.3.2.cmml" xref="Thmexample5.p1.6.m6.1.2.3.2"><csymbol cd="ambiguous" id="Thmexample5.p1.6.m6.1.2.3.2.1.cmml" xref="Thmexample5.p1.6.m6.1.2.3.2">subscript</csymbol><ci id="Thmexample5.p1.6.m6.1.2.3.2.2.cmml" xref="Thmexample5.p1.6.m6.1.2.3.2.2">𝖢𝖭𝖥</ci><ci id="Thmexample5.p1.6.m6.1.2.3.2.3.cmml" xref="Thmexample5.p1.6.m6.1.2.3.2.3">𝖯𝖦</ci></apply><ci id="Thmexample5.p1.6.m6.1.1.cmml" xref="Thmexample5.p1.6.m6.1.1">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.6.m6.1c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.6.m6.1d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math> instead. Since <math alttext="\psi" class="ltx_Math" display="inline" id="Thmexample5.p1.7.m7.1"><semantics id="Thmexample5.p1.7.m7.1a"><mi id="Thmexample5.p1.7.m7.1.1" xref="Thmexample5.p1.7.m7.1.1.cmml">ψ</mi><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.7.m7.1b"><ci id="Thmexample5.p1.7.m7.1.1.cmml" xref="Thmexample5.p1.7.m7.1.1">𝜓</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.7.m7.1c">\psi</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.7.m7.1d">italic_ψ</annotation></semantics></math> is <math alttext="\mathsf{CNF_{Ts}}" class="ltx_Math" display="inline" id="Thmexample5.p1.8.m8.1"><semantics id="Thmexample5.p1.8.m8.1a"><msub id="Thmexample5.p1.8.m8.1.1" xref="Thmexample5.p1.8.m8.1.1.cmml"><mi id="Thmexample5.p1.8.m8.1.1.2" xref="Thmexample5.p1.8.m8.1.1.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmexample5.p1.8.m8.1.1.3" xref="Thmexample5.p1.8.m8.1.1.3.cmml">𝖳𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.8.m8.1b"><apply id="Thmexample5.p1.8.m8.1.1.cmml" xref="Thmexample5.p1.8.m8.1.1"><csymbol cd="ambiguous" id="Thmexample5.p1.8.m8.1.1.1.cmml" xref="Thmexample5.p1.8.m8.1.1">subscript</csymbol><ci id="Thmexample5.p1.8.m8.1.1.2.cmml" xref="Thmexample5.p1.8.m8.1.1.2">𝖢𝖭𝖥</ci><ci id="Thmexample5.p1.8.m8.1.1.3.cmml" xref="Thmexample5.p1.8.m8.1.1.3">𝖳𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.8.m8.1c">\mathsf{CNF_{Ts}}</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.8.m8.1d">sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT</annotation></semantics></math>(<math alttext="\varphi" class="ltx_Math" display="inline" id="Thmexample5.p1.9.m9.1"><semantics id="Thmexample5.p1.9.m9.1a"><mi id="Thmexample5.p1.9.m9.1.1" xref="Thmexample5.p1.9.m9.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.9.m9.1b"><ci id="Thmexample5.p1.9.m9.1.1.cmml" xref="Thmexample5.p1.9.m9.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.9.m9.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.9.m9.1d">italic_φ</annotation></semantics></math>) minus its two ternary clauses, we also have that <math alttext="{\sf SE}[{\exists{\mathbf{B}}}.\psi]=\xcancel{(A_{1}\wedge A_{2}\wedge\neg A_{% 2})}\vee(A_{1}\wedge A_{2})\vee(A_{1}\wedge\neg A_{2})\vee\xcancel{\bot}." class="ltx_math_unparsed" display="inline" id="Thmexample5.p1.10.m10.1"><semantics id="Thmexample5.p1.10.m10.1a"><mrow id="Thmexample5.p1.10.m10.1b"><mi id="Thmexample5.p1.10.m10.1.2">𝖲𝖤</mi><mrow id="Thmexample5.p1.10.m10.1.3"><mo id="Thmexample5.p1.10.m10.1.3.1" stretchy="false">[</mo><mo id="Thmexample5.p1.10.m10.1.3.2" rspace="0.167em">∃</mo><mi id="Thmexample5.p1.10.m10.1.3.3">𝐁</mi><mo id="Thmexample5.p1.10.m10.1.3.4" lspace="0em" rspace="0.167em">.</mo><mi id="Thmexample5.p1.10.m10.1.3.5">ψ</mi><mo id="Thmexample5.p1.10.m10.1.3.6" stretchy="false">]</mo></mrow><mo id="Thmexample5.p1.10.m10.1.4">=</mo><menclose id="Thmexample5.p1.10.m10.1.1" notation="updiagonalstrike downdiagonalstrike"><mrow id="Thmexample5.p1.10.m10.1.1.1.1"><mo id="Thmexample5.p1.10.m10.1.1.1.1.2" stretchy="false">(</mo><mrow id="Thmexample5.p1.10.m10.1.1.1.1.1"><msub id="Thmexample5.p1.10.m10.1.1.1.1.1.2"><mi id="Thmexample5.p1.10.m10.1.1.1.1.1.2.2">A</mi><mn id="Thmexample5.p1.10.m10.1.1.1.1.1.2.3">1</mn></msub><mo id="Thmexample5.p1.10.m10.1.1.1.1.1.1">∧</mo><msub id="Thmexample5.p1.10.m10.1.1.1.1.1.3"><mi id="Thmexample5.p1.10.m10.1.1.1.1.1.3.2">A</mi><mn id="Thmexample5.p1.10.m10.1.1.1.1.1.3.3">2</mn></msub><mo id="Thmexample5.p1.10.m10.1.1.1.1.1.1a">∧</mo><mrow id="Thmexample5.p1.10.m10.1.1.1.1.1.4"><mo id="Thmexample5.p1.10.m10.1.1.1.1.1.4.1" rspace="0.167em">¬</mo><msub id="Thmexample5.p1.10.m10.1.1.1.1.1.4.2"><mi id="Thmexample5.p1.10.m10.1.1.1.1.1.4.2.2">A</mi><mn id="Thmexample5.p1.10.m10.1.1.1.1.1.4.2.3">2</mn></msub></mrow></mrow><mo id="Thmexample5.p1.10.m10.1.1.1.1.3" stretchy="false">)</mo></mrow></menclose><mo id="Thmexample5.p1.10.m10.1.5">∨</mo><mrow id="Thmexample5.p1.10.m10.1.6"><mo id="Thmexample5.p1.10.m10.1.6.1" stretchy="false">(</mo><msub id="Thmexample5.p1.10.m10.1.6.2"><mi id="Thmexample5.p1.10.m10.1.6.2.2">A</mi><mn id="Thmexample5.p1.10.m10.1.6.2.3">1</mn></msub><mo id="Thmexample5.p1.10.m10.1.6.3">∧</mo><msub id="Thmexample5.p1.10.m10.1.6.4"><mi id="Thmexample5.p1.10.m10.1.6.4.2">A</mi><mn id="Thmexample5.p1.10.m10.1.6.4.3">2</mn></msub><mo id="Thmexample5.p1.10.m10.1.6.5" stretchy="false">)</mo></mrow><mo id="Thmexample5.p1.10.m10.1.7">∨</mo><mrow id="Thmexample5.p1.10.m10.1.8"><mo id="Thmexample5.p1.10.m10.1.8.1" stretchy="false">(</mo><msub id="Thmexample5.p1.10.m10.1.8.2"><mi id="Thmexample5.p1.10.m10.1.8.2.2">A</mi><mn id="Thmexample5.p1.10.m10.1.8.2.3">1</mn></msub><mo id="Thmexample5.p1.10.m10.1.8.3">∧</mo><mo id="Thmexample5.p1.10.m10.1.8.4" rspace="0.167em">¬</mo><msub id="Thmexample5.p1.10.m10.1.8.5"><mi id="Thmexample5.p1.10.m10.1.8.5.2">A</mi><mn id="Thmexample5.p1.10.m10.1.8.5.3">2</mn></msub><mo id="Thmexample5.p1.10.m10.1.8.6" stretchy="false">)</mo></mrow><mo id="Thmexample5.p1.10.m10.1.9">∨</mo><menclose id="Thmexample5.p1.10.m10.1.10" notation="updiagonalstrike downdiagonalstrike"><mo id="Thmexample5.p1.10.m10.1.10.2">⊥</mo></menclose><mo id="Thmexample5.p1.10.m10.1.11" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="Thmexample5.p1.10.m10.1c">{\sf SE}[{\exists{\mathbf{B}}}.\psi]=\xcancel{(A_{1}\wedge A_{2}\wedge\neg A_{% 2})}\vee(A_{1}\wedge A_{2})\vee(A_{1}\wedge\neg A_{2})\vee\xcancel{\bot}.</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.10.m10.1d">sansserif_SE [ ∃ bold_B . italic_ψ ] = cancel ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ cancel ⊥ .</annotation></semantics></math> <br class="ltx_break"/>Thus <math alttext="\mu\models\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\varphi)" class="ltx_Math" display="inline" id="Thmexample5.p1.11.m11.3"><semantics id="Thmexample5.p1.11.m11.3a"><mrow id="Thmexample5.p1.11.m11.3.3.2" xref="Thmexample5.p1.11.m11.3.3.3.cmml"><mrow id="Thmexample5.p1.11.m11.2.2.1.1" xref="Thmexample5.p1.11.m11.2.2.1.1.cmml"><mi id="Thmexample5.p1.11.m11.2.2.1.1.2" xref="Thmexample5.p1.11.m11.2.2.1.1.2.cmml">μ</mi><mo id="Thmexample5.p1.11.m11.2.2.1.1.1" xref="Thmexample5.p1.11.m11.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmexample5.p1.11.m11.2.2.1.1.3" xref="Thmexample5.p1.11.m11.2.2.1.1.3.cmml"><mo id="Thmexample5.p1.11.m11.2.2.1.1.3.1" rspace="0.167em" xref="Thmexample5.p1.11.m11.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmexample5.p1.11.m11.2.2.1.1.3.2" xref="Thmexample5.p1.11.m11.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmexample5.p1.11.m11.3.3.2.3" lspace="0em" rspace="0.167em" xref="Thmexample5.p1.11.m11.3.3.3a.cmml">.</mo><mrow id="Thmexample5.p1.11.m11.3.3.2.2" xref="Thmexample5.p1.11.m11.3.3.2.2.cmml"><msub id="Thmexample5.p1.11.m11.3.3.2.2.2" xref="Thmexample5.p1.11.m11.3.3.2.2.2.cmml"><mi id="Thmexample5.p1.11.m11.3.3.2.2.2.2" xref="Thmexample5.p1.11.m11.3.3.2.2.2.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmexample5.p1.11.m11.3.3.2.2.2.3" xref="Thmexample5.p1.11.m11.3.3.2.2.2.3.cmml">𝖯𝖦</mi></msub><mo id="Thmexample5.p1.11.m11.3.3.2.2.1" xref="Thmexample5.p1.11.m11.3.3.2.2.1.cmml">⁢</mo><mrow id="Thmexample5.p1.11.m11.3.3.2.2.3.2" xref="Thmexample5.p1.11.m11.3.3.2.2.cmml"><mo id="Thmexample5.p1.11.m11.3.3.2.2.3.2.1" stretchy="false" xref="Thmexample5.p1.11.m11.3.3.2.2.cmml">(</mo><mi id="Thmexample5.p1.11.m11.1.1" xref="Thmexample5.p1.11.m11.1.1.cmml">φ</mi><mo id="Thmexample5.p1.11.m11.3.3.2.2.3.2.2" stretchy="false" xref="Thmexample5.p1.11.m11.3.3.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.11.m11.3b"><apply id="Thmexample5.p1.11.m11.3.3.3.cmml" xref="Thmexample5.p1.11.m11.3.3.2"><csymbol cd="ambiguous" id="Thmexample5.p1.11.m11.3.3.3a.cmml" xref="Thmexample5.p1.11.m11.3.3.2.3">formulae-sequence</csymbol><apply id="Thmexample5.p1.11.m11.2.2.1.1.cmml" xref="Thmexample5.p1.11.m11.2.2.1.1"><csymbol cd="latexml" id="Thmexample5.p1.11.m11.2.2.1.1.1.cmml" xref="Thmexample5.p1.11.m11.2.2.1.1.1">models</csymbol><ci id="Thmexample5.p1.11.m11.2.2.1.1.2.cmml" xref="Thmexample5.p1.11.m11.2.2.1.1.2">𝜇</ci><apply id="Thmexample5.p1.11.m11.2.2.1.1.3.cmml" xref="Thmexample5.p1.11.m11.2.2.1.1.3"><exists id="Thmexample5.p1.11.m11.2.2.1.1.3.1.cmml" xref="Thmexample5.p1.11.m11.2.2.1.1.3.1"></exists><ci id="Thmexample5.p1.11.m11.2.2.1.1.3.2.cmml" xref="Thmexample5.p1.11.m11.2.2.1.1.3.2">𝐁</ci></apply></apply><apply id="Thmexample5.p1.11.m11.3.3.2.2.cmml" xref="Thmexample5.p1.11.m11.3.3.2.2"><times id="Thmexample5.p1.11.m11.3.3.2.2.1.cmml" xref="Thmexample5.p1.11.m11.3.3.2.2.1"></times><apply id="Thmexample5.p1.11.m11.3.3.2.2.2.cmml" xref="Thmexample5.p1.11.m11.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmexample5.p1.11.m11.3.3.2.2.2.1.cmml" xref="Thmexample5.p1.11.m11.3.3.2.2.2">subscript</csymbol><ci id="Thmexample5.p1.11.m11.3.3.2.2.2.2.cmml" xref="Thmexample5.p1.11.m11.3.3.2.2.2.2">𝖢𝖭𝖥</ci><ci id="Thmexample5.p1.11.m11.3.3.2.2.2.3.cmml" xref="Thmexample5.p1.11.m11.3.3.2.2.2.3">𝖯𝖦</ci></apply><ci id="Thmexample5.p1.11.m11.1.1.cmml" xref="Thmexample5.p1.11.m11.1.1">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.11.m11.3c">\mu\models\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.11.m11.3d">italic_μ ⊧ ∃ bold_B . sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math> but <math alttext="\mu\not\mid\!\approx\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\varphi)" class="ltx_math_unparsed" display="inline" id="Thmexample5.p1.12.m12.1"><semantics id="Thmexample5.p1.12.m12.1a"><mrow id="Thmexample5.p1.12.m12.1b"><mi id="Thmexample5.p1.12.m12.1.1">μ</mi><mpadded id="Thmexample5.p1.12.m12.1c" width="0.969em"><mo id="Thmexample5.p1.12.m12.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample5.p1.12.m12.1.3">≈</mo><mo id="Thmexample5.p1.12.m12.1.4" rspace="0.167em">∃</mo><mi id="Thmexample5.p1.12.m12.1.5">𝐁</mi><mo id="Thmexample5.p1.12.m12.1.6" lspace="0em" rspace="0.167em">.</mo><msub id="Thmexample5.p1.12.m12.1.7"><mi id="Thmexample5.p1.12.m12.1.7.2">𝖢𝖭𝖥</mi><mi id="Thmexample5.p1.12.m12.1.7.3">𝖯𝖦</mi></msub><mrow id="Thmexample5.p1.12.m12.1.8"><mo id="Thmexample5.p1.12.m12.1.8.1" stretchy="false">(</mo><mi id="Thmexample5.p1.12.m12.1.8.2">φ</mi><mo id="Thmexample5.p1.12.m12.1.8.3" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmexample5.p1.12.m12.1d">\mu\not\mid\!\approx\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.12.m12.1e">italic_μ ∤ ≈ ∃ bold_B . sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math>. <math alttext="\diamond" class="ltx_Math" display="inline" id="Thmexample5.p1.13.m13.1"><semantics id="Thmexample5.p1.13.m13.1a"><mo id="Thmexample5.p1.13.m13.1.1" xref="Thmexample5.p1.13.m13.1.1.cmml">⋄</mo><annotation-xml encoding="MathML-Content" id="Thmexample5.p1.13.m13.1b"><ci id="Thmexample5.p1.13.m13.1.1.cmml" xref="Thmexample5.p1.13.m13.1.1">⋄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample5.p1.13.m13.1c">\diamond</annotation><annotation encoding="application/x-llamapun" id="Thmexample5.p1.13.m13.1d">⋄</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S3.SS4.p3"> <p class="ltx_p" id="S3.SS4.p3.8">Intuitively, the whole point is that both <math alttext="\mathsf{CNF_{Ts}}" class="ltx_Math" display="inline" id="S3.SS4.p3.1.m1.1"><semantics id="S3.SS4.p3.1.m1.1a"><msub id="S3.SS4.p3.1.m1.1.1" xref="S3.SS4.p3.1.m1.1.1.cmml"><mi id="S3.SS4.p3.1.m1.1.1.2" xref="S3.SS4.p3.1.m1.1.1.2.cmml">𝖢𝖭𝖥</mi><mi id="S3.SS4.p3.1.m1.1.1.3" xref="S3.SS4.p3.1.m1.1.1.3.cmml">𝖳𝗌</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.1.m1.1b"><apply id="S3.SS4.p3.1.m1.1.1.cmml" xref="S3.SS4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.1.m1.1.1.1.cmml" xref="S3.SS4.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS4.p3.1.m1.1.1.2.cmml" xref="S3.SS4.p3.1.m1.1.1.2">𝖢𝖭𝖥</ci><ci id="S3.SS4.p3.1.m1.1.1.3.cmml" xref="S3.SS4.p3.1.m1.1.1.3">𝖳𝗌</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.1.m1.1c">\mathsf{CNF_{Ts}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.1.m1.1d">sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT</annotation></semantics></math>(<math alttext="\varphi" class="ltx_Math" display="inline" id="S3.SS4.p3.2.m2.1"><semantics id="S3.SS4.p3.2.m2.1a"><mi id="S3.SS4.p3.2.m2.1.1" xref="S3.SS4.p3.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.2.m2.1b"><ci id="S3.SS4.p3.2.m2.1.1.cmml" xref="S3.SS4.p3.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.2.m2.1d">italic_φ</annotation></semantics></math>) and <math alttext="\mathsf{CNF_{PG}}" class="ltx_Math" display="inline" id="S3.SS4.p3.3.m3.1"><semantics id="S3.SS4.p3.3.m3.1a"><msub id="S3.SS4.p3.3.m3.1.1" xref="S3.SS4.p3.3.m3.1.1.cmml"><mi id="S3.SS4.p3.3.m3.1.1.2" xref="S3.SS4.p3.3.m3.1.1.2.cmml">𝖢𝖭𝖥</mi><mi id="S3.SS4.p3.3.m3.1.1.3" xref="S3.SS4.p3.3.m3.1.1.3.cmml">𝖯𝖦</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.3.m3.1b"><apply id="S3.SS4.p3.3.m3.1.1.cmml" xref="S3.SS4.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS4.p3.3.m3.1.1.1.cmml" xref="S3.SS4.p3.3.m3.1.1">subscript</csymbol><ci id="S3.SS4.p3.3.m3.1.1.2.cmml" xref="S3.SS4.p3.3.m3.1.1.2">𝖢𝖭𝖥</ci><ci id="S3.SS4.p3.3.m3.1.1.3.cmml" xref="S3.SS4.p3.3.m3.1.1.3">𝖯𝖦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.3.m3.1c">\mathsf{CNF_{PG}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.3.m3.1d">sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT</annotation></semantics></math>(<math alttext="\varphi" class="ltx_Math" display="inline" id="S3.SS4.p3.4.m4.1"><semantics id="S3.SS4.p3.4.m4.1a"><mi id="S3.SS4.p3.4.m4.1.1" xref="S3.SS4.p3.4.m4.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.4.m4.1b"><ci id="S3.SS4.p3.4.m4.1.1.cmml" xref="S3.SS4.p3.4.m4.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.4.m4.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.4.m4.1d">italic_φ</annotation></semantics></math>) introduce fresh variables <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="S3.SS4.p3.5.m5.1"><semantics id="S3.SS4.p3.5.m5.1a"><mi id="S3.SS4.p3.5.m5.1.1" xref="S3.SS4.p3.5.m5.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.5.m5.1b"><ci id="S3.SS4.p3.5.m5.1.1.cmml" xref="S3.SS4.p3.5.m5.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.5.m5.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.5.m5.1d">bold_B</annotation></semantics></math>, which must be implicitly existentially quantified away in order to preserve equivalence and thus produce the set of original satisfying assignments (property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty4" title="Property 4 ‣ CNF-ization. ‣ 2 Background ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">4</span></a>). Although these variable do not affect entailment, they may affect verification. In fact, since <math alttext="\varphi" class="ltx_Math" display="inline" id="S3.SS4.p3.6.m6.1"><semantics id="S3.SS4.p3.6.m6.1a"><mi id="S3.SS4.p3.6.m6.1.1" xref="S3.SS4.p3.6.m6.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.6.m6.1b"><ci id="S3.SS4.p3.6.m6.1.1.cmml" xref="S3.SS4.p3.6.m6.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p3.6.m6.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.6.m6.1d">italic_φ</annotation></semantics></math> and <math alttext="\exists{\mathbf{B}}.\mathsf{CNF_{Ts}}(\varphi)" class="ltx_Math" display="inline" id="S3.SS4.p3.7.m7.3"><semantics id="S3.SS4.p3.7.m7.3a"><mrow id="S3.SS4.p3.7.m7.3.3.2" xref="S3.SS4.p3.7.m7.3.3.3.cmml"><mrow id="S3.SS4.p3.7.m7.2.2.1.1" xref="S3.SS4.p3.7.m7.2.2.1.1.cmml"><mo id="S3.SS4.p3.7.m7.2.2.1.1.1" rspace="0.167em" xref="S3.SS4.p3.7.m7.2.2.1.1.1.cmml">∃</mo><mi id="S3.SS4.p3.7.m7.2.2.1.1.2" xref="S3.SS4.p3.7.m7.2.2.1.1.2.cmml">𝐁</mi></mrow><mo id="S3.SS4.p3.7.m7.3.3.2.3" lspace="0em" rspace="0.167em" xref="S3.SS4.p3.7.m7.3.3.3a.cmml">.</mo><mrow id="S3.SS4.p3.7.m7.3.3.2.2" xref="S3.SS4.p3.7.m7.3.3.2.2.cmml"><msub id="S3.SS4.p3.7.m7.3.3.2.2.2" xref="S3.SS4.p3.7.m7.3.3.2.2.2.cmml"><mi id="S3.SS4.p3.7.m7.3.3.2.2.2.2" xref="S3.SS4.p3.7.m7.3.3.2.2.2.2.cmml">𝖢𝖭𝖥</mi><mi id="S3.SS4.p3.7.m7.3.3.2.2.2.3" xref="S3.SS4.p3.7.m7.3.3.2.2.2.3.cmml">𝖳𝗌</mi></msub><mo id="S3.SS4.p3.7.m7.3.3.2.2.1" xref="S3.SS4.p3.7.m7.3.3.2.2.1.cmml">⁢</mo><mrow id="S3.SS4.p3.7.m7.3.3.2.2.3.2" xref="S3.SS4.p3.7.m7.3.3.2.2.cmml"><mo id="S3.SS4.p3.7.m7.3.3.2.2.3.2.1" stretchy="false" xref="S3.SS4.p3.7.m7.3.3.2.2.cmml">(</mo><mi id="S3.SS4.p3.7.m7.1.1" xref="S3.SS4.p3.7.m7.1.1.cmml">φ</mi><mo id="S3.SS4.p3.7.m7.3.3.2.2.3.2.2" stretchy="false" xref="S3.SS4.p3.7.m7.3.3.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p3.7.m7.3b"><apply id="S3.SS4.p3.7.m7.3.3.3.cmml" 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encoding="application/x-tex" id="S3.SS4.p3.7.m7.3c">\exists{\mathbf{B}}.\mathsf{CNF_{Ts}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p3.7.m7.3d">∃ bold_B . sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math> [resp. <math alttext="\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\varphi)" class="ltx_Math" display="inline" id="S3.SS4.p3.8.m8.3"><semantics id="S3.SS4.p3.8.m8.3a"><mrow id="S3.SS4.p3.8.m8.3.3.2" xref="S3.SS4.p3.8.m8.3.3.3.cmml"><mrow id="S3.SS4.p3.8.m8.2.2.1.1" xref="S3.SS4.p3.8.m8.2.2.1.1.cmml"><mo id="S3.SS4.p3.8.m8.2.2.1.1.1" rspace="0.167em" xref="S3.SS4.p3.8.m8.2.2.1.1.1.cmml">∃</mo><mi id="S3.SS4.p3.8.m8.2.2.1.1.2" xref="S3.SS4.p3.8.m8.2.2.1.1.2.cmml">𝐁</mi></mrow><mo id="S3.SS4.p3.8.m8.3.3.2.3" lspace="0em" rspace="0.167em" xref="S3.SS4.p3.8.m8.3.3.3a.cmml">.</mo><mrow id="S3.SS4.p3.8.m8.3.3.2.2" xref="S3.SS4.p3.8.m8.3.3.2.2.cmml"><msub id="S3.SS4.p3.8.m8.3.3.2.2.2" 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end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math>] are equivalent, they are entailed by the same partial assignments, but they are not verified by the same partial assignments.</p> </div> <div class="ltx_para" id="S3.SS4.p4"> <p class="ltx_p" id="S3.SS4.p4.6">To this extent, in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib25" title="">25</a>]</cite> we addressed a different though related problem: we proved that —unlike with <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{Ts}}(\varphi)" class="ltx_Math" display="inline" id="S3.SS4.p4.1.m1.1"><semantics id="S3.SS4.p4.1.m1.1a"><mrow id="S3.SS4.p4.1.m1.1.2" xref="S3.SS4.p4.1.m1.1.2.cmml"><mi id="S3.SS4.p4.1.m1.1.2.2" xref="S3.SS4.p4.1.m1.1.2.2.cmml">ψ</mi><mover id="S3.SS4.p4.1.m1.1.2.1" xref="S3.SS4.p4.1.m1.1.2.1.cmml"><mo id="S3.SS4.p4.1.m1.1.2.1.2" xref="S3.SS4.p4.1.m1.1.2.1.2.cmml">=</mo><mtext id="S3.SS4.p4.1.m1.1.2.1.3" mathsize="71%" xref="S3.SS4.p4.1.m1.1.2.1.3a.cmml">def</mtext></mover><mrow id="S3.SS4.p4.1.m1.1.2.3" xref="S3.SS4.p4.1.m1.1.2.3.cmml"><msub id="S3.SS4.p4.1.m1.1.2.3.2" xref="S3.SS4.p4.1.m1.1.2.3.2.cmml"><mi id="S3.SS4.p4.1.m1.1.2.3.2.2" xref="S3.SS4.p4.1.m1.1.2.3.2.2.cmml">𝖢𝖭𝖥</mi><mi id="S3.SS4.p4.1.m1.1.2.3.2.3" xref="S3.SS4.p4.1.m1.1.2.3.2.3.cmml">𝖳𝗌</mi></msub><mo id="S3.SS4.p4.1.m1.1.2.3.1" xref="S3.SS4.p4.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S3.SS4.p4.1.m1.1.2.3.3.2" xref="S3.SS4.p4.1.m1.1.2.3.cmml"><mo id="S3.SS4.p4.1.m1.1.2.3.3.2.1" stretchy="false" xref="S3.SS4.p4.1.m1.1.2.3.cmml">(</mo><mi id="S3.SS4.p4.1.m1.1.1" xref="S3.SS4.p4.1.m1.1.1.cmml">φ</mi><mo id="S3.SS4.p4.1.m1.1.2.3.3.2.2" stretchy="false" xref="S3.SS4.p4.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.1.m1.1b"><apply id="S3.SS4.p4.1.m1.1.2.cmml" xref="S3.SS4.p4.1.m1.1.2"><apply id="S3.SS4.p4.1.m1.1.2.1.cmml" xref="S3.SS4.p4.1.m1.1.2.1"><csymbol cd="ambiguous" 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encoding="application/x-tex" id="S3.SS4.p4.1.m1.1c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{Ts}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.1.m1.1d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math> or with <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\varphi)" class="ltx_Math" display="inline" id="S3.SS4.p4.2.m2.1"><semantics id="S3.SS4.p4.2.m2.1a"><mrow id="S3.SS4.p4.2.m2.1.2" xref="S3.SS4.p4.2.m2.1.2.cmml"><mi id="S3.SS4.p4.2.m2.1.2.2" xref="S3.SS4.p4.2.m2.1.2.2.cmml">ψ</mi><mover id="S3.SS4.p4.2.m2.1.2.1" xref="S3.SS4.p4.2.m2.1.2.1.cmml"><mo id="S3.SS4.p4.2.m2.1.2.1.2" xref="S3.SS4.p4.2.m2.1.2.1.2.cmml">=</mo><mtext id="S3.SS4.p4.2.m2.1.2.1.3" mathsize="71%" xref="S3.SS4.p4.2.m2.1.2.1.3a.cmml">def</mtext></mover><mrow 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id="S3.SS4.p4.2.m2.1c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.2.m2.1d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math>— if <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\mathsf{% NNF}(\varphi))" class="ltx_Math" display="inline" id="S3.SS4.p4.3.m3.2"><semantics id="S3.SS4.p4.3.m3.2a"><mrow id="S3.SS4.p4.3.m3.2.2" xref="S3.SS4.p4.3.m3.2.2.cmml"><mi id="S3.SS4.p4.3.m3.2.2.3" xref="S3.SS4.p4.3.m3.2.2.3.cmml">ψ</mi><mover id="S3.SS4.p4.3.m3.2.2.2" xref="S3.SS4.p4.3.m3.2.2.2.cmml"><mo id="S3.SS4.p4.3.m3.2.2.2.2" xref="S3.SS4.p4.3.m3.2.2.2.2.cmml">=</mo><mtext id="S3.SS4.p4.3.m3.2.2.2.3" mathsize="71%" xref="S3.SS4.p4.3.m3.2.2.2.3a.cmml">def</mtext></mover><mrow id="S3.SS4.p4.3.m3.2.2.1" xref="S3.SS4.p4.3.m3.2.2.1.cmml"><msub id="S3.SS4.p4.3.m3.2.2.1.3" xref="S3.SS4.p4.3.m3.2.2.1.3.cmml"><mi id="S3.SS4.p4.3.m3.2.2.1.3.2" xref="S3.SS4.p4.3.m3.2.2.1.3.2.cmml">𝖢𝖭𝖥</mi><mi id="S3.SS4.p4.3.m3.2.2.1.3.3" xref="S3.SS4.p4.3.m3.2.2.1.3.3.cmml">𝖯𝖦</mi></msub><mo id="S3.SS4.p4.3.m3.2.2.1.2" xref="S3.SS4.p4.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS4.p4.3.m3.2.2.1.1.1" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.cmml"><mo id="S3.SS4.p4.3.m3.2.2.1.1.1.2" stretchy="false" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p4.3.m3.2.2.1.1.1.1" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.cmml"><mi id="S3.SS4.p4.3.m3.2.2.1.1.1.1.2" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.2.cmml">𝖭𝖭𝖥</mi><mo id="S3.SS4.p4.3.m3.2.2.1.1.1.1.1" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.p4.3.m3.2.2.1.1.1.1.3.2" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.cmml"><mo id="S3.SS4.p4.3.m3.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.cmml">(</mo><mi id="S3.SS4.p4.3.m3.1.1" xref="S3.SS4.p4.3.m3.1.1.cmml">φ</mi><mo id="S3.SS4.p4.3.m3.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.p4.3.m3.2.2.1.1.1.3" stretchy="false" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.3.m3.2b"><apply id="S3.SS4.p4.3.m3.2.2.cmml" xref="S3.SS4.p4.3.m3.2.2"><apply id="S3.SS4.p4.3.m3.2.2.2.cmml" xref="S3.SS4.p4.3.m3.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.p4.3.m3.2.2.2.1.cmml" xref="S3.SS4.p4.3.m3.2.2.2">superscript</csymbol><eq id="S3.SS4.p4.3.m3.2.2.2.2.cmml" xref="S3.SS4.p4.3.m3.2.2.2.2"></eq><ci id="S3.SS4.p4.3.m3.2.2.2.3a.cmml" xref="S3.SS4.p4.3.m3.2.2.2.3"><mtext id="S3.SS4.p4.3.m3.2.2.2.3.cmml" mathsize="50%" xref="S3.SS4.p4.3.m3.2.2.2.3">def</mtext></ci></apply><ci id="S3.SS4.p4.3.m3.2.2.3.cmml" xref="S3.SS4.p4.3.m3.2.2.3">𝜓</ci><apply id="S3.SS4.p4.3.m3.2.2.1.cmml" xref="S3.SS4.p4.3.m3.2.2.1"><times id="S3.SS4.p4.3.m3.2.2.1.2.cmml" xref="S3.SS4.p4.3.m3.2.2.1.2"></times><apply id="S3.SS4.p4.3.m3.2.2.1.3.cmml" xref="S3.SS4.p4.3.m3.2.2.1.3"><csymbol cd="ambiguous" id="S3.SS4.p4.3.m3.2.2.1.3.1.cmml" xref="S3.SS4.p4.3.m3.2.2.1.3">subscript</csymbol><ci id="S3.SS4.p4.3.m3.2.2.1.3.2.cmml" xref="S3.SS4.p4.3.m3.2.2.1.3.2">𝖢𝖭𝖥</ci><ci id="S3.SS4.p4.3.m3.2.2.1.3.3.cmml" xref="S3.SS4.p4.3.m3.2.2.1.3.3">𝖯𝖦</ci></apply><apply id="S3.SS4.p4.3.m3.2.2.1.1.1.1.cmml" xref="S3.SS4.p4.3.m3.2.2.1.1.1"><times id="S3.SS4.p4.3.m3.2.2.1.1.1.1.1.cmml" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.1"></times><ci id="S3.SS4.p4.3.m3.2.2.1.1.1.1.2.cmml" xref="S3.SS4.p4.3.m3.2.2.1.1.1.1.2">𝖭𝖭𝖥</ci><ci id="S3.SS4.p4.3.m3.1.1.cmml" xref="S3.SS4.p4.3.m3.1.1">𝜑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.3.m3.2c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\mathsf{% NNF}(\varphi))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.3.m3.2d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( sansserif_NNF ( italic_φ ) )</annotation></semantics></math> and <math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S3.SS4.p4.4.m4.1"><semantics id="S3.SS4.p4.4.m4.1a"><mrow id="S3.SS4.p4.4.m4.1b"><mi id="S3.SS4.p4.4.m4.1.1">μ</mi><mpadded id="S3.SS4.p4.4.m4.1c" width="0.219em"><mo id="S3.SS4.p4.4.m4.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.SS4.p4.4.m4.1.3">≈</mo><mi id="S3.SS4.p4.4.m4.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S3.SS4.p4.4.m4.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.4.m4.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math>, then <math alttext="\mu\mid\!\approx\exists{\mathbf{B}}.\psi" class="ltx_math_unparsed" display="inline" id="S3.SS4.p4.5.m5.1"><semantics id="S3.SS4.p4.5.m5.1a"><mrow id="S3.SS4.p4.5.m5.1b"><mi id="S3.SS4.p4.5.m5.1.1">μ</mi><mpadded id="S3.SS4.p4.5.m5.1c" width="0.219em"><mo id="S3.SS4.p4.5.m5.1.2" lspace="0em">∣</mo></mpadded><mo id="S3.SS4.p4.5.m5.1.3">≈</mo><mo id="S3.SS4.p4.5.m5.1.4" rspace="0.167em">∃</mo><mi id="S3.SS4.p4.5.m5.1.5">𝐁</mi><mo id="S3.SS4.p4.5.m5.1.6" lspace="0em" rspace="0.167em">.</mo><mi id="S3.SS4.p4.5.m5.1.7">ψ</mi></mrow><annotation encoding="application/x-tex" id="S3.SS4.p4.5.m5.1d">\mu\mid\!\approx\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.5.m5.1e">italic_μ ∣ ≈ ∃ bold_B . italic_ψ</annotation></semantics></math>. Notice, however, that this fact does not apply to our problem, because <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.Thmtheorem6" title="Theorem 3.6 ‣ 3.4 Verification and entailment of CNF-ized non-CNF formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">theorem</span> <span class="ltx_text ltx_ref_tag">3.6</span></a> holds also for <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\mathsf{% NNF}(\varphi))" class="ltx_Math" display="inline" id="S3.SS4.p4.6.m6.2"><semantics id="S3.SS4.p4.6.m6.2a"><mrow id="S3.SS4.p4.6.m6.2.2" xref="S3.SS4.p4.6.m6.2.2.cmml"><mi id="S3.SS4.p4.6.m6.2.2.3" xref="S3.SS4.p4.6.m6.2.2.3.cmml">ψ</mi><mover id="S3.SS4.p4.6.m6.2.2.2" xref="S3.SS4.p4.6.m6.2.2.2.cmml"><mo id="S3.SS4.p4.6.m6.2.2.2.2" xref="S3.SS4.p4.6.m6.2.2.2.2.cmml">=</mo><mtext id="S3.SS4.p4.6.m6.2.2.2.3" mathsize="71%" xref="S3.SS4.p4.6.m6.2.2.2.3a.cmml">def</mtext></mover><mrow id="S3.SS4.p4.6.m6.2.2.1" xref="S3.SS4.p4.6.m6.2.2.1.cmml"><msub id="S3.SS4.p4.6.m6.2.2.1.3" xref="S3.SS4.p4.6.m6.2.2.1.3.cmml"><mi id="S3.SS4.p4.6.m6.2.2.1.3.2" xref="S3.SS4.p4.6.m6.2.2.1.3.2.cmml">𝖢𝖭𝖥</mi><mi id="S3.SS4.p4.6.m6.2.2.1.3.3" xref="S3.SS4.p4.6.m6.2.2.1.3.3.cmml">𝖯𝖦</mi></msub><mo id="S3.SS4.p4.6.m6.2.2.1.2" xref="S3.SS4.p4.6.m6.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS4.p4.6.m6.2.2.1.1.1" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.cmml"><mo id="S3.SS4.p4.6.m6.2.2.1.1.1.2" stretchy="false" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS4.p4.6.m6.2.2.1.1.1.1" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.cmml"><mi id="S3.SS4.p4.6.m6.2.2.1.1.1.1.2" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.2.cmml">𝖭𝖭𝖥</mi><mo id="S3.SS4.p4.6.m6.2.2.1.1.1.1.1" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS4.p4.6.m6.2.2.1.1.1.1.3.2" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.cmml"><mo id="S3.SS4.p4.6.m6.2.2.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.cmml">(</mo><mi id="S3.SS4.p4.6.m6.1.1" xref="S3.SS4.p4.6.m6.1.1.cmml">φ</mi><mo id="S3.SS4.p4.6.m6.2.2.1.1.1.1.3.2.2" stretchy="false" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS4.p4.6.m6.2.2.1.1.1.3" stretchy="false" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p4.6.m6.2b"><apply id="S3.SS4.p4.6.m6.2.2.cmml" xref="S3.SS4.p4.6.m6.2.2"><apply id="S3.SS4.p4.6.m6.2.2.2.cmml" xref="S3.SS4.p4.6.m6.2.2.2"><csymbol cd="ambiguous" id="S3.SS4.p4.6.m6.2.2.2.1.cmml" xref="S3.SS4.p4.6.m6.2.2.2">superscript</csymbol><eq id="S3.SS4.p4.6.m6.2.2.2.2.cmml" xref="S3.SS4.p4.6.m6.2.2.2.2"></eq><ci id="S3.SS4.p4.6.m6.2.2.2.3a.cmml" xref="S3.SS4.p4.6.m6.2.2.2.3"><mtext id="S3.SS4.p4.6.m6.2.2.2.3.cmml" mathsize="50%" xref="S3.SS4.p4.6.m6.2.2.2.3">def</mtext></ci></apply><ci id="S3.SS4.p4.6.m6.2.2.3.cmml" xref="S3.SS4.p4.6.m6.2.2.3">𝜓</ci><apply id="S3.SS4.p4.6.m6.2.2.1.cmml" xref="S3.SS4.p4.6.m6.2.2.1"><times id="S3.SS4.p4.6.m6.2.2.1.2.cmml" xref="S3.SS4.p4.6.m6.2.2.1.2"></times><apply id="S3.SS4.p4.6.m6.2.2.1.3.cmml" xref="S3.SS4.p4.6.m6.2.2.1.3"><csymbol cd="ambiguous" id="S3.SS4.p4.6.m6.2.2.1.3.1.cmml" xref="S3.SS4.p4.6.m6.2.2.1.3">subscript</csymbol><ci id="S3.SS4.p4.6.m6.2.2.1.3.2.cmml" xref="S3.SS4.p4.6.m6.2.2.1.3.2">𝖢𝖭𝖥</ci><ci id="S3.SS4.p4.6.m6.2.2.1.3.3.cmml" xref="S3.SS4.p4.6.m6.2.2.1.3.3">𝖯𝖦</ci></apply><apply id="S3.SS4.p4.6.m6.2.2.1.1.1.1.cmml" xref="S3.SS4.p4.6.m6.2.2.1.1.1"><times id="S3.SS4.p4.6.m6.2.2.1.1.1.1.1.cmml" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.1"></times><ci id="S3.SS4.p4.6.m6.2.2.1.1.1.1.2.cmml" xref="S3.SS4.p4.6.m6.2.2.1.1.1.1.2">𝖭𝖭𝖥</ci><ci id="S3.SS4.p4.6.m6.1.1.cmml" xref="S3.SS4.p4.6.m6.1.1">𝜑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p4.6.m6.2c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\mathsf{% NNF}(\varphi))</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p4.6.m6.2d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( sansserif_NNF ( italic_φ ) )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample6"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 6</span></h6> <div class="ltx_para" id="Thmexample6.p1"> <p class="ltx_p" id="Thmexample6.p1.7">Consider <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmexample6.p1.1.m1.1"><semantics id="Thmexample6.p1.1.m1.1a"><mi id="Thmexample6.p1.1.m1.1.1" xref="Thmexample6.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmexample6.p1.1.m1.1b"><ci id="Thmexample6.p1.1.m1.1.1.cmml" xref="Thmexample6.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample6.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample6.p1.1.m1.1d">italic_φ</annotation></semantics></math> and <math alttext="\mu" class="ltx_Math" display="inline" id="Thmexample6.p1.2.m2.1"><semantics id="Thmexample6.p1.2.m2.1a"><mi id="Thmexample6.p1.2.m2.1.1" xref="Thmexample6.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="Thmexample6.p1.2.m2.1b"><ci id="Thmexample6.p1.2.m2.1.1.cmml" xref="Thmexample6.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample6.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="Thmexample6.p1.2.m2.1d">italic_μ</annotation></semantics></math> as in <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample5" title="Example 5 ‣ 3.4 Verification and entailment of CNF-ized non-CNF formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">5</span></a>. Since <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmexample6.p1.3.m3.1"><semantics id="Thmexample6.p1.3.m3.1a"><mi id="Thmexample6.p1.3.m3.1.1" xref="Thmexample6.p1.3.m3.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmexample6.p1.3.m3.1b"><ci id="Thmexample6.p1.3.m3.1.1.cmml" xref="Thmexample6.p1.3.m3.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample6.p1.3.m3.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample6.p1.3.m3.1d">italic_φ</annotation></semantics></math> is already in NNF, then we have that <math alttext="\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\mathsf{% NNF}(\varphi))=\mathsf{CNF_{PG}}(\varphi)" class="ltx_Math" display="inline" id="Thmexample6.p1.4.m4.3"><semantics id="Thmexample6.p1.4.m4.3a"><mrow id="Thmexample6.p1.4.m4.3.3" xref="Thmexample6.p1.4.m4.3.3.cmml"><mi id="Thmexample6.p1.4.m4.3.3.3" xref="Thmexample6.p1.4.m4.3.3.3.cmml">ψ</mi><mover id="Thmexample6.p1.4.m4.3.3.4" xref="Thmexample6.p1.4.m4.3.3.4.cmml"><mo id="Thmexample6.p1.4.m4.3.3.4.2" xref="Thmexample6.p1.4.m4.3.3.4.2.cmml">=</mo><mtext id="Thmexample6.p1.4.m4.3.3.4.3" mathsize="71%" xref="Thmexample6.p1.4.m4.3.3.4.3a.cmml">def</mtext></mover><mrow id="Thmexample6.p1.4.m4.3.3.1" xref="Thmexample6.p1.4.m4.3.3.1.cmml"><msub id="Thmexample6.p1.4.m4.3.3.1.3" xref="Thmexample6.p1.4.m4.3.3.1.3.cmml"><mi id="Thmexample6.p1.4.m4.3.3.1.3.2" xref="Thmexample6.p1.4.m4.3.3.1.3.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmexample6.p1.4.m4.3.3.1.3.3" xref="Thmexample6.p1.4.m4.3.3.1.3.3.cmml">𝖯𝖦</mi></msub><mo id="Thmexample6.p1.4.m4.3.3.1.2" xref="Thmexample6.p1.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="Thmexample6.p1.4.m4.3.3.1.1.1" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.cmml"><mo id="Thmexample6.p1.4.m4.3.3.1.1.1.2" stretchy="false" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="Thmexample6.p1.4.m4.3.3.1.1.1.1" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.cmml"><mi id="Thmexample6.p1.4.m4.3.3.1.1.1.1.2" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.2.cmml">𝖭𝖭𝖥</mi><mo id="Thmexample6.p1.4.m4.3.3.1.1.1.1.1" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="Thmexample6.p1.4.m4.3.3.1.1.1.1.3.2" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.cmml"><mo id="Thmexample6.p1.4.m4.3.3.1.1.1.1.3.2.1" stretchy="false" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.cmml">(</mo><mi id="Thmexample6.p1.4.m4.1.1" xref="Thmexample6.p1.4.m4.1.1.cmml">φ</mi><mo id="Thmexample6.p1.4.m4.3.3.1.1.1.1.3.2.2" stretchy="false" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmexample6.p1.4.m4.3.3.1.1.1.3" stretchy="false" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmexample6.p1.4.m4.3.3.5" xref="Thmexample6.p1.4.m4.3.3.5.cmml">=</mo><mrow id="Thmexample6.p1.4.m4.3.3.6" xref="Thmexample6.p1.4.m4.3.3.6.cmml"><msub id="Thmexample6.p1.4.m4.3.3.6.2" xref="Thmexample6.p1.4.m4.3.3.6.2.cmml"><mi id="Thmexample6.p1.4.m4.3.3.6.2.2" xref="Thmexample6.p1.4.m4.3.3.6.2.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmexample6.p1.4.m4.3.3.6.2.3" xref="Thmexample6.p1.4.m4.3.3.6.2.3.cmml">𝖯𝖦</mi></msub><mo id="Thmexample6.p1.4.m4.3.3.6.1" xref="Thmexample6.p1.4.m4.3.3.6.1.cmml">⁢</mo><mrow id="Thmexample6.p1.4.m4.3.3.6.3.2" xref="Thmexample6.p1.4.m4.3.3.6.cmml"><mo id="Thmexample6.p1.4.m4.3.3.6.3.2.1" stretchy="false" xref="Thmexample6.p1.4.m4.3.3.6.cmml">(</mo><mi id="Thmexample6.p1.4.m4.2.2" xref="Thmexample6.p1.4.m4.2.2.cmml">φ</mi><mo id="Thmexample6.p1.4.m4.3.3.6.3.2.2" stretchy="false" xref="Thmexample6.p1.4.m4.3.3.6.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample6.p1.4.m4.3b"><apply id="Thmexample6.p1.4.m4.3.3.cmml" xref="Thmexample6.p1.4.m4.3.3"><and id="Thmexample6.p1.4.m4.3.3a.cmml" xref="Thmexample6.p1.4.m4.3.3"></and><apply id="Thmexample6.p1.4.m4.3.3b.cmml" xref="Thmexample6.p1.4.m4.3.3"><apply id="Thmexample6.p1.4.m4.3.3.4.cmml" xref="Thmexample6.p1.4.m4.3.3.4"><csymbol cd="ambiguous" id="Thmexample6.p1.4.m4.3.3.4.1.cmml" xref="Thmexample6.p1.4.m4.3.3.4">superscript</csymbol><eq id="Thmexample6.p1.4.m4.3.3.4.2.cmml" xref="Thmexample6.p1.4.m4.3.3.4.2"></eq><ci id="Thmexample6.p1.4.m4.3.3.4.3a.cmml" xref="Thmexample6.p1.4.m4.3.3.4.3"><mtext id="Thmexample6.p1.4.m4.3.3.4.3.cmml" mathsize="50%" xref="Thmexample6.p1.4.m4.3.3.4.3">def</mtext></ci></apply><ci id="Thmexample6.p1.4.m4.3.3.3.cmml" xref="Thmexample6.p1.4.m4.3.3.3">𝜓</ci><apply id="Thmexample6.p1.4.m4.3.3.1.cmml" xref="Thmexample6.p1.4.m4.3.3.1"><times id="Thmexample6.p1.4.m4.3.3.1.2.cmml" xref="Thmexample6.p1.4.m4.3.3.1.2"></times><apply id="Thmexample6.p1.4.m4.3.3.1.3.cmml" xref="Thmexample6.p1.4.m4.3.3.1.3"><csymbol cd="ambiguous" id="Thmexample6.p1.4.m4.3.3.1.3.1.cmml" xref="Thmexample6.p1.4.m4.3.3.1.3">subscript</csymbol><ci id="Thmexample6.p1.4.m4.3.3.1.3.2.cmml" xref="Thmexample6.p1.4.m4.3.3.1.3.2">𝖢𝖭𝖥</ci><ci id="Thmexample6.p1.4.m4.3.3.1.3.3.cmml" xref="Thmexample6.p1.4.m4.3.3.1.3.3">𝖯𝖦</ci></apply><apply id="Thmexample6.p1.4.m4.3.3.1.1.1.1.cmml" xref="Thmexample6.p1.4.m4.3.3.1.1.1"><times id="Thmexample6.p1.4.m4.3.3.1.1.1.1.1.cmml" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.1"></times><ci id="Thmexample6.p1.4.m4.3.3.1.1.1.1.2.cmml" xref="Thmexample6.p1.4.m4.3.3.1.1.1.1.2">𝖭𝖭𝖥</ci><ci id="Thmexample6.p1.4.m4.1.1.cmml" xref="Thmexample6.p1.4.m4.1.1">𝜑</ci></apply></apply></apply><apply id="Thmexample6.p1.4.m4.3.3c.cmml" xref="Thmexample6.p1.4.m4.3.3"><eq id="Thmexample6.p1.4.m4.3.3.5.cmml" xref="Thmexample6.p1.4.m4.3.3.5"></eq><share href="https://arxiv.org/html/2503.01536v1#Thmexample6.p1.4.m4.3.3.1.cmml" id="Thmexample6.p1.4.m4.3.3d.cmml" xref="Thmexample6.p1.4.m4.3.3"></share><apply id="Thmexample6.p1.4.m4.3.3.6.cmml" xref="Thmexample6.p1.4.m4.3.3.6"><times id="Thmexample6.p1.4.m4.3.3.6.1.cmml" xref="Thmexample6.p1.4.m4.3.3.6.1"></times><apply id="Thmexample6.p1.4.m4.3.3.6.2.cmml" xref="Thmexample6.p1.4.m4.3.3.6.2"><csymbol cd="ambiguous" id="Thmexample6.p1.4.m4.3.3.6.2.1.cmml" xref="Thmexample6.p1.4.m4.3.3.6.2">subscript</csymbol><ci id="Thmexample6.p1.4.m4.3.3.6.2.2.cmml" xref="Thmexample6.p1.4.m4.3.3.6.2.2">𝖢𝖭𝖥</ci><ci id="Thmexample6.p1.4.m4.3.3.6.2.3.cmml" xref="Thmexample6.p1.4.m4.3.3.6.2.3">𝖯𝖦</ci></apply><ci id="Thmexample6.p1.4.m4.2.2.cmml" xref="Thmexample6.p1.4.m4.2.2">𝜑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample6.p1.4.m4.3c">\psi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\mathsf{CNF_{PG}}(\mathsf{% NNF}(\varphi))=\mathsf{CNF_{PG}}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="Thmexample6.p1.4.m4.3d">italic_ψ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( sansserif_NNF ( italic_φ ) ) = sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( italic_φ )</annotation></semantics></math>. Therefore <math alttext="\mu\models\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\mathsf{NNF}(\varphi))" class="ltx_Math" display="inline" id="Thmexample6.p1.5.m5.3"><semantics id="Thmexample6.p1.5.m5.3a"><mrow id="Thmexample6.p1.5.m5.3.3.2" xref="Thmexample6.p1.5.m5.3.3.3.cmml"><mrow id="Thmexample6.p1.5.m5.2.2.1.1" xref="Thmexample6.p1.5.m5.2.2.1.1.cmml"><mi id="Thmexample6.p1.5.m5.2.2.1.1.2" xref="Thmexample6.p1.5.m5.2.2.1.1.2.cmml">μ</mi><mo id="Thmexample6.p1.5.m5.2.2.1.1.1" xref="Thmexample6.p1.5.m5.2.2.1.1.1.cmml">⊧</mo><mrow id="Thmexample6.p1.5.m5.2.2.1.1.3" xref="Thmexample6.p1.5.m5.2.2.1.1.3.cmml"><mo id="Thmexample6.p1.5.m5.2.2.1.1.3.1" rspace="0.167em" xref="Thmexample6.p1.5.m5.2.2.1.1.3.1.cmml">∃</mo><mi id="Thmexample6.p1.5.m5.2.2.1.1.3.2" xref="Thmexample6.p1.5.m5.2.2.1.1.3.2.cmml">𝐁</mi></mrow></mrow><mo id="Thmexample6.p1.5.m5.3.3.2.3" lspace="0em" rspace="0.167em" xref="Thmexample6.p1.5.m5.3.3.3a.cmml">.</mo><mrow id="Thmexample6.p1.5.m5.3.3.2.2" xref="Thmexample6.p1.5.m5.3.3.2.2.cmml"><msub id="Thmexample6.p1.5.m5.3.3.2.2.3" xref="Thmexample6.p1.5.m5.3.3.2.2.3.cmml"><mi id="Thmexample6.p1.5.m5.3.3.2.2.3.2" xref="Thmexample6.p1.5.m5.3.3.2.2.3.2.cmml">𝖢𝖭𝖥</mi><mi id="Thmexample6.p1.5.m5.3.3.2.2.3.3" xref="Thmexample6.p1.5.m5.3.3.2.2.3.3.cmml">𝖯𝖦</mi></msub><mo id="Thmexample6.p1.5.m5.3.3.2.2.2" xref="Thmexample6.p1.5.m5.3.3.2.2.2.cmml">⁢</mo><mrow id="Thmexample6.p1.5.m5.3.3.2.2.1.1" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.cmml"><mo id="Thmexample6.p1.5.m5.3.3.2.2.1.1.2" stretchy="false" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.cmml">(</mo><mrow id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.cmml"><mi id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.2" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.2.cmml">𝖭𝖭𝖥</mi><mo id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.1" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.1.cmml">⁢</mo><mrow id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.3.2" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.cmml"><mo id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.3.2.1" stretchy="false" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.cmml">(</mo><mi id="Thmexample6.p1.5.m5.1.1" xref="Thmexample6.p1.5.m5.1.1.cmml">φ</mi><mo id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.3.2.2" stretchy="false" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmexample6.p1.5.m5.3.3.2.2.1.1.3" stretchy="false" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample6.p1.5.m5.3b"><apply id="Thmexample6.p1.5.m5.3.3.3.cmml" xref="Thmexample6.p1.5.m5.3.3.2"><csymbol cd="ambiguous" id="Thmexample6.p1.5.m5.3.3.3a.cmml" xref="Thmexample6.p1.5.m5.3.3.2.3">formulae-sequence</csymbol><apply id="Thmexample6.p1.5.m5.2.2.1.1.cmml" xref="Thmexample6.p1.5.m5.2.2.1.1"><csymbol cd="latexml" id="Thmexample6.p1.5.m5.2.2.1.1.1.cmml" xref="Thmexample6.p1.5.m5.2.2.1.1.1">models</csymbol><ci id="Thmexample6.p1.5.m5.2.2.1.1.2.cmml" xref="Thmexample6.p1.5.m5.2.2.1.1.2">𝜇</ci><apply id="Thmexample6.p1.5.m5.2.2.1.1.3.cmml" xref="Thmexample6.p1.5.m5.2.2.1.1.3"><exists id="Thmexample6.p1.5.m5.2.2.1.1.3.1.cmml" xref="Thmexample6.p1.5.m5.2.2.1.1.3.1"></exists><ci id="Thmexample6.p1.5.m5.2.2.1.1.3.2.cmml" xref="Thmexample6.p1.5.m5.2.2.1.1.3.2">𝐁</ci></apply></apply><apply id="Thmexample6.p1.5.m5.3.3.2.2.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2"><times id="Thmexample6.p1.5.m5.3.3.2.2.2.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2.2"></times><apply id="Thmexample6.p1.5.m5.3.3.2.2.3.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2.3"><csymbol cd="ambiguous" id="Thmexample6.p1.5.m5.3.3.2.2.3.1.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2.3">subscript</csymbol><ci id="Thmexample6.p1.5.m5.3.3.2.2.3.2.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2.3.2">𝖢𝖭𝖥</ci><ci id="Thmexample6.p1.5.m5.3.3.2.2.3.3.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2.3.3">𝖯𝖦</ci></apply><apply id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1"><times id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.1.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.1"></times><ci id="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.2.cmml" xref="Thmexample6.p1.5.m5.3.3.2.2.1.1.1.2">𝖭𝖭𝖥</ci><ci id="Thmexample6.p1.5.m5.1.1.cmml" xref="Thmexample6.p1.5.m5.1.1">𝜑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample6.p1.5.m5.3c">\mu\models\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\mathsf{NNF}(\varphi))</annotation><annotation encoding="application/x-llamapun" id="Thmexample6.p1.5.m5.3d">italic_μ ⊧ ∃ bold_B . sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( sansserif_NNF ( italic_φ ) )</annotation></semantics></math> but <math alttext="\mu\not\mid\!\approx\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\mathsf{NNF}(\varphi))" class="ltx_math_unparsed" display="inline" id="Thmexample6.p1.6.m6.1"><semantics id="Thmexample6.p1.6.m6.1a"><mrow id="Thmexample6.p1.6.m6.1b"><mi id="Thmexample6.p1.6.m6.1.1">μ</mi><mpadded id="Thmexample6.p1.6.m6.1c" width="0.969em"><mo id="Thmexample6.p1.6.m6.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample6.p1.6.m6.1.3">≈</mo><mo id="Thmexample6.p1.6.m6.1.4" rspace="0.167em">∃</mo><mi id="Thmexample6.p1.6.m6.1.5">𝐁</mi><mo id="Thmexample6.p1.6.m6.1.6" lspace="0em" rspace="0.167em">.</mo><msub id="Thmexample6.p1.6.m6.1.7"><mi id="Thmexample6.p1.6.m6.1.7.2">𝖢𝖭𝖥</mi><mi id="Thmexample6.p1.6.m6.1.7.3">𝖯𝖦</mi></msub><mrow id="Thmexample6.p1.6.m6.1.8"><mo id="Thmexample6.p1.6.m6.1.8.1" stretchy="false">(</mo><mi id="Thmexample6.p1.6.m6.1.8.2">𝖭𝖭𝖥</mi><mrow id="Thmexample6.p1.6.m6.1.8.3"><mo id="Thmexample6.p1.6.m6.1.8.3.1" stretchy="false">(</mo><mi id="Thmexample6.p1.6.m6.1.8.3.2">φ</mi><mo id="Thmexample6.p1.6.m6.1.8.3.3" stretchy="false">)</mo></mrow><mo id="Thmexample6.p1.6.m6.1.8.4" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="Thmexample6.p1.6.m6.1d">\mu\not\mid\!\approx\exists{\mathbf{B}}.\mathsf{CNF_{PG}}(\mathsf{NNF}(\varphi))</annotation><annotation encoding="application/x-llamapun" id="Thmexample6.p1.6.m6.1e">italic_μ ∤ ≈ ∃ bold_B . sansserif_CNF start_POSTSUBSCRIPT sansserif_PG end_POSTSUBSCRIPT ( sansserif_NNF ( italic_φ ) )</annotation></semantics></math>. <math alttext="\diamond" class="ltx_Math" display="inline" id="Thmexample6.p1.7.m7.1"><semantics id="Thmexample6.p1.7.m7.1a"><mo id="Thmexample6.p1.7.m7.1.1" xref="Thmexample6.p1.7.m7.1.1.cmml">⋄</mo><annotation-xml encoding="MathML-Content" id="Thmexample6.p1.7.m7.1b"><ci id="Thmexample6.p1.7.m7.1.1.cmml" xref="Thmexample6.p1.7.m7.1.1">⋄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample6.p1.7.m7.1c">\diamond</annotation><annotation encoding="application/x-llamapun" id="Thmexample6.p1.7.m7.1d">⋄</annotation></semantics></math></p> </div> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Practical consequences of using verification or entailment</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section we analyze the practical consequences of adopting verification or entailment as a partial-assignment satisfiability tests in search procedures for satisfiability and enumeration and in formula compilation. In particular, when entailment and verification do not coincide, we analyze the tradeoff between the extra cost of entailment versus the benefits it may produce.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Verification vs. entailment in solving, enumeration and compilation</h3> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Verification vs. entailment for native CNF formulas.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.p1.1">When the input formula is natively in CNF, the distinction between verification and entailment is not relevant, because these two concepts coincide. (We conjecture that this is the main reason why the distinction between verification and entailment has been long overlooked in the literature.) For satisfiability of CNF formulas, CDCL SAT solvers directly produce <span class="ltx_text ltx_font_italic" id="S4.SS1.SSS0.Px1.p1.1.1">total</span> truth assignments, because it is not worth to perform intermediate satisfiability checks.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Verification vs. entailment in satisfiability.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px2.p1.3">In plain satisfiability we need finding only one total assignment <math alttext="\eta" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="S4.SS1.SSS0.Px2.p1.1.m1.1a"><mi id="S4.SS1.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.1.m1.1b"><ci id="S4.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.1.m1.1d">italic_η</annotation></semantics></math> extending <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="S4.SS1.SSS0.Px2.p1.2.m2.1a"><mi id="S4.SS1.SSS0.Px2.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.2.m2.1b"><ci id="S4.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.2.m2.1d">italic_μ</annotation></semantics></math> which satisfy <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px2.p1.3.m3.1"><semantics id="S4.SS1.SSS0.Px2.p1.3.m3.1a"><mi id="S4.SS1.SSS0.Px2.p1.3.m3.1.1" xref="S4.SS1.SSS0.Px2.p1.3.m3.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px2.p1.3.m3.1b"><ci id="S4.SS1.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS0.Px2.p1.3.m3.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px2.p1.3.m3.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px2.p1.3.m3.1d">italic_φ</annotation></semantics></math>. Therefore entailment produces no benefits wrt. verification and is more expensive, so that the latter is always preferred.</p> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px2.p2"> <p class="ltx_p" id="S4.SS1.SSS0.Px2.p2.1">The scenario changes when we deal with enumeration-based algorithms applied to non-CNF formulas, or to existentially-quantified formulas, or to CNF-ized formulas.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Verification vs. entailment for enumeration on non-CNF formulas.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px3.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px3.p1.5">The analysis in §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS1" title="3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.1</span></a> shows the following facts. On the one hand, the advantage of adopting verification rather than entailment for checking partial-assignment satisfiability is that it matches the intuition and practical need that satisfiability checking should be fast, since checking verification is polynomial (property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty5" title="Property 5 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">5</span></a>(iv)) and easier to implement, whereas checking entailment is co-NP-complete (property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty6" title="Property 6 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">6</span></a>(iv)), because it is equivalent to checking the validity of the residual <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.1.m1.2"><semantics id="S4.SS1.SSS0.Px3.p1.1.m1.2a"><msub id="S4.SS1.SSS0.Px3.p1.1.m1.2.3.2" xref="S4.SS1.SSS0.Px3.p1.1.m1.2.3.1.cmml"><mrow id="S4.SS1.SSS0.Px3.p1.1.m1.2.3.2.2" xref="S4.SS1.SSS0.Px3.p1.1.m1.2.3.1.cmml"><mi id="S4.SS1.SSS0.Px3.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px3.p1.1.m1.1.1.cmml">φ</mi><mo id="S4.SS1.SSS0.Px3.p1.1.m1.2.3.2.2.1" stretchy="false" xref="S4.SS1.SSS0.Px3.p1.1.m1.2.3.1.1.cmml">|</mo></mrow><mi id="S4.SS1.SSS0.Px3.p1.1.m1.2.2.1" xref="S4.SS1.SSS0.Px3.p1.1.m1.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.1.m1.2b"><apply id="S4.SS1.SSS0.Px3.p1.1.m1.2.3.1.cmml" xref="S4.SS1.SSS0.Px3.p1.1.m1.2.3.2"><csymbol cd="latexml" id="S4.SS1.SSS0.Px3.p1.1.m1.2.3.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.1.m1.2.3.2.2.1">evaluated-at</csymbol><ci id="S4.SS1.SSS0.Px3.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.1.m1.1.1">𝜑</ci><ci id="S4.SS1.SSS0.Px3.p1.1.m1.2.2.1.cmml" xref="S4.SS1.SSS0.Px3.p1.1.m1.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.1.m1.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.1.m1.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math>. We stress the fact, however, that if <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.2.m2.1"><semantics id="S4.SS1.SSS0.Px3.p1.2.m2.1a"><mrow id="S4.SS1.SSS0.Px3.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.cmml"><mi id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.2" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.2.cmml">μ</mi><mo id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.1" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.1.cmml">⊧</mo><mi id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.3" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.2.m2.1b"><apply id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1"><csymbol cd="latexml" id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.1">models</csymbol><ci id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.2">𝜇</ci><ci id="S4.SS1.SSS0.Px3.p1.2.m2.1.1.3.cmml" xref="S4.SS1.SSS0.Px3.p1.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.2.m2.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.2.m2.1d">italic_μ ⊧ italic_φ</annotation></semantics></math>, then the residual <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.3.m3.2"><semantics id="S4.SS1.SSS0.Px3.p1.3.m3.2a"><msub id="S4.SS1.SSS0.Px3.p1.3.m3.2.3.2" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.3.1.cmml"><mrow id="S4.SS1.SSS0.Px3.p1.3.m3.2.3.2.2" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.3.1.cmml"><mi id="S4.SS1.SSS0.Px3.p1.3.m3.1.1" xref="S4.SS1.SSS0.Px3.p1.3.m3.1.1.cmml">φ</mi><mo id="S4.SS1.SSS0.Px3.p1.3.m3.2.3.2.2.1" stretchy="false" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.3.1.1.cmml">|</mo></mrow><mi id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.3.m3.2b"><apply id="S4.SS1.SSS0.Px3.p1.3.m3.2.3.1.cmml" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.3.2"><csymbol cd="latexml" id="S4.SS1.SSS0.Px3.p1.3.m3.2.3.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.3.2.2.1">evaluated-at</csymbol><ci id="S4.SS1.SSS0.Px3.p1.3.m3.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.3.m3.1.1">𝜑</ci><ci id="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1.cmml" xref="S4.SS1.SSS0.Px3.p1.3.m3.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.3.m3.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.3.m3.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is typically much smaller than <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.4.m4.1"><semantics id="S4.SS1.SSS0.Px3.p1.4.m4.1a"><mi id="S4.SS1.SSS0.Px3.p1.4.m4.1.1" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.4.m4.1b"><ci id="S4.SS1.SSS0.Px3.p1.4.m4.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.4.m4.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.4.m4.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.4.m4.1d">italic_φ</annotation></semantics></math> with a much smaller search space, since its variables are only (a subset of) the variables which are not assigned by <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p1.5.m5.1"><semantics id="S4.SS1.SSS0.Px3.p1.5.m5.1a"><mi id="S4.SS1.SSS0.Px3.p1.5.m5.1.1" xref="S4.SS1.SSS0.Px3.p1.5.m5.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p1.5.m5.1b"><ci id="S4.SS1.SSS0.Px3.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS0.Px3.p1.5.m5.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p1.5.m5.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p1.5.m5.1d">italic_μ</annotation></semantics></math>, so that in many actual situations the above fact could not be a major issue.</p> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px3.p2"> <p class="ltx_p" id="S4.SS1.SSS0.Px3.p2.7">On the other hand, a main advantage of adopting entailment on enumeration is that, due to <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.Thmtheorem1" title="Theorem 3.1 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">theorem</span> <span class="ltx_text ltx_ref_tag">3.1</span></a>, every partial assignments entailing the input formula is a <em class="ltx_emph ltx_font_italic" id="S4.SS1.SSS0.Px3.p2.7.1">sub-assignment</em> of some other(s) verifying it. Consequently, as soon as one verification-based enumeration procedure produces (a branch corresponding to) an assignment <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p2.1.m1.1"><semantics id="S4.SS1.SSS0.Px3.p2.1.m1.1a"><mi id="S4.SS1.SSS0.Px3.p2.1.m1.1.1" xref="S4.SS1.SSS0.Px3.p2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p2.1.m1.1b"><ci id="S4.SS1.SSS0.Px3.p2.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p2.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p2.1.m1.1d">italic_μ</annotation></semantics></math> s.t. <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p2.2.m2.1"><semantics id="S4.SS1.SSS0.Px3.p2.2.m2.1a"><mrow id="S4.SS1.SSS0.Px3.p2.2.m2.1.1" xref="S4.SS1.SSS0.Px3.p2.2.m2.1.1.cmml"><mi id="S4.SS1.SSS0.Px3.p2.2.m2.1.1.2" xref="S4.SS1.SSS0.Px3.p2.2.m2.1.1.2.cmml">μ</mi><mo id="S4.SS1.SSS0.Px3.p2.2.m2.1.1.1" xref="S4.SS1.SSS0.Px3.p2.2.m2.1.1.1.cmml">⊧</mo><mi id="S4.SS1.SSS0.Px3.p2.2.m2.1.1.3" xref="S4.SS1.SSS0.Px3.p2.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p2.2.m2.1b"><apply id="S4.SS1.SSS0.Px3.p2.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.2.m2.1.1"><csymbol cd="latexml" id="S4.SS1.SSS0.Px3.p2.2.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.2.m2.1.1.1">models</csymbol><ci id="S4.SS1.SSS0.Px3.p2.2.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p2.2.m2.1.1.2">𝜇</ci><ci id="S4.SS1.SSS0.Px3.p2.2.m2.1.1.3.cmml" xref="S4.SS1.SSS0.Px3.p2.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p2.2.m2.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p2.2.m2.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> but <math alttext="\mu\not\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S4.SS1.SSS0.Px3.p2.3.m3.1"><semantics id="S4.SS1.SSS0.Px3.p2.3.m3.1a"><mrow id="S4.SS1.SSS0.Px3.p2.3.m3.1b"><mi id="S4.SS1.SSS0.Px3.p2.3.m3.1.1">μ</mi><mpadded id="S4.SS1.SSS0.Px3.p2.3.m3.1c" width="0.969em"><mo id="S4.SS1.SSS0.Px3.p2.3.m3.1.2" lspace="0em">∤</mo></mpadded><mo id="S4.SS1.SSS0.Px3.p2.3.m3.1.3">≈</mo><mi id="S4.SS1.SSS0.Px3.p2.3.m3.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p2.3.m3.1d">\mu\not\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p2.3.m3.1e">italic_μ ∤ ≈ italic_φ</annotation></semantics></math>, it cannot realize this fact and thus proceeds the search to extend <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p2.4.m4.1"><semantics id="S4.SS1.SSS0.Px3.p2.4.m4.1a"><mi id="S4.SS1.SSS0.Px3.p2.4.m4.1.1" xref="S4.SS1.SSS0.Px3.p2.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p2.4.m4.1b"><ci id="S4.SS1.SSS0.Px3.p2.4.m4.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p2.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p2.4.m4.1d">italic_μ</annotation></semantics></math> to some <math alttext="\mu^{\prime}\supset\mu" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p2.5.m5.1"><semantics id="S4.SS1.SSS0.Px3.p2.5.m5.1a"><mrow id="S4.SS1.SSS0.Px3.p2.5.m5.1.1" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.cmml"><msup id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.cmml"><mi id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.2" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.2.cmml">μ</mi><mo id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.3" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.1" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.cmml">⊃</mo><mi id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.3" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p2.5.m5.1b"><apply id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1"><subset id="S4.SS1.SSS0.Px3.p2.5.m5.1.1a.cmml" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1"></subset><ci id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.3.cmml" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.3">𝜇</ci><apply id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2">superscript</csymbol><ci id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.2">𝜇</ci><ci id="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.3.cmml" xref="S4.SS1.SSS0.Px3.p2.5.m5.1.1.2.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p2.5.m5.1c">\mu^{\prime}\supset\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p2.5.m5.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊃ italic_μ</annotation></semantics></math> s.t. <math alttext="\mu^{\prime}\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S4.SS1.SSS0.Px3.p2.6.m6.1"><semantics id="S4.SS1.SSS0.Px3.p2.6.m6.1a"><mrow id="S4.SS1.SSS0.Px3.p2.6.m6.1b"><msup id="S4.SS1.SSS0.Px3.p2.6.m6.1.1"><mi id="S4.SS1.SSS0.Px3.p2.6.m6.1.1.2">μ</mi><mo id="S4.SS1.SSS0.Px3.p2.6.m6.1.1.3">′</mo></msup><mpadded id="S4.SS1.SSS0.Px3.p2.6.m6.1c" width="0.219em"><mo id="S4.SS1.SSS0.Px3.p2.6.m6.1.2" lspace="0em">∣</mo></mpadded><mo id="S4.SS1.SSS0.Px3.p2.6.m6.1.3">≈</mo><mi id="S4.SS1.SSS0.Px3.p2.6.m6.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p2.6.m6.1d">\mu^{\prime}\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p2.6.m6.1e">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∣ ≈ italic_φ</annotation></semantics></math>. This causes an extension of the search tree of up to <math alttext="2^{|\mu^{\prime}|-|\mu|}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p2.7.m7.2"><semantics id="S4.SS1.SSS0.Px3.p2.7.m7.2a"><msup id="S4.SS1.SSS0.Px3.p2.7.m7.2.3" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.3.cmml"><mn id="S4.SS1.SSS0.Px3.p2.7.m7.2.3.2" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.3.2.cmml">2</mn><mrow id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.cmml"><mrow id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.2.cmml"><mo id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.2" stretchy="false" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.2.1.cmml">|</mo><msup id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.cmml"><mi id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.2" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.2.cmml">μ</mi><mo id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.3" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.3.cmml">′</mo></msup><mo id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.3" stretchy="false" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.2.1.cmml">|</mo></mrow><mo id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.3" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.3.cmml">−</mo><mrow id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.2" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.1.cmml"><mo id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.2.1" stretchy="false" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.1.1.cmml">|</mo><mi id="S4.SS1.SSS0.Px3.p2.7.m7.1.1.1.1" xref="S4.SS1.SSS0.Px3.p2.7.m7.1.1.1.1.cmml">μ</mi><mo id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.2.2" stretchy="false" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.1.1.cmml">|</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p2.7.m7.2b"><apply id="S4.SS1.SSS0.Px3.p2.7.m7.2.3.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.3"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px3.p2.7.m7.2.3.1.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.3">superscript</csymbol><cn id="S4.SS1.SSS0.Px3.p2.7.m7.2.3.2.cmml" type="integer" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.3.2">2</cn><apply id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2"><minus id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.3.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.3"></minus><apply id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.2.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1"><abs id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.2.1.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.2"></abs><apply id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1">superscript</csymbol><ci id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.2.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.2">𝜇</ci><ci id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.3.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.2.1.1.3">′</ci></apply></apply><apply id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.1.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.2"><abs id="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.2.2.2.4.2.1"></abs><ci id="S4.SS1.SSS0.Px3.p2.7.m7.1.1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p2.7.m7.1.1.1.1">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p2.7.m7.2c">2^{|\mu^{\prime}|-|\mu|}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p2.7.m7.2d">2 start_POSTSUPERSCRIPT | italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | - | italic_μ | end_POSTSUPERSCRIPT</annotation></semantics></math> branches. Therefore, for an assignment-enumeration algorithm, being able to enumerate partial assignments entailing the input formula rather than simply verifying it may (even drastically) reduce the number of the satisfying assignment enumerated.</p> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px3.p3"> <p class="ltx_p" id="S4.SS1.SSS0.Px3.p3.1">Also, the analysis in §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS2" title="3.2 Other candidate forms of partial-assignment satisfaction. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.2</span></a> shows that alternative forms of partial-assignment satisfiability extending verification —e.g., adding further simplifications for the residual <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px3.p3.1.m1.2"><semantics id="S4.SS1.SSS0.Px3.p3.1.m1.2a"><msub id="S4.SS1.SSS0.Px3.p3.1.m1.2.3.2" xref="S4.SS1.SSS0.Px3.p3.1.m1.2.3.1.cmml"><mrow id="S4.SS1.SSS0.Px3.p3.1.m1.2.3.2.2" xref="S4.SS1.SSS0.Px3.p3.1.m1.2.3.1.cmml"><mi id="S4.SS1.SSS0.Px3.p3.1.m1.1.1" xref="S4.SS1.SSS0.Px3.p3.1.m1.1.1.cmml">φ</mi><mo id="S4.SS1.SSS0.Px3.p3.1.m1.2.3.2.2.1" stretchy="false" xref="S4.SS1.SSS0.Px3.p3.1.m1.2.3.1.1.cmml">|</mo></mrow><mi id="S4.SS1.SSS0.Px3.p3.1.m1.2.2.1" xref="S4.SS1.SSS0.Px3.p3.1.m1.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px3.p3.1.m1.2b"><apply id="S4.SS1.SSS0.Px3.p3.1.m1.2.3.1.cmml" xref="S4.SS1.SSS0.Px3.p3.1.m1.2.3.2"><csymbol cd="latexml" id="S4.SS1.SSS0.Px3.p3.1.m1.2.3.1.1.cmml" xref="S4.SS1.SSS0.Px3.p3.1.m1.2.3.2.2.1">evaluated-at</csymbol><ci id="S4.SS1.SSS0.Px3.p3.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px3.p3.1.m1.1.1">𝜑</ci><ci id="S4.SS1.SSS0.Px3.p3.1.m1.2.2.1.cmml" xref="S4.SS1.SSS0.Px3.p3.1.m1.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px3.p3.1.m1.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px3.p3.1.m1.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> like unit-propagation or the removal of basic valid subformulas— cannot fill the gap wrt. entailment, whose theoretical properties are specific.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Verification vs. entailment for projected enumeration.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px4.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px4.p1.2">The analysis in §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS3" title="3.3 Verification and entailment of existentially-quantified formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.3</span></a> shows that the above considerations for non-CNF formulas extend automatically to procedures for projected enumeration (e.g. projected AllSAT/AllSMT, projected #SAT/#SMT), even on CNF formulas, because projected enumeration consists in enumeration on an existentially-quantified formula <math alttext="\exists{\mathbf{B}}.\psi" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px4.p1.1.m1.2"><semantics id="S4.SS1.SSS0.Px4.p1.1.m1.2a"><mrow id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.2.cmml"><mrow id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.cmml"><mo id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.1" rspace="0.167em" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.1.cmml">∃</mo><mi id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.2" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.2.cmml">𝐁</mi></mrow><mo id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.2" lspace="0em" rspace="0.167em" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.2a.cmml">.</mo><mi id="S4.SS1.SSS0.Px4.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px4.p1.1.m1.1.1.cmml">ψ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px4.p1.1.m1.2b"><apply id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.2.cmml" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.2a.cmml" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.2">formulae-sequence</csymbol><apply id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.cmml" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1"><exists id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.1.cmml" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.1"></exists><ci id="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.2.cmml" xref="S4.SS1.SSS0.Px4.p1.1.m1.2.2.1.1.2">𝐁</ci></apply><ci id="S4.SS1.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px4.p1.1.m1.1.1">𝜓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px4.p1.1.m1.2c">\exists{\mathbf{B}}.\psi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px4.p1.1.m1.2d">∃ bold_B . italic_ψ</annotation></semantics></math>, <math alttext="{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px4.p1.2.m2.1"><semantics id="S4.SS1.SSS0.Px4.p1.2.m2.1a"><mi id="S4.SS1.SSS0.Px4.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px4.p1.2.m2.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px4.p1.2.m2.1b"><ci id="S4.SS1.SSS0.Px4.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px4.p1.2.m2.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px4.p1.2.m2.1c">{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px4.p1.2.m2.1d">bold_B</annotation></semantics></math> being the non-relevant atoms to be projected away. The same considerations apply also to other applications like the computation of preimages in symbolic model checking <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib5" title="">5</a>]</cite> or predicate abstraction in SW verification <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib21" title="">21</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib24" title="">24</a>]</cite>.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px5"> <h4 class="ltx_title ltx_title_paragraph">Verification vs. entailment for CNF-ized non-CNF formulas.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px5.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px5.p1.2">The analysis in §<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S3.SS4" title="3.4 Verification and entailment of CNF-ized non-CNF formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">3.4</span></a> shows that, although verification and entailment coincide for CNF formulas, the problem with non-CNF formulas cannot be fixed by simply CNF-izing a formula upfront <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib32" title="">32</a>]</cite> and running a CNF enumeration procedure based on partial-assignment verification, projecting out the fresh variables introduced by the CNF-ization. In fact, a partial assignment over the original atoms may entail the existentially-quantified CNF-ized formula without verifying it. 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xref="S4.SS1.SSS0.Px5.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px5.p1.2.m2.1b"><apply id="S4.SS1.SSS0.Px5.p1.2.m2.1.2.cmml" xref="S4.SS1.SSS0.Px5.p1.2.m2.1.2"><times id="S4.SS1.SSS0.Px5.p1.2.m2.1.2.1.cmml" xref="S4.SS1.SSS0.Px5.p1.2.m2.1.2.1"></times><ci id="S4.SS1.SSS0.Px5.p1.2.m2.1.2.2.cmml" xref="S4.SS1.SSS0.Px5.p1.2.m2.1.2.2">𝖭𝖭𝖥</ci><ci id="S4.SS1.SSS0.Px5.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px5.p1.2.m2.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px5.p1.2.m2.1c">\mathsf{NNF}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px5.p1.2.m2.1d">sansserif_NNF ( italic_φ )</annotation></semantics></math>) we proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib25" title="">25</a>]</cite>, although improving the enumeration process, does not fill the gap between verification and entailment in this case.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px6"> <h4 class="ltx_title ltx_title_paragraph">Verification and entailment in formula compilation.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px6.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px6.p1.8">With formula compilers a formula <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p1.1.m1.1"><semantics id="S4.SS1.SSS0.Px6.p1.1.m1.1a"><mi id="S4.SS1.SSS0.Px6.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px6.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p1.1.m1.1b"><ci id="S4.SS1.SSS0.Px6.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p1.1.m1.1d">italic_φ</annotation></semantics></math> is encoded into a format which makes assignment enumeration straightforward 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not share atoms) and <span class="ltx_text ltx_font_italic" id="S4.SS1.SSS0.Px6.p1.8.2">deterministic</span> (i.e., all disjunctive subformulas <math alttext="\varphi_{1}\vee\varphi_{2}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p1.5.m5.1"><semantics id="S4.SS1.SSS0.Px6.p1.5.m5.1a"><mrow id="S4.SS1.SSS0.Px6.p1.5.m5.1.1" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.cmml"><msub id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.cmml"><mi id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.2" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.2.cmml">φ</mi><mn id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.3" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.1" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.1.cmml">∨</mo><msub id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.cmml"><mi id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.2" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.2.cmml">φ</mi><mn id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.3" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p1.5.m5.1b"><apply id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1"><or id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.1"></or><apply id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.cmml" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.1.cmml" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.2.cmml" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.2">𝜑</ci><cn id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.3.cmml" type="integer" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.2.3">1</cn></apply><apply id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.cmml" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.1.cmml" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.2.cmml" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.2">𝜑</ci><cn id="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S4.SS1.SSS0.Px6.p1.5.m5.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p1.5.m5.1c">\varphi_{1}\vee\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p1.5.m5.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are such that <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p1.6.m6.1"><semantics id="S4.SS1.SSS0.Px6.p1.6.m6.1a"><msub id="S4.SS1.SSS0.Px6.p1.6.m6.1.1" xref="S4.SS1.SSS0.Px6.p1.6.m6.1.1.cmml"><mi id="S4.SS1.SSS0.Px6.p1.6.m6.1.1.2" xref="S4.SS1.SSS0.Px6.p1.6.m6.1.1.2.cmml">φ</mi><mn id="S4.SS1.SSS0.Px6.p1.6.m6.1.1.3" xref="S4.SS1.SSS0.Px6.p1.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p1.6.m6.1b"><apply id="S4.SS1.SSS0.Px6.p1.6.m6.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px6.p1.6.m6.1.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.6.m6.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px6.p1.6.m6.1.1.2.cmml" xref="S4.SS1.SSS0.Px6.p1.6.m6.1.1.2">𝜑</ci><cn id="S4.SS1.SSS0.Px6.p1.6.m6.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px6.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p1.6.m6.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p1.6.m6.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p1.7.m7.1"><semantics id="S4.SS1.SSS0.Px6.p1.7.m7.1a"><msub id="S4.SS1.SSS0.Px6.p1.7.m7.1.1" xref="S4.SS1.SSS0.Px6.p1.7.m7.1.1.cmml"><mi id="S4.SS1.SSS0.Px6.p1.7.m7.1.1.2" xref="S4.SS1.SSS0.Px6.p1.7.m7.1.1.2.cmml">φ</mi><mn id="S4.SS1.SSS0.Px6.p1.7.m7.1.1.3" xref="S4.SS1.SSS0.Px6.p1.7.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p1.7.m7.1b"><apply id="S4.SS1.SSS0.Px6.p1.7.m7.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px6.p1.7.m7.1.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.7.m7.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px6.p1.7.m7.1.1.2.cmml" xref="S4.SS1.SSS0.Px6.p1.7.m7.1.1.2">𝜑</ci><cn id="S4.SS1.SSS0.Px6.p1.7.m7.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px6.p1.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p1.7.m7.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p1.7.m7.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are mutually inconsistent) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib11" title="">11</a>]</cite>. The d-DNNF compilers typically implement a top-down search, implicitly adopting verification (<math alttext="\varphi|_{\mu}=\top" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p1.8.m8.2"><semantics id="S4.SS1.SSS0.Px6.p1.8.m8.2a"><mrow id="S4.SS1.SSS0.Px6.p1.8.m8.2.3" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.cmml"><msub id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.2" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.1.cmml"><mrow id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.2.2" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.1.cmml"><mi id="S4.SS1.SSS0.Px6.p1.8.m8.1.1" xref="S4.SS1.SSS0.Px6.p1.8.m8.1.1.cmml">φ</mi><mo id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.2.2.1" stretchy="false" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.1.1.cmml">|</mo></mrow><mi id="S4.SS1.SSS0.Px6.p1.8.m8.2.2.1" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.2.1.cmml">μ</mi></msub><mo id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.1" rspace="0em" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.1.cmml">=</mo><mo id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.3" lspace="0em" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.3.cmml">⊤</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p1.8.m8.2b"><apply id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.cmml" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3"><eq id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.1.cmml" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.1"></eq><apply id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.1.cmml" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.2"><csymbol cd="latexml" id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.2.2.2.1">evaluated-at</csymbol><ci id="S4.SS1.SSS0.Px6.p1.8.m8.1.1.cmml" xref="S4.SS1.SSS0.Px6.p1.8.m8.1.1">𝜑</ci><ci id="S4.SS1.SSS0.Px6.p1.8.m8.2.2.1.cmml" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.2.1">𝜇</ci></apply><csymbol cd="latexml" id="S4.SS1.SSS0.Px6.p1.8.m8.2.3.3.cmml" xref="S4.SS1.SSS0.Px6.p1.8.m8.2.3.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p1.8.m8.2c">\varphi|_{\mu}=\top</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p1.8.m8.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ⊤</annotation></semantics></math>) as partial-assignment satisfiability test <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib10" title="">10</a>]</cite>.</p> </div> <div class="ltx_para" id="S4.SS1.SSS0.Px6.p2"> <p class="ltx_p" id="S4.SS1.SSS0.Px6.p2.5">OBDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib4" title="">4</a>]</cite> and SDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib12" title="">12</a>]</cite> are subcases of d-DNNFs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib13" title="">13</a>]</cite> which are canonical under some order condition —i.e., two equivalent subformulas <math alttext="\varphi_{1}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p2.1.m1.1"><semantics id="S4.SS1.SSS0.Px6.p2.1.m1.1a"><msub id="S4.SS1.SSS0.Px6.p2.1.m1.1.1" xref="S4.SS1.SSS0.Px6.p2.1.m1.1.1.cmml"><mi id="S4.SS1.SSS0.Px6.p2.1.m1.1.1.2" xref="S4.SS1.SSS0.Px6.p2.1.m1.1.1.2.cmml">φ</mi><mn id="S4.SS1.SSS0.Px6.p2.1.m1.1.1.3" xref="S4.SS1.SSS0.Px6.p2.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p2.1.m1.1b"><apply id="S4.SS1.SSS0.Px6.p2.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px6.p2.1.m1.1.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px6.p2.1.m1.1.1.2.cmml" xref="S4.SS1.SSS0.Px6.p2.1.m1.1.1.2">𝜑</ci><cn id="S4.SS1.SSS0.Px6.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px6.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p2.1.m1.1c">\varphi_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p2.1.m1.1d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\varphi_{2}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p2.2.m2.1"><semantics id="S4.SS1.SSS0.Px6.p2.2.m2.1a"><msub id="S4.SS1.SSS0.Px6.p2.2.m2.1.1" xref="S4.SS1.SSS0.Px6.p2.2.m2.1.1.cmml"><mi id="S4.SS1.SSS0.Px6.p2.2.m2.1.1.2" xref="S4.SS1.SSS0.Px6.p2.2.m2.1.1.2.cmml">φ</mi><mn id="S4.SS1.SSS0.Px6.p2.2.m2.1.1.3" xref="S4.SS1.SSS0.Px6.p2.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p2.2.m2.1b"><apply id="S4.SS1.SSS0.Px6.p2.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.SSS0.Px6.p2.2.m2.1.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.SSS0.Px6.p2.2.m2.1.1.2.cmml" xref="S4.SS1.SSS0.Px6.p2.2.m2.1.1.2">𝜑</ci><cn id="S4.SS1.SSS0.Px6.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS1.SSS0.Px6.p2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p2.2.m2.1c">\varphi_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p2.2.m2.1d">italic_φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are encoded into the same OBDD or SDD, and as such are shared inside the DAG representation. The OBDD and SDD compilers typically build the encoding bottom-up, and they are able to encode <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p2.3.m3.2"><semantics id="S4.SS1.SSS0.Px6.p2.3.m3.2a"><msub id="S4.SS1.SSS0.Px6.p2.3.m3.2.3.2" xref="S4.SS1.SSS0.Px6.p2.3.m3.2.3.1.cmml"><mrow id="S4.SS1.SSS0.Px6.p2.3.m3.2.3.2.2" xref="S4.SS1.SSS0.Px6.p2.3.m3.2.3.1.cmml"><mi id="S4.SS1.SSS0.Px6.p2.3.m3.1.1" xref="S4.SS1.SSS0.Px6.p2.3.m3.1.1.cmml">φ</mi><mo id="S4.SS1.SSS0.Px6.p2.3.m3.2.3.2.2.1" stretchy="false" xref="S4.SS1.SSS0.Px6.p2.3.m3.2.3.1.1.cmml">|</mo></mrow><mi id="S4.SS1.SSS0.Px6.p2.3.m3.2.2.1" xref="S4.SS1.SSS0.Px6.p2.3.m3.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p2.3.m3.2b"><apply id="S4.SS1.SSS0.Px6.p2.3.m3.2.3.1.cmml" xref="S4.SS1.SSS0.Px6.p2.3.m3.2.3.2"><csymbol cd="latexml" id="S4.SS1.SSS0.Px6.p2.3.m3.2.3.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.3.m3.2.3.2.2.1">evaluated-at</csymbol><ci id="S4.SS1.SSS0.Px6.p2.3.m3.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.3.m3.1.1">𝜑</ci><ci id="S4.SS1.SSS0.Px6.p2.3.m3.2.2.1.cmml" xref="S4.SS1.SSS0.Px6.p2.3.m3.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p2.3.m3.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p2.3.m3.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> into <math alttext="\top" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p2.4.m4.1"><semantics id="S4.SS1.SSS0.Px6.p2.4.m4.1a"><mo id="S4.SS1.SSS0.Px6.p2.4.m4.1.1" xref="S4.SS1.SSS0.Px6.p2.4.m4.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p2.4.m4.1b"><csymbol cd="latexml" id="S4.SS1.SSS0.Px6.p2.4.m4.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.4.m4.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p2.4.m4.1c">\top</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p2.4.m4.1d">⊤</annotation></semantics></math> as soon as <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px6.p2.5.m5.2"><semantics id="S4.SS1.SSS0.Px6.p2.5.m5.2a"><msub id="S4.SS1.SSS0.Px6.p2.5.m5.2.3.2" xref="S4.SS1.SSS0.Px6.p2.5.m5.2.3.1.cmml"><mrow id="S4.SS1.SSS0.Px6.p2.5.m5.2.3.2.2" xref="S4.SS1.SSS0.Px6.p2.5.m5.2.3.1.cmml"><mi id="S4.SS1.SSS0.Px6.p2.5.m5.1.1" xref="S4.SS1.SSS0.Px6.p2.5.m5.1.1.cmml">φ</mi><mo id="S4.SS1.SSS0.Px6.p2.5.m5.2.3.2.2.1" stretchy="false" xref="S4.SS1.SSS0.Px6.p2.5.m5.2.3.1.1.cmml">|</mo></mrow><mi id="S4.SS1.SSS0.Px6.p2.5.m5.2.2.1" xref="S4.SS1.SSS0.Px6.p2.5.m5.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px6.p2.5.m5.2b"><apply id="S4.SS1.SSS0.Px6.p2.5.m5.2.3.1.cmml" xref="S4.SS1.SSS0.Px6.p2.5.m5.2.3.2"><csymbol cd="latexml" id="S4.SS1.SSS0.Px6.p2.5.m5.2.3.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.5.m5.2.3.2.2.1">evaluated-at</csymbol><ci id="S4.SS1.SSS0.Px6.p2.5.m5.1.1.cmml" xref="S4.SS1.SSS0.Px6.p2.5.m5.1.1">𝜑</ci><ci id="S4.SS1.SSS0.Px6.p2.5.m5.2.2.1.cmml" xref="S4.SS1.SSS0.Px6.p2.5.m5.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px6.p2.5.m5.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px6.p2.5.m5.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> becomes valid. (That is, they implicitly use entailment as partial-assignment satisfiability test.)</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px7"> <h4 class="ltx_title ltx_title_paragraph">Verification and entailment for #SMT and WMI.</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px7.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px7.p1.2">With some enumeration-based techniques (e.g. #SMT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib31" title="">31</a>]</cite> and WMI <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib39" title="">39</a>]</cite>) the total cost of <span class="ltx_text ltx_font_italic" id="S4.SS1.SSS0.Px7.p1.2.1">processing</span> each enumerated partial assignment dominates that of the enumeration itself. (E.g., with WMI each satisfying assignment <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px7.p1.1.m1.1"><semantics id="S4.SS1.SSS0.Px7.p1.1.m1.1a"><mi id="S4.SS1.SSS0.Px7.p1.1.m1.1.1" xref="S4.SS1.SSS0.Px7.p1.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px7.p1.1.m1.1b"><ci id="S4.SS1.SSS0.Px7.p1.1.m1.1.1.cmml" xref="S4.SS1.SSS0.Px7.p1.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px7.p1.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px7.p1.1.m1.1d">italic_μ</annotation></semantics></math> corresponds to a polytope, and for each <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS1.SSS0.Px7.p1.2.m2.1"><semantics id="S4.SS1.SSS0.Px7.p1.2.m2.1a"><mi id="S4.SS1.SSS0.Px7.p1.2.m2.1.1" xref="S4.SS1.SSS0.Px7.p1.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.SSS0.Px7.p1.2.m2.1b"><ci id="S4.SS1.SSS0.Px7.p1.2.m2.1.1.cmml" xref="S4.SS1.SSS0.Px7.p1.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.SSS0.Px7.p1.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.SSS0.Px7.p1.2.m2.1d">italic_μ</annotation></semantics></math> it is necessary to compute the integral of some multivariate function over such polytope.) Therefore, in this context the <span class="ltx_text ltx_font_italic" id="S4.SS1.SSS0.Px7.p1.2.2">effectiveness</span> of enumeration in reducing the number of assignments generated is even more important that the <span class="ltx_text ltx_font_italic" id="S4.SS1.SSS0.Px7.p1.2.3">efficiency</span> in terms of CPU time required to enumerate them. Therefore, with entailment-based enumeration the reduced number of assignments produced, and hence of integrals computed, should definitely compensate the extra cost of entailment checking.</p> </div> </section> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Verification vs. entailment in search procedures and formula compilers.</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">When applied to satisfiable formulas, algorithms like <span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.1">Analytic Tableaux</span> <span class="ltx_note ltx_role_footnote" id="footnote3"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>Notice that Analytic Tableaux may generate duplicated or subsumed assignments (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib19" title="">19</a>]</cite>)</span></span></span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib38" title="">38</a>]</cite> or “classic” <span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.2">DPLL</span> <span class="ltx_note ltx_role_footnote" id="footnote4"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>Classic DPLL procedure <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib14" title="">14</a>]</cite> was designed to work for CNF formulas. Nevertheless it is easy to produce non-CNF a version of this procedure (see e.g. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib1" title="">1</a>]</cite>).</span></span></span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib14" title="">14</a>]</cite>, or even recent sophisticate AllSAT procedures for circuits <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>]</cite>, produce branches representing partial assignments which <span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.3">verify</span> the input formula. The same happens with enumeration algorithms for d-DNNF formulas, which resemble the tableaux enumeration algorithm, with the restriction that the formulas they are applied to are deterministic and decomposable NNFs, so that the enumerated assignment are pairwise disjoint and straighforward to compute, because the enumeration reduces to the traversal of the and-or DAG without encountering inconsistent branches <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib10" title="">10</a>]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample7"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 7</span></h6> <div class="ltx_para" id="Thmexample7.p1"> <p class="ltx_p" id="Thmexample7.p1.6">Consider <math alttext="\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}((A_{1}\wedge A_{2})\vee(% A_{1}\wedge\neg A_{2}))\wedge((\neg A_{3}\wedge A_{4})\vee(\neg A_{3}\wedge% \neg A_{4}))" class="ltx_Math" display="inline" id="Thmexample7.p1.1.m1.2"><semantics id="Thmexample7.p1.1.m1.2a"><mrow id="Thmexample7.p1.1.m1.2.2" xref="Thmexample7.p1.1.m1.2.2.cmml"><mi id="Thmexample7.p1.1.m1.2.2.4" xref="Thmexample7.p1.1.m1.2.2.4.cmml">φ</mi><mover id="Thmexample7.p1.1.m1.2.2.3" xref="Thmexample7.p1.1.m1.2.2.3.cmml"><mo id="Thmexample7.p1.1.m1.2.2.3.2" xref="Thmexample7.p1.1.m1.2.2.3.2.cmml">=</mo><mtext id="Thmexample7.p1.1.m1.2.2.3.3" mathsize="71%" xref="Thmexample7.p1.1.m1.2.2.3.3a.cmml">def</mtext></mover><mrow id="Thmexample7.p1.1.m1.2.2.2" xref="Thmexample7.p1.1.m1.2.2.2.cmml"><mrow id="Thmexample7.p1.1.m1.1.1.1.1.1" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample7.p1.1.m1.1.1.1.1.1.1" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.cmml"><mrow id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><msub id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.2" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.2.2" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.2.2.cmml">A</mi><mn id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.2.3" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.1" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">∧</mo><msub id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.3" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">A</mi><mn id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.1.3" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.3.cmml">∨</mo><mrow id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.cmml"><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.2" stretchy="false" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.cmml">(</mo><mrow id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.cmml"><msub id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.2" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.2.cmml"><mi id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.2.2" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.2.2.cmml">A</mi><mn id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.2.3" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.1" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.1.cmml">∧</mo><mrow id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.cmml"><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.1" rspace="0.167em" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.1.cmml">¬</mo><msub id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.cmml"><mi id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.2" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.2.cmml">A</mi><mn id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.3" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.3" stretchy="false" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="Thmexample7.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmexample7.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample7.p1.1.m1.2.2.2.3" xref="Thmexample7.p1.1.m1.2.2.2.3.cmml">∧</mo><mrow 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xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2"><not id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.1"></not><apply id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.2.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.2.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.2">subscript</csymbol><ci id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.2.2.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.2.2">𝐴</ci><cn id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.2.3.cmml" type="integer" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.2.2.3">3</cn></apply></apply><apply id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.3.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.3.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.3">subscript</csymbol><ci id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.3.2.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.3.2">𝐴</ci><cn id="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.3.3.cmml" type="integer" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.1.1.1.3.3">4</cn></apply></apply><apply id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1"><and id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.1"></and><apply id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2"><not id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.1"></not><apply id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2"><csymbol cd="ambiguous" id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2">subscript</csymbol><ci id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2.2.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2.2">𝐴</ci><cn id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2.3.cmml" type="integer" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2.3">3</cn></apply></apply><apply id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3"><not id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.1"></not><apply id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2"><csymbol cd="ambiguous" id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.1.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2">subscript</csymbol><ci id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.2.cmml" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.2">𝐴</ci><cn id="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.3.cmml" type="integer" xref="Thmexample7.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.3">4</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p1.1.m1.2c">\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}((A_{1}\wedge A_{2})\vee(% A_{1}\wedge\neg A_{2}))\wedge((\neg A_{3}\wedge A_{4})\vee(\neg A_{3}\wedge% \neg A_{4}))</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p1.1.m1.2d">italic_φ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ( ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) ∧ ( ( ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∨ ( ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) )</annotation></semantics></math>. <br class="ltx_break"/>Notice that the single assignment <math alttext="\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1},\neg A_{3}}\}{}" class="ltx_Math" display="inline" id="Thmexample7.p1.2.m2.2"><semantics id="Thmexample7.p1.2.m2.2a"><mrow id="Thmexample7.p1.2.m2.2.2" xref="Thmexample7.p1.2.m2.2.2.cmml"><mi id="Thmexample7.p1.2.m2.2.2.4" xref="Thmexample7.p1.2.m2.2.2.4.cmml">μ</mi><mover id="Thmexample7.p1.2.m2.2.2.3" xref="Thmexample7.p1.2.m2.2.2.3.cmml"><mo id="Thmexample7.p1.2.m2.2.2.3.2" xref="Thmexample7.p1.2.m2.2.2.3.2.cmml">=</mo><mtext id="Thmexample7.p1.2.m2.2.2.3.3" mathsize="71%" xref="Thmexample7.p1.2.m2.2.2.3.3a.cmml">def</mtext></mover><mrow id="Thmexample7.p1.2.m2.2.2.2.2" xref="Thmexample7.p1.2.m2.2.2.2.3.cmml"><mo id="Thmexample7.p1.2.m2.2.2.2.2.3" stretchy="false" xref="Thmexample7.p1.2.m2.2.2.2.3.cmml">{</mo><msub id="Thmexample7.p1.2.m2.1.1.1.1.1" xref="Thmexample7.p1.2.m2.1.1.1.1.1.cmml"><mi id="Thmexample7.p1.2.m2.1.1.1.1.1.2" xref="Thmexample7.p1.2.m2.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample7.p1.2.m2.1.1.1.1.1.3" xref="Thmexample7.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample7.p1.2.m2.2.2.2.2.4" xref="Thmexample7.p1.2.m2.2.2.2.3.cmml">,</mo><mrow id="Thmexample7.p1.2.m2.2.2.2.2.2" xref="Thmexample7.p1.2.m2.2.2.2.2.2.cmml"><mo id="Thmexample7.p1.2.m2.2.2.2.2.2.1" rspace="0.167em" xref="Thmexample7.p1.2.m2.2.2.2.2.2.1.cmml">¬</mo><msub id="Thmexample7.p1.2.m2.2.2.2.2.2.2" xref="Thmexample7.p1.2.m2.2.2.2.2.2.2.cmml"><mi id="Thmexample7.p1.2.m2.2.2.2.2.2.2.2" xref="Thmexample7.p1.2.m2.2.2.2.2.2.2.2.cmml">A</mi><mn id="Thmexample7.p1.2.m2.2.2.2.2.2.2.3" xref="Thmexample7.p1.2.m2.2.2.2.2.2.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample7.p1.2.m2.2.2.2.2.5" stretchy="false" xref="Thmexample7.p1.2.m2.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample7.p1.2.m2.2b"><apply id="Thmexample7.p1.2.m2.2.2.cmml" xref="Thmexample7.p1.2.m2.2.2"><apply id="Thmexample7.p1.2.m2.2.2.3.cmml" xref="Thmexample7.p1.2.m2.2.2.3"><csymbol cd="ambiguous" id="Thmexample7.p1.2.m2.2.2.3.1.cmml" xref="Thmexample7.p1.2.m2.2.2.3">superscript</csymbol><eq id="Thmexample7.p1.2.m2.2.2.3.2.cmml" xref="Thmexample7.p1.2.m2.2.2.3.2"></eq><ci id="Thmexample7.p1.2.m2.2.2.3.3a.cmml" xref="Thmexample7.p1.2.m2.2.2.3.3"><mtext id="Thmexample7.p1.2.m2.2.2.3.3.cmml" mathsize="50%" xref="Thmexample7.p1.2.m2.2.2.3.3">def</mtext></ci></apply><ci id="Thmexample7.p1.2.m2.2.2.4.cmml" xref="Thmexample7.p1.2.m2.2.2.4">𝜇</ci><set id="Thmexample7.p1.2.m2.2.2.2.3.cmml" xref="Thmexample7.p1.2.m2.2.2.2.2"><apply id="Thmexample7.p1.2.m2.1.1.1.1.1.cmml" xref="Thmexample7.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample7.p1.2.m2.1.1.1.1.1.1.cmml" xref="Thmexample7.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="Thmexample7.p1.2.m2.1.1.1.1.1.2.cmml" xref="Thmexample7.p1.2.m2.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample7.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample7.p1.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="Thmexample7.p1.2.m2.2.2.2.2.2.cmml" xref="Thmexample7.p1.2.m2.2.2.2.2.2"><not id="Thmexample7.p1.2.m2.2.2.2.2.2.1.cmml" xref="Thmexample7.p1.2.m2.2.2.2.2.2.1"></not><apply id="Thmexample7.p1.2.m2.2.2.2.2.2.2.cmml" xref="Thmexample7.p1.2.m2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample7.p1.2.m2.2.2.2.2.2.2.1.cmml" xref="Thmexample7.p1.2.m2.2.2.2.2.2.2">subscript</csymbol><ci id="Thmexample7.p1.2.m2.2.2.2.2.2.2.2.cmml" xref="Thmexample7.p1.2.m2.2.2.2.2.2.2.2">𝐴</ci><cn id="Thmexample7.p1.2.m2.2.2.2.2.2.2.3.cmml" type="integer" xref="Thmexample7.p1.2.m2.2.2.2.2.2.2.3">3</cn></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p1.2.m2.2c">\mu\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1},\neg A_{3}}\}{}</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p1.2.m2.2d">italic_μ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT }</annotation></semantics></math> subsumes all total truth assignments satisfying <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmexample7.p1.3.m3.1"><semantics id="Thmexample7.p1.3.m3.1a"><mi id="Thmexample7.p1.3.m3.1.1" xref="Thmexample7.p1.3.m3.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmexample7.p1.3.m3.1b"><ci id="Thmexample7.p1.3.m3.1.1.cmml" xref="Thmexample7.p1.3.m3.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p1.3.m3.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p1.3.m3.1d">italic_φ</annotation></semantics></math> and is such that <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="Thmexample7.p1.4.m4.1"><semantics id="Thmexample7.p1.4.m4.1a"><mrow id="Thmexample7.p1.4.m4.1.1" xref="Thmexample7.p1.4.m4.1.1.cmml"><mi id="Thmexample7.p1.4.m4.1.1.2" xref="Thmexample7.p1.4.m4.1.1.2.cmml">μ</mi><mo id="Thmexample7.p1.4.m4.1.1.1" xref="Thmexample7.p1.4.m4.1.1.1.cmml">⊧</mo><mi id="Thmexample7.p1.4.m4.1.1.3" xref="Thmexample7.p1.4.m4.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample7.p1.4.m4.1b"><apply id="Thmexample7.p1.4.m4.1.1.cmml" xref="Thmexample7.p1.4.m4.1.1"><csymbol cd="latexml" id="Thmexample7.p1.4.m4.1.1.1.cmml" xref="Thmexample7.p1.4.m4.1.1.1">models</csymbol><ci id="Thmexample7.p1.4.m4.1.1.2.cmml" xref="Thmexample7.p1.4.m4.1.1.2">𝜇</ci><ci id="Thmexample7.p1.4.m4.1.1.3.cmml" xref="Thmexample7.p1.4.m4.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p1.4.m4.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p1.4.m4.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> but <math alttext="\mu\not\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="Thmexample7.p1.5.m5.1"><semantics id="Thmexample7.p1.5.m5.1a"><mrow id="Thmexample7.p1.5.m5.1b"><mi id="Thmexample7.p1.5.m5.1.1">μ</mi><mpadded id="Thmexample7.p1.5.m5.1c" width="0.969em"><mo id="Thmexample7.p1.5.m5.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample7.p1.5.m5.1.3">≈</mo><mi id="Thmexample7.p1.5.m5.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmexample7.p1.5.m5.1d">\mu\not\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p1.5.m5.1e">italic_μ ∤ ≈ italic_φ</annotation></semantics></math>, since <math alttext="\varphi|_{\mu}=(A_{2}\vee\neg A_{2})\wedge(A_{4}\vee\neg A_{4})" class="ltx_Math" display="inline" id="Thmexample7.p1.6.m6.4"><semantics id="Thmexample7.p1.6.m6.4a"><mrow id="Thmexample7.p1.6.m6.4.4" xref="Thmexample7.p1.6.m6.4.4.cmml"><msub id="Thmexample7.p1.6.m6.4.4.4.2" xref="Thmexample7.p1.6.m6.4.4.4.1.cmml"><mrow id="Thmexample7.p1.6.m6.4.4.4.2.2" xref="Thmexample7.p1.6.m6.4.4.4.1.cmml"><mi id="Thmexample7.p1.6.m6.1.1" xref="Thmexample7.p1.6.m6.1.1.cmml">φ</mi><mo id="Thmexample7.p1.6.m6.4.4.4.2.2.1" stretchy="false" xref="Thmexample7.p1.6.m6.4.4.4.1.1.cmml">|</mo></mrow><mi id="Thmexample7.p1.6.m6.2.2.1" xref="Thmexample7.p1.6.m6.2.2.1.cmml">μ</mi></msub><mo id="Thmexample7.p1.6.m6.4.4.3" xref="Thmexample7.p1.6.m6.4.4.3.cmml">=</mo><mrow id="Thmexample7.p1.6.m6.4.4.2" xref="Thmexample7.p1.6.m6.4.4.2.cmml"><mrow id="Thmexample7.p1.6.m6.3.3.1.1.1" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.cmml"><mo id="Thmexample7.p1.6.m6.3.3.1.1.1.2" stretchy="false" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.cmml">(</mo><mrow id="Thmexample7.p1.6.m6.3.3.1.1.1.1" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.cmml"><msub id="Thmexample7.p1.6.m6.3.3.1.1.1.1.2" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.2.cmml"><mi id="Thmexample7.p1.6.m6.3.3.1.1.1.1.2.2" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.2.2.cmml">A</mi><mn id="Thmexample7.p1.6.m6.3.3.1.1.1.1.2.3" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.2.3.cmml">2</mn></msub><mo id="Thmexample7.p1.6.m6.3.3.1.1.1.1.1" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.1.cmml">∨</mo><mrow id="Thmexample7.p1.6.m6.3.3.1.1.1.1.3" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.cmml"><mo id="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.1" rspace="0.167em" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.1.cmml">¬</mo><msub id="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.2" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.2.cmml"><mi id="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.2.2" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.2.2.cmml">A</mi><mn id="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.2.3" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="Thmexample7.p1.6.m6.3.3.1.1.1.3" stretchy="false" xref="Thmexample7.p1.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample7.p1.6.m6.4.4.2.3" xref="Thmexample7.p1.6.m6.4.4.2.3.cmml">∧</mo><mrow id="Thmexample7.p1.6.m6.4.4.2.2.1" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.cmml"><mo id="Thmexample7.p1.6.m6.4.4.2.2.1.2" stretchy="false" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.cmml">(</mo><mrow id="Thmexample7.p1.6.m6.4.4.2.2.1.1" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.cmml"><msub id="Thmexample7.p1.6.m6.4.4.2.2.1.1.2" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.2.cmml"><mi id="Thmexample7.p1.6.m6.4.4.2.2.1.1.2.2" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.2.2.cmml">A</mi><mn id="Thmexample7.p1.6.m6.4.4.2.2.1.1.2.3" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.2.3.cmml">4</mn></msub><mo id="Thmexample7.p1.6.m6.4.4.2.2.1.1.1" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.1.cmml">∨</mo><mrow id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.cmml"><mo id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.1" rspace="0.167em" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.1.cmml">¬</mo><msub id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.cmml"><mi id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.2" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.2.cmml">A</mi><mn id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.3" 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id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.1.cmml" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.1"></not><apply id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.cmml" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2"><csymbol cd="ambiguous" id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.1.cmml" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2">subscript</csymbol><ci id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.2.cmml" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.2">𝐴</ci><cn id="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.3.cmml" type="integer" xref="Thmexample7.p1.6.m6.4.4.2.2.1.1.3.2.3">4</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p1.6.m6.4c">\varphi|_{\mu}=(A_{2}\vee\neg A_{2})\wedge(A_{4}\vee\neg A_{4})</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p1.6.m6.4d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = ( italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∨ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∧ ( italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ∨ ¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="Thmexample7.p2"> <p class="ltx_p" id="Thmexample7.p2.5">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.F4" title="Figure 4 ‣ 4.2 Verification vs. entailment in search procedures and formula compilers. ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">4</span></a> represents the search trees corresponding to AllSAT executions of Analytic Tableaux and (non-CNF) DPLL on <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmexample7.p2.1.m1.1"><semantics id="Thmexample7.p2.1.m1.1a"><mi id="Thmexample7.p2.1.m1.1.1" xref="Thmexample7.p2.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmexample7.p2.1.m1.1b"><ci id="Thmexample7.p2.1.m1.1.1.cmml" xref="Thmexample7.p2.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p2.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p2.1.m1.1d">italic_φ</annotation></semantics></math>,<span class="ltx_note ltx_role_footnote" id="footnote5"><sup class="ltx_note_mark">5</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">5</sup><span class="ltx_tag ltx_tag_note">5</span>Here in DPLL the pure-literal rule <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib14" title="">14</a>]</cite> is not used because in All-SAT it may hinder the enumeration of relevant models (see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib33" title="">33</a>]</cite>).</span></span></span> <span class="ltx_note ltx_role_footnote" id="footnote6"><sup class="ltx_note_mark">6</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">6</sup><span class="ltx_tag ltx_tag_note">6</span>The search trees may vary with the ordering by which variables and subformulas are analyzed.</span></span></span> Since (the DAG version of) <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmexample7.p2.2.m2.1"><semantics id="Thmexample7.p2.2.m2.1a"><mi id="Thmexample7.p2.2.m2.1.1" xref="Thmexample7.p2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmexample7.p2.2.m2.1b"><ci id="Thmexample7.p2.2.m2.1.1.cmml" xref="Thmexample7.p2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p2.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p2.2.m2.1d">italic_φ</annotation></semantics></math> is already in d-DNNF (<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.F4" title="In 4.2 Verification vs. entailment in search procedures and formula compilers. ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">fig.</span> <span class="ltx_text ltx_ref_tag">4</span></a>), in this case the d-DNNF enumeration resembles the tableaux enumeration. 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type="integer" xref="Thmexample7.p2.3.m3.2.2.2.2.4.4.2.3">4</cn></apply></apply></set><set id="Thmexample7.p2.3.m3.3.3.3.3.5.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.4"><apply id="Thmexample7.p2.3.m3.3.3.3.3.1.1.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.1.1"><csymbol cd="ambiguous" id="Thmexample7.p2.3.m3.3.3.3.3.1.1.1.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.1.1">subscript</csymbol><ci id="Thmexample7.p2.3.m3.3.3.3.3.1.1.2.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.1.1.2">𝐴</ci><cn id="Thmexample7.p2.3.m3.3.3.3.3.1.1.3.cmml" type="integer" xref="Thmexample7.p2.3.m3.3.3.3.3.1.1.3">1</cn></apply><apply id="Thmexample7.p2.3.m3.3.3.3.3.2.2.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.2.2"><not id="Thmexample7.p2.3.m3.3.3.3.3.2.2.1.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.2.2.1"></not><apply id="Thmexample7.p2.3.m3.3.3.3.3.2.2.2.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.2.2.2"><csymbol cd="ambiguous" id="Thmexample7.p2.3.m3.3.3.3.3.2.2.2.1.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.2.2.2">subscript</csymbol><ci id="Thmexample7.p2.3.m3.3.3.3.3.2.2.2.2.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.2.2.2.2">𝐴</ci><cn id="Thmexample7.p2.3.m3.3.3.3.3.2.2.2.3.cmml" type="integer" xref="Thmexample7.p2.3.m3.3.3.3.3.2.2.2.3">2</cn></apply></apply><apply id="Thmexample7.p2.3.m3.3.3.3.3.3.3.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.3.3"><not id="Thmexample7.p2.3.m3.3.3.3.3.3.3.1.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.3.3.1"></not><apply id="Thmexample7.p2.3.m3.3.3.3.3.3.3.2.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.3.3.2"><csymbol cd="ambiguous" id="Thmexample7.p2.3.m3.3.3.3.3.3.3.2.1.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.3.3.2">subscript</csymbol><ci id="Thmexample7.p2.3.m3.3.3.3.3.3.3.2.2.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.3.3.2.2">𝐴</ci><cn id="Thmexample7.p2.3.m3.3.3.3.3.3.3.2.3.cmml" type="integer" xref="Thmexample7.p2.3.m3.3.3.3.3.3.3.2.3">3</cn></apply></apply><apply id="Thmexample7.p2.3.m3.3.3.3.3.4.4.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.4.4"><csymbol cd="ambiguous" id="Thmexample7.p2.3.m3.3.3.3.3.4.4.1.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.4.4">subscript</csymbol><ci id="Thmexample7.p2.3.m3.3.3.3.3.4.4.2.cmml" xref="Thmexample7.p2.3.m3.3.3.3.3.4.4.2">𝐴</ci><cn id="Thmexample7.p2.3.m3.3.3.3.3.4.4.3.cmml" type="integer" xref="Thmexample7.p2.3.m3.3.3.3.3.4.4.3">4</cn></apply></set><set id="Thmexample7.p2.3.m3.4.4.4.4.5.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.4"><apply id="Thmexample7.p2.3.m3.4.4.4.4.1.1.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.1.1"><csymbol cd="ambiguous" id="Thmexample7.p2.3.m3.4.4.4.4.1.1.1.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.1.1">subscript</csymbol><ci id="Thmexample7.p2.3.m3.4.4.4.4.1.1.2.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.1.1.2">𝐴</ci><cn id="Thmexample7.p2.3.m3.4.4.4.4.1.1.3.cmml" type="integer" xref="Thmexample7.p2.3.m3.4.4.4.4.1.1.3">1</cn></apply><apply id="Thmexample7.p2.3.m3.4.4.4.4.2.2.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.2.2"><not id="Thmexample7.p2.3.m3.4.4.4.4.2.2.1.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.2.2.1"></not><apply id="Thmexample7.p2.3.m3.4.4.4.4.2.2.2.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.2.2.2"><csymbol cd="ambiguous" id="Thmexample7.p2.3.m3.4.4.4.4.2.2.2.1.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.2.2.2">subscript</csymbol><ci id="Thmexample7.p2.3.m3.4.4.4.4.2.2.2.2.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.2.2.2.2">𝐴</ci><cn id="Thmexample7.p2.3.m3.4.4.4.4.2.2.2.3.cmml" type="integer" xref="Thmexample7.p2.3.m3.4.4.4.4.2.2.2.3">2</cn></apply></apply><apply id="Thmexample7.p2.3.m3.4.4.4.4.3.3.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.3.3"><not id="Thmexample7.p2.3.m3.4.4.4.4.3.3.1.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.3.3.1"></not><apply id="Thmexample7.p2.3.m3.4.4.4.4.3.3.2.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.3.3.2"><csymbol cd="ambiguous" id="Thmexample7.p2.3.m3.4.4.4.4.3.3.2.1.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.3.3.2">subscript</csymbol><ci id="Thmexample7.p2.3.m3.4.4.4.4.3.3.2.2.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.3.3.2.2">𝐴</ci><cn id="Thmexample7.p2.3.m3.4.4.4.4.3.3.2.3.cmml" type="integer" xref="Thmexample7.p2.3.m3.4.4.4.4.3.3.2.3">3</cn></apply></apply><apply id="Thmexample7.p2.3.m3.4.4.4.4.4.4.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.4.4"><not id="Thmexample7.p2.3.m3.4.4.4.4.4.4.1.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.4.4.1"></not><apply id="Thmexample7.p2.3.m3.4.4.4.4.4.4.2.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.4.4.2"><csymbol cd="ambiguous" id="Thmexample7.p2.3.m3.4.4.4.4.4.4.2.1.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.4.4.2">subscript</csymbol><ci id="Thmexample7.p2.3.m3.4.4.4.4.4.4.2.2.cmml" xref="Thmexample7.p2.3.m3.4.4.4.4.4.4.2.2">𝐴</ci><cn id="Thmexample7.p2.3.m3.4.4.4.4.4.4.2.3.cmml" type="integer" xref="Thmexample7.p2.3.m3.4.4.4.4.4.4.2.3">4</cn></apply></apply></set></set></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p2.3.m3.4c">\{{\{{A_{1},A_{2},\neg A_{3},A_{4}}\},\{{A_{1},A_{2},\neg A_{3},\neg A_{4}}\},% \{{A_{1},\neg A_{2},\neg A_{3},A_{4}}\},\{{A_{1},\neg A_{2},\neg A_{3},\neg A_% {4}}\}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p2.3.m3.4d">{ { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } , { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } , { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } , { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } }</annotation></semantics></math>. Notice that all the assignments produced are total. Notice also that neither Analytic Tableaux nor DPLL nor d-DNNF enumeration in this case can produce <math alttext="\{{\{{A_{1},\neg A_{3}}\}}\}" class="ltx_Math" display="inline" id="Thmexample7.p2.4.m4.1"><semantics id="Thmexample7.p2.4.m4.1a"><mrow id="Thmexample7.p2.4.m4.1.1.1" xref="Thmexample7.p2.4.m4.1.1.2.cmml"><mo id="Thmexample7.p2.4.m4.1.1.1.2" stretchy="false" xref="Thmexample7.p2.4.m4.1.1.2.cmml">{</mo><mrow id="Thmexample7.p2.4.m4.1.1.1.1.2" xref="Thmexample7.p2.4.m4.1.1.1.1.3.cmml"><mo id="Thmexample7.p2.4.m4.1.1.1.1.2.3" stretchy="false" xref="Thmexample7.p2.4.m4.1.1.1.1.3.cmml">{</mo><msub id="Thmexample7.p2.4.m4.1.1.1.1.1.1" xref="Thmexample7.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="Thmexample7.p2.4.m4.1.1.1.1.1.1.2" xref="Thmexample7.p2.4.m4.1.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample7.p2.4.m4.1.1.1.1.1.1.3" xref="Thmexample7.p2.4.m4.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample7.p2.4.m4.1.1.1.1.2.4" xref="Thmexample7.p2.4.m4.1.1.1.1.3.cmml">,</mo><mrow id="Thmexample7.p2.4.m4.1.1.1.1.2.2" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.cmml"><mo id="Thmexample7.p2.4.m4.1.1.1.1.2.2.1" rspace="0.167em" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.1.cmml">¬</mo><msub id="Thmexample7.p2.4.m4.1.1.1.1.2.2.2" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.cmml"><mi id="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.2" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.2.cmml">A</mi><mn id="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.3" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample7.p2.4.m4.1.1.1.1.2.5" stretchy="false" xref="Thmexample7.p2.4.m4.1.1.1.1.3.cmml">}</mo></mrow><mo id="Thmexample7.p2.4.m4.1.1.1.3" stretchy="false" xref="Thmexample7.p2.4.m4.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample7.p2.4.m4.1b"><set id="Thmexample7.p2.4.m4.1.1.2.cmml" xref="Thmexample7.p2.4.m4.1.1.1"><set id="Thmexample7.p2.4.m4.1.1.1.1.3.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.2"><apply id="Thmexample7.p2.4.m4.1.1.1.1.1.1.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample7.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.1.1">subscript</csymbol><ci id="Thmexample7.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample7.p2.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample7.p2.4.m4.1.1.1.1.1.1.3">1</cn></apply><apply id="Thmexample7.p2.4.m4.1.1.1.1.2.2.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2"><not id="Thmexample7.p2.4.m4.1.1.1.1.2.2.1.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.1"></not><apply id="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.1.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.2">subscript</csymbol><ci id="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.2.cmml" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.2">𝐴</ci><cn id="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.3.cmml" type="integer" xref="Thmexample7.p2.4.m4.1.1.1.1.2.2.2.3">3</cn></apply></apply></set></set></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p2.4.m4.1c">\{{\{{A_{1},\neg A_{3}}\}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p2.4.m4.1d">{ { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT } }</annotation></semantics></math>, regardless the strategy adopted. The same applies also to the verification-based AllSAT circuit enumeration of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>]</cite>, see <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample9" title="Example 9 ‣ 4.3 Implementing entailment within enumeration procedures ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">9</span></a>. <math alttext="\diamond" class="ltx_Math" display="inline" id="Thmexample7.p2.5.m5.1"><semantics id="Thmexample7.p2.5.m5.1a"><mo id="Thmexample7.p2.5.m5.1.1" xref="Thmexample7.p2.5.m5.1.1.cmml">⋄</mo><annotation-xml encoding="MathML-Content" id="Thmexample7.p2.5.m5.1b"><ci id="Thmexample7.p2.5.m5.1.1.cmml" xref="Thmexample7.p2.5.m5.1.1">⋄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample7.p2.5.m5.1c">\diamond</annotation><annotation encoding="application/x-llamapun" id="Thmexample7.p2.5.m5.1d">⋄</annotation></semantics></math></p> </div> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.3">OBDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib4" title="">4</a>]</cite> contain branches representing partial assignments which <span class="ltx_text ltx_font_italic" id="S4.SS2.p2.3.1">entail</span> the input formula, because if <math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mrow id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.p2.1.m1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.2.cmml">μ</mi><mo id="S4.SS2.p2.1.m1.1.1.1" xref="S4.SS2.p2.1.m1.1.1.1.cmml">⊧</mo><mi id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1"><csymbol cd="latexml" id="S4.SS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1.1">models</csymbol><ci id="S4.SS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.2">𝜇</ci><ci id="S4.SS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_μ ⊧ italic_φ</annotation></semantics></math> then <math alttext="\varphi|_{\mu}" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.2"><semantics id="S4.SS2.p2.2.m2.2a"><msub id="S4.SS2.p2.2.m2.2.3.2" xref="S4.SS2.p2.2.m2.2.3.1.cmml"><mrow id="S4.SS2.p2.2.m2.2.3.2.2" xref="S4.SS2.p2.2.m2.2.3.1.cmml"><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">φ</mi><mo id="S4.SS2.p2.2.m2.2.3.2.2.1" stretchy="false" xref="S4.SS2.p2.2.m2.2.3.1.1.cmml">|</mo></mrow><mi id="S4.SS2.p2.2.m2.2.2.1" xref="S4.SS2.p2.2.m2.2.2.1.cmml">μ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.2b"><apply id="S4.SS2.p2.2.m2.2.3.1.cmml" xref="S4.SS2.p2.2.m2.2.3.2"><csymbol cd="latexml" id="S4.SS2.p2.2.m2.2.3.1.1.cmml" xref="S4.SS2.p2.2.m2.2.3.2.2.1">evaluated-at</csymbol><ci id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">𝜑</ci><ci id="S4.SS2.p2.2.m2.2.2.1.cmml" xref="S4.SS2.p2.2.m2.2.2.1">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.2c">\varphi|_{\mu}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.2d">italic_φ | start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT</annotation></semantics></math> is valid (property <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmproperty6" title="Property 6 ‣ Verification vs. Entailment. ‣ 3.1 Verification and entailment of plain formulas. ‣ 3 A theoretical Analysis of verificationand entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">6</span></a>(iii)), so that its corresponding sub-OBDD is reduced into the <math alttext="\top" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><mo id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml">⊤</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><csymbol cd="latexml" id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1">top</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">\top</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">⊤</annotation></semantics></math> node. (SDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib12" title="">12</a>]</cite> behave similarly.)</p> </div> <figure class="ltx_figure" id="S4.F4"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <table class="ltx_tabular ltx_figure_panel ltx_align_middle" id="S4.F4.4"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.F4.4.4"> <td class="ltx_td ltx_align_left" id="S4.F4.2.2.2"> <div class="ltx_inline-block ltx_transformed_outer" id="S4.F4.2.2.2.2" style="width:172.2pt;height:129pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-21.5pt,16.1pt) scale(0.8,0.8) ;"><svg fill="none" height="0" overflow="visible" stroke="none" version="1.1" width="0"><g transform="translate(0,0) scale(1,-1)"><g transform="translate(0,223) scale(1, -1)"><foreignobject height="223" overflow="visible" width="298"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" 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xref="S4.F4.2.2.2.2.pic2.4.4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F4.2.2.2.2.pic2.4.4.1.m1.1.1.1.cmml" xref="S4.F4.2.2.2.2.pic2.4.4.1.m1.1.1">subscript</csymbol><ci id="S4.F4.2.2.2.2.pic2.4.4.1.m1.1.1.2.cmml" xref="S4.F4.2.2.2.2.pic2.4.4.1.m1.1.1.2">𝐴</ci><cn id="S4.F4.2.2.2.2.pic2.4.4.1.m1.1.1.3.cmml" type="integer" xref="S4.F4.2.2.2.2.pic2.4.4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.2.2.2.2.pic2.4.4.1.m1.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.2.2.2.2.pic2.4.4.1.m1.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(334.75,-198.26)"><g class="makebox" transform="translate(-12.89,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="25.7368202573682"><math alttext="\neg A_{4}" class="ltx_Math" display="inline" 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id="S4.F4.2.2.2.2.pic2.5.5.1.m1.1.1.2.1.cmml" xref="S4.F4.2.2.2.2.pic2.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S4.F4.2.2.2.2.pic2.5.5.1.m1.1.1.2.2.cmml" xref="S4.F4.2.2.2.2.pic2.5.5.1.m1.1.1.2.2">𝐴</ci><cn id="S4.F4.2.2.2.2.pic2.5.5.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.2.2.2.2.pic2.5.5.1.m1.1.1.2.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.2.2.2.2.pic2.5.5.1.m1.1c">\neg A_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.2.2.2.2.pic2.5.5.1.m1.1d">¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(334.75,-166.76)"><g class="makebox" transform="translate(-12.89,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="25.7368202573682"><math alttext="\neg A_{2}" class="ltx_Math" display="inline" id="S4.F4.2.2.2.2.pic2.6.6.1.m1.1"><semantics 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xref="S4.F4.2.2.2.2.pic2.6.6.1.m1.1.1.2">subscript</csymbol><ci id="S4.F4.2.2.2.2.pic2.6.6.1.m1.1.1.2.2.cmml" xref="S4.F4.2.2.2.2.pic2.6.6.1.m1.1.1.2.2">𝐴</ci><cn id="S4.F4.2.2.2.2.pic2.6.6.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.2.2.2.2.pic2.6.6.1.m1.1.1.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.2.2.2.2.pic2.6.6.1.m1.1c">\neg A_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.2.2.2.2.pic2.6.6.1.m1.1d">¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(43.4,-151.02)"><g class="makebox" transform="translate(-7.13,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="14.2521101425211"><math alttext="A_{1}" class="ltx_Math" display="inline" id="S4.F4.2.2.2.2.pic2.7.7.1.m1.1"><semantics id="S4.F4.2.2.2.2.pic2.7.7.1.m1.1a"><msub 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start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(43.4,-198.26)"><g class="makebox" transform="translate(-7.13,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="14.2521101425211"><math alttext="A_{4}" class="ltx_Math" display="inline" id="S4.F4.2.2.2.2.pic2.8.8.1.m1.1"><semantics id="S4.F4.2.2.2.2.pic2.8.8.1.m1.1a"><msub id="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1" xref="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1.cmml"><mi id="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1.2" xref="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1.2.cmml">A</mi><mn id="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1.3" xref="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.2.2.2.2.pic2.8.8.1.m1.1b"><apply id="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1.cmml" xref="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F4.2.2.2.2.pic2.8.8.1.m1.1.1.1.cmml" 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y="0">F</text></g></g><g transform="translate(519.81,-151.02)"><g class="makebox" transform="translate(-5,0)"><text class="ltx_markedasmath" fill="black" transform="scale(1, -1)" x="0" y="0">T</text></g></g><g transform="translate(480.43,-147.08)"><g class="makebox" transform="translate(-7.13,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="14.2521101425211"><math alttext="A_{4}" class="ltx_Math" display="inline" id="S4.F4.4.4.4.2.pic2.16.16.1.m1.1"><semantics id="S4.F4.4.4.4.2.pic2.16.16.1.m1.1a"><msub id="S4.F4.4.4.4.2.pic2.16.16.1.m1.1.1" xref="S4.F4.4.4.4.2.pic2.16.16.1.m1.1.1.cmml"><mi id="S4.F4.4.4.4.2.pic2.16.16.1.m1.1.1.2" xref="S4.F4.4.4.4.2.pic2.16.16.1.m1.1.1.2.cmml">A</mi><mn id="S4.F4.4.4.4.2.pic2.16.16.1.m1.1.1.3" xref="S4.F4.4.4.4.2.pic2.16.16.1.m1.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.4.4.4.2.pic2.16.16.1.m1.1b"><apply 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transform="translate(-4.52,0)"><text class="ltx_markedasmath" fill="black" transform="scale(1, -1)" x="0" y="0">F</text></g></g></g></g></svg> </span></div> </td> </tr> </tbody> </table> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 3: </span> <math alttext="\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}((A_{1}\wedge A_{2})\vee(% A_{1}\wedge\neg A_{2}))\wedge((\neg A_{3}\wedge A_{4})\vee(\neg A_{3}\wedge% \neg A_{4})){}" class="ltx_Math" display="inline" id="S4.F4.6.m1.2"><semantics id="S4.F4.6.m1.2b"><mrow id="S4.F4.6.m1.2.2" xref="S4.F4.6.m1.2.2.cmml"><mi id="S4.F4.6.m1.2.2.4" xref="S4.F4.6.m1.2.2.4.cmml">φ</mi><mover id="S4.F4.6.m1.2.2.3" xref="S4.F4.6.m1.2.2.3.cmml"><mo id="S4.F4.6.m1.2.2.3.2" xref="S4.F4.6.m1.2.2.3.2.cmml">=</mo><mtext id="S4.F4.6.m1.2.2.3.3" mathsize="71%" xref="S4.F4.6.m1.2.2.3.3a.cmml">def</mtext></mover><mrow id="S4.F4.6.m1.2.2.2" xref="S4.F4.6.m1.2.2.2.cmml"><mrow id="S4.F4.6.m1.1.1.1.1.1" 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class="ltx_break"/>Enumeration by analytic tableaux (left) and (non-CNF) DPLL (right). </figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <table class="ltx_tabular ltx_figure_panel ltx_align_middle" id="S4.F4.12"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.F4.12.6"> <td class="ltx_td ltx_align_left" id="S4.F4.8.2.2"> <div class="ltx_inline-block ltx_transformed_outer" id="S4.F4.8.2.2.2" style="width:164.7pt;height:125.4pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-20.6pt,15.7pt) scale(0.8,0.8) ;"><svg fill="none" height="0" overflow="visible" stroke="none" version="1.1" width="0"><g transform="translate(0,0) scale(1,-1)"><g transform="translate(0,218) scale(1, -1)"><foreignobject height="218" overflow="visible" width="286"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="218" id="S4.F4.7.1.1.1.pic1.g1" src="x3.png" 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id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1a"><msub id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.cmml"><mi id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.2" xref="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.2.cmml">A</mi><mn id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.3" xref="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1b"><apply id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1">subscript</csymbol><ci id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.2.cmml" xref="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.2">𝐴</ci><cn id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.3.cmml" type="integer" xref="S4.F4.8.2.2.2.pic2.3.3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.2.2.pic2.3.3.1.m1.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(173.5,-193.01)"><g class="makebox" transform="translate(-12.89,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="25.7368202573682"><math alttext="\neg A_{2}" class="ltx_Math" display="inline" id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1a"><mrow id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.cmml"><mo id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.1" rspace="0.167em" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.1.cmml">¬</mo><msub id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.cmml"><mi id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.2" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.2.cmml">A</mi><mn id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.3" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1b"><apply id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1"><not id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.1"></not><apply id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.cmml" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.1.cmml" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2">subscript</csymbol><ci id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.2.cmml" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.2">𝐴</ci><cn id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.8.2.2.2.pic2.4.4.1.m1.1.1.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1c">\neg A_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.2.2.pic2.4.4.1.m1.1d">¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(90.56,-137.89)"><g class="makebox" transform="translate(-4.61,0)"><g transform="translate(0,7.7487200774872) scale(1, -1)"><foreignobject height="7.7487200774872" overflow="visible" width="9.2707900927079"><math alttext="\wedge" class="ltx_Math" display="inline" id="S4.F4.8.2.2.2.pic2.5.5.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.5.5.1.m1.1a"><mo id="S4.F4.8.2.2.2.pic2.5.5.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.5.5.1.m1.1.1.cmml">∧</mo><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.5.5.1.m1.1b"><and id="S4.F4.8.2.2.2.pic2.5.5.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.5.5.1.m1.1.1"></and></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.5.5.1.m1.1c">\wedge</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.2.2.pic2.5.5.1.m1.1d">∧</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(304.04,-137.89)"><g class="makebox" transform="translate(-4.61,0)"><g transform="translate(0,7.7487200774872) scale(1, -1)"><foreignobject height="7.7487200774872" overflow="visible" width="9.2707900927079"><math alttext="\wedge" class="ltx_Math" display="inline" id="S4.F4.8.2.2.2.pic2.6.6.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.6.6.1.m1.1a"><mo id="S4.F4.8.2.2.2.pic2.6.6.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.6.6.1.m1.1.1.cmml">∧</mo><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.6.6.1.m1.1b"><and id="S4.F4.8.2.2.2.pic2.6.6.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.6.6.1.m1.1.1"></and></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.6.6.1.m1.1c">\wedge</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.2.2.pic2.6.6.1.m1.1d">∧</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(216.81,-193.01)"><g class="makebox" transform="translate(-7.13,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="14.2521101425211"><math alttext="A_{4}" class="ltx_Math" display="inline" id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1a"><msub id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.cmml"><mi id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.2" xref="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.2.cmml">A</mi><mn id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.3" xref="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.3.cmml">4</mn></msub><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1b"><apply id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1">subscript</csymbol><ci id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.2.cmml" xref="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.2">𝐴</ci><cn id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.3.cmml" type="integer" xref="S4.F4.8.2.2.2.pic2.7.7.1.m1.1.1.3">4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1c">A_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.2.2.pic2.7.7.1.m1.1d">italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(268.52,-193.01)"><g class="makebox" transform="translate(-12.89,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="25.7368202573682"><math alttext="\neg A_{3}" class="ltx_Math" display="inline" id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1a"><mrow id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.cmml"><mo id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.1" rspace="0.167em" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.1.cmml">¬</mo><msub id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.cmml"><mi id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.2" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.2.cmml">A</mi><mn id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.3" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1b"><apply id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1"><not id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.1"></not><apply id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.cmml" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.1.cmml" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2">subscript</csymbol><ci id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.2.cmml" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.2">𝐴</ci><cn id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.8.2.2.2.pic2.8.8.1.m1.1.1.2.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1c">\neg A_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.2.2.pic2.8.8.1.m1.1d">¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(322.07,-193.01)"><g class="makebox" transform="translate(-12.89,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="25.7368202573682"><math alttext="\neg A_{4}" class="ltx_Math" display="inline" id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1a"><mrow id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.cmml"><mo id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.1" rspace="0.167em" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.1.cmml">¬</mo><msub id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.cmml"><mi id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.2" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.2.cmml">A</mi><mn id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.3" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.3.cmml">4</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1b"><apply id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1"><not id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.1"></not><apply id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.cmml" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.1.cmml" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2">subscript</csymbol><ci id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.2.cmml" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.2">𝐴</ci><cn id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.8.2.2.2.pic2.9.9.1.m1.1.1.2.3">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1c">\neg A_{4}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.2.2.pic2.9.9.1.m1.1d">¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(239.12,-137.89)"><g class="makebox" transform="translate(-4.61,0)"><g transform="translate(0,7.7487200774872) scale(1, -1)"><foreignobject height="7.7487200774872" overflow="visible" width="9.2707900927079"><math alttext="\wedge" class="ltx_Math" display="inline" id="S4.F4.8.2.2.2.pic2.10.10.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.10.10.1.m1.1a"><mo id="S4.F4.8.2.2.2.pic2.10.10.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.10.10.1.m1.1.1.cmml">∧</mo><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.10.10.1.m1.1b"><and id="S4.F4.8.2.2.2.pic2.10.10.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.10.10.1.m1.1.1"></and></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.10.10.1.m1.1c">\wedge</annotation><annotation encoding="application/x-llamapun" id="S4.F4.8.2.2.2.pic2.10.10.1.m1.1d">∧</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(122.41,-75.51)"><g class="makebox" transform="translate(-4.61,0)"><g transform="translate(0,7.7487200774872) scale(1, -1)"><foreignobject height="7.7487200774872" overflow="visible" width="9.2707900927079"><math alttext="\vee" class="ltx_Math" display="inline" id="S4.F4.8.2.2.2.pic2.11.11.1.m1.1"><semantics id="S4.F4.8.2.2.2.pic2.11.11.1.m1.1a"><mo id="S4.F4.8.2.2.2.pic2.11.11.1.m1.1.1" xref="S4.F4.8.2.2.2.pic2.11.11.1.m1.1.1.cmml">∨</mo><annotation-xml encoding="MathML-Content" id="S4.F4.8.2.2.2.pic2.11.11.1.m1.1b"><or id="S4.F4.8.2.2.2.pic2.11.11.1.m1.1.1.cmml" xref="S4.F4.8.2.2.2.pic2.11.11.1.m1.1.1"></or></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.8.2.2.2.pic2.11.11.1.m1.1c">\vee</annotation><annotation encoding="application/x-llamapun" 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id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.1" rspace="0.167em" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.1.cmml">¬</mo><msub id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.cmml"><mi id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.2" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.2.cmml">A</mi><mn id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.3" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1b"><apply id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.cmml" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1"><not id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.1.cmml" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.1"></not><apply id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.cmml" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.1.cmml" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.2.cmml" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.2">𝐴</ci><cn id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.12.6.6.2.pic2.5.5.1.m1.1.1.2.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1c">\neg A_{3}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.12.6.6.2.pic2.5.5.1.m1.1d">¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(406.76,-449.11)"><g class="makebox" transform="translate(-12.89,0)"><g transform="translate(0,11.8998201189982) scale(1, -1)"><foreignobject height="9.4091600940916" overflow="visible" width="25.7368202573682"><math alttext="\neg A_{1}" class="ltx_Math" display="inline" id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1"><semantics id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1a"><mrow id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.cmml"><mo id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.1" rspace="0.167em" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.1.cmml">¬</mo><msub id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.cmml"><mi id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.2" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.2.cmml">A</mi><mn id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.3" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1b"><apply id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.cmml" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1"><not id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.1.cmml" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.1"></not><apply id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.cmml" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.1.cmml" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2">subscript</csymbol><ci id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.2.cmml" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.2">𝐴</ci><cn id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.3.cmml" type="integer" xref="S4.F4.12.6.6.2.pic2.6.6.1.m1.1.1.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1c">\neg A_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.12.6.6.2.pic2.6.6.1.m1.1d">¬ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></foreignobject></g></g></g><g transform="translate(441.76,-448.67)"><g class="makebox" transform="translate(-4.52,0)"><text class="ltx_markedasmath" fill="black" transform="scale(1, -1)" x="0" y="0">F</text></g></g></g></g></svg> </span></div> </td> </tr> </tbody> </table> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 4: </span> <math alttext="\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}((A_{1}\wedge A_{2})\vee(% A_{1}\wedge\neg A_{2}))\wedge((\neg A_{3}\wedge A_{4})\vee(\neg A_{3}\wedge% \neg A_{4})){}" class="ltx_Math" display="inline" id="S4.F4.15.m1.2"><semantics id="S4.F4.15.m1.2a"><mrow id="S4.F4.15.m1.2.2" xref="S4.F4.15.m1.2.2.cmml"><mi id="S4.F4.15.m1.2.2.4" xref="S4.F4.15.m1.2.2.4.cmml">φ</mi><mover id="S4.F4.15.m1.2.2.3" xref="S4.F4.15.m1.2.2.3.cmml"><mo id="S4.F4.15.m1.2.2.3.2" xref="S4.F4.15.m1.2.2.3.2.cmml">=</mo><mtext id="S4.F4.15.m1.2.2.3.3" mathsize="71%" xref="S4.F4.15.m1.2.2.3.3a.cmml">def</mtext></mover><mrow id="S4.F4.15.m1.2.2.2" xref="S4.F4.15.m1.2.2.2.cmml"><mrow id="S4.F4.15.m1.1.1.1.1.1" xref="S4.F4.15.m1.1.1.1.1.1.1.cmml"><mo id="S4.F4.15.m1.1.1.1.1.1.2" stretchy="false" xref="S4.F4.15.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.F4.15.m1.1.1.1.1.1.1" xref="S4.F4.15.m1.1.1.1.1.1.1.cmml"><mrow id="S4.F4.15.m1.1.1.1.1.1.1.1.1" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S4.F4.15.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.F4.15.m1.1.1.1.1.1.1.1.1.1" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.2" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.2.2" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.2.2.cmml">A</mi><mn id="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.2.3" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.1" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.1.cmml">∧</mo><msub id="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.3" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.3.2" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">A</mi><mn id="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.3.3" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="S4.F4.15.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S4.F4.15.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.F4.15.m1.1.1.1.1.1.1.3" xref="S4.F4.15.m1.1.1.1.1.1.1.3.cmml">∨</mo><mrow id="S4.F4.15.m1.1.1.1.1.1.1.2.1" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.cmml"><mo id="S4.F4.15.m1.1.1.1.1.1.1.2.1.2" stretchy="false" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.cmml"><msub id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.2" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.2.cmml"><mi id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.2.2" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.2.2.cmml">A</mi><mn id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.2.3" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.2.3.cmml">1</mn></msub><mo id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.1" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.1.cmml">∧</mo><mrow id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.cmml"><mo id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.1" rspace="0.167em" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.1.cmml">¬</mo><msub id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.2" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.2.cmml"><mi id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.2.2" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.2.2.cmml">A</mi><mn id="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.2.3" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="S4.F4.15.m1.1.1.1.1.1.1.2.1.3" stretchy="false" xref="S4.F4.15.m1.1.1.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S4.F4.15.m1.1.1.1.1.1.3" stretchy="false" xref="S4.F4.15.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.F4.15.m1.2.2.2.3" xref="S4.F4.15.m1.2.2.2.3.cmml">∧</mo><mrow id="S4.F4.15.m1.2.2.2.2.1" xref="S4.F4.15.m1.2.2.2.2.1.1.cmml"><mo id="S4.F4.15.m1.2.2.2.2.1.2" stretchy="false" xref="S4.F4.15.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="S4.F4.15.m1.2.2.2.2.1.1" xref="S4.F4.15.m1.2.2.2.2.1.1.cmml"><mrow id="S4.F4.15.m1.2.2.2.2.1.1.1.1" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.cmml"><mo id="S4.F4.15.m1.2.2.2.2.1.1.1.1.2" stretchy="false" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.cmml"><mrow id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.cmml"><mo id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.1" rspace="0.167em" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.1.cmml">¬</mo><msub id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.2" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.2.cmml"><mi id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.2.2" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.2.2.cmml">A</mi><mn id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.2.3" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.2.2.3.cmml">3</mn></msub></mrow><mo id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.1" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.1.cmml">∧</mo><msub id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.3" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.3.cmml"><mi id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.3.2" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.3.2.cmml">A</mi><mn id="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.3.3" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.3.3.cmml">4</mn></msub></mrow><mo id="S4.F4.15.m1.2.2.2.2.1.1.1.1.3" stretchy="false" xref="S4.F4.15.m1.2.2.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.F4.15.m1.2.2.2.2.1.1.3" xref="S4.F4.15.m1.2.2.2.2.1.1.3.cmml">∨</mo><mrow id="S4.F4.15.m1.2.2.2.2.1.1.2.1" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.cmml"><mo id="S4.F4.15.m1.2.2.2.2.1.1.2.1.2" stretchy="false" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.cmml">(</mo><mrow id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.cmml"><mrow id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.cmml"><mo id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.1" rspace="0.167em" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.1.cmml">¬</mo><msub id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.2" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.2.cmml"><mi id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.2.2" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.2.2.cmml">A</mi><mn id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.2.3" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.2.2.3.cmml">3</mn></msub></mrow><mo id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.1" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.1.cmml">∧</mo><mrow id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.cmml"><mo id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.1" rspace="0.167em" 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id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.cmml" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3"><not id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.1.cmml" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.1"></not><apply id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.2.cmml" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.2"><csymbol cd="ambiguous" id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.2.1.cmml" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.2">subscript</csymbol><ci id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.2.2.cmml" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.2.2">𝐴</ci><cn id="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.2.3.cmml" type="integer" xref="S4.F4.15.m1.2.2.2.2.1.1.2.1.1.3.2.3">4</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.15.m1.2c">\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}((A_{1}\wedge A_{2})\vee(% A_{1}\wedge\neg A_{2}))\wedge((\neg A_{3}\wedge A_{4})\vee(\neg A_{3}\wedge% \neg A_{4})){}</annotation><annotation encoding="application/x-llamapun" id="S4.F4.15.m1.2d">italic_φ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ( ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) ∧ ( ( ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∨ ( ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) )</annotation></semantics></math>. <br class="ltx_break"/>Compilation of <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.F4.16.m2.1"><semantics id="S4.F4.16.m2.1a"><mi id="S4.F4.16.m2.1.1" xref="S4.F4.16.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.F4.16.m2.1b"><ci id="S4.F4.16.m2.1.1.cmml" xref="S4.F4.16.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F4.16.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.F4.16.m2.1d">italic_φ</annotation></semantics></math> into: d-DNNF (left), OBDD (center), SDD (right). <br class="ltx_break"/></figcaption> </figure> <div class="ltx_theorem ltx_theorem_example" id="Thmexample8"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 8</span></h6> <div class="ltx_para" id="Thmexample8.p1"> <p class="ltx_p" id="Thmexample8.p1.5">Consider <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmexample8.p1.1.m1.1"><semantics id="Thmexample8.p1.1.m1.1a"><mi id="Thmexample8.p1.1.m1.1.1" xref="Thmexample8.p1.1.m1.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmexample8.p1.1.m1.1b"><ci id="Thmexample8.p1.1.m1.1.1.cmml" xref="Thmexample8.p1.1.m1.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p1.1.m1.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p1.1.m1.1d">italic_φ</annotation></semantics></math> as in <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample7" title="Example 7 ‣ 4.2 Verification vs. entailment in search procedures and formula compilers. ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">7</span></a>. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#S4.F4" title="Figure 4 ‣ 4.2 Verification vs. entailment in search procedures and formula compilers. ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">4</span></a> represents possible compilations of <math alttext="\varphi" class="ltx_Math" display="inline" id="Thmexample8.p1.2.m2.1"><semantics id="Thmexample8.p1.2.m2.1a"><mi id="Thmexample8.p1.2.m2.1.1" xref="Thmexample8.p1.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="Thmexample8.p1.2.m2.1b"><ci id="Thmexample8.p1.2.m2.1.1.cmml" xref="Thmexample8.p1.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p1.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p1.2.m2.1d">italic_φ</annotation></semantics></math> into d-DNNF, OBDD and SDD respectively. <span class="ltx_note ltx_role_footnote" id="footnote7"><sup class="ltx_note_mark">7</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">7</sup><span class="ltx_tag ltx_tag_note">7</span>The outcome may vary with the ordering by which variables and subformulas are analyzed.</span></span></span> Notice that, unlike with OBDDs and SDDs which collapse equivalent subformulas into one, the two <math alttext="\vee" class="ltx_Math" display="inline" id="Thmexample8.p1.3.m3.1"><semantics id="Thmexample8.p1.3.m3.1a"><mo id="Thmexample8.p1.3.m3.1.1" xref="Thmexample8.p1.3.m3.1.1.cmml">∨</mo><annotation-xml encoding="MathML-Content" id="Thmexample8.p1.3.m3.1b"><or id="Thmexample8.p1.3.m3.1.1.cmml" xref="Thmexample8.p1.3.m3.1.1"></or></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p1.3.m3.1c">\vee</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p1.3.m3.1d">∨</annotation></semantics></math>-subformulas in the d-DNNF are not recognized to be equivalent to the nodes <math alttext="A_{1}" class="ltx_Math" display="inline" id="Thmexample8.p1.4.m4.1"><semantics id="Thmexample8.p1.4.m4.1a"><msub id="Thmexample8.p1.4.m4.1.1" xref="Thmexample8.p1.4.m4.1.1.cmml"><mi id="Thmexample8.p1.4.m4.1.1.2" xref="Thmexample8.p1.4.m4.1.1.2.cmml">A</mi><mn id="Thmexample8.p1.4.m4.1.1.3" xref="Thmexample8.p1.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="Thmexample8.p1.4.m4.1b"><apply id="Thmexample8.p1.4.m4.1.1.cmml" xref="Thmexample8.p1.4.m4.1.1"><csymbol cd="ambiguous" id="Thmexample8.p1.4.m4.1.1.1.cmml" xref="Thmexample8.p1.4.m4.1.1">subscript</csymbol><ci id="Thmexample8.p1.4.m4.1.1.2.cmml" xref="Thmexample8.p1.4.m4.1.1.2">𝐴</ci><cn id="Thmexample8.p1.4.m4.1.1.3.cmml" type="integer" xref="Thmexample8.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p1.4.m4.1c">A_{1}</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p1.4.m4.1d">italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\neg A_{3}" class="ltx_Math" display="inline" id="Thmexample8.p1.5.m5.1"><semantics id="Thmexample8.p1.5.m5.1a"><mrow id="Thmexample8.p1.5.m5.1.1" xref="Thmexample8.p1.5.m5.1.1.cmml"><mo id="Thmexample8.p1.5.m5.1.1.1" rspace="0.167em" xref="Thmexample8.p1.5.m5.1.1.1.cmml">¬</mo><msub id="Thmexample8.p1.5.m5.1.1.2" xref="Thmexample8.p1.5.m5.1.1.2.cmml"><mi id="Thmexample8.p1.5.m5.1.1.2.2" xref="Thmexample8.p1.5.m5.1.1.2.2.cmml">A</mi><mn id="Thmexample8.p1.5.m5.1.1.2.3" xref="Thmexample8.p1.5.m5.1.1.2.3.cmml">3</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmexample8.p1.5.m5.1b"><apply id="Thmexample8.p1.5.m5.1.1.cmml" xref="Thmexample8.p1.5.m5.1.1"><not id="Thmexample8.p1.5.m5.1.1.1.cmml" xref="Thmexample8.p1.5.m5.1.1.1"></not><apply id="Thmexample8.p1.5.m5.1.1.2.cmml" xref="Thmexample8.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmexample8.p1.5.m5.1.1.2.1.cmml" xref="Thmexample8.p1.5.m5.1.1.2">subscript</csymbol><ci id="Thmexample8.p1.5.m5.1.1.2.2.cmml" xref="Thmexample8.p1.5.m5.1.1.2.2">𝐴</ci><cn id="Thmexample8.p1.5.m5.1.1.2.3.cmml" type="integer" xref="Thmexample8.p1.5.m5.1.1.2.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p1.5.m5.1c">\neg A_{3}</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p1.5.m5.1d">¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> respectively.</p> </div> <div class="ltx_para" id="Thmexample8.p2"> <p class="ltx_p" id="Thmexample8.p2.5">Applying enumeration to the d-DNNF we obtain the total assignments of <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample7" title="Example 7 ‣ 4.2 Verification vs. entailment in search procedures and formula compilers. ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">7</span></a>: <br class="ltx_break"/> <math alttext="\{{\{{A_{1},A_{2},\neg A_{3},A_{4}}\},\{{A_{1},A_{2},\neg A_{3},\neg A_{4}}\},% \{{A_{1},\neg A_{2},\neg A_{3},A_{4}}\},\{{A_{1},\neg A_{2},\neg A_{3},\neg A_% {4}}\}}\}" class="ltx_Math" display="inline" id="Thmexample8.p2.1.m1.4"><semantics id="Thmexample8.p2.1.m1.4a"><mrow id="Thmexample8.p2.1.m1.4.4.4" xref="Thmexample8.p2.1.m1.4.4.5.cmml"><mo id="Thmexample8.p2.1.m1.4.4.4.5" stretchy="false" xref="Thmexample8.p2.1.m1.4.4.5.cmml">{</mo><mrow id="Thmexample8.p2.1.m1.1.1.1.1.4" xref="Thmexample8.p2.1.m1.1.1.1.1.5.cmml"><mo id="Thmexample8.p2.1.m1.1.1.1.1.4.5" stretchy="false" xref="Thmexample8.p2.1.m1.1.1.1.1.5.cmml">{</mo><msub id="Thmexample8.p2.1.m1.1.1.1.1.1.1" 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id="Thmexample8.p2.1.m1.3.3.3.3.2.2.2.3" xref="Thmexample8.p2.1.m1.3.3.3.3.2.2.2.3.cmml">2</mn></msub></mrow><mo id="Thmexample8.p2.1.m1.3.3.3.3.4.7" xref="Thmexample8.p2.1.m1.3.3.3.3.5.cmml">,</mo><mrow id="Thmexample8.p2.1.m1.3.3.3.3.3.3" xref="Thmexample8.p2.1.m1.3.3.3.3.3.3.cmml"><mo id="Thmexample8.p2.1.m1.3.3.3.3.3.3.1" rspace="0.167em" xref="Thmexample8.p2.1.m1.3.3.3.3.3.3.1.cmml">¬</mo><msub id="Thmexample8.p2.1.m1.3.3.3.3.3.3.2" xref="Thmexample8.p2.1.m1.3.3.3.3.3.3.2.cmml"><mi id="Thmexample8.p2.1.m1.3.3.3.3.3.3.2.2" xref="Thmexample8.p2.1.m1.3.3.3.3.3.3.2.2.cmml">A</mi><mn id="Thmexample8.p2.1.m1.3.3.3.3.3.3.2.3" xref="Thmexample8.p2.1.m1.3.3.3.3.3.3.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample8.p2.1.m1.3.3.3.3.4.8" xref="Thmexample8.p2.1.m1.3.3.3.3.5.cmml">,</mo><msub id="Thmexample8.p2.1.m1.3.3.3.3.4.4" xref="Thmexample8.p2.1.m1.3.3.3.3.4.4.cmml"><mi id="Thmexample8.p2.1.m1.3.3.3.3.4.4.2" xref="Thmexample8.p2.1.m1.3.3.3.3.4.4.2.cmml">A</mi><mn 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id="Thmexample8.p2.1.m1.3.3.3.3.4.4.cmml" xref="Thmexample8.p2.1.m1.3.3.3.3.4.4"><csymbol cd="ambiguous" id="Thmexample8.p2.1.m1.3.3.3.3.4.4.1.cmml" xref="Thmexample8.p2.1.m1.3.3.3.3.4.4">subscript</csymbol><ci id="Thmexample8.p2.1.m1.3.3.3.3.4.4.2.cmml" xref="Thmexample8.p2.1.m1.3.3.3.3.4.4.2">𝐴</ci><cn id="Thmexample8.p2.1.m1.3.3.3.3.4.4.3.cmml" type="integer" xref="Thmexample8.p2.1.m1.3.3.3.3.4.4.3">4</cn></apply></set><set id="Thmexample8.p2.1.m1.4.4.4.4.5.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.4"><apply id="Thmexample8.p2.1.m1.4.4.4.4.1.1.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="Thmexample8.p2.1.m1.4.4.4.4.1.1.1.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.1.1">subscript</csymbol><ci id="Thmexample8.p2.1.m1.4.4.4.4.1.1.2.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.1.1.2">𝐴</ci><cn id="Thmexample8.p2.1.m1.4.4.4.4.1.1.3.cmml" type="integer" xref="Thmexample8.p2.1.m1.4.4.4.4.1.1.3">1</cn></apply><apply id="Thmexample8.p2.1.m1.4.4.4.4.2.2.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.2.2"><not id="Thmexample8.p2.1.m1.4.4.4.4.2.2.1.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.2.2.1"></not><apply id="Thmexample8.p2.1.m1.4.4.4.4.2.2.2.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.2.2.2"><csymbol cd="ambiguous" id="Thmexample8.p2.1.m1.4.4.4.4.2.2.2.1.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.2.2.2">subscript</csymbol><ci id="Thmexample8.p2.1.m1.4.4.4.4.2.2.2.2.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.2.2.2.2">𝐴</ci><cn id="Thmexample8.p2.1.m1.4.4.4.4.2.2.2.3.cmml" type="integer" xref="Thmexample8.p2.1.m1.4.4.4.4.2.2.2.3">2</cn></apply></apply><apply id="Thmexample8.p2.1.m1.4.4.4.4.3.3.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.3.3"><not id="Thmexample8.p2.1.m1.4.4.4.4.3.3.1.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.3.3.1"></not><apply id="Thmexample8.p2.1.m1.4.4.4.4.3.3.2.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.3.3.2"><csymbol cd="ambiguous" id="Thmexample8.p2.1.m1.4.4.4.4.3.3.2.1.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.3.3.2">subscript</csymbol><ci id="Thmexample8.p2.1.m1.4.4.4.4.3.3.2.2.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.3.3.2.2">𝐴</ci><cn id="Thmexample8.p2.1.m1.4.4.4.4.3.3.2.3.cmml" type="integer" xref="Thmexample8.p2.1.m1.4.4.4.4.3.3.2.3">3</cn></apply></apply><apply id="Thmexample8.p2.1.m1.4.4.4.4.4.4.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.4.4"><not id="Thmexample8.p2.1.m1.4.4.4.4.4.4.1.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.4.4.1"></not><apply id="Thmexample8.p2.1.m1.4.4.4.4.4.4.2.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.4.4.2"><csymbol cd="ambiguous" id="Thmexample8.p2.1.m1.4.4.4.4.4.4.2.1.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.4.4.2">subscript</csymbol><ci id="Thmexample8.p2.1.m1.4.4.4.4.4.4.2.2.cmml" xref="Thmexample8.p2.1.m1.4.4.4.4.4.4.2.2">𝐴</ci><cn id="Thmexample8.p2.1.m1.4.4.4.4.4.4.2.3.cmml" type="integer" xref="Thmexample8.p2.1.m1.4.4.4.4.4.4.2.3">4</cn></apply></apply></set></set></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p2.1.m1.4c">\{{\{{A_{1},A_{2},\neg A_{3},A_{4}}\},\{{A_{1},A_{2},\neg A_{3},\neg A_{4}}\},% \{{A_{1},\neg A_{2},\neg A_{3},A_{4}}\},\{{A_{1},\neg A_{2},\neg A_{3},\neg A_% {4}}\}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p2.1.m1.4d">{ { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } , { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } , { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } , { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } }</annotation></semantics></math>. By applying enumeration to the OBDD or to the SDD, we obtain the single-assignment <math alttext="\{{\{{A_{1},\neg A_{3}}\}}\}" class="ltx_Math" display="inline" id="Thmexample8.p2.2.m2.1"><semantics id="Thmexample8.p2.2.m2.1a"><mrow id="Thmexample8.p2.2.m2.1.1.1" xref="Thmexample8.p2.2.m2.1.1.2.cmml"><mo id="Thmexample8.p2.2.m2.1.1.1.2" stretchy="false" xref="Thmexample8.p2.2.m2.1.1.2.cmml">{</mo><mrow id="Thmexample8.p2.2.m2.1.1.1.1.2" xref="Thmexample8.p2.2.m2.1.1.1.1.3.cmml"><mo id="Thmexample8.p2.2.m2.1.1.1.1.2.3" stretchy="false" xref="Thmexample8.p2.2.m2.1.1.1.1.3.cmml">{</mo><msub id="Thmexample8.p2.2.m2.1.1.1.1.1.1" xref="Thmexample8.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="Thmexample8.p2.2.m2.1.1.1.1.1.1.2" xref="Thmexample8.p2.2.m2.1.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample8.p2.2.m2.1.1.1.1.1.1.3" xref="Thmexample8.p2.2.m2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample8.p2.2.m2.1.1.1.1.2.4" xref="Thmexample8.p2.2.m2.1.1.1.1.3.cmml">,</mo><mrow id="Thmexample8.p2.2.m2.1.1.1.1.2.2" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.cmml"><mo id="Thmexample8.p2.2.m2.1.1.1.1.2.2.1" rspace="0.167em" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.1.cmml">¬</mo><msub id="Thmexample8.p2.2.m2.1.1.1.1.2.2.2" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.cmml"><mi id="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.2" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.2.cmml">A</mi><mn id="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.3" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample8.p2.2.m2.1.1.1.1.2.5" stretchy="false" xref="Thmexample8.p2.2.m2.1.1.1.1.3.cmml">}</mo></mrow><mo id="Thmexample8.p2.2.m2.1.1.1.3" stretchy="false" xref="Thmexample8.p2.2.m2.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample8.p2.2.m2.1b"><set id="Thmexample8.p2.2.m2.1.1.2.cmml" xref="Thmexample8.p2.2.m2.1.1.1"><set id="Thmexample8.p2.2.m2.1.1.1.1.3.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.2"><apply id="Thmexample8.p2.2.m2.1.1.1.1.1.1.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample8.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="Thmexample8.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample8.p2.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample8.p2.2.m2.1.1.1.1.1.1.3">1</cn></apply><apply id="Thmexample8.p2.2.m2.1.1.1.1.2.2.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2"><not id="Thmexample8.p2.2.m2.1.1.1.1.2.2.1.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.1"></not><apply id="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.1.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.2">subscript</csymbol><ci id="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.2.cmml" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.2">𝐴</ci><cn id="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.3.cmml" type="integer" xref="Thmexample8.p2.2.m2.1.1.1.1.2.2.2.3">3</cn></apply></apply></set></set></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p2.2.m2.1c">\{{\{{A_{1},\neg A_{3}}\}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p2.2.m2.1d">{ { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT } }</annotation></semantics></math>, so that <math alttext="\{{A_{1},\neg A_{3}}\}\models\varphi" class="ltx_Math" display="inline" id="Thmexample8.p2.3.m3.2"><semantics id="Thmexample8.p2.3.m3.2a"><mrow id="Thmexample8.p2.3.m3.2.2" xref="Thmexample8.p2.3.m3.2.2.cmml"><mrow id="Thmexample8.p2.3.m3.2.2.2.2" xref="Thmexample8.p2.3.m3.2.2.2.3.cmml"><mo id="Thmexample8.p2.3.m3.2.2.2.2.3" stretchy="false" xref="Thmexample8.p2.3.m3.2.2.2.3.cmml">{</mo><msub id="Thmexample8.p2.3.m3.1.1.1.1.1" xref="Thmexample8.p2.3.m3.1.1.1.1.1.cmml"><mi id="Thmexample8.p2.3.m3.1.1.1.1.1.2" xref="Thmexample8.p2.3.m3.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample8.p2.3.m3.1.1.1.1.1.3" xref="Thmexample8.p2.3.m3.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample8.p2.3.m3.2.2.2.2.4" xref="Thmexample8.p2.3.m3.2.2.2.3.cmml">,</mo><mrow id="Thmexample8.p2.3.m3.2.2.2.2.2" xref="Thmexample8.p2.3.m3.2.2.2.2.2.cmml"><mo id="Thmexample8.p2.3.m3.2.2.2.2.2.1" rspace="0.167em" xref="Thmexample8.p2.3.m3.2.2.2.2.2.1.cmml">¬</mo><msub id="Thmexample8.p2.3.m3.2.2.2.2.2.2" xref="Thmexample8.p2.3.m3.2.2.2.2.2.2.cmml"><mi id="Thmexample8.p2.3.m3.2.2.2.2.2.2.2" xref="Thmexample8.p2.3.m3.2.2.2.2.2.2.2.cmml">A</mi><mn id="Thmexample8.p2.3.m3.2.2.2.2.2.2.3" xref="Thmexample8.p2.3.m3.2.2.2.2.2.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample8.p2.3.m3.2.2.2.2.5" stretchy="false" xref="Thmexample8.p2.3.m3.2.2.2.3.cmml">}</mo></mrow><mo id="Thmexample8.p2.3.m3.2.2.3" xref="Thmexample8.p2.3.m3.2.2.3.cmml">⊧</mo><mi id="Thmexample8.p2.3.m3.2.2.4" xref="Thmexample8.p2.3.m3.2.2.4.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample8.p2.3.m3.2b"><apply id="Thmexample8.p2.3.m3.2.2.cmml" xref="Thmexample8.p2.3.m3.2.2"><csymbol cd="latexml" id="Thmexample8.p2.3.m3.2.2.3.cmml" xref="Thmexample8.p2.3.m3.2.2.3">models</csymbol><set id="Thmexample8.p2.3.m3.2.2.2.3.cmml" xref="Thmexample8.p2.3.m3.2.2.2.2"><apply id="Thmexample8.p2.3.m3.1.1.1.1.1.cmml" xref="Thmexample8.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample8.p2.3.m3.1.1.1.1.1.1.cmml" xref="Thmexample8.p2.3.m3.1.1.1.1.1">subscript</csymbol><ci id="Thmexample8.p2.3.m3.1.1.1.1.1.2.cmml" xref="Thmexample8.p2.3.m3.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample8.p2.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample8.p2.3.m3.1.1.1.1.1.3">1</cn></apply><apply id="Thmexample8.p2.3.m3.2.2.2.2.2.cmml" xref="Thmexample8.p2.3.m3.2.2.2.2.2"><not id="Thmexample8.p2.3.m3.2.2.2.2.2.1.cmml" xref="Thmexample8.p2.3.m3.2.2.2.2.2.1"></not><apply id="Thmexample8.p2.3.m3.2.2.2.2.2.2.cmml" xref="Thmexample8.p2.3.m3.2.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample8.p2.3.m3.2.2.2.2.2.2.1.cmml" xref="Thmexample8.p2.3.m3.2.2.2.2.2.2">subscript</csymbol><ci id="Thmexample8.p2.3.m3.2.2.2.2.2.2.2.cmml" xref="Thmexample8.p2.3.m3.2.2.2.2.2.2.2">𝐴</ci><cn id="Thmexample8.p2.3.m3.2.2.2.2.2.2.3.cmml" type="integer" xref="Thmexample8.p2.3.m3.2.2.2.2.2.2.3">3</cn></apply></apply></set><ci id="Thmexample8.p2.3.m3.2.2.4.cmml" xref="Thmexample8.p2.3.m3.2.2.4">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p2.3.m3.2c">\{{A_{1},\neg A_{3}}\}\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p2.3.m3.2d">{ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT } ⊧ italic_φ</annotation></semantics></math> but <math alttext="\{{A_{1},\neg A_{3}}\}\not\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="Thmexample8.p2.4.m4.1"><semantics id="Thmexample8.p2.4.m4.1a"><mrow id="Thmexample8.p2.4.m4.1b"><mrow id="Thmexample8.p2.4.m4.1.1"><mo id="Thmexample8.p2.4.m4.1.1.1" stretchy="false">{</mo><msub id="Thmexample8.p2.4.m4.1.1.2"><mi id="Thmexample8.p2.4.m4.1.1.2.2">A</mi><mn id="Thmexample8.p2.4.m4.1.1.2.3">1</mn></msub><mo id="Thmexample8.p2.4.m4.1.1.3">,</mo><mo id="Thmexample8.p2.4.m4.1.1.4" rspace="0.167em">¬</mo><msub id="Thmexample8.p2.4.m4.1.1.5"><mi id="Thmexample8.p2.4.m4.1.1.5.2">A</mi><mn id="Thmexample8.p2.4.m4.1.1.5.3">3</mn></msub><mo id="Thmexample8.p2.4.m4.1.1.6" stretchy="false">}</mo></mrow><mpadded id="Thmexample8.p2.4.m4.1c" width="0.969em"><mo id="Thmexample8.p2.4.m4.1.2" lspace="0em">∤</mo></mpadded><mo id="Thmexample8.p2.4.m4.1.3">≈</mo><mi id="Thmexample8.p2.4.m4.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmexample8.p2.4.m4.1d">\{{A_{1},\neg A_{3}}\}\not\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p2.4.m4.1e">{ italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT } ∤ ≈ italic_φ</annotation></semantics></math>. <math alttext="\diamond" class="ltx_Math" display="inline" id="Thmexample8.p2.5.m5.1"><semantics id="Thmexample8.p2.5.m5.1a"><mo id="Thmexample8.p2.5.m5.1.1" xref="Thmexample8.p2.5.m5.1.1.cmml">⋄</mo><annotation-xml encoding="MathML-Content" id="Thmexample8.p2.5.m5.1b"><ci id="Thmexample8.p2.5.m5.1.1.cmml" xref="Thmexample8.p2.5.m5.1.1">⋄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample8.p2.5.m5.1c">\diamond</annotation><annotation encoding="application/x-llamapun" id="Thmexample8.p2.5.m5.1d">⋄</annotation></semantics></math></p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Implementing entailment within enumeration procedures</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.1">Implementing efficiently entailment-based enumeration procedures can be tricky. One possible approach is to integrate AllSAT/AllSMT with OBDDs, exploiting the capability of OBDD to implicitly perform entailment-based reductions <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib6" title="">6</a>]</cite>. The main problem with this approach is the fact that OBDDs quickly blow up in memory as soon as the input formula increases in size.</p> </div> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.6"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib27" title="">27</a>]</cite> proposed a formal reasoning framework based on the idea of <span class="ltx_text ltx_font_italic" id="S4.SS3.p2.6.1">dualized search</span>: while the main enumerator produces partial assignments <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><mi id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><ci id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">italic_μ</annotation></semantics></math> for the input formula <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><mi id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><ci id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">italic_φ</annotation></semantics></math>, another “dual” SAT solver is incrementally called on <math alttext="\mathsf{CNF_{Ts}}(\neg\varphi)\wedge\bigwedge\!\mu" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m3.1"><semantics id="S4.SS3.p2.3.m3.1a"><mrow id="S4.SS3.p2.3.m3.1.1" xref="S4.SS3.p2.3.m3.1.1.cmml"><mrow id="S4.SS3.p2.3.m3.1.1.1" xref="S4.SS3.p2.3.m3.1.1.1.cmml"><msub id="S4.SS3.p2.3.m3.1.1.1.3" xref="S4.SS3.p2.3.m3.1.1.1.3.cmml"><mi id="S4.SS3.p2.3.m3.1.1.1.3.2" xref="S4.SS3.p2.3.m3.1.1.1.3.2.cmml">𝖢𝖭𝖥</mi><mi id="S4.SS3.p2.3.m3.1.1.1.3.3" xref="S4.SS3.p2.3.m3.1.1.1.3.3.cmml">𝖳𝗌</mi></msub><mo id="S4.SS3.p2.3.m3.1.1.1.2" xref="S4.SS3.p2.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS3.p2.3.m3.1.1.1.1.1" xref="S4.SS3.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.SS3.p2.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.SS3.p2.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS3.p2.3.m3.1.1.1.1.1.1" xref="S4.SS3.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="S4.SS3.p2.3.m3.1.1.1.1.1.1.1" rspace="0.167em" xref="S4.SS3.p2.3.m3.1.1.1.1.1.1.1.cmml">¬</mo><mi id="S4.SS3.p2.3.m3.1.1.1.1.1.1.2" xref="S4.SS3.p2.3.m3.1.1.1.1.1.1.2.cmml">φ</mi></mrow><mo id="S4.SS3.p2.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.SS3.p2.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS3.p2.3.m3.1.1.2" rspace="0.055em" xref="S4.SS3.p2.3.m3.1.1.2.cmml">∧</mo><mrow id="S4.SS3.p2.3.m3.1.1.3" xref="S4.SS3.p2.3.m3.1.1.3.cmml"><mpadded width="0.747em"><mo id="S4.SS3.p2.3.m3.1.1.3.1" xref="S4.SS3.p2.3.m3.1.1.3.1.cmml">⋀</mo></mpadded><mi id="S4.SS3.p2.3.m3.1.1.3.2" xref="S4.SS3.p2.3.m3.1.1.3.2.cmml">μ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m3.1b"><apply id="S4.SS3.p2.3.m3.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1"><and id="S4.SS3.p2.3.m3.1.1.2.cmml" xref="S4.SS3.p2.3.m3.1.1.2"></and><apply id="S4.SS3.p2.3.m3.1.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1.1"><times id="S4.SS3.p2.3.m3.1.1.1.2.cmml" xref="S4.SS3.p2.3.m3.1.1.1.2"></times><apply id="S4.SS3.p2.3.m3.1.1.1.3.cmml" xref="S4.SS3.p2.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.3.m3.1.1.1.3.1.cmml" xref="S4.SS3.p2.3.m3.1.1.1.3">subscript</csymbol><ci id="S4.SS3.p2.3.m3.1.1.1.3.2.cmml" xref="S4.SS3.p2.3.m3.1.1.1.3.2">𝖢𝖭𝖥</ci><ci id="S4.SS3.p2.3.m3.1.1.1.3.3.cmml" xref="S4.SS3.p2.3.m3.1.1.1.3.3">𝖳𝗌</ci></apply><apply id="S4.SS3.p2.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1.1.1.1"><not id="S4.SS3.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1.1.1.1.1.1"></not><ci id="S4.SS3.p2.3.m3.1.1.1.1.1.1.2.cmml" xref="S4.SS3.p2.3.m3.1.1.1.1.1.1.2">𝜑</ci></apply></apply><apply id="S4.SS3.p2.3.m3.1.1.3.cmml" xref="S4.SS3.p2.3.m3.1.1.3"><and id="S4.SS3.p2.3.m3.1.1.3.1.cmml" xref="S4.SS3.p2.3.m3.1.1.3.1"></and><ci id="S4.SS3.p2.3.m3.1.1.3.2.cmml" xref="S4.SS3.p2.3.m3.1.1.3.2">𝜇</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m3.1c">\mathsf{CNF_{Ts}}(\neg\varphi)\wedge\bigwedge\!\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m3.1d">sansserif_CNF start_POSTSUBSCRIPT sansserif_Ts end_POSTSUBSCRIPT ( ¬ italic_φ ) ∧ ⋀ italic_μ</annotation></semantics></math>: if it is unsatisfiable then the main enumerator produces <math alttext="\mu" class="ltx_Math" display="inline" id="S4.SS3.p2.4.m4.1"><semantics id="S4.SS3.p2.4.m4.1a"><mi id="S4.SS3.p2.4.m4.1.1" xref="S4.SS3.p2.4.m4.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.4.m4.1b"><ci id="S4.SS3.p2.4.m4.1.1.cmml" xref="S4.SS3.p2.4.m4.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.4.m4.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.4.m4.1d">italic_μ</annotation></semantics></math> and backtracks. Based on the analysis in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib35" title="">35</a>]</cite> and noticing that the above dual check in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib27" title="">27</a>]</cite> is an entailment (<math alttext="\mu\models\varphi" class="ltx_Math" display="inline" id="S4.SS3.p2.5.m5.1"><semantics id="S4.SS3.p2.5.m5.1a"><mrow id="S4.SS3.p2.5.m5.1.1" xref="S4.SS3.p2.5.m5.1.1.cmml"><mi id="S4.SS3.p2.5.m5.1.1.2" xref="S4.SS3.p2.5.m5.1.1.2.cmml">μ</mi><mo id="S4.SS3.p2.5.m5.1.1.1" xref="S4.SS3.p2.5.m5.1.1.1.cmml">⊧</mo><mi id="S4.SS3.p2.5.m5.1.1.3" xref="S4.SS3.p2.5.m5.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.5.m5.1b"><apply id="S4.SS3.p2.5.m5.1.1.cmml" xref="S4.SS3.p2.5.m5.1.1"><csymbol cd="latexml" id="S4.SS3.p2.5.m5.1.1.1.cmml" xref="S4.SS3.p2.5.m5.1.1.1">models</csymbol><ci id="S4.SS3.p2.5.m5.1.1.2.cmml" xref="S4.SS3.p2.5.m5.1.1.2">𝜇</ci><ci id="S4.SS3.p2.5.m5.1.1.3.cmml" xref="S4.SS3.p2.5.m5.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.5.m5.1c">\mu\models\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.5.m5.1d">italic_μ ⊧ italic_φ</annotation></semantics></math>) rather than a verification (<math alttext="\mu\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="S4.SS3.p2.6.m6.1"><semantics id="S4.SS3.p2.6.m6.1a"><mrow id="S4.SS3.p2.6.m6.1b"><mi id="S4.SS3.p2.6.m6.1.1">μ</mi><mpadded id="S4.SS3.p2.6.m6.1c" width="0.219em"><mo id="S4.SS3.p2.6.m6.1.2" lspace="0em">∣</mo></mpadded><mo id="S4.SS3.p2.6.m6.1.3">≈</mo><mi id="S4.SS3.p2.6.m6.1.4">φ</mi></mrow><annotation encoding="application/x-tex" id="S4.SS3.p2.6.m6.1d">\mu\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.6.m6.1e">italic_μ ∣ ≈ italic_φ</annotation></semantics></math>), in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib30" title="">30</a>]</cite> we proposed enumeration and model counting frameworks and algorithms based on dual search, enhanced with chronological backtracking. Unfortunately no implementation efficient enough to compete with s.o.a. enumeration procedures was produced.</p> </div> <div class="ltx_para" id="S4.SS3.p3"> <p class="ltx_p" id="S4.SS3.p3.4"><cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>]</cite> presented <span class="ltx_text ltx_font_sansserif" id="S4.SS3.p3.4.1">HALL</span>, a sophisticate AllSAT procedure for circuits which, among other techniques, exploited a form of (verification-based) <span class="ltx_text ltx_font_italic" id="S4.SS3.p3.4.2">generalization</span> (aka minimization or reduction): as soon as a total truth assignment <math alttext="\eta" class="ltx_Math" display="inline" id="S4.SS3.p3.1.m1.1"><semantics id="S4.SS3.p3.1.m1.1a"><mi id="S4.SS3.p3.1.m1.1.1" xref="S4.SS3.p3.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.1.m1.1b"><ci id="S4.SS3.p3.1.m1.1.1.cmml" xref="S4.SS3.p3.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.1.m1.1d">italic_η</annotation></semantics></math> satisfying <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.SS3.p3.2.m2.1"><semantics id="S4.SS3.p3.2.m2.1a"><mi id="S4.SS3.p3.2.m2.1.1" xref="S4.SS3.p3.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.2.m2.1b"><ci id="S4.SS3.p3.2.m2.1.1.cmml" xref="S4.SS3.p3.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.2.m2.1d">italic_φ</annotation></semantics></math> is produced, the literals in <math alttext="\eta" class="ltx_Math" display="inline" id="S4.SS3.p3.3.m3.1"><semantics id="S4.SS3.p3.3.m3.1a"><mi id="S4.SS3.p3.3.m3.1.1" xref="S4.SS3.p3.3.m3.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.3.m3.1b"><ci id="S4.SS3.p3.3.m3.1.1.cmml" xref="S4.SS3.p3.3.m3.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.3.m3.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.3.m3.1d">italic_η</annotation></semantics></math> are removed one by one and the resulting partial assignment is tested to verify <math alttext="\varphi" class="ltx_Math" display="inline" id="S4.SS3.p3.4.m4.1"><semantics id="S4.SS3.p3.4.m4.1a"><mi id="S4.SS3.p3.4.m4.1.1" xref="S4.SS3.p3.4.m4.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p3.4.m4.1b"><ci id="S4.SS3.p3.4.m4.1.1.cmml" xref="S4.SS3.p3.4.m4.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p3.4.m4.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p3.4.m4.1d">italic_φ</annotation></semantics></math>. In <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib16" title="">16</a>]</cite> we have enhanced <span class="ltx_text ltx_font_sansserif" id="S4.SS3.p3.4.3">HALL</span> generalization procedure by substituting verification checks with entailment checks based on dualized search, boosting its performance in terms of both time efficiency and size of partial-assignment sets.</p> </div> <div class="ltx_theorem ltx_theorem_example" id="Thmexample9"> <h6 class="ltx_title ltx_font_italic ltx_title_theorem"><span class="ltx_tag ltx_tag_theorem">Example 9</span></h6> <div class="ltx_para" id="Thmexample9.p1"> <p class="ltx_p" id="Thmexample9.p1.5">Consider <math alttext="\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}((A_{1}\wedge A_{2})\vee(% A_{1}\wedge\neg A_{2}))\wedge((\neg A_{3}\wedge A_{4})\vee(\neg A_{3}\wedge% \neg A_{4}))" class="ltx_Math" display="inline" id="Thmexample9.p1.1.m1.2"><semantics id="Thmexample9.p1.1.m1.2a"><mrow id="Thmexample9.p1.1.m1.2.2" xref="Thmexample9.p1.1.m1.2.2.cmml"><mi id="Thmexample9.p1.1.m1.2.2.4" xref="Thmexample9.p1.1.m1.2.2.4.cmml">φ</mi><mover id="Thmexample9.p1.1.m1.2.2.3" xref="Thmexample9.p1.1.m1.2.2.3.cmml"><mo id="Thmexample9.p1.1.m1.2.2.3.2" xref="Thmexample9.p1.1.m1.2.2.3.2.cmml">=</mo><mtext id="Thmexample9.p1.1.m1.2.2.3.3" mathsize="71%" xref="Thmexample9.p1.1.m1.2.2.3.3a.cmml">def</mtext></mover><mrow id="Thmexample9.p1.1.m1.2.2.2" xref="Thmexample9.p1.1.m1.2.2.2.cmml"><mrow id="Thmexample9.p1.1.m1.1.1.1.1.1" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample9.p1.1.m1.1.1.1.1.1.1" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.cmml"><mrow id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml"><msub id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.2" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.2.2" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.2.2.cmml">A</mi><mn id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.2.3" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.1" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.1.cmml">∧</mo><msub id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.3" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.3.2" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">A</mi><mn id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.3.3" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">2</mn></msub></mrow><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.1.3" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.3.cmml">∨</mo><mrow id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.cmml"><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.2" stretchy="false" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.cmml">(</mo><mrow id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.cmml"><msub id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.2" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.2.cmml"><mi id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.2.2" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.2.2.cmml">A</mi><mn id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.2.3" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.2.3.cmml">1</mn></msub><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.1" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.1.cmml">∧</mo><mrow id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.cmml"><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.1" rspace="0.167em" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.1.cmml">¬</mo><msub id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.cmml"><mi id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.2" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.2.cmml">A</mi><mn id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.3" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.3" stretchy="false" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="Thmexample9.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmexample9.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="Thmexample9.p1.1.m1.2.2.2.3" xref="Thmexample9.p1.1.m1.2.2.2.3.cmml">∧</mo><mrow id="Thmexample9.p1.1.m1.2.2.2.2.1" xref="Thmexample9.p1.1.m1.2.2.2.2.1.1.cmml"><mo id="Thmexample9.p1.1.m1.2.2.2.2.1.2" 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type="integer" xref="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.2.2.3">3</cn></apply></apply><apply id="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.cmml" xref="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3"><not id="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.1.cmml" xref="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.1"></not><apply id="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.cmml" xref="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2"><csymbol cd="ambiguous" id="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.1.cmml" xref="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2">subscript</csymbol><ci id="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.2.cmml" xref="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.2">𝐴</ci><cn id="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.3.cmml" type="integer" xref="Thmexample9.p1.1.m1.2.2.2.2.1.1.2.1.1.3.2.3">4</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p1.1.m1.2c">\varphi\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}((A_{1}\wedge A_{2})\vee(% A_{1}\wedge\neg A_{2}))\wedge((\neg A_{3}\wedge A_{4})\vee(\neg A_{3}\wedge% \neg A_{4}))</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p1.1.m1.2d">italic_φ start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP ( ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∨ ( italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ) ∧ ( ( ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∧ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ∨ ( ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ∧ ¬ italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) )</annotation></semantics></math> <br class="ltx_break"/>as in <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample7" title="Example 7 ‣ 4.2 Verification vs. entailment in search procedures and formula compilers. ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">7</span></a> and assume some total assignment satisfying it is produced, e.g. <math alttext="\eta\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1},A_{2},\neg A_{3},A_% {4}}\}" class="ltx_Math" display="inline" id="Thmexample9.p1.2.m2.4"><semantics id="Thmexample9.p1.2.m2.4a"><mrow id="Thmexample9.p1.2.m2.4.4" xref="Thmexample9.p1.2.m2.4.4.cmml"><mi id="Thmexample9.p1.2.m2.4.4.6" xref="Thmexample9.p1.2.m2.4.4.6.cmml">η</mi><mover id="Thmexample9.p1.2.m2.4.4.5" xref="Thmexample9.p1.2.m2.4.4.5.cmml"><mo id="Thmexample9.p1.2.m2.4.4.5.2" xref="Thmexample9.p1.2.m2.4.4.5.2.cmml">=</mo><mtext id="Thmexample9.p1.2.m2.4.4.5.3" mathsize="71%" xref="Thmexample9.p1.2.m2.4.4.5.3a.cmml">def</mtext></mover><mrow id="Thmexample9.p1.2.m2.4.4.4.4" xref="Thmexample9.p1.2.m2.4.4.4.5.cmml"><mo id="Thmexample9.p1.2.m2.4.4.4.4.5" stretchy="false" xref="Thmexample9.p1.2.m2.4.4.4.5.cmml">{</mo><msub id="Thmexample9.p1.2.m2.1.1.1.1.1" xref="Thmexample9.p1.2.m2.1.1.1.1.1.cmml"><mi id="Thmexample9.p1.2.m2.1.1.1.1.1.2" xref="Thmexample9.p1.2.m2.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample9.p1.2.m2.1.1.1.1.1.3" xref="Thmexample9.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample9.p1.2.m2.4.4.4.4.6" xref="Thmexample9.p1.2.m2.4.4.4.5.cmml">,</mo><msub id="Thmexample9.p1.2.m2.2.2.2.2.2" xref="Thmexample9.p1.2.m2.2.2.2.2.2.cmml"><mi id="Thmexample9.p1.2.m2.2.2.2.2.2.2" xref="Thmexample9.p1.2.m2.2.2.2.2.2.2.cmml">A</mi><mn id="Thmexample9.p1.2.m2.2.2.2.2.2.3" xref="Thmexample9.p1.2.m2.2.2.2.2.2.3.cmml">2</mn></msub><mo id="Thmexample9.p1.2.m2.4.4.4.4.7" xref="Thmexample9.p1.2.m2.4.4.4.5.cmml">,</mo><mrow id="Thmexample9.p1.2.m2.3.3.3.3.3" xref="Thmexample9.p1.2.m2.3.3.3.3.3.cmml"><mo id="Thmexample9.p1.2.m2.3.3.3.3.3.1" rspace="0.167em" xref="Thmexample9.p1.2.m2.3.3.3.3.3.1.cmml">¬</mo><msub id="Thmexample9.p1.2.m2.3.3.3.3.3.2" xref="Thmexample9.p1.2.m2.3.3.3.3.3.2.cmml"><mi id="Thmexample9.p1.2.m2.3.3.3.3.3.2.2" xref="Thmexample9.p1.2.m2.3.3.3.3.3.2.2.cmml">A</mi><mn id="Thmexample9.p1.2.m2.3.3.3.3.3.2.3" xref="Thmexample9.p1.2.m2.3.3.3.3.3.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample9.p1.2.m2.4.4.4.4.8" xref="Thmexample9.p1.2.m2.4.4.4.5.cmml">,</mo><msub id="Thmexample9.p1.2.m2.4.4.4.4.4" xref="Thmexample9.p1.2.m2.4.4.4.4.4.cmml"><mi id="Thmexample9.p1.2.m2.4.4.4.4.4.2" xref="Thmexample9.p1.2.m2.4.4.4.4.4.2.cmml">A</mi><mn id="Thmexample9.p1.2.m2.4.4.4.4.4.3" xref="Thmexample9.p1.2.m2.4.4.4.4.4.3.cmml">4</mn></msub><mo id="Thmexample9.p1.2.m2.4.4.4.4.9" stretchy="false" xref="Thmexample9.p1.2.m2.4.4.4.5.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmexample9.p1.2.m2.4b"><apply id="Thmexample9.p1.2.m2.4.4.cmml" xref="Thmexample9.p1.2.m2.4.4"><apply id="Thmexample9.p1.2.m2.4.4.5.cmml" 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xref="Thmexample9.p1.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="Thmexample9.p1.2.m2.2.2.2.2.2.cmml" xref="Thmexample9.p1.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample9.p1.2.m2.2.2.2.2.2.1.cmml" xref="Thmexample9.p1.2.m2.2.2.2.2.2">subscript</csymbol><ci id="Thmexample9.p1.2.m2.2.2.2.2.2.2.cmml" xref="Thmexample9.p1.2.m2.2.2.2.2.2.2">𝐴</ci><cn id="Thmexample9.p1.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="Thmexample9.p1.2.m2.2.2.2.2.2.3">2</cn></apply><apply id="Thmexample9.p1.2.m2.3.3.3.3.3.cmml" xref="Thmexample9.p1.2.m2.3.3.3.3.3"><not id="Thmexample9.p1.2.m2.3.3.3.3.3.1.cmml" xref="Thmexample9.p1.2.m2.3.3.3.3.3.1"></not><apply id="Thmexample9.p1.2.m2.3.3.3.3.3.2.cmml" xref="Thmexample9.p1.2.m2.3.3.3.3.3.2"><csymbol cd="ambiguous" id="Thmexample9.p1.2.m2.3.3.3.3.3.2.1.cmml" xref="Thmexample9.p1.2.m2.3.3.3.3.3.2">subscript</csymbol><ci id="Thmexample9.p1.2.m2.3.3.3.3.3.2.2.cmml" xref="Thmexample9.p1.2.m2.3.3.3.3.3.2.2">𝐴</ci><cn id="Thmexample9.p1.2.m2.3.3.3.3.3.2.3.cmml" type="integer" xref="Thmexample9.p1.2.m2.3.3.3.3.3.2.3">3</cn></apply></apply><apply id="Thmexample9.p1.2.m2.4.4.4.4.4.cmml" xref="Thmexample9.p1.2.m2.4.4.4.4.4"><csymbol cd="ambiguous" id="Thmexample9.p1.2.m2.4.4.4.4.4.1.cmml" xref="Thmexample9.p1.2.m2.4.4.4.4.4">subscript</csymbol><ci id="Thmexample9.p1.2.m2.4.4.4.4.4.2.cmml" xref="Thmexample9.p1.2.m2.4.4.4.4.4.2">𝐴</ci><cn id="Thmexample9.p1.2.m2.4.4.4.4.4.3.cmml" type="integer" xref="Thmexample9.p1.2.m2.4.4.4.4.4.3">4</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p1.2.m2.4c">\eta\stackrel{{\scriptstyle\text{\tiny def}}}{{=}}\{{A_{1},A_{2},\neg A_{3},A_% {4}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p1.2.m2.4d">italic_η start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG def end_ARG end_RELOP { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT }</annotation></semantics></math>. Since there is no literal <math alttext="l\in\eta" class="ltx_Math" display="inline" id="Thmexample9.p1.3.m3.1"><semantics id="Thmexample9.p1.3.m3.1a"><mrow id="Thmexample9.p1.3.m3.1.1" xref="Thmexample9.p1.3.m3.1.1.cmml"><mi id="Thmexample9.p1.3.m3.1.1.2" xref="Thmexample9.p1.3.m3.1.1.2.cmml">l</mi><mo id="Thmexample9.p1.3.m3.1.1.1" xref="Thmexample9.p1.3.m3.1.1.1.cmml">∈</mo><mi id="Thmexample9.p1.3.m3.1.1.3" xref="Thmexample9.p1.3.m3.1.1.3.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample9.p1.3.m3.1b"><apply id="Thmexample9.p1.3.m3.1.1.cmml" xref="Thmexample9.p1.3.m3.1.1"><in id="Thmexample9.p1.3.m3.1.1.1.cmml" xref="Thmexample9.p1.3.m3.1.1.1"></in><ci id="Thmexample9.p1.3.m3.1.1.2.cmml" xref="Thmexample9.p1.3.m3.1.1.2">𝑙</ci><ci id="Thmexample9.p1.3.m3.1.1.3.cmml" xref="Thmexample9.p1.3.m3.1.1.3">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p1.3.m3.1c">l\in\eta</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p1.3.m3.1d">italic_l ∈ italic_η</annotation></semantics></math> s.t. <math alttext="\eta\backslash\{{l}\}\mid\!\approx\varphi" class="ltx_math_unparsed" display="inline" id="Thmexample9.p1.4.m4.1"><semantics id="Thmexample9.p1.4.m4.1a"><mrow id="Thmexample9.p1.4.m4.1b"><mi id="Thmexample9.p1.4.m4.1.2">η</mi><mo id="Thmexample9.p1.4.m4.1.3" lspace="0.222em" rspace="0.222em">\</mo><mrow id="Thmexample9.p1.4.m4.1.4"><mo id="Thmexample9.p1.4.m4.1.4.1" stretchy="false">{</mo><mi id="Thmexample9.p1.4.m4.1.1">l</mi><mo id="Thmexample9.p1.4.m4.1.4.2" stretchy="false">}</mo></mrow><mpadded id="Thmexample9.p1.4.m4.1c" width="0.219em"><mo id="Thmexample9.p1.4.m4.1.5" lspace="0em">∣</mo></mpadded><mo id="Thmexample9.p1.4.m4.1.6">≈</mo><mi id="Thmexample9.p1.4.m4.1.7">φ</mi></mrow><annotation encoding="application/x-tex" id="Thmexample9.p1.4.m4.1d">\eta\backslash\{{l}\}\mid\!\approx\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p1.4.m4.1e">italic_η \ { italic_l } ∣ ≈ italic_φ</annotation></semantics></math>, no assignment reduction based on verification can be successfully applied to <math alttext="\eta" class="ltx_Math" display="inline" id="Thmexample9.p1.5.m5.1"><semantics id="Thmexample9.p1.5.m5.1a"><mi id="Thmexample9.p1.5.m5.1.1" xref="Thmexample9.p1.5.m5.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmexample9.p1.5.m5.1b"><ci id="Thmexample9.p1.5.m5.1.1.cmml" xref="Thmexample9.p1.5.m5.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p1.5.m5.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p1.5.m5.1d">italic_η</annotation></semantics></math>. Consequently, even the sophisticate verification-based circuit AllSAT procedure in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib17" title="">17</a>]</cite> produces the set of four un-reduced total assignments of <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#Thmexample7" title="Example 7 ‣ 4.2 Verification vs. entailment in search procedures and formula compilers. ‣ 4 Practical consequences of using verification or entailment ‣ Entailment vs. Verification for Partial-assignment Satisfiability and Enumeration"><span class="ltx_text ltx_ref_tag">example</span> <span class="ltx_text ltx_ref_tag">7</span></a>.</p> </div> <div class="ltx_para" id="Thmexample9.p2"> <p class="ltx_p" id="Thmexample9.p2.7">Instead, since <math alttext="\eta\backslash\{{A_{2},A_{4}}\}\models\varphi" class="ltx_Math" display="inline" id="Thmexample9.p2.1.m1.2"><semantics id="Thmexample9.p2.1.m1.2a"><mrow id="Thmexample9.p2.1.m1.2.2" xref="Thmexample9.p2.1.m1.2.2.cmml"><mrow id="Thmexample9.p2.1.m1.2.2.2" xref="Thmexample9.p2.1.m1.2.2.2.cmml"><mi id="Thmexample9.p2.1.m1.2.2.2.4" xref="Thmexample9.p2.1.m1.2.2.2.4.cmml">η</mi><mo id="Thmexample9.p2.1.m1.2.2.2.3" lspace="0.222em" rspace="0.222em" xref="Thmexample9.p2.1.m1.2.2.2.3.cmml">\</mo><mrow id="Thmexample9.p2.1.m1.2.2.2.2.2" xref="Thmexample9.p2.1.m1.2.2.2.2.3.cmml"><mo id="Thmexample9.p2.1.m1.2.2.2.2.2.3" stretchy="false" xref="Thmexample9.p2.1.m1.2.2.2.2.3.cmml">{</mo><msub id="Thmexample9.p2.1.m1.1.1.1.1.1.1" xref="Thmexample9.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmexample9.p2.1.m1.1.1.1.1.1.1.2" xref="Thmexample9.p2.1.m1.1.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample9.p2.1.m1.1.1.1.1.1.1.3" xref="Thmexample9.p2.1.m1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="Thmexample9.p2.1.m1.2.2.2.2.2.4" xref="Thmexample9.p2.1.m1.2.2.2.2.3.cmml">,</mo><msub id="Thmexample9.p2.1.m1.2.2.2.2.2.2" xref="Thmexample9.p2.1.m1.2.2.2.2.2.2.cmml"><mi id="Thmexample9.p2.1.m1.2.2.2.2.2.2.2" xref="Thmexample9.p2.1.m1.2.2.2.2.2.2.2.cmml">A</mi><mn id="Thmexample9.p2.1.m1.2.2.2.2.2.2.3" xref="Thmexample9.p2.1.m1.2.2.2.2.2.2.3.cmml">4</mn></msub><mo id="Thmexample9.p2.1.m1.2.2.2.2.2.5" stretchy="false" xref="Thmexample9.p2.1.m1.2.2.2.2.3.cmml">}</mo></mrow></mrow><mo id="Thmexample9.p2.1.m1.2.2.3" xref="Thmexample9.p2.1.m1.2.2.3.cmml">⊧</mo><mi id="Thmexample9.p2.1.m1.2.2.4" xref="Thmexample9.p2.1.m1.2.2.4.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample9.p2.1.m1.2b"><apply id="Thmexample9.p2.1.m1.2.2.cmml" xref="Thmexample9.p2.1.m1.2.2"><csymbol cd="latexml" id="Thmexample9.p2.1.m1.2.2.3.cmml" xref="Thmexample9.p2.1.m1.2.2.3">models</csymbol><apply id="Thmexample9.p2.1.m1.2.2.2.cmml" xref="Thmexample9.p2.1.m1.2.2.2"><ci id="Thmexample9.p2.1.m1.2.2.2.3.cmml" xref="Thmexample9.p2.1.m1.2.2.2.3">\</ci><ci id="Thmexample9.p2.1.m1.2.2.2.4.cmml" xref="Thmexample9.p2.1.m1.2.2.2.4">𝜂</ci><set id="Thmexample9.p2.1.m1.2.2.2.2.3.cmml" xref="Thmexample9.p2.1.m1.2.2.2.2.2"><apply id="Thmexample9.p2.1.m1.1.1.1.1.1.1.cmml" xref="Thmexample9.p2.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample9.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmexample9.p2.1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="Thmexample9.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmexample9.p2.1.m1.1.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample9.p2.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample9.p2.1.m1.1.1.1.1.1.1.3">2</cn></apply><apply id="Thmexample9.p2.1.m1.2.2.2.2.2.2.cmml" xref="Thmexample9.p2.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="Thmexample9.p2.1.m1.2.2.2.2.2.2.1.cmml" xref="Thmexample9.p2.1.m1.2.2.2.2.2.2">subscript</csymbol><ci id="Thmexample9.p2.1.m1.2.2.2.2.2.2.2.cmml" xref="Thmexample9.p2.1.m1.2.2.2.2.2.2.2">𝐴</ci><cn id="Thmexample9.p2.1.m1.2.2.2.2.2.2.3.cmml" type="integer" xref="Thmexample9.p2.1.m1.2.2.2.2.2.2.3">4</cn></apply></set></apply><ci id="Thmexample9.p2.1.m1.2.2.4.cmml" xref="Thmexample9.p2.1.m1.2.2.4">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p2.1.m1.2c">\eta\backslash\{{A_{2},A_{4}}\}\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p2.1.m1.2d">italic_η \ { italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_A start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT } ⊧ italic_φ</annotation></semantics></math> whereas <math alttext="\eta\backslash\{{A_{1}}\}\not\models\varphi" class="ltx_Math" display="inline" id="Thmexample9.p2.2.m2.1"><semantics id="Thmexample9.p2.2.m2.1a"><mrow id="Thmexample9.p2.2.m2.1.1" xref="Thmexample9.p2.2.m2.1.1.cmml"><mrow id="Thmexample9.p2.2.m2.1.1.1" xref="Thmexample9.p2.2.m2.1.1.1.cmml"><mi id="Thmexample9.p2.2.m2.1.1.1.3" xref="Thmexample9.p2.2.m2.1.1.1.3.cmml">η</mi><mo id="Thmexample9.p2.2.m2.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="Thmexample9.p2.2.m2.1.1.1.2.cmml">\</mo><mrow id="Thmexample9.p2.2.m2.1.1.1.1.1" xref="Thmexample9.p2.2.m2.1.1.1.1.2.cmml"><mo id="Thmexample9.p2.2.m2.1.1.1.1.1.2" stretchy="false" xref="Thmexample9.p2.2.m2.1.1.1.1.2.cmml">{</mo><msub id="Thmexample9.p2.2.m2.1.1.1.1.1.1" xref="Thmexample9.p2.2.m2.1.1.1.1.1.1.cmml"><mi id="Thmexample9.p2.2.m2.1.1.1.1.1.1.2" xref="Thmexample9.p2.2.m2.1.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample9.p2.2.m2.1.1.1.1.1.1.3" xref="Thmexample9.p2.2.m2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample9.p2.2.m2.1.1.1.1.1.3" stretchy="false" xref="Thmexample9.p2.2.m2.1.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="Thmexample9.p2.2.m2.1.1.2" xref="Thmexample9.p2.2.m2.1.1.2.cmml">⊧̸</mo><mi id="Thmexample9.p2.2.m2.1.1.3" xref="Thmexample9.p2.2.m2.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample9.p2.2.m2.1b"><apply id="Thmexample9.p2.2.m2.1.1.cmml" xref="Thmexample9.p2.2.m2.1.1"><csymbol cd="latexml" id="Thmexample9.p2.2.m2.1.1.2.cmml" xref="Thmexample9.p2.2.m2.1.1.2">not-models</csymbol><apply id="Thmexample9.p2.2.m2.1.1.1.cmml" xref="Thmexample9.p2.2.m2.1.1.1"><ci id="Thmexample9.p2.2.m2.1.1.1.2.cmml" xref="Thmexample9.p2.2.m2.1.1.1.2">\</ci><ci id="Thmexample9.p2.2.m2.1.1.1.3.cmml" xref="Thmexample9.p2.2.m2.1.1.1.3">𝜂</ci><set id="Thmexample9.p2.2.m2.1.1.1.1.2.cmml" xref="Thmexample9.p2.2.m2.1.1.1.1.1"><apply id="Thmexample9.p2.2.m2.1.1.1.1.1.1.cmml" xref="Thmexample9.p2.2.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample9.p2.2.m2.1.1.1.1.1.1.1.cmml" xref="Thmexample9.p2.2.m2.1.1.1.1.1.1">subscript</csymbol><ci id="Thmexample9.p2.2.m2.1.1.1.1.1.1.2.cmml" xref="Thmexample9.p2.2.m2.1.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample9.p2.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample9.p2.2.m2.1.1.1.1.1.1.3">1</cn></apply></set></apply><ci id="Thmexample9.p2.2.m2.1.1.3.cmml" xref="Thmexample9.p2.2.m2.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p2.2.m2.1c">\eta\backslash\{{A_{1}}\}\not\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p2.2.m2.1d">italic_η \ { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT } ⊧̸ italic_φ</annotation></semantics></math> and <math alttext="\eta\backslash\{{\neg A_{3}}\}\not\models\varphi" class="ltx_Math" display="inline" id="Thmexample9.p2.3.m3.1"><semantics id="Thmexample9.p2.3.m3.1a"><mrow id="Thmexample9.p2.3.m3.1.1" xref="Thmexample9.p2.3.m3.1.1.cmml"><mrow id="Thmexample9.p2.3.m3.1.1.1" xref="Thmexample9.p2.3.m3.1.1.1.cmml"><mi id="Thmexample9.p2.3.m3.1.1.1.3" xref="Thmexample9.p2.3.m3.1.1.1.3.cmml">η</mi><mo id="Thmexample9.p2.3.m3.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="Thmexample9.p2.3.m3.1.1.1.2.cmml">\</mo><mrow id="Thmexample9.p2.3.m3.1.1.1.1.1" xref="Thmexample9.p2.3.m3.1.1.1.1.2.cmml"><mo id="Thmexample9.p2.3.m3.1.1.1.1.1.2" stretchy="false" xref="Thmexample9.p2.3.m3.1.1.1.1.2.cmml">{</mo><mrow id="Thmexample9.p2.3.m3.1.1.1.1.1.1" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.cmml"><mo id="Thmexample9.p2.3.m3.1.1.1.1.1.1.1" rspace="0.167em" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.1.cmml">¬</mo><msub id="Thmexample9.p2.3.m3.1.1.1.1.1.1.2" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.cmml"><mi id="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.2" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.2.cmml">A</mi><mn id="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.3" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample9.p2.3.m3.1.1.1.1.1.3" stretchy="false" xref="Thmexample9.p2.3.m3.1.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="Thmexample9.p2.3.m3.1.1.2" xref="Thmexample9.p2.3.m3.1.1.2.cmml">⊧̸</mo><mi id="Thmexample9.p2.3.m3.1.1.3" xref="Thmexample9.p2.3.m3.1.1.3.cmml">φ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmexample9.p2.3.m3.1b"><apply id="Thmexample9.p2.3.m3.1.1.cmml" xref="Thmexample9.p2.3.m3.1.1"><csymbol cd="latexml" id="Thmexample9.p2.3.m3.1.1.2.cmml" xref="Thmexample9.p2.3.m3.1.1.2">not-models</csymbol><apply id="Thmexample9.p2.3.m3.1.1.1.cmml" xref="Thmexample9.p2.3.m3.1.1.1"><ci id="Thmexample9.p2.3.m3.1.1.1.2.cmml" xref="Thmexample9.p2.3.m3.1.1.1.2">\</ci><ci id="Thmexample9.p2.3.m3.1.1.1.3.cmml" xref="Thmexample9.p2.3.m3.1.1.1.3">𝜂</ci><set id="Thmexample9.p2.3.m3.1.1.1.1.2.cmml" xref="Thmexample9.p2.3.m3.1.1.1.1.1"><apply id="Thmexample9.p2.3.m3.1.1.1.1.1.1.cmml" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1"><not id="Thmexample9.p2.3.m3.1.1.1.1.1.1.1.cmml" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.1"></not><apply id="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.cmml" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.1.cmml" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.2">subscript</csymbol><ci id="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.2.cmml" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.2">𝐴</ci><cn id="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.3.cmml" type="integer" xref="Thmexample9.p2.3.m3.1.1.1.1.1.1.2.3">3</cn></apply></apply></set></apply><ci id="Thmexample9.p2.3.m3.1.1.3.cmml" xref="Thmexample9.p2.3.m3.1.1.3">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p2.3.m3.1c">\eta\backslash\{{\neg A_{3}}\}\not\models\varphi</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p2.3.m3.1d">italic_η \ { ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT } ⊧̸ italic_φ</annotation></semantics></math>, the entailment-based assignment-reduction technique of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib16" title="">16</a>]</cite> shrinks <math alttext="\eta" class="ltx_Math" display="inline" id="Thmexample9.p2.4.m4.1"><semantics id="Thmexample9.p2.4.m4.1a"><mi id="Thmexample9.p2.4.m4.1.1" xref="Thmexample9.p2.4.m4.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="Thmexample9.p2.4.m4.1b"><ci id="Thmexample9.p2.4.m4.1.1.cmml" xref="Thmexample9.p2.4.m4.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p2.4.m4.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p2.4.m4.1d">italic_η</annotation></semantics></math> into <math alttext="\{{A_{1},\neg A_{3}}\}" class="ltx_Math" display="inline" id="Thmexample9.p2.5.m5.2"><semantics id="Thmexample9.p2.5.m5.2a"><mrow id="Thmexample9.p2.5.m5.2.2.2" xref="Thmexample9.p2.5.m5.2.2.3.cmml"><mo id="Thmexample9.p2.5.m5.2.2.2.3" stretchy="false" xref="Thmexample9.p2.5.m5.2.2.3.cmml">{</mo><msub id="Thmexample9.p2.5.m5.1.1.1.1" xref="Thmexample9.p2.5.m5.1.1.1.1.cmml"><mi id="Thmexample9.p2.5.m5.1.1.1.1.2" xref="Thmexample9.p2.5.m5.1.1.1.1.2.cmml">A</mi><mn id="Thmexample9.p2.5.m5.1.1.1.1.3" xref="Thmexample9.p2.5.m5.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample9.p2.5.m5.2.2.2.4" xref="Thmexample9.p2.5.m5.2.2.3.cmml">,</mo><mrow id="Thmexample9.p2.5.m5.2.2.2.2" xref="Thmexample9.p2.5.m5.2.2.2.2.cmml"><mo id="Thmexample9.p2.5.m5.2.2.2.2.1" rspace="0.167em" xref="Thmexample9.p2.5.m5.2.2.2.2.1.cmml">¬</mo><msub id="Thmexample9.p2.5.m5.2.2.2.2.2" xref="Thmexample9.p2.5.m5.2.2.2.2.2.cmml"><mi id="Thmexample9.p2.5.m5.2.2.2.2.2.2" xref="Thmexample9.p2.5.m5.2.2.2.2.2.2.cmml">A</mi><mn id="Thmexample9.p2.5.m5.2.2.2.2.2.3" xref="Thmexample9.p2.5.m5.2.2.2.2.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample9.p2.5.m5.2.2.2.5" stretchy="false" xref="Thmexample9.p2.5.m5.2.2.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample9.p2.5.m5.2b"><set id="Thmexample9.p2.5.m5.2.2.3.cmml" xref="Thmexample9.p2.5.m5.2.2.2"><apply id="Thmexample9.p2.5.m5.1.1.1.1.cmml" xref="Thmexample9.p2.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample9.p2.5.m5.1.1.1.1.1.cmml" xref="Thmexample9.p2.5.m5.1.1.1.1">subscript</csymbol><ci id="Thmexample9.p2.5.m5.1.1.1.1.2.cmml" xref="Thmexample9.p2.5.m5.1.1.1.1.2">𝐴</ci><cn id="Thmexample9.p2.5.m5.1.1.1.1.3.cmml" type="integer" xref="Thmexample9.p2.5.m5.1.1.1.1.3">1</cn></apply><apply 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A_{3}}\}}\}" class="ltx_Math" display="inline" id="Thmexample9.p2.6.m6.1"><semantics id="Thmexample9.p2.6.m6.1a"><mrow id="Thmexample9.p2.6.m6.1.1.1" xref="Thmexample9.p2.6.m6.1.1.2.cmml"><mo id="Thmexample9.p2.6.m6.1.1.1.2" stretchy="false" xref="Thmexample9.p2.6.m6.1.1.2.cmml">{</mo><mrow id="Thmexample9.p2.6.m6.1.1.1.1.2" xref="Thmexample9.p2.6.m6.1.1.1.1.3.cmml"><mo id="Thmexample9.p2.6.m6.1.1.1.1.2.3" stretchy="false" xref="Thmexample9.p2.6.m6.1.1.1.1.3.cmml">{</mo><msub id="Thmexample9.p2.6.m6.1.1.1.1.1.1" xref="Thmexample9.p2.6.m6.1.1.1.1.1.1.cmml"><mi id="Thmexample9.p2.6.m6.1.1.1.1.1.1.2" xref="Thmexample9.p2.6.m6.1.1.1.1.1.1.2.cmml">A</mi><mn id="Thmexample9.p2.6.m6.1.1.1.1.1.1.3" xref="Thmexample9.p2.6.m6.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmexample9.p2.6.m6.1.1.1.1.2.4" xref="Thmexample9.p2.6.m6.1.1.1.1.3.cmml">,</mo><mrow id="Thmexample9.p2.6.m6.1.1.1.1.2.2" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.cmml"><mo id="Thmexample9.p2.6.m6.1.1.1.1.2.2.1" rspace="0.167em" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.1.cmml">¬</mo><msub id="Thmexample9.p2.6.m6.1.1.1.1.2.2.2" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.cmml"><mi id="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.2" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.2.cmml">A</mi><mn id="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.3" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.3.cmml">3</mn></msub></mrow><mo id="Thmexample9.p2.6.m6.1.1.1.1.2.5" stretchy="false" xref="Thmexample9.p2.6.m6.1.1.1.1.3.cmml">}</mo></mrow><mo id="Thmexample9.p2.6.m6.1.1.1.3" stretchy="false" xref="Thmexample9.p2.6.m6.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmexample9.p2.6.m6.1b"><set id="Thmexample9.p2.6.m6.1.1.2.cmml" xref="Thmexample9.p2.6.m6.1.1.1"><set id="Thmexample9.p2.6.m6.1.1.1.1.3.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.2"><apply id="Thmexample9.p2.6.m6.1.1.1.1.1.1.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmexample9.p2.6.m6.1.1.1.1.1.1.1.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.1.1">subscript</csymbol><ci id="Thmexample9.p2.6.m6.1.1.1.1.1.1.2.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.1.1.2">𝐴</ci><cn id="Thmexample9.p2.6.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="Thmexample9.p2.6.m6.1.1.1.1.1.1.3">1</cn></apply><apply id="Thmexample9.p2.6.m6.1.1.1.1.2.2.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2"><not id="Thmexample9.p2.6.m6.1.1.1.1.2.2.1.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.1"></not><apply id="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.1.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.2">subscript</csymbol><ci id="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.2.cmml" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.2">𝐴</ci><cn id="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.3.cmml" type="integer" xref="Thmexample9.p2.6.m6.1.1.1.1.2.2.2.3">3</cn></apply></apply></set></set></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p2.6.m6.1c">\{{\{{A_{1},\neg A_{3}}\}}\}</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p2.6.m6.1d">{ { italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ¬ italic_A start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT } }</annotation></semantics></math>. <math alttext="\diamond" class="ltx_Math" display="inline" id="Thmexample9.p2.7.m7.1"><semantics id="Thmexample9.p2.7.m7.1a"><mo id="Thmexample9.p2.7.m7.1.1" xref="Thmexample9.p2.7.m7.1.1.cmml">⋄</mo><annotation-xml encoding="MathML-Content" id="Thmexample9.p2.7.m7.1b"><ci id="Thmexample9.p2.7.m7.1.1.cmml" xref="Thmexample9.p2.7.m7.1.1">⋄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmexample9.p2.7.m7.1c">\diamond</annotation><annotation encoding="application/x-llamapun" id="Thmexample9.p2.7.m7.1d">⋄</annotation></semantics></math></p> </div> </div> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Conclusions and Future Work</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this paper we have analyzed in deep the issue of formula satisfaction by partial assignments. We have identified two alternative notions that are implicitly used in the literature, namely verification and entailment, and we have showed that, although the former is easier to check and as such is implicitly used by most current search procedures, the latter has better theoretical properties, and can improve the efficiency and effectiveness of enumeration procedures.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">From a theoretical viewpoint, we champion the idea that partial-assignment satisfaction should universally be defined as entailment, considering verification only an easy-to-check sufficient condition for it. From a practical viewpoint, we suggest that entailment should be adopted as partial-assignment satisfaction check inside enumeration-based procedures in place of verification.</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.3">The analysis presented in this paper opens several research avenues. First, we wish to investigate and evaluate empirically novel entailment-based techniques for both (projected) enumeration and (projected) model counting. In particular we wish to enhance our (projected) AllSAT procedure in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib40" title="">40</a>]</cite> with entailment tests. Second, we wish to implement entailment-based enumeration techniques to enhance our AllSMT procedures in <span class="ltx_text ltx_font_smallcaps" id="S5.p3.3.1">MathSAT</span> <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib8" title="">8</a>]</cite> and evaluate them empirically. In particular, we wish to embed and evaluate such entailment-based AllSMT procedure inside our WMI tools <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib29" title="">29</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib39" title="">39</a>]</cite>. Third, we wish to embed novel entailment-based AllSMT procedures inside our SMT compilers for <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><ci id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">caligraphic_T</annotation></semantics></math>-OBDDs and <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><ci id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">caligraphic_T</annotation></semantics></math>-SDDs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.01536v1#bib.bib26" title="">26</a>]</cite>, extending it also to general <math alttext="\mathcal{T}" class="ltx_Math" display="inline" id="S5.p3.3.m3.1"><semantics id="S5.p3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml">𝒯</mi><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.1b"><ci id="S5.p3.3.m3.1.1.cmml" xref="S5.p3.3.m3.1.1">𝒯</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.1c">\mathcal{T}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.1d">caligraphic_T</annotation></semantics></math>-d-DNNFs.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> A. 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