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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Direct Access for Answers to Conjunctive Queries with Aggregation</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> <meta content="aggregate queries, conjunctive queries, provenance semirings, commutative semirings, annotated databases, direct access, ranking function, answer orderings, query classification" lang="en" name="keywords"/> <base href="/html/2303.05327v2/"/></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/2303.05327v2#S1" title="In Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S2" title="In Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries</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/2303.05327v2#S2.SS0.SSS0.Px1" title="In 2. Preliminaries ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_title">Databases and conjunctive queries.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S3" title="In Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>The Direct-Access Problem</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S4" title="In Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">4</span> </span><span class="ltx_text ltx_font_italic">Incorporating Annotation and Aggregation in the Answers</span></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/2303.05327v2#S4.SS1" title="In 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">4.1</span> </span><span class="ltx_text ltx_font_italic">Generalized Dichotomies</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S4.SS2" title="In 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">4.2</span> </span><span class="ltx_text ltx_font_italic">Count Distinct</span></span></a></li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <p class="ltx_p" id="p1.1">[a] [b] [a]</p> </div> <h1 class="ltx_title ltx_title_document">Direct Access for Answers to Conjunctive Queries with Aggregation</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Idan Eldar<span class="ltx_ERROR undefined" id="id1.1.id1">\lmcsorcid</span>0009-0002-1664-8680 </span></span> <span class="ltx_author_before">, </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Nofar Carmeli<span class="ltx_ERROR undefined" id="id2.1.id1">\lmcsorcid</span>0000-0003-0673-5510 </span></span> <span class="ltx_author_before"> and </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Benny Kimelfeld<span class="ltx_ERROR undefined" id="id3.1.id1">\lmcsorcid</span>0000-0002-7156-1572 </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_address">Technion – Israel Institute of Technology, Haifa, Israel </span> <span class="ltx_contact ltx_role_email"><a href="mailto:idel@campus.technion.ac.il,%20bennyk@cs.technion.ac.il">idel@campus.technion.ac.il, bennyk@cs.technion.ac.il</a> </span> <span class="ltx_contact ltx_role_address">Inria, LIRMM, University of Montpellier, CNRS, France </span> <span class="ltx_contact ltx_role_email"><a href="mailto:nofar.carmeli@inria.fr">nofar.carmeli@inria.fr</a> </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract.</h6> <p class="ltx_p" id="id4.id1">We study the fine-grained complexity of conjunctive queries with grouping and aggregation. For common aggregate functions (e.g., min, max, count, sum), such a query can be phrased as an ordinary conjunctive query over a database annotated with a suitable commutative semiring. We investigate the ability to evaluate such queries by constructing in loglinear time a data structure that provides logarithmic-time direct access to the answers ordered by a given lexicographic order. This task is nontrivial since the number of answers might be larger than loglinear in the size of the input, so the data structure needs to provide a compact representation of the space of answers. In the absence of aggregation and annotation, past research established a sufficient tractability condition on queries and orders. For queries without self-joins, this condition is not just sufficient, but also necessary (under conventional lower-bound assumptions in fine-grained complexity).</p> <p class="ltx_p" id="id5.id2">We show that all past results continue to hold for annotated databases, assuming that the annotation itself does not participate in the lexicographic order. Yet, past algorithms do not apply to the count-distinct aggregation, which has no efficient representation as a commutative semiring; for this aggregation, we establish the corresponding tractability condition. We then show how the complexity of the problem changes when we include the aggregate and annotation value in the order. We also study the impact of having all relations but one annotated by the multiplicative identity (one), as happens when we translate aggregate queries into semiring annotations, and having a semiring with an idempotent addition, such as the case of min, max, and count-distinct over a logarithmic-size domain.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Key words and phrases: </h6>aggregate queries, conjunctive queries, provenance semirings, commutative semirings, annotated databases, direct access, ranking function, answer orderings, query classification </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.p1.15">Consider a query <math alttext="Q" 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">Q</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">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">italic_Q</annotation></semantics></math> that may have a large number of answers, say cubic in the number of tuples of the input database <math alttext="D" 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">D</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">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.2.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">italic_D</annotation></semantics></math>. By answering <math alttext="Q" 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">Q</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">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_Q</annotation></semantics></math> via <em class="ltx_emph ltx_font_italic" id="S1.p1.15.1">direct access</em>, we avoid the materialization of the list of answers, and instead, construct a compact data structure <math alttext="S" 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">S</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">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m4.1d">italic_S</annotation></semantics></math> that supports random access: given an index <math alttext="i" 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">i</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">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m5.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m5.1d">italic_i</annotation></semantics></math>, retrieve the <math alttext="i" class="ltx_Math" display="inline" id="S1.p1.6.m6.1"><semantics id="S1.p1.6.m6.1a"><mi id="S1.p1.6.m6.1.1" xref="S1.p1.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.p1.6.m6.1b"><ci id="S1.p1.6.m6.1.1.cmml" xref="S1.p1.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.p1.6.m6.1d">italic_i</annotation></semantics></math>th answer. Hence, direct access evaluation for a query <math alttext="Q" class="ltx_Math" display="inline" id="S1.p1.7.m7.1"><semantics id="S1.p1.7.m7.1a"><mi id="S1.p1.7.m7.1.1" xref="S1.p1.7.m7.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S1.p1.7.m7.1b"><ci id="S1.p1.7.m7.1.1.cmml" xref="S1.p1.7.m7.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.7.m7.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S1.p1.7.m7.1d">italic_Q</annotation></semantics></math> consists of two algorithms, one for the structure construction (with the input <math alttext="D" class="ltx_Math" display="inline" id="S1.p1.8.m8.1"><semantics id="S1.p1.8.m8.1a"><mi id="S1.p1.8.m8.1.1" xref="S1.p1.8.m8.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S1.p1.8.m8.1b"><ci id="S1.p1.8.m8.1.1.cmml" xref="S1.p1.8.m8.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.8.m8.1c">D</annotation><annotation encoding="application/x-llamapun" id="S1.p1.8.m8.1d">italic_D</annotation></semantics></math>), called <em class="ltx_emph ltx_font_italic" id="S1.p1.15.2">preprocessing</em>, and one for fast access to the answers (with the input <math alttext="S" class="ltx_Math" display="inline" id="S1.p1.9.m9.1"><semantics id="S1.p1.9.m9.1a"><mi id="S1.p1.9.m9.1.1" xref="S1.p1.9.m9.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p1.9.m9.1b"><ci id="S1.p1.9.m9.1.1.cmml" xref="S1.p1.9.m9.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.9.m9.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p1.9.m9.1d">italic_S</annotation></semantics></math> and <math alttext="i" class="ltx_Math" display="inline" id="S1.p1.10.m10.1"><semantics id="S1.p1.10.m10.1a"><mi id="S1.p1.10.m10.1.1" xref="S1.p1.10.m10.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.p1.10.m10.1b"><ci id="S1.p1.10.m10.1.1.cmml" xref="S1.p1.10.m10.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.10.m10.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.p1.10.m10.1d">italic_i</annotation></semantics></math>). This task is nontrivial when <math alttext="S" class="ltx_Math" display="inline" id="S1.p1.11.m11.1"><semantics id="S1.p1.11.m11.1a"><mi id="S1.p1.11.m11.1.1" xref="S1.p1.11.m11.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p1.11.m11.1b"><ci id="S1.p1.11.m11.1.1.cmml" xref="S1.p1.11.m11.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.11.m11.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p1.11.m11.1d">italic_S</annotation></semantics></math> is considerably cheaper to construct than <math alttext="Q(D)" class="ltx_Math" display="inline" id="S1.p1.12.m12.1"><semantics id="S1.p1.12.m12.1a"><mrow id="S1.p1.12.m12.1.2" xref="S1.p1.12.m12.1.2.cmml"><mi id="S1.p1.12.m12.1.2.2" xref="S1.p1.12.m12.1.2.2.cmml">Q</mi><mo id="S1.p1.12.m12.1.2.1" xref="S1.p1.12.m12.1.2.1.cmml">⁢</mo><mrow id="S1.p1.12.m12.1.2.3.2" xref="S1.p1.12.m12.1.2.cmml"><mo id="S1.p1.12.m12.1.2.3.2.1" stretchy="false" xref="S1.p1.12.m12.1.2.cmml">(</mo><mi id="S1.p1.12.m12.1.1" xref="S1.p1.12.m12.1.1.cmml">D</mi><mo id="S1.p1.12.m12.1.2.3.2.2" stretchy="false" xref="S1.p1.12.m12.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.12.m12.1b"><apply id="S1.p1.12.m12.1.2.cmml" xref="S1.p1.12.m12.1.2"><times id="S1.p1.12.m12.1.2.1.cmml" xref="S1.p1.12.m12.1.2.1"></times><ci id="S1.p1.12.m12.1.2.2.cmml" xref="S1.p1.12.m12.1.2.2">𝑄</ci><ci id="S1.p1.12.m12.1.1.cmml" xref="S1.p1.12.m12.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.12.m12.1c">Q(D)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.12.m12.1d">italic_Q ( italic_D )</annotation></semantics></math>. Similarly to past work on direct access <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/CarmeliTGKR21</span>]</cite>, we adopt the tractability requirement of linear or quasi-linear time to construct <math alttext="S" class="ltx_Math" display="inline" id="S1.p1.13.m13.1"><semantics id="S1.p1.13.m13.1a"><mi id="S1.p1.13.m13.1.1" xref="S1.p1.13.m13.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p1.13.m13.1b"><ci id="S1.p1.13.m13.1.1.cmml" xref="S1.p1.13.m13.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.13.m13.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p1.13.m13.1d">italic_S</annotation></semantics></math>, and logarithmic time per access. Hence, up to a poly-logarithmic factor, the required construction time is what it takes to read the database (i.e., linear time), and the access time is constant. The structure <math alttext="S" class="ltx_Math" display="inline" id="S1.p1.14.m14.1"><semantics id="S1.p1.14.m14.1a"><mi id="S1.p1.14.m14.1.1" xref="S1.p1.14.m14.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p1.14.m14.1b"><ci id="S1.p1.14.m14.1.1.cmml" xref="S1.p1.14.m14.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.14.m14.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p1.14.m14.1d">italic_S</annotation></semantics></math> can be viewed as a compact representation of <math alttext="Q(D)" class="ltx_Math" display="inline" id="S1.p1.15.m15.1"><semantics id="S1.p1.15.m15.1a"><mrow id="S1.p1.15.m15.1.2" xref="S1.p1.15.m15.1.2.cmml"><mi id="S1.p1.15.m15.1.2.2" xref="S1.p1.15.m15.1.2.2.cmml">Q</mi><mo id="S1.p1.15.m15.1.2.1" xref="S1.p1.15.m15.1.2.1.cmml">⁢</mo><mrow id="S1.p1.15.m15.1.2.3.2" xref="S1.p1.15.m15.1.2.cmml"><mo id="S1.p1.15.m15.1.2.3.2.1" stretchy="false" xref="S1.p1.15.m15.1.2.cmml">(</mo><mi id="S1.p1.15.m15.1.1" xref="S1.p1.15.m15.1.1.cmml">D</mi><mo id="S1.p1.15.m15.1.2.3.2.2" stretchy="false" xref="S1.p1.15.m15.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.15.m15.1b"><apply id="S1.p1.15.m15.1.2.cmml" xref="S1.p1.15.m15.1.2"><times id="S1.p1.15.m15.1.2.1.cmml" xref="S1.p1.15.m15.1.2.1"></times><ci id="S1.p1.15.m15.1.2.2.cmml" xref="S1.p1.15.m15.1.2.2">𝑄</ci><ci id="S1.p1.15.m15.1.1.cmml" xref="S1.p1.15.m15.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.15.m15.1c">Q(D)</annotation><annotation encoding="application/x-llamapun" id="S1.p1.15.m15.1d">italic_Q ( italic_D )</annotation></semantics></math>, in the general sense of Factorized Databases <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:journals/sigmod/OlteanuS16</span>]</cite>, since its size is necessarily quasi-linear and it provides fast access.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.6">Direct access solutions have been devised for Conjunctive Queries (CQs), first as a way to establish algorithms for enumerating the answers with linear preprocessing time and constant delay <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">braultbaron:tel-01081392</span>]</cite> (and FO queries with restrictions on the database <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">bagan:hal-00221730</span>]</cite>); the preprocessing phase constructs <math alttext="S" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_S</annotation></semantics></math>, and the enumeration phase retrieves the answers by accessing <math alttext="S" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_S</annotation></semantics></math> with increasing indices <math alttext="i" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><mi id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><ci id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">italic_i</annotation></semantics></math>. Later, direct access had a more crucial role within the task of enumerating the answers in a uniformly random order <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:journals/tods/CarmeliZBCKS22</span>]</cite>. As a notion of query evaluation, direct access is interesting in its own right, since we can view <math alttext="S" class="ltx_Math" display="inline" id="S1.p2.4.m4.1"><semantics id="S1.p2.4.m4.1a"><mi id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.1b"><ci id="S1.p2.4.m4.1.1.cmml" xref="S1.p2.4.m4.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.1d">italic_S</annotation></semantics></math> itself as the “result” of the query in the case where array-like access is sufficient for downstream processing (e.g., to produce a sample of answers, to return answers by pages, to answer <math alttext="q" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><ci id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">q</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_q</annotation></semantics></math>-quantile queries, etc.). But then <math alttext="S" class="ltx_Math" display="inline" id="S1.p2.6.m6.1"><semantics id="S1.p2.6.m6.1a"><mi id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.1b"><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.1d">italic_S</annotation></semantics></math> has the benefit that it is considerably smaller and faster to produce than the materialized set of answers. Indeed, recent work has studied the complexity of direct access independently (regardless of any enumeration context) <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/BringmannCM22</span>]</cite>, and specifically studied which <em class="ltx_emph ltx_font_italic" id="S1.p2.6.1">orders</em> over the answers allow for such evaluation <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/CarmeliTGKR21</span>]</cite>. In this paper, we continue with the line of work by Carmeli et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/CarmeliTGKR21</span>]</cite> and investigate the ability to support query evaluation via direct access for <em class="ltx_emph ltx_font_italic" id="S1.p2.6.2">aggregate queries</em>, while focusing on lexicographic orderings of answers.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.3">For illustration, consider the following example, inspired by the FIFA World Cup. Suppose that we have a database of players of teams (countries), sponsors of teams, and goals scored in different games. Specifically, we have three relations: <math alttext="\textsc{Teams}(\textit{player},\textit{country})" class="ltx_Math" display="inline" id="S1.p3.1.m1.2"><semantics id="S1.p3.1.m1.2a"><mrow id="S1.p3.1.m1.2.3" xref="S1.p3.1.m1.2.3.cmml"><mtext class="ltx_font_smallcaps" id="S1.p3.1.m1.2.3.2" xref="S1.p3.1.m1.2.3.2a.cmml">Teams</mtext><mo id="S1.p3.1.m1.2.3.1" xref="S1.p3.1.m1.2.3.1.cmml">⁢</mo><mrow id="S1.p3.1.m1.2.3.3.2" xref="S1.p3.1.m1.2.3.3.1.cmml"><mo id="S1.p3.1.m1.2.3.3.2.1" stretchy="false" xref="S1.p3.1.m1.2.3.3.1.cmml">(</mo><mtext class="ltx_mathvariant_italic" id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1a.cmml">player</mtext><mo id="S1.p3.1.m1.2.3.3.2.2" xref="S1.p3.1.m1.2.3.3.1.cmml">,</mo><mtext class="ltx_mathvariant_italic" id="S1.p3.1.m1.2.2" xref="S1.p3.1.m1.2.2a.cmml">country</mtext><mo id="S1.p3.1.m1.2.3.3.2.3" stretchy="false" xref="S1.p3.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.2b"><apply id="S1.p3.1.m1.2.3.cmml" xref="S1.p3.1.m1.2.3"><times id="S1.p3.1.m1.2.3.1.cmml" xref="S1.p3.1.m1.2.3.1"></times><ci id="S1.p3.1.m1.2.3.2a.cmml" xref="S1.p3.1.m1.2.3.2"><mtext class="ltx_font_smallcaps" id="S1.p3.1.m1.2.3.2.cmml" xref="S1.p3.1.m1.2.3.2">Teams</mtext></ci><interval closure="open" id="S1.p3.1.m1.2.3.3.1.cmml" xref="S1.p3.1.m1.2.3.3.2"><ci id="S1.p3.1.m1.1.1a.cmml" xref="S1.p3.1.m1.1.1"><mtext class="ltx_mathvariant_italic" id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">player</mtext></ci><ci id="S1.p3.1.m1.2.2a.cmml" xref="S1.p3.1.m1.2.2"><mtext class="ltx_mathvariant_italic" id="S1.p3.1.m1.2.2.cmml" xref="S1.p3.1.m1.2.2">country</mtext></ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.2c">\textsc{Teams}(\textit{player},\textit{country})</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.2d">Teams ( player , country )</annotation></semantics></math>, <math alttext="\textsc{Sponsors}(\textit{org},\textit{country})" class="ltx_Math" display="inline" id="S1.p3.2.m2.2"><semantics id="S1.p3.2.m2.2a"><mrow id="S1.p3.2.m2.2.3" xref="S1.p3.2.m2.2.3.cmml"><mtext class="ltx_font_smallcaps" id="S1.p3.2.m2.2.3.2" xref="S1.p3.2.m2.2.3.2a.cmml">Sponsors</mtext><mo id="S1.p3.2.m2.2.3.1" xref="S1.p3.2.m2.2.3.1.cmml">⁢</mo><mrow id="S1.p3.2.m2.2.3.3.2" xref="S1.p3.2.m2.2.3.3.1.cmml"><mo id="S1.p3.2.m2.2.3.3.2.1" stretchy="false" xref="S1.p3.2.m2.2.3.3.1.cmml">(</mo><mtext class="ltx_mathvariant_italic" id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1a.cmml">org</mtext><mo id="S1.p3.2.m2.2.3.3.2.2" xref="S1.p3.2.m2.2.3.3.1.cmml">,</mo><mtext class="ltx_mathvariant_italic" id="S1.p3.2.m2.2.2" xref="S1.p3.2.m2.2.2a.cmml">country</mtext><mo id="S1.p3.2.m2.2.3.3.2.3" stretchy="false" xref="S1.p3.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.2b"><apply id="S1.p3.2.m2.2.3.cmml" xref="S1.p3.2.m2.2.3"><times id="S1.p3.2.m2.2.3.1.cmml" xref="S1.p3.2.m2.2.3.1"></times><ci id="S1.p3.2.m2.2.3.2a.cmml" xref="S1.p3.2.m2.2.3.2"><mtext class="ltx_font_smallcaps" id="S1.p3.2.m2.2.3.2.cmml" xref="S1.p3.2.m2.2.3.2">Sponsors</mtext></ci><interval closure="open" id="S1.p3.2.m2.2.3.3.1.cmml" xref="S1.p3.2.m2.2.3.3.2"><ci id="S1.p3.2.m2.1.1a.cmml" xref="S1.p3.2.m2.1.1"><mtext class="ltx_mathvariant_italic" id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">org</mtext></ci><ci id="S1.p3.2.m2.2.2a.cmml" xref="S1.p3.2.m2.2.2"><mtext class="ltx_mathvariant_italic" id="S1.p3.2.m2.2.2.cmml" xref="S1.p3.2.m2.2.2">country</mtext></ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.2c">\textsc{Sponsors}(\textit{org},\textit{country})</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.2d">Sponsors ( org , country )</annotation></semantics></math>, and <math alttext="\textsc{Goals}(\textit{game},\textit{player},\textit{time})" class="ltx_Math" display="inline" id="S1.p3.3.m3.3"><semantics id="S1.p3.3.m3.3a"><mrow id="S1.p3.3.m3.3.4" xref="S1.p3.3.m3.3.4.cmml"><mtext class="ltx_font_smallcaps" id="S1.p3.3.m3.3.4.2" xref="S1.p3.3.m3.3.4.2a.cmml">Goals</mtext><mo id="S1.p3.3.m3.3.4.1" xref="S1.p3.3.m3.3.4.1.cmml">⁢</mo><mrow id="S1.p3.3.m3.3.4.3.2" xref="S1.p3.3.m3.3.4.3.1.cmml"><mo id="S1.p3.3.m3.3.4.3.2.1" stretchy="false" xref="S1.p3.3.m3.3.4.3.1.cmml">(</mo><mtext class="ltx_mathvariant_italic" id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1a.cmml">game</mtext><mo id="S1.p3.3.m3.3.4.3.2.2" xref="S1.p3.3.m3.3.4.3.1.cmml">,</mo><mtext class="ltx_mathvariant_italic" id="S1.p3.3.m3.2.2" xref="S1.p3.3.m3.2.2a.cmml">player</mtext><mo id="S1.p3.3.m3.3.4.3.2.3" xref="S1.p3.3.m3.3.4.3.1.cmml">,</mo><mtext class="ltx_mathvariant_italic" id="S1.p3.3.m3.3.3" xref="S1.p3.3.m3.3.3a.cmml">time</mtext><mo id="S1.p3.3.m3.3.4.3.2.4" stretchy="false" xref="S1.p3.3.m3.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.3b"><apply id="S1.p3.3.m3.3.4.cmml" xref="S1.p3.3.m3.3.4"><times id="S1.p3.3.m3.3.4.1.cmml" xref="S1.p3.3.m3.3.4.1"></times><ci id="S1.p3.3.m3.3.4.2a.cmml" xref="S1.p3.3.m3.3.4.2"><mtext class="ltx_font_smallcaps" id="S1.p3.3.m3.3.4.2.cmml" xref="S1.p3.3.m3.3.4.2">Goals</mtext></ci><vector id="S1.p3.3.m3.3.4.3.1.cmml" xref="S1.p3.3.m3.3.4.3.2"><ci id="S1.p3.3.m3.1.1a.cmml" xref="S1.p3.3.m3.1.1"><mtext class="ltx_mathvariant_italic" id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1">game</mtext></ci><ci id="S1.p3.3.m3.2.2a.cmml" xref="S1.p3.3.m3.2.2"><mtext class="ltx_mathvariant_italic" id="S1.p3.3.m3.2.2.cmml" xref="S1.p3.3.m3.2.2">player</mtext></ci><ci id="S1.p3.3.m3.3.3a.cmml" xref="S1.p3.3.m3.3.3"><mtext class="ltx_mathvariant_italic" id="S1.p3.3.m3.3.3.cmml" xref="S1.p3.3.m3.3.3">time</mtext></ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.3c">\textsc{Goals}(\textit{game},\textit{player},\textit{time})</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.3d">Goals ( game , player , time )</annotation></semantics></math>. The following CQ finds times when sponsors got exposure due to goals of supported teams:</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="Q_{1}(c,o,p,t)\mathrel{{:}{-}}\textsc{Teams}(p,c),\textsc{Sponsors}(o,c),% \textsc{Goals}(g,p,t)" class="ltx_Math" display="block" id="S1.Ex1.m1.14"><semantics id="S1.Ex1.m1.14a"><mrow id="S1.Ex1.m1.14.14" xref="S1.Ex1.m1.14.14.cmml"><mrow id="S1.Ex1.m1.14.14.5" xref="S1.Ex1.m1.14.14.5.cmml"><msub id="S1.Ex1.m1.14.14.5.2" xref="S1.Ex1.m1.14.14.5.2.cmml"><mi id="S1.Ex1.m1.14.14.5.2.2" xref="S1.Ex1.m1.14.14.5.2.2.cmml">Q</mi><mn id="S1.Ex1.m1.14.14.5.2.3" xref="S1.Ex1.m1.14.14.5.2.3.cmml">1</mn></msub><mo id="S1.Ex1.m1.14.14.5.1" xref="S1.Ex1.m1.14.14.5.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.14.14.5.3.2" xref="S1.Ex1.m1.14.14.5.3.1.cmml"><mo id="S1.Ex1.m1.14.14.5.3.2.1" stretchy="false" xref="S1.Ex1.m1.14.14.5.3.1.cmml">(</mo><mi id="S1.Ex1.m1.1.1" xref="S1.Ex1.m1.1.1.cmml">c</mi><mo id="S1.Ex1.m1.14.14.5.3.2.2" xref="S1.Ex1.m1.14.14.5.3.1.cmml">,</mo><mi id="S1.Ex1.m1.2.2" xref="S1.Ex1.m1.2.2.cmml">o</mi><mo id="S1.Ex1.m1.14.14.5.3.2.3" xref="S1.Ex1.m1.14.14.5.3.1.cmml">,</mo><mi id="S1.Ex1.m1.3.3" xref="S1.Ex1.m1.3.3.cmml">p</mi><mo id="S1.Ex1.m1.14.14.5.3.2.4" xref="S1.Ex1.m1.14.14.5.3.1.cmml">,</mo><mi id="S1.Ex1.m1.4.4" xref="S1.Ex1.m1.4.4.cmml">t</mi><mo id="S1.Ex1.m1.14.14.5.3.2.5" rspace="0.278em" stretchy="false" xref="S1.Ex1.m1.14.14.5.3.1.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_italic" id="S1.Ex1.m1.14.14.4" mathvariant="italic" rspace="0.278em" xref="S1.Ex1.m1.14.14.4.cmml">:-</mo><mrow id="S1.Ex1.m1.14.14.3.3" xref="S1.Ex1.m1.14.14.3.4.cmml"><mrow id="S1.Ex1.m1.12.12.1.1.1" xref="S1.Ex1.m1.12.12.1.1.1.cmml"><mtext class="ltx_font_smallcaps" id="S1.Ex1.m1.12.12.1.1.1.2" xref="S1.Ex1.m1.12.12.1.1.1.2a.cmml">Teams</mtext><mo 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xref="S1.Ex1.m1.10.10">𝑝</ci><ci id="S1.Ex1.m1.11.11.cmml" xref="S1.Ex1.m1.11.11">𝑡</ci></vector></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex1.m1.14c">Q_{1}(c,o,p,t)\mathrel{{:}{-}}\textsc{Teams}(p,c),\textsc{Sponsors}(o,c),% \textsc{Goals}(g,p,t)</annotation><annotation encoding="application/x-llamapun" id="S1.Ex1.m1.14d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_c , italic_o , italic_p , italic_t ) italic_:- Teams ( italic_p , italic_c ) , Sponsors ( italic_o , italic_c ) , Goals ( italic_g , italic_p , italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p3.13">Suppose also that we would like the answers to be ordered lexicographically by their order in the head: first by <math alttext="c" class="ltx_Math" display="inline" id="S1.p3.4.m1.1"><semantics id="S1.p3.4.m1.1a"><mi id="S1.p3.4.m1.1.1" xref="S1.p3.4.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.p3.4.m1.1b"><ci id="S1.p3.4.m1.1.1.cmml" xref="S1.p3.4.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m1.1d">italic_c</annotation></semantics></math> (country), then by <math alttext="o" class="ltx_Math" display="inline" id="S1.p3.5.m2.1"><semantics id="S1.p3.5.m2.1a"><mi id="S1.p3.5.m2.1.1" xref="S1.p3.5.m2.1.1.cmml">o</mi><annotation-xml encoding="MathML-Content" id="S1.p3.5.m2.1b"><ci id="S1.p3.5.m2.1.1.cmml" xref="S1.p3.5.m2.1.1">𝑜</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m2.1c">o</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m2.1d">italic_o</annotation></semantics></math> (organization), then by <math alttext="p" class="ltx_Math" display="inline" id="S1.p3.6.m3.1"><semantics id="S1.p3.6.m3.1a"><mi id="S1.p3.6.m3.1.1" xref="S1.p3.6.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p3.6.m3.1b"><ci id="S1.p3.6.m3.1.1.cmml" xref="S1.p3.6.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m3.1d">italic_p</annotation></semantics></math> (player), and lastly by <math alttext="t" class="ltx_Math" display="inline" id="S1.p3.7.m4.1"><semantics id="S1.p3.7.m4.1a"><mi id="S1.p3.7.m4.1.1" xref="S1.p3.7.m4.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.p3.7.m4.1b"><ci id="S1.p3.7.m4.1.1.cmml" xref="S1.p3.7.m4.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.7.m4.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.p3.7.m4.1d">italic_t</annotation></semantics></math> (time). Note that <math alttext="o" class="ltx_Math" display="inline" id="S1.p3.8.m5.1"><semantics id="S1.p3.8.m5.1a"><mi id="S1.p3.8.m5.1.1" xref="S1.p3.8.m5.1.1.cmml">o</mi><annotation-xml encoding="MathML-Content" id="S1.p3.8.m5.1b"><ci id="S1.p3.8.m5.1.1.cmml" xref="S1.p3.8.m5.1.1">𝑜</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.8.m5.1c">o</annotation><annotation encoding="application/x-llamapun" id="S1.p3.8.m5.1d">italic_o</annotation></semantics></math>, <math alttext="c" class="ltx_Math" display="inline" id="S1.p3.9.m6.1"><semantics id="S1.p3.9.m6.1a"><mi id="S1.p3.9.m6.1.1" xref="S1.p3.9.m6.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.p3.9.m6.1b"><ci id="S1.p3.9.m6.1.1.cmml" xref="S1.p3.9.m6.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.9.m6.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.p3.9.m6.1d">italic_c</annotation></semantics></math>, <math alttext="p" class="ltx_Math" display="inline" id="S1.p3.10.m7.1"><semantics id="S1.p3.10.m7.1a"><mi id="S1.p3.10.m7.1.1" xref="S1.p3.10.m7.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p3.10.m7.1b"><ci id="S1.p3.10.m7.1.1.cmml" xref="S1.p3.10.m7.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.10.m7.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p3.10.m7.1d">italic_p</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" id="S1.p3.11.m8.1"><semantics id="S1.p3.11.m8.1a"><mi id="S1.p3.11.m8.1.1" xref="S1.p3.11.m8.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S1.p3.11.m8.1b"><ci id="S1.p3.11.m8.1.1.cmml" xref="S1.p3.11.m8.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.11.m8.1c">t</annotation><annotation encoding="application/x-llamapun" id="S1.p3.11.m8.1d">italic_t</annotation></semantics></math> are <em class="ltx_emph ltx_font_italic" id="S1.p3.13.1">free</em> variables and <math alttext="g" class="ltx_Math" display="inline" id="S1.p3.12.m9.1"><semantics id="S1.p3.12.m9.1a"><mi id="S1.p3.12.m9.1.1" xref="S1.p3.12.m9.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S1.p3.12.m9.1b"><ci id="S1.p3.12.m9.1.1.cmml" xref="S1.p3.12.m9.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.12.m9.1c">g</annotation><annotation encoding="application/x-llamapun" id="S1.p3.12.m9.1d">italic_g</annotation></semantics></math> is an <em class="ltx_emph ltx_font_italic" id="S1.p3.13.2">existential</em> variable. Carmeli et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/CarmeliTGKR21</span>]</cite> studied the ability to evaluate such ordered queries with direct access. In the case of <math alttext="Q_{1}" class="ltx_Math" display="inline" id="S1.p3.13.m10.1"><semantics id="S1.p3.13.m10.1a"><msub id="S1.p3.13.m10.1.1" xref="S1.p3.13.m10.1.1.cmml"><mi id="S1.p3.13.m10.1.1.2" xref="S1.p3.13.m10.1.1.2.cmml">Q</mi><mn id="S1.p3.13.m10.1.1.3" xref="S1.p3.13.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p3.13.m10.1b"><apply id="S1.p3.13.m10.1.1.cmml" xref="S1.p3.13.m10.1.1"><csymbol cd="ambiguous" id="S1.p3.13.m10.1.1.1.cmml" xref="S1.p3.13.m10.1.1">subscript</csymbol><ci id="S1.p3.13.m10.1.1.2.cmml" xref="S1.p3.13.m10.1.1.2">𝑄</ci><cn id="S1.p3.13.m10.1.1.3.cmml" type="integer" xref="S1.p3.13.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.13.m10.1c">Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.13.m10.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, the results of Carmeli et al. show that there is an efficient direct access evaluation (since the query is free-connex and there is no “disruptive trio”).</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.12">Now, suppose that we would like to count the goals per sponsorship and player. In standard CQ notation (e.g., Cohen et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1145/1219092.1219093</span>]</cite>), we can phrase this query as follows.</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="Q_{2}(c,o,p,\mathsf{Count}())\mathrel{{:}{-}}\textsc{Teams}(p,c),\textsc{% Sponsors}(o,c),\textsc{Goals}(g,p,t)" class="ltx_Math" display="block" id="S1.Ex2.m1.14"><semantics id="S1.Ex2.m1.14a"><mrow id="S1.Ex2.m1.14.14" xref="S1.Ex2.m1.14.14.cmml"><mrow id="S1.Ex2.m1.11.11.1" xref="S1.Ex2.m1.11.11.1.cmml"><msub id="S1.Ex2.m1.11.11.1.3" xref="S1.Ex2.m1.11.11.1.3.cmml"><mi id="S1.Ex2.m1.11.11.1.3.2" xref="S1.Ex2.m1.11.11.1.3.2.cmml">Q</mi><mn id="S1.Ex2.m1.11.11.1.3.3" xref="S1.Ex2.m1.11.11.1.3.3.cmml">2</mn></msub><mo 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xref="S1.Ex2.m1.11.11.1.1.1.1.3.1.cmml">(</mo><mo id="S1.Ex2.m1.11.11.1.1.1.1.3.2.2" stretchy="false" xref="S1.Ex2.m1.11.11.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S1.Ex2.m1.11.11.1.1.1.6" rspace="0.278em" stretchy="false" xref="S1.Ex2.m1.11.11.1.1.2.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_italic" id="S1.Ex2.m1.14.14.5" mathvariant="italic" rspace="0.278em" xref="S1.Ex2.m1.14.14.5.cmml">:-</mo><mrow id="S1.Ex2.m1.14.14.4.3" xref="S1.Ex2.m1.14.14.4.4.cmml"><mrow id="S1.Ex2.m1.12.12.2.1.1" xref="S1.Ex2.m1.12.12.2.1.1.cmml"><mtext class="ltx_font_smallcaps" id="S1.Ex2.m1.12.12.2.1.1.2" xref="S1.Ex2.m1.12.12.2.1.1.2a.cmml">Teams</mtext><mo id="S1.Ex2.m1.12.12.2.1.1.1" xref="S1.Ex2.m1.12.12.2.1.1.1.cmml">⁢</mo><mrow id="S1.Ex2.m1.12.12.2.1.1.3.2" xref="S1.Ex2.m1.12.12.2.1.1.3.1.cmml"><mo id="S1.Ex2.m1.12.12.2.1.1.3.2.1" stretchy="false" xref="S1.Ex2.m1.12.12.2.1.1.3.1.cmml">(</mo><mi id="S1.Ex2.m1.4.4" xref="S1.Ex2.m1.4.4.cmml">p</mi><mo id="S1.Ex2.m1.12.12.2.1.1.3.2.2" xref="S1.Ex2.m1.12.12.2.1.1.3.1.cmml">,</mo><mi id="S1.Ex2.m1.5.5" xref="S1.Ex2.m1.5.5.cmml">c</mi><mo id="S1.Ex2.m1.12.12.2.1.1.3.2.3" stretchy="false" xref="S1.Ex2.m1.12.12.2.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S1.Ex2.m1.14.14.4.3.4" xref="S1.Ex2.m1.14.14.4.4.cmml">,</mo><mrow id="S1.Ex2.m1.13.13.3.2.2" xref="S1.Ex2.m1.13.13.3.2.2.cmml"><mtext class="ltx_font_smallcaps" id="S1.Ex2.m1.13.13.3.2.2.2" xref="S1.Ex2.m1.13.13.3.2.2.2a.cmml">Sponsors</mtext><mo id="S1.Ex2.m1.13.13.3.2.2.1" xref="S1.Ex2.m1.13.13.3.2.2.1.cmml">⁢</mo><mrow id="S1.Ex2.m1.13.13.3.2.2.3.2" xref="S1.Ex2.m1.13.13.3.2.2.3.1.cmml"><mo id="S1.Ex2.m1.13.13.3.2.2.3.2.1" stretchy="false" xref="S1.Ex2.m1.13.13.3.2.2.3.1.cmml">(</mo><mi id="S1.Ex2.m1.6.6" xref="S1.Ex2.m1.6.6.cmml">o</mi><mo id="S1.Ex2.m1.13.13.3.2.2.3.2.2" xref="S1.Ex2.m1.13.13.3.2.2.3.1.cmml">,</mo><mi id="S1.Ex2.m1.7.7" xref="S1.Ex2.m1.7.7.cmml">c</mi><mo id="S1.Ex2.m1.13.13.3.2.2.3.2.3" stretchy="false" 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xref="S1.Ex2.m1.14.14.4.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex2.m1.14b"><apply id="S1.Ex2.m1.14.14.cmml" xref="S1.Ex2.m1.14.14"><ci id="S1.Ex2.m1.14.14.5.cmml" xref="S1.Ex2.m1.14.14.5">italic-:-</ci><apply id="S1.Ex2.m1.11.11.1.cmml" xref="S1.Ex2.m1.11.11.1"><times id="S1.Ex2.m1.11.11.1.2.cmml" xref="S1.Ex2.m1.11.11.1.2"></times><apply id="S1.Ex2.m1.11.11.1.3.cmml" xref="S1.Ex2.m1.11.11.1.3"><csymbol cd="ambiguous" id="S1.Ex2.m1.11.11.1.3.1.cmml" xref="S1.Ex2.m1.11.11.1.3">subscript</csymbol><ci id="S1.Ex2.m1.11.11.1.3.2.cmml" xref="S1.Ex2.m1.11.11.1.3.2">𝑄</ci><cn id="S1.Ex2.m1.11.11.1.3.3.cmml" type="integer" xref="S1.Ex2.m1.11.11.1.3.3">2</cn></apply><vector id="S1.Ex2.m1.11.11.1.1.2.cmml" xref="S1.Ex2.m1.11.11.1.1.1"><ci id="S1.Ex2.m1.1.1.cmml" xref="S1.Ex2.m1.1.1">𝑐</ci><ci id="S1.Ex2.m1.2.2.cmml" xref="S1.Ex2.m1.2.2">𝑜</ci><ci id="S1.Ex2.m1.3.3.cmml" xref="S1.Ex2.m1.3.3">𝑝</ci><apply id="S1.Ex2.m1.11.11.1.1.1.1.cmml" 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id="S1.Ex2.m1.9.9.cmml" xref="S1.Ex2.m1.9.9">𝑝</ci><ci id="S1.Ex2.m1.10.10.cmml" xref="S1.Ex2.m1.10.10">𝑡</ci></vector></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex2.m1.14c">Q_{2}(c,o,p,\mathsf{Count}())\mathrel{{:}{-}}\textsc{Teams}(p,c),\textsc{% Sponsors}(o,c),\textsc{Goals}(g,p,t)</annotation><annotation encoding="application/x-llamapun" id="S1.Ex2.m1.14d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_c , italic_o , italic_p , sansserif_Count ( ) ) italic_:- Teams ( italic_p , italic_c ) , Sponsors ( italic_o , italic_c ) , Goals ( italic_g , italic_p , italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p4.11">Here, the free variables <math alttext="c" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mi id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_c</annotation></semantics></math>, <math alttext="o" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">o</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">𝑜</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">o</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">italic_o</annotation></semantics></math>, and <math alttext="p" class="ltx_Math" display="inline" id="S1.p4.3.m3.1"><semantics id="S1.p4.3.m3.1a"><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.m3.1b"><ci id="S1.p4.3.m3.1.1.cmml" xref="S1.p4.3.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.m3.1d">italic_p</annotation></semantics></math> are treated as the <em class="ltx_emph ltx_font_italic" id="S1.p4.11.1">grouping variables</em>, where each combination of values defines a group of tuples over <math alttext="(c,o,p,g,t)" class="ltx_Math" display="inline" id="S1.p4.4.m4.5"><semantics id="S1.p4.4.m4.5a"><mrow id="S1.p4.4.m4.5.6.2" xref="S1.p4.4.m4.5.6.1.cmml"><mo id="S1.p4.4.m4.5.6.2.1" stretchy="false" xref="S1.p4.4.m4.5.6.1.cmml">(</mo><mi id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml">c</mi><mo id="S1.p4.4.m4.5.6.2.2" xref="S1.p4.4.m4.5.6.1.cmml">,</mo><mi id="S1.p4.4.m4.2.2" xref="S1.p4.4.m4.2.2.cmml">o</mi><mo id="S1.p4.4.m4.5.6.2.3" xref="S1.p4.4.m4.5.6.1.cmml">,</mo><mi id="S1.p4.4.m4.3.3" xref="S1.p4.4.m4.3.3.cmml">p</mi><mo id="S1.p4.4.m4.5.6.2.4" xref="S1.p4.4.m4.5.6.1.cmml">,</mo><mi id="S1.p4.4.m4.4.4" xref="S1.p4.4.m4.4.4.cmml">g</mi><mo id="S1.p4.4.m4.5.6.2.5" xref="S1.p4.4.m4.5.6.1.cmml">,</mo><mi id="S1.p4.4.m4.5.5" xref="S1.p4.4.m4.5.5.cmml">t</mi><mo id="S1.p4.4.m4.5.6.2.6" stretchy="false" xref="S1.p4.4.m4.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.4.m4.5b"><vector id="S1.p4.4.m4.5.6.1.cmml" xref="S1.p4.4.m4.5.6.2"><ci id="S1.p4.4.m4.1.1.cmml" xref="S1.p4.4.m4.1.1">𝑐</ci><ci id="S1.p4.4.m4.2.2.cmml" xref="S1.p4.4.m4.2.2">𝑜</ci><ci id="S1.p4.4.m4.3.3.cmml" xref="S1.p4.4.m4.3.3">𝑝</ci><ci id="S1.p4.4.m4.4.4.cmml" xref="S1.p4.4.m4.4.4">𝑔</ci><ci id="S1.p4.4.m4.5.5.cmml" xref="S1.p4.4.m4.5.5">𝑡</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m4.5c">(c,o,p,g,t)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m4.5d">( italic_c , italic_o , italic_p , italic_g , italic_t )</annotation></semantics></math> and <math alttext="\mathsf{Count}()" class="ltx_Math" display="inline" id="S1.p4.5.m5.1"><semantics id="S1.p4.5.m5.1a"><mrow id="S1.p4.5.m5.1.1" xref="S1.p4.5.m5.1.1.cmml"><mi id="S1.p4.5.m5.1.1.2" xref="S1.p4.5.m5.1.1.2.cmml">𝖢𝗈𝗎𝗇𝗍</mi><mo id="S1.p4.5.m5.1.1.1" xref="S1.p4.5.m5.1.1.1.cmml">⁢</mo><mrow id="S1.p4.5.m5.1.1.3.2" xref="S1.p4.5.m5.1.1.cmml"><mo id="S1.p4.5.m5.1.1.3.2.1" stretchy="false" xref="S1.p4.5.m5.1.1.3.1.cmml">(</mo><mo id="S1.p4.5.m5.1.1.3.2.2" stretchy="false" xref="S1.p4.5.m5.1.1.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.5.m5.1b"><apply id="S1.p4.5.m5.1.1.cmml" xref="S1.p4.5.m5.1.1"><times id="S1.p4.5.m5.1.1.1.cmml" xref="S1.p4.5.m5.1.1.1"></times><ci id="S1.p4.5.m5.1.1.2.cmml" xref="S1.p4.5.m5.1.1.2">𝖢𝗈𝗎𝗇𝗍</ci><list id="S1.p4.5.m5.1.1.3.1.cmml" xref="S1.p4.5.m5.1.1.3.2.1"></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.5.m5.1c">\mathsf{Count}()</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m5.1d">sansserif_Count ( )</annotation></semantics></math> simply counts the tuples in the group. Again, we would like to answer this query via direct access. This introduces two challenges. The first challenge is <em class="ltx_emph ltx_font_italic" id="S1.p4.11.2">aggregate construction</em>: when we access a tuple using <math alttext="S" class="ltx_Math" display="inline" id="S1.p4.6.m6.1"><semantics id="S1.p4.6.m6.1a"><mi id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S1.p4.6.m6.1b"><ci id="S1.p4.6.m6.1.1.cmml" xref="S1.p4.6.m6.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m6.1c">S</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m6.1d">italic_S</annotation></semantics></math>, the aggregate value should be quickly produced as well. The second challenge is <em class="ltx_emph ltx_font_italic" id="S1.p4.11.3">ordering by aggregation</em>: how can we incorporate the aggregation in the lexicographic order of the answers if so desired by the query? As an example, we may wish to order the answers first by <math alttext="c" class="ltx_Math" display="inline" id="S1.p4.7.m7.1"><semantics id="S1.p4.7.m7.1a"><mi id="S1.p4.7.m7.1.1" xref="S1.p4.7.m7.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.p4.7.m7.1b"><ci id="S1.p4.7.m7.1.1.cmml" xref="S1.p4.7.m7.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.7.m7.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.p4.7.m7.1d">italic_c</annotation></semantics></math>, then by <math alttext="\mathsf{Count}()" class="ltx_Math" display="inline" id="S1.p4.8.m8.1"><semantics id="S1.p4.8.m8.1a"><mrow id="S1.p4.8.m8.1.1" xref="S1.p4.8.m8.1.1.cmml"><mi id="S1.p4.8.m8.1.1.2" xref="S1.p4.8.m8.1.1.2.cmml">𝖢𝗈𝗎𝗇𝗍</mi><mo id="S1.p4.8.m8.1.1.1" xref="S1.p4.8.m8.1.1.1.cmml">⁢</mo><mrow id="S1.p4.8.m8.1.1.3.2" xref="S1.p4.8.m8.1.1.cmml"><mo id="S1.p4.8.m8.1.1.3.2.1" stretchy="false" xref="S1.p4.8.m8.1.1.3.1.cmml">(</mo><mo id="S1.p4.8.m8.1.1.3.2.2" stretchy="false" xref="S1.p4.8.m8.1.1.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.8.m8.1b"><apply id="S1.p4.8.m8.1.1.cmml" xref="S1.p4.8.m8.1.1"><times id="S1.p4.8.m8.1.1.1.cmml" xref="S1.p4.8.m8.1.1.1"></times><ci id="S1.p4.8.m8.1.1.2.cmml" xref="S1.p4.8.m8.1.1.2">𝖢𝗈𝗎𝗇𝗍</ci><list id="S1.p4.8.m8.1.1.3.1.cmml" xref="S1.p4.8.m8.1.1.3.2.1"></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.8.m8.1c">\mathsf{Count}()</annotation><annotation encoding="application/x-llamapun" id="S1.p4.8.m8.1d">sansserif_Count ( )</annotation></semantics></math>, and then by <math alttext="o" class="ltx_Math" display="inline" id="S1.p4.9.m9.1"><semantics id="S1.p4.9.m9.1a"><mi id="S1.p4.9.m9.1.1" xref="S1.p4.9.m9.1.1.cmml">o</mi><annotation-xml encoding="MathML-Content" id="S1.p4.9.m9.1b"><ci id="S1.p4.9.m9.1.1.cmml" xref="S1.p4.9.m9.1.1">𝑜</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.9.m9.1c">o</annotation><annotation encoding="application/x-llamapun" id="S1.p4.9.m9.1d">italic_o</annotation></semantics></math> and <math alttext="p" class="ltx_Math" display="inline" id="S1.p4.10.m10.1"><semantics id="S1.p4.10.m10.1a"><mi id="S1.p4.10.m10.1.1" xref="S1.p4.10.m10.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p4.10.m10.1b"><ci id="S1.p4.10.m10.1.1.cmml" xref="S1.p4.10.m10.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.10.m10.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p4.10.m10.1d">italic_p</annotation></semantics></math>; in this case, we would phrase the head accordingly as <math alttext="Q_{2}(c,\mathsf{Count}(),o,p)" class="ltx_Math" display="inline" id="S1.p4.11.m11.4"><semantics id="S1.p4.11.m11.4a"><mrow id="S1.p4.11.m11.4.4" xref="S1.p4.11.m11.4.4.cmml"><msub id="S1.p4.11.m11.4.4.3" xref="S1.p4.11.m11.4.4.3.cmml"><mi id="S1.p4.11.m11.4.4.3.2" xref="S1.p4.11.m11.4.4.3.2.cmml">Q</mi><mn id="S1.p4.11.m11.4.4.3.3" xref="S1.p4.11.m11.4.4.3.3.cmml">2</mn></msub><mo id="S1.p4.11.m11.4.4.2" xref="S1.p4.11.m11.4.4.2.cmml">⁢</mo><mrow id="S1.p4.11.m11.4.4.1.1" xref="S1.p4.11.m11.4.4.1.2.cmml"><mo id="S1.p4.11.m11.4.4.1.1.2" stretchy="false" xref="S1.p4.11.m11.4.4.1.2.cmml">(</mo><mi id="S1.p4.11.m11.1.1" xref="S1.p4.11.m11.1.1.cmml">c</mi><mo id="S1.p4.11.m11.4.4.1.1.3" xref="S1.p4.11.m11.4.4.1.2.cmml">,</mo><mrow id="S1.p4.11.m11.4.4.1.1.1" xref="S1.p4.11.m11.4.4.1.1.1.cmml"><mi id="S1.p4.11.m11.4.4.1.1.1.2" xref="S1.p4.11.m11.4.4.1.1.1.2.cmml">𝖢𝗈𝗎𝗇𝗍</mi><mo id="S1.p4.11.m11.4.4.1.1.1.1" xref="S1.p4.11.m11.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S1.p4.11.m11.4.4.1.1.1.3.2" xref="S1.p4.11.m11.4.4.1.1.1.cmml"><mo id="S1.p4.11.m11.4.4.1.1.1.3.2.1" stretchy="false" xref="S1.p4.11.m11.4.4.1.1.1.3.1.cmml">(</mo><mo id="S1.p4.11.m11.4.4.1.1.1.3.2.2" stretchy="false" xref="S1.p4.11.m11.4.4.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S1.p4.11.m11.4.4.1.1.4" xref="S1.p4.11.m11.4.4.1.2.cmml">,</mo><mi id="S1.p4.11.m11.2.2" xref="S1.p4.11.m11.2.2.cmml">o</mi><mo id="S1.p4.11.m11.4.4.1.1.5" xref="S1.p4.11.m11.4.4.1.2.cmml">,</mo><mi id="S1.p4.11.m11.3.3" xref="S1.p4.11.m11.3.3.cmml">p</mi><mo id="S1.p4.11.m11.4.4.1.1.6" stretchy="false" xref="S1.p4.11.m11.4.4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.11.m11.4b"><apply id="S1.p4.11.m11.4.4.cmml" xref="S1.p4.11.m11.4.4"><times id="S1.p4.11.m11.4.4.2.cmml" xref="S1.p4.11.m11.4.4.2"></times><apply id="S1.p4.11.m11.4.4.3.cmml" xref="S1.p4.11.m11.4.4.3"><csymbol cd="ambiguous" id="S1.p4.11.m11.4.4.3.1.cmml" xref="S1.p4.11.m11.4.4.3">subscript</csymbol><ci id="S1.p4.11.m11.4.4.3.2.cmml" xref="S1.p4.11.m11.4.4.3.2">𝑄</ci><cn id="S1.p4.11.m11.4.4.3.3.cmml" type="integer" xref="S1.p4.11.m11.4.4.3.3">2</cn></apply><vector id="S1.p4.11.m11.4.4.1.2.cmml" xref="S1.p4.11.m11.4.4.1.1"><ci id="S1.p4.11.m11.1.1.cmml" xref="S1.p4.11.m11.1.1">𝑐</ci><apply id="S1.p4.11.m11.4.4.1.1.1.cmml" xref="S1.p4.11.m11.4.4.1.1.1"><times id="S1.p4.11.m11.4.4.1.1.1.1.cmml" xref="S1.p4.11.m11.4.4.1.1.1.1"></times><ci id="S1.p4.11.m11.4.4.1.1.1.2.cmml" xref="S1.p4.11.m11.4.4.1.1.1.2">𝖢𝗈𝗎𝗇𝗍</ci><list id="S1.p4.11.m11.4.4.1.1.1.3.1.cmml" xref="S1.p4.11.m11.4.4.1.1.1.3.2.1"></list></apply><ci id="S1.p4.11.m11.2.2.cmml" xref="S1.p4.11.m11.2.2">𝑜</ci><ci id="S1.p4.11.m11.3.3.cmml" xref="S1.p4.11.m11.3.3">𝑝</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.11.m11.4c">Q_{2}(c,\mathsf{Count}(),o,p)</annotation><annotation encoding="application/x-llamapun" id="S1.p4.11.m11.4d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_c , sansserif_Count ( ) , italic_o , italic_p )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.2">As previously done in the context of algorithms for aggregate queries <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:journals/vldb/ReS09</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:journals/tods/KhamisCMNNOS20</span>]</cite>, we also study a semiring alternative to the above formalism of aggregate queries. Specifically, we can adopt the well-known framework of <em class="ltx_emph ltx_font_italic" id="S1.p5.2.1">provenance semiring</em> of Green, Karvounarakis and Tannen <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1145/1265530.1265535</span>]</cite> and phrase the query as an ordinary CQ with the annotation carrying the aggregate value (e.g., the number of goals in our example). To reason about random-access evaluation, we found it more elegant, general, and insightful to support CQs over annotated databases rather than SQL-like aggregate functions. For illustration, we can phrase the above aggregate query <math alttext="Q_{2}" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><msub id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mi id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2.cmml">Q</mi><mn id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1">subscript</csymbol><ci id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.2">𝑄</ci><cn id="S1.p5.1.m1.1.1.3.cmml" type="integer" xref="S1.p5.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> as the following CQ <math alttext="Q_{3}" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><msub id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><mi id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2.cmml">Q</mi><mn id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><apply id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p5.2.m2.1.1.1.cmml" xref="S1.p5.2.m2.1.1">subscript</csymbol><ci id="S1.p5.2.m2.1.1.2.cmml" xref="S1.p5.2.m2.1.1.2">𝑄</ci><cn id="S1.p5.2.m2.1.1.3.cmml" type="integer" xref="S1.p5.2.m2.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">Q_{3}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">italic_Q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>, but for a database that is annotated using a specific commutative semiring.</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="Q_{3}(c,o,p)\mathrel{{:}{-}}\textsc{Teams}(p,c),\textsc{Sponsors}(o,c),\textsc% {Goals}(g,p,t)" class="ltx_Math" display="block" id="S1.Ex3.m1.13"><semantics id="S1.Ex3.m1.13a"><mrow id="S1.Ex3.m1.13.13" xref="S1.Ex3.m1.13.13.cmml"><mrow id="S1.Ex3.m1.13.13.5" xref="S1.Ex3.m1.13.13.5.cmml"><msub id="S1.Ex3.m1.13.13.5.2" xref="S1.Ex3.m1.13.13.5.2.cmml"><mi id="S1.Ex3.m1.13.13.5.2.2" xref="S1.Ex3.m1.13.13.5.2.2.cmml">Q</mi><mn id="S1.Ex3.m1.13.13.5.2.3" xref="S1.Ex3.m1.13.13.5.2.3.cmml">3</mn></msub><mo id="S1.Ex3.m1.13.13.5.1" xref="S1.Ex3.m1.13.13.5.1.cmml">⁢</mo><mrow id="S1.Ex3.m1.13.13.5.3.2" xref="S1.Ex3.m1.13.13.5.3.1.cmml"><mo id="S1.Ex3.m1.13.13.5.3.2.1" stretchy="false" xref="S1.Ex3.m1.13.13.5.3.1.cmml">(</mo><mi id="S1.Ex3.m1.1.1" xref="S1.Ex3.m1.1.1.cmml">c</mi><mo id="S1.Ex3.m1.13.13.5.3.2.2" xref="S1.Ex3.m1.13.13.5.3.1.cmml">,</mo><mi id="S1.Ex3.m1.2.2" xref="S1.Ex3.m1.2.2.cmml">o</mi><mo id="S1.Ex3.m1.13.13.5.3.2.3" xref="S1.Ex3.m1.13.13.5.3.1.cmml">,</mo><mi id="S1.Ex3.m1.3.3" xref="S1.Ex3.m1.3.3.cmml">p</mi><mo id="S1.Ex3.m1.13.13.5.3.2.4" rspace="0.278em" stretchy="false" xref="S1.Ex3.m1.13.13.5.3.1.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_italic" id="S1.Ex3.m1.13.13.4" mathvariant="italic" rspace="0.278em" xref="S1.Ex3.m1.13.13.4.cmml">:-</mo><mrow id="S1.Ex3.m1.13.13.3.3" xref="S1.Ex3.m1.13.13.3.4.cmml"><mrow id="S1.Ex3.m1.11.11.1.1.1" xref="S1.Ex3.m1.11.11.1.1.1.cmml"><mtext class="ltx_font_smallcaps" id="S1.Ex3.m1.11.11.1.1.1.2" xref="S1.Ex3.m1.11.11.1.1.1.2a.cmml">Teams</mtext><mo id="S1.Ex3.m1.11.11.1.1.1.1" xref="S1.Ex3.m1.11.11.1.1.1.1.cmml">⁢</mo><mrow id="S1.Ex3.m1.11.11.1.1.1.3.2" xref="S1.Ex3.m1.11.11.1.1.1.3.1.cmml"><mo id="S1.Ex3.m1.11.11.1.1.1.3.2.1" stretchy="false" xref="S1.Ex3.m1.11.11.1.1.1.3.1.cmml">(</mo><mi id="S1.Ex3.m1.4.4" 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xref="S1.Ex3.m1.13.13.3.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex3.m1.13b"><apply id="S1.Ex3.m1.13.13.cmml" xref="S1.Ex3.m1.13.13"><ci id="S1.Ex3.m1.13.13.4.cmml" xref="S1.Ex3.m1.13.13.4">italic-:-</ci><apply id="S1.Ex3.m1.13.13.5.cmml" xref="S1.Ex3.m1.13.13.5"><times id="S1.Ex3.m1.13.13.5.1.cmml" xref="S1.Ex3.m1.13.13.5.1"></times><apply id="S1.Ex3.m1.13.13.5.2.cmml" xref="S1.Ex3.m1.13.13.5.2"><csymbol cd="ambiguous" id="S1.Ex3.m1.13.13.5.2.1.cmml" xref="S1.Ex3.m1.13.13.5.2">subscript</csymbol><ci id="S1.Ex3.m1.13.13.5.2.2.cmml" xref="S1.Ex3.m1.13.13.5.2.2">𝑄</ci><cn id="S1.Ex3.m1.13.13.5.2.3.cmml" type="integer" xref="S1.Ex3.m1.13.13.5.2.3">3</cn></apply><vector id="S1.Ex3.m1.13.13.5.3.1.cmml" xref="S1.Ex3.m1.13.13.5.3.2"><ci id="S1.Ex3.m1.1.1.cmml" xref="S1.Ex3.m1.1.1">𝑐</ci><ci id="S1.Ex3.m1.2.2.cmml" xref="S1.Ex3.m1.2.2">𝑜</ci><ci id="S1.Ex3.m1.3.3.cmml" xref="S1.Ex3.m1.3.3">𝑝</ci></vector></apply><list 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id="S1.Ex3.m1.12.12.2.2.2.3.1.cmml" xref="S1.Ex3.m1.12.12.2.2.2.3.2"><ci id="S1.Ex3.m1.6.6.cmml" xref="S1.Ex3.m1.6.6">𝑜</ci><ci id="S1.Ex3.m1.7.7.cmml" xref="S1.Ex3.m1.7.7">𝑐</ci></interval></apply><apply id="S1.Ex3.m1.13.13.3.3.3.cmml" xref="S1.Ex3.m1.13.13.3.3.3"><times id="S1.Ex3.m1.13.13.3.3.3.1.cmml" xref="S1.Ex3.m1.13.13.3.3.3.1"></times><ci id="S1.Ex3.m1.13.13.3.3.3.2a.cmml" xref="S1.Ex3.m1.13.13.3.3.3.2"><mtext class="ltx_font_smallcaps" id="S1.Ex3.m1.13.13.3.3.3.2.cmml" xref="S1.Ex3.m1.13.13.3.3.3.2">Goals</mtext></ci><vector id="S1.Ex3.m1.13.13.3.3.3.3.1.cmml" xref="S1.Ex3.m1.13.13.3.3.3.3.2"><ci id="S1.Ex3.m1.8.8.cmml" xref="S1.Ex3.m1.8.8">𝑔</ci><ci id="S1.Ex3.m1.9.9.cmml" xref="S1.Ex3.m1.9.9">𝑝</ci><ci id="S1.Ex3.m1.10.10.cmml" xref="S1.Ex3.m1.10.10">𝑡</ci></vector></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex3.m1.13c">Q_{3}(c,o,p)\mathrel{{:}{-}}\textsc{Teams}(p,c),\textsc{Sponsors}(o,c),\textsc% {Goals}(g,p,t)</annotation><annotation encoding="application/x-llamapun" id="S1.Ex3.m1.13d">italic_Q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_c , italic_o , italic_p ) italic_:- Teams ( italic_p , italic_c ) , Sponsors ( italic_o , italic_c ) , Goals ( italic_g , italic_p , italic_t )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p5.11">In a nutshell (the formal definition is in Section <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S2" title="2. Preliminaries ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">2</span></a>), the idea is that each tuple is annotated with an element of the semiring, the annotation of each tuple in the group is the product of the participating tuple annotations, and the annotation of the whole group is the sum of all tuple annotations in the group’s tuples. In the case of our example with <math alttext="Q_{3}" class="ltx_Math" display="inline" id="S1.p5.3.m1.1"><semantics id="S1.p5.3.m1.1a"><msub id="S1.p5.3.m1.1.1" xref="S1.p5.3.m1.1.1.cmml"><mi id="S1.p5.3.m1.1.1.2" xref="S1.p5.3.m1.1.1.2.cmml">Q</mi><mn id="S1.p5.3.m1.1.1.3" xref="S1.p5.3.m1.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p5.3.m1.1b"><apply id="S1.p5.3.m1.1.1.cmml" xref="S1.p5.3.m1.1.1"><csymbol cd="ambiguous" id="S1.p5.3.m1.1.1.1.cmml" xref="S1.p5.3.m1.1.1">subscript</csymbol><ci id="S1.p5.3.m1.1.1.2.cmml" xref="S1.p5.3.m1.1.1.2">𝑄</ci><cn id="S1.p5.3.m1.1.1.3.cmml" type="integer" xref="S1.p5.3.m1.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m1.1c">Q_{3}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m1.1d">italic_Q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math>, we use the counting semiring <math alttext="(\mathbb{N},+,\cdot,0,1)" class="ltx_Math" display="inline" id="S1.p5.4.m2.5"><semantics id="S1.p5.4.m2.5a"><mrow id="S1.p5.4.m2.5.6.2" xref="S1.p5.4.m2.5.6.1.cmml"><mo id="S1.p5.4.m2.5.6.2.1" stretchy="false" xref="S1.p5.4.m2.5.6.1.cmml">(</mo><mi id="S1.p5.4.m2.1.1" xref="S1.p5.4.m2.1.1.cmml">ℕ</mi><mo id="S1.p5.4.m2.5.6.2.2" rspace="0em" xref="S1.p5.4.m2.5.6.1.cmml">,</mo><mo id="S1.p5.4.m2.2.2" lspace="0em" rspace="0em" xref="S1.p5.4.m2.2.2.cmml">+</mo><mo id="S1.p5.4.m2.5.6.2.3" rspace="0em" xref="S1.p5.4.m2.5.6.1.cmml">,</mo><mo id="S1.p5.4.m2.3.3" lspace="0em" rspace="0em" xref="S1.p5.4.m2.3.3.cmml">⋅</mo><mo id="S1.p5.4.m2.5.6.2.4" xref="S1.p5.4.m2.5.6.1.cmml">,</mo><mn id="S1.p5.4.m2.4.4" xref="S1.p5.4.m2.4.4.cmml">0</mn><mo id="S1.p5.4.m2.5.6.2.5" xref="S1.p5.4.m2.5.6.1.cmml">,</mo><mn id="S1.p5.4.m2.5.5" xref="S1.p5.4.m2.5.5.cmml">1</mn><mo id="S1.p5.4.m2.5.6.2.6" stretchy="false" xref="S1.p5.4.m2.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.4.m2.5b"><vector id="S1.p5.4.m2.5.6.1.cmml" xref="S1.p5.4.m2.5.6.2"><ci id="S1.p5.4.m2.1.1.cmml" xref="S1.p5.4.m2.1.1">ℕ</ci><plus id="S1.p5.4.m2.2.2.cmml" xref="S1.p5.4.m2.2.2"></plus><ci id="S1.p5.4.m2.3.3.cmml" xref="S1.p5.4.m2.3.3">⋅</ci><cn id="S1.p5.4.m2.4.4.cmml" type="integer" xref="S1.p5.4.m2.4.4">0</cn><cn id="S1.p5.4.m2.5.5.cmml" type="integer" xref="S1.p5.4.m2.5.5">1</cn></vector></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m2.5c">(\mathbb{N},+,\cdot,0,1)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m2.5d">( blackboard_N , + , ⋅ , 0 , 1 )</annotation></semantics></math>, and each tuple is annotated simply with the number <math alttext="1" class="ltx_Math" display="inline" id="S1.p5.5.m3.1"><semantics id="S1.p5.5.m3.1a"><mn id="S1.p5.5.m3.1.1" xref="S1.p5.5.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.p5.5.m3.1b"><cn id="S1.p5.5.m3.1.1.cmml" type="integer" xref="S1.p5.5.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.5.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.p5.5.m3.1d">1</annotation></semantics></math>. We can use different semirings and annotations to compute different aggregate functions like sum, min, and max. Here again, we have challenges analogous to the aggregate case: <em class="ltx_emph ltx_font_italic" id="S1.p5.11.1">annotation construction</em> and <em class="ltx_emph ltx_font_italic" id="S1.p5.11.2">ordering by annotation</em>. The previous example becomes ordering by <math alttext="c" class="ltx_Math" display="inline" id="S1.p5.6.m4.1"><semantics id="S1.p5.6.m4.1a"><mi id="S1.p5.6.m4.1.1" xref="S1.p5.6.m4.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.p5.6.m4.1b"><ci id="S1.p5.6.m4.1.1.cmml" xref="S1.p5.6.m4.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.6.m4.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.p5.6.m4.1d">italic_c</annotation></semantics></math>, then by the annotation, and then by <math alttext="o" class="ltx_Math" display="inline" id="S1.p5.7.m5.1"><semantics id="S1.p5.7.m5.1a"><mi id="S1.p5.7.m5.1.1" xref="S1.p5.7.m5.1.1.cmml">o</mi><annotation-xml encoding="MathML-Content" id="S1.p5.7.m5.1b"><ci id="S1.p5.7.m5.1.1.cmml" xref="S1.p5.7.m5.1.1">𝑜</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.7.m5.1c">o</annotation><annotation encoding="application/x-llamapun" id="S1.p5.7.m5.1d">italic_o</annotation></semantics></math> and <math alttext="p" class="ltx_Math" display="inline" id="S1.p5.8.m6.1"><semantics id="S1.p5.8.m6.1a"><mi id="S1.p5.8.m6.1.1" xref="S1.p5.8.m6.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p5.8.m6.1b"><ci id="S1.p5.8.m6.1.1.cmml" xref="S1.p5.8.m6.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.8.m6.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p5.8.m6.1d">italic_p</annotation></semantics></math>. Notationally, we specify the annotation position by the symbol <math alttext="\star" class="ltx_Math" display="inline" id="S1.p5.9.m7.1"><semantics id="S1.p5.9.m7.1a"><mo id="S1.p5.9.m7.1.1" xref="S1.p5.9.m7.1.1.cmml">⋆</mo><annotation-xml encoding="MathML-Content" id="S1.p5.9.m7.1b"><ci id="S1.p5.9.m7.1.1.cmml" xref="S1.p5.9.m7.1.1">⋆</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.9.m7.1c">\star</annotation><annotation encoding="application/x-llamapun" id="S1.p5.9.m7.1d">⋆</annotation></semantics></math> and phrase the query as <math alttext="Q_{3}(c,\star,o,p)\mathrel{{:}{-}}\textsc{Teams}(p,c),\textsc{Sponsors}(o,c),% \textsc{Goals}(g,p,t)" class="ltx_Math" display="inline" id="S1.p5.10.m8.14"><semantics id="S1.p5.10.m8.14a"><mrow id="S1.p5.10.m8.14.14" xref="S1.p5.10.m8.14.14.cmml"><mrow id="S1.p5.10.m8.14.14.5" xref="S1.p5.10.m8.14.14.5.cmml"><msub id="S1.p5.10.m8.14.14.5.2" xref="S1.p5.10.m8.14.14.5.2.cmml"><mi id="S1.p5.10.m8.14.14.5.2.2" xref="S1.p5.10.m8.14.14.5.2.2.cmml">Q</mi><mn id="S1.p5.10.m8.14.14.5.2.3" xref="S1.p5.10.m8.14.14.5.2.3.cmml">3</mn></msub><mo id="S1.p5.10.m8.14.14.5.1" xref="S1.p5.10.m8.14.14.5.1.cmml">⁢</mo><mrow id="S1.p5.10.m8.14.14.5.3.2" xref="S1.p5.10.m8.14.14.5.3.1.cmml"><mo id="S1.p5.10.m8.14.14.5.3.2.1" stretchy="false" xref="S1.p5.10.m8.14.14.5.3.1.cmml">(</mo><mi id="S1.p5.10.m8.1.1" xref="S1.p5.10.m8.1.1.cmml">c</mi><mo id="S1.p5.10.m8.14.14.5.3.2.2" rspace="0em" xref="S1.p5.10.m8.14.14.5.3.1.cmml">,</mo><mo id="S1.p5.10.m8.2.2" lspace="0em" rspace="0em" xref="S1.p5.10.m8.2.2.cmml">⋆</mo><mo id="S1.p5.10.m8.14.14.5.3.2.3" xref="S1.p5.10.m8.14.14.5.3.1.cmml">,</mo><mi id="S1.p5.10.m8.3.3" xref="S1.p5.10.m8.3.3.cmml">o</mi><mo id="S1.p5.10.m8.14.14.5.3.2.4" xref="S1.p5.10.m8.14.14.5.3.1.cmml">,</mo><mi id="S1.p5.10.m8.4.4" xref="S1.p5.10.m8.4.4.cmml">p</mi><mo id="S1.p5.10.m8.14.14.5.3.2.5" rspace="0.278em" stretchy="false" xref="S1.p5.10.m8.14.14.5.3.1.cmml">)</mo></mrow></mrow><mo class="ltx_mathvariant_italic" id="S1.p5.10.m8.14.14.4" mathvariant="italic" rspace="0.278em" xref="S1.p5.10.m8.14.14.4.cmml">:-</mo><mrow id="S1.p5.10.m8.14.14.3.3" xref="S1.p5.10.m8.14.14.3.4.cmml"><mrow id="S1.p5.10.m8.12.12.1.1.1" xref="S1.p5.10.m8.12.12.1.1.1.cmml"><mtext class="ltx_font_smallcaps" id="S1.p5.10.m8.12.12.1.1.1.2" xref="S1.p5.10.m8.12.12.1.1.1.2a.cmml">Teams</mtext><mo id="S1.p5.10.m8.12.12.1.1.1.1" xref="S1.p5.10.m8.12.12.1.1.1.1.cmml">⁢</mo><mrow id="S1.p5.10.m8.12.12.1.1.1.3.2" xref="S1.p5.10.m8.12.12.1.1.1.3.1.cmml"><mo id="S1.p5.10.m8.12.12.1.1.1.3.2.1" stretchy="false" xref="S1.p5.10.m8.12.12.1.1.1.3.1.cmml">(</mo><mi id="S1.p5.10.m8.5.5" xref="S1.p5.10.m8.5.5.cmml">p</mi><mo id="S1.p5.10.m8.12.12.1.1.1.3.2.2" xref="S1.p5.10.m8.12.12.1.1.1.3.1.cmml">,</mo><mi id="S1.p5.10.m8.6.6" xref="S1.p5.10.m8.6.6.cmml">c</mi><mo id="S1.p5.10.m8.12.12.1.1.1.3.2.3" stretchy="false" xref="S1.p5.10.m8.12.12.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S1.p5.10.m8.14.14.3.3.4" xref="S1.p5.10.m8.14.14.3.4.cmml">,</mo><mrow id="S1.p5.10.m8.13.13.2.2.2" xref="S1.p5.10.m8.13.13.2.2.2.cmml"><mtext class="ltx_font_smallcaps" id="S1.p5.10.m8.13.13.2.2.2.2" xref="S1.p5.10.m8.13.13.2.2.2.2a.cmml">Sponsors</mtext><mo id="S1.p5.10.m8.13.13.2.2.2.1" xref="S1.p5.10.m8.13.13.2.2.2.1.cmml">⁢</mo><mrow id="S1.p5.10.m8.13.13.2.2.2.3.2" xref="S1.p5.10.m8.13.13.2.2.2.3.1.cmml"><mo id="S1.p5.10.m8.13.13.2.2.2.3.2.1" stretchy="false" xref="S1.p5.10.m8.13.13.2.2.2.3.1.cmml">(</mo><mi id="S1.p5.10.m8.7.7" xref="S1.p5.10.m8.7.7.cmml">o</mi><mo id="S1.p5.10.m8.13.13.2.2.2.3.2.2" xref="S1.p5.10.m8.13.13.2.2.2.3.1.cmml">,</mo><mi id="S1.p5.10.m8.8.8" xref="S1.p5.10.m8.8.8.cmml">c</mi><mo id="S1.p5.10.m8.13.13.2.2.2.3.2.3" stretchy="false" xref="S1.p5.10.m8.13.13.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S1.p5.10.m8.14.14.3.3.5" xref="S1.p5.10.m8.14.14.3.4.cmml">,</mo><mrow id="S1.p5.10.m8.14.14.3.3.3" xref="S1.p5.10.m8.14.14.3.3.3.cmml"><mtext class="ltx_font_smallcaps" id="S1.p5.10.m8.14.14.3.3.3.2" xref="S1.p5.10.m8.14.14.3.3.3.2a.cmml">Goals</mtext><mo id="S1.p5.10.m8.14.14.3.3.3.1" xref="S1.p5.10.m8.14.14.3.3.3.1.cmml">⁢</mo><mrow id="S1.p5.10.m8.14.14.3.3.3.3.2" xref="S1.p5.10.m8.14.14.3.3.3.3.1.cmml"><mo id="S1.p5.10.m8.14.14.3.3.3.3.2.1" stretchy="false" xref="S1.p5.10.m8.14.14.3.3.3.3.1.cmml">(</mo><mi id="S1.p5.10.m8.9.9" xref="S1.p5.10.m8.9.9.cmml">g</mi><mo id="S1.p5.10.m8.14.14.3.3.3.3.2.2" xref="S1.p5.10.m8.14.14.3.3.3.3.1.cmml">,</mo><mi id="S1.p5.10.m8.10.10" xref="S1.p5.10.m8.10.10.cmml">p</mi><mo id="S1.p5.10.m8.14.14.3.3.3.3.2.3" xref="S1.p5.10.m8.14.14.3.3.3.3.1.cmml">,</mo><mi id="S1.p5.10.m8.11.11" xref="S1.p5.10.m8.11.11.cmml">t</mi><mo id="S1.p5.10.m8.14.14.3.3.3.3.2.4" stretchy="false" xref="S1.p5.10.m8.14.14.3.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.10.m8.14b"><apply id="S1.p5.10.m8.14.14.cmml" xref="S1.p5.10.m8.14.14"><ci id="S1.p5.10.m8.14.14.4.cmml" xref="S1.p5.10.m8.14.14.4">italic-:-</ci><apply id="S1.p5.10.m8.14.14.5.cmml" xref="S1.p5.10.m8.14.14.5"><times id="S1.p5.10.m8.14.14.5.1.cmml" xref="S1.p5.10.m8.14.14.5.1"></times><apply id="S1.p5.10.m8.14.14.5.2.cmml" xref="S1.p5.10.m8.14.14.5.2"><csymbol cd="ambiguous" id="S1.p5.10.m8.14.14.5.2.1.cmml" xref="S1.p5.10.m8.14.14.5.2">subscript</csymbol><ci id="S1.p5.10.m8.14.14.5.2.2.cmml" xref="S1.p5.10.m8.14.14.5.2.2">𝑄</ci><cn id="S1.p5.10.m8.14.14.5.2.3.cmml" type="integer" xref="S1.p5.10.m8.14.14.5.2.3">3</cn></apply><vector id="S1.p5.10.m8.14.14.5.3.1.cmml" xref="S1.p5.10.m8.14.14.5.3.2"><ci id="S1.p5.10.m8.1.1.cmml" xref="S1.p5.10.m8.1.1">𝑐</ci><ci id="S1.p5.10.m8.2.2.cmml" xref="S1.p5.10.m8.2.2">⋆</ci><ci id="S1.p5.10.m8.3.3.cmml" xref="S1.p5.10.m8.3.3">𝑜</ci><ci id="S1.p5.10.m8.4.4.cmml" xref="S1.p5.10.m8.4.4">𝑝</ci></vector></apply><list id="S1.p5.10.m8.14.14.3.4.cmml" xref="S1.p5.10.m8.14.14.3.3"><apply id="S1.p5.10.m8.12.12.1.1.1.cmml" xref="S1.p5.10.m8.12.12.1.1.1"><times id="S1.p5.10.m8.12.12.1.1.1.1.cmml" xref="S1.p5.10.m8.12.12.1.1.1.1"></times><ci id="S1.p5.10.m8.12.12.1.1.1.2a.cmml" xref="S1.p5.10.m8.12.12.1.1.1.2"><mtext class="ltx_font_smallcaps" id="S1.p5.10.m8.12.12.1.1.1.2.cmml" xref="S1.p5.10.m8.12.12.1.1.1.2">Teams</mtext></ci><interval closure="open" id="S1.p5.10.m8.12.12.1.1.1.3.1.cmml" xref="S1.p5.10.m8.12.12.1.1.1.3.2"><ci id="S1.p5.10.m8.5.5.cmml" xref="S1.p5.10.m8.5.5">𝑝</ci><ci id="S1.p5.10.m8.6.6.cmml" xref="S1.p5.10.m8.6.6">𝑐</ci></interval></apply><apply id="S1.p5.10.m8.13.13.2.2.2.cmml" xref="S1.p5.10.m8.13.13.2.2.2"><times id="S1.p5.10.m8.13.13.2.2.2.1.cmml" xref="S1.p5.10.m8.13.13.2.2.2.1"></times><ci id="S1.p5.10.m8.13.13.2.2.2.2a.cmml" xref="S1.p5.10.m8.13.13.2.2.2.2"><mtext class="ltx_font_smallcaps" id="S1.p5.10.m8.13.13.2.2.2.2.cmml" xref="S1.p5.10.m8.13.13.2.2.2.2">Sponsors</mtext></ci><interval closure="open" id="S1.p5.10.m8.13.13.2.2.2.3.1.cmml" xref="S1.p5.10.m8.13.13.2.2.2.3.2"><ci id="S1.p5.10.m8.7.7.cmml" xref="S1.p5.10.m8.7.7">𝑜</ci><ci id="S1.p5.10.m8.8.8.cmml" xref="S1.p5.10.m8.8.8">𝑐</ci></interval></apply><apply id="S1.p5.10.m8.14.14.3.3.3.cmml" xref="S1.p5.10.m8.14.14.3.3.3"><times id="S1.p5.10.m8.14.14.3.3.3.1.cmml" xref="S1.p5.10.m8.14.14.3.3.3.1"></times><ci id="S1.p5.10.m8.14.14.3.3.3.2a.cmml" xref="S1.p5.10.m8.14.14.3.3.3.2"><mtext class="ltx_font_smallcaps" id="S1.p5.10.m8.14.14.3.3.3.2.cmml" xref="S1.p5.10.m8.14.14.3.3.3.2">Goals</mtext></ci><vector id="S1.p5.10.m8.14.14.3.3.3.3.1.cmml" xref="S1.p5.10.m8.14.14.3.3.3.3.2"><ci id="S1.p5.10.m8.9.9.cmml" xref="S1.p5.10.m8.9.9">𝑔</ci><ci id="S1.p5.10.m8.10.10.cmml" xref="S1.p5.10.m8.10.10">𝑝</ci><ci id="S1.p5.10.m8.11.11.cmml" xref="S1.p5.10.m8.11.11">𝑡</ci></vector></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.10.m8.14c">Q_{3}(c,\star,o,p)\mathrel{{:}{-}}\textsc{Teams}(p,c),\textsc{Sponsors}(o,c),% \textsc{Goals}(g,p,t)</annotation><annotation encoding="application/x-llamapun" id="S1.p5.10.m8.14d">italic_Q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_c , ⋆ , italic_o , italic_p ) italic_:- Teams ( italic_p , italic_c ) , Sponsors ( italic_o , italic_c ) , Goals ( italic_g , italic_p , italic_t )</annotation></semantics></math>. We refer to such a query as a CQ<sup class="ltx_sup" id="S1.p5.11.3">⋆</sup>.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">In this paper, we study queries in both formalisms—CQs enhanced with aggregate functions and ordinary CQ<sup class="ltx_sup" id="S1.p6.1.1">⋆</sup>s over annotated databases. We usually devise algorithms and upper bounds on general commutative semirings (possibly with additional conditions), as positive results carry over to the aggregate formalism, and we prove cases of specific intractable queries with specific aggregate functions over ordinary (non-annotated) databases.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">Our analysis is done in two parts. In Section <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S4" title="4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">4</span></a>, we study the case where the annotation or aggregation is <em class="ltx_emph ltx_font_italic" id="S1.p7.1.1">not</em> a part of the lexicographic order; we show that under reasonable assumptions about the complexity of the semiring operations, the known dichotomy for ordinary databases <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/CarmeliTGKR21</span>]</cite> continues to hold in the presence of annotation (hence, we can efficiently solve the aforementioned first challenge, namely annotation construction). We conclude the analogous tractability frontier for the common aggregate functions (count, sum, min, max, average). A notable exception is the count-distinct aggregation, which cannot be expressed efficiently as a semiring annotation; we show that the class of tractable queries for count-distinct is indeed more restricted, and we establish the precise tractability condition (dichotomy) for this aggregation.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.6">In <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:sec:inside</span>, we study the ability to include the annotation or aggregation in nontrivial positions within the lexicographic order (i.e., the second challenge). We establish a sufficient tractability condition for CQ<sup class="ltx_sup" id="S1.p8.6.1">⋆</sup>s. Moreover, we prove that this tractability condition is also necessary (under standard assumptions in fine-grained complexity theory) for commonly studied semirings, when the CQ<sup class="ltx_sup" id="S1.p8.6.2">⋆</sup> has no self-joins. We also investigate special cases of aggregate CQs where the complexity is not resolved by the dichotomy on CQ<sup class="ltx_sup" id="S1.p8.6.3">⋆</sup>s. Interestingly, the translation to CQ<sup class="ltx_sup" id="S1.p8.6.4">⋆</sup>s of CQs with the aforementioned aggregate functions has a special property: in all relations but one, the annotation is constantly the multiplicative identity (i.e., the “one” element). We refer to annotated databases with this property as <em class="ltx_emph ltx_font_italic" id="S1.p8.6.5">locally annotated</em>. In <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:sec:locally-annotated</span>, we examine the implication of local annotation on the complexity of direct access, that is, whether this property can be utilized to establish tractable queries that are hard in the absence of this property, and to what extent. We answer the first question affirmatively. Moreover, we establish a dichotomy similarly to the previous ones (i.e., the hardness side assumes that there are no self-joins) for the class of full CQ<sup class="ltx_sup" id="S1.p8.6.6">⋆</sup>s. We also study the implication of local annotation in conjunction of another property, namely that the addition operation is <em class="ltx_emph ltx_font_italic" id="S1.p8.6.7">idempotent</em>. This covers the min and max aggregations, as well as count-distinct in the case of a small (logarithmic-size) domain of counted entities. We establish the corresponding dichotomy for the class of CQ<sup class="ltx_sup" id="S1.p8.6.8">⋆</sup>s (with projection).</p> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.1">As closely related work, we note that Keppeler <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:phd/dnb/Keppeler20</span>]</cite> has proposed algorithms for direct access with aggregation, where he considers the class of q-hierarchical CQs (that support efficient updates of the direct-access structure <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/BerkholzKS17</span>]</cite>). Yet, he does not discuss the dependence of tractability on the lexicographic order. Moreover, our positive results (e.g., <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm11" title="Theorem 11. ‣ 4.1. Generalized Dichotomies ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Theorems</span> <span class="ltx_text ltx_ref_tag">11</span></a> and <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:thm:dichotomy-general-annotation</span>) apply to the class of free-connex CQs, which is more general than that of the q-hierarchical CQs.</p> </div> <div class="ltx_para" id="S1.p10"> <p class="ltx_p" id="S1.p10.3">An abridged version of this manuscript appeared in a conference proceedings <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/icdt/EldarCK24</span>]</cite>. Compared to the conference version, this manuscript has significant additions and generalizations. First, we include here the full proofs of all results. Second, we generalized two results into full dichotomy theorems: the lower bound for a specific count-distinct query <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/icdt/EldarCK24</span>, Theorem 8]</cite> is generalized to the dichotomy of <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:thm:countd-dichotomy</span> here, and the tractability condition for annotated databases (with the computed annotation taking part of the order) <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/icdt/EldarCK24</span>, Theorem 12]</cite> is now <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:cor:general-annotations-order-tractability</span> of the new <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:thm:dichotomy-general-annotation</span> here (that extends the tractability criterion and shows its necessity). Third, we reorganized the results and proofs by providing new concepts (e.g., the restriction <math alttext="Q_{|V}" class="ltx_math_unparsed" display="inline" id="S1.p10.1.m1.1"><semantics id="S1.p10.1.m1.1a"><msub id="S1.p10.1.m1.1.1"><mi id="S1.p10.1.m1.1.1.2">Q</mi><mrow id="S1.p10.1.m1.1.1.3"><mo fence="false" id="S1.p10.1.m1.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S1.p10.1.m1.1.1.3.2">V</mi></mrow></msub><annotation encoding="application/x-tex" id="S1.p10.1.m1.1b">Q_{|V}</annotation><annotation encoding="application/x-llamapun" id="S1.p10.1.m1.1c">italic_Q start_POSTSUBSCRIPT | italic_V end_POSTSUBSCRIPT</annotation></semantics></math> of a query <math alttext="Q" class="ltx_Math" display="inline" id="S1.p10.2.m2.1"><semantics id="S1.p10.2.m2.1a"><mi id="S1.p10.2.m2.1.1" xref="S1.p10.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S1.p10.2.m2.1b"><ci id="S1.p10.2.m2.1.1.cmml" xref="S1.p10.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S1.p10.2.m2.1d">italic_Q</annotation></semantics></math> to a subset <math alttext="V" class="ltx_Math" display="inline" id="S1.p10.3.m3.1"><semantics id="S1.p10.3.m3.1a"><mi id="S1.p10.3.m3.1.1" xref="S1.p10.3.m3.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S1.p10.3.m3.1b"><ci id="S1.p10.3.m3.1.1.cmml" xref="S1.p10.3.m3.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.3.m3.1c">V</annotation><annotation encoding="application/x-llamapun" id="S1.p10.3.m3.1d">italic_V</annotation></semantics></math> of its variables) and machinery (e.g., <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm6" title="Lemma 6. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm9" title="Theorem 9. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">9</span></a> that give general reducibility conditions that are used throughout the article and are of interest independently of this work). Fourth, we rephrased the results on locally annotated databases (<span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:sec:locally-annotated</span>) to state the precise tractability condition rather than a procedure for its detection <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/icdt/EldarCK24</span>, Theorem 20]</cite>. In particular, this rephrasing allowed us to add <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:cor:min-max-tractability</span> for the min and max aggregations.</p> </div> <div class="ltx_para" id="S1.p11"> <p class="ltx_p" id="S1.p11.1">The remainder of the manuscript is organized as follows. After preliminary concepts and notation in Section <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S2" title="2. Preliminaries ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">2</span></a>, Section <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S3" title="3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">3</span></a> defines the challenge of direct access with an underlying order and recalls the state of affairs for ordinary CQs over ordinary databases. In <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S4" title="4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">4</span></a>, we analyze queries where the aggregation or annotation does not participate in the lexicographic order. We study the incorporation of the aggregation and annotation in the order in <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:sec:inside</span>, and in <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:sec:locally-annotated</span> we study the implication of local annotation (with a general addition and an idempotent addition) on the complexity of direct access. We conclude in Section <span class="ltx_ref ltx_missing_label ltx_ref_self">LABEL:sec:conclusions</span>.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2. </span>Preliminaries</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">We begin with preliminary notation and terminology that we use throughout the paper.</p> </div> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Databases and conjunctive queries.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p1.15">A <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p1.15.1">schema</em> <math alttext="\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p1.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.1.m1.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.1.m1.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.1.m1.1d">bold_S</annotation></semantics></math> is a finite set <math alttext="\{R_{1},\dots,R_{k}\}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.2.m2.3"><semantics id="S2.SS0.SSS0.Px1.p1.2.m2.3a"><mrow id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.3.cmml"><mo id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.3.cmml">{</mo><msub id="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1" xref="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.2" xref="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.2.cmml">R</mi><mn id="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.3" xref="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.4" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p1.2.m2.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.5" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.2.cmml">R</mi><mi id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.3" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.3.cmml">k</mi></msub><mo id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.2.m2.3b"><set id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2"><apply id="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.2">𝑅</ci><cn id="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p1.2.m2.2.2.1.1.3">1</cn></apply><ci id="S2.SS0.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.1.1">…</ci><apply id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.2">𝑅</ci><ci id="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.2.m2.3.3.2.2.3">𝑘</ci></apply></set></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.2.m2.3c">\{R_{1},\dots,R_{k}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.2.m2.3d">{ italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_R start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }</annotation></semantics></math> of relation symbols. Each relation symbol <math alttext="R" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p1.3.m3.1a"><mi id="S2.SS0.SSS0.Px1.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.3.m3.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.3.m3.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.3.m3.1d">italic_R</annotation></semantics></math> is associated with an arity <math alttext="\texttt{ar}(R)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.p1.4.m4.1a"><mrow id="S2.SS0.SSS0.Px1.p1.4.m4.1.2" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.cmml"><mtext class="ltx_mathvariant_monospace" id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.2" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.2a.cmml">ar</mtext><mo id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.1" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.3.2" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p1.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.4.m4.1b"><apply id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2"><times id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.2a.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.2"><mtext class="ltx_mathvariant_monospace" id="S2.SS0.SSS0.Px1.p1.4.m4.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.2.2">ar</mtext></ci><ci id="S2.SS0.SSS0.Px1.p1.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.4.m4.1.1">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.4.m4.1c">\texttt{ar}(R)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.4.m4.1d">ar ( italic_R )</annotation></semantics></math>, which is a natural number. We assume a countably infinite set <math alttext="\mathsf{Const}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.p1.5.m5.1a"><mi id="S2.SS0.SSS0.Px1.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1.cmml">𝖢𝗈𝗇𝗌𝗍</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.5.m5.1b"><ci id="S2.SS0.SSS0.Px1.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.5.m5.1.1">𝖢𝗈𝗇𝗌𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.5.m5.1c">\mathsf{Const}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.5.m5.1d">sansserif_Const</annotation></semantics></math> of <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p1.15.2">constants</em> that appear as values of databases. A <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p1.15.3">database</em> <math alttext="D" 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">D</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">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.6.m6.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.6.m6.1d">italic_D</annotation></semantics></math> over a schema <math alttext="\mathord{\mathbf{S}}" 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">𝐒</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">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.7.m7.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.7.m7.1d">bold_S</annotation></semantics></math> maps every relation symbol <math alttext="R" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.p1.8.m8.1a"><mi id="S2.SS0.SSS0.Px1.p1.8.m8.1.1" xref="S2.SS0.SSS0.Px1.p1.8.m8.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.8.m8.1b"><ci id="S2.SS0.SSS0.Px1.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.8.m8.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.8.m8.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.8.m8.1d">italic_R</annotation></semantics></math> of <math alttext="\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.p1.9.m9.1a"><mi id="S2.SS0.SSS0.Px1.p1.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p1.9.m9.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.9.m9.1b"><ci id="S2.SS0.SSS0.Px1.p1.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.9.m9.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.9.m9.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.9.m9.1d">bold_S</annotation></semantics></math> to a finite relation <math alttext="R^{D}\subseteq\mathsf{Const}^{\texttt{ar}(R)}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.p1.10.m10.1a"><mrow id="S2.SS0.SSS0.Px1.p1.10.m10.1.2" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.cmml"><msup id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.2" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.2.cmml">R</mi><mi id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.3" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.3.cmml">D</mi></msup><mo id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.1" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.1.cmml">⊆</mo><msup id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3.2" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3.2.cmml">𝖢𝗈𝗇𝗌𝗍</mi><mrow id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.3" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.3a.cmml">ar</mtext><mo id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.4.2" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.4.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.1" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.1.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.4.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.cmml">)</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.10.m10.1b"><apply id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2"><subset id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.1"></subset><apply id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.2">𝑅</ci><ci id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.2.3">𝐷</ci></apply><apply id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.2.3.2">𝖢𝗈𝗇𝗌𝗍</ci><apply id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1"><times id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.2"></times><ci id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.3a.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.3.cmml" mathsize="70%" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.3">ar</mtext></ci><ci id="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.10.m10.1.1.1.1">𝑅</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.10.m10.1c">R^{D}\subseteq\mathsf{Const}^{\texttt{ar}(R)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.10.m10.1d">italic_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT ⊆ sansserif_Const start_POSTSUPERSCRIPT ar ( italic_R ) end_POSTSUPERSCRIPT</annotation></semantics></math>. If <math alttext="(c_{1},\dots,c_{k})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.11.m11.3"><semantics id="S2.SS0.SSS0.Px1.p1.11.m11.3a"><mrow id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.3.cmml"><mo id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1" xref="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.2" xref="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.2.cmml">c</mi><mn id="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.3" xref="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.4" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p1.11.m11.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p1.11.m11.1.1.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.5" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2.cmml">c</mi><mi id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.3" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.3.cmml">k</mi></msub><mo id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.11.m11.3b"><vector id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2"><apply id="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.2">𝑐</ci><cn id="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p1.11.m11.2.2.1.1.3">1</cn></apply><ci id="S2.SS0.SSS0.Px1.p1.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.1.1">…</ci><apply id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.2">𝑐</ci><ci id="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.11.m11.3.3.2.2.3">𝑘</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.11.m11.3c">(c_{1},\dots,c_{k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.11.m11.3d">( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> is a tuple of <math alttext="R^{D}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.12.m12.1"><semantics id="S2.SS0.SSS0.Px1.p1.12.m12.1a"><msup id="S2.SS0.SSS0.Px1.p1.12.m12.1.1" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.12.m12.1.1.2" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1.2.cmml">R</mi><mi id="S2.SS0.SSS0.Px1.p1.12.m12.1.1.3" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1.3.cmml">D</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.12.m12.1b"><apply id="S2.SS0.SSS0.Px1.p1.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.12.m12.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.12.m12.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1.2">𝑅</ci><ci id="S2.SS0.SSS0.Px1.p1.12.m12.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p1.12.m12.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.12.m12.1c">R^{D}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.12.m12.1d">italic_R start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT</annotation></semantics></math> (where <math alttext="k=\texttt{ar}(R)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.13.m13.1"><semantics id="S2.SS0.SSS0.Px1.p1.13.m13.1a"><mrow id="S2.SS0.SSS0.Px1.p1.13.m13.1.2" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.2" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.2.cmml">k</mi><mo id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.1" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.cmml"><mtext class="ltx_mathvariant_monospace" id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.2" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.2a.cmml">ar</mtext><mo id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.1" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.3.2" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p1.13.m13.1.1" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.1.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.13.m13.1b"><apply id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2"><eq id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.1"></eq><ci id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.2">𝑘</ci><apply id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3"><times id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.2a.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.2"><mtext class="ltx_mathvariant_monospace" id="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.2.3.2">ar</mtext></ci><ci id="S2.SS0.SSS0.Px1.p1.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.13.m13.1.1">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.13.m13.1c">k=\texttt{ar}(R)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.13.m13.1d">italic_k = ar ( italic_R )</annotation></semantics></math>), then we call the expression <math alttext="R(c_{1},\dots,c_{k})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p1.14.m14.3"><semantics id="S2.SS0.SSS0.Px1.p1.14.m14.3a"><mrow id="S2.SS0.SSS0.Px1.p1.14.m14.3.3" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.4" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.4.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.3" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.2.cmml">c</mi><mn id="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.4" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p1.14.m14.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p1.14.m14.1.1.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.5" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.2" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.2.cmml">c</mi><mi id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.3" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p1.14.m14.3b"><apply id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3"><times id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.3"></times><ci id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.4.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.4">𝑅</ci><vector id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2"><apply id="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.2">𝑐</ci><cn id="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p1.14.m14.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS0.SSS0.Px1.p1.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.1.1">…</ci><apply id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.2">𝑐</ci><ci id="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p1.14.m14.3.3.2.2.2.3">𝑘</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.14.m14.3c">R(c_{1},\dots,c_{k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.14.m14.3d">italic_R ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> a <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p1.15.4">fact</em> of <math alttext="D" 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">D</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">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p1.15.m15.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p1.15.m15.1d">italic_D</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p2.29">A <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.29.1">Conjunctive Query (CQ)</em> over the schema <math alttext="\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p2.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p2.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.1.m1.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.1.m1.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.1.m1.1d">bold_S</annotation></semantics></math> has the form <math alttext="Q(\vec{x}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{\ell}(\vec{x% },\vec{y})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.2.m2.8"><semantics id="S2.SS0.SSS0.Px1.p2.2.m2.8a"><mrow id="S2.SS0.SSS0.Px1.p2.2.m2.8.8" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.cmml"><mrow id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.cmml"><mi id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.2.cmml">Q</mi><mo 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id="S2.SS0.SSS0.Px1.p2.2.m2.2.2.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.2.2.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.2.m2.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.2.m2.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.3.2.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.2.m2.3.3" xref="S2.SS0.SSS0.Px1.p2.2.m2.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p2.2.m2.3.3.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.3.3.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.2.m2.3.3.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.2.m2.3.3.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.3" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p2.2.m2.6.6" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p2.2.m2.6.6.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.4" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.3.cmml">,</mo><mrow id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.cmml"><msub id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.2.cmml">φ</mi><mi id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.3" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.3.cmml">ℓ</mi></msub><mo id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.1" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.2.m2.4.4" xref="S2.SS0.SSS0.Px1.p2.2.m2.4.4.cmml"><mi id="S2.SS0.SSS0.Px1.p2.2.m2.4.4.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.4.4.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.2.m2.4.4.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.2.m2.4.4.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.2.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.2.m2.5.5" xref="S2.SS0.SSS0.Px1.p2.2.m2.5.5.cmml"><mi id="S2.SS0.SSS0.Px1.p2.2.m2.5.5.2" xref="S2.SS0.SSS0.Px1.p2.2.m2.5.5.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.2.m2.5.5.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.2.m2.5.5.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.2.m2.8b"><apply id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8"><ci id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.3.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.3">:</ci><apply id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4"><times id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.1"></times><ci id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.2">𝑄</ci><apply id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.4.3.2"><ci id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.1.1.2">𝑥</ci></apply></apply><list id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2"><apply id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1"><minus id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1"></minus><apply id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2"><times id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.1"></times><apply id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.2.3">1</cn></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.7.7.1.1.1.2.3.2"><apply id="S2.SS0.SSS0.Px1.p2.2.m2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.2.2"><ci id="S2.SS0.SSS0.Px1.p2.2.m2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.2.m2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.2.2.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p2.2.m2.3.3.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.3.3"><ci id="S2.SS0.SSS0.Px1.p2.2.m2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.3.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.2.m2.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.3.3.2">𝑦</ci></apply></interval></apply></apply><ci id="S2.SS0.SSS0.Px1.p2.2.m2.6.6.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.6.6">…</ci><apply id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2"><times id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.1"></times><apply id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.2.3">ℓ</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.8.8.2.2.2.3.2"><apply id="S2.SS0.SSS0.Px1.p2.2.m2.4.4.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.4.4"><ci id="S2.SS0.SSS0.Px1.p2.2.m2.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.4.4.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.2.m2.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.4.4.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p2.2.m2.5.5.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.5.5"><ci id="S2.SS0.SSS0.Px1.p2.2.m2.5.5.1.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.5.5.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.2.m2.5.5.2.cmml" xref="S2.SS0.SSS0.Px1.p2.2.m2.5.5.2">𝑦</ci></apply></interval></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.2.m2.8c">Q(\vec{x}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{\ell}(\vec{x% },\vec{y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.2.m2.8d">italic_Q ( over→ start_ARG italic_x end_ARG ) : - italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) , … , italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math> where <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p2.3.m3.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p2.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1"><ci id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.3.m3.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.3.m3.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.3.m3.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> and <math alttext="\vec{y}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.p2.4.m4.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p2.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p2.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.p2.4.m4.1.1.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.4.m4.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.4.m4.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.4.m4.1b"><apply id="S2.SS0.SSS0.Px1.p2.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.4.m4.1.1"><ci id="S2.SS0.SSS0.Px1.p2.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.4.m4.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.4.m4.1.1.2">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.4.m4.1c">\vec{y}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.4.m4.1d">over→ start_ARG italic_y end_ARG</annotation></semantics></math> are disjoint sequences of variables, and each <math alttext="\varphi_{i}(\vec{x},\vec{y})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.5.m5.2"><semantics id="S2.SS0.SSS0.Px1.p2.5.m5.2a"><mrow id="S2.SS0.SSS0.Px1.p2.5.m5.2.3" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.cmml"><msub id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.2" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.2.cmml">φ</mi><mi id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.3" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.1" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.2" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p2.5.m5.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.5.m5.1.1.2" xref="S2.SS0.SSS0.Px1.p2.5.m5.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.5.m5.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.5.m5.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.5.m5.2.2" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.5.m5.2.2.2" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.2.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.5.m5.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.5.m5.2b"><apply id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3"><times id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.1"></times><apply id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.2.3">𝑖</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.3.3.2"><apply id="S2.SS0.SSS0.Px1.p2.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.1.1"><ci id="S2.SS0.SSS0.Px1.p2.5.m5.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.1.1.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p2.5.m5.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.2"><ci id="S2.SS0.SSS0.Px1.p2.5.m5.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.5.m5.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.5.m5.2.2.2">𝑦</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.5.m5.2c">\varphi_{i}(\vec{x},\vec{y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.5.m5.2d">italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math> is an <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.29.2">atomic query</em> <math alttext="R(z_{1},\dots,z_{k})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.6.m6.3"><semantics id="S2.SS0.SSS0.Px1.p2.6.m6.3a"><mrow id="S2.SS0.SSS0.Px1.p2.6.m6.3.3" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.4" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.4.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.3" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.2.cmml">z</mi><mn id="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.4" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p2.6.m6.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p2.6.m6.1.1.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.5" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.2" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.2.cmml">z</mi><mi id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.3" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.6.m6.3b"><apply id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3"><times id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.3"></times><ci id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.4.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.4">𝑅</ci><vector id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2"><apply id="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.2">𝑧</ci><cn id="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p2.6.m6.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS0.SSS0.Px1.p2.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.1.1">…</ci><apply id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.2">𝑧</ci><ci id="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.6.m6.3.3.2.2.2.3">𝑘</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.6.m6.3c">R(z_{1},\dots,z_{k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.6.m6.3d">italic_R ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> such that <math alttext="R\in\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.p2.7.m7.1a"><mrow id="S2.SS0.SSS0.Px1.p2.7.m7.1.1" xref="S2.SS0.SSS0.Px1.p2.7.m7.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.7.m7.1.1.2" xref="S2.SS0.SSS0.Px1.p2.7.m7.1.1.2.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p2.7.m7.1.1.1" xref="S2.SS0.SSS0.Px1.p2.7.m7.1.1.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.p2.7.m7.1.1.3" xref="S2.SS0.SSS0.Px1.p2.7.m7.1.1.3.cmml">𝐒</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.7.m7.1b"><apply id="S2.SS0.SSS0.Px1.p2.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.7.m7.1.1"><in id="S2.SS0.SSS0.Px1.p2.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.7.m7.1.1.1"></in><ci id="S2.SS0.SSS0.Px1.p2.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.7.m7.1.1.2">𝑅</ci><ci id="S2.SS0.SSS0.Px1.p2.7.m7.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p2.7.m7.1.1.3">𝐒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.7.m7.1c">R\in\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.7.m7.1d">italic_R ∈ bold_S</annotation></semantics></math> with <math alttext="\texttt{ar}(R)=k" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.p2.8.m8.1a"><mrow id="S2.SS0.SSS0.Px1.p2.8.m8.1.2" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.cmml"><mrow id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.cmml"><mtext class="ltx_mathvariant_monospace" id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.2" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.2a.cmml">ar</mtext><mo id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.1" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.3.2" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.cmml"><mo id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p2.8.m8.1.1" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.1.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.1" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.1.cmml">=</mo><mi id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.3" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.8.m8.1b"><apply id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2"><eq id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.1"></eq><apply id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2"><times id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.1"></times><ci id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.2a.cmml" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.2"><mtext class="ltx_mathvariant_monospace" id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.2.2">ar</mtext></ci><ci id="S2.SS0.SSS0.Px1.p2.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.1">𝑅</ci></apply><ci id="S2.SS0.SSS0.Px1.p2.8.m8.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.8.m8.1.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.8.m8.1c">\texttt{ar}(R)=k</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.8.m8.1d">ar ( italic_R ) = italic_k</annotation></semantics></math> and each <math alttext="z_{i}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.p2.9.m9.1a"><msub id="S2.SS0.SSS0.Px1.p2.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p2.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.9.m9.1.1.2" xref="S2.SS0.SSS0.Px1.p2.9.m9.1.1.2.cmml">z</mi><mi id="S2.SS0.SSS0.Px1.p2.9.m9.1.1.3" xref="S2.SS0.SSS0.Px1.p2.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.9.m9.1b"><apply id="S2.SS0.SSS0.Px1.p2.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.9.m9.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.9.m9.1.1.2">𝑧</ci><ci id="S2.SS0.SSS0.Px1.p2.9.m9.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p2.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.9.m9.1c">z_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.9.m9.1d">italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a variable in <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.p2.10.m10.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p2.10.m10.1.1" xref="S2.SS0.SSS0.Px1.p2.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.10.m10.1.1.2" xref="S2.SS0.SSS0.Px1.p2.10.m10.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.10.m10.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.10.m10.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.10.m10.1b"><apply id="S2.SS0.SSS0.Px1.p2.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.10.m10.1.1"><ci id="S2.SS0.SSS0.Px1.p2.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.10.m10.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.10.m10.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.10.m10.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.10.m10.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> or <math alttext="\vec{y}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.p2.11.m11.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p2.11.m11.1.1" xref="S2.SS0.SSS0.Px1.p2.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.11.m11.1.1.2" xref="S2.SS0.SSS0.Px1.p2.11.m11.1.1.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.11.m11.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.11.m11.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.11.m11.1b"><apply id="S2.SS0.SSS0.Px1.p2.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.11.m11.1.1"><ci id="S2.SS0.SSS0.Px1.p2.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.11.m11.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.11.m11.1.1.2">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.11.m11.1c">\vec{y}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.11.m11.1d">over→ start_ARG italic_y end_ARG</annotation></semantics></math>. Each <math alttext="\varphi_{i}(\vec{x},\vec{y})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.12.m12.2"><semantics id="S2.SS0.SSS0.Px1.p2.12.m12.2a"><mrow id="S2.SS0.SSS0.Px1.p2.12.m12.2.3" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.cmml"><msub id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.2" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.2.cmml">φ</mi><mi id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.3" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.1" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.2" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.12.m12.1.1" xref="S2.SS0.SSS0.Px1.p2.12.m12.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.12.m12.1.1.2" xref="S2.SS0.SSS0.Px1.p2.12.m12.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.12.m12.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.12.m12.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.12.m12.2.2" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.12.m12.2.2.2" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.2.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.12.m12.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.12.m12.2b"><apply id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3"><times id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.1"></times><apply id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.2.3">𝑖</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.3.3.2"><apply id="S2.SS0.SSS0.Px1.p2.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.1.1"><ci id="S2.SS0.SSS0.Px1.p2.12.m12.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.12.m12.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.1.1.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p2.12.m12.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.2"><ci id="S2.SS0.SSS0.Px1.p2.12.m12.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.12.m12.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.12.m12.2.2.2">𝑦</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.12.m12.2c">\varphi_{i}(\vec{x},\vec{y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.12.m12.2d">italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math> is an <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.29.3">atom</em> of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.13.m13.1"><semantics id="S2.SS0.SSS0.Px1.p2.13.m13.1a"><mi id="S2.SS0.SSS0.Px1.p2.13.m13.1.1" xref="S2.SS0.SSS0.Px1.p2.13.m13.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.13.m13.1b"><ci id="S2.SS0.SSS0.Px1.p2.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.13.m13.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.13.m13.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.13.m13.1d">italic_Q</annotation></semantics></math>, and we denote by <math alttext="\mathrm{atoms}(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.14.m14.1"><semantics id="S2.SS0.SSS0.Px1.p2.14.m14.1a"><mrow id="S2.SS0.SSS0.Px1.p2.14.m14.1.2" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.14.m14.1.2.2" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2.2.cmml">atoms</mi><mo id="S2.SS0.SSS0.Px1.p2.14.m14.1.2.1" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.14.m14.1.2.3.2" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p2.14.m14.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p2.14.m14.1.1" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p2.14.m14.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.14.m14.1b"><apply id="S2.SS0.SSS0.Px1.p2.14.m14.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2"><times id="S2.SS0.SSS0.Px1.p2.14.m14.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p2.14.m14.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.2.2">atoms</ci><ci id="S2.SS0.SSS0.Px1.p2.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.14.m14.1.1">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.14.m14.1c">\mathrm{atoms}(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.14.m14.1d">roman_atoms ( italic_Q )</annotation></semantics></math> the set of atoms of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.15.m15.1"><semantics id="S2.SS0.SSS0.Px1.p2.15.m15.1a"><mi id="S2.SS0.SSS0.Px1.p2.15.m15.1.1" xref="S2.SS0.SSS0.Px1.p2.15.m15.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.15.m15.1b"><ci id="S2.SS0.SSS0.Px1.p2.15.m15.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.15.m15.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.15.m15.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.15.m15.1d">italic_Q</annotation></semantics></math>. We call <math alttext="Q(\vec{x})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.16.m16.1"><semantics id="S2.SS0.SSS0.Px1.p2.16.m16.1a"><mrow id="S2.SS0.SSS0.Px1.p2.16.m16.1.2" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.16.m16.1.2.2" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.2.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p2.16.m16.1.2.1" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.16.m16.1.2.3.2" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p2.16.m16.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.16.m16.1.1" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.16.m16.1.1.2" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.16.m16.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.16.m16.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.16.m16.1b"><apply id="S2.SS0.SSS0.Px1.p2.16.m16.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.2"><times id="S2.SS0.SSS0.Px1.p2.16.m16.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p2.16.m16.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.2.2">𝑄</ci><apply id="S2.SS0.SSS0.Px1.p2.16.m16.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.p2.16.m16.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.16.m16.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.16.m16.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.16.m16.1c">Q(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.16.m16.1d">italic_Q ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math> the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.29.4">head</em> of the query and <math alttext="\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{\ell}(\vec{x},\vec{y})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.17.m17.7"><semantics id="S2.SS0.SSS0.Px1.p2.17.m17.7a"><mrow id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.3.cmml"><mrow id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.cmml"><msub id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.3" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.1" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.17.m17.1.1" xref="S2.SS0.SSS0.Px1.p2.17.m17.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.17.m17.1.1.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.17.m17.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.17.m17.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.2.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.17.m17.2.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.17.m17.2.2.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.2.2.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.17.m17.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.17.m17.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.3" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p2.17.m17.5.5" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p2.17.m17.5.5.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.4" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.3.cmml">,</mo><mrow id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.cmml"><msub id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.2.cmml">φ</mi><mi id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.3" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.3.cmml">ℓ</mi></msub><mo id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.1" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.17.m17.3.3" xref="S2.SS0.SSS0.Px1.p2.17.m17.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p2.17.m17.3.3.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.3.3.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.17.m17.3.3.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.17.m17.3.3.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.2.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p2.17.m17.4.4" xref="S2.SS0.SSS0.Px1.p2.17.m17.4.4.cmml"><mi id="S2.SS0.SSS0.Px1.p2.17.m17.4.4.2" xref="S2.SS0.SSS0.Px1.p2.17.m17.4.4.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.17.m17.4.4.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.17.m17.4.4.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.17.m17.7b"><list id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.3.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2"><apply id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1"><times id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.1"></times><apply id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.2.3">1</cn></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.6.6.1.1.3.2"><apply id="S2.SS0.SSS0.Px1.p2.17.m17.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.1.1"><ci id="S2.SS0.SSS0.Px1.p2.17.m17.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.17.m17.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.1.1.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p2.17.m17.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.2.2"><ci id="S2.SS0.SSS0.Px1.p2.17.m17.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.17.m17.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.2.2.2">𝑦</ci></apply></interval></apply><ci id="S2.SS0.SSS0.Px1.p2.17.m17.5.5.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.5.5">…</ci><apply id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2"><times id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.1"></times><apply id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.2.3">ℓ</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.7.7.2.2.3.2"><apply id="S2.SS0.SSS0.Px1.p2.17.m17.3.3.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.3.3"><ci id="S2.SS0.SSS0.Px1.p2.17.m17.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.3.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.17.m17.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.3.3.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p2.17.m17.4.4.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.4.4"><ci id="S2.SS0.SSS0.Px1.p2.17.m17.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.4.4.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.17.m17.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.p2.17.m17.4.4.2">𝑦</ci></apply></interval></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.17.m17.7c">\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{\ell}(\vec{x},\vec{y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.17.m17.7d">italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) , … , italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math> the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.29.5">body</em> of the query. The variables of <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.18.m18.1"><semantics id="S2.SS0.SSS0.Px1.p2.18.m18.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p2.18.m18.1.1" xref="S2.SS0.SSS0.Px1.p2.18.m18.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.18.m18.1.1.2" xref="S2.SS0.SSS0.Px1.p2.18.m18.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p2.18.m18.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.18.m18.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.18.m18.1b"><apply id="S2.SS0.SSS0.Px1.p2.18.m18.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.18.m18.1.1"><ci id="S2.SS0.SSS0.Px1.p2.18.m18.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.18.m18.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.18.m18.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.18.m18.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.18.m18.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.18.m18.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> are the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.29.6">free</em> variables of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.19.m19.1"><semantics id="S2.SS0.SSS0.Px1.p2.19.m19.1a"><mi id="S2.SS0.SSS0.Px1.p2.19.m19.1.1" xref="S2.SS0.SSS0.Px1.p2.19.m19.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.19.m19.1b"><ci id="S2.SS0.SSS0.Px1.p2.19.m19.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.19.m19.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.19.m19.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.19.m19.1d">italic_Q</annotation></semantics></math>, and those of <math alttext="\vec{y}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.20.m20.1"><semantics id="S2.SS0.SSS0.Px1.p2.20.m20.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p2.20.m20.1.1" xref="S2.SS0.SSS0.Px1.p2.20.m20.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p2.20.m20.1.1.2" xref="S2.SS0.SSS0.Px1.p2.20.m20.1.1.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p2.20.m20.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.20.m20.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.20.m20.1b"><apply id="S2.SS0.SSS0.Px1.p2.20.m20.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.20.m20.1.1"><ci id="S2.SS0.SSS0.Px1.p2.20.m20.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.20.m20.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p2.20.m20.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.20.m20.1.1.2">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.20.m20.1c">\vec{y}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.20.m20.1d">over→ start_ARG italic_y end_ARG</annotation></semantics></math> are the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.29.7">existential</em> variables of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.21.m21.1"><semantics id="S2.SS0.SSS0.Px1.p2.21.m21.1a"><mi id="S2.SS0.SSS0.Px1.p2.21.m21.1.1" xref="S2.SS0.SSS0.Px1.p2.21.m21.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.21.m21.1b"><ci id="S2.SS0.SSS0.Px1.p2.21.m21.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.21.m21.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.21.m21.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.21.m21.1d">italic_Q</annotation></semantics></math>, and every variable occurs at least once in the body. We use <math alttext="\mathrm{vars}(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.22.m22.1"><semantics id="S2.SS0.SSS0.Px1.p2.22.m22.1a"><mrow id="S2.SS0.SSS0.Px1.p2.22.m22.1.2" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.22.m22.1.2.2" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2.2.cmml">vars</mi><mo id="S2.SS0.SSS0.Px1.p2.22.m22.1.2.1" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.22.m22.1.2.3.2" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p2.22.m22.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p2.22.m22.1.1" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p2.22.m22.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.22.m22.1b"><apply id="S2.SS0.SSS0.Px1.p2.22.m22.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2"><times id="S2.SS0.SSS0.Px1.p2.22.m22.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p2.22.m22.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.2.2">vars</ci><ci id="S2.SS0.SSS0.Px1.p2.22.m22.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.22.m22.1.1">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.22.m22.1c">\mathrm{vars}(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.22.m22.1d">roman_vars ( italic_Q )</annotation></semantics></math> and <math alttext="\mathrm{free}(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.23.m23.1"><semantics id="S2.SS0.SSS0.Px1.p2.23.m23.1a"><mrow id="S2.SS0.SSS0.Px1.p2.23.m23.1.2" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.23.m23.1.2.2" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2.2.cmml">free</mi><mo id="S2.SS0.SSS0.Px1.p2.23.m23.1.2.1" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.23.m23.1.2.3.2" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p2.23.m23.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p2.23.m23.1.1" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p2.23.m23.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.23.m23.1b"><apply id="S2.SS0.SSS0.Px1.p2.23.m23.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2"><times id="S2.SS0.SSS0.Px1.p2.23.m23.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p2.23.m23.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.2.2">free</ci><ci id="S2.SS0.SSS0.Px1.p2.23.m23.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.23.m23.1.1">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.23.m23.1c">\mathrm{free}(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.23.m23.1d">roman_free ( italic_Q )</annotation></semantics></math> to denote the set of all variables and all free variables of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.24.m24.1"><semantics id="S2.SS0.SSS0.Px1.p2.24.m24.1a"><mi id="S2.SS0.SSS0.Px1.p2.24.m24.1.1" xref="S2.SS0.SSS0.Px1.p2.24.m24.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.24.m24.1b"><ci id="S2.SS0.SSS0.Px1.p2.24.m24.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.24.m24.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.24.m24.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.24.m24.1d">italic_Q</annotation></semantics></math>, respectively. If <math alttext="\varphi\in\mathrm{atoms}(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.25.m25.1"><semantics id="S2.SS0.SSS0.Px1.p2.25.m25.1a"><mrow id="S2.SS0.SSS0.Px1.p2.25.m25.1.2" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.2" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.2.cmml">φ</mi><mo id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.1" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.1.cmml">∈</mo><mrow id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.2" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.2.cmml">atoms</mi><mo id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.1" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.3.2" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p2.25.m25.1.1" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.25.m25.1b"><apply id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2"><in id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.1"></in><ci id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.2">𝜑</ci><apply id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3"><times id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.2.3.2">atoms</ci><ci id="S2.SS0.SSS0.Px1.p2.25.m25.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.25.m25.1.1">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.25.m25.1c">\varphi\in\mathrm{atoms}(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.25.m25.1d">italic_φ ∈ roman_atoms ( italic_Q )</annotation></semantics></math>, then <math alttext="\mathrm{vars}(\varphi)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.26.m26.1"><semantics id="S2.SS0.SSS0.Px1.p2.26.m26.1a"><mrow id="S2.SS0.SSS0.Px1.p2.26.m26.1.2" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.26.m26.1.2.2" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2.2.cmml">vars</mi><mo id="S2.SS0.SSS0.Px1.p2.26.m26.1.2.1" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.26.m26.1.2.3.2" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p2.26.m26.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p2.26.m26.1.1" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.1.cmml">φ</mi><mo id="S2.SS0.SSS0.Px1.p2.26.m26.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.26.m26.1b"><apply id="S2.SS0.SSS0.Px1.p2.26.m26.1.2.cmml" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2"><times id="S2.SS0.SSS0.Px1.p2.26.m26.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p2.26.m26.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.2.2">vars</ci><ci id="S2.SS0.SSS0.Px1.p2.26.m26.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.26.m26.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.26.m26.1c">\mathrm{vars}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.26.m26.1d">roman_vars ( italic_φ )</annotation></semantics></math> is the set of variables in <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.27.m27.1"><semantics id="S2.SS0.SSS0.Px1.p2.27.m27.1a"><mi id="S2.SS0.SSS0.Px1.p2.27.m27.1.1" xref="S2.SS0.SSS0.Px1.p2.27.m27.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.27.m27.1b"><ci id="S2.SS0.SSS0.Px1.p2.27.m27.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.27.m27.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.27.m27.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.27.m27.1d">italic_φ</annotation></semantics></math>. We say that <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.28.m28.1"><semantics id="S2.SS0.SSS0.Px1.p2.28.m28.1a"><mi id="S2.SS0.SSS0.Px1.p2.28.m28.1.1" xref="S2.SS0.SSS0.Px1.p2.28.m28.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.28.m28.1b"><ci id="S2.SS0.SSS0.Px1.p2.28.m28.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.28.m28.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.28.m28.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.28.m28.1d">italic_Q</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p2.29.8">full</em> if it has no existential variables, that is <math alttext="\mathrm{vars}(Q)=\mathrm{free}(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p2.29.m29.2"><semantics id="S2.SS0.SSS0.Px1.p2.29.m29.2a"><mrow id="S2.SS0.SSS0.Px1.p2.29.m29.2.3" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.cmml"><mrow id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.cmml"><mi id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.2" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.2.cmml">vars</mi><mo id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.1" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.3.2" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.cmml"><mo id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p2.29.m29.1.1" xref="S2.SS0.SSS0.Px1.p2.29.m29.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.1" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.2" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.2.cmml">free</mi><mo id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.1" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.3.2" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.cmml"><mo id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p2.29.m29.2.2" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p2.29.m29.2b"><apply id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3"><eq id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.1"></eq><apply id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2"><times id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.1"></times><ci id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.2.2">vars</ci><ci id="S2.SS0.SSS0.Px1.p2.29.m29.1.1.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.1.1">𝑄</ci></apply><apply id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3"><times id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.1"></times><ci id="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.3.3.2">free</ci><ci id="S2.SS0.SSS0.Px1.p2.29.m29.2.2.cmml" xref="S2.SS0.SSS0.Px1.p2.29.m29.2.2">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p2.29.m29.2c">\mathrm{vars}(Q)=\mathrm{free}(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p2.29.m29.2d">roman_vars ( italic_Q ) = roman_free ( italic_Q )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p3"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p3.33">We refer to a database <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p3.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p3.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p3.1.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p3.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.1.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.1.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.1.m1.1d">italic_D</annotation></semantics></math> over the schema <math alttext="\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.p3.2.m2.1a"><mi id="S2.SS0.SSS0.Px1.p3.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p3.2.m2.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.2.m2.1b"><ci id="S2.SS0.SSS0.Px1.p3.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.2.m2.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.2.m2.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.2.m2.1d">bold_S</annotation></semantics></math> of the CQ <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p3.3.m3.1a"><mi id="S2.SS0.SSS0.Px1.p3.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p3.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.p3.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.3.m3.1d">italic_Q</annotation></semantics></math> as a <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p3.4.1">database over <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.4.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p3.4.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p3.4.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p3.4.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.4.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p3.4.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.4.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.4.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.4.1.m1.1d">italic_Q</annotation></semantics></math></em>. A <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p3.33.2">homomorphism</em> from a CQ <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.5.m4.1"><semantics id="S2.SS0.SSS0.Px1.p3.5.m4.1a"><mi id="S2.SS0.SSS0.Px1.p3.5.m4.1.1" xref="S2.SS0.SSS0.Px1.p3.5.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.5.m4.1b"><ci id="S2.SS0.SSS0.Px1.p3.5.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.5.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.5.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.5.m4.1d">italic_Q</annotation></semantics></math> to a database <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.6.m5.1"><semantics id="S2.SS0.SSS0.Px1.p3.6.m5.1a"><mi id="S2.SS0.SSS0.Px1.p3.6.m5.1.1" xref="S2.SS0.SSS0.Px1.p3.6.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.6.m5.1b"><ci id="S2.SS0.SSS0.Px1.p3.6.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.6.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.6.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.6.m5.1d">italic_D</annotation></semantics></math> over <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.7.m6.1"><semantics id="S2.SS0.SSS0.Px1.p3.7.m6.1a"><mi id="S2.SS0.SSS0.Px1.p3.7.m6.1.1" xref="S2.SS0.SSS0.Px1.p3.7.m6.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.7.m6.1b"><ci id="S2.SS0.SSS0.Px1.p3.7.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.7.m6.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.7.m6.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.7.m6.1d">italic_Q</annotation></semantics></math> is a mapping <math alttext="h" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.8.m7.1"><semantics id="S2.SS0.SSS0.Px1.p3.8.m7.1a"><mi id="S2.SS0.SSS0.Px1.p3.8.m7.1.1" xref="S2.SS0.SSS0.Px1.p3.8.m7.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.8.m7.1b"><ci id="S2.SS0.SSS0.Px1.p3.8.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.8.m7.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.8.m7.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.8.m7.1d">italic_h</annotation></semantics></math> from the variables of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.9.m8.1"><semantics id="S2.SS0.SSS0.Px1.p3.9.m8.1a"><mi id="S2.SS0.SSS0.Px1.p3.9.m8.1.1" xref="S2.SS0.SSS0.Px1.p3.9.m8.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.9.m8.1b"><ci id="S2.SS0.SSS0.Px1.p3.9.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.9.m8.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.9.m8.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.9.m8.1d">italic_Q</annotation></semantics></math> into values of <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.10.m9.1"><semantics id="S2.SS0.SSS0.Px1.p3.10.m9.1a"><mi id="S2.SS0.SSS0.Px1.p3.10.m9.1.1" xref="S2.SS0.SSS0.Px1.p3.10.m9.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.10.m9.1b"><ci id="S2.SS0.SSS0.Px1.p3.10.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.10.m9.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.10.m9.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.10.m9.1d">italic_D</annotation></semantics></math> such that for each atom <math alttext="R(z_{1},\dots,z_{k})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.11.m10.3"><semantics id="S2.SS0.SSS0.Px1.p3.11.m10.3a"><mrow id="S2.SS0.SSS0.Px1.p3.11.m10.3.3" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.4" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.4.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.3" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.2.cmml">z</mi><mn id="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.4" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p3.11.m10.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p3.11.m10.1.1.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.5" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.2" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.2.cmml">z</mi><mi id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.3" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.3.cmml">k</mi></msub><mo id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.11.m10.3b"><apply id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3"><times id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.3"></times><ci id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.4.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.4">𝑅</ci><vector id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2"><apply id="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.2">𝑧</ci><cn id="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p3.11.m10.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS0.SSS0.Px1.p3.11.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.1.1">…</ci><apply id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.2">𝑧</ci><ci id="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.11.m10.3.3.2.2.2.3">𝑘</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.11.m10.3c">R(z_{1},\dots,z_{k})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.11.m10.3d">italic_R ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.12.m11.1"><semantics id="S2.SS0.SSS0.Px1.p3.12.m11.1a"><mi id="S2.SS0.SSS0.Px1.p3.12.m11.1.1" xref="S2.SS0.SSS0.Px1.p3.12.m11.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.12.m11.1b"><ci id="S2.SS0.SSS0.Px1.p3.12.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.12.m11.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.12.m11.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.12.m11.1d">italic_Q</annotation></semantics></math> it holds that <math alttext="R(h(z_{1}),\dots,h(z_{k}))" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.13.m12.3"><semantics id="S2.SS0.SSS0.Px1.p3.13.m12.3a"><mrow id="S2.SS0.SSS0.Px1.p3.13.m12.3.3" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.4" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.4.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.3" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.3.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.3.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.2.cmml">z</mi><mn id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.4" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p3.13.m12.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px1.p3.13.m12.1.1.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.5" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.3.cmml">,</mo><mrow id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.3" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.3.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.2" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.2.cmml">z</mi><mi id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.3.cmml">k</mi></msub><mo id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.13.m12.3b"><apply id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3"><times id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.3"></times><ci id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.4.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.4">𝑅</ci><vector id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2"><apply id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1"><times id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.2"></times><ci id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.3">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.2">𝑧</ci><cn id="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p3.13.m12.2.2.1.1.1.1.1.1.3">1</cn></apply></apply><ci id="S2.SS0.SSS0.Px1.p3.13.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.1.1">…</ci><apply id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2"><times id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.2"></times><ci id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.3">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.2">𝑧</ci><ci id="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p3.13.m12.3.3.2.2.2.1.1.1.3">𝑘</ci></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.13.m12.3c">R(h(z_{1}),\dots,h(z_{k}))</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.13.m12.3d">italic_R ( italic_h ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , … , italic_h ( italic_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math> is a fact of <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.14.m13.1"><semantics id="S2.SS0.SSS0.Px1.p3.14.m13.1a"><mi id="S2.SS0.SSS0.Px1.p3.14.m13.1.1" xref="S2.SS0.SSS0.Px1.p3.14.m13.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.14.m13.1b"><ci id="S2.SS0.SSS0.Px1.p3.14.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.14.m13.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.14.m13.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.14.m13.1d">italic_D</annotation></semantics></math>. We denote by <math alttext="\mathsf{Hom}(Q,D)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.15.m14.2"><semantics id="S2.SS0.SSS0.Px1.p3.15.m14.2a"><mrow id="S2.SS0.SSS0.Px1.p3.15.m14.2.3" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.2" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.2.cmml">𝖧𝗈𝗆</mi><mo id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.1" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.2" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p3.15.m14.1.1" xref="S2.SS0.SSS0.Px1.p3.15.m14.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p3.15.m14.2.2" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.2.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.15.m14.2b"><apply id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3"><times id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.2">𝖧𝗈𝗆</ci><interval closure="open" id="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.3.3.2"><ci id="S2.SS0.SSS0.Px1.p3.15.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.15.m14.1.1">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p3.15.m14.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.15.m14.2.2">𝐷</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.15.m14.2c">\mathsf{Hom}(Q,D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.15.m14.2d">sansserif_Hom ( italic_Q , italic_D )</annotation></semantics></math> the set of all homomorphisms from <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.16.m15.1"><semantics id="S2.SS0.SSS0.Px1.p3.16.m15.1a"><mi id="S2.SS0.SSS0.Px1.p3.16.m15.1.1" xref="S2.SS0.SSS0.Px1.p3.16.m15.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.16.m15.1b"><ci id="S2.SS0.SSS0.Px1.p3.16.m15.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.16.m15.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.16.m15.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.16.m15.1d">italic_Q</annotation></semantics></math> to <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.17.m16.1"><semantics id="S2.SS0.SSS0.Px1.p3.17.m16.1a"><mi id="S2.SS0.SSS0.Px1.p3.17.m16.1.1" xref="S2.SS0.SSS0.Px1.p3.17.m16.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.17.m16.1b"><ci id="S2.SS0.SSS0.Px1.p3.17.m16.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.17.m16.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.17.m16.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.17.m16.1d">italic_D</annotation></semantics></math>. If <math alttext="h\in\mathsf{Hom}(Q,D)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.18.m17.2"><semantics id="S2.SS0.SSS0.Px1.p3.18.m17.2a"><mrow id="S2.SS0.SSS0.Px1.p3.18.m17.2.3" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.2" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.1" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.1.cmml">∈</mo><mrow id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.2" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.2.cmml">𝖧𝗈𝗆</mi><mo id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.1" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.2" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p3.18.m17.1.1" xref="S2.SS0.SSS0.Px1.p3.18.m17.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.2.2" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p3.18.m17.2.2" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.2.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.18.m17.2b"><apply id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3"><in id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.1"></in><ci id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3"><times id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.1"></times><ci id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.2">𝖧𝗈𝗆</ci><interval closure="open" id="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.3.3.3.2"><ci id="S2.SS0.SSS0.Px1.p3.18.m17.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.1.1">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p3.18.m17.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.18.m17.2.2">𝐷</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.18.m17.2c">h\in\mathsf{Hom}(Q,D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.18.m17.2d">italic_h ∈ sansserif_Hom ( italic_Q , italic_D )</annotation></semantics></math> then we denote by <math alttext="h(\vec{x})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.19.m18.1"><semantics id="S2.SS0.SSS0.Px1.p3.19.m18.1a"><mrow id="S2.SS0.SSS0.Px1.p3.19.m18.1.2" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.19.m18.1.2.2" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.2.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.19.m18.1.2.1" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.19.m18.1.2.3.2" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.19.m18.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p3.19.m18.1.1" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.19.m18.1.1.2" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p3.19.m18.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p3.19.m18.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.19.m18.1b"><apply id="S2.SS0.SSS0.Px1.p3.19.m18.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.2"><times id="S2.SS0.SSS0.Px1.p3.19.m18.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p3.19.m18.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.2.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.19.m18.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.p3.19.m18.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p3.19.m18.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.19.m18.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.19.m18.1c">h(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.19.m18.1d">italic_h ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math> the tuple obtained by replacing every variable <math alttext="x" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.20.m19.1"><semantics id="S2.SS0.SSS0.Px1.p3.20.m19.1a"><mi id="S2.SS0.SSS0.Px1.p3.20.m19.1.1" xref="S2.SS0.SSS0.Px1.p3.20.m19.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.20.m19.1b"><ci id="S2.SS0.SSS0.Px1.p3.20.m19.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.20.m19.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.20.m19.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.20.m19.1d">italic_x</annotation></semantics></math> with the constant <math alttext="h(x)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.21.m20.1"><semantics id="S2.SS0.SSS0.Px1.p3.21.m20.1a"><mrow id="S2.SS0.SSS0.Px1.p3.21.m20.1.2" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.21.m20.1.2.2" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.21.m20.1.2.1" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.21.m20.1.2.3.2" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p3.21.m20.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p3.21.m20.1.1" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.1.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p3.21.m20.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.21.m20.1b"><apply id="S2.SS0.SSS0.Px1.p3.21.m20.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2"><times id="S2.SS0.SSS0.Px1.p3.21.m20.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p3.21.m20.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.2.2">ℎ</ci><ci id="S2.SS0.SSS0.Px1.p3.21.m20.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.21.m20.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.21.m20.1c">h(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.21.m20.1d">italic_h ( italic_x )</annotation></semantics></math>, and we denote by <math alttext="h(\varphi_{i}(\vec{x},\vec{y}))" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.22.m21.3"><semantics id="S2.SS0.SSS0.Px1.p3.22.m21.3a"><mrow id="S2.SS0.SSS0.Px1.p3.22.m21.3.3" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.3" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.3.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.2" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.cmml"><msub id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.2" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.2.cmml">φ</mi><mi id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.3" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.1" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.2" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p3.22.m21.1.1" xref="S2.SS0.SSS0.Px1.p3.22.m21.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.22.m21.1.1.2" xref="S2.SS0.SSS0.Px1.p3.22.m21.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p3.22.m21.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.22.m21.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.2.2" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p3.22.m21.2.2" xref="S2.SS0.SSS0.Px1.p3.22.m21.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.22.m21.2.2.2" xref="S2.SS0.SSS0.Px1.p3.22.m21.2.2.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p3.22.m21.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.22.m21.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.22.m21.3b"><apply id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3"><times id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.2"></times><ci id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.3">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1"><times id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.1"></times><apply id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.2.3">𝑖</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.3.3.1.1.1.3.2"><apply id="S2.SS0.SSS0.Px1.p3.22.m21.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.1.1"><ci id="S2.SS0.SSS0.Px1.p3.22.m21.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p3.22.m21.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.1.1.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p3.22.m21.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.2.2"><ci id="S2.SS0.SSS0.Px1.p3.22.m21.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.p3.22.m21.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.22.m21.2.2.2">𝑦</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.22.m21.3c">h(\varphi_{i}(\vec{x},\vec{y}))</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.22.m21.3d">italic_h ( italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) )</annotation></semantics></math> the fact that is obtained from the atom <math alttext="\varphi_{i}(\vec{x},\vec{y})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.23.m22.2"><semantics id="S2.SS0.SSS0.Px1.p3.23.m22.2a"><mrow id="S2.SS0.SSS0.Px1.p3.23.m22.2.3" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.cmml"><msub id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.2" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.2.cmml">φ</mi><mi id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.3" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.1" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.2" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p3.23.m22.1.1" xref="S2.SS0.SSS0.Px1.p3.23.m22.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.23.m22.1.1.2" xref="S2.SS0.SSS0.Px1.p3.23.m22.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p3.23.m22.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.23.m22.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p3.23.m22.2.2" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.23.m22.2.2.2" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.2.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.p3.23.m22.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.23.m22.2b"><apply id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3"><times id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.1"></times><apply id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.2.3">𝑖</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.3.3.2"><apply id="S2.SS0.SSS0.Px1.p3.23.m22.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.1.1"><ci id="S2.SS0.SSS0.Px1.p3.23.m22.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p3.23.m22.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.1.1.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p3.23.m22.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.2"><ci id="S2.SS0.SSS0.Px1.p3.23.m22.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.p3.23.m22.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.23.m22.2.2.2">𝑦</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.23.m22.2c">\varphi_{i}(\vec{x},\vec{y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.23.m22.2d">italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math> by replacing every variable <math alttext="z" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.24.m23.1"><semantics id="S2.SS0.SSS0.Px1.p3.24.m23.1a"><mi id="S2.SS0.SSS0.Px1.p3.24.m23.1.1" xref="S2.SS0.SSS0.Px1.p3.24.m23.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.24.m23.1b"><ci id="S2.SS0.SSS0.Px1.p3.24.m23.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.24.m23.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.24.m23.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.24.m23.1d">italic_z</annotation></semantics></math> with the constant <math alttext="h(z)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.25.m24.1"><semantics id="S2.SS0.SSS0.Px1.p3.25.m24.1a"><mrow id="S2.SS0.SSS0.Px1.p3.25.m24.1.2" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.25.m24.1.2.2" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.25.m24.1.2.1" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.25.m24.1.2.3.2" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p3.25.m24.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p3.25.m24.1.1" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.1.cmml">z</mi><mo id="S2.SS0.SSS0.Px1.p3.25.m24.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.25.m24.1b"><apply id="S2.SS0.SSS0.Px1.p3.25.m24.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2"><times id="S2.SS0.SSS0.Px1.p3.25.m24.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p3.25.m24.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.2.2">ℎ</ci><ci id="S2.SS0.SSS0.Px1.p3.25.m24.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.25.m24.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.25.m24.1c">h(z)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.25.m24.1d">italic_h ( italic_z )</annotation></semantics></math>. An <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p3.33.3">answer</em> to <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.26.m25.1"><semantics id="S2.SS0.SSS0.Px1.p3.26.m25.1a"><mi id="S2.SS0.SSS0.Px1.p3.26.m25.1.1" xref="S2.SS0.SSS0.Px1.p3.26.m25.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.26.m25.1b"><ci id="S2.SS0.SSS0.Px1.p3.26.m25.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.26.m25.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.26.m25.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.26.m25.1d">italic_Q</annotation></semantics></math> over <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.27.m26.1"><semantics id="S2.SS0.SSS0.Px1.p3.27.m26.1a"><mi id="S2.SS0.SSS0.Px1.p3.27.m26.1.1" xref="S2.SS0.SSS0.Px1.p3.27.m26.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.27.m26.1b"><ci id="S2.SS0.SSS0.Px1.p3.27.m26.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.27.m26.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.27.m26.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.27.m26.1d">italic_D</annotation></semantics></math> is a tuple of the form <math alttext="h(\vec{x})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.28.m27.1"><semantics id="S2.SS0.SSS0.Px1.p3.28.m27.1a"><mrow id="S2.SS0.SSS0.Px1.p3.28.m27.1.2" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.28.m27.1.2.2" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.2.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.28.m27.1.2.1" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.28.m27.1.2.3.2" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.28.m27.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p3.28.m27.1.1" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.28.m27.1.1.2" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p3.28.m27.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p3.28.m27.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.28.m27.1b"><apply id="S2.SS0.SSS0.Px1.p3.28.m27.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.2"><times id="S2.SS0.SSS0.Px1.p3.28.m27.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p3.28.m27.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.2.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.28.m27.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.p3.28.m27.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p3.28.m27.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.28.m27.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.28.m27.1c">h(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.28.m27.1d">italic_h ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math> where <math alttext="h\in\mathsf{Hom}(Q,D)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.29.m28.2"><semantics id="S2.SS0.SSS0.Px1.p3.29.m28.2a"><mrow id="S2.SS0.SSS0.Px1.p3.29.m28.2.3" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.2" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.1" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.1.cmml">∈</mo><mrow id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.2" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.2.cmml">𝖧𝗈𝗆</mi><mo id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.1" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.2" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p3.29.m28.1.1" xref="S2.SS0.SSS0.Px1.p3.29.m28.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.2.2" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p3.29.m28.2.2" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.2.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.29.m28.2b"><apply id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3"><in id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.1"></in><ci id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3"><times id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.1"></times><ci id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.2">𝖧𝗈𝗆</ci><interval closure="open" id="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.3.3.3.2"><ci id="S2.SS0.SSS0.Px1.p3.29.m28.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.1.1">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p3.29.m28.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.29.m28.2.2">𝐷</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.29.m28.2c">h\in\mathsf{Hom}(Q,D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.29.m28.2d">italic_h ∈ sansserif_Hom ( italic_Q , italic_D )</annotation></semantics></math>. The <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p3.33.4">result</em> of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.30.m29.1"><semantics id="S2.SS0.SSS0.Px1.p3.30.m29.1a"><mi id="S2.SS0.SSS0.Px1.p3.30.m29.1.1" xref="S2.SS0.SSS0.Px1.p3.30.m29.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.30.m29.1b"><ci id="S2.SS0.SSS0.Px1.p3.30.m29.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.30.m29.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.30.m29.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.30.m29.1d">italic_Q</annotation></semantics></math> over <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.31.m30.1"><semantics id="S2.SS0.SSS0.Px1.p3.31.m30.1a"><mi id="S2.SS0.SSS0.Px1.p3.31.m30.1.1" xref="S2.SS0.SSS0.Px1.p3.31.m30.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.31.m30.1b"><ci id="S2.SS0.SSS0.Px1.p3.31.m30.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.31.m30.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.31.m30.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.31.m30.1d">italic_D</annotation></semantics></math>, denoted <math alttext="Q(D)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.32.m31.1"><semantics id="S2.SS0.SSS0.Px1.p3.32.m31.1a"><mrow id="S2.SS0.SSS0.Px1.p3.32.m31.1.2" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.32.m31.1.2.2" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p3.32.m31.1.2.1" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.32.m31.1.2.3.2" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p3.32.m31.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p3.32.m31.1.1" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.p3.32.m31.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.32.m31.1b"><apply id="S2.SS0.SSS0.Px1.p3.32.m31.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2"><times id="S2.SS0.SSS0.Px1.p3.32.m31.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p3.32.m31.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.2.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p3.32.m31.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.32.m31.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.32.m31.1c">Q(D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.32.m31.1d">italic_Q ( italic_D )</annotation></semantics></math>, is <math alttext="Q(D)\vcentcolon=\{h(\vec{x})\mid h\in\mathsf{Hom}(Q,D)\}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p3.33.m32.6"><semantics id="S2.SS0.SSS0.Px1.p3.33.m32.6a"><mrow id="S2.SS0.SSS0.Px1.p3.33.m32.6.6" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.cmml"><mrow id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.cmml"><mi id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.2" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.1" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.3.2" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.cmml"><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p3.33.m32.1.1" xref="S2.SS0.SSS0.Px1.p3.33.m32.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.3" rspace="0.278em" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.3.cmml">:=</mo><mrow id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.3.1.cmml">{</mo><mrow id="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1" xref="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.2" xref="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.1" xref="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.3.2" xref="S2.SS0.SSS0.Px1.p3.33.m32.2.2.cmml"><mo id="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.33.m32.2.2.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p3.33.m32.2.2" xref="S2.SS0.SSS0.Px1.p3.33.m32.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p3.33.m32.2.2.2" xref="S2.SS0.SSS0.Px1.p3.33.m32.2.2.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p3.33.m32.2.2.1" stretchy="false" 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xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p3.33.m32.3.3" xref="S2.SS0.SSS0.Px1.p3.33.m32.3.3.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p3.33.m32.4.4" xref="S2.SS0.SSS0.Px1.p3.33.m32.4.4.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.5" stretchy="false" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p3.33.m32.6b"><apply id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.3.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.3">assign</csymbol><apply id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4"><times id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.1"></times><ci id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.2.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.4.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p3.33.m32.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.1.1">𝐷</ci></apply><apply id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.3">conditional-set</csymbol><apply id="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1"><times id="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.1"></times><ci id="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.33.m32.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.5.5.1.1.1.3.2"><ci id="S2.SS0.SSS0.Px1.p3.33.m32.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.p3.33.m32.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.2.2.2">𝑥</ci></apply></apply><apply id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2"><in id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.1"></in><ci id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3"><times id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.2">𝖧𝗈𝗆</ci><interval closure="open" id="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.6.6.2.2.2.3.3.2"><ci id="S2.SS0.SSS0.Px1.p3.33.m32.3.3.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.3.3">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p3.33.m32.4.4.cmml" xref="S2.SS0.SSS0.Px1.p3.33.m32.4.4">𝐷</ci></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p3.33.m32.6c">Q(D)\vcentcolon=\{h(\vec{x})\mid h\in\mathsf{Hom}(Q,D)\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p3.33.m32.6d">italic_Q ( italic_D ) := { italic_h ( over→ start_ARG italic_x end_ARG ) ∣ italic_h ∈ sansserif_Hom ( italic_Q , italic_D ) }</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p4"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p4.27">In our proofs, we will use the following definition of when two facts <math alttext="f" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p4.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p4.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p4.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p4.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.1.m1.1d">italic_f</annotation></semantics></math> and <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.p4.2.m2.1a"><msup id="S2.SS0.SSS0.Px1.p4.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p4.2.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.2.m2.1.1.2" xref="S2.SS0.SSS0.Px1.p4.2.m2.1.1.2.cmml">f</mi><mo id="S2.SS0.SSS0.Px1.p4.2.m2.1.1.3" xref="S2.SS0.SSS0.Px1.p4.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.2.m2.1b"><apply id="S2.SS0.SSS0.Px1.p4.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.2.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.2.m2.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.2.m2.1.1.2">𝑓</ci><ci id="S2.SS0.SSS0.Px1.p4.2.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.2.m2.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.2.m2.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> agree in the context of two queries. Intuitively, facts agree if they assign the same variables with the same values. Let <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p4.3.m3.1a"><mi id="S2.SS0.SSS0.Px1.p4.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p4.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.p4.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.3.m3.1d">italic_Q</annotation></semantics></math> and <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.p4.4.m4.1a"><msup id="S2.SS0.SSS0.Px1.p4.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p4.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.p4.4.m4.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p4.4.m4.1.1.3" xref="S2.SS0.SSS0.Px1.p4.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.4.m4.1b"><apply id="S2.SS0.SSS0.Px1.p4.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.4.m4.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.4.m4.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p4.4.m4.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.4.m4.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.4.m4.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be CQs over the schemas <math alttext="\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.p4.5.m5.1a"><mi id="S2.SS0.SSS0.Px1.p4.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p4.5.m5.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.5.m5.1b"><ci id="S2.SS0.SSS0.Px1.p4.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.5.m5.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.5.m5.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.5.m5.1d">bold_S</annotation></semantics></math> and <math alttext="\mathord{\mathbf{S}}^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.p4.6.m6.1a"><msup id="S2.SS0.SSS0.Px1.p4.6.m6.1.1" xref="S2.SS0.SSS0.Px1.p4.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.6.m6.1.1.2" xref="S2.SS0.SSS0.Px1.p4.6.m6.1.1.2.cmml">𝐒</mi><mo id="S2.SS0.SSS0.Px1.p4.6.m6.1.1.3" xref="S2.SS0.SSS0.Px1.p4.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.p4.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.6.m6.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.6.m6.1.1.2">𝐒</ci><ci id="S2.SS0.SSS0.Px1.p4.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.6.m6.1c">\mathord{\mathbf{S}}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.6.m6.1d">bold_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, respectively. Let <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.p4.7.m7.1a"><mi id="S2.SS0.SSS0.Px1.p4.7.m7.1.1" xref="S2.SS0.SSS0.Px1.p4.7.m7.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.7.m7.1b"><ci id="S2.SS0.SSS0.Px1.p4.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.7.m7.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.7.m7.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.7.m7.1d">italic_φ</annotation></semantics></math> and <math alttext="\varphi^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.p4.8.m8.1a"><msup id="S2.SS0.SSS0.Px1.p4.8.m8.1.1" xref="S2.SS0.SSS0.Px1.p4.8.m8.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.8.m8.1.1.2" xref="S2.SS0.SSS0.Px1.p4.8.m8.1.1.2.cmml">φ</mi><mo id="S2.SS0.SSS0.Px1.p4.8.m8.1.1.3" xref="S2.SS0.SSS0.Px1.p4.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.8.m8.1b"><apply id="S2.SS0.SSS0.Px1.p4.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.8.m8.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.8.m8.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.8.m8.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.8.m8.1.1.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p4.8.m8.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.8.m8.1c">\varphi^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.8.m8.1d">italic_φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be atoms of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.p4.9.m9.1a"><mi id="S2.SS0.SSS0.Px1.p4.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p4.9.m9.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.9.m9.1b"><ci id="S2.SS0.SSS0.Px1.p4.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.9.m9.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.9.m9.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.9.m9.1d">italic_Q</annotation></semantics></math> and <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.p4.10.m10.1a"><msup id="S2.SS0.SSS0.Px1.p4.10.m10.1.1" xref="S2.SS0.SSS0.Px1.p4.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.10.m10.1.1.2" xref="S2.SS0.SSS0.Px1.p4.10.m10.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p4.10.m10.1.1.3" xref="S2.SS0.SSS0.Px1.p4.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.10.m10.1b"><apply id="S2.SS0.SSS0.Px1.p4.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.10.m10.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.10.m10.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p4.10.m10.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.10.m10.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.10.m10.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> over the relation symbols <math alttext="R" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.p4.11.m11.1a"><mi id="S2.SS0.SSS0.Px1.p4.11.m11.1.1" xref="S2.SS0.SSS0.Px1.p4.11.m11.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.11.m11.1b"><ci id="S2.SS0.SSS0.Px1.p4.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.11.m11.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.11.m11.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.11.m11.1d">italic_R</annotation></semantics></math> and <math alttext="R^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.12.m12.1"><semantics id="S2.SS0.SSS0.Px1.p4.12.m12.1a"><msup id="S2.SS0.SSS0.Px1.p4.12.m12.1.1" xref="S2.SS0.SSS0.Px1.p4.12.m12.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.12.m12.1.1.2" xref="S2.SS0.SSS0.Px1.p4.12.m12.1.1.2.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p4.12.m12.1.1.3" xref="S2.SS0.SSS0.Px1.p4.12.m12.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.12.m12.1b"><apply id="S2.SS0.SSS0.Px1.p4.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.12.m12.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.12.m12.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.12.m12.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.12.m12.1.1.2">𝑅</ci><ci id="S2.SS0.SSS0.Px1.p4.12.m12.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.12.m12.1c">R^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.12.m12.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, respectively. Let <math alttext="f" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.13.m13.1"><semantics id="S2.SS0.SSS0.Px1.p4.13.m13.1a"><mi id="S2.SS0.SSS0.Px1.p4.13.m13.1.1" xref="S2.SS0.SSS0.Px1.p4.13.m13.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.13.m13.1b"><ci id="S2.SS0.SSS0.Px1.p4.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.13.m13.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.13.m13.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.13.m13.1d">italic_f</annotation></semantics></math> and <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.14.m14.1"><semantics id="S2.SS0.SSS0.Px1.p4.14.m14.1a"><msup id="S2.SS0.SSS0.Px1.p4.14.m14.1.1" xref="S2.SS0.SSS0.Px1.p4.14.m14.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.14.m14.1.1.2" xref="S2.SS0.SSS0.Px1.p4.14.m14.1.1.2.cmml">f</mi><mo id="S2.SS0.SSS0.Px1.p4.14.m14.1.1.3" xref="S2.SS0.SSS0.Px1.p4.14.m14.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.14.m14.1b"><apply id="S2.SS0.SSS0.Px1.p4.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.14.m14.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.14.m14.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.14.m14.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.14.m14.1.1.2">𝑓</ci><ci id="S2.SS0.SSS0.Px1.p4.14.m14.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.14.m14.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.14.m14.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be facts over <math alttext="R" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.15.m15.1"><semantics id="S2.SS0.SSS0.Px1.p4.15.m15.1a"><mi id="S2.SS0.SSS0.Px1.p4.15.m15.1.1" xref="S2.SS0.SSS0.Px1.p4.15.m15.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.15.m15.1b"><ci id="S2.SS0.SSS0.Px1.p4.15.m15.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.15.m15.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.15.m15.1c">R</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.15.m15.1d">italic_R</annotation></semantics></math> and <math alttext="R^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.16.m16.1"><semantics id="S2.SS0.SSS0.Px1.p4.16.m16.1a"><msup id="S2.SS0.SSS0.Px1.p4.16.m16.1.1" xref="S2.SS0.SSS0.Px1.p4.16.m16.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.16.m16.1.1.2" xref="S2.SS0.SSS0.Px1.p4.16.m16.1.1.2.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p4.16.m16.1.1.3" xref="S2.SS0.SSS0.Px1.p4.16.m16.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.16.m16.1b"><apply id="S2.SS0.SSS0.Px1.p4.16.m16.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.16.m16.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.16.m16.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.16.m16.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.16.m16.1.1.2">𝑅</ci><ci id="S2.SS0.SSS0.Px1.p4.16.m16.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.16.m16.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.16.m16.1c">R^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.16.m16.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, respectively. We say that <math alttext="f" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.17.m17.1"><semantics id="S2.SS0.SSS0.Px1.p4.17.m17.1a"><mi id="S2.SS0.SSS0.Px1.p4.17.m17.1.1" xref="S2.SS0.SSS0.Px1.p4.17.m17.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.17.m17.1b"><ci id="S2.SS0.SSS0.Px1.p4.17.m17.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.17.m17.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.17.m17.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.17.m17.1d">italic_f</annotation></semantics></math> and <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.18.m18.1"><semantics id="S2.SS0.SSS0.Px1.p4.18.m18.1a"><msup id="S2.SS0.SSS0.Px1.p4.18.m18.1.1" xref="S2.SS0.SSS0.Px1.p4.18.m18.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.18.m18.1.1.2" xref="S2.SS0.SSS0.Px1.p4.18.m18.1.1.2.cmml">f</mi><mo id="S2.SS0.SSS0.Px1.p4.18.m18.1.1.3" xref="S2.SS0.SSS0.Px1.p4.18.m18.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.18.m18.1b"><apply id="S2.SS0.SSS0.Px1.p4.18.m18.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.18.m18.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.18.m18.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.18.m18.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.18.m18.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.18.m18.1.1.2">𝑓</ci><ci id="S2.SS0.SSS0.Px1.p4.18.m18.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.18.m18.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.18.m18.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.18.m18.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p4.27.1">agree</em> (w.r.t. <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.19.m19.1"><semantics id="S2.SS0.SSS0.Px1.p4.19.m19.1a"><mi id="S2.SS0.SSS0.Px1.p4.19.m19.1.1" xref="S2.SS0.SSS0.Px1.p4.19.m19.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.19.m19.1b"><ci id="S2.SS0.SSS0.Px1.p4.19.m19.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.19.m19.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.19.m19.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.19.m19.1d">italic_φ</annotation></semantics></math> and <math alttext="\varphi^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.20.m20.1"><semantics id="S2.SS0.SSS0.Px1.p4.20.m20.1a"><msup id="S2.SS0.SSS0.Px1.p4.20.m20.1.1" xref="S2.SS0.SSS0.Px1.p4.20.m20.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.20.m20.1.1.2" xref="S2.SS0.SSS0.Px1.p4.20.m20.1.1.2.cmml">φ</mi><mo id="S2.SS0.SSS0.Px1.p4.20.m20.1.1.3" xref="S2.SS0.SSS0.Px1.p4.20.m20.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.20.m20.1b"><apply id="S2.SS0.SSS0.Px1.p4.20.m20.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.20.m20.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.20.m20.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.20.m20.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.20.m20.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.20.m20.1.1.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p4.20.m20.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.20.m20.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.20.m20.1c">\varphi^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.20.m20.1d">italic_φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>) if there exists a homomorphism <math alttext="h:\mathrm{vars}(\phi)\cup\mathrm{vars}(\phi^{\prime})\rightarrow\mathsf{Const}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.21.m21.2"><semantics id="S2.SS0.SSS0.Px1.p4.21.m21.2a"><mrow id="S2.SS0.SSS0.Px1.p4.21.m21.2.2" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.3" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.3.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.2" lspace="0.278em" rspace="0.278em" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.2.cmml">:</mo><mrow id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.cmml"><mrow id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.cmml"><mrow id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.2" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.2.cmml">vars</mi><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.1" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.3.2" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.cmml"><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p4.21.m21.1.1" xref="S2.SS0.SSS0.Px1.p4.21.m21.1.1.cmml">ϕ</mi><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.2" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.2.cmml">∪</mo><mrow id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.3.cmml">vars</mi><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.2.cmml">ϕ</mi><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.2.cmml">→</mo><mi id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.3" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.3.cmml">𝖢𝗈𝗇𝗌𝗍</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.21.m21.2b"><apply id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2"><ci id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.2">:</ci><ci id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.3">ℎ</ci><apply id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1"><ci id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.2">→</ci><apply id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1"><union id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.2"></union><apply id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3"><times id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.1"></times><ci id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.3.2">vars</ci><ci id="S2.SS0.SSS0.Px1.p4.21.m21.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.1.1">italic-ϕ</ci></apply><apply id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1"><times id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.2"></times><ci id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.3">vars</ci><apply id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.2">italic-ϕ</ci><ci id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.1.1.1.1.1.3">′</ci></apply></apply></apply><ci id="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.21.m21.2.2.1.3">𝖢𝗈𝗇𝗌𝗍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.21.m21.2c">h:\mathrm{vars}(\phi)\cup\mathrm{vars}(\phi^{\prime})\rightarrow\mathsf{Const}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.21.m21.2d">italic_h : roman_vars ( italic_ϕ ) ∪ roman_vars ( italic_ϕ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) → sansserif_Const</annotation></semantics></math> such that <math alttext="f" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.22.m22.1"><semantics id="S2.SS0.SSS0.Px1.p4.22.m22.1a"><mi id="S2.SS0.SSS0.Px1.p4.22.m22.1.1" xref="S2.SS0.SSS0.Px1.p4.22.m22.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.22.m22.1b"><ci id="S2.SS0.SSS0.Px1.p4.22.m22.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.22.m22.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.22.m22.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.22.m22.1d">italic_f</annotation></semantics></math> and <math alttext="f^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.23.m23.1"><semantics id="S2.SS0.SSS0.Px1.p4.23.m23.1a"><msup id="S2.SS0.SSS0.Px1.p4.23.m23.1.1" xref="S2.SS0.SSS0.Px1.p4.23.m23.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.23.m23.1.1.2" xref="S2.SS0.SSS0.Px1.p4.23.m23.1.1.2.cmml">f</mi><mo id="S2.SS0.SSS0.Px1.p4.23.m23.1.1.3" xref="S2.SS0.SSS0.Px1.p4.23.m23.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.23.m23.1b"><apply id="S2.SS0.SSS0.Px1.p4.23.m23.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.23.m23.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.23.m23.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.23.m23.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.23.m23.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.23.m23.1.1.2">𝑓</ci><ci id="S2.SS0.SSS0.Px1.p4.23.m23.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.23.m23.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.23.m23.1c">f^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.23.m23.1d">italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are obtained from <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.24.m24.1"><semantics id="S2.SS0.SSS0.Px1.p4.24.m24.1a"><mi id="S2.SS0.SSS0.Px1.p4.24.m24.1.1" xref="S2.SS0.SSS0.Px1.p4.24.m24.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.24.m24.1b"><ci id="S2.SS0.SSS0.Px1.p4.24.m24.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.24.m24.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.24.m24.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.24.m24.1d">italic_φ</annotation></semantics></math> and <math alttext="\varphi^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.25.m25.1"><semantics id="S2.SS0.SSS0.Px1.p4.25.m25.1a"><msup id="S2.SS0.SSS0.Px1.p4.25.m25.1.1" xref="S2.SS0.SSS0.Px1.p4.25.m25.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p4.25.m25.1.1.2" xref="S2.SS0.SSS0.Px1.p4.25.m25.1.1.2.cmml">φ</mi><mo id="S2.SS0.SSS0.Px1.p4.25.m25.1.1.3" xref="S2.SS0.SSS0.Px1.p4.25.m25.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.25.m25.1b"><apply id="S2.SS0.SSS0.Px1.p4.25.m25.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.25.m25.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p4.25.m25.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.25.m25.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p4.25.m25.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.25.m25.1.1.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p4.25.m25.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p4.25.m25.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.25.m25.1c">\varphi^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.25.m25.1d">italic_φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, respectively, by replacing each variable <math alttext="x" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.26.m26.1"><semantics id="S2.SS0.SSS0.Px1.p4.26.m26.1a"><mi id="S2.SS0.SSS0.Px1.p4.26.m26.1.1" xref="S2.SS0.SSS0.Px1.p4.26.m26.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.26.m26.1b"><ci id="S2.SS0.SSS0.Px1.p4.26.m26.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.26.m26.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.26.m26.1c">x</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.26.m26.1d">italic_x</annotation></semantics></math> with the constant <math alttext="h(x)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p4.27.m27.1"><semantics id="S2.SS0.SSS0.Px1.p4.27.m27.1a"><mrow id="S2.SS0.SSS0.Px1.p4.27.m27.1.2" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p4.27.m27.1.2.2" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.p4.27.m27.1.2.1" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p4.27.m27.1.2.3.2" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p4.27.m27.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p4.27.m27.1.1" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.1.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p4.27.m27.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p4.27.m27.1b"><apply id="S2.SS0.SSS0.Px1.p4.27.m27.1.2.cmml" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2"><times id="S2.SS0.SSS0.Px1.p4.27.m27.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p4.27.m27.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.2.2">ℎ</ci><ci id="S2.SS0.SSS0.Px1.p4.27.m27.1.1.cmml" xref="S2.SS0.SSS0.Px1.p4.27.m27.1.1">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p4.27.m27.1c">h(x)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p4.27.m27.1d">italic_h ( italic_x )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p5"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p5.13">Consider a CQ <math alttext="Q(\vec{x}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{\ell}(\vec{x% },\vec{y})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.1.m1.8"><semantics id="S2.SS0.SSS0.Px1.p5.1.m1.8a"><mrow id="S2.SS0.SSS0.Px1.p5.1.m1.8.8" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.cmml"><mrow id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.cmml"><mi id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.1" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.3.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.1.m1.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p5.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p5.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.1.m1.1.1.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p5.1.m1.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.1.m1.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.3.2.2" rspace="0.448em" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.1.m1.1.1.cmml">)</mo></mrow></mrow><mpadded width="0.226em"><mo id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.3" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.3.cmml">:</mo></mpadded><mrow id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.3.cmml"><mrow id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1a" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.cmml">−</mo><mrow id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.cmml"><msub id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.3" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.1" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.3.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p5.1.m1.2.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.1.m1.2.2.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.2.2.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p5.1.m1.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.1.m1.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.3.2.2" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p5.1.m1.3.3" xref="S2.SS0.SSS0.Px1.p5.1.m1.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.p5.1.m1.3.3.2" 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stretchy="false" xref="S2.SS0.SSS0.Px1.p5.1.m1.5.5.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.1.m1.8b"><apply id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8"><ci id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.3.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.3">:</ci><apply id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4"><times id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.1"></times><ci id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.2">𝑄</ci><apply id="S2.SS0.SSS0.Px1.p5.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.4.3.2"><ci id="S2.SS0.SSS0.Px1.p5.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p5.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.1.1.2">𝑥</ci></apply></apply><list id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.3.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2"><apply id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1"><minus id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1"></minus><apply id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2"><times id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.1"></times><apply id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.2.3">1</cn></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.7.7.1.1.1.2.3.2"><apply id="S2.SS0.SSS0.Px1.p5.1.m1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.2.2"><ci id="S2.SS0.SSS0.Px1.p5.1.m1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.p5.1.m1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.2.2.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p5.1.m1.3.3.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.3.3"><ci id="S2.SS0.SSS0.Px1.p5.1.m1.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.3.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.p5.1.m1.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.3.3.2">𝑦</ci></apply></interval></apply></apply><ci id="S2.SS0.SSS0.Px1.p5.1.m1.6.6.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.6.6">…</ci><apply id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2"><times id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.1"></times><apply id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.2.3">ℓ</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.8.8.2.2.2.3.2"><apply id="S2.SS0.SSS0.Px1.p5.1.m1.4.4.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.4.4"><ci id="S2.SS0.SSS0.Px1.p5.1.m1.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.4.4.1">→</ci><ci id="S2.SS0.SSS0.Px1.p5.1.m1.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.4.4.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.p5.1.m1.5.5.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.5.5"><ci id="S2.SS0.SSS0.Px1.p5.1.m1.5.5.1.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.5.5.1">→</ci><ci id="S2.SS0.SSS0.Px1.p5.1.m1.5.5.2.cmml" xref="S2.SS0.SSS0.Px1.p5.1.m1.5.5.2">𝑦</ci></apply></interval></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.1.m1.8c">Q(\vec{x}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{\ell}(\vec{x% },\vec{y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.1.m1.8d">italic_Q ( over→ start_ARG italic_x end_ARG ) : - italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) , … , italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math>. We may refer to <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.p5.2.m2.1a"><mi id="S2.SS0.SSS0.Px1.p5.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p5.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.2.m2.1b"><ci id="S2.SS0.SSS0.Px1.p5.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.2.m2.1d">italic_Q</annotation></semantics></math> as <math alttext="Q(\vec{x})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p5.3.m3.1a"><mrow id="S2.SS0.SSS0.Px1.p5.3.m3.1.2" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.3.m3.1.2.2" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.2.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p5.3.m3.1.2.1" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p5.3.m3.1.2.3.2" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p5.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p5.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p5.3.m3.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p5.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.p5.3.m3.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.2"><times id="S2.SS0.SSS0.Px1.p5.3.m3.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p5.3.m3.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.2.2">𝑄</ci><apply id="S2.SS0.SSS0.Px1.p5.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.p5.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p5.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.3.m3.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.3.m3.1c">Q(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.3.m3.1d">italic_Q ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math> to specify the sequence of free variables in the head. In this work, the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p5.13.1">order</em> of the sequence <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.p5.4.m4.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p5.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p5.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.p5.4.m4.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p5.4.m4.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.4.m4.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.4.m4.1b"><apply id="S2.SS0.SSS0.Px1.p5.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.4.m4.1.1"><ci id="S2.SS0.SSS0.Px1.p5.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.4.m4.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p5.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.4.m4.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.4.m4.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.4.m4.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> has a crucial role, since it determines the desired order of answers. Specifically, we will assume that the desired order of answers is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p5.13.2">lexicographic</em> in the left-to-right order of <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.p5.5.m5.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p5.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p5.5.m5.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.5.m5.1.1.2" xref="S2.SS0.SSS0.Px1.p5.5.m5.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p5.5.m5.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.5.m5.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.5.m5.1b"><apply id="S2.SS0.SSS0.Px1.p5.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.5.m5.1.1"><ci id="S2.SS0.SSS0.Px1.p5.5.m5.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.5.m5.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p5.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.5.m5.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.5.m5.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.5.m5.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math>. For example, the CQ <math alttext="Q(x_{1},x_{2}){\,:\!\!-\,}R(x_{1},x_{2}),S(x_{2},y)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.6.m6.5"><semantics id="S2.SS0.SSS0.Px1.p5.6.m6.5a"><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.5.5" xref="S2.SS0.SSS0.Px1.p5.6.m6.5.5.cmml"><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.4" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.4.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.3" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p5.6.m6.2.2.1.1.1.1" xref="S2.SS0.SSS0.Px1.p5.6.m6.2.2.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.6.m6.2.2.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.2.2.1.1.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.6.m6.2.2.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p5.6.m6.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.4" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.2.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.2.3" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.2.5" rspace="0.448em" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.6.m6.3.3.2.2.3.cmml">)</mo></mrow></mrow><mpadded width="0.226em"><mo id="S2.SS0.SSS0.Px1.p5.6.m6.5.5.5" xref="S2.SS0.SSS0.Px1.p5.6.m6.5.5.5.cmml">:</mo></mpadded><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.3.cmml"><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1a" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.cmml">−</mo><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.4" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.4.cmml">R</mi><mo id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.3" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.1.1.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.4" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.2.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.2.3" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.2.5" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.6.m6.4.4.3.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2.3" xref="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.3.cmml">,</mo><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2.2.3" xref="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2.2.3.cmml">S</mi><mo id="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2.2.2" xref="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2.2.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p5.6.m6.5.5.4.2.2.1.1" 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end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , italic_S ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y )</annotation></semantics></math> not only in the order of values within each answer tuple but also in the order over the answers. For <math alttext="Q(x_{1},x_{2})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.8.m8.2"><semantics id="S2.SS0.SSS0.Px1.p5.8.m8.2a"><mrow id="S2.SS0.SSS0.Px1.p5.8.m8.2.2" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.4" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.4.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.3" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.4" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.3" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.5" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.8.m8.2b"><apply id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2"><times id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.3"></times><ci id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.4.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.4">𝑄</ci><interval closure="open" id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2"><apply id="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.8.m8.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.8.m8.2.2.2.2.2.3">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.8.m8.2c">Q(x_{1},x_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.8.m8.2d">italic_Q ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> we order the answers first by <math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.p5.9.m9.1a"><msub id="S2.SS0.SSS0.Px1.p5.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p5.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.9.m9.1.1.2" xref="S2.SS0.SSS0.Px1.p5.9.m9.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.9.m9.1.1.3" xref="S2.SS0.SSS0.Px1.p5.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.9.m9.1b"><apply id="S2.SS0.SSS0.Px1.p5.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.9.m9.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.9.m9.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p5.9.m9.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.9.m9.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.9.m9.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and then by <math alttext="x_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.p5.10.m10.1a"><msub id="S2.SS0.SSS0.Px1.p5.10.m10.1.1" xref="S2.SS0.SSS0.Px1.p5.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.10.m10.1.1.2" xref="S2.SS0.SSS0.Px1.p5.10.m10.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.10.m10.1.1.3" xref="S2.SS0.SSS0.Px1.p5.10.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.10.m10.1b"><apply id="S2.SS0.SSS0.Px1.p5.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.10.m10.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.10.m10.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p5.10.m10.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.10.m10.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.10.m10.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and for <math alttext="Q^{\prime}(x_{2},x_{1})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.11.m11.2"><semantics id="S2.SS0.SSS0.Px1.p5.11.m11.2a"><mrow id="S2.SS0.SSS0.Px1.p5.11.m11.2.2" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.cmml"><msup id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.cmml"><mi id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.2" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.3" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.3.cmml">′</mo></msup><mo id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.3" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.4" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.3" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.5" stretchy="false" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.11.m11.2b"><apply id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2"><times id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.3"></times><apply id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.1.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.2.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.3.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.4.3">′</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2"><apply id="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.11.m11.1.1.1.1.1.3">2</cn></apply><apply id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.11.m11.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.11.m11.2c">Q^{\prime}(x_{2},x_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.11.m11.2d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> we order first by <math alttext="x_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.12.m12.1"><semantics id="S2.SS0.SSS0.Px1.p5.12.m12.1a"><msub id="S2.SS0.SSS0.Px1.p5.12.m12.1.1" xref="S2.SS0.SSS0.Px1.p5.12.m12.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.12.m12.1.1.2" xref="S2.SS0.SSS0.Px1.p5.12.m12.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.12.m12.1.1.3" xref="S2.SS0.SSS0.Px1.p5.12.m12.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.12.m12.1b"><apply id="S2.SS0.SSS0.Px1.p5.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.12.m12.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.12.m12.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.12.m12.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.12.m12.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p5.12.m12.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.12.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.12.m12.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.12.m12.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and then by <math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p5.13.m13.1"><semantics id="S2.SS0.SSS0.Px1.p5.13.m13.1a"><msub id="S2.SS0.SSS0.Px1.p5.13.m13.1.1" xref="S2.SS0.SSS0.Px1.p5.13.m13.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p5.13.m13.1.1.2" xref="S2.SS0.SSS0.Px1.p5.13.m13.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p5.13.m13.1.1.3" xref="S2.SS0.SSS0.Px1.p5.13.m13.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p5.13.m13.1b"><apply id="S2.SS0.SSS0.Px1.p5.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.13.m13.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p5.13.m13.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p5.13.m13.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p5.13.m13.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p5.13.m13.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p5.13.m13.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p5.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p5.13.m13.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p5.13.m13.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p6"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p6.22">As usual, we associate a CQ <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p6.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p6.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p6.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p6.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.1.m1.1d">italic_Q</annotation></semantics></math> with the hypergraph <math alttext="H(Q)=(V_{Q},E_{Q})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.2.m2.3"><semantics id="S2.SS0.SSS0.Px1.p6.2.m2.3a"><mrow id="S2.SS0.SSS0.Px1.p6.2.m2.3.3" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.cmml"><mrow id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.cmml"><mi id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.2" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.2.cmml">H</mi><mo id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.1" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.3.2" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.cmml"><mo id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p6.2.m2.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.3" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.3.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.2.cmml">V</mi><mi id="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.3.cmml">Q</mi></msub><mo id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.4" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.2" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.2.cmml">E</mi><mi id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.3" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.3.cmml">Q</mi></msub><mo id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.5" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.2.m2.3b"><apply id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3"><eq id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.3"></eq><apply id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4"><times id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.1.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.1"></times><ci id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.2.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.4.2">𝐻</ci><ci id="S2.SS0.SSS0.Px1.p6.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.1.1">𝑄</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2"><apply id="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.2">𝑉</ci><ci id="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.2.2.1.1.1.3">𝑄</ci></apply><apply id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.2">𝐸</ci><ci id="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p6.2.m2.3.3.2.2.2.3">𝑄</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.2.m2.3c">H(Q)=(V_{Q},E_{Q})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.2.m2.3d">italic_H ( italic_Q ) = ( italic_V start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT )</annotation></semantics></math> where <math alttext="V_{Q}=\mathrm{vars}(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p6.3.m3.1a"><mrow id="S2.SS0.SSS0.Px1.p6.3.m3.1.2" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.cmml"><msub id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.2" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.2.cmml">V</mi><mi id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.3" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.3.cmml">Q</mi></msub><mo id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.1" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.2" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.2.cmml">vars</mi><mo id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.1" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.3.2" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2"><eq id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.1"></eq><apply id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.2">𝑉</ci><ci id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.2.3">𝑄</ci></apply><apply id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3"><times id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.2.3.2">vars</ci><ci id="S2.SS0.SSS0.Px1.p6.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.3.m3.1.1">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.3.m3.1c">V_{Q}=\mathrm{vars}(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.3.m3.1d">italic_V start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = roman_vars ( italic_Q )</annotation></semantics></math> and <math alttext="E_{Q}=\{\mathrm{vars}(\varphi)|\varphi\in\mathrm{atoms}(Q)\}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.4.m4.4"><semantics id="S2.SS0.SSS0.Px1.p6.4.m4.4a"><mrow id="S2.SS0.SSS0.Px1.p6.4.m4.4.4" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.cmml"><msub id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.cmml"><mi id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.2.cmml">E</mi><mi id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.3" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.3.cmml">Q</mi></msub><mo id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.3" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.3.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.3.1.cmml">{</mo><mrow id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.2.cmml">vars</mi><mo id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.1" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.3.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p6.4.m4.1.1.cmml">φ</mi><mo id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.4" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.3.1.cmml">|</mo><mrow id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.2.cmml">φ</mi><mo id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.1" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.1.cmml">∈</mo><mrow id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.2.cmml">atoms</mi><mo id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.1" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.3.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.4.m4.2.2" xref="S2.SS0.SSS0.Px1.p6.4.m4.2.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.5" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.4.m4.4b"><apply id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4"><eq id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.3.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.3"></eq><apply id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.2">𝐸</ci><ci id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.3.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.4.3">𝑄</ci></apply><apply id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.3.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.3">conditional-set</csymbol><apply id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1"><times id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.1"></times><ci id="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.3.3.1.1.1.2">vars</ci><ci id="S2.SS0.SSS0.Px1.p6.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.1.1">𝜑</ci></apply><apply id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2"><in id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.1"></in><ci id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.2">𝜑</ci><apply id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3"><times id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.4.4.2.2.2.3.2">atoms</ci><ci id="S2.SS0.SSS0.Px1.p6.4.m4.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.4.m4.2.2">𝑄</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.4.m4.4c">E_{Q}=\{\mathrm{vars}(\varphi)|\varphi\in\mathrm{atoms}(Q)\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.4.m4.4d">italic_E start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT = { roman_vars ( italic_φ ) | italic_φ ∈ roman_atoms ( italic_Q ) }</annotation></semantics></math>. We say that <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.p6.5.m5.1a"><mi id="S2.SS0.SSS0.Px1.p6.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p6.5.m5.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.5.m5.1b"><ci id="S2.SS0.SSS0.Px1.p6.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.5.m5.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.5.m5.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.5.m5.1d">italic_Q</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p6.22.2">acyclic</em> if <math alttext="H(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.p6.6.m6.1a"><mrow id="S2.SS0.SSS0.Px1.p6.6.m6.1.2" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p6.6.m6.1.2.2" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2.2.cmml">H</mi><mo id="S2.SS0.SSS0.Px1.p6.6.m6.1.2.1" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p6.6.m6.1.2.3.2" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p6.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.6.m6.1.1" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p6.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.p6.6.m6.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2"><times id="S2.SS0.SSS0.Px1.p6.6.m6.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p6.6.m6.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.2.2">𝐻</ci><ci id="S2.SS0.SSS0.Px1.p6.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.6.m6.1.1">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.6.m6.1c">H(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.6.m6.1d">italic_H ( italic_Q )</annotation></semantics></math> is an acyclic (<math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.p6.7.m7.1a"><mi id="S2.SS0.SSS0.Px1.p6.7.m7.1.1" xref="S2.SS0.SSS0.Px1.p6.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.7.m7.1b"><ci id="S2.SS0.SSS0.Px1.p6.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.7.m7.1d">italic_α</annotation></semantics></math>-acyclic) hypergraph. Recall that a hypergraph <math alttext="H=(V,E)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.8.m8.2"><semantics id="S2.SS0.SSS0.Px1.p6.8.m8.2a"><mrow id="S2.SS0.SSS0.Px1.p6.8.m8.2.3" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.2" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.2.cmml">H</mi><mo id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.1" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.2" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.8.m8.1.1" xref="S2.SS0.SSS0.Px1.p6.8.m8.1.1.cmml">V</mi><mo id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p6.8.m8.2.2" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.2.cmml">E</mi><mo id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.8.m8.2b"><apply id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.cmml" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3"><eq id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.1"></eq><ci id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.2">𝐻</ci><interval closure="open" id="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.3.3.2"><ci id="S2.SS0.SSS0.Px1.p6.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.8.m8.1.1">𝑉</ci><ci id="S2.SS0.SSS0.Px1.p6.8.m8.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.8.m8.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.8.m8.2c">H=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.8.m8.2d">italic_H = ( italic_V , italic_E )</annotation></semantics></math> is acyclic if there is a tree <math alttext="T=(E,E_{T})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.9.m9.2"><semantics id="S2.SS0.SSS0.Px1.p6.9.m9.2a"><mrow id="S2.SS0.SSS0.Px1.p6.9.m9.2.2" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.3" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.3.cmml">T</mi><mo id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.2" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.2.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p6.9.m9.1.1.cmml">E</mi><mo id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.3" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.2.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.2.cmml">E</mi><mi id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.3.cmml">T</mi></msub><mo id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.4" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.9.m9.2b"><apply id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2"><eq id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.2"></eq><ci id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.3">𝑇</ci><interval closure="open" id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1"><ci id="S2.SS0.SSS0.Px1.p6.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.1.1">𝐸</ci><apply id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.2">𝐸</ci><ci id="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p6.9.m9.2.2.1.1.1.3">𝑇</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.9.m9.2c">T=(E,E_{T})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.9.m9.2d">italic_T = ( italic_E , italic_E start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT )</annotation></semantics></math>, called a <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p6.22.3">join tree</em> of <math alttext="H" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.p6.10.m10.1a"><mi id="S2.SS0.SSS0.Px1.p6.10.m10.1.1" xref="S2.SS0.SSS0.Px1.p6.10.m10.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.10.m10.1b"><ci id="S2.SS0.SSS0.Px1.p6.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.10.m10.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.10.m10.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.10.m10.1d">italic_H</annotation></semantics></math>, with the running intersection property: for each vertex <math alttext="v\in V" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.p6.11.m11.1a"><mrow id="S2.SS0.SSS0.Px1.p6.11.m11.1.1" xref="S2.SS0.SSS0.Px1.p6.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p6.11.m11.1.1.2" xref="S2.SS0.SSS0.Px1.p6.11.m11.1.1.2.cmml">v</mi><mo id="S2.SS0.SSS0.Px1.p6.11.m11.1.1.1" xref="S2.SS0.SSS0.Px1.p6.11.m11.1.1.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.p6.11.m11.1.1.3" xref="S2.SS0.SSS0.Px1.p6.11.m11.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.11.m11.1b"><apply id="S2.SS0.SSS0.Px1.p6.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.11.m11.1.1"><in id="S2.SS0.SSS0.Px1.p6.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.11.m11.1.1.1"></in><ci id="S2.SS0.SSS0.Px1.p6.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.11.m11.1.1.2">𝑣</ci><ci id="S2.SS0.SSS0.Px1.p6.11.m11.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p6.11.m11.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.11.m11.1c">v\in V</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.11.m11.1d">italic_v ∈ italic_V</annotation></semantics></math>, the set of hyperedges that contain <math alttext="v" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.12.m12.1"><semantics id="S2.SS0.SSS0.Px1.p6.12.m12.1a"><mi id="S2.SS0.SSS0.Px1.p6.12.m12.1.1" xref="S2.SS0.SSS0.Px1.p6.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.12.m12.1b"><ci id="S2.SS0.SSS0.Px1.p6.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.12.m12.1d">italic_v</annotation></semantics></math> induces a connected subtree of <math alttext="T" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.13.m13.1"><semantics id="S2.SS0.SSS0.Px1.p6.13.m13.1a"><mi id="S2.SS0.SSS0.Px1.p6.13.m13.1.1" xref="S2.SS0.SSS0.Px1.p6.13.m13.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.13.m13.1b"><ci id="S2.SS0.SSS0.Px1.p6.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.13.m13.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.13.m13.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.13.m13.1d">italic_T</annotation></semantics></math>. If <math alttext="H" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.14.m14.1"><semantics id="S2.SS0.SSS0.Px1.p6.14.m14.1a"><mi id="S2.SS0.SSS0.Px1.p6.14.m14.1.1" xref="S2.SS0.SSS0.Px1.p6.14.m14.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.14.m14.1b"><ci id="S2.SS0.SSS0.Px1.p6.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.14.m14.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.14.m14.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.14.m14.1d">italic_H</annotation></semantics></math> is acyclic and <math alttext="S\subseteq V" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.15.m15.1"><semantics id="S2.SS0.SSS0.Px1.p6.15.m15.1a"><mrow id="S2.SS0.SSS0.Px1.p6.15.m15.1.1" xref="S2.SS0.SSS0.Px1.p6.15.m15.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p6.15.m15.1.1.2" xref="S2.SS0.SSS0.Px1.p6.15.m15.1.1.2.cmml">S</mi><mo id="S2.SS0.SSS0.Px1.p6.15.m15.1.1.1" xref="S2.SS0.SSS0.Px1.p6.15.m15.1.1.1.cmml">⊆</mo><mi id="S2.SS0.SSS0.Px1.p6.15.m15.1.1.3" xref="S2.SS0.SSS0.Px1.p6.15.m15.1.1.3.cmml">V</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.15.m15.1b"><apply id="S2.SS0.SSS0.Px1.p6.15.m15.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.15.m15.1.1"><subset id="S2.SS0.SSS0.Px1.p6.15.m15.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.15.m15.1.1.1"></subset><ci id="S2.SS0.SSS0.Px1.p6.15.m15.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.15.m15.1.1.2">𝑆</ci><ci id="S2.SS0.SSS0.Px1.p6.15.m15.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p6.15.m15.1.1.3">𝑉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.15.m15.1c">S\subseteq V</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.15.m15.1d">italic_S ⊆ italic_V</annotation></semantics></math>, then we say that <math alttext="H" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.16.m16.1"><semantics id="S2.SS0.SSS0.Px1.p6.16.m16.1a"><mi id="S2.SS0.SSS0.Px1.p6.16.m16.1.1" xref="S2.SS0.SSS0.Px1.p6.16.m16.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.16.m16.1b"><ci id="S2.SS0.SSS0.Px1.p6.16.m16.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.16.m16.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.16.m16.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.16.m16.1d">italic_H</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p6.17.1"><math alttext="S" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.17.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p6.17.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p6.17.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p6.17.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.17.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p6.17.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.17.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.17.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.17.1.m1.1d">italic_S</annotation></semantics></math>-connex</em> if <math alttext="H" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.18.m17.1"><semantics id="S2.SS0.SSS0.Px1.p6.18.m17.1a"><mi id="S2.SS0.SSS0.Px1.p6.18.m17.1.1" xref="S2.SS0.SSS0.Px1.p6.18.m17.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.18.m17.1b"><ci id="S2.SS0.SSS0.Px1.p6.18.m17.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.18.m17.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.18.m17.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.18.m17.1d">italic_H</annotation></semantics></math> remains acyclic even if we add <math alttext="S" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.19.m18.1"><semantics id="S2.SS0.SSS0.Px1.p6.19.m18.1a"><mi id="S2.SS0.SSS0.Px1.p6.19.m18.1.1" xref="S2.SS0.SSS0.Px1.p6.19.m18.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.19.m18.1b"><ci id="S2.SS0.SSS0.Px1.p6.19.m18.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.19.m18.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.19.m18.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.19.m18.1d">italic_S</annotation></semantics></math> to the set of hyperedges <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">braultbaron:tel-01081392</span>]</cite>. An acyclic CQ <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.20.m19.1"><semantics id="S2.SS0.SSS0.Px1.p6.20.m19.1a"><mi id="S2.SS0.SSS0.Px1.p6.20.m19.1.1" xref="S2.SS0.SSS0.Px1.p6.20.m19.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.20.m19.1b"><ci id="S2.SS0.SSS0.Px1.p6.20.m19.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.20.m19.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.20.m19.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.20.m19.1d">italic_Q</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p6.22.4">free-connex</em> if <math alttext="H(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.21.m20.1"><semantics id="S2.SS0.SSS0.Px1.p6.21.m20.1a"><mrow id="S2.SS0.SSS0.Px1.p6.21.m20.1.2" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p6.21.m20.1.2.2" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2.2.cmml">H</mi><mo id="S2.SS0.SSS0.Px1.p6.21.m20.1.2.1" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p6.21.m20.1.2.3.2" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p6.21.m20.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.21.m20.1.1" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p6.21.m20.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.21.m20.1b"><apply id="S2.SS0.SSS0.Px1.p6.21.m20.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2"><times id="S2.SS0.SSS0.Px1.p6.21.m20.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p6.21.m20.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.2.2">𝐻</ci><ci id="S2.SS0.SSS0.Px1.p6.21.m20.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.21.m20.1.1">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.21.m20.1c">H(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.21.m20.1d">italic_H ( italic_Q )</annotation></semantics></math> is acyclic and <math alttext="\mathrm{free}(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p6.22.m21.1"><semantics id="S2.SS0.SSS0.Px1.p6.22.m21.1a"><mrow id="S2.SS0.SSS0.Px1.p6.22.m21.1.2" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p6.22.m21.1.2.2" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2.2.cmml">free</mi><mo id="S2.SS0.SSS0.Px1.p6.22.m21.1.2.1" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p6.22.m21.1.2.3.2" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p6.22.m21.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p6.22.m21.1.1" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p6.22.m21.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p6.22.m21.1b"><apply id="S2.SS0.SSS0.Px1.p6.22.m21.1.2.cmml" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2"><times id="S2.SS0.SSS0.Px1.p6.22.m21.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p6.22.m21.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.2.2">free</ci><ci id="S2.SS0.SSS0.Px1.p6.22.m21.1.1.cmml" xref="S2.SS0.SSS0.Px1.p6.22.m21.1.1">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p6.22.m21.1c">\mathrm{free}(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p6.22.m21.1d">roman_free ( italic_Q )</annotation></semantics></math>-connex.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p7"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p7.14">A hypergraph <math alttext="H^{\prime}=(V,E^{\prime})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.1.m1.2"><semantics id="S2.SS0.SSS0.Px1.p7.1.m1.2a"><mrow id="S2.SS0.SSS0.Px1.p7.1.m1.2.2" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.cmml"><msup id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.2" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.2.cmml">H</mi><mo id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.3" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.3.cmml">′</mo></msup><mo id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.2" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.2.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p7.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p7.1.m1.1.1.cmml">V</mi><mo id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.3" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.2.cmml">,</mo><msup id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.2.cmml">E</mi><mo id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.4" stretchy="false" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.1.m1.2b"><apply id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2"><eq id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.2"></eq><apply id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.2">𝐻</ci><ci id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.3.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.3.3">′</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.2.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1"><ci id="S2.SS0.SSS0.Px1.p7.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.1.1">𝑉</ci><apply id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.2">𝐸</ci><ci id="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p7.1.m1.2.2.1.1.1.3">′</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.1.m1.2c">H^{\prime}=(V,E^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.1.m1.2d">italic_H start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_V , italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> is an <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p7.14.1">inclusive extension</em> of a hypergraph <math alttext="H=(V,E)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.2.m2.2"><semantics id="S2.SS0.SSS0.Px1.p7.2.m2.2a"><mrow id="S2.SS0.SSS0.Px1.p7.2.m2.2.3" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.2" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.2.cmml">H</mi><mo id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.1" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.2" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p7.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p7.2.m2.1.1.cmml">V</mi><mo id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.p7.2.m2.2.2" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.2.cmml">E</mi><mo id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.2.m2.2b"><apply id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3"><eq id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.1"></eq><ci id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.2">𝐻</ci><interval closure="open" id="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.3.3.2"><ci id="S2.SS0.SSS0.Px1.p7.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.2.m2.1.1">𝑉</ci><ci id="S2.SS0.SSS0.Px1.p7.2.m2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p7.2.m2.2.2">𝐸</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.2.m2.2c">H=(V,E)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.2.m2.2d">italic_H = ( italic_V , italic_E )</annotation></semantics></math> if <math alttext="E\subseteq E^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p7.3.m3.1a"><mrow id="S2.SS0.SSS0.Px1.p7.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.2.cmml">E</mi><mo id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.1" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.1.cmml">⊆</mo><msup id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.2" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.2.cmml">E</mi><mo id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.3" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1"><subset id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.1"></subset><ci id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.2">𝐸</ci><apply id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.2">𝐸</ci><ci id="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px1.p7.3.m3.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.3.m3.1c">E\subseteq E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.3.m3.1d">italic_E ⊆ italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and for every edge <math alttext="e^{\prime}\in E^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.p7.4.m4.1a"><mrow id="S2.SS0.SSS0.Px1.p7.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.cmml"><msup id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.2" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.2.cmml">e</mi><mo id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.3" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.1" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.1.cmml">∈</mo><msup id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.2" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.2.cmml">E</mi><mo id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.3" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.4.m4.1b"><apply id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1"><in id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.1"></in><apply id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.2">𝑒</ci><ci id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.2.3">′</ci></apply><apply id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.2">𝐸</ci><ci id="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px1.p7.4.m4.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.4.m4.1c">e^{\prime}\in E^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.4.m4.1d">italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∈ italic_E start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> there is an edge <math alttext="e\in E" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.p7.5.m5.1a"><mrow id="S2.SS0.SSS0.Px1.p7.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p7.5.m5.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p7.5.m5.1.1.2" xref="S2.SS0.SSS0.Px1.p7.5.m5.1.1.2.cmml">e</mi><mo id="S2.SS0.SSS0.Px1.p7.5.m5.1.1.1" xref="S2.SS0.SSS0.Px1.p7.5.m5.1.1.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.p7.5.m5.1.1.3" xref="S2.SS0.SSS0.Px1.p7.5.m5.1.1.3.cmml">E</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.5.m5.1b"><apply id="S2.SS0.SSS0.Px1.p7.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.5.m5.1.1"><in id="S2.SS0.SSS0.Px1.p7.5.m5.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.5.m5.1.1.1"></in><ci id="S2.SS0.SSS0.Px1.p7.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p7.5.m5.1.1.2">𝑒</ci><ci id="S2.SS0.SSS0.Px1.p7.5.m5.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p7.5.m5.1.1.3">𝐸</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.5.m5.1c">e\in E</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.5.m5.1d">italic_e ∈ italic_E</annotation></semantics></math> such that <math alttext="e^{\prime}\subseteq e" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.p7.6.m6.1a"><mrow id="S2.SS0.SSS0.Px1.p7.6.m6.1.1" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.cmml"><msup id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.2" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.2.cmml">e</mi><mo id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.3" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.1" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.1.cmml">⊆</mo><mi id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.3" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.3.cmml">e</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1"><subset id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.1"></subset><apply id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.2">𝑒</ci><ci id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.2.3">′</ci></apply><ci id="S2.SS0.SSS0.Px1.p7.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.p7.6.m6.1.1.3">𝑒</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.6.m6.1c">e^{\prime}\subseteq e</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.6.m6.1d">italic_e start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊆ italic_e</annotation></semantics></math>. It is known that <math alttext="H" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.p7.7.m7.1a"><mi id="S2.SS0.SSS0.Px1.p7.7.m7.1.1" xref="S2.SS0.SSS0.Px1.p7.7.m7.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.7.m7.1b"><ci id="S2.SS0.SSS0.Px1.p7.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.7.m7.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.7.m7.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.7.m7.1d">italic_H</annotation></semantics></math> is acyclic <math alttext="S" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.p7.8.m8.1a"><mi id="S2.SS0.SSS0.Px1.p7.8.m8.1.1" xref="S2.SS0.SSS0.Px1.p7.8.m8.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.8.m8.1b"><ci id="S2.SS0.SSS0.Px1.p7.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.8.m8.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.8.m8.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.8.m8.1d">italic_S</annotation></semantics></math>-connex if and only if <math alttext="H" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.p7.9.m9.1a"><mi id="S2.SS0.SSS0.Px1.p7.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p7.9.m9.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.9.m9.1b"><ci id="S2.SS0.SSS0.Px1.p7.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.9.m9.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.9.m9.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.9.m9.1d">italic_H</annotation></semantics></math> has an inclusive extension with a join tree <math alttext="T" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.p7.10.m10.1a"><mi id="S2.SS0.SSS0.Px1.p7.10.m10.1.1" xref="S2.SS0.SSS0.Px1.p7.10.m10.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.10.m10.1b"><ci id="S2.SS0.SSS0.Px1.p7.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.10.m10.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.10.m10.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.10.m10.1d">italic_T</annotation></semantics></math> such that <math alttext="S" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.p7.11.m11.1a"><mi id="S2.SS0.SSS0.Px1.p7.11.m11.1.1" xref="S2.SS0.SSS0.Px1.p7.11.m11.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.11.m11.1b"><ci id="S2.SS0.SSS0.Px1.p7.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.11.m11.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.11.m11.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.11.m11.1d">italic_S</annotation></semantics></math> is precisely the set of all variables contained in the vertices of some subtree of <math alttext="T" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.12.m12.1"><semantics id="S2.SS0.SSS0.Px1.p7.12.m12.1a"><mi id="S2.SS0.SSS0.Px1.p7.12.m12.1.1" xref="S2.SS0.SSS0.Px1.p7.12.m12.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.12.m12.1b"><ci id="S2.SS0.SSS0.Px1.p7.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.12.m12.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.12.m12.1c">T</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.12.m12.1d">italic_T</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1007/978-3-540-74915-8_18</span>]</cite>. We call such a tree ext-<math alttext="S" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.13.m13.1"><semantics id="S2.SS0.SSS0.Px1.p7.13.m13.1a"><mi id="S2.SS0.SSS0.Px1.p7.13.m13.1.1" xref="S2.SS0.SSS0.Px1.p7.13.m13.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.13.m13.1b"><ci id="S2.SS0.SSS0.Px1.p7.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.13.m13.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.13.m13.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.13.m13.1d">italic_S</annotation></semantics></math>-connex tree. When <math alttext="S" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p7.14.m14.1"><semantics id="S2.SS0.SSS0.Px1.p7.14.m14.1a"><mi id="S2.SS0.SSS0.Px1.p7.14.m14.1.1" xref="S2.SS0.SSS0.Px1.p7.14.m14.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p7.14.m14.1b"><ci id="S2.SS0.SSS0.Px1.p7.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.p7.14.m14.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p7.14.m14.1c">S</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p7.14.m14.1d">italic_S</annotation></semantics></math> is the set of free variables of the CQ, and the CQ is clear from the context, we call such a tree <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p7.14.2">ext-free-connex</em>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p8"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p8.11">Let <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p8.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.p8.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p8.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.p8.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.1.m1.1d">italic_Q</annotation></semantics></math> be a CQ. Two variables of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.p8.2.m2.1a"><mi id="S2.SS0.SSS0.Px1.p8.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p8.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.2.m2.1b"><ci id="S2.SS0.SSS0.Px1.p8.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.2.m2.1d">italic_Q</annotation></semantics></math> are <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p8.11.1">neighbors</em> (in <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p8.3.m3.1a"><mi id="S2.SS0.SSS0.Px1.p8.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p8.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.p8.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.3.m3.1d">italic_Q</annotation></semantics></math>) if they occur jointly in at least one of the atoms. If <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.p8.4.m4.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p8.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p8.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p8.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.p8.4.m4.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p8.4.m4.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.4.m4.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.4.m4.1b"><apply id="S2.SS0.SSS0.Px1.p8.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.4.m4.1.1"><ci id="S2.SS0.SSS0.Px1.p8.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.4.m4.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p8.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.4.m4.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.4.m4.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.4.m4.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> is a set of variables in <math alttext="\mathrm{vars}(Q)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.p8.5.m5.1a"><mrow id="S2.SS0.SSS0.Px1.p8.5.m5.1.2" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p8.5.m5.1.2.2" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2.2.cmml">vars</mi><mo id="S2.SS0.SSS0.Px1.p8.5.m5.1.2.1" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p8.5.m5.1.2.3.2" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.p8.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.p8.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.1.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p8.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.5.m5.1b"><apply id="S2.SS0.SSS0.Px1.p8.5.m5.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2"><times id="S2.SS0.SSS0.Px1.p8.5.m5.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p8.5.m5.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.2.2">vars</ci><ci id="S2.SS0.SSS0.Px1.p8.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.5.m5.1.1">𝑄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.5.m5.1c">\mathrm{vars}(Q)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.5.m5.1d">roman_vars ( italic_Q )</annotation></semantics></math>, then we denote by <math alttext="\mathsf{N}_{Q}(\vec{x})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.p8.6.m6.1a"><mrow id="S2.SS0.SSS0.Px1.p8.6.m6.1.2" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.cmml"><msub id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.2" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.2.cmml">𝖭</mi><mi id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.3" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.3.cmml">Q</mi></msub><mo id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.1" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.3.2" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p8.6.m6.1.1" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p8.6.m6.1.1.2" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p8.6.m6.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2"><times id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.1"></times><apply id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.2">𝖭</ci><ci id="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.2.3">𝑄</ci></apply><apply id="S2.SS0.SSS0.Px1.p8.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.p8.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p8.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.6.m6.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.6.m6.1c">\mathsf{N}_{Q}(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.6.m6.1d">sansserif_N start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math> the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p8.11.2">neighborhood</em> of <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.p8.7.m7.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p8.7.m7.1.1" xref="S2.SS0.SSS0.Px1.p8.7.m7.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p8.7.m7.1.1.2" xref="S2.SS0.SSS0.Px1.p8.7.m7.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p8.7.m7.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.7.m7.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.7.m7.1b"><apply id="S2.SS0.SSS0.Px1.p8.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.7.m7.1.1"><ci id="S2.SS0.SSS0.Px1.p8.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.7.m7.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p8.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.7.m7.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.7.m7.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.7.m7.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> that consists of every variable in <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.p8.8.m8.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p8.8.m8.1.1" xref="S2.SS0.SSS0.Px1.p8.8.m8.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p8.8.m8.1.1.2" xref="S2.SS0.SSS0.Px1.p8.8.m8.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p8.8.m8.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.8.m8.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.8.m8.1b"><apply id="S2.SS0.SSS0.Px1.p8.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.8.m8.1.1"><ci id="S2.SS0.SSS0.Px1.p8.8.m8.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.8.m8.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p8.8.m8.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.8.m8.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.8.m8.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.8.m8.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> and every neighbor of every variable in <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.p8.9.m9.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p8.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p8.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p8.9.m9.1.1.2" xref="S2.SS0.SSS0.Px1.p8.9.m9.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p8.9.m9.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.9.m9.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.9.m9.1b"><apply id="S2.SS0.SSS0.Px1.p8.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.9.m9.1.1"><ci id="S2.SS0.SSS0.Px1.p8.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.9.m9.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p8.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.9.m9.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.9.m9.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.9.m9.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math>. If <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.p8.10.m10.1a"><mi id="S2.SS0.SSS0.Px1.p8.10.m10.1.1" xref="S2.SS0.SSS0.Px1.p8.10.m10.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.10.m10.1b"><ci id="S2.SS0.SSS0.Px1.p8.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.10.m10.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.10.m10.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.10.m10.1d">italic_Q</annotation></semantics></math> is clear from the context, then we may omit it and write just <math alttext="\mathsf{N}(\vec{x})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p8.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.p8.11.m11.1a"><mrow id="S2.SS0.SSS0.Px1.p8.11.m11.1.2" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p8.11.m11.1.2.2" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.2.2.cmml">𝖭</mi><mo id="S2.SS0.SSS0.Px1.p8.11.m11.1.2.1" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p8.11.m11.1.2.3.2" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p8.11.m11.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p8.11.m11.1.1" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p8.11.m11.1.1.2" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p8.11.m11.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p8.11.m11.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p8.11.m11.1b"><apply id="S2.SS0.SSS0.Px1.p8.11.m11.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.2"><times id="S2.SS0.SSS0.Px1.p8.11.m11.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p8.11.m11.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.2.2">𝖭</ci><apply id="S2.SS0.SSS0.Px1.p8.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.p8.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p8.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p8.11.m11.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p8.11.m11.1c">\mathsf{N}(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p8.11.m11.1d">sansserif_N ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p9"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p9.11">The notion of a <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.p9.11.1">disruptive trio</em> has been introduced previously in the context of direct access to the answers of CQs <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/CarmeliTGKR21</span>]</cite>. A disruptive trio of a CQ <math alttext="Q(\vec{x})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.p9.1.m1.1a"><mrow id="S2.SS0.SSS0.Px1.p9.1.m1.1.2" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.p9.1.m1.1.2.2" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.2.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.p9.1.m1.1.2.1" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.p9.1.m1.1.2.3.2" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.p9.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.p9.1.m1.1.1" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.1.m1.1.1.2" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p9.1.m1.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.p9.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.1.m1.1b"><apply id="S2.SS0.SSS0.Px1.p9.1.m1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.2"><times id="S2.SS0.SSS0.Px1.p9.1.m1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.p9.1.m1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.2.2">𝑄</ci><apply id="S2.SS0.SSS0.Px1.p9.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.p9.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p9.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.1.m1.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.1.m1.1c">Q(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.1.m1.1d">italic_Q ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math> is a set of three distinct free variables <math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.p9.2.m2.1a"><msub id="S2.SS0.SSS0.Px1.p9.2.m2.1.1" xref="S2.SS0.SSS0.Px1.p9.2.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.2.m2.1.1.2" xref="S2.SS0.SSS0.Px1.p9.2.m2.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.2.m2.1.1.3" xref="S2.SS0.SSS0.Px1.p9.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.2.m2.1b"><apply id="S2.SS0.SSS0.Px1.p9.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.2.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.2.m2.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.2.m2.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.2.m2.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.2.m2.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.2.m2.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="x_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.p9.3.m3.1a"><msub id="S2.SS0.SSS0.Px1.p9.3.m3.1.1" xref="S2.SS0.SSS0.Px1.p9.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.p9.3.m3.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.3.m3.1.1.3" xref="S2.SS0.SSS0.Px1.p9.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.p9.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.3.m3.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.3.m3.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.3.m3.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.3.m3.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.3.m3.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="x_{3}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.p9.4.m4.1a"><msub id="S2.SS0.SSS0.Px1.p9.4.m4.1.1" xref="S2.SS0.SSS0.Px1.p9.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.p9.4.m4.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.4.m4.1.1.3" xref="S2.SS0.SSS0.Px1.p9.4.m4.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.4.m4.1b"><apply id="S2.SS0.SSS0.Px1.p9.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.4.m4.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.4.m4.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.4.m4.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.4.m4.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.4.m4.1c">x_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.4.m4.1d">italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.p9.5.m5.1a"><msub id="S2.SS0.SSS0.Px1.p9.5.m5.1.1" xref="S2.SS0.SSS0.Px1.p9.5.m5.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.5.m5.1.1.2" xref="S2.SS0.SSS0.Px1.p9.5.m5.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.5.m5.1.1.3" xref="S2.SS0.SSS0.Px1.p9.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.5.m5.1b"><apply id="S2.SS0.SSS0.Px1.p9.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.5.m5.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.5.m5.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.5.m5.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.5.m5.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.5.m5.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.5.m5.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.5.m5.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.p9.6.m6.1a"><msub id="S2.SS0.SSS0.Px1.p9.6.m6.1.1" xref="S2.SS0.SSS0.Px1.p9.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.6.m6.1.1.2" xref="S2.SS0.SSS0.Px1.p9.6.m6.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.6.m6.1.1.3" xref="S2.SS0.SSS0.Px1.p9.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.p9.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.6.m6.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.6.m6.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.6.m6.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.6.m6.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.6.m6.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> neighbor <math alttext="x_{3}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.p9.7.m7.1a"><msub id="S2.SS0.SSS0.Px1.p9.7.m7.1.1" xref="S2.SS0.SSS0.Px1.p9.7.m7.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.7.m7.1.1.2" xref="S2.SS0.SSS0.Px1.p9.7.m7.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.7.m7.1.1.3" xref="S2.SS0.SSS0.Px1.p9.7.m7.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.7.m7.1b"><apply id="S2.SS0.SSS0.Px1.p9.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.7.m7.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.7.m7.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.7.m7.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.7.m7.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.7.m7.1c">x_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.7.m7.1d">italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> but not each other, and <math alttext="x_{3}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.p9.8.m8.1a"><msub id="S2.SS0.SSS0.Px1.p9.8.m8.1.1" xref="S2.SS0.SSS0.Px1.p9.8.m8.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.8.m8.1.1.2" xref="S2.SS0.SSS0.Px1.p9.8.m8.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.8.m8.1.1.3" xref="S2.SS0.SSS0.Px1.p9.8.m8.1.1.3.cmml">3</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.8.m8.1b"><apply id="S2.SS0.SSS0.Px1.p9.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.8.m8.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.8.m8.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.8.m8.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.8.m8.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.8.m8.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.8.m8.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.8.m8.1c">x_{3}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.8.m8.1d">italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT</annotation></semantics></math> succeeds both <math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.p9.9.m9.1a"><msub id="S2.SS0.SSS0.Px1.p9.9.m9.1.1" xref="S2.SS0.SSS0.Px1.p9.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.9.m9.1.1.2" xref="S2.SS0.SSS0.Px1.p9.9.m9.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.9.m9.1.1.3" xref="S2.SS0.SSS0.Px1.p9.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.9.m9.1b"><apply id="S2.SS0.SSS0.Px1.p9.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.9.m9.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.9.m9.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.9.m9.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.9.m9.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.9.m9.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.p9.10.m10.1a"><msub id="S2.SS0.SSS0.Px1.p9.10.m10.1.1" xref="S2.SS0.SSS0.Px1.p9.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.10.m10.1.1.2" xref="S2.SS0.SSS0.Px1.p9.10.m10.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.p9.10.m10.1.1.3" xref="S2.SS0.SSS0.Px1.p9.10.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.10.m10.1b"><apply id="S2.SS0.SSS0.Px1.p9.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.p9.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.10.m10.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.p9.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.10.m10.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.p9.10.m10.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.p9.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.10.m10.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.10.m10.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.p9.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.p9.11.m11.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.p9.11.m11.1.1" xref="S2.SS0.SSS0.Px1.p9.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.p9.11.m11.1.1.2" xref="S2.SS0.SSS0.Px1.p9.11.m11.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.p9.11.m11.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.p9.11.m11.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.p9.11.m11.1b"><apply id="S2.SS0.SSS0.Px1.p9.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.11.m11.1.1"><ci id="S2.SS0.SSS0.Px1.p9.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.p9.11.m11.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.p9.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.p9.11.m11.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.p9.11.m11.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.p9.11.m11.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math>.</p> </div> <section class="ltx_subparagraph" id="S2.SS0.SSS0.Px1.SPx1"> <h5 class="ltx_title ltx_title_subparagraph">Aggregate queries.</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx1.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx1.p1.2">By an <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx1.p1.2.1">aggregate function</em> we refer to a function that takes as input a bag of tuples over <math alttext="\mathsf{Const}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1.1.cmml">𝖢𝗈𝗇𝗌𝗍</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1.1">𝖢𝗈𝗇𝗌𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1c">\mathsf{Const}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.1.m1.1d">sansserif_Const</annotation></semantics></math> and returns a single value in <math alttext="\mathsf{Const}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1.1.cmml">𝖢𝗈𝗇𝗌𝗍</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1.1">𝖢𝗈𝗇𝗌𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1c">\mathsf{Const}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.2.m2.1d">sansserif_Const</annotation></semantics></math>. We adopt the notation of Cohen et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1145/1219092.1219093</span>]</cite>, as follows. An <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx1.p1.2.2">aggregate query</em> here is an expression of the form</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="Q(\vec{x},\alpha(\vec{w}),\vec{z}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y},\vec% {z}),\dots,\varphi_{\ell}(\vec{x},\vec{y},\vec{z})" class="ltx_Math" display="block" id="S2.Ex4.m1.13"><semantics id="S2.Ex4.m1.13a"><mrow id="S2.Ex4.m1.13.13" xref="S2.Ex4.m1.13.13.cmml"><mrow id="S2.Ex4.m1.11.11.1" xref="S2.Ex4.m1.11.11.1.cmml"><mi id="S2.Ex4.m1.11.11.1.3" xref="S2.Ex4.m1.11.11.1.3.cmml">Q</mi><mo id="S2.Ex4.m1.11.11.1.2" xref="S2.Ex4.m1.11.11.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m1.11.11.1.1.1" xref="S2.Ex4.m1.11.11.1.1.2.cmml"><mo id="S2.Ex4.m1.11.11.1.1.1.2" stretchy="false" xref="S2.Ex4.m1.11.11.1.1.2.cmml">(</mo><mover accent="true" id="S2.Ex4.m1.2.2" xref="S2.Ex4.m1.2.2.cmml"><mi id="S2.Ex4.m1.2.2.2" xref="S2.Ex4.m1.2.2.2.cmml">x</mi><mo id="S2.Ex4.m1.2.2.1" stretchy="false" xref="S2.Ex4.m1.2.2.1.cmml">→</mo></mover><mo id="S2.Ex4.m1.11.11.1.1.1.3" xref="S2.Ex4.m1.11.11.1.1.2.cmml">,</mo><mrow id="S2.Ex4.m1.11.11.1.1.1.1" xref="S2.Ex4.m1.11.11.1.1.1.1.cmml"><mi id="S2.Ex4.m1.11.11.1.1.1.1.2" xref="S2.Ex4.m1.11.11.1.1.1.1.2.cmml">α</mi><mo id="S2.Ex4.m1.11.11.1.1.1.1.1" xref="S2.Ex4.m1.11.11.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex4.m1.11.11.1.1.1.1.3.2" xref="S2.Ex4.m1.1.1.cmml"><mo id="S2.Ex4.m1.11.11.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex4.m1.1.1.cmml">(</mo><mover accent="true" id="S2.Ex4.m1.1.1" xref="S2.Ex4.m1.1.1.cmml"><mi id="S2.Ex4.m1.1.1.2" xref="S2.Ex4.m1.1.1.2.cmml">w</mi><mo id="S2.Ex4.m1.1.1.1" stretchy="false" xref="S2.Ex4.m1.1.1.1.cmml">→</mo></mover><mo id="S2.Ex4.m1.11.11.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex4.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.11.11.1.1.1.4" xref="S2.Ex4.m1.11.11.1.1.2.cmml">,</mo><mover accent="true" id="S2.Ex4.m1.3.3" xref="S2.Ex4.m1.3.3.cmml"><mi id="S2.Ex4.m1.3.3.2" xref="S2.Ex4.m1.3.3.2.cmml">z</mi><mo id="S2.Ex4.m1.3.3.1" stretchy="false" xref="S2.Ex4.m1.3.3.1.cmml">→</mo></mover><mo id="S2.Ex4.m1.11.11.1.1.1.5" rspace="0.448em" stretchy="false" xref="S2.Ex4.m1.11.11.1.1.2.cmml">)</mo></mrow></mrow><mpadded width="0.226em"><mo id="S2.Ex4.m1.13.13.4" xref="S2.Ex4.m1.13.13.4.cmml">:</mo></mpadded><mrow id="S2.Ex4.m1.13.13.3.2" xref="S2.Ex4.m1.13.13.3.3.cmml"><mrow id="S2.Ex4.m1.12.12.2.1.1" xref="S2.Ex4.m1.12.12.2.1.1.cmml"><mo id="S2.Ex4.m1.12.12.2.1.1a" xref="S2.Ex4.m1.12.12.2.1.1.cmml">−</mo><mrow id="S2.Ex4.m1.12.12.2.1.1.2" xref="S2.Ex4.m1.12.12.2.1.1.2.cmml"><msub id="S2.Ex4.m1.12.12.2.1.1.2.2" xref="S2.Ex4.m1.12.12.2.1.1.2.2.cmml"><mi id="S2.Ex4.m1.12.12.2.1.1.2.2.2" xref="S2.Ex4.m1.12.12.2.1.1.2.2.2.cmml">φ</mi><mn id="S2.Ex4.m1.12.12.2.1.1.2.2.3" xref="S2.Ex4.m1.12.12.2.1.1.2.2.3.cmml">1</mn></msub><mo 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encoding="application/x-llamapun" id="S2.Ex4.m1.13d">italic_Q ( over→ start_ARG italic_x end_ARG , italic_α ( over→ start_ARG italic_w end_ARG ) , over→ start_ARG italic_z end_ARG ) : - italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG , over→ start_ARG italic_z end_ARG ) , … , italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG , over→ start_ARG italic_z end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx1.p1.16">such that <math alttext="Q^{\prime}(\vec{x},\vec{z}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y},\vec{z}),% \dots,\varphi_{\ell}(\vec{x},\vec{y},\vec{z})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11" 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xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1"><minus id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1"></minus><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2"><times id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.1"></times><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.2.3">1</cn></apply><vector id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.10.10.1.1.1.2.3.2"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.3.3"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.3.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.3.3.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.4.4.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.4.4"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.4.4.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.4.4.2">𝑦</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.5.5.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.5.5"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.5.5.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.5.5.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.5.5.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.5.5.2">𝑧</ci></apply></vector></apply></apply><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.9.9.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.9.9">…</ci><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2"><times id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.1"></times><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.2.3">ℓ</ci></apply><vector id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11.11.2.2.2.3.2"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.6.6.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.6.6"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.6.6.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.6.6.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.6.6.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.6.6.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.7.7.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.7.7"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.7.7.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.7.7.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.7.7.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.7.7.2">𝑦</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.8.8.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.8.8"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.8.8.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.8.8.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.8.8.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.8.8.2">𝑧</ci></apply></vector></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11c">Q^{\prime}(\vec{x},\vec{z}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y},\vec{z}),% \dots,\varphi_{\ell}(\vec{x},\vec{y},\vec{z})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.3.m1.11d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_z end_ARG ) : - italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG , over→ start_ARG italic_z end_ARG ) , … , italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG , over→ start_ARG italic_z end_ARG )</annotation></semantics></math> is a CQ, <math alttext="\alpha" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.4.m2.1d">italic_α</annotation></semantics></math> an aggregate function, and <math alttext="\vec{w}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.2.cmml">w</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1.1.2">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1c">\vec{w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.5.m3.1d">over→ start_ARG italic_w end_ARG</annotation></semantics></math> a sequence of variables from <math alttext="\vec{y}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1.1.2">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1c">\vec{y}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.6.m4.1d">over→ start_ARG italic_y end_ARG</annotation></semantics></math>. An example is <math alttext="Q(x_{1},x_{2},\mathsf{Sum}(y_{2}),z){\,:\!\!-\,}R(x_{1},x_{2},y_{1}),S(y_{1},y% _{2},z)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.5" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.5.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.4" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.4.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.4.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.3.4" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.4.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.3.3.1.1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.3.3.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.3.3.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.3.3.1.1.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.3.3.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.3.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.3.5" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.4.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.4.4.2.2.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.4.4.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.4.4.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.4.4.2.2.2.2.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.4.4.2.2.2.2.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.4.4.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.3.6" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.4.cmml">,</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.3.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.5.5.3.3.3.3.3" 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id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.2.2.2.2.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.2.2.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.2.2.2.2.3">2</cn></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.3.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.3.3.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.3.3.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.3.3.3.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.3.3.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.3.3.3.3.2">𝑦</ci><cn id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.3.3.3.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.6.6.4.1.1.3.3.3.3.3">1</cn></apply></vector></apply></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2"><times id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.3"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.4.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.4">𝑆</ci><vector id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.1.1.1.2">𝑦</ci><cn id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.1.1.1.3">1</cn></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2.2.2">𝑦</ci><cn id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7.7.5.2.2.2.2.2.3">2</cn></apply><ci id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.2.2">𝑧</ci></vector></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7c">Q(x_{1},x_{2},\mathsf{Sum}(y_{2}),z){\,:\!\!-\,}R(x_{1},x_{2},y_{1}),S(y_{1},y% _{2},z)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.7.m5.7d">italic_Q ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , sansserif_Sum ( italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , italic_z ) : - italic_R ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , italic_S ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z )</annotation></semantics></math>. We refer to such a query as an <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx1.p1.16.1">Aggregate CQ</em> or <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx1.p1.16.2">AggCQ</em> for short. Given a database <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.8.m6.1d">italic_D</annotation></semantics></math> over <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1a"><msup id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.9.m7.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, the result <math alttext="Q(D)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2"><times id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.2.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1c">Q(D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.10.m8.1d">italic_Q ( italic_D )</annotation></semantics></math> is defined by <math alttext="Q(D)\vcentcolon=\{(\vec{a},\alpha(B(\vec{a},\vec{b})),\vec{b})\mid(\vec{a},% \vec{b})\in Q^{\prime}(D)\}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.4.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.3" rspace="0.278em" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.3.cmml">:=</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.3.1.cmml">{</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.2.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.2.cmml">,</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.3" 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id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.4.4.2">𝑎</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1"><times id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.2"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.3">𝛼</ci><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.1.1"><times id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.1.1.1.1"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.1.1.1.2">𝐵</ci><interval closure="open" id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.1.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.9.9.1.1.1.1.1.1.1.1.3.2"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.2.2"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.2.2.2">𝑎</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.3.3"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.3.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.3.3.2">𝑏</ci></apply></interval></apply></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.5.5.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.5.5"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.5.5.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.5.5.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.5.5.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.5.5.2">𝑏</ci></apply></vector><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2"><in id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.1"></in><interval closure="open" id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.2.2"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.6.6.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.6.6"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.6.6.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.6.6.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.6.6.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.6.6.2">𝑎</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.7.7.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.7.7"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.7.7.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.7.7.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.7.7.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.7.7.2">𝑏</ci></apply></interval><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3"><times id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.1"></times><apply id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.2">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.2.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10.10.2.2.2.3.2.3">′</ci></apply><ci id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.8.8.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.8.8">𝐷</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10c">Q(D)\vcentcolon=\{(\vec{a},\alpha(B(\vec{a},\vec{b})),\vec{b})\mid(\vec{a},% \vec{b})\in Q^{\prime}(D)\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.11.m9.10d">italic_Q ( italic_D ) := { ( over→ start_ARG italic_a end_ARG , italic_α ( italic_B ( over→ start_ARG italic_a end_ARG , over→ start_ARG italic_b end_ARG ) ) , over→ start_ARG italic_b end_ARG ) ∣ ( over→ start_ARG italic_a end_ARG , over→ start_ARG italic_b end_ARG ) ∈ italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_D ) }</annotation></semantics></math> where <math alttext="B(\vec{a},\vec{b})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.2.cmml">B</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.2.cmml">b</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3"><times id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.2">𝐵</ci><interval closure="open" id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.3.3.2"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.1.1.2">𝑎</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2.2.2">𝑏</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2c">B(\vec{a},\vec{b})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.12.m10.2d">italic_B ( over→ start_ARG italic_a end_ARG , over→ start_ARG italic_b end_ARG )</annotation></semantics></math> is the bag that is obtained by collecting the tuples <math alttext="h(\vec{w})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.2.cmml">w</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2"><times id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1.1.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1c">h(\vec{w})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.13.m11.1d">italic_h ( over→ start_ARG italic_w end_ARG )</annotation></semantics></math> from every <math alttext="h\in\mathsf{Hom}(Q^{\prime},D)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.3.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.2.cmml">∈</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.3.cmml">𝖧𝗈𝗆</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.2.cmml">(</mo><msup id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.2.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.4" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2"><in id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.2"></in><ci id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.3">ℎ</ci><apply id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1"><times id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.2"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.3">𝖧𝗈𝗆</ci><interval closure="open" id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2.2.1.1.1.1.3">′</ci></apply><ci id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.1.1">𝐷</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2c">h\in\mathsf{Hom}(Q^{\prime},D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.14.m12.2d">italic_h ∈ sansserif_Hom ( italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_D )</annotation></semantics></math> with <math alttext="h(\vec{x})=\vec{a}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.1.cmml">=</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.1.cmml">→</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2"><eq id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.1"></eq><apply id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2"><times id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.1"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.1.2">𝑥</ci></apply></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1.2.3.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1c">h(\vec{x})=\vec{a}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.15.m13.1d">italic_h ( over→ start_ARG italic_x end_ARG ) = over→ start_ARG italic_a end_ARG</annotation></semantics></math> and <math alttext="h(\vec{z})=\vec{b}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.2.cmml">h</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.2.cmml">z</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.1" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.1.cmml">=</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.2.cmml">b</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.1.cmml">→</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2"><eq id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.1"></eq><apply id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2"><times id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.1"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.2">ℎ</ci><apply id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.1.2">𝑧</ci></apply></apply><apply id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3"><ci id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1.2.3.2">𝑏</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1c">h(\vec{z})=\vec{b}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p1.16.m14.1d">italic_h ( over→ start_ARG italic_z end_ARG ) = over→ start_ARG italic_b end_ARG</annotation></semantics></math>. Note that our database and query model use set semantics, and we use bag semantics only to define the aggregate functions (in order to capture important functions such as count and sum).</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx1.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx1.p2.6">We say that <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p2.1.m1.1d">italic_Q</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx1.p2.6.1">acyclic</em> if <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1a"><msup id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p2.2.m2.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is acyclic. Similarly, <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p2.3.m3.1d">italic_Q</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx1.p2.6.2">free-connex</em> if <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1a"><msup id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p2.4.m4.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is free-connex. A <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx1.p2.6.3">disruptive trio</em> of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p2.5.m5.1d">italic_Q</annotation></semantics></math> is a disruptive trio of <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1a"><msup id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p2.6.m6.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>; in other words, the definition of a disruptive trio remains unchanged when introducing aggregates, while we consider only the free variables and not the aggregate function.</p> </div> <div class="ltx_theorem ltx_theorem_rem" id="Thmthm1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm1.1.1.1">Remark 1</span></span><span class="ltx_text ltx_font_bold" id="Thmthm1.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm1.p1"> <p class="ltx_p" id="Thmthm1.p1.5"><span class="ltx_text ltx_font_italic" id="Thmthm1.p1.5.5">We remark on two aspects in our definition of AggCQs. First, the reason for using both <math alttext="\vec{x}" class="ltx_Math" display="inline" id="Thmthm1.p1.1.1.m1.1"><semantics id="Thmthm1.p1.1.1.m1.1a"><mover accent="true" id="Thmthm1.p1.1.1.m1.1.1" xref="Thmthm1.p1.1.1.m1.1.1.cmml"><mi id="Thmthm1.p1.1.1.m1.1.1.2" xref="Thmthm1.p1.1.1.m1.1.1.2.cmml">x</mi><mo id="Thmthm1.p1.1.1.m1.1.1.1" stretchy="false" xref="Thmthm1.p1.1.1.m1.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="Thmthm1.p1.1.1.m1.1b"><apply id="Thmthm1.p1.1.1.m1.1.1.cmml" xref="Thmthm1.p1.1.1.m1.1.1"><ci id="Thmthm1.p1.1.1.m1.1.1.1.cmml" xref="Thmthm1.p1.1.1.m1.1.1.1">→</ci><ci id="Thmthm1.p1.1.1.m1.1.1.2.cmml" xref="Thmthm1.p1.1.1.m1.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm1.p1.1.1.m1.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="Thmthm1.p1.1.1.m1.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> and <math alttext="\vec{z}" class="ltx_Math" display="inline" id="Thmthm1.p1.2.2.m2.1"><semantics id="Thmthm1.p1.2.2.m2.1a"><mover accent="true" id="Thmthm1.p1.2.2.m2.1.1" xref="Thmthm1.p1.2.2.m2.1.1.cmml"><mi id="Thmthm1.p1.2.2.m2.1.1.2" xref="Thmthm1.p1.2.2.m2.1.1.2.cmml">z</mi><mo id="Thmthm1.p1.2.2.m2.1.1.1" stretchy="false" xref="Thmthm1.p1.2.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="Thmthm1.p1.2.2.m2.1b"><apply id="Thmthm1.p1.2.2.m2.1.1.cmml" xref="Thmthm1.p1.2.2.m2.1.1"><ci id="Thmthm1.p1.2.2.m2.1.1.1.cmml" xref="Thmthm1.p1.2.2.m2.1.1.1">→</ci><ci id="Thmthm1.p1.2.2.m2.1.1.2.cmml" xref="Thmthm1.p1.2.2.m2.1.1.2">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm1.p1.2.2.m2.1c">\vec{z}</annotation><annotation encoding="application/x-llamapun" id="Thmthm1.p1.2.2.m2.1d">over→ start_ARG italic_z end_ARG</annotation></semantics></math> as sequences of free variables is to determine a position for aggregate value <math alttext="\alpha(\vec{w})" class="ltx_Math" display="inline" id="Thmthm1.p1.3.3.m3.1"><semantics id="Thmthm1.p1.3.3.m3.1a"><mrow id="Thmthm1.p1.3.3.m3.1.2" xref="Thmthm1.p1.3.3.m3.1.2.cmml"><mi id="Thmthm1.p1.3.3.m3.1.2.2" xref="Thmthm1.p1.3.3.m3.1.2.2.cmml">α</mi><mo id="Thmthm1.p1.3.3.m3.1.2.1" xref="Thmthm1.p1.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="Thmthm1.p1.3.3.m3.1.2.3.2" xref="Thmthm1.p1.3.3.m3.1.1.cmml"><mo id="Thmthm1.p1.3.3.m3.1.2.3.2.1" stretchy="false" xref="Thmthm1.p1.3.3.m3.1.1.cmml">(</mo><mover accent="true" id="Thmthm1.p1.3.3.m3.1.1" xref="Thmthm1.p1.3.3.m3.1.1.cmml"><mi id="Thmthm1.p1.3.3.m3.1.1.2" xref="Thmthm1.p1.3.3.m3.1.1.2.cmml">w</mi><mo id="Thmthm1.p1.3.3.m3.1.1.1" stretchy="false" xref="Thmthm1.p1.3.3.m3.1.1.1.cmml">→</mo></mover><mo id="Thmthm1.p1.3.3.m3.1.2.3.2.2" stretchy="false" xref="Thmthm1.p1.3.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm1.p1.3.3.m3.1b"><apply id="Thmthm1.p1.3.3.m3.1.2.cmml" xref="Thmthm1.p1.3.3.m3.1.2"><times id="Thmthm1.p1.3.3.m3.1.2.1.cmml" xref="Thmthm1.p1.3.3.m3.1.2.1"></times><ci id="Thmthm1.p1.3.3.m3.1.2.2.cmml" xref="Thmthm1.p1.3.3.m3.1.2.2">𝛼</ci><apply id="Thmthm1.p1.3.3.m3.1.1.cmml" xref="Thmthm1.p1.3.3.m3.1.2.3.2"><ci id="Thmthm1.p1.3.3.m3.1.1.1.cmml" xref="Thmthm1.p1.3.3.m3.1.1.1">→</ci><ci id="Thmthm1.p1.3.3.m3.1.1.2.cmml" xref="Thmthm1.p1.3.3.m3.1.1.2">𝑤</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm1.p1.3.3.m3.1c">\alpha(\vec{w})</annotation><annotation encoding="application/x-llamapun" id="Thmthm1.p1.3.3.m3.1d">italic_α ( over→ start_ARG italic_w end_ARG )</annotation></semantics></math> and, consequently, define its position in the lexicographic order over the answers. Second, the reader should note that, in our notation, an AggCQ has a single aggregate function. While this is important for some of our results, other results can be easily extended to multiple aggregate functions <math alttext="\alpha(\vec{w_{1}})" class="ltx_Math" display="inline" id="Thmthm1.p1.4.4.m4.1"><semantics id="Thmthm1.p1.4.4.m4.1a"><mrow id="Thmthm1.p1.4.4.m4.1.2" xref="Thmthm1.p1.4.4.m4.1.2.cmml"><mi id="Thmthm1.p1.4.4.m4.1.2.2" xref="Thmthm1.p1.4.4.m4.1.2.2.cmml">α</mi><mo id="Thmthm1.p1.4.4.m4.1.2.1" xref="Thmthm1.p1.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="Thmthm1.p1.4.4.m4.1.2.3.2" xref="Thmthm1.p1.4.4.m4.1.1.cmml"><mo id="Thmthm1.p1.4.4.m4.1.2.3.2.1" stretchy="false" xref="Thmthm1.p1.4.4.m4.1.1.cmml">(</mo><mover accent="true" id="Thmthm1.p1.4.4.m4.1.1" xref="Thmthm1.p1.4.4.m4.1.1.cmml"><msub id="Thmthm1.p1.4.4.m4.1.1.2" xref="Thmthm1.p1.4.4.m4.1.1.2.cmml"><mi id="Thmthm1.p1.4.4.m4.1.1.2.2" xref="Thmthm1.p1.4.4.m4.1.1.2.2.cmml">w</mi><mn id="Thmthm1.p1.4.4.m4.1.1.2.3" xref="Thmthm1.p1.4.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="Thmthm1.p1.4.4.m4.1.1.1" stretchy="false" xref="Thmthm1.p1.4.4.m4.1.1.1.cmml">→</mo></mover><mo id="Thmthm1.p1.4.4.m4.1.2.3.2.2" stretchy="false" xref="Thmthm1.p1.4.4.m4.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm1.p1.4.4.m4.1b"><apply id="Thmthm1.p1.4.4.m4.1.2.cmml" xref="Thmthm1.p1.4.4.m4.1.2"><times id="Thmthm1.p1.4.4.m4.1.2.1.cmml" xref="Thmthm1.p1.4.4.m4.1.2.1"></times><ci id="Thmthm1.p1.4.4.m4.1.2.2.cmml" xref="Thmthm1.p1.4.4.m4.1.2.2">𝛼</ci><apply id="Thmthm1.p1.4.4.m4.1.1.cmml" xref="Thmthm1.p1.4.4.m4.1.2.3.2"><ci id="Thmthm1.p1.4.4.m4.1.1.1.cmml" xref="Thmthm1.p1.4.4.m4.1.1.1">→</ci><apply id="Thmthm1.p1.4.4.m4.1.1.2.cmml" xref="Thmthm1.p1.4.4.m4.1.1.2"><csymbol cd="ambiguous" id="Thmthm1.p1.4.4.m4.1.1.2.1.cmml" xref="Thmthm1.p1.4.4.m4.1.1.2">subscript</csymbol><ci id="Thmthm1.p1.4.4.m4.1.1.2.2.cmml" xref="Thmthm1.p1.4.4.m4.1.1.2.2">𝑤</ci><cn id="Thmthm1.p1.4.4.m4.1.1.2.3.cmml" type="integer" xref="Thmthm1.p1.4.4.m4.1.1.2.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm1.p1.4.4.m4.1c">\alpha(\vec{w_{1}})</annotation><annotation encoding="application/x-llamapun" id="Thmthm1.p1.4.4.m4.1d">italic_α ( over→ start_ARG italic_w start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math>, …, <math alttext="\alpha(\vec{w_{k}})" class="ltx_Math" display="inline" id="Thmthm1.p1.5.5.m5.1"><semantics id="Thmthm1.p1.5.5.m5.1a"><mrow id="Thmthm1.p1.5.5.m5.1.2" xref="Thmthm1.p1.5.5.m5.1.2.cmml"><mi id="Thmthm1.p1.5.5.m5.1.2.2" xref="Thmthm1.p1.5.5.m5.1.2.2.cmml">α</mi><mo id="Thmthm1.p1.5.5.m5.1.2.1" xref="Thmthm1.p1.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="Thmthm1.p1.5.5.m5.1.2.3.2" xref="Thmthm1.p1.5.5.m5.1.1.cmml"><mo id="Thmthm1.p1.5.5.m5.1.2.3.2.1" stretchy="false" xref="Thmthm1.p1.5.5.m5.1.1.cmml">(</mo><mover accent="true" id="Thmthm1.p1.5.5.m5.1.1" xref="Thmthm1.p1.5.5.m5.1.1.cmml"><msub id="Thmthm1.p1.5.5.m5.1.1.2" xref="Thmthm1.p1.5.5.m5.1.1.2.cmml"><mi id="Thmthm1.p1.5.5.m5.1.1.2.2" xref="Thmthm1.p1.5.5.m5.1.1.2.2.cmml">w</mi><mi id="Thmthm1.p1.5.5.m5.1.1.2.3" xref="Thmthm1.p1.5.5.m5.1.1.2.3.cmml">k</mi></msub><mo id="Thmthm1.p1.5.5.m5.1.1.1" stretchy="false" xref="Thmthm1.p1.5.5.m5.1.1.1.cmml">→</mo></mover><mo id="Thmthm1.p1.5.5.m5.1.2.3.2.2" stretchy="false" xref="Thmthm1.p1.5.5.m5.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm1.p1.5.5.m5.1b"><apply id="Thmthm1.p1.5.5.m5.1.2.cmml" xref="Thmthm1.p1.5.5.m5.1.2"><times id="Thmthm1.p1.5.5.m5.1.2.1.cmml" xref="Thmthm1.p1.5.5.m5.1.2.1"></times><ci id="Thmthm1.p1.5.5.m5.1.2.2.cmml" xref="Thmthm1.p1.5.5.m5.1.2.2">𝛼</ci><apply id="Thmthm1.p1.5.5.m5.1.1.cmml" xref="Thmthm1.p1.5.5.m5.1.2.3.2"><ci id="Thmthm1.p1.5.5.m5.1.1.1.cmml" xref="Thmthm1.p1.5.5.m5.1.1.1">→</ci><apply id="Thmthm1.p1.5.5.m5.1.1.2.cmml" xref="Thmthm1.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmthm1.p1.5.5.m5.1.1.2.1.cmml" xref="Thmthm1.p1.5.5.m5.1.1.2">subscript</csymbol><ci id="Thmthm1.p1.5.5.m5.1.1.2.2.cmml" xref="Thmthm1.p1.5.5.m5.1.1.2.2">𝑤</ci><ci id="Thmthm1.p1.5.5.m5.1.1.2.3.cmml" xref="Thmthm1.p1.5.5.m5.1.1.2.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm1.p1.5.5.m5.1c">\alpha(\vec{w_{k}})</annotation><annotation encoding="application/x-llamapun" id="Thmthm1.p1.5.5.m5.1d">italic_α ( over→ start_ARG italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG )</annotation></semantics></math>. We will mention this extension when relevant.∎</span></p> </div> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx1.p3"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx1.p3.11">In this work, we restrict the discussion to the common aggregate functions <math alttext="\mathsf{Count}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1.1.cmml">𝖢𝗈𝗎𝗇𝗍</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1.1">𝖢𝗈𝗎𝗇𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1c">\mathsf{Count}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.1.m1.1d">sansserif_Count</annotation></semantics></math>, <math alttext="\mathsf{CountD}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1.1.cmml">𝖢𝗈𝗎𝗇𝗍𝖣</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1.1">𝖢𝗈𝗎𝗇𝗍𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1c">\mathsf{CountD}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.2.m2.1d">sansserif_CountD</annotation></semantics></math> (count distinct), <math alttext="\mathsf{Sum}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1.1.cmml">𝖲𝗎𝗆</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1.1">𝖲𝗎𝗆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1c">\mathsf{Sum}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.3.m3.1d">sansserif_Sum</annotation></semantics></math>, <math alttext="\mathsf{Avg}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1.1.cmml">𝖠𝗏𝗀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1.1">𝖠𝗏𝗀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1c">\mathsf{Avg}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.4.m4.1d">sansserif_Avg</annotation></semantics></math> (average), <math alttext="\mathsf{Min}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1.1.cmml">𝖬𝗂𝗇</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1.1">𝖬𝗂𝗇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1c">\mathsf{Min}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.5.m5.1d">sansserif_Min</annotation></semantics></math> and <math alttext="\mathsf{Max}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1.1.cmml">𝖬𝖺𝗑</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1.1">𝖬𝖺𝗑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1c">\mathsf{Max}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.6.m6.1d">sansserif_Max</annotation></semantics></math>. All aggregate functions take a single column as input (i.e., <math alttext="\vec{w}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.2.cmml">w</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1"><ci id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1.1.2">𝑤</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1c">\vec{w}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.7.m7.1d">over→ start_ARG italic_w end_ARG</annotation></semantics></math> is of length one) except for <math alttext="\mathsf{Count}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1a"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1.1.cmml">𝖢𝗈𝗎𝗇𝗍</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1b"><ci id="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1.1">𝖢𝗈𝗎𝗇𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1c">\mathsf{Count}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.8.m8.1d">sansserif_Count</annotation></semantics></math> that counts the tuples in the group and takes no argument. For instance, the query <math alttext="Q_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1a"><msub id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.2.cmml">Q</mi><mn id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.3" xref="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.2">𝑄</ci><cn id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1c">Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.9.m9.1d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in the Introduction uses <math alttext="\mathsf{Count}()" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.2.cmml">𝖢𝗈𝗎𝗇𝗍</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.3.1.cmml">(</mo><mo id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1"><times id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.1"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.2">𝖢𝗈𝗎𝗇𝗍</ci><list id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1.1.3.2.1"></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1c">\mathsf{Count}()</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.10.m10.1d">sansserif_Count ( )</annotation></semantics></math> and it could also use <math alttext="\mathsf{CountD}(g)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1a"><mrow id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.2" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.2.cmml">𝖢𝗈𝗎𝗇𝗍𝖣</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.1" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.1.cmml">g</mi><mo id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1b"><apply id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2"><times id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.2.2">𝖢𝗈𝗎𝗇𝗍𝖣</ci><ci id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1.1">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1c">\mathsf{CountD}(g)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx1.p3.11.m11.1d">sansserif_CountD ( italic_g )</annotation></semantics></math> for counting the distinct games with scored goals.</p> </div> </section> <section class="ltx_subparagraph" id="S2.SS0.SSS0.Px1.SPx2"> <h5 class="ltx_title ltx_title_subparagraph">Commutative semirings.</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx2.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx2.p1.25">A <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p1.25.1">commutative monoid</em> is an algebraic structure <math alttext="(\mathbb{K},\cdot)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.1.1.cmml">𝕂</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.2.2" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.2" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.2.cmml">⋅</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.1.1">𝕂</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2.2">⋅</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2c">(\mathbb{K},\cdot)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.1.m1.2d">( blackboard_K , ⋅ )</annotation></semantics></math> over a domain <math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1a"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.2.m2.1d">blackboard_K</annotation></semantics></math>, with a binary operation <math alttext="\cdot" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1a"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1.1.cmml">⋅</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1.1">⋅</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1c">\cdot</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.3.m3.1d">⋅</annotation></semantics></math> that satisfies <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p1.25.2">associativity</em>: <math alttext="(a\cdot b)\cdot c=a\cdot(b\cdot 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id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.2.cmml">b</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.1.cmml">⋅</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.3.cmml">c</mi></mrow><mo id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2"><eq id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.3"></eq><apply id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.2">⋅</ci><apply id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.1.1"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.1.1.1.1">⋅</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.1.1.1.2">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.1.1.1.3">𝑏</ci></apply><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.1.1.1.3">𝑐</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.2">⋅</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.3">𝑎</ci><apply id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.1">⋅</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.2">𝑏</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2.2.2.1.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2c">(a\cdot b)\cdot c=a\cdot(b\cdot c)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.4.m4.2d">( italic_a ⋅ italic_b ) ⋅ italic_c = italic_a ⋅ ( italic_b ⋅ italic_c )</annotation></semantics></math> for any <math alttext="a,b,c\in\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.2.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.2.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.1.1.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.2.2.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.2.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.2.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.2.2.cmml">b</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.2.2.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.2.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.3.cmml">c</mi></mrow><mo id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.3.cmml">𝕂</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4"><in id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.1"></in><list id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.2.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.1.1">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.2.2">𝑏</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.3">𝑐</ci></list><ci id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3.4.3">𝕂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3c">a,b,c\in\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.5.m5.3d">italic_a , italic_b , italic_c ∈ blackboard_K</annotation></semantics></math>, <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p1.25.3">commutativity</em>: <math alttext="a\cdot b=b\cdot a" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.1.cmml">⋅</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.3.cmml">b</mi></mrow><mo id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.2.cmml">b</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.1.cmml">⋅</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.3.cmml">a</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1"><eq id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.1"></eq><apply id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.1">⋅</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.2">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.2.3">𝑏</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.1">⋅</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.2">𝑏</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1.1.3.3">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1c">a\cdot b=b\cdot a</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.6.m6.1d">italic_a ⋅ italic_b = italic_b ⋅ italic_a</annotation></semantics></math> for any <math alttext="a,b\in\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.2.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.1.1.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.2.2.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.2.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.2.cmml">b</mi></mrow><mo id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.3.cmml">𝕂</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3"><in id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.1"></in><list id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.2.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.1.1">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.2">𝑏</ci></list><ci id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2.3.3">𝕂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2c">a,b\in\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.7.m7.2d">italic_a , italic_b ∈ blackboard_K</annotation></semantics></math>, and <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p1.25.4">identity element</em>: there exists an element <math alttext="\varnothing\in\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.2" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.2.cmml">∅</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.3.cmml">𝕂</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1"><in id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.1"></in><emptyset id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.2"></emptyset><ci id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1.1.3">𝕂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1c">\varnothing\in\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.8.m8.1d">∅ ∈ blackboard_K</annotation></semantics></math> such that <math alttext="a\cdot\varnothing=a" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.1.cmml">⋅</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.3" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.3.cmml">∅</mi></mrow><mo id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.1.cmml">=</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.3.cmml">a</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1"><eq id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.1"></eq><apply id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.1">⋅</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.2">𝑎</ci><emptyset id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.2.3"></emptyset></apply><ci id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1c">a\cdot\varnothing=a</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.9.m9.1d">italic_a ⋅ ∅ = italic_a</annotation></semantics></math> for any <math alttext="a\in\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.3.cmml">𝕂</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1"><in id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.1"></in><ci id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.2">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1.1.3">𝕂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1c">a\in\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.10.m10.1d">italic_a ∈ blackboard_K</annotation></semantics></math>. A <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p1.25.5">commutative semiring</em> is an algebraic structure <math alttext="(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.1.1.cmml">𝕂</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.2.2" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.2.2" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.2.2.cmml">⊕</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.2.3" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.3.3" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.3.3.cmml">⊗</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.2.4" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.2.cmml">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.1.cmml">¯</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.2.5" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.2.cmml">1</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.1.cmml">¯</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5b"><vector id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.6.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.1.1">𝕂</ci><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.2.2">direct-sum</csymbol><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.3.3">tensor-product</csymbol><ci id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.2">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.4.4.1">¯</mo></mover></ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.2">1</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5.5.1">¯</mo></mover></ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5c">(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.11.m11.5d">( blackboard_K , ⊕ , ⊗ , start_ID over¯ start_ARG 0 end_ARG end_ID , start_ID over¯ start_ARG 1 end_ARG end_ID )</annotation></semantics></math> over a domain <math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1a"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.12.m12.1d">blackboard_K</annotation></semantics></math>, with two binary operations <math alttext="\oplus" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1a"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.13.m13.1d">⊕</annotation></semantics></math> and <math alttext="\otimes" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1a"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.14.m14.1d">⊗</annotation></semantics></math> and two distinguished elements <math alttext="\mathord{\bar{0}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.2.cmml">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.2">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1c">\mathord{\bar{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.15.m15.1d">start_ID over¯ start_ARG 0 end_ARG end_ID</annotation></semantics></math> and <math alttext="\mathord{\bar{1}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.2.cmml">1</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.2">1</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1c">\mathord{\bar{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.16.m16.1d">start_ID over¯ start_ARG 1 end_ARG end_ID</annotation></semantics></math> in <math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1a"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.17.m17.1d">blackboard_K</annotation></semantics></math> that satisfy the following conditions: <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p1.25.6">(a)</em> <math alttext="(\mathbb{K},\oplus)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.1.1.cmml">𝕂</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.2.2" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.2" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.2.cmml">⊕</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.1.1">𝕂</ci><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2.2">direct-sum</csymbol></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2c">(\mathbb{K},\oplus)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.18.m18.2d">( blackboard_K , ⊕ )</annotation></semantics></math> is a commutative monoid with the identity element <math alttext="\mathord{\bar{0}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.2.cmml">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.2">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1c">\mathord{\bar{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.19.m19.1d">start_ID over¯ start_ARG 0 end_ARG end_ID</annotation></semantics></math>; <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p1.25.7">(b)</em> <math alttext="(\mathbb{K},\otimes)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.1.1.cmml">𝕂</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.2.2" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.2" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.2.cmml">⊗</mo><mo id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.1.1">𝕂</ci><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2.2">tensor-product</csymbol></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2c">(\mathbb{K},\otimes)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.20.m20.2d">( blackboard_K , ⊗ )</annotation></semantics></math> is a commutative monoid with the identity element <math alttext="\mathord{\bar{1}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.2.cmml">1</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.2">1</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1c">\mathord{\bar{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.21.m21.1d">start_ID over¯ start_ARG 1 end_ARG end_ID</annotation></semantics></math>; <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p1.25.8">(c)</em> <math alttext="a\otimes(b\oplus c)=(a\otimes b)\oplus(a\otimes c)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.3.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.2.cmml">⊗</mo><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.2.cmml">b</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.1.cmml">⊕</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.3.cmml">c</mi></mrow><mo id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.4" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.4.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.1.cmml">⊗</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.3" 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id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.2.2.2.1.1.1.3">𝑏</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.2.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.2.1.1.1">tensor-product</csymbol><ci id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.2.1.1.2">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.2.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3.3.3.2.1.1.3">𝑐</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3c">a\otimes(b\oplus c)=(a\otimes b)\oplus(a\otimes c)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.22.m22.3d">italic_a ⊗ ( italic_b ⊕ italic_c ) = ( italic_a ⊗ italic_b ) ⊕ ( italic_a ⊗ italic_c 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xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.3.cmml">𝕂</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4"><in id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.1"></in><list id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.2.2"><ci id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.1.1">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.2.2">𝑏</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.3">𝑐</ci></list><ci id="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.23.m23.3.4.3">𝕂</ci></apply></annotation-xml><annotation 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id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.2.cmml">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.1.cmml">¯</mo></mover></mrow><mo id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.1.cmml">=</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.2.cmml">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1"><eq id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.1"></eq><apply id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.1">tensor-product</csymbol><ci id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.2">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.2">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.2.3.1">¯</mo></mover></ci></apply><ci id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3"><mn id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.2">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1.1.3.1">¯</mo></mover></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1c">a\otimes\mathord{\bar{0}}=\mathord{\bar{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.24.m24.1d">italic_a ⊗ start_ID over¯ start_ARG 0 end_ARG end_ID = start_ID over¯ start_ARG 0 end_ARG end_ID</annotation></semantics></math> for all <math alttext="a\in\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1a"><mrow id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.1.cmml">∈</mo><mi id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.3" xref="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.3.cmml">𝕂</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1b"><apply id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1"><in id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.1"></in><ci id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.2">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1.1.3">𝕂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1c">a\in\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p1.25.m25.1d">italic_a ∈ blackboard_K</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx2.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx2.p2.4">We refer to <math alttext="\oplus" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1a"><mo id="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p2.1.m1.1d">⊕</annotation></semantics></math> as the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p2.4.1">addition</em> operation, <math alttext="\otimes" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1a"><mo id="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p2.2.m2.1d">⊗</annotation></semantics></math> as the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p2.4.2">multiplication</em> operation, <math alttext="\mathord{\bar{0}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.2.cmml">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1"><mn id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.2">0</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1c">\mathord{\bar{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p2.3.m3.1d">start_ID over¯ start_ARG 0 end_ARG end_ID</annotation></semantics></math> as the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p2.4.3">additive identity</em> and <math alttext="\mathord{\bar{1}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.2.cmml">1</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.1" xref="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1b"><ci id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1a.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1"><mn id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.2">1</mn><mo id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1c">\mathord{\bar{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx2.p2.4.m4.1d">start_ID over¯ start_ARG 1 end_ARG end_ID</annotation></semantics></math> as the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx2.p2.4.4">multiplicative identity</em>. We give examples of commutative semirings at the end of this section.</p> </div> </section> <section class="ltx_subparagraph" id="S2.SS0.SSS0.Px1.SPx3"> <h5 class="ltx_title ltx_title_subparagraph">Annotated databases and query answers.</h5> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx3.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p1.13">Let <math alttext="\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.1.m1.1d">bold_S</annotation></semantics></math> be a schema and <math alttext="(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.1.1.cmml">𝕂</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.2.2" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.2.2" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.2.2.cmml">⊕</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.2.3" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.3.3" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.3.3.cmml">⊗</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.2.4" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.2.cmml">0</mn><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.1.cmml">¯</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.2.5" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5a.cmml"><mn id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.2.cmml">1</mn><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.1.cmml">¯</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5b"><vector id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.6.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.1.1">𝕂</ci><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.2.2">direct-sum</csymbol><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.3.3">tensor-product</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4a.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4"><mn id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.2">0</mn><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.4.4.1">¯</mo></mover></ci><ci id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5a.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5"><mn id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.2">1</mn><mo id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5.5.1">¯</mo></mover></ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5c">(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.2.m2.5d">( blackboard_K , ⊕ , ⊗ , start_ID over¯ start_ARG 0 end_ARG end_ID , start_ID over¯ start_ARG 1 end_ARG end_ID )</annotation></semantics></math> a commutative semiring. A <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p1.3.1"><math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.3.1.m1.1d">blackboard_K</annotation></semantics></math>-database</em> (<em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p1.4.2">over <math alttext="\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.4.2.m1.1d">bold_S</annotation></semantics></math></em>) is a pair <math alttext="(D,\tau)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.2.cmml">τ</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.1.1">𝐷</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2c">(D,\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.5.m3.2d">( italic_D , italic_τ )</annotation></semantics></math> where <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.6.m4.1d">italic_D</annotation></semantics></math> is a database over <math alttext="\mathord{\mathbf{S}}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1c">\mathord{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.7.m5.1d">bold_S</annotation></semantics></math> and <math alttext="\tau:D\rightarrow\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.2.cmml">τ</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.1.cmml">:</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.2.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.1.cmml">→</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.3" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.3.cmml">𝕂</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.1">:</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.2">𝜏</ci><apply id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.2">𝐷</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1.1.3.3">𝕂</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1c">\tau:D\rightarrow\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.8.m6.1d">italic_τ : italic_D → blackboard_K</annotation></semantics></math> is function that maps every fact <math alttext="f" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.9.m7.1d">italic_f</annotation></semantics></math> of <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.10.m8.1d">italic_D</annotation></semantics></math> to an element <math alttext="\tau(f)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.2.cmml">τ</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.1.cmml">f</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2"><times id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.2.2">𝜏</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1.1">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1c">\tau(f)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.11.m9.1d">italic_τ ( italic_f )</annotation></semantics></math> of <math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.12.m10.1d">blackboard_K</annotation></semantics></math>, called the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p1.13.3">annotation</em> of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p1.13.m11.1d">italic_f</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx3.p2"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p2.1">The annotation of a database propagates to the query answers by associating a semiring operation with each algebraic operation <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1145/1265530.1265535</span>]</cite>. In the case of CQs, the relevant operations are <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p2.1.1">joins</em> and <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p2.1.2">projection</em>. For join, the annotation of the result is the product of the annotation of the operands. For projection, the annotation is the sum of the annotations of the tuples that give rise to the answer. In our terminology, we have the following.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx3.p3"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p3.12">Let <math alttext="Q(\vec{x}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{\ell}(\vec{x% },\vec{y})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.3.2.2" rspace="0.448em" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.cmml">)</mo></mrow></mrow><mpadded width="0.226em"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.3.cmml">:</mo></mpadded><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.3.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1a" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.cmml">−</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.cmml"><msub id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.2.cmml">φ</mi><mn id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.6.6" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.6.6.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.4" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.3.cmml">,</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.cmml"><msub id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.2.cmml">φ</mi><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.3" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.3.cmml">ℓ</mi></msub><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.2.cmml">y</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8b"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.3">:</ci><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4"><times id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.1"></times><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.2">𝑄</ci><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.4.3.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.1.1.2">𝑥</ci></apply></apply><list id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1"><minus id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1"></minus><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2"><times id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.1"></times><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.2">𝜑</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.2.3">1</cn></apply><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.7.7.1.1.1.2.3.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.2.2.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.3.3.2">𝑦</ci></apply></interval></apply></apply><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.6.6.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.6.6">…</ci><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2"><times id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.1"></times><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.2">𝜑</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.2.3">ℓ</ci></apply><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8.8.2.2.2.3.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.4.4.2">𝑥</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.5.5.2">𝑦</ci></apply></interval></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8c">Q(\vec{x}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{\ell}(\vec{x% },\vec{y})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.1.m1.8d">italic_Q ( over→ start_ARG italic_x end_ARG ) : - italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) , … , italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math> be a CQ and <math alttext="(D,\tau)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.2.cmml">τ</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.1.1">𝐷</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2c">(D,\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.2.m2.2d">( italic_D , italic_τ )</annotation></semantics></math> an annotated database. For a homomorphism <math alttext="h" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.3.m3.1d">italic_h</annotation></semantics></math> from <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.4.m4.1d">italic_Q</annotation></semantics></math> to <math alttext="D" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.5.m5.1d">italic_D</annotation></semantics></math>, we denote by <math alttext="\otimes h" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.2.cmml"></mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.1.cmml">⊗</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.3.cmml">h</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.1">tensor-product</csymbol><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.2">absent</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1.1.3">ℎ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1c">\otimes h</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.6.m6.1d">⊗ italic_h</annotation></semantics></math> the product of the annotations of the facts in the range of <math alttext="h" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1.1.cmml">h</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1.1">ℎ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1c">h</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.7.m7.1d">italic_h</annotation></semantics></math>, that is <math alttext="\otimes h\vcentcolon=\tau(h(\varphi_{1}(\vec{x},\vec{y})))\otimes\dots\otimes% \tau(h(\varphi_{\ell}(\vec{x},\vec{y})))" class="ltx_math_unparsed" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1b"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.1" rspace="0.222em">⊗</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.2">h</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.3" lspace="0.278em" rspace="0.278em">:=</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.4">τ</mi><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.1" stretchy="false">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.2">h</mi><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.1" stretchy="false">(</mo><msub id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.2"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.2.2">φ</mi><mn id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.2.3">1</mn></msub><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.1" stretchy="false">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.2"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.2.2">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.2.1" stretchy="false">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.3">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.4"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.4.2">y</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.4.1" stretchy="false">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.3.5" stretchy="false">)</mo></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.3.4" stretchy="false">)</mo></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.5.4" rspace="0.055em" stretchy="false">)</mo></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.6" rspace="0.222em">⊗</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.7" mathvariant="normal">⋯</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.8" lspace="0.222em" rspace="0.222em">⊗</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.9">τ</mi><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.1" stretchy="false">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.2">h</mi><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.1" stretchy="false">(</mo><msub id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.2"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.2.2">φ</mi><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.2.3" mathvariant="normal">ℓ</mi></msub><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.1" stretchy="false">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.2"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.2.2">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.2.1" stretchy="false">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.3">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.4"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.4.2">y</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.4.1" stretchy="false">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.3.5" stretchy="false">)</mo></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.3.4" stretchy="false">)</mo></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1.10.4" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1c">\otimes h\vcentcolon=\tau(h(\varphi_{1}(\vec{x},\vec{y})))\otimes\dots\otimes% \tau(h(\varphi_{\ell}(\vec{x},\vec{y})))</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.8.m8.1d">⊗ italic_h := italic_τ ( italic_h ( italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) ) ) ⊗ ⋯ ⊗ italic_τ ( italic_h ( italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) ) )</annotation></semantics></math>. An <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p3.12.1">answer</em> to <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.9.m9.1d">italic_Q</annotation></semantics></math> over <math alttext="(D,\tau)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.2.cmml">τ</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.1.1">𝐷</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2c">(D,\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.10.m10.2d">( italic_D , italic_τ )</annotation></semantics></math> is a pair <math alttext="(\vec{c},a)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.2.cmml">c</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.3.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.1.1.2">𝑐</ci></apply><ci id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2.2">𝑎</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2c">(\vec{c},a)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.11.m11.2d">( over→ start_ARG italic_c end_ARG , italic_a )</annotation></semantics></math> such that <math alttext="\vec{c}\in Q(D)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.cmml"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.2.cmml">c</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.1.cmml">∈</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2"><in id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.1"></in><apply id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.2.2">𝑐</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3"><times id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.2.3.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1.1">𝐷</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1c">\vec{c}\in Q(D)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.12.m12.1d">over→ start_ARG italic_c end_ARG ∈ italic_Q ( italic_D )</annotation></semantics></math> and</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="a=\oplus\{\otimes h\mid h\in\mathsf{Hom}(Q,D)\land h(\vec{x})=\vec{c}\}" class="ltx_math_unparsed" display="block" id="S2.Ex5.m1.1"><semantics id="S2.Ex5.m1.1a"><mrow id="S2.Ex5.m1.1b"><mi id="S2.Ex5.m1.1.1">a</mi><mo id="S2.Ex5.m1.1.2" rspace="0em">=</mo><mo id="S2.Ex5.m1.1.3" lspace="0em">⊕</mo><mrow id="S2.Ex5.m1.1.4"><mo id="S2.Ex5.m1.1.4.1" stretchy="false">{</mo><mo id="S2.Ex5.m1.1.4.2" lspace="0em" rspace="0.222em">⊗</mo><mi id="S2.Ex5.m1.1.4.3">h</mi><mo id="S2.Ex5.m1.1.4.4" lspace="0em" rspace="0.167em">∣</mo><mi id="S2.Ex5.m1.1.4.5">h</mi><mo id="S2.Ex5.m1.1.4.6">∈</mo><mi id="S2.Ex5.m1.1.4.7">𝖧𝗈𝗆</mi><mrow id="S2.Ex5.m1.1.4.8"><mo id="S2.Ex5.m1.1.4.8.1" stretchy="false">(</mo><mi id="S2.Ex5.m1.1.4.8.2">Q</mi><mo id="S2.Ex5.m1.1.4.8.3">,</mo><mi id="S2.Ex5.m1.1.4.8.4">D</mi><mo id="S2.Ex5.m1.1.4.8.5" stretchy="false">)</mo></mrow><mo id="S2.Ex5.m1.1.4.9">∧</mo><mi id="S2.Ex5.m1.1.4.10">h</mi><mrow id="S2.Ex5.m1.1.4.11"><mo id="S2.Ex5.m1.1.4.11.1" stretchy="false">(</mo><mover accent="true" id="S2.Ex5.m1.1.4.11.2"><mi id="S2.Ex5.m1.1.4.11.2.2">x</mi><mo id="S2.Ex5.m1.1.4.11.2.1" stretchy="false">→</mo></mover><mo id="S2.Ex5.m1.1.4.11.3" stretchy="false">)</mo></mrow><mo id="S2.Ex5.m1.1.4.12">=</mo><mover accent="true" id="S2.Ex5.m1.1.4.13"><mi id="S2.Ex5.m1.1.4.13.2">c</mi><mo id="S2.Ex5.m1.1.4.13.1" stretchy="false">→</mo></mover><mo id="S2.Ex5.m1.1.4.14" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.Ex5.m1.1c">a=\oplus\{\otimes h\mid h\in\mathsf{Hom}(Q,D)\land h(\vec{x})=\vec{c}\}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.1d">italic_a = ⊕ { ⊗ italic_h ∣ italic_h ∈ sansserif_Hom ( italic_Q , italic_D ) ∧ italic_h ( over→ start_ARG italic_x end_ARG ) = over→ start_ARG italic_c end_ARG }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p3.20">where, for <math alttext="A=\{a_{1},\dots,a_{n}\}\subseteq\mathbb{K}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.4" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.4.cmml">A</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.5" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.5.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.3.cmml">{</mo><msub id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.2.cmml">a</mi><mn id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.4" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.3.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.1.1.cmml">…</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.5" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.2.cmml">a</mi><mi id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.6" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.3.cmml">}</mo></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.6" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.6.cmml">⊆</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.7" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.7.cmml">𝕂</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3b"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3"><and id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3a.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3"></and><apply id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3b.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3"><eq id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.5.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.5"></eq><ci id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.4.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.4">𝐴</ci><set id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.2">𝑎</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.2.2.1.1.1.3">1</cn></apply><ci id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.1.1">…</ci><apply id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.2">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.2.2.3">𝑛</ci></apply></set></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3c.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3"><subset id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.6.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.6"></subset><share href="https://arxiv.org/html/2303.05327v2#S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.2.cmml" id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3d.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3"></share><ci id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.7.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3.3.7">𝕂</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3c">A=\{a_{1},\dots,a_{n}\}\subseteq\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.13.m1.3d">italic_A = { italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } ⊆ blackboard_K</annotation></semantics></math>, we define <math alttext="\oplus A=a_{1}\oplus\dots\oplus a_{n}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2a" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2.cmml">⊕</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2.2.cmml">A</mi></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.1.cmml">=</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.cmml"><msub id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.2.cmml">a</mi><mn id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.1.cmml">⊕</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.3" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.3.cmml">⋯</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.1a" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.1.cmml">⊕</mo><msub id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.2.cmml">a</mi><mi id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.3.cmml">n</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1"><eq id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.1"></eq><apply id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2">direct-sum</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.2.2">𝐴</ci></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.1">direct-sum</csymbol><apply id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.2">𝑎</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.2.3">1</cn></apply><ci id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.3">⋯</ci><apply id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.2">𝑎</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1.1.3.4.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1c">\oplus A=a_{1}\oplus\dots\oplus a_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.14.m2.1d">⊕ italic_A = italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ ⋯ ⊕ italic_a start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>. As before, the <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p3.20.1">result</em> of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.15.m3.1d">italic_Q</annotation></semantics></math> over <math alttext="(D,\tau)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.2.cmml">τ</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.1.1">𝐷</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2c">(D,\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.16.m4.2d">( italic_D , italic_τ )</annotation></semantics></math>, denoted <math alttext="Q(D,\tau)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.2.cmml">τ</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2b"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3"><times id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.1"></times><ci id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.2">𝑄</ci><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.3.3.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.1.1">𝐷</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2.2">𝜏</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2c">Q(D,\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.17.m5.2d">italic_Q ( italic_D , italic_τ )</annotation></semantics></math>, is the set of answers <math alttext="(\vec{c},a)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.2.cmml">c</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.2.cmml">a</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.3.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.1.1.2">𝑐</ci></apply><ci id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2.2">𝑎</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2c">(\vec{c},a)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.18.m6.2d">( over→ start_ARG italic_c end_ARG , italic_a )</annotation></semantics></math> to <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.19.m7.1d">italic_Q</annotation></semantics></math> over <math alttext="(D,\tau)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2"><semantics id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.1.1.cmml">D</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.1.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.2.cmml">τ</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2b"><interval closure="open" id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.1.1">𝐷</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2c">(D,\tau)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p3.20.m8.2d">( italic_D , italic_τ )</annotation></semantics></math>. We will make use of the fact that, over commutative semirings, projections and joins are commutative <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1145/1265530.1265535</span>]</cite>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx3.p4"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p4.11">In this work, we study the ability to incorporate the annotation in the order over the answers. More precisely, we will investigate the complexity of involving the annotation in the lexicographic order over the answers, as if it were another value in the tuple. So, when we consider a CQ <math alttext="Q(\vec{x})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2"><times id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.1"></times><ci id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.2">𝑄</ci><apply id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.2.3.2"><ci id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1c">Q(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.1.m1.1d">italic_Q ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math>, we need to specify where the annotation goes inside <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1"><ci id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.2.m2.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math>. Similarly to the AggCQ notation, we do so by replacing <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1a"><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1"><ci id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.3.m3.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math> with a sequence <math alttext="(\vec{x},\star,\vec{z})" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.1.cmml">(</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.2.cmml">x</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.2.2" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.1.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.2.2" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.2.2.cmml">⋆</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.1.cmml">,</mo><mover accent="true" id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.2.cmml">z</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.1.cmml">→</mo></mover><mo id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.2.4" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3b"><vector id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.4.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1"><ci id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.1.1.2">𝑥</ci></apply><ci id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.2.2">⋆</ci><apply id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3"><ci id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.1">→</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3.3.2">𝑧</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3c">(\vec{x},\star,\vec{z})</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.4.m4.3d">( over→ start_ARG italic_x end_ARG , ⋆ , over→ start_ARG italic_z end_ARG )</annotation></semantics></math> where <math alttext="\star" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1a"><mo id="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1.1.cmml">⋆</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1.1">⋆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1c">\star</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.5.m5.1d">⋆</annotation></semantics></math> represents the annotation value. We refer to a CQ of this form as a CQ<sup class="ltx_sup" id="S2.SS0.SSS0.Px1.SPx3.p4.11.1">⋆</sup>. An example of a CQ<sup class="ltx_sup" id="S2.SS0.SSS0.Px1.SPx3.p4.11.2">⋆</sup> is <math alttext="Q(x_{1},x_{2},\star,z){\,:\!\!-\,}R(x_{1},x_{2},y_{1}),S(y_{1},y_{2},z)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.cmml"><mrow id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.4" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.4.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.3.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.3.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.3" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.3.cmml">(</mo><msub id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.4.4.1.1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.4.4.1.1.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.4.4.1.1.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.4.4.1.1.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.4.4.1.1.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.4.4.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.4" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.3.cmml">,</mo><msub id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.3" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.5" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.3.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.1.1" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.1.1.cmml">⋆</mo><mo 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id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.4.4.1.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.4.4.1.1.1.1.3">1</cn></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.5.5.2.2.2.2.3">2</cn></apply><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.1.1">⋆</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.2.2">𝑧</ci></vector></apply><list id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.cmml" 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xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.1.1.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.2.2.2.2.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.2.2.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.2.2.2.2.3">2</cn></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.3.3.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.3.3.3.3"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.3.3.3.3.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.3.3.3.3">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.3.3.3.3.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.3.3.3.3.2">𝑦</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.3.3.3.3.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.6.6.3.1.1.3.3.3.3.3">1</cn></apply></vector></apply></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2"><times id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.3"></times><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.4.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.4">𝑆</ci><vector id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2"><apply id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.1.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.1.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.1.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.1.1.1.2">𝑦</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.1.1.1.3">1</cn></apply><apply id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2.2.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2.2">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2.2.2">𝑦</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7.7.4.2.2.2.2.2.3">2</cn></apply><ci id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.3.3">𝑧</ci></vector></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7c">Q(x_{1},x_{2},\star,z){\,:\!\!-\,}R(x_{1},x_{2},y_{1}),S(y_{1},y_{2},z)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.8.m8.7d">italic_Q ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , ⋆ , italic_z ) : - italic_R ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , italic_S ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z )</annotation></semantics></math> where the lexicographic order is by <math alttext="x_{1}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1a"><msub id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1c">x_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.9.m9.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, then by <math alttext="x_{2}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1a"><msub id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.2.cmml">x</mi><mn id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1">subscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.2">𝑥</ci><cn id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.3.cmml" type="integer" xref="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1c">x_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.10.m10.1d">italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, then by the annotation, and then by <math alttext="z" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p4.11.m11.1d">italic_z</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx3.p5"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p5.11">Let <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.1.m1.1d">italic_Q</annotation></semantics></math> be a CQ<sup class="ltx_sup" id="S2.SS0.SSS0.Px1.SPx3.p5.11.1">⋆</sup>, and let <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1a"><msup id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.3.m3.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be the CQ obtained from <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.4.m4.1d">italic_Q</annotation></semantics></math> by removing <math alttext="\star" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1a"><mo id="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1.1.cmml">⋆</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1.1">⋆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1c">\star</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.5.m5.1d">⋆</annotation></semantics></math> from the head. As in the case of AggCQs, <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.6.m6.1d">italic_Q</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p5.11.2">acyclic</em> if <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1a"><msup id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.7.m7.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is acyclic, <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.8.m8.1d">italic_Q</annotation></semantics></math> is <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p5.11.3">free-connex</em> if <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1a"><msup id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.9.m9.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is free-connex, and a <em class="ltx_emph ltx_font_italic" id="S2.SS0.SSS0.Px1.SPx3.p5.11.4">disruptive trio</em> of <math alttext="Q" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.10.m10.1d">italic_Q</annotation></semantics></math> is a disruptive trio of <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1a"><msup id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.cmml"><mi id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.2.cmml">Q</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.3" xref="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1b"><apply id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1">superscript</csymbol><ci id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.2">𝑄</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p5.11.m11.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx3.p6"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p6.1">Aggregate functions can often be captured by annotations of answers in annotated databases, where each aggregate function might require a different commutative semiring:</p> <ul class="ltx_itemize" id="S2.I1"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.2"><math alttext="\mathsf{Sum}" class="ltx_Math" display="inline" id="S2.I1.i1.p1.1.m1.1"><semantics id="S2.I1.i1.p1.1.m1.1a"><mi id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml">𝖲𝗎𝗆</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><ci id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1">𝖲𝗎𝗆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">\mathsf{Sum}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.1d">sansserif_Sum</annotation></semantics></math>: the numeric semiring <math alttext="(\mathbb{Q},+,\cdot,0,1)" class="ltx_Math" display="inline" id="S2.I1.i1.p1.2.m2.5"><semantics id="S2.I1.i1.p1.2.m2.5a"><mrow id="S2.I1.i1.p1.2.m2.5.6.2" xref="S2.I1.i1.p1.2.m2.5.6.1.cmml"><mo id="S2.I1.i1.p1.2.m2.5.6.2.1" stretchy="false" xref="S2.I1.i1.p1.2.m2.5.6.1.cmml">(</mo><mi id="S2.I1.i1.p1.2.m2.1.1" xref="S2.I1.i1.p1.2.m2.1.1.cmml">ℚ</mi><mo id="S2.I1.i1.p1.2.m2.5.6.2.2" rspace="0em" xref="S2.I1.i1.p1.2.m2.5.6.1.cmml">,</mo><mo id="S2.I1.i1.p1.2.m2.2.2" lspace="0em" rspace="0em" xref="S2.I1.i1.p1.2.m2.2.2.cmml">+</mo><mo id="S2.I1.i1.p1.2.m2.5.6.2.3" rspace="0em" xref="S2.I1.i1.p1.2.m2.5.6.1.cmml">,</mo><mo id="S2.I1.i1.p1.2.m2.3.3" lspace="0em" rspace="0em" xref="S2.I1.i1.p1.2.m2.3.3.cmml">⋅</mo><mo id="S2.I1.i1.p1.2.m2.5.6.2.4" xref="S2.I1.i1.p1.2.m2.5.6.1.cmml">,</mo><mn id="S2.I1.i1.p1.2.m2.4.4" xref="S2.I1.i1.p1.2.m2.4.4.cmml">0</mn><mo id="S2.I1.i1.p1.2.m2.5.6.2.5" xref="S2.I1.i1.p1.2.m2.5.6.1.cmml">,</mo><mn id="S2.I1.i1.p1.2.m2.5.5" xref="S2.I1.i1.p1.2.m2.5.5.cmml">1</mn><mo id="S2.I1.i1.p1.2.m2.5.6.2.6" stretchy="false" xref="S2.I1.i1.p1.2.m2.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.2.m2.5b"><vector id="S2.I1.i1.p1.2.m2.5.6.1.cmml" xref="S2.I1.i1.p1.2.m2.5.6.2"><ci id="S2.I1.i1.p1.2.m2.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1">ℚ</ci><plus id="S2.I1.i1.p1.2.m2.2.2.cmml" xref="S2.I1.i1.p1.2.m2.2.2"></plus><ci id="S2.I1.i1.p1.2.m2.3.3.cmml" xref="S2.I1.i1.p1.2.m2.3.3">⋅</ci><cn id="S2.I1.i1.p1.2.m2.4.4.cmml" type="integer" xref="S2.I1.i1.p1.2.m2.4.4">0</cn><cn id="S2.I1.i1.p1.2.m2.5.5.cmml" type="integer" xref="S2.I1.i1.p1.2.m2.5.5">1</cn></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.2.m2.5c">(\mathbb{Q},+,\cdot,0,1)</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.2.m2.5d">( blackboard_Q , + , ⋅ , 0 , 1 )</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> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.2"><math alttext="\mathsf{Count}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.1.m1.1"><semantics id="S2.I1.i2.p1.1.m1.1a"><mi id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml">𝖢𝗈𝗎𝗇𝗍</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.1b"><ci id="S2.I1.i2.p1.1.m1.1.1.cmml" xref="S2.I1.i2.p1.1.m1.1.1">𝖢𝗈𝗎𝗇𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.1c">\mathsf{Count}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.1.m1.1d">sansserif_Count</annotation></semantics></math>: the counting semiring <math alttext="(\mathbb{N},+,\cdot,0,1)" class="ltx_Math" display="inline" id="S2.I1.i2.p1.2.m2.5"><semantics id="S2.I1.i2.p1.2.m2.5a"><mrow id="S2.I1.i2.p1.2.m2.5.6.2" xref="S2.I1.i2.p1.2.m2.5.6.1.cmml"><mo id="S2.I1.i2.p1.2.m2.5.6.2.1" stretchy="false" xref="S2.I1.i2.p1.2.m2.5.6.1.cmml">(</mo><mi id="S2.I1.i2.p1.2.m2.1.1" xref="S2.I1.i2.p1.2.m2.1.1.cmml">ℕ</mi><mo id="S2.I1.i2.p1.2.m2.5.6.2.2" rspace="0em" xref="S2.I1.i2.p1.2.m2.5.6.1.cmml">,</mo><mo id="S2.I1.i2.p1.2.m2.2.2" lspace="0em" rspace="0em" xref="S2.I1.i2.p1.2.m2.2.2.cmml">+</mo><mo id="S2.I1.i2.p1.2.m2.5.6.2.3" rspace="0em" xref="S2.I1.i2.p1.2.m2.5.6.1.cmml">,</mo><mo id="S2.I1.i2.p1.2.m2.3.3" lspace="0em" rspace="0em" xref="S2.I1.i2.p1.2.m2.3.3.cmml">⋅</mo><mo id="S2.I1.i2.p1.2.m2.5.6.2.4" xref="S2.I1.i2.p1.2.m2.5.6.1.cmml">,</mo><mn id="S2.I1.i2.p1.2.m2.4.4" xref="S2.I1.i2.p1.2.m2.4.4.cmml">0</mn><mo id="S2.I1.i2.p1.2.m2.5.6.2.5" xref="S2.I1.i2.p1.2.m2.5.6.1.cmml">,</mo><mn id="S2.I1.i2.p1.2.m2.5.5" xref="S2.I1.i2.p1.2.m2.5.5.cmml">1</mn><mo id="S2.I1.i2.p1.2.m2.5.6.2.6" stretchy="false" xref="S2.I1.i2.p1.2.m2.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.5b"><vector id="S2.I1.i2.p1.2.m2.5.6.1.cmml" xref="S2.I1.i2.p1.2.m2.5.6.2"><ci id="S2.I1.i2.p1.2.m2.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1">ℕ</ci><plus id="S2.I1.i2.p1.2.m2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2"></plus><ci id="S2.I1.i2.p1.2.m2.3.3.cmml" xref="S2.I1.i2.p1.2.m2.3.3">⋅</ci><cn id="S2.I1.i2.p1.2.m2.4.4.cmml" type="integer" xref="S2.I1.i2.p1.2.m2.4.4">0</cn><cn id="S2.I1.i2.p1.2.m2.5.5.cmml" type="integer" xref="S2.I1.i2.p1.2.m2.5.5">1</cn></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.5c">(\mathbb{N},+,\cdot,0,1)</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.2.m2.5d">( blackboard_N , + , ⋅ , 0 , 1 )</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> <div class="ltx_para" id="S2.I1.i3.p1"> <p class="ltx_p" id="S2.I1.i3.p1.2"><math alttext="\mathsf{Max}" class="ltx_Math" display="inline" id="S2.I1.i3.p1.1.m1.1"><semantics id="S2.I1.i3.p1.1.m1.1a"><mi id="S2.I1.i3.p1.1.m1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.cmml">𝖬𝖺𝗑</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.1.m1.1b"><ci id="S2.I1.i3.p1.1.m1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1">𝖬𝖺𝗑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.1.m1.1c">\mathsf{Max}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.1.m1.1d">sansserif_Max</annotation></semantics></math>: the max tropical semiring <math alttext="(\mathbb{Q}\cup\{-\infty\},\mathsf{max},+,-\infty,0)" class="ltx_Math" display="inline" id="S2.I1.i3.p1.2.m2.5"><semantics id="S2.I1.i3.p1.2.m2.5a"><mrow id="S2.I1.i3.p1.2.m2.5.5.2" xref="S2.I1.i3.p1.2.m2.5.5.3.cmml"><mo id="S2.I1.i3.p1.2.m2.5.5.2.3" stretchy="false" xref="S2.I1.i3.p1.2.m2.5.5.3.cmml">(</mo><mrow id="S2.I1.i3.p1.2.m2.4.4.1.1" xref="S2.I1.i3.p1.2.m2.4.4.1.1.cmml"><mi id="S2.I1.i3.p1.2.m2.4.4.1.1.3" xref="S2.I1.i3.p1.2.m2.4.4.1.1.3.cmml">ℚ</mi><mo id="S2.I1.i3.p1.2.m2.4.4.1.1.2" xref="S2.I1.i3.p1.2.m2.4.4.1.1.2.cmml">∪</mo><mrow id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.2.cmml"><mo id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.2" stretchy="false" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.2.cmml">{</mo><mrow id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1.cmml"><mo id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1a" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1.cmml">−</mo><mi id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1.2" mathvariant="normal" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1.2.cmml">∞</mi></mrow><mo id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.3" stretchy="false" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="S2.I1.i3.p1.2.m2.5.5.2.4" xref="S2.I1.i3.p1.2.m2.5.5.3.cmml">,</mo><mi id="S2.I1.i3.p1.2.m2.1.1" xref="S2.I1.i3.p1.2.m2.1.1.cmml">𝗆𝖺𝗑</mi><mo id="S2.I1.i3.p1.2.m2.5.5.2.5" rspace="0em" xref="S2.I1.i3.p1.2.m2.5.5.3.cmml">,</mo><mo id="S2.I1.i3.p1.2.m2.2.2" lspace="0em" rspace="0em" xref="S2.I1.i3.p1.2.m2.2.2.cmml">+</mo><mo id="S2.I1.i3.p1.2.m2.5.5.2.6" xref="S2.I1.i3.p1.2.m2.5.5.3.cmml">,</mo><mrow id="S2.I1.i3.p1.2.m2.5.5.2.2" xref="S2.I1.i3.p1.2.m2.5.5.2.2.cmml"><mo id="S2.I1.i3.p1.2.m2.5.5.2.2a" xref="S2.I1.i3.p1.2.m2.5.5.2.2.cmml">−</mo><mi id="S2.I1.i3.p1.2.m2.5.5.2.2.2" mathvariant="normal" xref="S2.I1.i3.p1.2.m2.5.5.2.2.2.cmml">∞</mi></mrow><mo id="S2.I1.i3.p1.2.m2.5.5.2.7" xref="S2.I1.i3.p1.2.m2.5.5.3.cmml">,</mo><mn id="S2.I1.i3.p1.2.m2.3.3" xref="S2.I1.i3.p1.2.m2.3.3.cmml">0</mn><mo id="S2.I1.i3.p1.2.m2.5.5.2.8" stretchy="false" xref="S2.I1.i3.p1.2.m2.5.5.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.2.m2.5b"><vector id="S2.I1.i3.p1.2.m2.5.5.3.cmml" xref="S2.I1.i3.p1.2.m2.5.5.2"><apply id="S2.I1.i3.p1.2.m2.4.4.1.1.cmml" xref="S2.I1.i3.p1.2.m2.4.4.1.1"><union id="S2.I1.i3.p1.2.m2.4.4.1.1.2.cmml" xref="S2.I1.i3.p1.2.m2.4.4.1.1.2"></union><ci id="S2.I1.i3.p1.2.m2.4.4.1.1.3.cmml" xref="S2.I1.i3.p1.2.m2.4.4.1.1.3">ℚ</ci><set id="S2.I1.i3.p1.2.m2.4.4.1.1.1.2.cmml" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.1"><apply id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1.cmml" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1"><minus id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1.1.cmml" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1"></minus><infinity id="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1.2.cmml" xref="S2.I1.i3.p1.2.m2.4.4.1.1.1.1.1.2"></infinity></apply></set></apply><ci id="S2.I1.i3.p1.2.m2.1.1.cmml" xref="S2.I1.i3.p1.2.m2.1.1">𝗆𝖺𝗑</ci><plus id="S2.I1.i3.p1.2.m2.2.2.cmml" xref="S2.I1.i3.p1.2.m2.2.2"></plus><apply id="S2.I1.i3.p1.2.m2.5.5.2.2.cmml" xref="S2.I1.i3.p1.2.m2.5.5.2.2"><minus id="S2.I1.i3.p1.2.m2.5.5.2.2.1.cmml" xref="S2.I1.i3.p1.2.m2.5.5.2.2"></minus><infinity id="S2.I1.i3.p1.2.m2.5.5.2.2.2.cmml" xref="S2.I1.i3.p1.2.m2.5.5.2.2.2"></infinity></apply><cn id="S2.I1.i3.p1.2.m2.3.3.cmml" type="integer" xref="S2.I1.i3.p1.2.m2.3.3">0</cn></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.2.m2.5c">(\mathbb{Q}\cup\{-\infty\},\mathsf{max},+,-\infty,0)</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.2.m2.5d">( blackboard_Q ∪ { - ∞ } , sansserif_max , + , - ∞ , 0 )</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> <div class="ltx_para" id="S2.I1.i4.p1"> <p class="ltx_p" id="S2.I1.i4.p1.2"><math alttext="\mathsf{Min}" class="ltx_Math" display="inline" id="S2.I1.i4.p1.1.m1.1"><semantics id="S2.I1.i4.p1.1.m1.1a"><mi id="S2.I1.i4.p1.1.m1.1.1" xref="S2.I1.i4.p1.1.m1.1.1.cmml">𝖬𝗂𝗇</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i4.p1.1.m1.1b"><ci id="S2.I1.i4.p1.1.m1.1.1.cmml" xref="S2.I1.i4.p1.1.m1.1.1">𝖬𝗂𝗇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i4.p1.1.m1.1c">\mathsf{Min}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i4.p1.1.m1.1d">sansserif_Min</annotation></semantics></math>: the min tropical semiring <math alttext="(\mathbb{Q}\cup\{\infty\},\mathsf{min},+,\infty,0)" class="ltx_Math" display="inline" id="S2.I1.i4.p1.2.m2.6"><semantics id="S2.I1.i4.p1.2.m2.6a"><mrow id="S2.I1.i4.p1.2.m2.6.6.1" xref="S2.I1.i4.p1.2.m2.6.6.2.cmml"><mo id="S2.I1.i4.p1.2.m2.6.6.1.2" stretchy="false" xref="S2.I1.i4.p1.2.m2.6.6.2.cmml">(</mo><mrow id="S2.I1.i4.p1.2.m2.6.6.1.1" xref="S2.I1.i4.p1.2.m2.6.6.1.1.cmml"><mi id="S2.I1.i4.p1.2.m2.6.6.1.1.2" xref="S2.I1.i4.p1.2.m2.6.6.1.1.2.cmml">ℚ</mi><mo id="S2.I1.i4.p1.2.m2.6.6.1.1.1" xref="S2.I1.i4.p1.2.m2.6.6.1.1.1.cmml">∪</mo><mrow id="S2.I1.i4.p1.2.m2.6.6.1.1.3.2" xref="S2.I1.i4.p1.2.m2.6.6.1.1.3.1.cmml"><mo id="S2.I1.i4.p1.2.m2.6.6.1.1.3.2.1" stretchy="false" xref="S2.I1.i4.p1.2.m2.6.6.1.1.3.1.cmml">{</mo><mi id="S2.I1.i4.p1.2.m2.1.1" mathvariant="normal" xref="S2.I1.i4.p1.2.m2.1.1.cmml">∞</mi><mo id="S2.I1.i4.p1.2.m2.6.6.1.1.3.2.2" stretchy="false" xref="S2.I1.i4.p1.2.m2.6.6.1.1.3.1.cmml">}</mo></mrow></mrow><mo id="S2.I1.i4.p1.2.m2.6.6.1.3" xref="S2.I1.i4.p1.2.m2.6.6.2.cmml">,</mo><mi id="S2.I1.i4.p1.2.m2.2.2" xref="S2.I1.i4.p1.2.m2.2.2.cmml">𝗆𝗂𝗇</mi><mo id="S2.I1.i4.p1.2.m2.6.6.1.4" rspace="0em" xref="S2.I1.i4.p1.2.m2.6.6.2.cmml">,</mo><mo id="S2.I1.i4.p1.2.m2.3.3" lspace="0em" rspace="0em" xref="S2.I1.i4.p1.2.m2.3.3.cmml">+</mo><mo id="S2.I1.i4.p1.2.m2.6.6.1.5" xref="S2.I1.i4.p1.2.m2.6.6.2.cmml">,</mo><mi id="S2.I1.i4.p1.2.m2.4.4" mathvariant="normal" xref="S2.I1.i4.p1.2.m2.4.4.cmml">∞</mi><mo id="S2.I1.i4.p1.2.m2.6.6.1.6" xref="S2.I1.i4.p1.2.m2.6.6.2.cmml">,</mo><mn id="S2.I1.i4.p1.2.m2.5.5" xref="S2.I1.i4.p1.2.m2.5.5.cmml">0</mn><mo id="S2.I1.i4.p1.2.m2.6.6.1.7" stretchy="false" xref="S2.I1.i4.p1.2.m2.6.6.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i4.p1.2.m2.6b"><vector id="S2.I1.i4.p1.2.m2.6.6.2.cmml" xref="S2.I1.i4.p1.2.m2.6.6.1"><apply id="S2.I1.i4.p1.2.m2.6.6.1.1.cmml" xref="S2.I1.i4.p1.2.m2.6.6.1.1"><union id="S2.I1.i4.p1.2.m2.6.6.1.1.1.cmml" xref="S2.I1.i4.p1.2.m2.6.6.1.1.1"></union><ci id="S2.I1.i4.p1.2.m2.6.6.1.1.2.cmml" xref="S2.I1.i4.p1.2.m2.6.6.1.1.2">ℚ</ci><set id="S2.I1.i4.p1.2.m2.6.6.1.1.3.1.cmml" xref="S2.I1.i4.p1.2.m2.6.6.1.1.3.2"><infinity id="S2.I1.i4.p1.2.m2.1.1.cmml" xref="S2.I1.i4.p1.2.m2.1.1"></infinity></set></apply><ci id="S2.I1.i4.p1.2.m2.2.2.cmml" xref="S2.I1.i4.p1.2.m2.2.2">𝗆𝗂𝗇</ci><plus id="S2.I1.i4.p1.2.m2.3.3.cmml" xref="S2.I1.i4.p1.2.m2.3.3"></plus><infinity id="S2.I1.i4.p1.2.m2.4.4.cmml" xref="S2.I1.i4.p1.2.m2.4.4"></infinity><cn id="S2.I1.i4.p1.2.m2.5.5.cmml" type="integer" xref="S2.I1.i4.p1.2.m2.5.5">0</cn></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i4.p1.2.m2.6c">(\mathbb{Q}\cup\{\infty\},\mathsf{min},+,\infty,0)</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i4.p1.2.m2.6d">( blackboard_Q ∪ { ∞ } , sansserif_min , + , ∞ , 0 )</annotation></semantics></math>.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx3.p7"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p7.1">The translation is straightforward (and known, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:journals/vldb/ReS09</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:journals/tods/KhamisCMNNOS20</span>]</cite>), as we illustrate in <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S2.F1" title="In Annotated databases and query answers. ‣ Databases and conjunctive queries. ‣ 2. Preliminaries ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Figure</span> <span class="ltx_text ltx_ref_tag">1</span></a>: the aggregated value becomes the annotation on one of the relations, the annotation outside of this relation is the multiplicative identity (as we later term “locally annotated”), and the addition operation captures the aggregate function. Note that in the case of the numeric and min/max tropical semirings, we are using the domain <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1c">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p7.1.m1.1d">blackboard_Q</annotation></semantics></math> of rational numbers rather than all real numbers to avoid issues of numerical presentation in the computational model.</p> </div> <div class="ltx_para" id="S2.SS0.SSS0.Px1.SPx3.p8"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.SPx3.p8.6"><math alttext="\mathsf{Avg}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1.1.cmml">𝖠𝗏𝗀</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1.1">𝖠𝗏𝗀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1c">\mathsf{Avg}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p8.1.m1.1d">sansserif_Avg</annotation></semantics></math> can be computed using <math alttext="\mathsf{Sum}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1.1.cmml">𝖲𝗎𝗆</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1.1">𝖲𝗎𝗆</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1c">\mathsf{Sum}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p8.2.m2.1d">sansserif_Sum</annotation></semantics></math> and <math alttext="\mathsf{Count}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1.1.cmml">𝖢𝗈𝗎𝗇𝗍</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1.1">𝖢𝗈𝗎𝗇𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1c">\mathsf{Count}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p8.3.m3.1d">sansserif_Count</annotation></semantics></math>. <math alttext="\mathsf{CountD}" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1a"><mi id="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1.1.cmml">𝖢𝗈𝗎𝗇𝗍𝖣</mi><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1b"><ci id="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1.1">𝖢𝗈𝗎𝗇𝗍𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1c">\mathsf{CountD}</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p8.4.m4.1d">sansserif_CountD</annotation></semantics></math> (count distinct) cannot be captured by a semiring, as the result of <math alttext="\oplus" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1"><semantics id="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1a"><mo id="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p8.5.m5.1d">⊕</annotation></semantics></math> cannot be computed from two intermediary annotations in the domain. We can, however, capture a semantically similar concept with the set semiring <math alttext="({\mathcal{P}}(\Omega),\cup,\cap,\varnothing,\Omega)" class="ltx_Math" display="inline" id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6"><semantics id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6a"><mrow id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.2.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.2.cmml">(</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.2" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.2.cmml">𝒫</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.1" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.1.cmml">⁢</mo><mrow id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.3.2" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.cmml"><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.3.2.1" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.cmml">(</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.1.1" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.1.1.cmml">Ω</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.3.2.2" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.3" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.2.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.2.2" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.2.2.cmml">∪</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.4" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.2.cmml">,</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.3.3" lspace="0em" rspace="0em" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.3.3.cmml">∩</mo><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.5" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.2.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.4.4" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.4.4.cmml">∅</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.6" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.2.cmml">,</mo><mi id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.5.5" mathvariant="normal" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.5.5.cmml">Ω</mi><mo id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.7" stretchy="false" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6b"><vector id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1"><apply id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1"><times id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.1"></times><ci id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6.6.1.1.2">𝒫</ci><ci id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.1.1.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.1.1">Ω</ci></apply><union id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.2.2.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.2.2"></union><intersect id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.3.3.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.3.3"></intersect><emptyset id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.4.4.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.4.4"></emptyset><ci id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.5.5.cmml" xref="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.5.5">Ω</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6c">({\mathcal{P}}(\Omega),\cup,\cap,\varnothing,\Omega)</annotation><annotation encoding="application/x-llamapun" id="S2.SS0.SSS0.Px1.SPx3.p8.6.m6.6d">( caligraphic_P ( roman_Ω ) , ∪ , ∩ , ∅ , roman_Ω )</annotation></semantics></math> by annotating each fact with the actual set of distinct elements. However, in such cases, we will need our complexity analysis to be aware of the cost of the operations.</p> </div> <figure class="ltx_figure" id="S2.F1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_3"> <table class="ltx_tabular ltx_figure_panel ltx_align_middle" id="S2.F1.2"> <tr class="ltx_tr" id="S2.F1.2.3"> <td class="ltx_td ltx_align_left" colspan="2" id="S2.F1.2.3.1"><span class="ltx_text ltx_font_smallcaps" id="S2.F1.2.3.1.1" style="font-size:90%;">Teams</span></td> </tr> <tr class="ltx_tr" id="S2.F1.2.2"> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.1.1.1" style="background-color:#D9D9D9;"><math alttext="p" class="ltx_Math" display="inline" id="S2.F1.1.1.1.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.1.1.1.m1.1a"><mi id="S2.F1.1.1.1.m1.1.1" mathbackground="#D9D9D9" mathsize="90%" xref="S2.F1.1.1.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.F1.1.1.1.m1.1b"><ci id="S2.F1.1.1.1.m1.1.1.cmml" xref="S2.F1.1.1.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.1.1.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.F1.1.1.1.m1.1d">italic_p</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.2.2.2" style="background-color:#D9D9D9;"><math alttext="c" class="ltx_Math" display="inline" id="S2.F1.2.2.2.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.2.2.2.m1.1a"><mi id="S2.F1.2.2.2.m1.1.1" mathbackground="#D9D9D9" mathsize="90%" xref="S2.F1.2.2.2.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.F1.2.2.2.m1.1b"><ci id="S2.F1.2.2.2.m1.1.1.cmml" xref="S2.F1.2.2.2.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.2.2.2.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.F1.2.2.2.m1.1d">italic_c</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S2.F1.2.4"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_t" id="S2.F1.2.4.1"><span class="ltx_text" id="S2.F1.2.4.1.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.2.4.2"><span class="ltx_text" id="S2.F1.2.4.2.1" style="font-size:90%;">5</span></td> </tr> <tr class="ltx_tr" id="S2.F1.2.5"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.2.5.1"><span class="ltx_text" id="S2.F1.2.5.1.1" style="font-size:90%;">2</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.2.5.2"><span class="ltx_text" id="S2.F1.2.5.2.1" style="font-size:90%;">5</span></td> </tr> <tr class="ltx_tr" id="S2.F1.2.6"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.2.6.1"><span class="ltx_text" id="S2.F1.2.6.1.1" style="font-size:90%;">3</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.2.6.2"><span class="ltx_text" id="S2.F1.2.6.2.1" style="font-size:90%;">6</span></td> </tr> <tr class="ltx_tr" id="S2.F1.2.7"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.2.7.1"><span class="ltx_text" id="S2.F1.2.7.1.1" style="font-size:90%;">4</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.2.7.2"><span class="ltx_text" id="S2.F1.2.7.2.1" style="font-size:90%;">7</span></td> </tr> <tr class="ltx_tr" id="S2.F1.2.8"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l" id="S2.F1.2.8.1"><span class="ltx_text" id="S2.F1.2.8.1.1" style="font-size:90%;">5</span></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.2.8.2"><span class="ltx_text" id="S2.F1.2.8.2.1" style="font-size:90%;">8</span></td> </tr> </table> </div> <div class="ltx_flex_cell ltx_flex_size_3"> <table class="ltx_tabular ltx_figure_panel ltx_align_middle" id="S2.F1.5"> <tr class="ltx_tr" id="S2.F1.5.4"> <td class="ltx_td ltx_align_left" colspan="3" id="S2.F1.5.4.1"><span class="ltx_text ltx_font_smallcaps" id="S2.F1.5.4.1.1" style="font-size:90%;">Goals</span></td> </tr> <tr class="ltx_tr" id="S2.F1.5.3"> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.3.1.1" style="background-color:#D9D9D9;"><math alttext="g" class="ltx_Math" display="inline" id="S2.F1.3.1.1.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.3.1.1.m1.1a"><mi id="S2.F1.3.1.1.m1.1.1" mathbackground="#D9D9D9" mathsize="90%" xref="S2.F1.3.1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.F1.3.1.1.m1.1b"><ci id="S2.F1.3.1.1.m1.1.1.cmml" xref="S2.F1.3.1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.3.1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.F1.3.1.1.m1.1d">italic_g</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.4.2.2" style="background-color:#D9D9D9;"><math alttext="p" class="ltx_Math" display="inline" id="S2.F1.4.2.2.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.4.2.2.m1.1a"><mi id="S2.F1.4.2.2.m1.1.1" mathbackground="#D9D9D9" mathsize="90%" xref="S2.F1.4.2.2.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.F1.4.2.2.m1.1b"><ci id="S2.F1.4.2.2.m1.1.1.cmml" xref="S2.F1.4.2.2.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.4.2.2.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.F1.4.2.2.m1.1d">italic_p</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.5.3.3" style="background-color:#D9D9D9;"><math alttext="t" class="ltx_Math" display="inline" id="S2.F1.5.3.3.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.5.3.3.m1.1a"><mi id="S2.F1.5.3.3.m1.1.1" mathbackground="#D9D9D9" mathsize="90%" xref="S2.F1.5.3.3.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.F1.5.3.3.m1.1b"><ci id="S2.F1.5.3.3.m1.1.1.cmml" xref="S2.F1.5.3.3.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.5.3.3.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.F1.5.3.3.m1.1d">italic_t</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S2.F1.5.5"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_t" id="S2.F1.5.5.1"><span class="ltx_text" id="S2.F1.5.5.1.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.5.5.2"><span class="ltx_text" id="S2.F1.5.5.2.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.5.5.3"><span class="ltx_text" id="S2.F1.5.5.3.1" style="font-size:90%;">31</span></td> </tr> <tr class="ltx_tr" id="S2.F1.5.6"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.5.6.1"><span class="ltx_text" id="S2.F1.5.6.1.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center" id="S2.F1.5.6.2"><span class="ltx_text" id="S2.F1.5.6.2.1" style="font-size:90%;">3</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.5.6.3"><span class="ltx_text" id="S2.F1.5.6.3.1" style="font-size:90%;">50</span></td> </tr> <tr class="ltx_tr" id="S2.F1.5.7"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.5.7.1"><span class="ltx_text" id="S2.F1.5.7.1.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center" id="S2.F1.5.7.2"><span class="ltx_text" id="S2.F1.5.7.2.1" style="font-size:90%;">3</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.5.7.3"><span class="ltx_text" id="S2.F1.5.7.3.1" style="font-size:90%;">75</span></td> </tr> <tr class="ltx_tr" id="S2.F1.5.8"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.5.8.1"><span class="ltx_text" id="S2.F1.5.8.1.1" style="font-size:90%;">2</span></td> <td class="ltx_td ltx_align_center" id="S2.F1.5.8.2"><span class="ltx_text" id="S2.F1.5.8.2.1" style="font-size:90%;">4</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.5.8.3"><span class="ltx_text" id="S2.F1.5.8.3.1" style="font-size:90%;">90</span></td> </tr> <tr class="ltx_tr" id="S2.F1.5.9"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l" id="S2.F1.5.9.1"><span class="ltx_text" id="S2.F1.5.9.1.1" style="font-size:90%;">2</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S2.F1.5.9.2"><span class="ltx_text" id="S2.F1.5.9.2.1" style="font-size:90%;">4</span></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.5.9.3"><span class="ltx_text" id="S2.F1.5.9.3.1" style="font-size:90%;">9</span></td> </tr> </table> </div> <div class="ltx_flex_cell ltx_flex_size_3"> <table class="ltx_tabular ltx_figure_panel ltx_align_middle" id="S2.F1.7"> <tr class="ltx_tr" id="S2.F1.7.3"> <td class="ltx_td ltx_align_left" colspan="2" id="S2.F1.7.3.1"><span class="ltx_text ltx_font_smallcaps" id="S2.F1.7.3.1.1" style="font-size:90%;">Replays</span></td> </tr> <tr class="ltx_tr" id="S2.F1.7.2"> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.6.1.1" style="background-color:#D9D9D9;"><math alttext="g" class="ltx_Math" display="inline" id="S2.F1.6.1.1.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.6.1.1.m1.1a"><mi id="S2.F1.6.1.1.m1.1.1" mathbackground="#D9D9D9" mathsize="90%" xref="S2.F1.6.1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.F1.6.1.1.m1.1b"><ci id="S2.F1.6.1.1.m1.1.1.cmml" xref="S2.F1.6.1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.6.1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.F1.6.1.1.m1.1d">italic_g</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.7.2.2" style="background-color:#D9D9D9;"><math alttext="t" class="ltx_Math" display="inline" id="S2.F1.7.2.2.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.7.2.2.m1.1a"><mi id="S2.F1.7.2.2.m1.1.1" mathbackground="#D9D9D9" mathsize="90%" xref="S2.F1.7.2.2.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.F1.7.2.2.m1.1b"><ci id="S2.F1.7.2.2.m1.1.1.cmml" xref="S2.F1.7.2.2.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.7.2.2.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.F1.7.2.2.m1.1d">italic_t</annotation></semantics></math></td> </tr> <tr class="ltx_tr" id="S2.F1.7.4"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_t" id="S2.F1.7.4.1"><span class="ltx_text" id="S2.F1.7.4.1.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.7.4.2"><span class="ltx_text" id="S2.F1.7.4.2.1" style="font-size:90%;">1</span></td> </tr> <tr class="ltx_tr" id="S2.F1.7.5"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.7.5.1"><span class="ltx_text" id="S2.F1.7.5.1.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.7.5.2"><span class="ltx_text" id="S2.F1.7.5.2.1" style="font-size:90%;">31</span></td> </tr> <tr class="ltx_tr" id="S2.F1.7.6"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.7.6.1"><span class="ltx_text" id="S2.F1.7.6.1.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.7.6.2"><span class="ltx_text" id="S2.F1.7.6.2.1" style="font-size:90%;">50</span></td> </tr> <tr class="ltx_tr" id="S2.F1.7.7"> <td class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.7.7.1"><span class="ltx_text" id="S2.F1.7.7.1.1" style="font-size:90%;">2</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.7.7.2"><span class="ltx_text" id="S2.F1.7.7.2.1" style="font-size:90%;">5</span></td> </tr> <tr class="ltx_tr" id="S2.F1.7.8"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l" id="S2.F1.7.8.1"><span class="ltx_text" id="S2.F1.7.8.1.1" style="font-size:90%;">1</span></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.7.8.2"><span class="ltx_text" id="S2.F1.7.8.2.1" style="font-size:90%;">90</span></td> </tr> </table> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel" id="S2.F1.18"><math alttext="\Rightarrow" class="ltx_Math" display="inline" id="S2.F1.8.m1.1"><semantics id="S2.F1.8.m1.1a"><mo id="S2.F1.8.m1.1.1" mathsize="90%" stretchy="false" xref="S2.F1.8.m1.1.1.cmml">⇒</mo><annotation-xml encoding="MathML-Content" id="S2.F1.8.m1.1b"><ci id="S2.F1.8.m1.1.1.cmml" xref="S2.F1.8.m1.1.1">⇒</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.8.m1.1c">\Rightarrow</annotation><annotation encoding="application/x-llamapun" id="S2.F1.8.m1.1d">⇒</annotation></semantics></math><span class="ltx_text" id="S2.F1.18.10" style="font-size:90%;"> <span class="ltx_tabular ltx_align_middle" id="S2.F1.11.3.3"> <span class="ltx_tr" id="S2.F1.11.3.3.4"> <span class="ltx_td ltx_align_left ltx_colspan ltx_colspan_3" id="S2.F1.11.3.3.4.1"><span class="ltx_text ltx_font_smallcaps" id="S2.F1.11.3.3.4.1.1">Team</span></span></span> <span class="ltx_tr" id="S2.F1.11.3.3.3"> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.9.1.1.1.1" style="background-color:#D9D9D9;"><math alttext="p" class="ltx_Math" display="inline" id="S2.F1.9.1.1.1.1.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.9.1.1.1.1.m1.1a"><mi id="S2.F1.9.1.1.1.1.m1.1.1" mathbackground="#D9D9D9" xref="S2.F1.9.1.1.1.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.F1.9.1.1.1.1.m1.1b"><ci id="S2.F1.9.1.1.1.1.m1.1.1.cmml" xref="S2.F1.9.1.1.1.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.9.1.1.1.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.F1.9.1.1.1.1.m1.1d">italic_p</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.10.2.2.2.2" style="background-color:#D9D9D9;"><math alttext="c" class="ltx_Math" display="inline" id="S2.F1.10.2.2.2.2.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.10.2.2.2.2.m1.1a"><mi id="S2.F1.10.2.2.2.2.m1.1.1" mathbackground="#D9D9D9" xref="S2.F1.10.2.2.2.2.m1.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.F1.10.2.2.2.2.m1.1b"><ci id="S2.F1.10.2.2.2.2.m1.1.1.cmml" xref="S2.F1.10.2.2.2.2.m1.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.10.2.2.2.2.m1.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.F1.10.2.2.2.2.m1.1d">italic_c</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.11.3.3.3.3" style="background-color:#D9D9D9;"><math alttext="\tau_{+}" class="ltx_Math" display="inline" id="S2.F1.11.3.3.3.3.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.11.3.3.3.3.m1.1a"><msub id="S2.F1.11.3.3.3.3.m1.1.1" xref="S2.F1.11.3.3.3.3.m1.1.1.cmml"><mi id="S2.F1.11.3.3.3.3.m1.1.1.2" mathbackground="#D9D9D9" xref="S2.F1.11.3.3.3.3.m1.1.1.2.cmml">τ</mi><mo id="S2.F1.11.3.3.3.3.m1.1.1.3" mathbackground="#D9D9D9" xref="S2.F1.11.3.3.3.3.m1.1.1.3.cmml">+</mo></msub><annotation-xml encoding="MathML-Content" id="S2.F1.11.3.3.3.3.m1.1b"><apply id="S2.F1.11.3.3.3.3.m1.1.1.cmml" xref="S2.F1.11.3.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S2.F1.11.3.3.3.3.m1.1.1.1.cmml" xref="S2.F1.11.3.3.3.3.m1.1.1">subscript</csymbol><ci id="S2.F1.11.3.3.3.3.m1.1.1.2.cmml" xref="S2.F1.11.3.3.3.3.m1.1.1.2">𝜏</ci><plus id="S2.F1.11.3.3.3.3.m1.1.1.3.cmml" xref="S2.F1.11.3.3.3.3.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.11.3.3.3.3.m1.1c">\tau_{+}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.11.3.3.3.3.m1.1d">italic_τ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math></span></span> <span class="ltx_tr" id="S2.F1.11.3.3.5"> <span class="ltx_td ltx_align_center ltx_border_l ltx_border_t" id="S2.F1.11.3.3.5.1">1</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.11.3.3.5.2">5</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.11.3.3.5.3">1</span></span> <span class="ltx_tr" id="S2.F1.11.3.3.6"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.11.3.3.6.1">2</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.11.3.3.6.2">5</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.11.3.3.6.3">1</span></span> <span class="ltx_tr" id="S2.F1.11.3.3.7"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.11.3.3.7.1">3</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.11.3.3.7.2">6</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.11.3.3.7.3">1</span></span> <span class="ltx_tr" id="S2.F1.11.3.3.8"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.11.3.3.8.1">4</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.11.3.3.8.2">7</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.11.3.3.8.3">1</span></span> <span class="ltx_tr" id="S2.F1.11.3.3.9"> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_l" id="S2.F1.11.3.3.9.1">5</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.11.3.3.9.2">8</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.11.3.3.9.3">1</span></span> </span> <span class="ltx_tabular ltx_align_middle" id="S2.F1.15.7.7"> <span class="ltx_tr" id="S2.F1.15.7.7.5"> <span class="ltx_td ltx_align_left ltx_colspan ltx_colspan_4" id="S2.F1.15.7.7.5.1"><span class="ltx_text ltx_font_smallcaps" id="S2.F1.15.7.7.5.1.1">Goals</span></span></span> <span class="ltx_tr" id="S2.F1.15.7.7.4"> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.12.4.4.1.1" style="background-color:#D9D9D9;"><math alttext="g" class="ltx_Math" display="inline" id="S2.F1.12.4.4.1.1.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.12.4.4.1.1.m1.1a"><mi id="S2.F1.12.4.4.1.1.m1.1.1" mathbackground="#D9D9D9" xref="S2.F1.12.4.4.1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.F1.12.4.4.1.1.m1.1b"><ci id="S2.F1.12.4.4.1.1.m1.1.1.cmml" xref="S2.F1.12.4.4.1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.12.4.4.1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.F1.12.4.4.1.1.m1.1d">italic_g</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.13.5.5.2.2" style="background-color:#D9D9D9;"><math alttext="p" class="ltx_Math" display="inline" id="S2.F1.13.5.5.2.2.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.13.5.5.2.2.m1.1a"><mi id="S2.F1.13.5.5.2.2.m1.1.1" mathbackground="#D9D9D9" xref="S2.F1.13.5.5.2.2.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.F1.13.5.5.2.2.m1.1b"><ci id="S2.F1.13.5.5.2.2.m1.1.1.cmml" xref="S2.F1.13.5.5.2.2.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.13.5.5.2.2.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.F1.13.5.5.2.2.m1.1d">italic_p</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.14.6.6.3.3" style="background-color:#D9D9D9;"><math alttext="t" class="ltx_Math" display="inline" id="S2.F1.14.6.6.3.3.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.14.6.6.3.3.m1.1a"><mi id="S2.F1.14.6.6.3.3.m1.1.1" mathbackground="#D9D9D9" xref="S2.F1.14.6.6.3.3.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.F1.14.6.6.3.3.m1.1b"><ci id="S2.F1.14.6.6.3.3.m1.1.1.cmml" xref="S2.F1.14.6.6.3.3.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.14.6.6.3.3.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.F1.14.6.6.3.3.m1.1d">italic_t</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.15.7.7.4.4" style="background-color:#D9D9D9;"><math alttext="\tau_{+}" class="ltx_Math" display="inline" id="S2.F1.15.7.7.4.4.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.15.7.7.4.4.m1.1a"><msub id="S2.F1.15.7.7.4.4.m1.1.1" xref="S2.F1.15.7.7.4.4.m1.1.1.cmml"><mi id="S2.F1.15.7.7.4.4.m1.1.1.2" mathbackground="#D9D9D9" xref="S2.F1.15.7.7.4.4.m1.1.1.2.cmml">τ</mi><mo id="S2.F1.15.7.7.4.4.m1.1.1.3" mathbackground="#D9D9D9" xref="S2.F1.15.7.7.4.4.m1.1.1.3.cmml">+</mo></msub><annotation-xml encoding="MathML-Content" id="S2.F1.15.7.7.4.4.m1.1b"><apply id="S2.F1.15.7.7.4.4.m1.1.1.cmml" xref="S2.F1.15.7.7.4.4.m1.1.1"><csymbol cd="ambiguous" id="S2.F1.15.7.7.4.4.m1.1.1.1.cmml" xref="S2.F1.15.7.7.4.4.m1.1.1">subscript</csymbol><ci id="S2.F1.15.7.7.4.4.m1.1.1.2.cmml" xref="S2.F1.15.7.7.4.4.m1.1.1.2">𝜏</ci><plus id="S2.F1.15.7.7.4.4.m1.1.1.3.cmml" xref="S2.F1.15.7.7.4.4.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.15.7.7.4.4.m1.1c">\tau_{+}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.15.7.7.4.4.m1.1d">italic_τ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math></span></span> <span class="ltx_tr" id="S2.F1.15.7.7.6"> <span class="ltx_td ltx_align_center ltx_border_l ltx_border_t" id="S2.F1.15.7.7.6.1">1</span> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.15.7.7.6.2">1</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.15.7.7.6.3">31</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.15.7.7.6.4">1</span></span> <span class="ltx_tr" id="S2.F1.15.7.7.7"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.15.7.7.7.1">1</span> <span class="ltx_td ltx_align_center" id="S2.F1.15.7.7.7.2">3</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.15.7.7.7.3">50</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.15.7.7.7.4">1</span></span> <span class="ltx_tr" id="S2.F1.15.7.7.8"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.15.7.7.8.1">1</span> <span class="ltx_td ltx_align_center" id="S2.F1.15.7.7.8.2">3</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.15.7.7.8.3">75</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.15.7.7.8.4">1</span></span> <span class="ltx_tr" id="S2.F1.15.7.7.9"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.15.7.7.9.1">2</span> <span class="ltx_td ltx_align_center" id="S2.F1.15.7.7.9.2">4</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.15.7.7.9.3">90</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.15.7.7.9.4">1</span></span> <span class="ltx_tr" id="S2.F1.15.7.7.10"> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_l" id="S2.F1.15.7.7.10.1">2</span> <span class="ltx_td ltx_align_center ltx_border_b" id="S2.F1.15.7.7.10.2">4</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.15.7.7.10.3">9</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.15.7.7.10.4">1</span></span> </span> <span class="ltx_tabular ltx_align_middle" id="S2.F1.18.10.10"> <span class="ltx_tr" id="S2.F1.18.10.10.4"> <span class="ltx_td ltx_align_left ltx_colspan ltx_colspan_3" id="S2.F1.18.10.10.4.1"><span class="ltx_text ltx_font_smallcaps" id="S2.F1.18.10.10.4.1.1">Replays</span></span></span> <span class="ltx_tr" id="S2.F1.18.10.10.3"> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.16.8.8.1.1" style="background-color:#D9D9D9;"><math alttext="g" class="ltx_Math" display="inline" id="S2.F1.16.8.8.1.1.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.16.8.8.1.1.m1.1a"><mi id="S2.F1.16.8.8.1.1.m1.1.1" mathbackground="#D9D9D9" xref="S2.F1.16.8.8.1.1.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S2.F1.16.8.8.1.1.m1.1b"><ci id="S2.F1.16.8.8.1.1.m1.1.1.cmml" xref="S2.F1.16.8.8.1.1.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.16.8.8.1.1.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S2.F1.16.8.8.1.1.m1.1d">italic_g</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.17.9.9.2.2" style="background-color:#D9D9D9;"><math alttext="t" class="ltx_Math" display="inline" id="S2.F1.17.9.9.2.2.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.17.9.9.2.2.m1.1a"><mi id="S2.F1.17.9.9.2.2.m1.1.1" mathbackground="#D9D9D9" xref="S2.F1.17.9.9.2.2.m1.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.F1.17.9.9.2.2.m1.1b"><ci id="S2.F1.17.9.9.2.2.m1.1.1.cmml" xref="S2.F1.17.9.9.2.2.m1.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.17.9.9.2.2.m1.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.F1.17.9.9.2.2.m1.1d">italic_t</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_t" id="S2.F1.18.10.10.3.3" style="background-color:#D9D9D9;"><math alttext="\tau_{+}" class="ltx_Math" display="inline" id="S2.F1.18.10.10.3.3.m1.1" style="background-color:#D9D9D9;"><semantics id="S2.F1.18.10.10.3.3.m1.1a"><msub id="S2.F1.18.10.10.3.3.m1.1.1" xref="S2.F1.18.10.10.3.3.m1.1.1.cmml"><mi id="S2.F1.18.10.10.3.3.m1.1.1.2" mathbackground="#D9D9D9" xref="S2.F1.18.10.10.3.3.m1.1.1.2.cmml">τ</mi><mo id="S2.F1.18.10.10.3.3.m1.1.1.3" mathbackground="#D9D9D9" xref="S2.F1.18.10.10.3.3.m1.1.1.3.cmml">+</mo></msub><annotation-xml encoding="MathML-Content" id="S2.F1.18.10.10.3.3.m1.1b"><apply id="S2.F1.18.10.10.3.3.m1.1.1.cmml" xref="S2.F1.18.10.10.3.3.m1.1.1"><csymbol cd="ambiguous" id="S2.F1.18.10.10.3.3.m1.1.1.1.cmml" xref="S2.F1.18.10.10.3.3.m1.1.1">subscript</csymbol><ci id="S2.F1.18.10.10.3.3.m1.1.1.2.cmml" xref="S2.F1.18.10.10.3.3.m1.1.1.2">𝜏</ci><plus id="S2.F1.18.10.10.3.3.m1.1.1.3.cmml" xref="S2.F1.18.10.10.3.3.m1.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.18.10.10.3.3.m1.1c">\tau_{+}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.18.10.10.3.3.m1.1d">italic_τ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math></span></span> <span class="ltx_tr" id="S2.F1.18.10.10.5"> <span class="ltx_td ltx_align_center ltx_border_l ltx_border_t" id="S2.F1.18.10.10.5.1">1</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.18.10.10.5.2">1</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S2.F1.18.10.10.5.3">1</span></span> <span class="ltx_tr" id="S2.F1.18.10.10.6"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.18.10.10.6.1">1</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.18.10.10.6.2">31</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.18.10.10.6.3">31</span></span> <span class="ltx_tr" id="S2.F1.18.10.10.7"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.18.10.10.7.1">1</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.18.10.10.7.2">50</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.18.10.10.7.3">50</span></span> <span class="ltx_tr" id="S2.F1.18.10.10.8"> <span class="ltx_td ltx_align_center ltx_border_l" id="S2.F1.18.10.10.8.1">2</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.18.10.10.8.2">5</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S2.F1.18.10.10.8.3">5</span></span> <span class="ltx_tr" id="S2.F1.18.10.10.9"> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_l" id="S2.F1.18.10.10.9.1">1</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.18.10.10.9.2">90</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S2.F1.18.10.10.9.3">90</span></span> </span></span></p> </div> </div> <figcaption class="ltx_caption" style="font-size:90%;"><span class="ltx_tag ltx_tag_figure">Figure 1. </span>An example of a <math alttext="\mathbb{Q}" class="ltx_Math" display="inline" id="S2.F1.21.m1.1"><semantics id="S2.F1.21.m1.1b"><mi id="S2.F1.21.m1.1.1" xref="S2.F1.21.m1.1.1.cmml">ℚ</mi><annotation-xml encoding="MathML-Content" id="S2.F1.21.m1.1c"><ci id="S2.F1.21.m1.1.1.cmml" xref="S2.F1.21.m1.1.1">ℚ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.21.m1.1d">\mathbb{Q}</annotation><annotation encoding="application/x-llamapun" id="S2.F1.21.m1.1e">blackboard_Q</annotation></semantics></math>-database over the numerical semiring constructed to evaluate the AggCQ <math alttext="Q(c,\mathsf{Sum}(t)){\,:\!\!-\,}\textsc{Teams}(p,c),\textsc{Goals}(g,p,t),% \textsc{Replays}(g,t)" class="ltx_Math" display="inline" id="S2.F1.22.m2.13"><semantics id="S2.F1.22.m2.13b"><mrow id="S2.F1.22.m2.13.13" xref="S2.F1.22.m2.13.13.cmml"><mrow id="S2.F1.22.m2.10.10.1" xref="S2.F1.22.m2.10.10.1.cmml"><mi id="S2.F1.22.m2.10.10.1.3" xref="S2.F1.22.m2.10.10.1.3.cmml">Q</mi><mo id="S2.F1.22.m2.10.10.1.2" xref="S2.F1.22.m2.10.10.1.2.cmml">⁢</mo><mrow id="S2.F1.22.m2.10.10.1.1.1" xref="S2.F1.22.m2.10.10.1.1.2.cmml"><mo id="S2.F1.22.m2.10.10.1.1.1.2" stretchy="false" xref="S2.F1.22.m2.10.10.1.1.2.cmml">(</mo><mi id="S2.F1.22.m2.2.2" xref="S2.F1.22.m2.2.2.cmml">c</mi><mo id="S2.F1.22.m2.10.10.1.1.1.3" xref="S2.F1.22.m2.10.10.1.1.2.cmml">,</mo><mrow id="S2.F1.22.m2.10.10.1.1.1.1" xref="S2.F1.22.m2.10.10.1.1.1.1.cmml"><mi id="S2.F1.22.m2.10.10.1.1.1.1.2" xref="S2.F1.22.m2.10.10.1.1.1.1.2.cmml">𝖲𝗎𝗆</mi><mo id="S2.F1.22.m2.10.10.1.1.1.1.1" xref="S2.F1.22.m2.10.10.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.F1.22.m2.10.10.1.1.1.1.3.2" xref="S2.F1.22.m2.10.10.1.1.1.1.cmml"><mo id="S2.F1.22.m2.10.10.1.1.1.1.3.2.1" stretchy="false" xref="S2.F1.22.m2.10.10.1.1.1.1.cmml">(</mo><mi id="S2.F1.22.m2.1.1" xref="S2.F1.22.m2.1.1.cmml">t</mi><mo id="S2.F1.22.m2.10.10.1.1.1.1.3.2.2" stretchy="false" xref="S2.F1.22.m2.10.10.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.F1.22.m2.10.10.1.1.1.4" rspace="0.448em" stretchy="false" xref="S2.F1.22.m2.10.10.1.1.2.cmml">)</mo></mrow></mrow><mpadded width="0.226em"><mo id="S2.F1.22.m2.13.13.5" xref="S2.F1.22.m2.13.13.5.cmml">:</mo></mpadded><mrow id="S2.F1.22.m2.13.13.4.3" xref="S2.F1.22.m2.13.13.4.4.cmml"><mrow id="S2.F1.22.m2.11.11.2.1.1" xref="S2.F1.22.m2.11.11.2.1.1.cmml"><mo id="S2.F1.22.m2.11.11.2.1.1b" xref="S2.F1.22.m2.11.11.2.1.1.cmml">−</mo><mrow id="S2.F1.22.m2.11.11.2.1.1.2" xref="S2.F1.22.m2.11.11.2.1.1.2.cmml"><mtext class="ltx_font_smallcaps" id="S2.F1.22.m2.11.11.2.1.1.2.2" xref="S2.F1.22.m2.11.11.2.1.1.2.2a.cmml">Teams</mtext><mo id="S2.F1.22.m2.11.11.2.1.1.2.1" xref="S2.F1.22.m2.11.11.2.1.1.2.1.cmml">⁢</mo><mrow id="S2.F1.22.m2.11.11.2.1.1.2.3.2" xref="S2.F1.22.m2.11.11.2.1.1.2.3.1.cmml"><mo id="S2.F1.22.m2.11.11.2.1.1.2.3.2.1" stretchy="false" 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id="S2.F1.22.m2.13.13.4.3.3.3.1.cmml" xref="S2.F1.22.m2.13.13.4.3.3.3.2"><ci id="S2.F1.22.m2.8.8.cmml" xref="S2.F1.22.m2.8.8">𝑔</ci><ci id="S2.F1.22.m2.9.9.cmml" xref="S2.F1.22.m2.9.9">𝑡</ci></interval></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F1.22.m2.13d">Q(c,\mathsf{Sum}(t)){\,:\!\!-\,}\textsc{Teams}(p,c),\textsc{Goals}(g,p,t),% \textsc{Replays}(g,t)</annotation><annotation encoding="application/x-llamapun" id="S2.F1.22.m2.13e">italic_Q ( italic_c , sansserif_Sum ( italic_t ) ) : - Teams ( italic_p , italic_c ) , Goals ( italic_g , italic_p , italic_t ) , Replays ( italic_g , italic_t )</annotation></semantics></math>. </figcaption> </figure> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3. </span>The Direct-Access Problem</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.5">In this paper, we study CQs with lexicographic orders over the answers. As said earlier, the lexicographic order for the CQ <math alttext="Q(\vec{x})" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mrow id="S3.p1.1.m1.1.2" xref="S3.p1.1.m1.1.2.cmml"><mi id="S3.p1.1.m1.1.2.2" xref="S3.p1.1.m1.1.2.2.cmml">Q</mi><mo id="S3.p1.1.m1.1.2.1" xref="S3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.p1.1.m1.1.2.3.2" xref="S3.p1.1.m1.1.1.cmml"><mo id="S3.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S3.p1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mi id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml">x</mi><mo id="S3.p1.1.m1.1.1.1" stretchy="false" xref="S3.p1.1.m1.1.1.1.cmml">→</mo></mover><mo id="S3.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S3.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.2.cmml" xref="S3.p1.1.m1.1.2"><times id="S3.p1.1.m1.1.2.1.cmml" xref="S3.p1.1.m1.1.2.1"></times><ci id="S3.p1.1.m1.1.2.2.cmml" xref="S3.p1.1.m1.1.2.2">𝑄</ci><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.2.3.2"><ci id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1.1">→</ci><ci id="S3.p1.1.m1.1.1.2.cmml" xref="S3.p1.1.m1.1.1.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">Q(\vec{x})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">italic_Q ( over→ start_ARG italic_x end_ARG )</annotation></semantics></math> is left to right according to <math alttext="\vec{x}" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mover accent="true" id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">x</mi><mo id="S3.p1.2.m2.1.1.1" stretchy="false" xref="S3.p1.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><ci id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1.1">→</ci><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\vec{x}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">over→ start_ARG italic_x end_ARG</annotation></semantics></math>. We will also investigate lexicographic orders that involve the annotation or aggregation when the query is a CQ<sup class="ltx_sup" id="S3.p1.5.1">⋆</sup> <math alttext="Q(\vec{x},\star,\vec{z})" class="ltx_Math" display="inline" id="S3.p1.4.m4.3"><semantics id="S3.p1.4.m4.3a"><mrow id="S3.p1.4.m4.3.4" xref="S3.p1.4.m4.3.4.cmml"><mi id="S3.p1.4.m4.3.4.2" xref="S3.p1.4.m4.3.4.2.cmml">Q</mi><mo id="S3.p1.4.m4.3.4.1" xref="S3.p1.4.m4.3.4.1.cmml">⁢</mo><mrow id="S3.p1.4.m4.3.4.3.2" xref="S3.p1.4.m4.3.4.3.1.cmml"><mo id="S3.p1.4.m4.3.4.3.2.1" stretchy="false" xref="S3.p1.4.m4.3.4.3.1.cmml">(</mo><mover accent="true" id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><mi id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml">x</mi><mo id="S3.p1.4.m4.1.1.1" stretchy="false" xref="S3.p1.4.m4.1.1.1.cmml">→</mo></mover><mo id="S3.p1.4.m4.3.4.3.2.2" rspace="0em" xref="S3.p1.4.m4.3.4.3.1.cmml">,</mo><mo id="S3.p1.4.m4.2.2" lspace="0em" rspace="0em" xref="S3.p1.4.m4.2.2.cmml">⋆</mo><mo id="S3.p1.4.m4.3.4.3.2.3" xref="S3.p1.4.m4.3.4.3.1.cmml">,</mo><mover accent="true" id="S3.p1.4.m4.3.3" xref="S3.p1.4.m4.3.3.cmml"><mi id="S3.p1.4.m4.3.3.2" xref="S3.p1.4.m4.3.3.2.cmml">z</mi><mo id="S3.p1.4.m4.3.3.1" stretchy="false" xref="S3.p1.4.m4.3.3.1.cmml">→</mo></mover><mo id="S3.p1.4.m4.3.4.3.2.4" stretchy="false" xref="S3.p1.4.m4.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.3b"><apply id="S3.p1.4.m4.3.4.cmml" xref="S3.p1.4.m4.3.4"><times id="S3.p1.4.m4.3.4.1.cmml" xref="S3.p1.4.m4.3.4.1"></times><ci id="S3.p1.4.m4.3.4.2.cmml" xref="S3.p1.4.m4.3.4.2">𝑄</ci><vector id="S3.p1.4.m4.3.4.3.1.cmml" xref="S3.p1.4.m4.3.4.3.2"><apply id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1"><ci id="S3.p1.4.m4.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1">→</ci><ci id="S3.p1.4.m4.1.1.2.cmml" xref="S3.p1.4.m4.1.1.2">𝑥</ci></apply><ci id="S3.p1.4.m4.2.2.cmml" xref="S3.p1.4.m4.2.2">⋆</ci><apply id="S3.p1.4.m4.3.3.cmml" xref="S3.p1.4.m4.3.3"><ci id="S3.p1.4.m4.3.3.1.cmml" xref="S3.p1.4.m4.3.3.1">→</ci><ci id="S3.p1.4.m4.3.3.2.cmml" xref="S3.p1.4.m4.3.3.2">𝑧</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.3c">Q(\vec{x},\star,\vec{z})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.3d">italic_Q ( over→ start_ARG italic_x end_ARG , ⋆ , over→ start_ARG italic_z end_ARG )</annotation></semantics></math> or an AggCQ <math alttext="Q(\vec{x},\alpha(\vec{w}),\vec{z})" class="ltx_Math" display="inline" id="S3.p1.5.m5.4"><semantics id="S3.p1.5.m5.4a"><mrow id="S3.p1.5.m5.4.4" xref="S3.p1.5.m5.4.4.cmml"><mi id="S3.p1.5.m5.4.4.3" xref="S3.p1.5.m5.4.4.3.cmml">Q</mi><mo id="S3.p1.5.m5.4.4.2" xref="S3.p1.5.m5.4.4.2.cmml">⁢</mo><mrow id="S3.p1.5.m5.4.4.1.1" xref="S3.p1.5.m5.4.4.1.2.cmml"><mo id="S3.p1.5.m5.4.4.1.1.2" stretchy="false" xref="S3.p1.5.m5.4.4.1.2.cmml">(</mo><mover accent="true" id="S3.p1.5.m5.2.2" xref="S3.p1.5.m5.2.2.cmml"><mi id="S3.p1.5.m5.2.2.2" xref="S3.p1.5.m5.2.2.2.cmml">x</mi><mo id="S3.p1.5.m5.2.2.1" stretchy="false" xref="S3.p1.5.m5.2.2.1.cmml">→</mo></mover><mo id="S3.p1.5.m5.4.4.1.1.3" xref="S3.p1.5.m5.4.4.1.2.cmml">,</mo><mrow id="S3.p1.5.m5.4.4.1.1.1" xref="S3.p1.5.m5.4.4.1.1.1.cmml"><mi id="S3.p1.5.m5.4.4.1.1.1.2" xref="S3.p1.5.m5.4.4.1.1.1.2.cmml">α</mi><mo id="S3.p1.5.m5.4.4.1.1.1.1" xref="S3.p1.5.m5.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S3.p1.5.m5.4.4.1.1.1.3.2" xref="S3.p1.5.m5.1.1.cmml"><mo id="S3.p1.5.m5.4.4.1.1.1.3.2.1" stretchy="false" xref="S3.p1.5.m5.1.1.cmml">(</mo><mover accent="true" id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml"><mi id="S3.p1.5.m5.1.1.2" xref="S3.p1.5.m5.1.1.2.cmml">w</mi><mo id="S3.p1.5.m5.1.1.1" stretchy="false" xref="S3.p1.5.m5.1.1.1.cmml">→</mo></mover><mo id="S3.p1.5.m5.4.4.1.1.1.3.2.2" stretchy="false" xref="S3.p1.5.m5.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p1.5.m5.4.4.1.1.4" xref="S3.p1.5.m5.4.4.1.2.cmml">,</mo><mover accent="true" id="S3.p1.5.m5.3.3" xref="S3.p1.5.m5.3.3.cmml"><mi id="S3.p1.5.m5.3.3.2" xref="S3.p1.5.m5.3.3.2.cmml">z</mi><mo id="S3.p1.5.m5.3.3.1" stretchy="false" xref="S3.p1.5.m5.3.3.1.cmml">→</mo></mover><mo id="S3.p1.5.m5.4.4.1.1.5" stretchy="false" xref="S3.p1.5.m5.4.4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.4b"><apply id="S3.p1.5.m5.4.4.cmml" xref="S3.p1.5.m5.4.4"><times id="S3.p1.5.m5.4.4.2.cmml" xref="S3.p1.5.m5.4.4.2"></times><ci id="S3.p1.5.m5.4.4.3.cmml" xref="S3.p1.5.m5.4.4.3">𝑄</ci><vector id="S3.p1.5.m5.4.4.1.2.cmml" xref="S3.p1.5.m5.4.4.1.1"><apply id="S3.p1.5.m5.2.2.cmml" xref="S3.p1.5.m5.2.2"><ci id="S3.p1.5.m5.2.2.1.cmml" xref="S3.p1.5.m5.2.2.1">→</ci><ci id="S3.p1.5.m5.2.2.2.cmml" xref="S3.p1.5.m5.2.2.2">𝑥</ci></apply><apply id="S3.p1.5.m5.4.4.1.1.1.cmml" xref="S3.p1.5.m5.4.4.1.1.1"><times id="S3.p1.5.m5.4.4.1.1.1.1.cmml" xref="S3.p1.5.m5.4.4.1.1.1.1"></times><ci id="S3.p1.5.m5.4.4.1.1.1.2.cmml" xref="S3.p1.5.m5.4.4.1.1.1.2">𝛼</ci><apply id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.4.4.1.1.1.3.2"><ci id="S3.p1.5.m5.1.1.1.cmml" xref="S3.p1.5.m5.1.1.1">→</ci><ci id="S3.p1.5.m5.1.1.2.cmml" xref="S3.p1.5.m5.1.1.2">𝑤</ci></apply></apply><apply id="S3.p1.5.m5.3.3.cmml" xref="S3.p1.5.m5.3.3"><ci id="S3.p1.5.m5.3.3.1.cmml" xref="S3.p1.5.m5.3.3.1">→</ci><ci id="S3.p1.5.m5.3.3.2.cmml" xref="S3.p1.5.m5.3.3.2">𝑧</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.4c">Q(\vec{x},\alpha(\vec{w}),\vec{z})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.4d">italic_Q ( over→ start_ARG italic_x end_ARG , italic_α ( over→ start_ARG italic_w end_ARG ) , over→ start_ARG italic_z end_ARG )</annotation></semantics></math>, respectively. We refer uniformly to the annotation of an answer (over an annotated database) and to the aggregate value of the answer’s group (over an ordinary database) as the <em class="ltx_emph ltx_font_italic" id="S3.p1.5.2">computed value</em>.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.3">Let <math alttext="Q" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mi id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><ci id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">italic_Q</annotation></semantics></math> be a CQ, CQ<sup class="ltx_sup" id="S3.p2.3.1">⋆</sup>, or an AggCQ. A <em class="ltx_emph ltx_font_italic" id="S3.p2.3.2">direct access</em> solution for <math alttext="Q" class="ltx_Math" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><mi id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.1b"><ci id="S3.p2.3.m3.1.1.cmml" xref="S3.p2.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">italic_Q</annotation></semantics></math> consists of two algorithms: one for <em class="ltx_emph ltx_font_italic" id="S3.p2.3.3">preprocessing</em> and one for <em class="ltx_emph ltx_font_italic" id="S3.p2.3.4">access</em>.</p> <ul class="ltx_itemize" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.3">The preprocessing algorithm takes as input a database <math alttext="D" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><mi id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><ci id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_D</annotation></semantics></math> over <math alttext="Q" class="ltx_Math" display="inline" id="S3.I1.i1.p1.2.m2.1"><semantics id="S3.I1.i1.p1.2.m2.1a"><mi id="S3.I1.i1.p1.2.m2.1.1" xref="S3.I1.i1.p1.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.2.m2.1b"><ci id="S3.I1.i1.p1.2.m2.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">italic_Q</annotation></semantics></math> and constructs a data structure <math alttext="S_{D}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.3.m3.1"><semantics id="S3.I1.i1.p1.3.m3.1a"><msub id="S3.I1.i1.p1.3.m3.1.1" xref="S3.I1.i1.p1.3.m3.1.1.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.2" xref="S3.I1.i1.p1.3.m3.1.1.2.cmml">S</mi><mi id="S3.I1.i1.p1.3.m3.1.1.3" xref="S3.I1.i1.p1.3.m3.1.1.3.cmml">D</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.3.m3.1b"><apply id="S3.I1.i1.p1.3.m3.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.3.m3.1.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.2">𝑆</ci><ci id="S3.I1.i1.p1.3.m3.1.1.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.3.m3.1c">S_{D}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.3.m3.1d">italic_S start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.5">The access algorithm takes as input <math alttext="S_{D}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><msub id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml">S</mi><mi id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml">D</mi></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2">𝑆</ci><ci id="S3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">S_{D}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_S start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT</annotation></semantics></math> and an index <math alttext="i" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><mi id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><ci id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">italic_i</annotation></semantics></math>, and returns the <math alttext="i" class="ltx_Math" display="inline" id="S3.I1.i2.p1.3.m3.1"><semantics id="S3.I1.i2.p1.3.m3.1a"><mi id="S3.I1.i2.p1.3.m3.1.1" xref="S3.I1.i2.p1.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.3.m3.1b"><ci id="S3.I1.i2.p1.3.m3.1.1.cmml" xref="S3.I1.i2.p1.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.3.m3.1d">italic_i</annotation></semantics></math>th answer of <math alttext="Q(D)" class="ltx_Math" display="inline" id="S3.I1.i2.p1.4.m4.1"><semantics id="S3.I1.i2.p1.4.m4.1a"><mrow id="S3.I1.i2.p1.4.m4.1.2" xref="S3.I1.i2.p1.4.m4.1.2.cmml"><mi id="S3.I1.i2.p1.4.m4.1.2.2" xref="S3.I1.i2.p1.4.m4.1.2.2.cmml">Q</mi><mo id="S3.I1.i2.p1.4.m4.1.2.1" xref="S3.I1.i2.p1.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.I1.i2.p1.4.m4.1.2.3.2" xref="S3.I1.i2.p1.4.m4.1.2.cmml"><mo id="S3.I1.i2.p1.4.m4.1.2.3.2.1" stretchy="false" xref="S3.I1.i2.p1.4.m4.1.2.cmml">(</mo><mi id="S3.I1.i2.p1.4.m4.1.1" xref="S3.I1.i2.p1.4.m4.1.1.cmml">D</mi><mo id="S3.I1.i2.p1.4.m4.1.2.3.2.2" stretchy="false" xref="S3.I1.i2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.4.m4.1b"><apply id="S3.I1.i2.p1.4.m4.1.2.cmml" xref="S3.I1.i2.p1.4.m4.1.2"><times id="S3.I1.i2.p1.4.m4.1.2.1.cmml" xref="S3.I1.i2.p1.4.m4.1.2.1"></times><ci id="S3.I1.i2.p1.4.m4.1.2.2.cmml" xref="S3.I1.i2.p1.4.m4.1.2.2">𝑄</ci><ci id="S3.I1.i2.p1.4.m4.1.1.cmml" xref="S3.I1.i2.p1.4.m4.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.4.m4.1c">Q(D)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.4.m4.1d">italic_Q ( italic_D )</annotation></semantics></math> in the lexicographic order. Note that this answer includes the computed value, when it exists. If <math alttext="i>|Q(D)|" class="ltx_Math" display="inline" id="S3.I1.i2.p1.5.m5.2"><semantics id="S3.I1.i2.p1.5.m5.2a"><mrow id="S3.I1.i2.p1.5.m5.2.2" xref="S3.I1.i2.p1.5.m5.2.2.cmml"><mi id="S3.I1.i2.p1.5.m5.2.2.3" xref="S3.I1.i2.p1.5.m5.2.2.3.cmml">i</mi><mo id="S3.I1.i2.p1.5.m5.2.2.2" xref="S3.I1.i2.p1.5.m5.2.2.2.cmml">></mo><mrow id="S3.I1.i2.p1.5.m5.2.2.1.1" xref="S3.I1.i2.p1.5.m5.2.2.1.2.cmml"><mo id="S3.I1.i2.p1.5.m5.2.2.1.1.2" stretchy="false" xref="S3.I1.i2.p1.5.m5.2.2.1.2.1.cmml">|</mo><mrow id="S3.I1.i2.p1.5.m5.2.2.1.1.1" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1.cmml"><mi id="S3.I1.i2.p1.5.m5.2.2.1.1.1.2" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1.2.cmml">Q</mi><mo id="S3.I1.i2.p1.5.m5.2.2.1.1.1.1" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.I1.i2.p1.5.m5.2.2.1.1.1.3.2" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1.cmml"><mo id="S3.I1.i2.p1.5.m5.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1.cmml">(</mo><mi id="S3.I1.i2.p1.5.m5.1.1" xref="S3.I1.i2.p1.5.m5.1.1.cmml">D</mi><mo id="S3.I1.i2.p1.5.m5.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.I1.i2.p1.5.m5.2.2.1.1.3" stretchy="false" xref="S3.I1.i2.p1.5.m5.2.2.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.5.m5.2b"><apply id="S3.I1.i2.p1.5.m5.2.2.cmml" xref="S3.I1.i2.p1.5.m5.2.2"><gt id="S3.I1.i2.p1.5.m5.2.2.2.cmml" xref="S3.I1.i2.p1.5.m5.2.2.2"></gt><ci id="S3.I1.i2.p1.5.m5.2.2.3.cmml" xref="S3.I1.i2.p1.5.m5.2.2.3">𝑖</ci><apply id="S3.I1.i2.p1.5.m5.2.2.1.2.cmml" xref="S3.I1.i2.p1.5.m5.2.2.1.1"><abs id="S3.I1.i2.p1.5.m5.2.2.1.2.1.cmml" xref="S3.I1.i2.p1.5.m5.2.2.1.1.2"></abs><apply id="S3.I1.i2.p1.5.m5.2.2.1.1.1.cmml" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1"><times id="S3.I1.i2.p1.5.m5.2.2.1.1.1.1.cmml" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1.1"></times><ci id="S3.I1.i2.p1.5.m5.2.2.1.1.1.2.cmml" xref="S3.I1.i2.p1.5.m5.2.2.1.1.1.2">𝑄</ci><ci id="S3.I1.i2.p1.5.m5.1.1.cmml" xref="S3.I1.i2.p1.5.m5.1.1">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.5.m5.2c">i>|Q(D)|</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.5.m5.2d">italic_i > | italic_Q ( italic_D ) |</annotation></semantics></math> then the algorithm should return <em class="ltx_emph ltx_font_italic" id="S3.I1.i2.p1.5.1">null</em>.</p> </div> </li> </ul> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.2">To define the complexity requirements of <em class="ltx_emph ltx_font_italic" id="S3.p3.2.1">efficient</em> direct access, we first describe the complexity model that we adopt. We use <em class="ltx_emph ltx_font_italic" id="S3.p3.2.2">data complexity</em> as a yardstick of tractability. Hence, complexity is measured in terms of the size of the database, while the size of the query is fixed (and every query is a separate computational problem). Assuming the input is of size <math alttext="n" class="ltx_Math" display="inline" id="S3.p3.1.m1.1"><semantics id="S3.p3.1.m1.1a"><mi id="S3.p3.1.m1.1.1" xref="S3.p3.1.m1.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.1b"><ci id="S3.p3.1.m1.1.1.cmml" xref="S3.p3.1.m1.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.1d">italic_n</annotation></semantics></math>, we use the RAM model of computation with <math alttext="O(\log n)" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><mrow id="S3.p3.2.m2.1.1" xref="S3.p3.2.m2.1.1.cmml"><mi id="S3.p3.2.m2.1.1.3" xref="S3.p3.2.m2.1.1.3.cmml">O</mi><mo id="S3.p3.2.m2.1.1.2" xref="S3.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="S3.p3.2.m2.1.1.1.1" xref="S3.p3.2.m2.1.1.1.1.1.cmml"><mo id="S3.p3.2.m2.1.1.1.1.2" stretchy="false" xref="S3.p3.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.p3.2.m2.1.1.1.1.1" xref="S3.p3.2.m2.1.1.1.1.1.cmml"><mi id="S3.p3.2.m2.1.1.1.1.1.1" xref="S3.p3.2.m2.1.1.1.1.1.1.cmml">log</mi><mo id="S3.p3.2.m2.1.1.1.1.1a" lspace="0.167em" xref="S3.p3.2.m2.1.1.1.1.1.cmml">⁡</mo><mi id="S3.p3.2.m2.1.1.1.1.1.2" xref="S3.p3.2.m2.1.1.1.1.1.2.cmml">n</mi></mrow><mo id="S3.p3.2.m2.1.1.1.1.3" stretchy="false" xref="S3.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><apply id="S3.p3.2.m2.1.1.cmml" xref="S3.p3.2.m2.1.1"><times id="S3.p3.2.m2.1.1.2.cmml" xref="S3.p3.2.m2.1.1.2"></times><ci id="S3.p3.2.m2.1.1.3.cmml" xref="S3.p3.2.m2.1.1.3">𝑂</ci><apply id="S3.p3.2.m2.1.1.1.1.1.cmml" xref="S3.p3.2.m2.1.1.1.1"><log id="S3.p3.2.m2.1.1.1.1.1.1.cmml" xref="S3.p3.2.m2.1.1.1.1.1.1"></log><ci id="S3.p3.2.m2.1.1.1.1.1.2.cmml" xref="S3.p3.2.m2.1.1.1.1.1.2">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">O(\log n)</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">italic_O ( roman_log italic_n )</annotation></semantics></math>-bit words and uniform-cost operations. Notably, this model allows us to assume perfect hash tables can be constructed in linear time, and they provide access in constant time <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">louis-model</span>]</cite>.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.3">Regarding the complexity of the semiring operations, the RAM model allows us to assume that the numeric, counting, min tropical, and max tropical semirings use constant space for representing values and constant time for the operations <math alttext="\oplus" class="ltx_Math" display="inline" id="S3.p4.1.m1.1"><semantics id="S3.p4.1.m1.1a"><mo id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><csymbol cd="latexml" id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.1d">⊕</annotation></semantics></math> and <math alttext="\otimes" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><mo id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><csymbol cd="latexml" id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">⊗</annotation></semantics></math>. In fact, it suffices for our results to assume that the operations take logarithmic time, and later we will make use of this relaxed assumption (within a special case of <math alttext="\mathsf{CountD}" class="ltx_Math" display="inline" id="S3.p4.3.m3.1"><semantics id="S3.p4.3.m3.1a"><mi id="S3.p4.3.m3.1.1" xref="S3.p4.3.m3.1.1.cmml">𝖢𝗈𝗎𝗇𝗍𝖣</mi><annotation-xml encoding="MathML-Content" id="S3.p4.3.m3.1b"><ci id="S3.p4.3.m3.1.1.cmml" xref="S3.p4.3.m3.1.1">𝖢𝗈𝗎𝗇𝗍𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m3.1c">\mathsf{CountD}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m3.1d">sansserif_CountD</annotation></semantics></math>). We refer to a semiring with this property as a <em class="ltx_emph ltx_font_italic" id="S3.p4.3.1">logarithmic-time (commutative) semiring</em>.</p> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.11">Let <math alttext="T_{p}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><msub id="S3.p5.1.m1.1.1" xref="S3.p5.1.m1.1.1.cmml"><mi id="S3.p5.1.m1.1.1.2" xref="S3.p5.1.m1.1.1.2.cmml">T</mi><mi id="S3.p5.1.m1.1.1.3" xref="S3.p5.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><apply id="S3.p5.1.m1.1.1.cmml" xref="S3.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p5.1.m1.1.1.1.cmml" xref="S3.p5.1.m1.1.1">subscript</csymbol><ci id="S3.p5.1.m1.1.1.2.cmml" xref="S3.p5.1.m1.1.1.2">𝑇</ci><ci id="S3.p5.1.m1.1.1.3.cmml" xref="S3.p5.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.1.m1.1c">T_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.1.m1.1d">italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="T_{a}" class="ltx_Math" display="inline" id="S3.p5.2.m2.1"><semantics id="S3.p5.2.m2.1a"><msub id="S3.p5.2.m2.1.1" xref="S3.p5.2.m2.1.1.cmml"><mi id="S3.p5.2.m2.1.1.2" xref="S3.p5.2.m2.1.1.2.cmml">T</mi><mi id="S3.p5.2.m2.1.1.3" xref="S3.p5.2.m2.1.1.3.cmml">a</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p5.2.m2.1b"><apply id="S3.p5.2.m2.1.1.cmml" xref="S3.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p5.2.m2.1.1.1.cmml" xref="S3.p5.2.m2.1.1">subscript</csymbol><ci id="S3.p5.2.m2.1.1.2.cmml" xref="S3.p5.2.m2.1.1.2">𝑇</ci><ci id="S3.p5.2.m2.1.1.3.cmml" xref="S3.p5.2.m2.1.1.3">𝑎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.2.m2.1c">T_{a}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.2.m2.1d">italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT</annotation></semantics></math> be numeric functions. A direct-access algorithm is said to be <em class="ltx_emph ltx_font_italic" id="S3.p5.3.1">in <math alttext="\mathord{\langle T_{p},T_{a}\rangle}" class="ltx_Math" display="inline" id="S3.p5.3.1.m1.2"><semantics id="S3.p5.3.1.m1.2a"><mrow id="S3.p5.3.1.m1.2.2.2" xref="S3.p5.3.1.m1.2.2.3.cmml"><mo id="S3.p5.3.1.m1.2.2.2.3" stretchy="false" xref="S3.p5.3.1.m1.2.2.3.cmml">⟨</mo><msub id="S3.p5.3.1.m1.1.1.1.1" xref="S3.p5.3.1.m1.1.1.1.1.cmml"><mi id="S3.p5.3.1.m1.1.1.1.1.2" xref="S3.p5.3.1.m1.1.1.1.1.2.cmml">T</mi><mi id="S3.p5.3.1.m1.1.1.1.1.3" xref="S3.p5.3.1.m1.1.1.1.1.3.cmml">p</mi></msub><mo id="S3.p5.3.1.m1.2.2.2.4" xref="S3.p5.3.1.m1.2.2.3.cmml">,</mo><msub id="S3.p5.3.1.m1.2.2.2.2" xref="S3.p5.3.1.m1.2.2.2.2.cmml"><mi id="S3.p5.3.1.m1.2.2.2.2.2" xref="S3.p5.3.1.m1.2.2.2.2.2.cmml">T</mi><mi id="S3.p5.3.1.m1.2.2.2.2.3" xref="S3.p5.3.1.m1.2.2.2.2.3.cmml">a</mi></msub><mo id="S3.p5.3.1.m1.2.2.2.5" stretchy="false" xref="S3.p5.3.1.m1.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.3.1.m1.2b"><list id="S3.p5.3.1.m1.2.2.3.cmml" xref="S3.p5.3.1.m1.2.2.2"><apply id="S3.p5.3.1.m1.1.1.1.1.cmml" xref="S3.p5.3.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p5.3.1.m1.1.1.1.1.1.cmml" xref="S3.p5.3.1.m1.1.1.1.1">subscript</csymbol><ci id="S3.p5.3.1.m1.1.1.1.1.2.cmml" xref="S3.p5.3.1.m1.1.1.1.1.2">𝑇</ci><ci id="S3.p5.3.1.m1.1.1.1.1.3.cmml" xref="S3.p5.3.1.m1.1.1.1.1.3">𝑝</ci></apply><apply id="S3.p5.3.1.m1.2.2.2.2.cmml" xref="S3.p5.3.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.p5.3.1.m1.2.2.2.2.1.cmml" xref="S3.p5.3.1.m1.2.2.2.2">subscript</csymbol><ci id="S3.p5.3.1.m1.2.2.2.2.2.cmml" xref="S3.p5.3.1.m1.2.2.2.2.2">𝑇</ci><ci id="S3.p5.3.1.m1.2.2.2.2.3.cmml" xref="S3.p5.3.1.m1.2.2.2.2.3">𝑎</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.3.1.m1.2c">\mathord{\langle T_{p},T_{a}\rangle}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.3.1.m1.2d">⟨ italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟩</annotation></semantics></math></em> if the preprocessing phase takes <math alttext="O(T_{p}(|D|))" class="ltx_Math" display="inline" id="S3.p5.4.m3.2"><semantics id="S3.p5.4.m3.2a"><mrow id="S3.p5.4.m3.2.2" xref="S3.p5.4.m3.2.2.cmml"><mi id="S3.p5.4.m3.2.2.3" xref="S3.p5.4.m3.2.2.3.cmml">O</mi><mo id="S3.p5.4.m3.2.2.2" xref="S3.p5.4.m3.2.2.2.cmml">⁢</mo><mrow id="S3.p5.4.m3.2.2.1.1" xref="S3.p5.4.m3.2.2.1.1.1.cmml"><mo id="S3.p5.4.m3.2.2.1.1.2" stretchy="false" xref="S3.p5.4.m3.2.2.1.1.1.cmml">(</mo><mrow id="S3.p5.4.m3.2.2.1.1.1" xref="S3.p5.4.m3.2.2.1.1.1.cmml"><msub id="S3.p5.4.m3.2.2.1.1.1.3" xref="S3.p5.4.m3.2.2.1.1.1.3.cmml"><mi id="S3.p5.4.m3.2.2.1.1.1.3.2" xref="S3.p5.4.m3.2.2.1.1.1.3.2.cmml">T</mi><mi id="S3.p5.4.m3.2.2.1.1.1.3.3" xref="S3.p5.4.m3.2.2.1.1.1.3.3.cmml">p</mi></msub><mo id="S3.p5.4.m3.2.2.1.1.1.2" xref="S3.p5.4.m3.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.p5.4.m3.2.2.1.1.1.1.1" xref="S3.p5.4.m3.2.2.1.1.1.cmml"><mo id="S3.p5.4.m3.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.p5.4.m3.2.2.1.1.1.cmml">(</mo><mrow id="S3.p5.4.m3.2.2.1.1.1.1.1.1.2" xref="S3.p5.4.m3.2.2.1.1.1.1.1.1.1.cmml"><mo id="S3.p5.4.m3.2.2.1.1.1.1.1.1.2.1" stretchy="false" xref="S3.p5.4.m3.2.2.1.1.1.1.1.1.1.1.cmml">|</mo><mi id="S3.p5.4.m3.1.1" xref="S3.p5.4.m3.1.1.cmml">D</mi><mo id="S3.p5.4.m3.2.2.1.1.1.1.1.1.2.2" stretchy="false" xref="S3.p5.4.m3.2.2.1.1.1.1.1.1.1.1.cmml">|</mo></mrow><mo id="S3.p5.4.m3.2.2.1.1.1.1.1.3" stretchy="false" xref="S3.p5.4.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p5.4.m3.2.2.1.1.3" stretchy="false" xref="S3.p5.4.m3.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m3.2b"><apply id="S3.p5.4.m3.2.2.cmml" xref="S3.p5.4.m3.2.2"><times id="S3.p5.4.m3.2.2.2.cmml" xref="S3.p5.4.m3.2.2.2"></times><ci id="S3.p5.4.m3.2.2.3.cmml" xref="S3.p5.4.m3.2.2.3">𝑂</ci><apply id="S3.p5.4.m3.2.2.1.1.1.cmml" xref="S3.p5.4.m3.2.2.1.1"><times id="S3.p5.4.m3.2.2.1.1.1.2.cmml" xref="S3.p5.4.m3.2.2.1.1.1.2"></times><apply id="S3.p5.4.m3.2.2.1.1.1.3.cmml" xref="S3.p5.4.m3.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.p5.4.m3.2.2.1.1.1.3.1.cmml" xref="S3.p5.4.m3.2.2.1.1.1.3">subscript</csymbol><ci id="S3.p5.4.m3.2.2.1.1.1.3.2.cmml" xref="S3.p5.4.m3.2.2.1.1.1.3.2">𝑇</ci><ci id="S3.p5.4.m3.2.2.1.1.1.3.3.cmml" xref="S3.p5.4.m3.2.2.1.1.1.3.3">𝑝</ci></apply><apply id="S3.p5.4.m3.2.2.1.1.1.1.1.1.1.cmml" xref="S3.p5.4.m3.2.2.1.1.1.1.1.1.2"><abs id="S3.p5.4.m3.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.p5.4.m3.2.2.1.1.1.1.1.1.2.1"></abs><ci id="S3.p5.4.m3.1.1.cmml" xref="S3.p5.4.m3.1.1">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m3.2c">O(T_{p}(|D|))</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m3.2d">italic_O ( italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( | italic_D | ) )</annotation></semantics></math> time and each access takes <math alttext="O(T_{a}(|D|))" class="ltx_Math" display="inline" id="S3.p5.5.m4.2"><semantics id="S3.p5.5.m4.2a"><mrow id="S3.p5.5.m4.2.2" xref="S3.p5.5.m4.2.2.cmml"><mi id="S3.p5.5.m4.2.2.3" xref="S3.p5.5.m4.2.2.3.cmml">O</mi><mo id="S3.p5.5.m4.2.2.2" xref="S3.p5.5.m4.2.2.2.cmml">⁢</mo><mrow id="S3.p5.5.m4.2.2.1.1" xref="S3.p5.5.m4.2.2.1.1.1.cmml"><mo id="S3.p5.5.m4.2.2.1.1.2" stretchy="false" xref="S3.p5.5.m4.2.2.1.1.1.cmml">(</mo><mrow id="S3.p5.5.m4.2.2.1.1.1" xref="S3.p5.5.m4.2.2.1.1.1.cmml"><msub id="S3.p5.5.m4.2.2.1.1.1.3" xref="S3.p5.5.m4.2.2.1.1.1.3.cmml"><mi id="S3.p5.5.m4.2.2.1.1.1.3.2" xref="S3.p5.5.m4.2.2.1.1.1.3.2.cmml">T</mi><mi id="S3.p5.5.m4.2.2.1.1.1.3.3" xref="S3.p5.5.m4.2.2.1.1.1.3.3.cmml">a</mi></msub><mo id="S3.p5.5.m4.2.2.1.1.1.2" xref="S3.p5.5.m4.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.p5.5.m4.2.2.1.1.1.1.1" xref="S3.p5.5.m4.2.2.1.1.1.cmml"><mo id="S3.p5.5.m4.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.p5.5.m4.2.2.1.1.1.cmml">(</mo><mrow id="S3.p5.5.m4.2.2.1.1.1.1.1.1.2" xref="S3.p5.5.m4.2.2.1.1.1.1.1.1.1.cmml"><mo id="S3.p5.5.m4.2.2.1.1.1.1.1.1.2.1" stretchy="false" xref="S3.p5.5.m4.2.2.1.1.1.1.1.1.1.1.cmml">|</mo><mi id="S3.p5.5.m4.1.1" xref="S3.p5.5.m4.1.1.cmml">D</mi><mo id="S3.p5.5.m4.2.2.1.1.1.1.1.1.2.2" stretchy="false" xref="S3.p5.5.m4.2.2.1.1.1.1.1.1.1.1.cmml">|</mo></mrow><mo id="S3.p5.5.m4.2.2.1.1.1.1.1.3" stretchy="false" xref="S3.p5.5.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p5.5.m4.2.2.1.1.3" stretchy="false" xref="S3.p5.5.m4.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m4.2b"><apply id="S3.p5.5.m4.2.2.cmml" xref="S3.p5.5.m4.2.2"><times id="S3.p5.5.m4.2.2.2.cmml" xref="S3.p5.5.m4.2.2.2"></times><ci id="S3.p5.5.m4.2.2.3.cmml" xref="S3.p5.5.m4.2.2.3">𝑂</ci><apply id="S3.p5.5.m4.2.2.1.1.1.cmml" xref="S3.p5.5.m4.2.2.1.1"><times id="S3.p5.5.m4.2.2.1.1.1.2.cmml" xref="S3.p5.5.m4.2.2.1.1.1.2"></times><apply id="S3.p5.5.m4.2.2.1.1.1.3.cmml" xref="S3.p5.5.m4.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.p5.5.m4.2.2.1.1.1.3.1.cmml" xref="S3.p5.5.m4.2.2.1.1.1.3">subscript</csymbol><ci id="S3.p5.5.m4.2.2.1.1.1.3.2.cmml" xref="S3.p5.5.m4.2.2.1.1.1.3.2">𝑇</ci><ci id="S3.p5.5.m4.2.2.1.1.1.3.3.cmml" xref="S3.p5.5.m4.2.2.1.1.1.3.3">𝑎</ci></apply><apply id="S3.p5.5.m4.2.2.1.1.1.1.1.1.1.cmml" xref="S3.p5.5.m4.2.2.1.1.1.1.1.1.2"><abs id="S3.p5.5.m4.2.2.1.1.1.1.1.1.1.1.cmml" xref="S3.p5.5.m4.2.2.1.1.1.1.1.1.2.1"></abs><ci id="S3.p5.5.m4.1.1.cmml" xref="S3.p5.5.m4.1.1">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m4.2c">O(T_{a}(|D|))</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m4.2d">italic_O ( italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ( | italic_D | ) )</annotation></semantics></math> time. For example, <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S3.p5.6.m5.2"><semantics id="S3.p5.6.m5.2a"><mrow id="S3.p5.6.m5.2.2.4" xref="S3.p5.6.m5.2.2.3.cmml"><mo id="S3.p5.6.m5.2.2.4.1" stretchy="false" xref="S3.p5.6.m5.2.2.3.cmml">⟨</mo><mi id="S3.p5.6.m5.1.1.1" xref="S3.p5.6.m5.1.1.1.cmml">loglinear</mi><mo id="S3.p5.6.m5.2.2.4.2" xref="S3.p5.6.m5.2.2.3.cmml">,</mo><mi id="S3.p5.6.m5.2.2.2" xref="S3.p5.6.m5.2.2.2.cmml">log</mi><mo id="S3.p5.6.m5.2.2.4.3" stretchy="false" xref="S3.p5.6.m5.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.6.m5.2b"><list id="S3.p5.6.m5.2.2.3.cmml" xref="S3.p5.6.m5.2.2.4"><ci id="S3.p5.6.m5.1.1.1.cmml" xref="S3.p5.6.m5.1.1.1">loglinear</ci><log id="S3.p5.6.m5.2.2.2.cmml" xref="S3.p5.6.m5.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.6.m5.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.6.m5.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math> states that preprocessing constructs in <math alttext="O(|D|\log|D|)" class="ltx_Math" display="inline" id="S3.p5.7.m6.3"><semantics id="S3.p5.7.m6.3a"><mrow id="S3.p5.7.m6.3.3" xref="S3.p5.7.m6.3.3.cmml"><mi id="S3.p5.7.m6.3.3.3" xref="S3.p5.7.m6.3.3.3.cmml">O</mi><mo id="S3.p5.7.m6.3.3.2" xref="S3.p5.7.m6.3.3.2.cmml">⁢</mo><mrow id="S3.p5.7.m6.3.3.1.1" xref="S3.p5.7.m6.3.3.1.1.1.cmml"><mo id="S3.p5.7.m6.3.3.1.1.2" stretchy="false" xref="S3.p5.7.m6.3.3.1.1.1.cmml">(</mo><mrow id="S3.p5.7.m6.3.3.1.1.1" xref="S3.p5.7.m6.3.3.1.1.1.cmml"><mrow id="S3.p5.7.m6.3.3.1.1.1.2.2" xref="S3.p5.7.m6.3.3.1.1.1.2.1.cmml"><mo id="S3.p5.7.m6.3.3.1.1.1.2.2.1" stretchy="false" xref="S3.p5.7.m6.3.3.1.1.1.2.1.1.cmml">|</mo><mi id="S3.p5.7.m6.1.1" xref="S3.p5.7.m6.1.1.cmml">D</mi><mo id="S3.p5.7.m6.3.3.1.1.1.2.2.2" stretchy="false" xref="S3.p5.7.m6.3.3.1.1.1.2.1.1.cmml">|</mo></mrow><mo id="S3.p5.7.m6.3.3.1.1.1.1" lspace="0.167em" xref="S3.p5.7.m6.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S3.p5.7.m6.3.3.1.1.1.3" xref="S3.p5.7.m6.3.3.1.1.1.3.cmml"><mi id="S3.p5.7.m6.3.3.1.1.1.3.1" xref="S3.p5.7.m6.3.3.1.1.1.3.1.cmml">log</mi><mo id="S3.p5.7.m6.3.3.1.1.1.3a" xref="S3.p5.7.m6.3.3.1.1.1.3.cmml">⁡</mo><mrow id="S3.p5.7.m6.3.3.1.1.1.3.2.2" xref="S3.p5.7.m6.3.3.1.1.1.3.2.1.cmml"><mo id="S3.p5.7.m6.3.3.1.1.1.3.2.2.1" stretchy="false" xref="S3.p5.7.m6.3.3.1.1.1.3.2.1.1.cmml">|</mo><mi id="S3.p5.7.m6.2.2" xref="S3.p5.7.m6.2.2.cmml">D</mi><mo id="S3.p5.7.m6.3.3.1.1.1.3.2.2.2" stretchy="false" xref="S3.p5.7.m6.3.3.1.1.1.3.2.1.1.cmml">|</mo></mrow></mrow></mrow><mo id="S3.p5.7.m6.3.3.1.1.3" stretchy="false" xref="S3.p5.7.m6.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.7.m6.3b"><apply id="S3.p5.7.m6.3.3.cmml" xref="S3.p5.7.m6.3.3"><times id="S3.p5.7.m6.3.3.2.cmml" xref="S3.p5.7.m6.3.3.2"></times><ci id="S3.p5.7.m6.3.3.3.cmml" xref="S3.p5.7.m6.3.3.3">𝑂</ci><apply id="S3.p5.7.m6.3.3.1.1.1.cmml" xref="S3.p5.7.m6.3.3.1.1"><times id="S3.p5.7.m6.3.3.1.1.1.1.cmml" xref="S3.p5.7.m6.3.3.1.1.1.1"></times><apply id="S3.p5.7.m6.3.3.1.1.1.2.1.cmml" xref="S3.p5.7.m6.3.3.1.1.1.2.2"><abs id="S3.p5.7.m6.3.3.1.1.1.2.1.1.cmml" xref="S3.p5.7.m6.3.3.1.1.1.2.2.1"></abs><ci id="S3.p5.7.m6.1.1.cmml" xref="S3.p5.7.m6.1.1">𝐷</ci></apply><apply id="S3.p5.7.m6.3.3.1.1.1.3.cmml" xref="S3.p5.7.m6.3.3.1.1.1.3"><log id="S3.p5.7.m6.3.3.1.1.1.3.1.cmml" xref="S3.p5.7.m6.3.3.1.1.1.3.1"></log><apply id="S3.p5.7.m6.3.3.1.1.1.3.2.1.cmml" xref="S3.p5.7.m6.3.3.1.1.1.3.2.2"><abs id="S3.p5.7.m6.3.3.1.1.1.3.2.1.1.cmml" xref="S3.p5.7.m6.3.3.1.1.1.3.2.2.1"></abs><ci id="S3.p5.7.m6.2.2.cmml" xref="S3.p5.7.m6.2.2">𝐷</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.7.m6.3c">O(|D|\log|D|)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.7.m6.3d">italic_O ( | italic_D | roman_log | italic_D | )</annotation></semantics></math> time a data structure that provides <math alttext="O(\log|D|)" class="ltx_Math" display="inline" id="S3.p5.8.m7.2"><semantics id="S3.p5.8.m7.2a"><mrow id="S3.p5.8.m7.2.2" xref="S3.p5.8.m7.2.2.cmml"><mi id="S3.p5.8.m7.2.2.3" xref="S3.p5.8.m7.2.2.3.cmml">O</mi><mo id="S3.p5.8.m7.2.2.2" xref="S3.p5.8.m7.2.2.2.cmml">⁢</mo><mrow id="S3.p5.8.m7.2.2.1.1" xref="S3.p5.8.m7.2.2.1.1.1.cmml"><mo id="S3.p5.8.m7.2.2.1.1.2" stretchy="false" xref="S3.p5.8.m7.2.2.1.1.1.cmml">(</mo><mrow id="S3.p5.8.m7.2.2.1.1.1" xref="S3.p5.8.m7.2.2.1.1.1.cmml"><mi id="S3.p5.8.m7.2.2.1.1.1.1" xref="S3.p5.8.m7.2.2.1.1.1.1.cmml">log</mi><mo id="S3.p5.8.m7.2.2.1.1.1a" xref="S3.p5.8.m7.2.2.1.1.1.cmml">⁡</mo><mrow id="S3.p5.8.m7.2.2.1.1.1.2.2" xref="S3.p5.8.m7.2.2.1.1.1.2.1.cmml"><mo id="S3.p5.8.m7.2.2.1.1.1.2.2.1" stretchy="false" xref="S3.p5.8.m7.2.2.1.1.1.2.1.1.cmml">|</mo><mi id="S3.p5.8.m7.1.1" xref="S3.p5.8.m7.1.1.cmml">D</mi><mo id="S3.p5.8.m7.2.2.1.1.1.2.2.2" stretchy="false" xref="S3.p5.8.m7.2.2.1.1.1.2.1.1.cmml">|</mo></mrow></mrow><mo id="S3.p5.8.m7.2.2.1.1.3" stretchy="false" xref="S3.p5.8.m7.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.8.m7.2b"><apply id="S3.p5.8.m7.2.2.cmml" xref="S3.p5.8.m7.2.2"><times id="S3.p5.8.m7.2.2.2.cmml" xref="S3.p5.8.m7.2.2.2"></times><ci id="S3.p5.8.m7.2.2.3.cmml" xref="S3.p5.8.m7.2.2.3">𝑂</ci><apply id="S3.p5.8.m7.2.2.1.1.1.cmml" xref="S3.p5.8.m7.2.2.1.1"><log id="S3.p5.8.m7.2.2.1.1.1.1.cmml" xref="S3.p5.8.m7.2.2.1.1.1.1"></log><apply id="S3.p5.8.m7.2.2.1.1.1.2.1.cmml" xref="S3.p5.8.m7.2.2.1.1.1.2.2"><abs id="S3.p5.8.m7.2.2.1.1.1.2.1.1.cmml" xref="S3.p5.8.m7.2.2.1.1.1.2.2.1"></abs><ci id="S3.p5.8.m7.1.1.cmml" xref="S3.p5.8.m7.1.1">𝐷</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.8.m7.2c">O(\log|D|)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.8.m7.2d">italic_O ( roman_log | italic_D | )</annotation></semantics></math>-time access. In this work, a query <math alttext="Q" class="ltx_Math" display="inline" id="S3.p5.9.m8.1"><semantics id="S3.p5.9.m8.1a"><mi id="S3.p5.9.m8.1.1" xref="S3.p5.9.m8.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.p5.9.m8.1b"><ci id="S3.p5.9.m8.1.1.cmml" xref="S3.p5.9.m8.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.9.m8.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.p5.9.m8.1d">italic_Q</annotation></semantics></math> has <em class="ltx_emph ltx_font_italic" id="S3.p5.11.2">efficient</em> direct access (and <math alttext="Q" class="ltx_Math" display="inline" id="S3.p5.10.m9.1"><semantics id="S3.p5.10.m9.1a"><mi id="S3.p5.10.m9.1.1" xref="S3.p5.10.m9.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.p5.10.m9.1b"><ci id="S3.p5.10.m9.1.1.cmml" xref="S3.p5.10.m9.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.10.m9.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.p5.10.m9.1d">italic_Q</annotation></semantics></math> is deemed tractable) if it has a direct access algorithm in <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S3.p5.11.m10.2"><semantics id="S3.p5.11.m10.2a"><mrow id="S3.p5.11.m10.2.2.4" xref="S3.p5.11.m10.2.2.3.cmml"><mo id="S3.p5.11.m10.2.2.4.1" stretchy="false" xref="S3.p5.11.m10.2.2.3.cmml">⟨</mo><mi id="S3.p5.11.m10.1.1.1" xref="S3.p5.11.m10.1.1.1.cmml">loglinear</mi><mo id="S3.p5.11.m10.2.2.4.2" xref="S3.p5.11.m10.2.2.3.cmml">,</mo><mi id="S3.p5.11.m10.2.2.2" xref="S3.p5.11.m10.2.2.2.cmml">log</mi><mo id="S3.p5.11.m10.2.2.4.3" stretchy="false" xref="S3.p5.11.m10.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.11.m10.2b"><list id="S3.p5.11.m10.2.2.3.cmml" xref="S3.p5.11.m10.2.2.4"><ci id="S3.p5.11.m10.1.1.1.cmml" xref="S3.p5.11.m10.1.1.1">loglinear</ci><log id="S3.p5.11.m10.2.2.2.cmml" xref="S3.p5.11.m10.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.11.m10.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.11.m10.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.1">Carmeli et al. <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/CarmeliTGKR21</span>]</cite> established a dichotomy in the tractability of the CQs and lexicographic orders. This dichotomy relies on the following hypotheses.</p> <ul class="ltx_itemize" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> </li> </ul> </div> <div class="ltx_para" id="S3.p7"> <span class="ltx_ERROR undefined" id="S3.p7.2">{thmC}</span> <p class="ltx_p" id="S3.p7.1">[<cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">DBLP:conf/pods/CarmeliTGKR21</span>]</cite>] Let <math alttext="Q" class="ltx_Math" display="inline" id="S3.p7.1.m1.1"><semantics id="S3.p7.1.m1.1a"><mi id="S3.p7.1.m1.1.1" xref="S3.p7.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.p7.1.m1.1b"><ci id="S3.p7.1.m1.1.1.cmml" xref="S3.p7.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.p7.1.m1.1d">italic_Q</annotation></semantics></math> be a CQ.</p> <ol class="ltx_enumerate" id="S3.I3"> <li class="ltx_item" id="S3.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S3.I3.i1.p1"> <p class="ltx_p" id="S3.I3.i1.p1.3">If <math alttext="Q" 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" xref="S3.I3.i1.p1.1.m1.1.1.cmml">Q</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">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.1.m1.1d">italic_Q</annotation></semantics></math> is free-connex with no disruptive trio, then direct access for <math alttext="Q" class="ltx_Math" display="inline" id="S3.I3.i1.p1.2.m2.1"><semantics id="S3.I3.i1.p1.2.m2.1a"><mi id="S3.I3.i1.p1.2.m2.1.1" xref="S3.I3.i1.p1.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.2.m2.1b"><ci id="S3.I3.i1.p1.2.m2.1.1.cmml" xref="S3.I3.i1.p1.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.2.m2.1d">italic_Q</annotation></semantics></math> is in <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S3.I3.i1.p1.3.m3.2"><semantics id="S3.I3.i1.p1.3.m3.2a"><mrow id="S3.I3.i1.p1.3.m3.2.2.4" xref="S3.I3.i1.p1.3.m3.2.2.3.cmml"><mo id="S3.I3.i1.p1.3.m3.2.2.4.1" stretchy="false" xref="S3.I3.i1.p1.3.m3.2.2.3.cmml">⟨</mo><mi id="S3.I3.i1.p1.3.m3.1.1.1" xref="S3.I3.i1.p1.3.m3.1.1.1.cmml">loglinear</mi><mo id="S3.I3.i1.p1.3.m3.2.2.4.2" xref="S3.I3.i1.p1.3.m3.2.2.3.cmml">,</mo><mi id="S3.I3.i1.p1.3.m3.2.2.2" xref="S3.I3.i1.p1.3.m3.2.2.2.cmml">log</mi><mo id="S3.I3.i1.p1.3.m3.2.2.4.3" stretchy="false" xref="S3.I3.i1.p1.3.m3.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.3.m3.2b"><list id="S3.I3.i1.p1.3.m3.2.2.3.cmml" xref="S3.I3.i1.p1.3.m3.2.2.4"><ci id="S3.I3.i1.p1.3.m3.1.1.1.cmml" xref="S3.I3.i1.p1.3.m3.1.1.1">loglinear</ci><log id="S3.I3.i1.p1.3.m3.2.2.2.cmml" xref="S3.I3.i1.p1.3.m3.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.3.m3.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.3.m3.2d">⟨ roman_loglinear , roman_log ⟩</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">(2)</span> <div class="ltx_para" id="S3.I3.i2.p1"> <p class="ltx_p" id="S3.I3.i2.p1.3">Otherwise, if <math alttext="Q" 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" xref="S3.I3.i2.p1.1.m1.1.1.cmml">Q</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">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.1.m1.1d">italic_Q</annotation></semantics></math> is also self-join-free, then direct access for <math alttext="Q" class="ltx_Math" display="inline" id="S3.I3.i2.p1.2.m2.1"><semantics id="S3.I3.i2.p1.2.m2.1a"><mi id="S3.I3.i2.p1.2.m2.1.1" xref="S3.I3.i2.p1.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.2.m2.1b"><ci id="S3.I3.i2.p1.2.m2.1.1.cmml" xref="S3.I3.i2.p1.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.2.m2.1d">italic_Q</annotation></semantics></math> is not in <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.3.m3.2"><semantics id="S3.I3.i2.p1.3.m3.2a"><mrow id="S3.I3.i2.p1.3.m3.2.2.4" xref="S3.I3.i2.p1.3.m3.2.2.3.cmml"><mo id="S3.I3.i2.p1.3.m3.2.2.4.1" stretchy="false" xref="S3.I3.i2.p1.3.m3.2.2.3.cmml">⟨</mo><mi id="S3.I3.i2.p1.3.m3.1.1.1" xref="S3.I3.i2.p1.3.m3.1.1.1.cmml">loglinear</mi><mo id="S3.I3.i2.p1.3.m3.2.2.4.2" xref="S3.I3.i2.p1.3.m3.2.2.3.cmml">,</mo><mi id="S3.I3.i2.p1.3.m3.2.2.2" xref="S3.I3.i2.p1.3.m3.2.2.2.cmml">log</mi><mo id="S3.I3.i2.p1.3.m3.2.2.4.3" stretchy="false" xref="S3.I3.i2.p1.3.m3.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.3.m3.2b"><list id="S3.I3.i2.p1.3.m3.2.2.3.cmml" xref="S3.I3.i2.p1.3.m3.2.2.4"><ci id="S3.I3.i2.p1.3.m3.1.1.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.1">loglinear</ci><log id="S3.I3.i2.p1.3.m3.2.2.2.cmml" xref="S3.I3.i2.p1.3.m3.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.3.m3.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.3.m3.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math>, assuming the</p> </div> </li> </ol> </div> <div class="ltx_para" id="S3.p8"> <p class="ltx_p" id="S3.p8.1">The following lemma claims that introducing new variables into a hard query cannot make it easier (as long as the connections between the original variables remain unchanged).</p> </div> <div class="ltx_para" id="S3.p9"> <p class="ltx_p" id="S3.p9.15">To formalize this claim, we use the following definitions. Given a self-join-free query <math alttext="Q" class="ltx_Math" display="inline" id="S3.p9.1.m1.1"><semantics id="S3.p9.1.m1.1a"><mi id="S3.p9.1.m1.1.1" xref="S3.p9.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.p9.1.m1.1b"><ci id="S3.p9.1.m1.1.1.cmml" xref="S3.p9.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.p9.1.m1.1d">italic_Q</annotation></semantics></math> and a subset <math alttext="V" class="ltx_Math" display="inline" id="S3.p9.2.m2.1"><semantics id="S3.p9.2.m2.1a"><mi id="S3.p9.2.m2.1.1" xref="S3.p9.2.m2.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p9.2.m2.1b"><ci id="S3.p9.2.m2.1.1.cmml" xref="S3.p9.2.m2.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.2.m2.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p9.2.m2.1d">italic_V</annotation></semantics></math> of its variables, we denote by <math alttext="Q_{|V}" class="ltx_math_unparsed" display="inline" id="S3.p9.3.m3.1"><semantics id="S3.p9.3.m3.1a"><msub id="S3.p9.3.m3.1.1"><mi id="S3.p9.3.m3.1.1.2">Q</mi><mrow id="S3.p9.3.m3.1.1.3"><mo fence="false" id="S3.p9.3.m3.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.p9.3.m3.1.1.3.2">V</mi></mrow></msub><annotation encoding="application/x-tex" id="S3.p9.3.m3.1b">Q_{|V}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.3.m3.1c">italic_Q start_POSTSUBSCRIPT | italic_V end_POSTSUBSCRIPT</annotation></semantics></math> the query obtained by removing all variables that are not in <math alttext="V" class="ltx_Math" display="inline" id="S3.p9.4.m4.1"><semantics id="S3.p9.4.m4.1a"><mi id="S3.p9.4.m4.1.1" xref="S3.p9.4.m4.1.1.cmml">V</mi><annotation-xml encoding="MathML-Content" id="S3.p9.4.m4.1b"><ci id="S3.p9.4.m4.1.1.cmml" xref="S3.p9.4.m4.1.1">𝑉</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.4.m4.1c">V</annotation><annotation encoding="application/x-llamapun" id="S3.p9.4.m4.1d">italic_V</annotation></semantics></math> from <math alttext="Q" class="ltx_Math" display="inline" id="S3.p9.5.m5.1"><semantics id="S3.p9.5.m5.1a"><mi id="S3.p9.5.m5.1.1" xref="S3.p9.5.m5.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.p9.5.m5.1b"><ci id="S3.p9.5.m5.1.1.cmml" xref="S3.p9.5.m5.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.5.m5.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.p9.5.m5.1d">italic_Q</annotation></semantics></math>. We say that two queries <math alttext="Q_{1}" class="ltx_Math" display="inline" id="S3.p9.6.m6.1"><semantics id="S3.p9.6.m6.1a"><msub id="S3.p9.6.m6.1.1" xref="S3.p9.6.m6.1.1.cmml"><mi id="S3.p9.6.m6.1.1.2" xref="S3.p9.6.m6.1.1.2.cmml">Q</mi><mn id="S3.p9.6.m6.1.1.3" xref="S3.p9.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p9.6.m6.1b"><apply id="S3.p9.6.m6.1.1.cmml" xref="S3.p9.6.m6.1.1"><csymbol cd="ambiguous" id="S3.p9.6.m6.1.1.1.cmml" xref="S3.p9.6.m6.1.1">subscript</csymbol><ci id="S3.p9.6.m6.1.1.2.cmml" xref="S3.p9.6.m6.1.1.2">𝑄</ci><cn id="S3.p9.6.m6.1.1.3.cmml" type="integer" xref="S3.p9.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.6.m6.1c">Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.6.m6.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Q_{2}" class="ltx_Math" display="inline" id="S3.p9.7.m7.1"><semantics id="S3.p9.7.m7.1a"><msub id="S3.p9.7.m7.1.1" xref="S3.p9.7.m7.1.1.cmml"><mi id="S3.p9.7.m7.1.1.2" xref="S3.p9.7.m7.1.1.2.cmml">Q</mi><mn id="S3.p9.7.m7.1.1.3" xref="S3.p9.7.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p9.7.m7.1b"><apply id="S3.p9.7.m7.1.1.cmml" xref="S3.p9.7.m7.1.1"><csymbol cd="ambiguous" id="S3.p9.7.m7.1.1.1.cmml" xref="S3.p9.7.m7.1.1">subscript</csymbol><ci id="S3.p9.7.m7.1.1.2.cmml" xref="S3.p9.7.m7.1.1.2">𝑄</ci><cn id="S3.p9.7.m7.1.1.3.cmml" type="integer" xref="S3.p9.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.7.m7.1c">Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.7.m7.1d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are <em class="ltx_emph ltx_font_italic" id="S3.p9.15.1">structurally equivalent</em> if there exists a query <math alttext="Q_{1}^{\prime}" class="ltx_Math" display="inline" id="S3.p9.8.m8.1"><semantics id="S3.p9.8.m8.1a"><msubsup id="S3.p9.8.m8.1.1" xref="S3.p9.8.m8.1.1.cmml"><mi id="S3.p9.8.m8.1.1.2.2" xref="S3.p9.8.m8.1.1.2.2.cmml">Q</mi><mn id="S3.p9.8.m8.1.1.2.3" xref="S3.p9.8.m8.1.1.2.3.cmml">1</mn><mo id="S3.p9.8.m8.1.1.3" xref="S3.p9.8.m8.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.p9.8.m8.1b"><apply id="S3.p9.8.m8.1.1.cmml" xref="S3.p9.8.m8.1.1"><csymbol cd="ambiguous" id="S3.p9.8.m8.1.1.1.cmml" xref="S3.p9.8.m8.1.1">superscript</csymbol><apply id="S3.p9.8.m8.1.1.2.cmml" xref="S3.p9.8.m8.1.1"><csymbol cd="ambiguous" id="S3.p9.8.m8.1.1.2.1.cmml" xref="S3.p9.8.m8.1.1">subscript</csymbol><ci id="S3.p9.8.m8.1.1.2.2.cmml" xref="S3.p9.8.m8.1.1.2.2">𝑄</ci><cn id="S3.p9.8.m8.1.1.2.3.cmml" type="integer" xref="S3.p9.8.m8.1.1.2.3">1</cn></apply><ci id="S3.p9.8.m8.1.1.3.cmml" xref="S3.p9.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.8.m8.1c">Q_{1}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.8.m8.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> that is isomorphic to <math alttext="Q_{1}" class="ltx_Math" display="inline" id="S3.p9.9.m9.1"><semantics id="S3.p9.9.m9.1a"><msub id="S3.p9.9.m9.1.1" xref="S3.p9.9.m9.1.1.cmml"><mi id="S3.p9.9.m9.1.1.2" xref="S3.p9.9.m9.1.1.2.cmml">Q</mi><mn id="S3.p9.9.m9.1.1.3" xref="S3.p9.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p9.9.m9.1b"><apply id="S3.p9.9.m9.1.1.cmml" xref="S3.p9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.p9.9.m9.1.1.1.cmml" xref="S3.p9.9.m9.1.1">subscript</csymbol><ci id="S3.p9.9.m9.1.1.2.cmml" xref="S3.p9.9.m9.1.1.2">𝑄</ci><cn id="S3.p9.9.m9.1.1.3.cmml" type="integer" xref="S3.p9.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.9.m9.1c">Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.9.m9.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> (i.e., <math alttext="Q_{1}^{\prime}" class="ltx_Math" display="inline" id="S3.p9.10.m10.1"><semantics id="S3.p9.10.m10.1a"><msubsup id="S3.p9.10.m10.1.1" xref="S3.p9.10.m10.1.1.cmml"><mi id="S3.p9.10.m10.1.1.2.2" xref="S3.p9.10.m10.1.1.2.2.cmml">Q</mi><mn id="S3.p9.10.m10.1.1.2.3" xref="S3.p9.10.m10.1.1.2.3.cmml">1</mn><mo id="S3.p9.10.m10.1.1.3" xref="S3.p9.10.m10.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.p9.10.m10.1b"><apply id="S3.p9.10.m10.1.1.cmml" xref="S3.p9.10.m10.1.1"><csymbol cd="ambiguous" id="S3.p9.10.m10.1.1.1.cmml" xref="S3.p9.10.m10.1.1">superscript</csymbol><apply id="S3.p9.10.m10.1.1.2.cmml" xref="S3.p9.10.m10.1.1"><csymbol cd="ambiguous" id="S3.p9.10.m10.1.1.2.1.cmml" xref="S3.p9.10.m10.1.1">subscript</csymbol><ci id="S3.p9.10.m10.1.1.2.2.cmml" xref="S3.p9.10.m10.1.1.2.2">𝑄</ci><cn id="S3.p9.10.m10.1.1.2.3.cmml" type="integer" xref="S3.p9.10.m10.1.1.2.3">1</cn></apply><ci id="S3.p9.10.m10.1.1.3.cmml" xref="S3.p9.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.10.m10.1c">Q_{1}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.10.m10.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is obtained by renaming the variables of <math alttext="Q_{1}" class="ltx_Math" display="inline" id="S3.p9.11.m11.1"><semantics id="S3.p9.11.m11.1a"><msub id="S3.p9.11.m11.1.1" xref="S3.p9.11.m11.1.1.cmml"><mi id="S3.p9.11.m11.1.1.2" xref="S3.p9.11.m11.1.1.2.cmml">Q</mi><mn id="S3.p9.11.m11.1.1.3" xref="S3.p9.11.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p9.11.m11.1b"><apply id="S3.p9.11.m11.1.1.cmml" xref="S3.p9.11.m11.1.1"><csymbol cd="ambiguous" id="S3.p9.11.m11.1.1.1.cmml" xref="S3.p9.11.m11.1.1">subscript</csymbol><ci id="S3.p9.11.m11.1.1.2.cmml" xref="S3.p9.11.m11.1.1.2">𝑄</ci><cn id="S3.p9.11.m11.1.1.3.cmml" type="integer" xref="S3.p9.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.11.m11.1c">Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.11.m11.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>) such that: (1) the variables of every atom of <math alttext="Q_{1}^{\prime}" class="ltx_Math" display="inline" id="S3.p9.12.m12.1"><semantics id="S3.p9.12.m12.1a"><msubsup id="S3.p9.12.m12.1.1" xref="S3.p9.12.m12.1.1.cmml"><mi id="S3.p9.12.m12.1.1.2.2" xref="S3.p9.12.m12.1.1.2.2.cmml">Q</mi><mn id="S3.p9.12.m12.1.1.2.3" xref="S3.p9.12.m12.1.1.2.3.cmml">1</mn><mo id="S3.p9.12.m12.1.1.3" xref="S3.p9.12.m12.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.p9.12.m12.1b"><apply id="S3.p9.12.m12.1.1.cmml" xref="S3.p9.12.m12.1.1"><csymbol cd="ambiguous" id="S3.p9.12.m12.1.1.1.cmml" xref="S3.p9.12.m12.1.1">superscript</csymbol><apply id="S3.p9.12.m12.1.1.2.cmml" xref="S3.p9.12.m12.1.1"><csymbol cd="ambiguous" id="S3.p9.12.m12.1.1.2.1.cmml" xref="S3.p9.12.m12.1.1">subscript</csymbol><ci id="S3.p9.12.m12.1.1.2.2.cmml" xref="S3.p9.12.m12.1.1.2.2">𝑄</ci><cn id="S3.p9.12.m12.1.1.2.3.cmml" type="integer" xref="S3.p9.12.m12.1.1.2.3">1</cn></apply><ci id="S3.p9.12.m12.1.1.3.cmml" xref="S3.p9.12.m12.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.12.m12.1c">Q_{1}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.12.m12.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are contained in an atom of <math alttext="Q_{2}" class="ltx_Math" display="inline" id="S3.p9.13.m13.1"><semantics id="S3.p9.13.m13.1a"><msub id="S3.p9.13.m13.1.1" xref="S3.p9.13.m13.1.1.cmml"><mi id="S3.p9.13.m13.1.1.2" xref="S3.p9.13.m13.1.1.2.cmml">Q</mi><mn id="S3.p9.13.m13.1.1.3" xref="S3.p9.13.m13.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p9.13.m13.1b"><apply id="S3.p9.13.m13.1.1.cmml" xref="S3.p9.13.m13.1.1"><csymbol cd="ambiguous" id="S3.p9.13.m13.1.1.1.cmml" xref="S3.p9.13.m13.1.1">subscript</csymbol><ci id="S3.p9.13.m13.1.1.2.cmml" xref="S3.p9.13.m13.1.1.2">𝑄</ci><cn id="S3.p9.13.m13.1.1.3.cmml" type="integer" xref="S3.p9.13.m13.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.13.m13.1c">Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.13.m13.1d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and vice versa; and (2) <math alttext="Q_{1}^{\prime}" class="ltx_Math" display="inline" id="S3.p9.14.m14.1"><semantics id="S3.p9.14.m14.1a"><msubsup id="S3.p9.14.m14.1.1" xref="S3.p9.14.m14.1.1.cmml"><mi id="S3.p9.14.m14.1.1.2.2" xref="S3.p9.14.m14.1.1.2.2.cmml">Q</mi><mn id="S3.p9.14.m14.1.1.2.3" xref="S3.p9.14.m14.1.1.2.3.cmml">1</mn><mo id="S3.p9.14.m14.1.1.3" xref="S3.p9.14.m14.1.1.3.cmml">′</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.p9.14.m14.1b"><apply id="S3.p9.14.m14.1.1.cmml" xref="S3.p9.14.m14.1.1"><csymbol cd="ambiguous" id="S3.p9.14.m14.1.1.1.cmml" xref="S3.p9.14.m14.1.1">superscript</csymbol><apply id="S3.p9.14.m14.1.1.2.cmml" xref="S3.p9.14.m14.1.1"><csymbol cd="ambiguous" id="S3.p9.14.m14.1.1.2.1.cmml" xref="S3.p9.14.m14.1.1">subscript</csymbol><ci id="S3.p9.14.m14.1.1.2.2.cmml" xref="S3.p9.14.m14.1.1.2.2">𝑄</ci><cn id="S3.p9.14.m14.1.1.2.3.cmml" type="integer" xref="S3.p9.14.m14.1.1.2.3">1</cn></apply><ci id="S3.p9.14.m14.1.1.3.cmml" xref="S3.p9.14.m14.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.14.m14.1c">Q_{1}^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.14.m14.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="Q_{2}" class="ltx_Math" display="inline" id="S3.p9.15.m15.1"><semantics id="S3.p9.15.m15.1a"><msub id="S3.p9.15.m15.1.1" xref="S3.p9.15.m15.1.1.cmml"><mi id="S3.p9.15.m15.1.1.2" xref="S3.p9.15.m15.1.1.2.cmml">Q</mi><mn id="S3.p9.15.m15.1.1.3" xref="S3.p9.15.m15.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p9.15.m15.1b"><apply id="S3.p9.15.m15.1.1.cmml" xref="S3.p9.15.m15.1.1"><csymbol cd="ambiguous" id="S3.p9.15.m15.1.1.1.cmml" xref="S3.p9.15.m15.1.1">subscript</csymbol><ci id="S3.p9.15.m15.1.1.2.cmml" xref="S3.p9.15.m15.1.1.2">𝑄</ci><cn id="S3.p9.15.m15.1.1.3.cmml" type="integer" xref="S3.p9.15.m15.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p9.15.m15.1c">Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p9.15.m15.1d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> have the same head.</p> </div> <div class="ltx_theorem ltx_theorem_lem" id="Thmthm6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm6.1.1.1">Lemma 6</span></span><span class="ltx_text ltx_font_bold" id="Thmthm6.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm6.p1"> <p class="ltx_p" id="Thmthm6.p1.11"><span class="ltx_text ltx_font_italic" id="Thmthm6.p1.11.11">Let <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm6.p1.1.1.m1.1"><semantics id="Thmthm6.p1.1.1.m1.1a"><mi id="Thmthm6.p1.1.1.m1.1.1" xref="Thmthm6.p1.1.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.1.1.m1.1b"><ci id="Thmthm6.p1.1.1.m1.1.1.cmml" xref="Thmthm6.p1.1.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.1.1.m1.1d">italic_Q</annotation></semantics></math> and <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="Thmthm6.p1.2.2.m2.1"><semantics id="Thmthm6.p1.2.2.m2.1a"><msup id="Thmthm6.p1.2.2.m2.1.1" xref="Thmthm6.p1.2.2.m2.1.1.cmml"><mi id="Thmthm6.p1.2.2.m2.1.1.2" xref="Thmthm6.p1.2.2.m2.1.1.2.cmml">Q</mi><mo id="Thmthm6.p1.2.2.m2.1.1.3" xref="Thmthm6.p1.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.2.2.m2.1b"><apply id="Thmthm6.p1.2.2.m2.1.1.cmml" xref="Thmthm6.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmthm6.p1.2.2.m2.1.1.1.cmml" xref="Thmthm6.p1.2.2.m2.1.1">superscript</csymbol><ci id="Thmthm6.p1.2.2.m2.1.1.2.cmml" xref="Thmthm6.p1.2.2.m2.1.1.2">𝑄</ci><ci id="Thmthm6.p1.2.2.m2.1.1.3.cmml" xref="Thmthm6.p1.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.2.2.m2.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.2.2.m2.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> be queries such that they are both CQs, both CQ<sup class="ltx_sup" id="Thmthm6.p1.11.11.1"><span class="ltx_text ltx_font_upright" id="Thmthm6.p1.11.11.1.1">⋆</span></sup>s over a logarithmic-time semiring, or both AggCQs, and <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm6.p1.4.4.m4.1"><semantics id="Thmthm6.p1.4.4.m4.1a"><mi id="Thmthm6.p1.4.4.m4.1.1" xref="Thmthm6.p1.4.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.4.4.m4.1b"><ci id="Thmthm6.p1.4.4.m4.1.1.cmml" xref="Thmthm6.p1.4.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.4.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.4.4.m4.1d">italic_Q</annotation></semantics></math> has no self-joins. If there exists <math alttext="V\subseteq\mathrm{vars}{(Q)}" class="ltx_Math" display="inline" id="Thmthm6.p1.5.5.m5.1"><semantics id="Thmthm6.p1.5.5.m5.1a"><mrow id="Thmthm6.p1.5.5.m5.1.2" xref="Thmthm6.p1.5.5.m5.1.2.cmml"><mi id="Thmthm6.p1.5.5.m5.1.2.2" xref="Thmthm6.p1.5.5.m5.1.2.2.cmml">V</mi><mo id="Thmthm6.p1.5.5.m5.1.2.1" xref="Thmthm6.p1.5.5.m5.1.2.1.cmml">⊆</mo><mrow id="Thmthm6.p1.5.5.m5.1.2.3" xref="Thmthm6.p1.5.5.m5.1.2.3.cmml"><mi id="Thmthm6.p1.5.5.m5.1.2.3.2" xref="Thmthm6.p1.5.5.m5.1.2.3.2.cmml">vars</mi><mo id="Thmthm6.p1.5.5.m5.1.2.3.1" xref="Thmthm6.p1.5.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="Thmthm6.p1.5.5.m5.1.2.3.3.2" xref="Thmthm6.p1.5.5.m5.1.2.3.cmml"><mo id="Thmthm6.p1.5.5.m5.1.2.3.3.2.1" stretchy="false" xref="Thmthm6.p1.5.5.m5.1.2.3.cmml">(</mo><mi id="Thmthm6.p1.5.5.m5.1.1" xref="Thmthm6.p1.5.5.m5.1.1.cmml">Q</mi><mo id="Thmthm6.p1.5.5.m5.1.2.3.3.2.2" stretchy="false" xref="Thmthm6.p1.5.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.5.5.m5.1b"><apply id="Thmthm6.p1.5.5.m5.1.2.cmml" xref="Thmthm6.p1.5.5.m5.1.2"><subset id="Thmthm6.p1.5.5.m5.1.2.1.cmml" xref="Thmthm6.p1.5.5.m5.1.2.1"></subset><ci id="Thmthm6.p1.5.5.m5.1.2.2.cmml" xref="Thmthm6.p1.5.5.m5.1.2.2">𝑉</ci><apply id="Thmthm6.p1.5.5.m5.1.2.3.cmml" xref="Thmthm6.p1.5.5.m5.1.2.3"><times id="Thmthm6.p1.5.5.m5.1.2.3.1.cmml" xref="Thmthm6.p1.5.5.m5.1.2.3.1"></times><ci id="Thmthm6.p1.5.5.m5.1.2.3.2.cmml" xref="Thmthm6.p1.5.5.m5.1.2.3.2">vars</ci><ci id="Thmthm6.p1.5.5.m5.1.1.cmml" xref="Thmthm6.p1.5.5.m5.1.1">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.5.5.m5.1c">V\subseteq\mathrm{vars}{(Q)}</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.5.5.m5.1d">italic_V ⊆ roman_vars ( italic_Q )</annotation></semantics></math> such that <math alttext="Q_{|V}" class="ltx_math_unparsed" display="inline" id="Thmthm6.p1.6.6.m6.1"><semantics id="Thmthm6.p1.6.6.m6.1a"><msub id="Thmthm6.p1.6.6.m6.1.1"><mi id="Thmthm6.p1.6.6.m6.1.1.2">Q</mi><mrow id="Thmthm6.p1.6.6.m6.1.1.3"><mo fence="false" id="Thmthm6.p1.6.6.m6.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="Thmthm6.p1.6.6.m6.1.1.3.2">V</mi></mrow></msub><annotation encoding="application/x-tex" id="Thmthm6.p1.6.6.m6.1b">Q_{|V}</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.6.6.m6.1c">italic_Q start_POSTSUBSCRIPT | italic_V end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="Thmthm6.p1.7.7.m7.1"><semantics id="Thmthm6.p1.7.7.m7.1a"><msup id="Thmthm6.p1.7.7.m7.1.1" xref="Thmthm6.p1.7.7.m7.1.1.cmml"><mi id="Thmthm6.p1.7.7.m7.1.1.2" xref="Thmthm6.p1.7.7.m7.1.1.2.cmml">Q</mi><mo id="Thmthm6.p1.7.7.m7.1.1.3" xref="Thmthm6.p1.7.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.7.7.m7.1b"><apply id="Thmthm6.p1.7.7.m7.1.1.cmml" xref="Thmthm6.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="Thmthm6.p1.7.7.m7.1.1.1.cmml" xref="Thmthm6.p1.7.7.m7.1.1">superscript</csymbol><ci id="Thmthm6.p1.7.7.m7.1.1.2.cmml" xref="Thmthm6.p1.7.7.m7.1.1.2">𝑄</ci><ci id="Thmthm6.p1.7.7.m7.1.1.3.cmml" xref="Thmthm6.p1.7.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.7.7.m7.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.7.7.m7.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> are structurally equivalent, and direct access for <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm6.p1.8.8.m8.1"><semantics id="Thmthm6.p1.8.8.m8.1a"><mi id="Thmthm6.p1.8.8.m8.1.1" xref="Thmthm6.p1.8.8.m8.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.8.8.m8.1b"><ci id="Thmthm6.p1.8.8.m8.1.1.cmml" xref="Thmthm6.p1.8.8.m8.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.8.8.m8.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.8.8.m8.1d">italic_Q</annotation></semantics></math> is in <math alttext="\mathord{\langle T_{p},T_{a}\rangle}" class="ltx_Math" display="inline" id="Thmthm6.p1.9.9.m9.2"><semantics id="Thmthm6.p1.9.9.m9.2a"><mrow id="Thmthm6.p1.9.9.m9.2.2.2" xref="Thmthm6.p1.9.9.m9.2.2.3.cmml"><mo id="Thmthm6.p1.9.9.m9.2.2.2.3" stretchy="false" xref="Thmthm6.p1.9.9.m9.2.2.3.cmml">⟨</mo><msub id="Thmthm6.p1.9.9.m9.1.1.1.1" xref="Thmthm6.p1.9.9.m9.1.1.1.1.cmml"><mi id="Thmthm6.p1.9.9.m9.1.1.1.1.2" xref="Thmthm6.p1.9.9.m9.1.1.1.1.2.cmml">T</mi><mi id="Thmthm6.p1.9.9.m9.1.1.1.1.3" xref="Thmthm6.p1.9.9.m9.1.1.1.1.3.cmml">p</mi></msub><mo id="Thmthm6.p1.9.9.m9.2.2.2.4" xref="Thmthm6.p1.9.9.m9.2.2.3.cmml">,</mo><msub id="Thmthm6.p1.9.9.m9.2.2.2.2" xref="Thmthm6.p1.9.9.m9.2.2.2.2.cmml"><mi id="Thmthm6.p1.9.9.m9.2.2.2.2.2" xref="Thmthm6.p1.9.9.m9.2.2.2.2.2.cmml">T</mi><mi id="Thmthm6.p1.9.9.m9.2.2.2.2.3" xref="Thmthm6.p1.9.9.m9.2.2.2.2.3.cmml">a</mi></msub><mo id="Thmthm6.p1.9.9.m9.2.2.2.5" stretchy="false" xref="Thmthm6.p1.9.9.m9.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.9.9.m9.2b"><list id="Thmthm6.p1.9.9.m9.2.2.3.cmml" xref="Thmthm6.p1.9.9.m9.2.2.2"><apply id="Thmthm6.p1.9.9.m9.1.1.1.1.cmml" xref="Thmthm6.p1.9.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm6.p1.9.9.m9.1.1.1.1.1.cmml" xref="Thmthm6.p1.9.9.m9.1.1.1.1">subscript</csymbol><ci id="Thmthm6.p1.9.9.m9.1.1.1.1.2.cmml" xref="Thmthm6.p1.9.9.m9.1.1.1.1.2">𝑇</ci><ci id="Thmthm6.p1.9.9.m9.1.1.1.1.3.cmml" xref="Thmthm6.p1.9.9.m9.1.1.1.1.3">𝑝</ci></apply><apply id="Thmthm6.p1.9.9.m9.2.2.2.2.cmml" xref="Thmthm6.p1.9.9.m9.2.2.2.2"><csymbol cd="ambiguous" id="Thmthm6.p1.9.9.m9.2.2.2.2.1.cmml" xref="Thmthm6.p1.9.9.m9.2.2.2.2">subscript</csymbol><ci id="Thmthm6.p1.9.9.m9.2.2.2.2.2.cmml" xref="Thmthm6.p1.9.9.m9.2.2.2.2.2">𝑇</ci><ci id="Thmthm6.p1.9.9.m9.2.2.2.2.3.cmml" xref="Thmthm6.p1.9.9.m9.2.2.2.2.3">𝑎</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.9.9.m9.2c">\mathord{\langle T_{p},T_{a}\rangle}</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.9.9.m9.2d">⟨ italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟩</annotation></semantics></math>, then direct access for <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="Thmthm6.p1.10.10.m10.1"><semantics id="Thmthm6.p1.10.10.m10.1a"><msup id="Thmthm6.p1.10.10.m10.1.1" xref="Thmthm6.p1.10.10.m10.1.1.cmml"><mi id="Thmthm6.p1.10.10.m10.1.1.2" xref="Thmthm6.p1.10.10.m10.1.1.2.cmml">Q</mi><mo id="Thmthm6.p1.10.10.m10.1.1.3" xref="Thmthm6.p1.10.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.10.10.m10.1b"><apply id="Thmthm6.p1.10.10.m10.1.1.cmml" xref="Thmthm6.p1.10.10.m10.1.1"><csymbol cd="ambiguous" id="Thmthm6.p1.10.10.m10.1.1.1.cmml" xref="Thmthm6.p1.10.10.m10.1.1">superscript</csymbol><ci id="Thmthm6.p1.10.10.m10.1.1.2.cmml" xref="Thmthm6.p1.10.10.m10.1.1.2">𝑄</ci><ci id="Thmthm6.p1.10.10.m10.1.1.3.cmml" xref="Thmthm6.p1.10.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.10.10.m10.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.10.10.m10.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is in <math alttext="\mathord{\langle\mathrm{loglinear}+T_{p},T_{a}\rangle}" class="ltx_Math" display="inline" id="Thmthm6.p1.11.11.m11.2"><semantics id="Thmthm6.p1.11.11.m11.2a"><mrow id="Thmthm6.p1.11.11.m11.2.2.2" xref="Thmthm6.p1.11.11.m11.2.2.3.cmml"><mo id="Thmthm6.p1.11.11.m11.2.2.2.3" stretchy="false" xref="Thmthm6.p1.11.11.m11.2.2.3.cmml">⟨</mo><mrow id="Thmthm6.p1.11.11.m11.1.1.1.1" xref="Thmthm6.p1.11.11.m11.1.1.1.1.cmml"><mi id="Thmthm6.p1.11.11.m11.1.1.1.1.2" xref="Thmthm6.p1.11.11.m11.1.1.1.1.2.cmml">loglinear</mi><mo id="Thmthm6.p1.11.11.m11.1.1.1.1.1" xref="Thmthm6.p1.11.11.m11.1.1.1.1.1.cmml">+</mo><msub id="Thmthm6.p1.11.11.m11.1.1.1.1.3" xref="Thmthm6.p1.11.11.m11.1.1.1.1.3.cmml"><mi id="Thmthm6.p1.11.11.m11.1.1.1.1.3.2" xref="Thmthm6.p1.11.11.m11.1.1.1.1.3.2.cmml">T</mi><mi id="Thmthm6.p1.11.11.m11.1.1.1.1.3.3" xref="Thmthm6.p1.11.11.m11.1.1.1.1.3.3.cmml">p</mi></msub></mrow><mo id="Thmthm6.p1.11.11.m11.2.2.2.4" xref="Thmthm6.p1.11.11.m11.2.2.3.cmml">,</mo><msub id="Thmthm6.p1.11.11.m11.2.2.2.2" xref="Thmthm6.p1.11.11.m11.2.2.2.2.cmml"><mi id="Thmthm6.p1.11.11.m11.2.2.2.2.2" xref="Thmthm6.p1.11.11.m11.2.2.2.2.2.cmml">T</mi><mi id="Thmthm6.p1.11.11.m11.2.2.2.2.3" xref="Thmthm6.p1.11.11.m11.2.2.2.2.3.cmml">a</mi></msub><mo id="Thmthm6.p1.11.11.m11.2.2.2.5" stretchy="false" xref="Thmthm6.p1.11.11.m11.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm6.p1.11.11.m11.2b"><list id="Thmthm6.p1.11.11.m11.2.2.3.cmml" xref="Thmthm6.p1.11.11.m11.2.2.2"><apply id="Thmthm6.p1.11.11.m11.1.1.1.1.cmml" xref="Thmthm6.p1.11.11.m11.1.1.1.1"><plus id="Thmthm6.p1.11.11.m11.1.1.1.1.1.cmml" xref="Thmthm6.p1.11.11.m11.1.1.1.1.1"></plus><ci id="Thmthm6.p1.11.11.m11.1.1.1.1.2.cmml" xref="Thmthm6.p1.11.11.m11.1.1.1.1.2">loglinear</ci><apply id="Thmthm6.p1.11.11.m11.1.1.1.1.3.cmml" xref="Thmthm6.p1.11.11.m11.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmthm6.p1.11.11.m11.1.1.1.1.3.1.cmml" xref="Thmthm6.p1.11.11.m11.1.1.1.1.3">subscript</csymbol><ci id="Thmthm6.p1.11.11.m11.1.1.1.1.3.2.cmml" xref="Thmthm6.p1.11.11.m11.1.1.1.1.3.2">𝑇</ci><ci id="Thmthm6.p1.11.11.m11.1.1.1.1.3.3.cmml" xref="Thmthm6.p1.11.11.m11.1.1.1.1.3.3">𝑝</ci></apply></apply><apply id="Thmthm6.p1.11.11.m11.2.2.2.2.cmml" xref="Thmthm6.p1.11.11.m11.2.2.2.2"><csymbol cd="ambiguous" id="Thmthm6.p1.11.11.m11.2.2.2.2.1.cmml" xref="Thmthm6.p1.11.11.m11.2.2.2.2">subscript</csymbol><ci id="Thmthm6.p1.11.11.m11.2.2.2.2.2.cmml" xref="Thmthm6.p1.11.11.m11.2.2.2.2.2">𝑇</ci><ci id="Thmthm6.p1.11.11.m11.2.2.2.2.3.cmml" xref="Thmthm6.p1.11.11.m11.2.2.2.2.3">𝑎</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm6.p1.11.11.m11.2c">\mathord{\langle\mathrm{loglinear}+T_{p},T_{a}\rangle}</annotation><annotation encoding="application/x-llamapun" id="Thmthm6.p1.11.11.m11.2d">⟨ roman_loglinear + italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟩</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.p10"> <p class="ltx_p" id="S3.p10.1">As an example for the use of <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm6" title="Lemma 6. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a>, it can reprove the fact that, given that <math alttext="Q_{\text{trio}}(x_{1},x_{2},x_{3}){\,:\!\!-\,}R_{1}(x_{1},x_{3}),R_{2}(x_{2},x% _{3})" class="ltx_Math" display="inline" id="S3.p10.1.m1.5"><semantics id="S3.p10.1.m1.5a"><mrow id="S3.p10.1.m1.5.5" xref="S3.p10.1.m1.5.5.cmml"><mrow id="S3.p10.1.m1.3.3.3" xref="S3.p10.1.m1.3.3.3.cmml"><msub id="S3.p10.1.m1.3.3.3.5" xref="S3.p10.1.m1.3.3.3.5.cmml"><mi id="S3.p10.1.m1.3.3.3.5.2" xref="S3.p10.1.m1.3.3.3.5.2.cmml">Q</mi><mtext id="S3.p10.1.m1.3.3.3.5.3" xref="S3.p10.1.m1.3.3.3.5.3a.cmml">trio</mtext></msub><mo id="S3.p10.1.m1.3.3.3.4" xref="S3.p10.1.m1.3.3.3.4.cmml">⁢</mo><mrow id="S3.p10.1.m1.3.3.3.3.3" xref="S3.p10.1.m1.3.3.3.3.4.cmml"><mo id="S3.p10.1.m1.3.3.3.3.3.4" stretchy="false" xref="S3.p10.1.m1.3.3.3.3.4.cmml">(</mo><msub 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xref="S3.p10.1.m1.5.5.5.2.2.4">subscript</csymbol><ci id="S3.p10.1.m1.5.5.5.2.2.4.2.cmml" xref="S3.p10.1.m1.5.5.5.2.2.4.2">𝑅</ci><cn id="S3.p10.1.m1.5.5.5.2.2.4.3.cmml" type="integer" xref="S3.p10.1.m1.5.5.5.2.2.4.3">2</cn></apply><interval closure="open" id="S3.p10.1.m1.5.5.5.2.2.2.3.cmml" xref="S3.p10.1.m1.5.5.5.2.2.2.2"><apply id="S3.p10.1.m1.5.5.5.2.2.1.1.1.cmml" xref="S3.p10.1.m1.5.5.5.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p10.1.m1.5.5.5.2.2.1.1.1.1.cmml" xref="S3.p10.1.m1.5.5.5.2.2.1.1.1">subscript</csymbol><ci id="S3.p10.1.m1.5.5.5.2.2.1.1.1.2.cmml" xref="S3.p10.1.m1.5.5.5.2.2.1.1.1.2">𝑥</ci><cn id="S3.p10.1.m1.5.5.5.2.2.1.1.1.3.cmml" type="integer" xref="S3.p10.1.m1.5.5.5.2.2.1.1.1.3">2</cn></apply><apply id="S3.p10.1.m1.5.5.5.2.2.2.2.2.cmml" xref="S3.p10.1.m1.5.5.5.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.p10.1.m1.5.5.5.2.2.2.2.2.1.cmml" xref="S3.p10.1.m1.5.5.5.2.2.2.2.2">subscript</csymbol><ci id="S3.p10.1.m1.5.5.5.2.2.2.2.2.2.cmml" xref="S3.p10.1.m1.5.5.5.2.2.2.2.2.2">𝑥</ci><cn id="S3.p10.1.m1.5.5.5.2.2.2.2.2.3.cmml" type="integer" xref="S3.p10.1.m1.5.5.5.2.2.2.2.2.3">3</cn></apply></interval></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p10.1.m1.5c">Q_{\text{trio}}(x_{1},x_{2},x_{3}){\,:\!\!-\,}R_{1}(x_{1},x_{3}),R_{2}(x_{2},x% _{3})</annotation><annotation encoding="application/x-llamapun" id="S3.p10.1.m1.5d">italic_Q start_POSTSUBSCRIPT trio end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) : - italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math> does admit efficient direct access (assuming the</p> </div> <div class="ltx_theorem ltx_theorem_hypo" id="Thmthm7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm7.1.1.1">Hypothesis 7</span></span><span class="ltx_text ltx_font_bold" id="Thmthm7.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm7.p1"> <p class="ltx_p" id="Thmthm7.p1.1"><span class="ltx_text ltx_font_italic" id="Thmthm7.p1.1.1">SparseBMM hypothesis), neither does any other CQ with a disruptive trio. We will later use <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm6" title="Lemma 6. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a> similarly with different hard queries instead of <math alttext="Q_{\text{trio}}" class="ltx_Math" display="inline" id="Thmthm7.p1.1.1.m1.1"><semantics id="Thmthm7.p1.1.1.m1.1a"><msub id="Thmthm7.p1.1.1.m1.1.1" xref="Thmthm7.p1.1.1.m1.1.1.cmml"><mi id="Thmthm7.p1.1.1.m1.1.1.2" xref="Thmthm7.p1.1.1.m1.1.1.2.cmml">Q</mi><mtext class="ltx_mathvariant_italic" id="Thmthm7.p1.1.1.m1.1.1.3" xref="Thmthm7.p1.1.1.m1.1.1.3a.cmml">trio</mtext></msub><annotation-xml encoding="MathML-Content" id="Thmthm7.p1.1.1.m1.1b"><apply id="Thmthm7.p1.1.1.m1.1.1.cmml" xref="Thmthm7.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmthm7.p1.1.1.m1.1.1.1.cmml" xref="Thmthm7.p1.1.1.m1.1.1">subscript</csymbol><ci id="Thmthm7.p1.1.1.m1.1.1.2.cmml" xref="Thmthm7.p1.1.1.m1.1.1.2">𝑄</ci><ci id="Thmthm7.p1.1.1.m1.1.1.3a.cmml" xref="Thmthm7.p1.1.1.m1.1.1.3"><mtext class="ltx_mathvariant_italic" id="Thmthm7.p1.1.1.m1.1.1.3.cmml" mathsize="70%" xref="Thmthm7.p1.1.1.m1.1.1.3">trio</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p1.1.1.m1.1c">Q_{\text{trio}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p1.1.1.m1.1d">italic_Q start_POSTSUBSCRIPT trio end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_theorem ltx_theorem_proof" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.1.1.1">Proof 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2"> </span>(Proof of <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm6" title="Lemma 6. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a>)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p1.5.5">For the simplicity of presentation, we assume that the variables of <math alttext="Q" 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">Q</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">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.1d">italic_Q</annotation></semantics></math> are renamed such that every atom of <math alttext="Q_{|\mathrm{vars}{(Q^{\prime})}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem1.p1.2.2.m2.1"><semantics id="S3.Thmtheorem1.p1.2.2.m2.1a"><msub id="S3.Thmtheorem1.p1.2.2.m2.1.1"><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1.2">Q</mi><mrow id="S3.Thmtheorem1.p1.2.2.m2.1.1.3"><mo fence="false" id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.2">vars</mi><mrow id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.3"><mo id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.3.1" stretchy="false">(</mo><msup id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.3.2"><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.3.2.2">Q</mi><mo id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.3.2.3">′</mo></msup><mo id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.3.3" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.2.m2.1b">Q_{|\mathrm{vars}{(Q^{\prime})}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.2.m2.1c">italic_Q start_POSTSUBSCRIPT | roman_vars ( italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math> is contained in an atom of <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.3.m3.1"><semantics id="S3.Thmtheorem1.p1.3.3.m3.1a"><msup id="S3.Thmtheorem1.p1.3.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.3.m3.1b"><apply id="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.3.m3.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.3.m3.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and vice versa, and <math alttext="Q_{|\mathrm{vars}{(Q^{\prime})}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem1.p1.4.4.m4.1"><semantics id="S3.Thmtheorem1.p1.4.4.m4.1a"><msub id="S3.Thmtheorem1.p1.4.4.m4.1.1"><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1.2">Q</mi><mrow id="S3.Thmtheorem1.p1.4.4.m4.1.1.3"><mo fence="false" id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.2">vars</mi><mrow id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3"><mo id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.1" stretchy="false">(</mo><msup id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.2"><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.2.2">Q</mi><mo id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.2.3">′</mo></msup><mo id="S3.Thmtheorem1.p1.4.4.m4.1.1.3.3.3" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.4.4.m4.1b">Q_{|\mathrm{vars}{(Q^{\prime})}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.4.4.m4.1c">italic_Q start_POSTSUBSCRIPT | roman_vars ( italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.5.5.m5.1"><semantics id="S3.Thmtheorem1.p1.5.5.m5.1a"><msup id="S3.Thmtheorem1.p1.5.5.m5.1.1" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem1.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.5.5.m5.1b"><apply id="S3.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.5.5.m5.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.5.5.m5.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> have the same head.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem1.p2"> <p class="ltx_p" id="S3.Thmtheorem1.p2.10"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p2.10.10">We show an <em class="ltx_emph ltx_font_upright" id="S3.Thmtheorem1.p2.10.10.1">order-preserving exact reduction</em> from <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.1.1.m1.1"><semantics id="S3.Thmtheorem1.p2.1.1.m1.1a"><msup id="S3.Thmtheorem1.p2.1.1.m1.1.1" xref="S3.Thmtheorem1.p2.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem1.p2.1.1.m1.1.1.2" xref="S3.Thmtheorem1.p2.1.1.m1.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p2.1.1.m1.1.1.3" xref="S3.Thmtheorem1.p2.1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.1.1.m1.1b"><apply id="S3.Thmtheorem1.p2.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p2.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p2.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem1.p2.1.1.m1.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p2.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem1.p2.1.1.m1.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p2.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem1.p2.1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.1.1.m1.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.1.1.m1.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.2.2.m2.1"><semantics id="S3.Thmtheorem1.p2.2.2.m2.1a"><mi id="S3.Thmtheorem1.p2.2.2.m2.1.1" xref="S3.Thmtheorem1.p2.2.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.2.2.m2.1b"><ci id="S3.Thmtheorem1.p2.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p2.2.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.2.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.2.2.m2.1d">italic_Q</annotation></semantics></math>. That is, given a database <math alttext="D^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.3.3.m3.1"><semantics id="S3.Thmtheorem1.p2.3.3.m3.1a"><msup id="S3.Thmtheorem1.p2.3.3.m3.1.1" xref="S3.Thmtheorem1.p2.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem1.p2.3.3.m3.1.1.2" xref="S3.Thmtheorem1.p2.3.3.m3.1.1.2.cmml">D</mi><mo id="S3.Thmtheorem1.p2.3.3.m3.1.1.3" xref="S3.Thmtheorem1.p2.3.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.3.3.m3.1b"><apply id="S3.Thmtheorem1.p2.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p2.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p2.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p2.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p2.3.3.m3.1.1.2">𝐷</ci><ci id="S3.Thmtheorem1.p2.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem1.p2.3.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.3.3.m3.1c">D^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.3.3.m3.1d">italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.4.4.m4.1"><semantics id="S3.Thmtheorem1.p2.4.4.m4.1a"><msup id="S3.Thmtheorem1.p2.4.4.m4.1.1" xref="S3.Thmtheorem1.p2.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem1.p2.4.4.m4.1.1.2" xref="S3.Thmtheorem1.p2.4.4.m4.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p2.4.4.m4.1.1.3" xref="S3.Thmtheorem1.p2.4.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.4.4.m4.1b"><apply id="S3.Thmtheorem1.p2.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p2.4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p2.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem1.p2.4.4.m4.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p2.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem1.p2.4.4.m4.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p2.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem1.p2.4.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.4.4.m4.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.4.4.m4.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> we show how to construct a database <math alttext="D" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.5.5.m5.1"><semantics id="S3.Thmtheorem1.p2.5.5.m5.1a"><mi id="S3.Thmtheorem1.p2.5.5.m5.1.1" xref="S3.Thmtheorem1.p2.5.5.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.5.5.m5.1b"><ci id="S3.Thmtheorem1.p2.5.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p2.5.5.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.5.5.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.5.5.m5.1d">italic_D</annotation></semantics></math> for <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.6.6.m6.1"><semantics id="S3.Thmtheorem1.p2.6.6.m6.1a"><mi id="S3.Thmtheorem1.p2.6.6.m6.1.1" xref="S3.Thmtheorem1.p2.6.6.m6.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.6.6.m6.1b"><ci id="S3.Thmtheorem1.p2.6.6.m6.1.1.cmml" xref="S3.Thmtheorem1.p2.6.6.m6.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.6.6.m6.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.6.6.m6.1d">italic_Q</annotation></semantics></math> in loglinear time such that there is an order-preserving bijection from <math alttext="Q(D)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.7.7.m7.1"><semantics id="S3.Thmtheorem1.p2.7.7.m7.1a"><mrow id="S3.Thmtheorem1.p2.7.7.m7.1.2" xref="S3.Thmtheorem1.p2.7.7.m7.1.2.cmml"><mi id="S3.Thmtheorem1.p2.7.7.m7.1.2.2" xref="S3.Thmtheorem1.p2.7.7.m7.1.2.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p2.7.7.m7.1.2.1" xref="S3.Thmtheorem1.p2.7.7.m7.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p2.7.7.m7.1.2.3.2" xref="S3.Thmtheorem1.p2.7.7.m7.1.2.cmml"><mo id="S3.Thmtheorem1.p2.7.7.m7.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p2.7.7.m7.1.2.cmml">(</mo><mi id="S3.Thmtheorem1.p2.7.7.m7.1.1" xref="S3.Thmtheorem1.p2.7.7.m7.1.1.cmml">D</mi><mo id="S3.Thmtheorem1.p2.7.7.m7.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p2.7.7.m7.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.7.7.m7.1b"><apply id="S3.Thmtheorem1.p2.7.7.m7.1.2.cmml" xref="S3.Thmtheorem1.p2.7.7.m7.1.2"><times id="S3.Thmtheorem1.p2.7.7.m7.1.2.1.cmml" xref="S3.Thmtheorem1.p2.7.7.m7.1.2.1"></times><ci id="S3.Thmtheorem1.p2.7.7.m7.1.2.2.cmml" xref="S3.Thmtheorem1.p2.7.7.m7.1.2.2">𝑄</ci><ci id="S3.Thmtheorem1.p2.7.7.m7.1.1.cmml" xref="S3.Thmtheorem1.p2.7.7.m7.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.7.7.m7.1c">Q(D)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.7.7.m7.1d">italic_Q ( italic_D )</annotation></semantics></math> to <math alttext="Q^{\prime}(D^{\prime})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.8.8.m8.1"><semantics id="S3.Thmtheorem1.p2.8.8.m8.1a"><mrow id="S3.Thmtheorem1.p2.8.8.m8.1.1" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.cmml"><msup id="S3.Thmtheorem1.p2.8.8.m8.1.1.3" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.3.cmml"><mi id="S3.Thmtheorem1.p2.8.8.m8.1.1.3.2" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.3.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p2.8.8.m8.1.1.3.3" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.3.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p2.8.8.m8.1.1.2" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.2" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.2.cmml">D</mi><mo id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.3" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.8.8.m8.1b"><apply id="S3.Thmtheorem1.p2.8.8.m8.1.1.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1"><times id="S3.Thmtheorem1.p2.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.2"></times><apply id="S3.Thmtheorem1.p2.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p2.8.8.m8.1.1.3.1.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem1.p2.8.8.m8.1.1.3.2.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.3.2">𝑄</ci><ci id="S3.Thmtheorem1.p2.8.8.m8.1.1.3.3.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.3.3">′</ci></apply><apply id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.2">𝐷</ci><ci id="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem1.p2.8.8.m8.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.8.8.m8.1c">Q^{\prime}(D^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.8.8.m8.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> that can be computed in constant time. This proves that direct access for <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.9.9.m9.1"><semantics id="S3.Thmtheorem1.p2.9.9.m9.1a"><mi id="S3.Thmtheorem1.p2.9.9.m9.1.1" xref="S3.Thmtheorem1.p2.9.9.m9.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.9.9.m9.1b"><ci id="S3.Thmtheorem1.p2.9.9.m9.1.1.cmml" xref="S3.Thmtheorem1.p2.9.9.m9.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.9.9.m9.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.9.9.m9.1d">italic_Q</annotation></semantics></math> implies direct access for <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p2.10.10.m10.1"><semantics id="S3.Thmtheorem1.p2.10.10.m10.1a"><msup id="S3.Thmtheorem1.p2.10.10.m10.1.1" xref="S3.Thmtheorem1.p2.10.10.m10.1.1.cmml"><mi id="S3.Thmtheorem1.p2.10.10.m10.1.1.2" xref="S3.Thmtheorem1.p2.10.10.m10.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p2.10.10.m10.1.1.3" xref="S3.Thmtheorem1.p2.10.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p2.10.10.m10.1b"><apply id="S3.Thmtheorem1.p2.10.10.m10.1.1.cmml" xref="S3.Thmtheorem1.p2.10.10.m10.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p2.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem1.p2.10.10.m10.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p2.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem1.p2.10.10.m10.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p2.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem1.p2.10.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p2.10.10.m10.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p2.10.10.m10.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> with the same time guarantees after loglinear preprocessing.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem1.p3"> <p class="ltx_p" id="S3.Thmtheorem1.p3.15"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p3.15.15">For every atom <math alttext="R(\vec{u})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.1.1.m1.1"><semantics id="S3.Thmtheorem1.p3.1.1.m1.1a"><mrow id="S3.Thmtheorem1.p3.1.1.m1.1.2" xref="S3.Thmtheorem1.p3.1.1.m1.1.2.cmml"><mi id="S3.Thmtheorem1.p3.1.1.m1.1.2.2" xref="S3.Thmtheorem1.p3.1.1.m1.1.2.2.cmml">R</mi><mo id="S3.Thmtheorem1.p3.1.1.m1.1.2.1" xref="S3.Thmtheorem1.p3.1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p3.1.1.m1.1.2.3.2" xref="S3.Thmtheorem1.p3.1.1.m1.1.1.cmml"><mo id="S3.Thmtheorem1.p3.1.1.m1.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p3.1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p3.1.1.m1.1.1" xref="S3.Thmtheorem1.p3.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem1.p3.1.1.m1.1.1.2" xref="S3.Thmtheorem1.p3.1.1.m1.1.1.2.cmml">u</mi><mo id="S3.Thmtheorem1.p3.1.1.m1.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p3.1.1.m1.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p3.1.1.m1.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p3.1.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.1.1.m1.1b"><apply id="S3.Thmtheorem1.p3.1.1.m1.1.2.cmml" xref="S3.Thmtheorem1.p3.1.1.m1.1.2"><times id="S3.Thmtheorem1.p3.1.1.m1.1.2.1.cmml" xref="S3.Thmtheorem1.p3.1.1.m1.1.2.1"></times><ci id="S3.Thmtheorem1.p3.1.1.m1.1.2.2.cmml" xref="S3.Thmtheorem1.p3.1.1.m1.1.2.2">𝑅</ci><apply id="S3.Thmtheorem1.p3.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p3.1.1.m1.1.2.3.2"><ci id="S3.Thmtheorem1.p3.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem1.p3.1.1.m1.1.1.1">→</ci><ci id="S3.Thmtheorem1.p3.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem1.p3.1.1.m1.1.1.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.1.1.m1.1c">R(\vec{u})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.1.1.m1.1d">italic_R ( over→ start_ARG italic_u end_ARG )</annotation></semantics></math> of <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.2.2.m2.1"><semantics id="S3.Thmtheorem1.p3.2.2.m2.1a"><mi id="S3.Thmtheorem1.p3.2.2.m2.1.1" xref="S3.Thmtheorem1.p3.2.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.2.2.m2.1b"><ci id="S3.Thmtheorem1.p3.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p3.2.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.2.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.2.2.m2.1d">italic_Q</annotation></semantics></math>, take an atom <math alttext="R^{\prime}(\vec{v})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.3.3.m3.1"><semantics id="S3.Thmtheorem1.p3.3.3.m3.1a"><mrow id="S3.Thmtheorem1.p3.3.3.m3.1.2" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.cmml"><msup id="S3.Thmtheorem1.p3.3.3.m3.1.2.2" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.2.cmml"><mi id="S3.Thmtheorem1.p3.3.3.m3.1.2.2.2" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.2.2.cmml">R</mi><mo id="S3.Thmtheorem1.p3.3.3.m3.1.2.2.3" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p3.3.3.m3.1.2.1" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p3.3.3.m3.1.2.3.2" xref="S3.Thmtheorem1.p3.3.3.m3.1.1.cmml"><mo id="S3.Thmtheorem1.p3.3.3.m3.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p3.3.3.m3.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p3.3.3.m3.1.1" xref="S3.Thmtheorem1.p3.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem1.p3.3.3.m3.1.1.2" xref="S3.Thmtheorem1.p3.3.3.m3.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem1.p3.3.3.m3.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p3.3.3.m3.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p3.3.3.m3.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p3.3.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.3.3.m3.1b"><apply id="S3.Thmtheorem1.p3.3.3.m3.1.2.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.2"><times id="S3.Thmtheorem1.p3.3.3.m3.1.2.1.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.1"></times><apply id="S3.Thmtheorem1.p3.3.3.m3.1.2.2.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p3.3.3.m3.1.2.2.1.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.2">superscript</csymbol><ci id="S3.Thmtheorem1.p3.3.3.m3.1.2.2.2.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.2.2">𝑅</ci><ci id="S3.Thmtheorem1.p3.3.3.m3.1.2.2.3.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.2.3">′</ci></apply><apply id="S3.Thmtheorem1.p3.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.2.3.2"><ci id="S3.Thmtheorem1.p3.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.1.1">→</ci><ci id="S3.Thmtheorem1.p3.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p3.3.3.m3.1.1.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.3.3.m3.1c">R^{\prime}(\vec{v})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.3.3.m3.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( over→ start_ARG italic_v end_ARG )</annotation></semantics></math> of <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.4.4.m4.1"><semantics id="S3.Thmtheorem1.p3.4.4.m4.1a"><msup id="S3.Thmtheorem1.p3.4.4.m4.1.1" xref="S3.Thmtheorem1.p3.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem1.p3.4.4.m4.1.1.2" xref="S3.Thmtheorem1.p3.4.4.m4.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p3.4.4.m4.1.1.3" xref="S3.Thmtheorem1.p3.4.4.m4.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.4.4.m4.1b"><apply id="S3.Thmtheorem1.p3.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p3.4.4.m4.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p3.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem1.p3.4.4.m4.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p3.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem1.p3.4.4.m4.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p3.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem1.p3.4.4.m4.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.4.4.m4.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.4.4.m4.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> that contains <math alttext="\vec{u}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.5.5.m5.1"><semantics id="S3.Thmtheorem1.p3.5.5.m5.1a"><mover accent="true" id="S3.Thmtheorem1.p3.5.5.m5.1.1" xref="S3.Thmtheorem1.p3.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem1.p3.5.5.m5.1.1.2" xref="S3.Thmtheorem1.p3.5.5.m5.1.1.2.cmml">u</mi><mo id="S3.Thmtheorem1.p3.5.5.m5.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p3.5.5.m5.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.5.5.m5.1b"><apply id="S3.Thmtheorem1.p3.5.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p3.5.5.m5.1.1"><ci id="S3.Thmtheorem1.p3.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem1.p3.5.5.m5.1.1.1">→</ci><ci id="S3.Thmtheorem1.p3.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem1.p3.5.5.m5.1.1.2">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.5.5.m5.1c">\vec{u}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.5.5.m5.1d">over→ start_ARG italic_u end_ARG</annotation></semantics></math> restricted to <math alttext="\mathrm{vars}{(Q^{\prime})}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.6.6.m6.1"><semantics id="S3.Thmtheorem1.p3.6.6.m6.1a"><mrow id="S3.Thmtheorem1.p3.6.6.m6.1.1" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.cmml"><mi id="S3.Thmtheorem1.p3.6.6.m6.1.1.3" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.3.cmml">vars</mi><mo id="S3.Thmtheorem1.p3.6.6.m6.1.1.2" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.2" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.3" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.6.6.m6.1b"><apply id="S3.Thmtheorem1.p3.6.6.m6.1.1.cmml" xref="S3.Thmtheorem1.p3.6.6.m6.1.1"><times id="S3.Thmtheorem1.p3.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.2"></times><ci id="S3.Thmtheorem1.p3.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.3">vars</ci><apply id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem1.p3.6.6.m6.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.6.6.m6.1c">\mathrm{vars}{(Q^{\prime})}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.6.6.m6.1d">roman_vars ( italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math>. Construct a relation for <math alttext="R" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.7.7.m7.1"><semantics id="S3.Thmtheorem1.p3.7.7.m7.1a"><mi id="S3.Thmtheorem1.p3.7.7.m7.1.1" xref="S3.Thmtheorem1.p3.7.7.m7.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.7.7.m7.1b"><ci id="S3.Thmtheorem1.p3.7.7.m7.1.1.cmml" xref="S3.Thmtheorem1.p3.7.7.m7.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.7.7.m7.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.7.7.m7.1d">italic_R</annotation></semantics></math> from <math alttext="R^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.8.8.m8.1"><semantics id="S3.Thmtheorem1.p3.8.8.m8.1a"><msup id="S3.Thmtheorem1.p3.8.8.m8.1.1" xref="S3.Thmtheorem1.p3.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem1.p3.8.8.m8.1.1.2" xref="S3.Thmtheorem1.p3.8.8.m8.1.1.2.cmml">R</mi><mo id="S3.Thmtheorem1.p3.8.8.m8.1.1.3" xref="S3.Thmtheorem1.p3.8.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.8.8.m8.1b"><apply id="S3.Thmtheorem1.p3.8.8.m8.1.1.cmml" xref="S3.Thmtheorem1.p3.8.8.m8.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p3.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem1.p3.8.8.m8.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p3.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem1.p3.8.8.m8.1.1.2">𝑅</ci><ci id="S3.Thmtheorem1.p3.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem1.p3.8.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.8.8.m8.1c">R^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.8.8.m8.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> according to these atoms while padding facts with the constant <math alttext="c" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.9.9.m9.1"><semantics id="S3.Thmtheorem1.p3.9.9.m9.1a"><mi id="S3.Thmtheorem1.p3.9.9.m9.1.1" xref="S3.Thmtheorem1.p3.9.9.m9.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.9.9.m9.1b"><ci id="S3.Thmtheorem1.p3.9.9.m9.1.1.cmml" xref="S3.Thmtheorem1.p3.9.9.m9.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.9.9.m9.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.9.9.m9.1d">italic_c</annotation></semantics></math> where a value is missing. That is, treat every fact <math alttext="R^{\prime}(\vec{a})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.10.10.m10.1"><semantics id="S3.Thmtheorem1.p3.10.10.m10.1a"><mrow id="S3.Thmtheorem1.p3.10.10.m10.1.2" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.cmml"><msup id="S3.Thmtheorem1.p3.10.10.m10.1.2.2" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.2.cmml"><mi id="S3.Thmtheorem1.p3.10.10.m10.1.2.2.2" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.2.2.cmml">R</mi><mo id="S3.Thmtheorem1.p3.10.10.m10.1.2.2.3" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p3.10.10.m10.1.2.1" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p3.10.10.m10.1.2.3.2" xref="S3.Thmtheorem1.p3.10.10.m10.1.1.cmml"><mo id="S3.Thmtheorem1.p3.10.10.m10.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p3.10.10.m10.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p3.10.10.m10.1.1" xref="S3.Thmtheorem1.p3.10.10.m10.1.1.cmml"><mi id="S3.Thmtheorem1.p3.10.10.m10.1.1.2" xref="S3.Thmtheorem1.p3.10.10.m10.1.1.2.cmml">a</mi><mo id="S3.Thmtheorem1.p3.10.10.m10.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p3.10.10.m10.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p3.10.10.m10.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p3.10.10.m10.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.10.10.m10.1b"><apply id="S3.Thmtheorem1.p3.10.10.m10.1.2.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.2"><times id="S3.Thmtheorem1.p3.10.10.m10.1.2.1.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.1"></times><apply id="S3.Thmtheorem1.p3.10.10.m10.1.2.2.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p3.10.10.m10.1.2.2.1.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.2">superscript</csymbol><ci id="S3.Thmtheorem1.p3.10.10.m10.1.2.2.2.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.2.2">𝑅</ci><ci id="S3.Thmtheorem1.p3.10.10.m10.1.2.2.3.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.2.3">′</ci></apply><apply id="S3.Thmtheorem1.p3.10.10.m10.1.1.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.2.3.2"><ci id="S3.Thmtheorem1.p3.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.1.1">→</ci><ci id="S3.Thmtheorem1.p3.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem1.p3.10.10.m10.1.1.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.10.10.m10.1c">R^{\prime}(\vec{a})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.10.10.m10.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( over→ start_ARG italic_a end_ARG )</annotation></semantics></math> as a function <math alttext="f" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.11.11.m11.1"><semantics id="S3.Thmtheorem1.p3.11.11.m11.1a"><mi id="S3.Thmtheorem1.p3.11.11.m11.1.1" xref="S3.Thmtheorem1.p3.11.11.m11.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.11.11.m11.1b"><ci id="S3.Thmtheorem1.p3.11.11.m11.1.1.cmml" xref="S3.Thmtheorem1.p3.11.11.m11.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.11.11.m11.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.11.11.m11.1d">italic_f</annotation></semantics></math> mapping <math alttext="v_{i}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.12.12.m12.1"><semantics id="S3.Thmtheorem1.p3.12.12.m12.1a"><msub id="S3.Thmtheorem1.p3.12.12.m12.1.1" xref="S3.Thmtheorem1.p3.12.12.m12.1.1.cmml"><mi id="S3.Thmtheorem1.p3.12.12.m12.1.1.2" xref="S3.Thmtheorem1.p3.12.12.m12.1.1.2.cmml">v</mi><mi id="S3.Thmtheorem1.p3.12.12.m12.1.1.3" xref="S3.Thmtheorem1.p3.12.12.m12.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.12.12.m12.1b"><apply id="S3.Thmtheorem1.p3.12.12.m12.1.1.cmml" xref="S3.Thmtheorem1.p3.12.12.m12.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p3.12.12.m12.1.1.1.cmml" xref="S3.Thmtheorem1.p3.12.12.m12.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p3.12.12.m12.1.1.2.cmml" xref="S3.Thmtheorem1.p3.12.12.m12.1.1.2">𝑣</ci><ci id="S3.Thmtheorem1.p3.12.12.m12.1.1.3.cmml" xref="S3.Thmtheorem1.p3.12.12.m12.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.12.12.m12.1c">v_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.12.12.m12.1d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="a_{i}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.13.13.m13.1"><semantics id="S3.Thmtheorem1.p3.13.13.m13.1a"><msub id="S3.Thmtheorem1.p3.13.13.m13.1.1" xref="S3.Thmtheorem1.p3.13.13.m13.1.1.cmml"><mi id="S3.Thmtheorem1.p3.13.13.m13.1.1.2" xref="S3.Thmtheorem1.p3.13.13.m13.1.1.2.cmml">a</mi><mi id="S3.Thmtheorem1.p3.13.13.m13.1.1.3" xref="S3.Thmtheorem1.p3.13.13.m13.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.13.13.m13.1b"><apply id="S3.Thmtheorem1.p3.13.13.m13.1.1.cmml" xref="S3.Thmtheorem1.p3.13.13.m13.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p3.13.13.m13.1.1.1.cmml" xref="S3.Thmtheorem1.p3.13.13.m13.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p3.13.13.m13.1.1.2.cmml" xref="S3.Thmtheorem1.p3.13.13.m13.1.1.2">𝑎</ci><ci id="S3.Thmtheorem1.p3.13.13.m13.1.1.3.cmml" xref="S3.Thmtheorem1.p3.13.13.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.13.13.m13.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.13.13.m13.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, extend this function to map all other variables to <math alttext="c" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.14.14.m14.1"><semantics id="S3.Thmtheorem1.p3.14.14.m14.1a"><mi id="S3.Thmtheorem1.p3.14.14.m14.1.1" xref="S3.Thmtheorem1.p3.14.14.m14.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.14.14.m14.1b"><ci id="S3.Thmtheorem1.p3.14.14.m14.1.1.cmml" xref="S3.Thmtheorem1.p3.14.14.m14.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.14.14.m14.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.14.14.m14.1d">italic_c</annotation></semantics></math>, and add the fact <math alttext="R(f(\vec{u}))" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p3.15.15.m15.2"><semantics id="S3.Thmtheorem1.p3.15.15.m15.2a"><mrow id="S3.Thmtheorem1.p3.15.15.m15.2.2" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.cmml"><mi id="S3.Thmtheorem1.p3.15.15.m15.2.2.3" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.3.cmml">R</mi><mo id="S3.Thmtheorem1.p3.15.15.m15.2.2.2" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.cmml"><mi id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.2" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.2.cmml">f</mi><mo id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.1" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.3.2" xref="S3.Thmtheorem1.p3.15.15.m15.1.1.cmml"><mo id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p3.15.15.m15.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p3.15.15.m15.1.1" xref="S3.Thmtheorem1.p3.15.15.m15.1.1.cmml"><mi id="S3.Thmtheorem1.p3.15.15.m15.1.1.2" xref="S3.Thmtheorem1.p3.15.15.m15.1.1.2.cmml">u</mi><mo id="S3.Thmtheorem1.p3.15.15.m15.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p3.15.15.m15.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p3.15.15.m15.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p3.15.15.m15.2b"><apply id="S3.Thmtheorem1.p3.15.15.m15.2.2.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.2.2"><times id="S3.Thmtheorem1.p3.15.15.m15.2.2.2.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.2"></times><ci id="S3.Thmtheorem1.p3.15.15.m15.2.2.3.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.3">𝑅</ci><apply id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1"><times id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.1"></times><ci id="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.2">𝑓</ci><apply id="S3.Thmtheorem1.p3.15.15.m15.1.1.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.2.2.1.1.1.3.2"><ci id="S3.Thmtheorem1.p3.15.15.m15.1.1.1.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.1.1.1">→</ci><ci id="S3.Thmtheorem1.p3.15.15.m15.1.1.2.cmml" xref="S3.Thmtheorem1.p3.15.15.m15.1.1.2">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p3.15.15.m15.2c">R(f(\vec{u}))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p3.15.15.m15.2d">italic_R ( italic_f ( over→ start_ARG italic_u end_ARG ) )</annotation></semantics></math> to the construction.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem1.p4"> <p class="ltx_p" id="S3.Thmtheorem1.p4.12"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p4.12.12">For every atom <math alttext="R^{\prime}(\vec{v})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.1.1.m1.1"><semantics id="S3.Thmtheorem1.p4.1.1.m1.1a"><mrow id="S3.Thmtheorem1.p4.1.1.m1.1.2" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.cmml"><msup id="S3.Thmtheorem1.p4.1.1.m1.1.2.2" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.2.cmml"><mi id="S3.Thmtheorem1.p4.1.1.m1.1.2.2.2" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.2.2.cmml">R</mi><mo id="S3.Thmtheorem1.p4.1.1.m1.1.2.2.3" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p4.1.1.m1.1.2.1" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p4.1.1.m1.1.2.3.2" xref="S3.Thmtheorem1.p4.1.1.m1.1.1.cmml"><mo id="S3.Thmtheorem1.p4.1.1.m1.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p4.1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p4.1.1.m1.1.1" xref="S3.Thmtheorem1.p4.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem1.p4.1.1.m1.1.1.2" xref="S3.Thmtheorem1.p4.1.1.m1.1.1.2.cmml">v</mi><mo id="S3.Thmtheorem1.p4.1.1.m1.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p4.1.1.m1.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p4.1.1.m1.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p4.1.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.1.1.m1.1b"><apply id="S3.Thmtheorem1.p4.1.1.m1.1.2.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.2"><times id="S3.Thmtheorem1.p4.1.1.m1.1.2.1.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.1"></times><apply id="S3.Thmtheorem1.p4.1.1.m1.1.2.2.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p4.1.1.m1.1.2.2.1.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.2">superscript</csymbol><ci id="S3.Thmtheorem1.p4.1.1.m1.1.2.2.2.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.2.2">𝑅</ci><ci id="S3.Thmtheorem1.p4.1.1.m1.1.2.2.3.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.2.3">′</ci></apply><apply id="S3.Thmtheorem1.p4.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.2.3.2"><ci id="S3.Thmtheorem1.p4.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.1.1">→</ci><ci id="S3.Thmtheorem1.p4.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem1.p4.1.1.m1.1.1.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.1.1.m1.1c">R^{\prime}(\vec{v})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.1.1.m1.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( over→ start_ARG italic_v end_ARG )</annotation></semantics></math> of <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.2.2.m2.1"><semantics id="S3.Thmtheorem1.p4.2.2.m2.1a"><msup id="S3.Thmtheorem1.p4.2.2.m2.1.1" xref="S3.Thmtheorem1.p4.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem1.p4.2.2.m2.1.1.2" xref="S3.Thmtheorem1.p4.2.2.m2.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p4.2.2.m2.1.1.3" xref="S3.Thmtheorem1.p4.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.2.2.m2.1b"><apply id="S3.Thmtheorem1.p4.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p4.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p4.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem1.p4.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p4.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem1.p4.2.2.m2.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p4.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem1.p4.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.2.2.m2.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.2.2.m2.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> that was not used in the above process, take an atom <math alttext="R(\vec{u})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.3.3.m3.1"><semantics id="S3.Thmtheorem1.p4.3.3.m3.1a"><mrow id="S3.Thmtheorem1.p4.3.3.m3.1.2" xref="S3.Thmtheorem1.p4.3.3.m3.1.2.cmml"><mi id="S3.Thmtheorem1.p4.3.3.m3.1.2.2" xref="S3.Thmtheorem1.p4.3.3.m3.1.2.2.cmml">R</mi><mo id="S3.Thmtheorem1.p4.3.3.m3.1.2.1" xref="S3.Thmtheorem1.p4.3.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p4.3.3.m3.1.2.3.2" xref="S3.Thmtheorem1.p4.3.3.m3.1.1.cmml"><mo id="S3.Thmtheorem1.p4.3.3.m3.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p4.3.3.m3.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p4.3.3.m3.1.1" xref="S3.Thmtheorem1.p4.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem1.p4.3.3.m3.1.1.2" xref="S3.Thmtheorem1.p4.3.3.m3.1.1.2.cmml">u</mi><mo id="S3.Thmtheorem1.p4.3.3.m3.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p4.3.3.m3.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p4.3.3.m3.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p4.3.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.3.3.m3.1b"><apply id="S3.Thmtheorem1.p4.3.3.m3.1.2.cmml" xref="S3.Thmtheorem1.p4.3.3.m3.1.2"><times id="S3.Thmtheorem1.p4.3.3.m3.1.2.1.cmml" xref="S3.Thmtheorem1.p4.3.3.m3.1.2.1"></times><ci id="S3.Thmtheorem1.p4.3.3.m3.1.2.2.cmml" xref="S3.Thmtheorem1.p4.3.3.m3.1.2.2">𝑅</ci><apply id="S3.Thmtheorem1.p4.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p4.3.3.m3.1.2.3.2"><ci id="S3.Thmtheorem1.p4.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p4.3.3.m3.1.1.1">→</ci><ci id="S3.Thmtheorem1.p4.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p4.3.3.m3.1.1.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.3.3.m3.1c">R(\vec{u})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.3.3.m3.1d">italic_R ( over→ start_ARG italic_u end_ARG )</annotation></semantics></math> of <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.4.4.m4.1"><semantics id="S3.Thmtheorem1.p4.4.4.m4.1a"><mi id="S3.Thmtheorem1.p4.4.4.m4.1.1" xref="S3.Thmtheorem1.p4.4.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.4.4.m4.1b"><ci id="S3.Thmtheorem1.p4.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p4.4.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.4.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.4.4.m4.1d">italic_Q</annotation></semantics></math> that contains its variables, and filter its relation in <math alttext="D" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.5.5.m5.1"><semantics id="S3.Thmtheorem1.p4.5.5.m5.1a"><mi id="S3.Thmtheorem1.p4.5.5.m5.1.1" xref="S3.Thmtheorem1.p4.5.5.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.5.5.m5.1b"><ci id="S3.Thmtheorem1.p4.5.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p4.5.5.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.5.5.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.5.5.m5.1d">italic_D</annotation></semantics></math> according to <math alttext="R^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.6.6.m6.1"><semantics id="S3.Thmtheorem1.p4.6.6.m6.1a"><msup id="S3.Thmtheorem1.p4.6.6.m6.1.1" xref="S3.Thmtheorem1.p4.6.6.m6.1.1.cmml"><mi id="S3.Thmtheorem1.p4.6.6.m6.1.1.2" xref="S3.Thmtheorem1.p4.6.6.m6.1.1.2.cmml">R</mi><mo id="S3.Thmtheorem1.p4.6.6.m6.1.1.3" xref="S3.Thmtheorem1.p4.6.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.6.6.m6.1b"><apply id="S3.Thmtheorem1.p4.6.6.m6.1.1.cmml" xref="S3.Thmtheorem1.p4.6.6.m6.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p4.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem1.p4.6.6.m6.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p4.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem1.p4.6.6.m6.1.1.2">𝑅</ci><ci id="S3.Thmtheorem1.p4.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem1.p4.6.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.6.6.m6.1c">R^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.6.6.m6.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. That is, for every fact <math alttext="R(\vec{a})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.7.7.m7.1"><semantics id="S3.Thmtheorem1.p4.7.7.m7.1a"><mrow id="S3.Thmtheorem1.p4.7.7.m7.1.2" xref="S3.Thmtheorem1.p4.7.7.m7.1.2.cmml"><mi id="S3.Thmtheorem1.p4.7.7.m7.1.2.2" xref="S3.Thmtheorem1.p4.7.7.m7.1.2.2.cmml">R</mi><mo id="S3.Thmtheorem1.p4.7.7.m7.1.2.1" xref="S3.Thmtheorem1.p4.7.7.m7.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p4.7.7.m7.1.2.3.2" xref="S3.Thmtheorem1.p4.7.7.m7.1.1.cmml"><mo id="S3.Thmtheorem1.p4.7.7.m7.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p4.7.7.m7.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p4.7.7.m7.1.1" xref="S3.Thmtheorem1.p4.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem1.p4.7.7.m7.1.1.2" xref="S3.Thmtheorem1.p4.7.7.m7.1.1.2.cmml">a</mi><mo id="S3.Thmtheorem1.p4.7.7.m7.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p4.7.7.m7.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p4.7.7.m7.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p4.7.7.m7.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.7.7.m7.1b"><apply id="S3.Thmtheorem1.p4.7.7.m7.1.2.cmml" xref="S3.Thmtheorem1.p4.7.7.m7.1.2"><times id="S3.Thmtheorem1.p4.7.7.m7.1.2.1.cmml" xref="S3.Thmtheorem1.p4.7.7.m7.1.2.1"></times><ci id="S3.Thmtheorem1.p4.7.7.m7.1.2.2.cmml" xref="S3.Thmtheorem1.p4.7.7.m7.1.2.2">𝑅</ci><apply id="S3.Thmtheorem1.p4.7.7.m7.1.1.cmml" xref="S3.Thmtheorem1.p4.7.7.m7.1.2.3.2"><ci id="S3.Thmtheorem1.p4.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem1.p4.7.7.m7.1.1.1">→</ci><ci id="S3.Thmtheorem1.p4.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem1.p4.7.7.m7.1.1.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.7.7.m7.1c">R(\vec{a})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.7.7.m7.1d">italic_R ( over→ start_ARG italic_a end_ARG )</annotation></semantics></math> in our constructed instance, treat it as a function <math alttext="f" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.8.8.m8.1"><semantics id="S3.Thmtheorem1.p4.8.8.m8.1a"><mi id="S3.Thmtheorem1.p4.8.8.m8.1.1" xref="S3.Thmtheorem1.p4.8.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.8.8.m8.1b"><ci id="S3.Thmtheorem1.p4.8.8.m8.1.1.cmml" xref="S3.Thmtheorem1.p4.8.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.8.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.8.8.m8.1d">italic_f</annotation></semantics></math> mapping <math alttext="v_{i}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.9.9.m9.1"><semantics id="S3.Thmtheorem1.p4.9.9.m9.1a"><msub id="S3.Thmtheorem1.p4.9.9.m9.1.1" xref="S3.Thmtheorem1.p4.9.9.m9.1.1.cmml"><mi id="S3.Thmtheorem1.p4.9.9.m9.1.1.2" xref="S3.Thmtheorem1.p4.9.9.m9.1.1.2.cmml">v</mi><mi id="S3.Thmtheorem1.p4.9.9.m9.1.1.3" xref="S3.Thmtheorem1.p4.9.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.9.9.m9.1b"><apply id="S3.Thmtheorem1.p4.9.9.m9.1.1.cmml" xref="S3.Thmtheorem1.p4.9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p4.9.9.m9.1.1.1.cmml" xref="S3.Thmtheorem1.p4.9.9.m9.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p4.9.9.m9.1.1.2.cmml" xref="S3.Thmtheorem1.p4.9.9.m9.1.1.2">𝑣</ci><ci id="S3.Thmtheorem1.p4.9.9.m9.1.1.3.cmml" xref="S3.Thmtheorem1.p4.9.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.9.9.m9.1c">v_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.9.9.m9.1d">italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="a_{i}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.10.10.m10.1"><semantics id="S3.Thmtheorem1.p4.10.10.m10.1a"><msub id="S3.Thmtheorem1.p4.10.10.m10.1.1" xref="S3.Thmtheorem1.p4.10.10.m10.1.1.cmml"><mi id="S3.Thmtheorem1.p4.10.10.m10.1.1.2" xref="S3.Thmtheorem1.p4.10.10.m10.1.1.2.cmml">a</mi><mi id="S3.Thmtheorem1.p4.10.10.m10.1.1.3" xref="S3.Thmtheorem1.p4.10.10.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.10.10.m10.1b"><apply id="S3.Thmtheorem1.p4.10.10.m10.1.1.cmml" xref="S3.Thmtheorem1.p4.10.10.m10.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p4.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem1.p4.10.10.m10.1.1">subscript</csymbol><ci id="S3.Thmtheorem1.p4.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem1.p4.10.10.m10.1.1.2">𝑎</ci><ci id="S3.Thmtheorem1.p4.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem1.p4.10.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.10.10.m10.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.10.10.m10.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, and check whether there is a corresponding fact <math alttext="R^{\prime}(f(\vec{u}))" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.11.11.m11.2"><semantics id="S3.Thmtheorem1.p4.11.11.m11.2a"><mrow id="S3.Thmtheorem1.p4.11.11.m11.2.2" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.cmml"><msup id="S3.Thmtheorem1.p4.11.11.m11.2.2.3" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.3.cmml"><mi id="S3.Thmtheorem1.p4.11.11.m11.2.2.3.2" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.3.2.cmml">R</mi><mo id="S3.Thmtheorem1.p4.11.11.m11.2.2.3.3" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.3.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p4.11.11.m11.2.2.2" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.cmml"><mi id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.2" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.2.cmml">f</mi><mo id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.1" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.3.2" xref="S3.Thmtheorem1.p4.11.11.m11.1.1.cmml"><mo id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p4.11.11.m11.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p4.11.11.m11.1.1" xref="S3.Thmtheorem1.p4.11.11.m11.1.1.cmml"><mi id="S3.Thmtheorem1.p4.11.11.m11.1.1.2" xref="S3.Thmtheorem1.p4.11.11.m11.1.1.2.cmml">u</mi><mo id="S3.Thmtheorem1.p4.11.11.m11.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p4.11.11.m11.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p4.11.11.m11.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.11.11.m11.2b"><apply id="S3.Thmtheorem1.p4.11.11.m11.2.2.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2"><times id="S3.Thmtheorem1.p4.11.11.m11.2.2.2.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.2"></times><apply id="S3.Thmtheorem1.p4.11.11.m11.2.2.3.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p4.11.11.m11.2.2.3.1.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.3">superscript</csymbol><ci id="S3.Thmtheorem1.p4.11.11.m11.2.2.3.2.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.3.2">𝑅</ci><ci id="S3.Thmtheorem1.p4.11.11.m11.2.2.3.3.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.3.3">′</ci></apply><apply id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1"><times id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.1"></times><ci id="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.2">𝑓</ci><apply id="S3.Thmtheorem1.p4.11.11.m11.1.1.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.2.2.1.1.1.3.2"><ci id="S3.Thmtheorem1.p4.11.11.m11.1.1.1.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.1.1.1">→</ci><ci id="S3.Thmtheorem1.p4.11.11.m11.1.1.2.cmml" xref="S3.Thmtheorem1.p4.11.11.m11.1.1.2">𝑢</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.11.11.m11.2c">R^{\prime}(f(\vec{u}))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.11.11.m11.2d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_f ( over→ start_ARG italic_u end_ARG ) )</annotation></semantics></math> in the input; if none exist, then remove the fact <math alttext="R(\vec{a})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p4.12.12.m12.1"><semantics id="S3.Thmtheorem1.p4.12.12.m12.1a"><mrow id="S3.Thmtheorem1.p4.12.12.m12.1.2" xref="S3.Thmtheorem1.p4.12.12.m12.1.2.cmml"><mi id="S3.Thmtheorem1.p4.12.12.m12.1.2.2" xref="S3.Thmtheorem1.p4.12.12.m12.1.2.2.cmml">R</mi><mo id="S3.Thmtheorem1.p4.12.12.m12.1.2.1" xref="S3.Thmtheorem1.p4.12.12.m12.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p4.12.12.m12.1.2.3.2" xref="S3.Thmtheorem1.p4.12.12.m12.1.1.cmml"><mo id="S3.Thmtheorem1.p4.12.12.m12.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p4.12.12.m12.1.1.cmml">(</mo><mover accent="true" id="S3.Thmtheorem1.p4.12.12.m12.1.1" xref="S3.Thmtheorem1.p4.12.12.m12.1.1.cmml"><mi id="S3.Thmtheorem1.p4.12.12.m12.1.1.2" xref="S3.Thmtheorem1.p4.12.12.m12.1.1.2.cmml">a</mi><mo id="S3.Thmtheorem1.p4.12.12.m12.1.1.1" stretchy="false" xref="S3.Thmtheorem1.p4.12.12.m12.1.1.1.cmml">→</mo></mover><mo id="S3.Thmtheorem1.p4.12.12.m12.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p4.12.12.m12.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p4.12.12.m12.1b"><apply id="S3.Thmtheorem1.p4.12.12.m12.1.2.cmml" xref="S3.Thmtheorem1.p4.12.12.m12.1.2"><times id="S3.Thmtheorem1.p4.12.12.m12.1.2.1.cmml" xref="S3.Thmtheorem1.p4.12.12.m12.1.2.1"></times><ci id="S3.Thmtheorem1.p4.12.12.m12.1.2.2.cmml" xref="S3.Thmtheorem1.p4.12.12.m12.1.2.2">𝑅</ci><apply id="S3.Thmtheorem1.p4.12.12.m12.1.1.cmml" xref="S3.Thmtheorem1.p4.12.12.m12.1.2.3.2"><ci id="S3.Thmtheorem1.p4.12.12.m12.1.1.1.cmml" xref="S3.Thmtheorem1.p4.12.12.m12.1.1.1">→</ci><ci id="S3.Thmtheorem1.p4.12.12.m12.1.1.2.cmml" xref="S3.Thmtheorem1.p4.12.12.m12.1.1.2">𝑎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p4.12.12.m12.1c">R(\vec{a})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p4.12.12.m12.1d">italic_R ( over→ start_ARG italic_a end_ARG )</annotation></semantics></math> from the construction.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem1.p5"> <p class="ltx_p" id="S3.Thmtheorem1.p5.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p5.5.5">In case the queries are CQ<sup class="ltx_sup" id="S3.Thmtheorem1.p5.5.5.1"><span class="ltx_text ltx_font_upright" id="S3.Thmtheorem1.p5.5.5.1.1">⋆</span></sup>s, the database <math alttext="D^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p5.2.2.m2.1"><semantics id="S3.Thmtheorem1.p5.2.2.m2.1a"><msup id="S3.Thmtheorem1.p5.2.2.m2.1.1" xref="S3.Thmtheorem1.p5.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem1.p5.2.2.m2.1.1.2" xref="S3.Thmtheorem1.p5.2.2.m2.1.1.2.cmml">D</mi><mo id="S3.Thmtheorem1.p5.2.2.m2.1.1.3" xref="S3.Thmtheorem1.p5.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p5.2.2.m2.1b"><apply id="S3.Thmtheorem1.p5.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p5.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p5.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem1.p5.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p5.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem1.p5.2.2.m2.1.1.2">𝐷</ci><ci id="S3.Thmtheorem1.p5.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem1.p5.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p5.2.2.m2.1c">D^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p5.2.2.m2.1d">italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is accompanied by the annotations <math alttext="\tau^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p5.3.3.m3.1"><semantics id="S3.Thmtheorem1.p5.3.3.m3.1a"><msup id="S3.Thmtheorem1.p5.3.3.m3.1.1" xref="S3.Thmtheorem1.p5.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem1.p5.3.3.m3.1.1.2" xref="S3.Thmtheorem1.p5.3.3.m3.1.1.2.cmml">τ</mi><mo id="S3.Thmtheorem1.p5.3.3.m3.1.1.3" xref="S3.Thmtheorem1.p5.3.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p5.3.3.m3.1b"><apply id="S3.Thmtheorem1.p5.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p5.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p5.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p5.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p5.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p5.3.3.m3.1.1.2">𝜏</ci><ci id="S3.Thmtheorem1.p5.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem1.p5.3.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p5.3.3.m3.1c">\tau^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p5.3.3.m3.1d">italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and we construct the annotations <math alttext="\tau" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p5.4.4.m4.1"><semantics id="S3.Thmtheorem1.p5.4.4.m4.1a"><mi id="S3.Thmtheorem1.p5.4.4.m4.1.1" xref="S3.Thmtheorem1.p5.4.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p5.4.4.m4.1b"><ci id="S3.Thmtheorem1.p5.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p5.4.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p5.4.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p5.4.4.m4.1d">italic_τ</annotation></semantics></math> for <math alttext="D" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p5.5.5.m5.1"><semantics id="S3.Thmtheorem1.p5.5.5.m5.1a"><mi id="S3.Thmtheorem1.p5.5.5.m5.1.1" xref="S3.Thmtheorem1.p5.5.5.m5.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p5.5.5.m5.1b"><ci id="S3.Thmtheorem1.p5.5.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p5.5.5.m5.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p5.5.5.m5.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p5.5.5.m5.1d">italic_D</annotation></semantics></math> as follows:</span></p> <ol class="ltx_enumerate" id="S3.I4"> <li class="ltx_item" id="S3.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S3.I4.i1.p1"> <p class="ltx_p" id="S3.I4.i1.p1.1"><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.1.1">We first annotate all facts in the constructed instance with </span><math alttext="\mathord{\bar{1}}" class="ltx_Math" display="inline" id="S3.I4.i1.p1.1.m1.1"><semantics id="S3.I4.i1.p1.1.m1.1a"><mover accent="true" id="S3.I4.i1.p1.1.m1.1.1" xref="S3.I4.i1.p1.1.m1.1.1a.cmml"><mn id="S3.I4.i1.p1.1.m1.1.1.2" xref="S3.I4.i1.p1.1.m1.1.1.2.cmml">1</mn><mo id="S3.I4.i1.p1.1.m1.1.1.1" xref="S3.I4.i1.p1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.I4.i1.p1.1.m1.1b"><ci id="S3.I4.i1.p1.1.m1.1.1a.cmml" xref="S3.I4.i1.p1.1.m1.1.1"><mover accent="true" id="S3.I4.i1.p1.1.m1.1.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1"><mn id="S3.I4.i1.p1.1.m1.1.1.2.cmml" xref="S3.I4.i1.p1.1.m1.1.1.2">1</mn><mo id="S3.I4.i1.p1.1.m1.1.1.1.cmml" xref="S3.I4.i1.p1.1.m1.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i1.p1.1.m1.1c">\mathord{\bar{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i1.p1.1.m1.1d">start_ID over¯ start_ARG 1 end_ARG end_ID</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i1.p1.1.2">.</span></p> </div> </li> <li class="ltx_item" id="S3.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S3.I4.i2.p1"> <p class="ltx_p" id="S3.I4.i2.p1.7"><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.7.1">Then, for every atom </span><math alttext="R^{\prime}(\vec{v})" class="ltx_Math" display="inline" id="S3.I4.i2.p1.1.m1.1"><semantics id="S3.I4.i2.p1.1.m1.1a"><mrow id="S3.I4.i2.p1.1.m1.1.2" xref="S3.I4.i2.p1.1.m1.1.2.cmml"><msup id="S3.I4.i2.p1.1.m1.1.2.2" xref="S3.I4.i2.p1.1.m1.1.2.2.cmml"><mi id="S3.I4.i2.p1.1.m1.1.2.2.2" xref="S3.I4.i2.p1.1.m1.1.2.2.2.cmml">R</mi><mo id="S3.I4.i2.p1.1.m1.1.2.2.3" xref="S3.I4.i2.p1.1.m1.1.2.2.3.cmml">′</mo></msup><mo id="S3.I4.i2.p1.1.m1.1.2.1" xref="S3.I4.i2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.1.m1.1.2.3.2" xref="S3.I4.i2.p1.1.m1.1.1.cmml"><mo id="S3.I4.i2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S3.I4.i2.p1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S3.I4.i2.p1.1.m1.1.1" xref="S3.I4.i2.p1.1.m1.1.1.cmml"><mi id="S3.I4.i2.p1.1.m1.1.1.2" xref="S3.I4.i2.p1.1.m1.1.1.2.cmml">v</mi><mo id="S3.I4.i2.p1.1.m1.1.1.1" stretchy="false" xref="S3.I4.i2.p1.1.m1.1.1.1.cmml">→</mo></mover><mo id="S3.I4.i2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S3.I4.i2.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.1.m1.1b"><apply id="S3.I4.i2.p1.1.m1.1.2.cmml" xref="S3.I4.i2.p1.1.m1.1.2"><times id="S3.I4.i2.p1.1.m1.1.2.1.cmml" xref="S3.I4.i2.p1.1.m1.1.2.1"></times><apply id="S3.I4.i2.p1.1.m1.1.2.2.cmml" xref="S3.I4.i2.p1.1.m1.1.2.2"><csymbol cd="ambiguous" id="S3.I4.i2.p1.1.m1.1.2.2.1.cmml" xref="S3.I4.i2.p1.1.m1.1.2.2">superscript</csymbol><ci id="S3.I4.i2.p1.1.m1.1.2.2.2.cmml" xref="S3.I4.i2.p1.1.m1.1.2.2.2">𝑅</ci><ci id="S3.I4.i2.p1.1.m1.1.2.2.3.cmml" xref="S3.I4.i2.p1.1.m1.1.2.2.3">′</ci></apply><apply id="S3.I4.i2.p1.1.m1.1.1.cmml" xref="S3.I4.i2.p1.1.m1.1.2.3.2"><ci id="S3.I4.i2.p1.1.m1.1.1.1.cmml" xref="S3.I4.i2.p1.1.m1.1.1.1">→</ci><ci id="S3.I4.i2.p1.1.m1.1.1.2.cmml" xref="S3.I4.i2.p1.1.m1.1.1.2">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.1.m1.1c">R^{\prime}(\vec{v})</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.1.m1.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( over→ start_ARG italic_v end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.7.2"> in </span><math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.2.m2.1"><semantics id="S3.I4.i2.p1.2.m2.1a"><msup id="S3.I4.i2.p1.2.m2.1.1" xref="S3.I4.i2.p1.2.m2.1.1.cmml"><mi id="S3.I4.i2.p1.2.m2.1.1.2" xref="S3.I4.i2.p1.2.m2.1.1.2.cmml">Q</mi><mo id="S3.I4.i2.p1.2.m2.1.1.3" xref="S3.I4.i2.p1.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.2.m2.1b"><apply id="S3.I4.i2.p1.2.m2.1.1.cmml" xref="S3.I4.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I4.i2.p1.2.m2.1.1.1.cmml" xref="S3.I4.i2.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I4.i2.p1.2.m2.1.1.2.cmml" xref="S3.I4.i2.p1.2.m2.1.1.2">𝑄</ci><ci id="S3.I4.i2.p1.2.m2.1.1.3.cmml" xref="S3.I4.i2.p1.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.2.m2.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.2.m2.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.7.3">, we take an atom </span><math alttext="R(\vec{u})" class="ltx_Math" display="inline" id="S3.I4.i2.p1.3.m3.1"><semantics id="S3.I4.i2.p1.3.m3.1a"><mrow id="S3.I4.i2.p1.3.m3.1.2" xref="S3.I4.i2.p1.3.m3.1.2.cmml"><mi id="S3.I4.i2.p1.3.m3.1.2.2" xref="S3.I4.i2.p1.3.m3.1.2.2.cmml">R</mi><mo id="S3.I4.i2.p1.3.m3.1.2.1" xref="S3.I4.i2.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.I4.i2.p1.3.m3.1.2.3.2" xref="S3.I4.i2.p1.3.m3.1.1.cmml"><mo id="S3.I4.i2.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S3.I4.i2.p1.3.m3.1.1.cmml">(</mo><mover accent="true" id="S3.I4.i2.p1.3.m3.1.1" xref="S3.I4.i2.p1.3.m3.1.1.cmml"><mi id="S3.I4.i2.p1.3.m3.1.1.2" xref="S3.I4.i2.p1.3.m3.1.1.2.cmml">u</mi><mo id="S3.I4.i2.p1.3.m3.1.1.1" stretchy="false" xref="S3.I4.i2.p1.3.m3.1.1.1.cmml">→</mo></mover><mo id="S3.I4.i2.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S3.I4.i2.p1.3.m3.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.3.m3.1b"><apply id="S3.I4.i2.p1.3.m3.1.2.cmml" xref="S3.I4.i2.p1.3.m3.1.2"><times id="S3.I4.i2.p1.3.m3.1.2.1.cmml" xref="S3.I4.i2.p1.3.m3.1.2.1"></times><ci id="S3.I4.i2.p1.3.m3.1.2.2.cmml" xref="S3.I4.i2.p1.3.m3.1.2.2">𝑅</ci><apply id="S3.I4.i2.p1.3.m3.1.1.cmml" xref="S3.I4.i2.p1.3.m3.1.2.3.2"><ci id="S3.I4.i2.p1.3.m3.1.1.1.cmml" xref="S3.I4.i2.p1.3.m3.1.1.1">→</ci><ci id="S3.I4.i2.p1.3.m3.1.1.2.cmml" xref="S3.I4.i2.p1.3.m3.1.1.2">𝑢</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.3.m3.1c">R(\vec{u})</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.3.m3.1d">italic_R ( over→ start_ARG italic_u end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.7.4"> in </span><math alttext="Q" 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">Q</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">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.4.m4.1d">italic_Q</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.7.5"> containing its variables, and multiply the annotation of every fact in </span><math alttext="R" class="ltx_Math" display="inline" id="S3.I4.i2.p1.5.m5.1"><semantics id="S3.I4.i2.p1.5.m5.1a"><mi id="S3.I4.i2.p1.5.m5.1.1" xref="S3.I4.i2.p1.5.m5.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.5.m5.1b"><ci id="S3.I4.i2.p1.5.m5.1.1.cmml" xref="S3.I4.i2.p1.5.m5.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.5.m5.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.5.m5.1d">italic_R</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.7.6"> with the annotation of the corresponding fact in </span><math alttext="R^{\prime}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.6.m6.1"><semantics id="S3.I4.i2.p1.6.m6.1a"><msup 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">R</mi><mo id="S3.I4.i2.p1.6.m6.1.1.3" xref="S3.I4.i2.p1.6.m6.1.1.3.cmml">′</mo></msup><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="ambiguous" id="S3.I4.i2.p1.6.m6.1.1.1.cmml" xref="S3.I4.i2.p1.6.m6.1.1">superscript</csymbol><ci id="S3.I4.i2.p1.6.m6.1.1.2.cmml" xref="S3.I4.i2.p1.6.m6.1.1.2">𝑅</ci><ci id="S3.I4.i2.p1.6.m6.1.1.3.cmml" xref="S3.I4.i2.p1.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.6.m6.1c">R^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.6.m6.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.7.7"> (if a corresponding fact exists; if none exist, then this fact of </span><math alttext="R^{\prime}" class="ltx_Math" display="inline" id="S3.I4.i2.p1.7.m7.1"><semantics id="S3.I4.i2.p1.7.m7.1a"><msup id="S3.I4.i2.p1.7.m7.1.1" xref="S3.I4.i2.p1.7.m7.1.1.cmml"><mi id="S3.I4.i2.p1.7.m7.1.1.2" xref="S3.I4.i2.p1.7.m7.1.1.2.cmml">R</mi><mo id="S3.I4.i2.p1.7.m7.1.1.3" xref="S3.I4.i2.p1.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.I4.i2.p1.7.m7.1b"><apply id="S3.I4.i2.p1.7.m7.1.1.cmml" xref="S3.I4.i2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.I4.i2.p1.7.m7.1.1.1.cmml" xref="S3.I4.i2.p1.7.m7.1.1">superscript</csymbol><ci id="S3.I4.i2.p1.7.m7.1.1.2.cmml" xref="S3.I4.i2.p1.7.m7.1.1.2">𝑅</ci><ci id="S3.I4.i2.p1.7.m7.1.1.3.cmml" xref="S3.I4.i2.p1.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I4.i2.p1.7.m7.1c">R^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.I4.i2.p1.7.m7.1d">italic_R start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I4.i2.p1.7.8"> would not be used in an answer, and the annotation we choose for this fact does not matter).</span></p> </div> </li> </ol> <p class="ltx_p" id="S3.Thmtheorem1.p5.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p5.6.1">Note that the construction uses a linear number of operations. As the semiring multiplication takes logarithmic time, and the other operations take constant time, the construction takes <math alttext="O(|D^{\prime}|\log|D^{\prime}|)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p5.6.1.m1.1"><semantics id="S3.Thmtheorem1.p5.6.1.m1.1a"><mrow id="S3.Thmtheorem1.p5.6.1.m1.1.1" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.cmml"><mi id="S3.Thmtheorem1.p5.6.1.m1.1.1.3" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.3.cmml">O</mi><mo id="S3.Thmtheorem1.p5.6.1.m1.1.1.2" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.cmml"><mrow id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.2.cmml"><mo id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.2.1.cmml">|</mo><msup id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.1" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.1.2" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.1.2.cmml">D</mi><mo id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.1.3" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.3" lspace="0.167em" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.3.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.cmml"><mi id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.2" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.2.cmml">log</mi><mo id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2a" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.cmml">⁡</mo><mrow 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id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.1.1.1.3">′</ci></apply></apply><apply id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2"><log id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.2.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.2"></log><apply id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.2.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1"><abs id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.2.1.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.2"></abs><apply id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.1.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.1.1.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.1.2.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.1.2">𝐷</ci><ci id="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.1.3.cmml" xref="S3.Thmtheorem1.p5.6.1.m1.1.1.1.1.1.2.1.1.1.3">′</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p5.6.1.m1.1c">O(|D^{\prime}|\log|D^{\prime}|)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p5.6.1.m1.1d">italic_O ( | italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | roman_log | italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | )</annotation></semantics></math> time in total.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem1.p6"> <p class="ltx_p" id="S3.Thmtheorem1.p6.12"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p6.12.12">We claim that there is a bijection between homomorphisms from <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.1.1.m1.1"><semantics id="S3.Thmtheorem1.p6.1.1.m1.1a"><msup id="S3.Thmtheorem1.p6.1.1.m1.1.1" xref="S3.Thmtheorem1.p6.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem1.p6.1.1.m1.1.1.2" xref="S3.Thmtheorem1.p6.1.1.m1.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p6.1.1.m1.1.1.3" xref="S3.Thmtheorem1.p6.1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.1.1.m1.1b"><apply id="S3.Thmtheorem1.p6.1.1.m1.1.1.cmml" xref="S3.Thmtheorem1.p6.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p6.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem1.p6.1.1.m1.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p6.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem1.p6.1.1.m1.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p6.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem1.p6.1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.1.1.m1.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.1.1.m1.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="D^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.2.2.m2.1"><semantics id="S3.Thmtheorem1.p6.2.2.m2.1a"><msup id="S3.Thmtheorem1.p6.2.2.m2.1.1" xref="S3.Thmtheorem1.p6.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem1.p6.2.2.m2.1.1.2" xref="S3.Thmtheorem1.p6.2.2.m2.1.1.2.cmml">D</mi><mo id="S3.Thmtheorem1.p6.2.2.m2.1.1.3" xref="S3.Thmtheorem1.p6.2.2.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.2.2.m2.1b"><apply id="S3.Thmtheorem1.p6.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p6.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p6.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem1.p6.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p6.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem1.p6.2.2.m2.1.1.2">𝐷</ci><ci id="S3.Thmtheorem1.p6.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem1.p6.2.2.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.2.2.m2.1c">D^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.2.2.m2.1d">italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and homomorphisms from <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.3.3.m3.1"><semantics id="S3.Thmtheorem1.p6.3.3.m3.1a"><mi id="S3.Thmtheorem1.p6.3.3.m3.1.1" xref="S3.Thmtheorem1.p6.3.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.3.3.m3.1b"><ci id="S3.Thmtheorem1.p6.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p6.3.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.3.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.3.3.m3.1d">italic_Q</annotation></semantics></math> to <math alttext="D" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.4.4.m4.1"><semantics id="S3.Thmtheorem1.p6.4.4.m4.1a"><mi id="S3.Thmtheorem1.p6.4.4.m4.1.1" xref="S3.Thmtheorem1.p6.4.4.m4.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.4.4.m4.1b"><ci id="S3.Thmtheorem1.p6.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p6.4.4.m4.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.4.4.m4.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.4.4.m4.1d">italic_D</annotation></semantics></math> given by extending the homomorphism by mapping all other variables to <math alttext="c" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.5.5.m5.1"><semantics id="S3.Thmtheorem1.p6.5.5.m5.1a"><mi id="S3.Thmtheorem1.p6.5.5.m5.1.1" xref="S3.Thmtheorem1.p6.5.5.m5.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.5.5.m5.1b"><ci id="S3.Thmtheorem1.p6.5.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p6.5.5.m5.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.5.5.m5.1c">c</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.5.5.m5.1d">italic_c</annotation></semantics></math>. In addition, in case the queries are CQ<sup class="ltx_sup" id="S3.Thmtheorem1.p6.12.12.1"><span class="ltx_text ltx_font_upright" id="S3.Thmtheorem1.p6.12.12.1.1">⋆</span></sup>s, multiplying all annotations of facts matching the source homomorphism and doing the same for the target homomorphism gives the same result. In case the queries are AggCQs, this implies that the aggregation is applied to the same set of assignments, and therefore the aggregates computed in both cases are the same. Together with the fact that <math alttext="Q_{|\mathrm{vars}{(Q^{\prime})}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem1.p6.7.7.m7.1"><semantics id="S3.Thmtheorem1.p6.7.7.m7.1a"><msub id="S3.Thmtheorem1.p6.7.7.m7.1.1"><mi id="S3.Thmtheorem1.p6.7.7.m7.1.1.2">Q</mi><mrow id="S3.Thmtheorem1.p6.7.7.m7.1.1.3"><mo fence="false" id="S3.Thmtheorem1.p6.7.7.m7.1.1.3.1" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem1.p6.7.7.m7.1.1.3.2">vars</mi><mrow id="S3.Thmtheorem1.p6.7.7.m7.1.1.3.3"><mo id="S3.Thmtheorem1.p6.7.7.m7.1.1.3.3.1" stretchy="false">(</mo><msup id="S3.Thmtheorem1.p6.7.7.m7.1.1.3.3.2"><mi id="S3.Thmtheorem1.p6.7.7.m7.1.1.3.3.2.2">Q</mi><mo id="S3.Thmtheorem1.p6.7.7.m7.1.1.3.3.2.3">′</mo></msup><mo id="S3.Thmtheorem1.p6.7.7.m7.1.1.3.3.3" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.7.7.m7.1b">Q_{|\mathrm{vars}{(Q^{\prime})}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.7.7.m7.1c">italic_Q start_POSTSUBSCRIPT | roman_vars ( italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.8.8.m8.1"><semantics id="S3.Thmtheorem1.p6.8.8.m8.1a"><msup id="S3.Thmtheorem1.p6.8.8.m8.1.1" xref="S3.Thmtheorem1.p6.8.8.m8.1.1.cmml"><mi id="S3.Thmtheorem1.p6.8.8.m8.1.1.2" xref="S3.Thmtheorem1.p6.8.8.m8.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p6.8.8.m8.1.1.3" xref="S3.Thmtheorem1.p6.8.8.m8.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.8.8.m8.1b"><apply id="S3.Thmtheorem1.p6.8.8.m8.1.1.cmml" xref="S3.Thmtheorem1.p6.8.8.m8.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p6.8.8.m8.1.1.1.cmml" xref="S3.Thmtheorem1.p6.8.8.m8.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p6.8.8.m8.1.1.2.cmml" xref="S3.Thmtheorem1.p6.8.8.m8.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p6.8.8.m8.1.1.3.cmml" xref="S3.Thmtheorem1.p6.8.8.m8.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.8.8.m8.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.8.8.m8.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> have the same heads, this means that projecting <math alttext="Q(D)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.9.9.m9.1"><semantics id="S3.Thmtheorem1.p6.9.9.m9.1a"><mrow id="S3.Thmtheorem1.p6.9.9.m9.1.2" xref="S3.Thmtheorem1.p6.9.9.m9.1.2.cmml"><mi id="S3.Thmtheorem1.p6.9.9.m9.1.2.2" xref="S3.Thmtheorem1.p6.9.9.m9.1.2.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p6.9.9.m9.1.2.1" xref="S3.Thmtheorem1.p6.9.9.m9.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p6.9.9.m9.1.2.3.2" xref="S3.Thmtheorem1.p6.9.9.m9.1.2.cmml"><mo id="S3.Thmtheorem1.p6.9.9.m9.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p6.9.9.m9.1.2.cmml">(</mo><mi id="S3.Thmtheorem1.p6.9.9.m9.1.1" xref="S3.Thmtheorem1.p6.9.9.m9.1.1.cmml">D</mi><mo id="S3.Thmtheorem1.p6.9.9.m9.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem1.p6.9.9.m9.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.9.9.m9.1b"><apply id="S3.Thmtheorem1.p6.9.9.m9.1.2.cmml" xref="S3.Thmtheorem1.p6.9.9.m9.1.2"><times id="S3.Thmtheorem1.p6.9.9.m9.1.2.1.cmml" xref="S3.Thmtheorem1.p6.9.9.m9.1.2.1"></times><ci id="S3.Thmtheorem1.p6.9.9.m9.1.2.2.cmml" xref="S3.Thmtheorem1.p6.9.9.m9.1.2.2">𝑄</ci><ci id="S3.Thmtheorem1.p6.9.9.m9.1.1.cmml" xref="S3.Thmtheorem1.p6.9.9.m9.1.1">𝐷</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.9.9.m9.1c">Q(D)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.9.9.m9.1d">italic_Q ( italic_D )</annotation></semantics></math> to the variables of <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.10.10.m10.1"><semantics id="S3.Thmtheorem1.p6.10.10.m10.1a"><msup id="S3.Thmtheorem1.p6.10.10.m10.1.1" xref="S3.Thmtheorem1.p6.10.10.m10.1.1.cmml"><mi id="S3.Thmtheorem1.p6.10.10.m10.1.1.2" xref="S3.Thmtheorem1.p6.10.10.m10.1.1.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p6.10.10.m10.1.1.3" xref="S3.Thmtheorem1.p6.10.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.10.10.m10.1b"><apply id="S3.Thmtheorem1.p6.10.10.m10.1.1.cmml" xref="S3.Thmtheorem1.p6.10.10.m10.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p6.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem1.p6.10.10.m10.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p6.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem1.p6.10.10.m10.1.1.2">𝑄</ci><ci id="S3.Thmtheorem1.p6.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem1.p6.10.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.10.10.m10.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.10.10.m10.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> results in <math alttext="Q^{\prime}(D^{\prime})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p6.11.11.m11.1"><semantics id="S3.Thmtheorem1.p6.11.11.m11.1a"><mrow id="S3.Thmtheorem1.p6.11.11.m11.1.1" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.cmml"><msup id="S3.Thmtheorem1.p6.11.11.m11.1.1.3" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.3.cmml"><mi id="S3.Thmtheorem1.p6.11.11.m11.1.1.3.2" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.3.2.cmml">Q</mi><mo id="S3.Thmtheorem1.p6.11.11.m11.1.1.3.3" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.3.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p6.11.11.m11.1.1.2" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.2" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.2.cmml">D</mi><mo id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.3" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p6.11.11.m11.1b"><apply id="S3.Thmtheorem1.p6.11.11.m11.1.1.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1"><times id="S3.Thmtheorem1.p6.11.11.m11.1.1.2.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.2"></times><apply id="S3.Thmtheorem1.p6.11.11.m11.1.1.3.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p6.11.11.m11.1.1.3.1.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem1.p6.11.11.m11.1.1.3.2.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.3.2">𝑄</ci><ci id="S3.Thmtheorem1.p6.11.11.m11.1.1.3.3.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.3.3">′</ci></apply><apply id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.2">𝐷</ci><ci id="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem1.p6.11.11.m11.1.1.1.1.1.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p6.11.11.m11.1c">Q^{\prime}(D^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p6.11.11.m11.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> with the same answer order (and the same annotations as in case of CQ<sup class="ltx_sup" id="S3.Thmtheorem1.p6.12.12.2"><span class="ltx_text ltx_font_upright" id="S3.Thmtheorem1.p6.12.12.2.1">⋆</span></sup>s), and this projection can be done in constant time.</span></p> </div> </div> <div class="ltx_para" id="Thmthm7.p2"> <p class="ltx_p" id="Thmthm7.p2.2"><span class="ltx_text ltx_font_italic" id="Thmthm7.p2.2.2">We will next also use <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm6" title="Lemma 6. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a> to eliminate the existential variables from our queries by setting <math alttext="V=\mathrm{free}{(Q)}" class="ltx_Math" display="inline" id="Thmthm7.p2.1.1.m1.1"><semantics id="Thmthm7.p2.1.1.m1.1a"><mrow id="Thmthm7.p2.1.1.m1.1.2" xref="Thmthm7.p2.1.1.m1.1.2.cmml"><mi id="Thmthm7.p2.1.1.m1.1.2.2" xref="Thmthm7.p2.1.1.m1.1.2.2.cmml">V</mi><mo id="Thmthm7.p2.1.1.m1.1.2.1" xref="Thmthm7.p2.1.1.m1.1.2.1.cmml">=</mo><mrow id="Thmthm7.p2.1.1.m1.1.2.3" xref="Thmthm7.p2.1.1.m1.1.2.3.cmml"><mi id="Thmthm7.p2.1.1.m1.1.2.3.2" xref="Thmthm7.p2.1.1.m1.1.2.3.2.cmml">free</mi><mo id="Thmthm7.p2.1.1.m1.1.2.3.1" xref="Thmthm7.p2.1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="Thmthm7.p2.1.1.m1.1.2.3.3.2" xref="Thmthm7.p2.1.1.m1.1.2.3.cmml"><mo id="Thmthm7.p2.1.1.m1.1.2.3.3.2.1" stretchy="false" xref="Thmthm7.p2.1.1.m1.1.2.3.cmml">(</mo><mi id="Thmthm7.p2.1.1.m1.1.1" xref="Thmthm7.p2.1.1.m1.1.1.cmml">Q</mi><mo id="Thmthm7.p2.1.1.m1.1.2.3.3.2.2" stretchy="false" xref="Thmthm7.p2.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p2.1.1.m1.1b"><apply id="Thmthm7.p2.1.1.m1.1.2.cmml" xref="Thmthm7.p2.1.1.m1.1.2"><eq id="Thmthm7.p2.1.1.m1.1.2.1.cmml" xref="Thmthm7.p2.1.1.m1.1.2.1"></eq><ci id="Thmthm7.p2.1.1.m1.1.2.2.cmml" xref="Thmthm7.p2.1.1.m1.1.2.2">𝑉</ci><apply id="Thmthm7.p2.1.1.m1.1.2.3.cmml" xref="Thmthm7.p2.1.1.m1.1.2.3"><times id="Thmthm7.p2.1.1.m1.1.2.3.1.cmml" xref="Thmthm7.p2.1.1.m1.1.2.3.1"></times><ci id="Thmthm7.p2.1.1.m1.1.2.3.2.cmml" xref="Thmthm7.p2.1.1.m1.1.2.3.2">free</ci><ci id="Thmthm7.p2.1.1.m1.1.1.cmml" xref="Thmthm7.p2.1.1.m1.1.1">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p2.1.1.m1.1c">V=\mathrm{free}{(Q)}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p2.1.1.m1.1d">italic_V = roman_free ( italic_Q )</annotation></semantics></math>. Regarding the opposite direction, it is folklore that free-connex CQs (over non-annotated databases) can be transformed into <em class="ltx_emph ltx_font_upright" id="Thmthm7.p2.2.2.1">full</em> acyclic CQs in linear time <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1145/3035918.3064027</span>, <span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1145/2656335</span>]</cite>. The following lemma states that the same holds for free-connex CQ<sup class="ltx_sup" id="Thmthm7.p2.2.2.2"><span class="ltx_text ltx_font_upright" id="Thmthm7.p2.2.2.2.1">⋆</span></sup>s over annotated databases.</span></p> </div> <div class="ltx_theorem ltx_theorem_lem" id="Thmthm8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm8.1.1.1">Lemma 8</span></span><span class="ltx_text ltx_font_bold" id="Thmthm8.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm8.p1"> <p class="ltx_p" id="Thmthm8.p1.11"><span class="ltx_text ltx_font_italic" id="Thmthm8.p1.11.11">Let <math alttext="(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})" class="ltx_Math" display="inline" id="Thmthm8.p1.1.1.m1.5"><semantics id="Thmthm8.p1.1.1.m1.5a"><mrow id="Thmthm8.p1.1.1.m1.5.6.2" xref="Thmthm8.p1.1.1.m1.5.6.1.cmml"><mo id="Thmthm8.p1.1.1.m1.5.6.2.1" stretchy="false" xref="Thmthm8.p1.1.1.m1.5.6.1.cmml">(</mo><mi id="Thmthm8.p1.1.1.m1.1.1" xref="Thmthm8.p1.1.1.m1.1.1.cmml">𝕂</mi><mo id="Thmthm8.p1.1.1.m1.5.6.2.2" rspace="0em" xref="Thmthm8.p1.1.1.m1.5.6.1.cmml">,</mo><mo id="Thmthm8.p1.1.1.m1.2.2" lspace="0em" rspace="0em" xref="Thmthm8.p1.1.1.m1.2.2.cmml">⊕</mo><mo id="Thmthm8.p1.1.1.m1.5.6.2.3" rspace="0em" xref="Thmthm8.p1.1.1.m1.5.6.1.cmml">,</mo><mo id="Thmthm8.p1.1.1.m1.3.3" lspace="0em" rspace="0em" xref="Thmthm8.p1.1.1.m1.3.3.cmml">⊗</mo><mo id="Thmthm8.p1.1.1.m1.5.6.2.4" xref="Thmthm8.p1.1.1.m1.5.6.1.cmml">,</mo><mover accent="true" id="Thmthm8.p1.1.1.m1.4.4" xref="Thmthm8.p1.1.1.m1.4.4a.cmml"><mn id="Thmthm8.p1.1.1.m1.4.4.2" xref="Thmthm8.p1.1.1.m1.4.4.2.cmml">0</mn><mo id="Thmthm8.p1.1.1.m1.4.4.1" xref="Thmthm8.p1.1.1.m1.4.4.1.cmml">¯</mo></mover><mo id="Thmthm8.p1.1.1.m1.5.6.2.5" xref="Thmthm8.p1.1.1.m1.5.6.1.cmml">,</mo><mover accent="true" id="Thmthm8.p1.1.1.m1.5.5" xref="Thmthm8.p1.1.1.m1.5.5a.cmml"><mn id="Thmthm8.p1.1.1.m1.5.5.2" xref="Thmthm8.p1.1.1.m1.5.5.2.cmml">1</mn><mo id="Thmthm8.p1.1.1.m1.5.5.1" xref="Thmthm8.p1.1.1.m1.5.5.1.cmml">¯</mo></mover><mo id="Thmthm8.p1.1.1.m1.5.6.2.6" stretchy="false" xref="Thmthm8.p1.1.1.m1.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.1.1.m1.5b"><vector id="Thmthm8.p1.1.1.m1.5.6.1.cmml" xref="Thmthm8.p1.1.1.m1.5.6.2"><ci id="Thmthm8.p1.1.1.m1.1.1.cmml" xref="Thmthm8.p1.1.1.m1.1.1">𝕂</ci><csymbol cd="latexml" id="Thmthm8.p1.1.1.m1.2.2.cmml" xref="Thmthm8.p1.1.1.m1.2.2">direct-sum</csymbol><csymbol cd="latexml" id="Thmthm8.p1.1.1.m1.3.3.cmml" xref="Thmthm8.p1.1.1.m1.3.3">tensor-product</csymbol><ci id="Thmthm8.p1.1.1.m1.4.4a.cmml" xref="Thmthm8.p1.1.1.m1.4.4"><mover accent="true" id="Thmthm8.p1.1.1.m1.4.4.cmml" xref="Thmthm8.p1.1.1.m1.4.4"><mn id="Thmthm8.p1.1.1.m1.4.4.2.cmml" xref="Thmthm8.p1.1.1.m1.4.4.2">0</mn><mo id="Thmthm8.p1.1.1.m1.4.4.1.cmml" xref="Thmthm8.p1.1.1.m1.4.4.1">¯</mo></mover></ci><ci id="Thmthm8.p1.1.1.m1.5.5a.cmml" xref="Thmthm8.p1.1.1.m1.5.5"><mover accent="true" id="Thmthm8.p1.1.1.m1.5.5.cmml" xref="Thmthm8.p1.1.1.m1.5.5"><mn id="Thmthm8.p1.1.1.m1.5.5.2.cmml" xref="Thmthm8.p1.1.1.m1.5.5.2">1</mn><mo id="Thmthm8.p1.1.1.m1.5.5.1.cmml" xref="Thmthm8.p1.1.1.m1.5.5.1">¯</mo></mover></ci></vector></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.1.1.m1.5c">(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.1.1.m1.5d">( blackboard_K , ⊕ , ⊗ , start_ID over¯ start_ARG 0 end_ARG end_ID , start_ID over¯ start_ARG 1 end_ARG end_ID )</annotation></semantics></math> be a logarithmic-time commutative semiring, and let <math alttext="Q(\vec{x},\star,\vec{z})" class="ltx_Math" display="inline" id="Thmthm8.p1.2.2.m2.3"><semantics id="Thmthm8.p1.2.2.m2.3a"><mrow id="Thmthm8.p1.2.2.m2.3.4" xref="Thmthm8.p1.2.2.m2.3.4.cmml"><mi id="Thmthm8.p1.2.2.m2.3.4.2" xref="Thmthm8.p1.2.2.m2.3.4.2.cmml">Q</mi><mo id="Thmthm8.p1.2.2.m2.3.4.1" xref="Thmthm8.p1.2.2.m2.3.4.1.cmml">⁢</mo><mrow id="Thmthm8.p1.2.2.m2.3.4.3.2" xref="Thmthm8.p1.2.2.m2.3.4.3.1.cmml"><mo id="Thmthm8.p1.2.2.m2.3.4.3.2.1" stretchy="false" xref="Thmthm8.p1.2.2.m2.3.4.3.1.cmml">(</mo><mover accent="true" id="Thmthm8.p1.2.2.m2.1.1" xref="Thmthm8.p1.2.2.m2.1.1.cmml"><mi id="Thmthm8.p1.2.2.m2.1.1.2" xref="Thmthm8.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="Thmthm8.p1.2.2.m2.1.1.1" stretchy="false" xref="Thmthm8.p1.2.2.m2.1.1.1.cmml">→</mo></mover><mo id="Thmthm8.p1.2.2.m2.3.4.3.2.2" rspace="0em" xref="Thmthm8.p1.2.2.m2.3.4.3.1.cmml">,</mo><mo id="Thmthm8.p1.2.2.m2.2.2" lspace="0em" rspace="0em" xref="Thmthm8.p1.2.2.m2.2.2.cmml">⋆</mo><mo id="Thmthm8.p1.2.2.m2.3.4.3.2.3" xref="Thmthm8.p1.2.2.m2.3.4.3.1.cmml">,</mo><mover accent="true" id="Thmthm8.p1.2.2.m2.3.3" xref="Thmthm8.p1.2.2.m2.3.3.cmml"><mi id="Thmthm8.p1.2.2.m2.3.3.2" xref="Thmthm8.p1.2.2.m2.3.3.2.cmml">z</mi><mo id="Thmthm8.p1.2.2.m2.3.3.1" stretchy="false" xref="Thmthm8.p1.2.2.m2.3.3.1.cmml">→</mo></mover><mo id="Thmthm8.p1.2.2.m2.3.4.3.2.4" stretchy="false" xref="Thmthm8.p1.2.2.m2.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.2.2.m2.3b"><apply id="Thmthm8.p1.2.2.m2.3.4.cmml" xref="Thmthm8.p1.2.2.m2.3.4"><times id="Thmthm8.p1.2.2.m2.3.4.1.cmml" xref="Thmthm8.p1.2.2.m2.3.4.1"></times><ci id="Thmthm8.p1.2.2.m2.3.4.2.cmml" xref="Thmthm8.p1.2.2.m2.3.4.2">𝑄</ci><vector id="Thmthm8.p1.2.2.m2.3.4.3.1.cmml" xref="Thmthm8.p1.2.2.m2.3.4.3.2"><apply id="Thmthm8.p1.2.2.m2.1.1.cmml" xref="Thmthm8.p1.2.2.m2.1.1"><ci id="Thmthm8.p1.2.2.m2.1.1.1.cmml" xref="Thmthm8.p1.2.2.m2.1.1.1">→</ci><ci id="Thmthm8.p1.2.2.m2.1.1.2.cmml" xref="Thmthm8.p1.2.2.m2.1.1.2">𝑥</ci></apply><ci id="Thmthm8.p1.2.2.m2.2.2.cmml" xref="Thmthm8.p1.2.2.m2.2.2">⋆</ci><apply id="Thmthm8.p1.2.2.m2.3.3.cmml" xref="Thmthm8.p1.2.2.m2.3.3"><ci id="Thmthm8.p1.2.2.m2.3.3.1.cmml" xref="Thmthm8.p1.2.2.m2.3.3.1">→</ci><ci id="Thmthm8.p1.2.2.m2.3.3.2.cmml" xref="Thmthm8.p1.2.2.m2.3.3.2">𝑧</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.2.2.m2.3c">Q(\vec{x},\star,\vec{z})</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.2.2.m2.3d">italic_Q ( over→ start_ARG italic_x end_ARG , ⋆ , over→ start_ARG italic_z end_ARG )</annotation></semantics></math> be a self-join-free free-connex CQ<sup class="ltx_sup" id="Thmthm8.p1.11.11.1"><span class="ltx_text ltx_font_upright" id="Thmthm8.p1.11.11.1.1">⋆</span></sup>. Then, <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="Thmthm8.p1.4.4.m4.1"><semantics id="Thmthm8.p1.4.4.m4.1a"><msub id="Thmthm8.p1.4.4.m4.1.2"><mi id="Thmthm8.p1.4.4.m4.1.2.2">Q</mi><mrow id="Thmthm8.p1.4.4.m4.1.1.1"><mo fence="false" id="Thmthm8.p1.4.4.m4.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="Thmthm8.p1.4.4.m4.1.1.1.3">free</mi><mrow id="Thmthm8.p1.4.4.m4.1.1.1.4"><mo id="Thmthm8.p1.4.4.m4.1.1.1.4.1" stretchy="false">(</mo><mi id="Thmthm8.p1.4.4.m4.1.1.1.1">Q</mi><mo id="Thmthm8.p1.4.4.m4.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="Thmthm8.p1.4.4.m4.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.4.4.m4.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> is a full acyclic CQ<sup class="ltx_sup" id="Thmthm8.p1.11.11.2"><span class="ltx_text ltx_font_upright" id="Thmthm8.p1.11.11.2.1">⋆</span></sup>, and if direct access for <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="Thmthm8.p1.6.6.m6.1"><semantics id="Thmthm8.p1.6.6.m6.1a"><msub id="Thmthm8.p1.6.6.m6.1.2"><mi id="Thmthm8.p1.6.6.m6.1.2.2">Q</mi><mrow id="Thmthm8.p1.6.6.m6.1.1.1"><mo fence="false" id="Thmthm8.p1.6.6.m6.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="Thmthm8.p1.6.6.m6.1.1.1.3">free</mi><mrow id="Thmthm8.p1.6.6.m6.1.1.1.4"><mo id="Thmthm8.p1.6.6.m6.1.1.1.4.1" stretchy="false">(</mo><mi id="Thmthm8.p1.6.6.m6.1.1.1.1">Q</mi><mo id="Thmthm8.p1.6.6.m6.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="Thmthm8.p1.6.6.m6.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.6.6.m6.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> over <math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="Thmthm8.p1.7.7.m7.1"><semantics id="Thmthm8.p1.7.7.m7.1a"><mi id="Thmthm8.p1.7.7.m7.1.1" xref="Thmthm8.p1.7.7.m7.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.7.7.m7.1b"><ci id="Thmthm8.p1.7.7.m7.1.1.cmml" xref="Thmthm8.p1.7.7.m7.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.7.7.m7.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.7.7.m7.1d">blackboard_K</annotation></semantics></math>-databases is in <math alttext="\mathord{\langle T_{p},T_{a}\rangle}" class="ltx_Math" display="inline" id="Thmthm8.p1.8.8.m8.2"><semantics id="Thmthm8.p1.8.8.m8.2a"><mrow id="Thmthm8.p1.8.8.m8.2.2.2" xref="Thmthm8.p1.8.8.m8.2.2.3.cmml"><mo id="Thmthm8.p1.8.8.m8.2.2.2.3" stretchy="false" xref="Thmthm8.p1.8.8.m8.2.2.3.cmml">⟨</mo><msub id="Thmthm8.p1.8.8.m8.1.1.1.1" xref="Thmthm8.p1.8.8.m8.1.1.1.1.cmml"><mi id="Thmthm8.p1.8.8.m8.1.1.1.1.2" xref="Thmthm8.p1.8.8.m8.1.1.1.1.2.cmml">T</mi><mi id="Thmthm8.p1.8.8.m8.1.1.1.1.3" xref="Thmthm8.p1.8.8.m8.1.1.1.1.3.cmml">p</mi></msub><mo id="Thmthm8.p1.8.8.m8.2.2.2.4" xref="Thmthm8.p1.8.8.m8.2.2.3.cmml">,</mo><msub id="Thmthm8.p1.8.8.m8.2.2.2.2" xref="Thmthm8.p1.8.8.m8.2.2.2.2.cmml"><mi id="Thmthm8.p1.8.8.m8.2.2.2.2.2" xref="Thmthm8.p1.8.8.m8.2.2.2.2.2.cmml">T</mi><mi id="Thmthm8.p1.8.8.m8.2.2.2.2.3" xref="Thmthm8.p1.8.8.m8.2.2.2.2.3.cmml">a</mi></msub><mo id="Thmthm8.p1.8.8.m8.2.2.2.5" stretchy="false" xref="Thmthm8.p1.8.8.m8.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.8.8.m8.2b"><list id="Thmthm8.p1.8.8.m8.2.2.3.cmml" xref="Thmthm8.p1.8.8.m8.2.2.2"><apply id="Thmthm8.p1.8.8.m8.1.1.1.1.cmml" xref="Thmthm8.p1.8.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm8.p1.8.8.m8.1.1.1.1.1.cmml" xref="Thmthm8.p1.8.8.m8.1.1.1.1">subscript</csymbol><ci id="Thmthm8.p1.8.8.m8.1.1.1.1.2.cmml" xref="Thmthm8.p1.8.8.m8.1.1.1.1.2">𝑇</ci><ci id="Thmthm8.p1.8.8.m8.1.1.1.1.3.cmml" xref="Thmthm8.p1.8.8.m8.1.1.1.1.3">𝑝</ci></apply><apply id="Thmthm8.p1.8.8.m8.2.2.2.2.cmml" xref="Thmthm8.p1.8.8.m8.2.2.2.2"><csymbol cd="ambiguous" id="Thmthm8.p1.8.8.m8.2.2.2.2.1.cmml" xref="Thmthm8.p1.8.8.m8.2.2.2.2">subscript</csymbol><ci id="Thmthm8.p1.8.8.m8.2.2.2.2.2.cmml" xref="Thmthm8.p1.8.8.m8.2.2.2.2.2">𝑇</ci><ci id="Thmthm8.p1.8.8.m8.2.2.2.2.3.cmml" xref="Thmthm8.p1.8.8.m8.2.2.2.2.3">𝑎</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.8.8.m8.2c">\mathord{\langle T_{p},T_{a}\rangle}</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.8.8.m8.2d">⟨ italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟩</annotation></semantics></math>, then direct access for <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm8.p1.9.9.m9.1"><semantics id="Thmthm8.p1.9.9.m9.1a"><mi id="Thmthm8.p1.9.9.m9.1.1" xref="Thmthm8.p1.9.9.m9.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.9.9.m9.1b"><ci id="Thmthm8.p1.9.9.m9.1.1.cmml" xref="Thmthm8.p1.9.9.m9.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.9.9.m9.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.9.9.m9.1d">italic_Q</annotation></semantics></math> over <math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="Thmthm8.p1.10.10.m10.1"><semantics id="Thmthm8.p1.10.10.m10.1a"><mi id="Thmthm8.p1.10.10.m10.1.1" xref="Thmthm8.p1.10.10.m10.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.10.10.m10.1b"><ci id="Thmthm8.p1.10.10.m10.1.1.cmml" xref="Thmthm8.p1.10.10.m10.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.10.10.m10.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.10.10.m10.1d">blackboard_K</annotation></semantics></math>-databases is in <math alttext="\mathord{\langle\mathrm{loglinear}+T_{p},T_{a}\rangle}" class="ltx_Math" display="inline" id="Thmthm8.p1.11.11.m11.2"><semantics id="Thmthm8.p1.11.11.m11.2a"><mrow id="Thmthm8.p1.11.11.m11.2.2.2" xref="Thmthm8.p1.11.11.m11.2.2.3.cmml"><mo id="Thmthm8.p1.11.11.m11.2.2.2.3" stretchy="false" xref="Thmthm8.p1.11.11.m11.2.2.3.cmml">⟨</mo><mrow id="Thmthm8.p1.11.11.m11.1.1.1.1" xref="Thmthm8.p1.11.11.m11.1.1.1.1.cmml"><mi id="Thmthm8.p1.11.11.m11.1.1.1.1.2" xref="Thmthm8.p1.11.11.m11.1.1.1.1.2.cmml">loglinear</mi><mo id="Thmthm8.p1.11.11.m11.1.1.1.1.1" xref="Thmthm8.p1.11.11.m11.1.1.1.1.1.cmml">+</mo><msub id="Thmthm8.p1.11.11.m11.1.1.1.1.3" xref="Thmthm8.p1.11.11.m11.1.1.1.1.3.cmml"><mi id="Thmthm8.p1.11.11.m11.1.1.1.1.3.2" xref="Thmthm8.p1.11.11.m11.1.1.1.1.3.2.cmml">T</mi><mi id="Thmthm8.p1.11.11.m11.1.1.1.1.3.3" xref="Thmthm8.p1.11.11.m11.1.1.1.1.3.3.cmml">p</mi></msub></mrow><mo id="Thmthm8.p1.11.11.m11.2.2.2.4" xref="Thmthm8.p1.11.11.m11.2.2.3.cmml">,</mo><msub id="Thmthm8.p1.11.11.m11.2.2.2.2" xref="Thmthm8.p1.11.11.m11.2.2.2.2.cmml"><mi id="Thmthm8.p1.11.11.m11.2.2.2.2.2" xref="Thmthm8.p1.11.11.m11.2.2.2.2.2.cmml">T</mi><mi id="Thmthm8.p1.11.11.m11.2.2.2.2.3" xref="Thmthm8.p1.11.11.m11.2.2.2.2.3.cmml">a</mi></msub><mo id="Thmthm8.p1.11.11.m11.2.2.2.5" stretchy="false" xref="Thmthm8.p1.11.11.m11.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm8.p1.11.11.m11.2b"><list id="Thmthm8.p1.11.11.m11.2.2.3.cmml" xref="Thmthm8.p1.11.11.m11.2.2.2"><apply id="Thmthm8.p1.11.11.m11.1.1.1.1.cmml" xref="Thmthm8.p1.11.11.m11.1.1.1.1"><plus id="Thmthm8.p1.11.11.m11.1.1.1.1.1.cmml" xref="Thmthm8.p1.11.11.m11.1.1.1.1.1"></plus><ci id="Thmthm8.p1.11.11.m11.1.1.1.1.2.cmml" xref="Thmthm8.p1.11.11.m11.1.1.1.1.2">loglinear</ci><apply id="Thmthm8.p1.11.11.m11.1.1.1.1.3.cmml" xref="Thmthm8.p1.11.11.m11.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmthm8.p1.11.11.m11.1.1.1.1.3.1.cmml" xref="Thmthm8.p1.11.11.m11.1.1.1.1.3">subscript</csymbol><ci id="Thmthm8.p1.11.11.m11.1.1.1.1.3.2.cmml" xref="Thmthm8.p1.11.11.m11.1.1.1.1.3.2">𝑇</ci><ci id="Thmthm8.p1.11.11.m11.1.1.1.1.3.3.cmml" xref="Thmthm8.p1.11.11.m11.1.1.1.1.3.3">𝑝</ci></apply></apply><apply id="Thmthm8.p1.11.11.m11.2.2.2.2.cmml" xref="Thmthm8.p1.11.11.m11.2.2.2.2"><csymbol cd="ambiguous" id="Thmthm8.p1.11.11.m11.2.2.2.2.1.cmml" xref="Thmthm8.p1.11.11.m11.2.2.2.2">subscript</csymbol><ci id="Thmthm8.p1.11.11.m11.2.2.2.2.2.cmml" xref="Thmthm8.p1.11.11.m11.2.2.2.2.2">𝑇</ci><ci id="Thmthm8.p1.11.11.m11.2.2.2.2.3.cmml" xref="Thmthm8.p1.11.11.m11.2.2.2.2.3">𝑎</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm8.p1.11.11.m11.2c">\mathord{\langle\mathrm{loglinear}+T_{p},T_{a}\rangle}</annotation><annotation encoding="application/x-llamapun" id="Thmthm8.p1.11.11.m11.2d">⟨ roman_loglinear + italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟩</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.1.1.1">Proof 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.8"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.8.8">We show an <math alttext="O(|D|\log|D|)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.1.m1.3"><semantics id="S3.Thmtheorem2.p1.1.1.m1.3a"><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.3" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.cmml"><mi id="S3.Thmtheorem2.p1.1.1.m1.3.3.3" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.3.cmml">O</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.2" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.2.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.cmml"><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.2" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.cmml">(</mo><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.cmml"><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.2" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.1.cmml"><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.1.1.cmml">|</mo><mi id="S3.Thmtheorem2.p1.1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml">D</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.2.2" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.1.1.cmml">|</mo></mrow><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1" lspace="0.167em" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.1" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.1.cmml">log</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3a" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.cmml">⁡</mo><mrow id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.2" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.1.cmml"><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.1.1.cmml">|</mo><mi id="S3.Thmtheorem2.p1.1.1.m1.2.2" xref="S3.Thmtheorem2.p1.1.1.m1.2.2.cmml">D</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.2.2" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.1.1.cmml">|</mo></mrow></mrow></mrow><mo id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.3" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.1.m1.3b"><apply id="S3.Thmtheorem2.p1.1.1.m1.3.3.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3"><times id="S3.Thmtheorem2.p1.1.1.m1.3.3.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.2"></times><ci id="S3.Thmtheorem2.p1.1.1.m1.3.3.3.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.3">𝑂</ci><apply id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1"><times id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.1"></times><apply id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.2"><abs id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.2.2.1"></abs><ci id="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.1.1">𝐷</ci></apply><apply id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3"><log id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.1"></log><apply id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.2"><abs id="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.1.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.3.3.1.1.1.3.2.2.1"></abs><ci id="S3.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.2.2">𝐷</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.1.m1.3c">O(|D|\log|D|)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.1.m1.3d">italic_O ( | italic_D | roman_log | italic_D | )</annotation></semantics></math>-time construction that maps <math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.2.m2.1a"><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.2.m2.1b"><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.2.m2.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.2.m2.1d">blackboard_K</annotation></semantics></math>-databases <math alttext="(D,\tau)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.3.m3.2"><semantics id="S3.Thmtheorem2.p1.3.3.m3.2a"><mrow id="S3.Thmtheorem2.p1.3.3.m3.2.3.2" xref="S3.Thmtheorem2.p1.3.3.m3.2.3.1.cmml"><mo id="S3.Thmtheorem2.p1.3.3.m3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.3.3.m3.2.3.1.cmml">(</mo><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml">D</mi><mo id="S3.Thmtheorem2.p1.3.3.m3.2.3.2.2" xref="S3.Thmtheorem2.p1.3.3.m3.2.3.1.cmml">,</mo><mi id="S3.Thmtheorem2.p1.3.3.m3.2.2" xref="S3.Thmtheorem2.p1.3.3.m3.2.2.cmml">τ</mi><mo id="S3.Thmtheorem2.p1.3.3.m3.2.3.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.3.3.m3.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.3.m3.2b"><interval closure="open" id="S3.Thmtheorem2.p1.3.3.m3.2.3.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.2.3.2"><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1">𝐷</ci><ci id="S3.Thmtheorem2.p1.3.3.m3.2.2.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.2.2">𝜏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.3.m3.2c">(D,\tau)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.3.m3.2d">( italic_D , italic_τ )</annotation></semantics></math> of <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.4.m4.1"><semantics id="S3.Thmtheorem2.p1.4.4.m4.1a"><mi id="S3.Thmtheorem2.p1.4.4.m4.1.1" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.4.4.m4.1b"><ci id="S3.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.4.m4.1d">italic_Q</annotation></semantics></math> to <math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.5.5.m5.1"><semantics id="S3.Thmtheorem2.p1.5.5.m5.1a"><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.5.5.m5.1b"><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.5.m5.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.5.m5.1d">blackboard_K</annotation></semantics></math>-databases <math alttext="(D^{\prime},\tau^{\prime})" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.6.6.m6.2"><semantics id="S3.Thmtheorem2.p1.6.6.m6.2a"><mrow id="S3.Thmtheorem2.p1.6.6.m6.2.2.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.3.cmml"><mo id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.3.cmml">(</mo><msup id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.cmml"><mi id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.2" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.2.cmml">D</mi><mo id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.3" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.4" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.3.cmml">,</mo><msup id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.cmml"><mi id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.2" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.2.cmml">τ</mi><mo id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.3" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.3.cmml">′</mo></msup><mo id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.5" stretchy="false" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.6.6.m6.2b"><interval closure="open" id="S3.Thmtheorem2.p1.6.6.m6.2.2.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2"><apply id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.2">𝐷</ci><ci id="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.1.1.1.1.3">′</ci></apply><apply id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.1.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2">superscript</csymbol><ci id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.2.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.2">𝜏</ci><ci id="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.3.cmml" xref="S3.Thmtheorem2.p1.6.6.m6.2.2.2.2.3">′</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.6.6.m6.2c">(D^{\prime},\tau^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.6.6.m6.2d">( italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> of <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p1.7.7.m7.1"><semantics id="S3.Thmtheorem2.p1.7.7.m7.1a"><msub id="S3.Thmtheorem2.p1.7.7.m7.1.2"><mi id="S3.Thmtheorem2.p1.7.7.m7.1.2.2">Q</mi><mrow id="S3.Thmtheorem2.p1.7.7.m7.1.1.1"><mo fence="false" id="S3.Thmtheorem2.p1.7.7.m7.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p1.7.7.m7.1.1.1.3">free</mi><mrow id="S3.Thmtheorem2.p1.7.7.m7.1.1.1.4"><mo id="S3.Thmtheorem2.p1.7.7.m7.1.1.1.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem2.p1.7.7.m7.1.1.1.1">Q</mi><mo id="S3.Thmtheorem2.p1.7.7.m7.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.7.7.m7.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.7.7.m7.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="Q_{|\mathrm{free}{(Q)}}(D^{\prime},\tau^{\prime})=Q(D,\tau)" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p1.8.8.m8.5"><semantics id="S3.Thmtheorem2.p1.8.8.m8.5a"><mrow id="S3.Thmtheorem2.p1.8.8.m8.5.5"><mrow id="S3.Thmtheorem2.p1.8.8.m8.5.5.2"><msub id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.4"><mi id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.4.2">Q</mi><mrow id="S3.Thmtheorem2.p1.8.8.m8.1.1.1"><mo fence="false" id="S3.Thmtheorem2.p1.8.8.m8.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p1.8.8.m8.1.1.1.3">free</mi><mrow id="S3.Thmtheorem2.p1.8.8.m8.1.1.1.4"><mo id="S3.Thmtheorem2.p1.8.8.m8.1.1.1.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem2.p1.8.8.m8.1.1.1.1">Q</mi><mo id="S3.Thmtheorem2.p1.8.8.m8.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.3">⁢</mo><mrow id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.2.2"><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.2.2.3" stretchy="false">(</mo><msup id="S3.Thmtheorem2.p1.8.8.m8.4.4.1.1.1.1"><mi id="S3.Thmtheorem2.p1.8.8.m8.4.4.1.1.1.1.2">D</mi><mo id="S3.Thmtheorem2.p1.8.8.m8.4.4.1.1.1.1.3">′</mo></msup><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.2.2.4">,</mo><msup id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.2.2.2"><mi id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.2.2.2.2">τ</mi><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.2.2.2.3">′</mo></msup><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.2.2.2.5" stretchy="false">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.3">=</mo><mrow id="S3.Thmtheorem2.p1.8.8.m8.5.5.4"><mi id="S3.Thmtheorem2.p1.8.8.m8.5.5.4.2">Q</mi><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.4.1">⁢</mo><mrow id="S3.Thmtheorem2.p1.8.8.m8.5.5.4.3.2"><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.4.3.2.1" stretchy="false">(</mo><mi id="S3.Thmtheorem2.p1.8.8.m8.2.2">D</mi><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.4.3.2.2">,</mo><mi id="S3.Thmtheorem2.p1.8.8.m8.3.3">τ</mi><mo id="S3.Thmtheorem2.p1.8.8.m8.5.5.4.3.2.3" stretchy="false">)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.8.8.m8.5b">Q_{|\mathrm{free}{(Q)}}(D^{\prime},\tau^{\prime})=Q(D,\tau)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.8.8.m8.5c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT ( italic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_Q ( italic_D , italic_τ )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem2.p2"> <p class="ltx_p" id="S3.Thmtheorem2.p2.8"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p2.8.8">Since <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.1.1.m1.1"><semantics id="S3.Thmtheorem2.p2.1.1.m1.1a"><mi id="S3.Thmtheorem2.p2.1.1.m1.1.1" xref="S3.Thmtheorem2.p2.1.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.1.1.m1.1b"><ci id="S3.Thmtheorem2.p2.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p2.1.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.1.1.m1.1d">italic_Q</annotation></semantics></math> is free-connex, it has an ext-free-connex join tree <math alttext="T" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.2.2.m2.1"><semantics id="S3.Thmtheorem2.p2.2.2.m2.1a"><mi id="S3.Thmtheorem2.p2.2.2.m2.1.1" xref="S3.Thmtheorem2.p2.2.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.2.2.m2.1b"><ci id="S3.Thmtheorem2.p2.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p2.2.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.2.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.2.2.m2.1d">italic_T</annotation></semantics></math> with a subtree <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.3.3.m3.1"><semantics id="S3.Thmtheorem2.p2.3.3.m3.1a"><msup id="S3.Thmtheorem2.p2.3.3.m3.1.1" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem2.p2.3.3.m3.1.1.2" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p2.3.3.m3.1.1.3" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.3.3.m3.1b"><apply id="S3.Thmtheorem2.p2.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p2.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem2.p2.3.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.3.3.m3.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.3.3.m3.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> which contains precisely the free variables of <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.4.4.m4.1"><semantics id="S3.Thmtheorem2.p2.4.4.m4.1a"><mi id="S3.Thmtheorem2.p2.4.4.m4.1.1" xref="S3.Thmtheorem2.p2.4.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.4.4.m4.1b"><ci id="S3.Thmtheorem2.p2.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p2.4.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.4.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.4.4.m4.1d">italic_Q</annotation></semantics></math>. We call the vertices of <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.5.5.m5.1"><semantics id="S3.Thmtheorem2.p2.5.5.m5.1a"><msup id="S3.Thmtheorem2.p2.5.5.m5.1.1" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem2.p2.5.5.m5.1.1.2" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p2.5.5.m5.1.1.3" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.5.5.m5.1b"><apply id="S3.Thmtheorem2.p2.5.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p2.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem2.p2.5.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.5.5.m5.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.5.5.m5.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> the <em class="ltx_emph ltx_font_upright" id="S3.Thmtheorem2.p2.8.8.1">free vertices</em>. The proof idea is as follows: we first adapt the database to have a relation for each vertex of <math alttext="T" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.6.6.m6.1"><semantics id="S3.Thmtheorem2.p2.6.6.m6.1a"><mi id="S3.Thmtheorem2.p2.6.6.m6.1.1" xref="S3.Thmtheorem2.p2.6.6.m6.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.6.6.m6.1b"><ci id="S3.Thmtheorem2.p2.6.6.m6.1.1.cmml" xref="S3.Thmtheorem2.p2.6.6.m6.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.6.6.m6.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.6.6.m6.1d">italic_T</annotation></semantics></math>, we then build a database with the same answers using only the vertices of <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p2.7.7.m7.1"><semantics id="S3.Thmtheorem2.p2.7.7.m7.1a"><msup id="S3.Thmtheorem2.p2.7.7.m7.1.1" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem2.p2.7.7.m7.1.1.2" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p2.7.7.m7.1.1.3" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p2.7.7.m7.1b"><apply id="S3.Thmtheorem2.p2.7.7.m7.1.1.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p2.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p2.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p2.7.7.m7.1.1.3.cmml" xref="S3.Thmtheorem2.p2.7.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.7.7.m7.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.7.7.m7.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, and finally we translate this into a database for <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p2.8.8.m8.1"><semantics id="S3.Thmtheorem2.p2.8.8.m8.1a"><msub id="S3.Thmtheorem2.p2.8.8.m8.1.2"><mi id="S3.Thmtheorem2.p2.8.8.m8.1.2.2">Q</mi><mrow id="S3.Thmtheorem2.p2.8.8.m8.1.1.1"><mo fence="false" id="S3.Thmtheorem2.p2.8.8.m8.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p2.8.8.m8.1.1.1.3">free</mi><mrow id="S3.Thmtheorem2.p2.8.8.m8.1.1.1.4"><mo id="S3.Thmtheorem2.p2.8.8.m8.1.1.1.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem2.p2.8.8.m8.1.1.1.1">Q</mi><mo id="S3.Thmtheorem2.p2.8.8.m8.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p2.8.8.m8.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p2.8.8.m8.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem2.p3"> <p class="ltx_p" id="S3.Thmtheorem2.p3.1"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p3.1.1">The first step is to make it so that each vertex of the tree has a unique associated atom in the query and a unique associated relation in the database. For every vertex of the tree that does not correspond to a relation, we take the relation of a vertex that contains it (without the annotations), project it accordingly, and annotate the facts with <math alttext="\mathord{\bar{1}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p3.1.1.m1.1"><semantics id="S3.Thmtheorem2.p3.1.1.m1.1a"><mover accent="true" id="S3.Thmtheorem2.p3.1.1.m1.1.1" xref="S3.Thmtheorem2.p3.1.1.m1.1.1a.cmml"><mn id="S3.Thmtheorem2.p3.1.1.m1.1.1.2" xref="S3.Thmtheorem2.p3.1.1.m1.1.1.2.cmml">1</mn><mo id="S3.Thmtheorem2.p3.1.1.m1.1.1.1" xref="S3.Thmtheorem2.p3.1.1.m1.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p3.1.1.m1.1b"><ci id="S3.Thmtheorem2.p3.1.1.m1.1.1a.cmml" xref="S3.Thmtheorem2.p3.1.1.m1.1.1"><mover accent="true" id="S3.Thmtheorem2.p3.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p3.1.1.m1.1.1"><mn id="S3.Thmtheorem2.p3.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem2.p3.1.1.m1.1.1.2">1</mn><mo id="S3.Thmtheorem2.p3.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem2.p3.1.1.m1.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p3.1.1.m1.1c">\mathord{\bar{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p3.1.1.m1.1d">start_ID over¯ start_ARG 1 end_ARG end_ID</annotation></semantics></math>. The query answers are unchanged by the addition of these annotated relations as they contain projections of existing relations annotated with the multiplicative identity.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem2.p4"> <p class="ltx_p" id="S3.Thmtheorem2.p4.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p4.2.2">Next, our goal is to only use relations matching the free vertices and maintain the same query answers. We claim that as long as <math alttext="T" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p4.1.1.m1.1"><semantics id="S3.Thmtheorem2.p4.1.1.m1.1a"><mi id="S3.Thmtheorem2.p4.1.1.m1.1.1" xref="S3.Thmtheorem2.p4.1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p4.1.1.m1.1b"><ci id="S3.Thmtheorem2.p4.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p4.1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p4.1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p4.1.1.m1.1d">italic_T</annotation></semantics></math> has vertices that are not in the set of free vertices, we can remove such a vertex from <math alttext="T" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p4.2.2.m2.1"><semantics id="S3.Thmtheorem2.p4.2.2.m2.1a"><mi id="S3.Thmtheorem2.p4.2.2.m2.1.1" xref="S3.Thmtheorem2.p4.2.2.m2.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p4.2.2.m2.1b"><ci id="S3.Thmtheorem2.p4.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p4.2.2.m2.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p4.2.2.m2.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p4.2.2.m2.1d">italic_T</annotation></semantics></math> and remove its associated atom from the query while adapting the database so that the query results remain unchanged.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem2.p5"> <p class="ltx_p" id="S3.Thmtheorem2.p5.1"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p5.1.1">As long as <math alttext="T" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p5.1.1.m1.1"><semantics id="S3.Thmtheorem2.p5.1.1.m1.1a"><mi id="S3.Thmtheorem2.p5.1.1.m1.1.1" xref="S3.Thmtheorem2.p5.1.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p5.1.1.m1.1b"><ci id="S3.Thmtheorem2.p5.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p5.1.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p5.1.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p5.1.1.m1.1d">italic_T</annotation></semantics></math> has a vertex not in the set of free vertices, eliminate such a leaf vertex as follows:</span></p> <ol class="ltx_enumerate" id="S3.I5"> <li class="ltx_item" id="S3.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S3.I5.i1.p1"> <p class="ltx_p" id="S3.I5.i1.p1.1"><span class="ltx_text ltx_font_italic" id="S3.I5.i1.p1.1.1">In the relation associated with the vertex, project out every variable not shared between the vertex and its neighbor while summing (</span><math alttext="\oplus" class="ltx_Math" display="inline" id="S3.I5.i1.p1.1.m1.1"><semantics id="S3.I5.i1.p1.1.m1.1a"><mo id="S3.I5.i1.p1.1.m1.1.1" xref="S3.I5.i1.p1.1.m1.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S3.I5.i1.p1.1.m1.1b"><csymbol cd="latexml" id="S3.I5.i1.p1.1.m1.1.1.cmml" xref="S3.I5.i1.p1.1.m1.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.I5.i1.p1.1.m1.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S3.I5.i1.p1.1.m1.1d">⊕</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I5.i1.p1.1.2">) annotations.</span></p> </div> </li> <li class="ltx_item" id="S3.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S3.I5.i2.p1"> <p class="ltx_p" id="S3.I5.i2.p1.1"><span class="ltx_text ltx_font_italic" id="S3.I5.i2.p1.1.1">Join the obtained relation into the relation of the neighbor vertex while multiplying (</span><math alttext="\otimes" class="ltx_Math" display="inline" id="S3.I5.i2.p1.1.m1.1"><semantics id="S3.I5.i2.p1.1.m1.1a"><mo id="S3.I5.i2.p1.1.m1.1.1" xref="S3.I5.i2.p1.1.m1.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S3.I5.i2.p1.1.m1.1b"><csymbol cd="latexml" id="S3.I5.i2.p1.1.m1.1.1.cmml" xref="S3.I5.i2.p1.1.m1.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.I5.i2.p1.1.m1.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S3.I5.i2.p1.1.m1.1d">⊗</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S3.I5.i2.p1.1.2">) annotations.</span></p> </div> </li> <li class="ltx_item" id="S3.I5.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(3)</span> <div class="ltx_para" id="S3.I5.i3.p1"> <p class="ltx_p" id="S3.I5.i3.p1.1"><span class="ltx_text ltx_font_italic" id="S3.I5.i3.p1.1.1">Remove the leaf from the tree, its associated atom from the query, and its associated relation from the database.</span></p> </div> </li> </ol> </div> <div class="ltx_para" id="S3.Thmtheorem2.p6"> <p class="ltx_p" id="S3.Thmtheorem2.p6.13"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p6.13.13">More specifically, denote by <math alttext="S" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.1.1.m1.1"><semantics id="S3.Thmtheorem2.p6.1.1.m1.1a"><mi id="S3.Thmtheorem2.p6.1.1.m1.1.1" xref="S3.Thmtheorem2.p6.1.1.m1.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.1.1.m1.1b"><ci id="S3.Thmtheorem2.p6.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p6.1.1.m1.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.1.1.m1.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.1.1.m1.1d">italic_S</annotation></semantics></math> the variables shared between the treated leaf vertex <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.2.2.m2.1"><semantics id="S3.Thmtheorem2.p6.2.2.m2.1a"><mi id="S3.Thmtheorem2.p6.2.2.m2.1.1" xref="S3.Thmtheorem2.p6.2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.2.2.m2.1b"><ci id="S3.Thmtheorem2.p6.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p6.2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.2.2.m2.1d">italic_v</annotation></semantics></math> and its neighbor. During step (<a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S3.I5.i1" title="Item 1 ‣ Proof 3.2. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">1</span></a>), we initialize a lookup table where for each unique assignment of <math alttext="S" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.3.3.m3.1"><semantics id="S3.Thmtheorem2.p6.3.3.m3.1a"><mi id="S3.Thmtheorem2.p6.3.3.m3.1.1" xref="S3.Thmtheorem2.p6.3.3.m3.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.3.3.m3.1b"><ci id="S3.Thmtheorem2.p6.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p6.3.3.m3.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.3.3.m3.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.3.3.m3.1d">italic_S</annotation></semantics></math> matching a fact in the relation of <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.4.4.m4.1"><semantics id="S3.Thmtheorem2.p6.4.4.m4.1a"><mi id="S3.Thmtheorem2.p6.4.4.m4.1.1" xref="S3.Thmtheorem2.p6.4.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.4.4.m4.1b"><ci id="S3.Thmtheorem2.p6.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p6.4.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.4.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.4.4.m4.1d">italic_v</annotation></semantics></math>, we set the semiring value <math alttext="\mathord{\bar{0}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.5.5.m5.1"><semantics id="S3.Thmtheorem2.p6.5.5.m5.1a"><mover accent="true" id="S3.Thmtheorem2.p6.5.5.m5.1.1" xref="S3.Thmtheorem2.p6.5.5.m5.1.1a.cmml"><mn id="S3.Thmtheorem2.p6.5.5.m5.1.1.2" xref="S3.Thmtheorem2.p6.5.5.m5.1.1.2.cmml">0</mn><mo id="S3.Thmtheorem2.p6.5.5.m5.1.1.1" xref="S3.Thmtheorem2.p6.5.5.m5.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.5.5.m5.1b"><ci id="S3.Thmtheorem2.p6.5.5.m5.1.1a.cmml" xref="S3.Thmtheorem2.p6.5.5.m5.1.1"><mover accent="true" id="S3.Thmtheorem2.p6.5.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p6.5.5.m5.1.1"><mn id="S3.Thmtheorem2.p6.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem2.p6.5.5.m5.1.1.2">0</mn><mo id="S3.Thmtheorem2.p6.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem2.p6.5.5.m5.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.5.5.m5.1c">\mathord{\bar{0}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.5.5.m5.1d">start_ID over¯ start_ARG 0 end_ARG end_ID</annotation></semantics></math>. Then, we iterate over each fact of the relation of <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.6.6.m6.1"><semantics id="S3.Thmtheorem2.p6.6.6.m6.1a"><mi id="S3.Thmtheorem2.p6.6.6.m6.1.1" xref="S3.Thmtheorem2.p6.6.6.m6.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.6.6.m6.1b"><ci id="S3.Thmtheorem2.p6.6.6.m6.1.1.cmml" xref="S3.Thmtheorem2.p6.6.6.m6.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.6.6.m6.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.6.6.m6.1d">italic_v</annotation></semantics></math> and add its annotation (using the semiring <math alttext="\oplus" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.7.7.m7.1"><semantics id="S3.Thmtheorem2.p6.7.7.m7.1a"><mo id="S3.Thmtheorem2.p6.7.7.m7.1.1" xref="S3.Thmtheorem2.p6.7.7.m7.1.1.cmml">⊕</mo><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.7.7.m7.1b"><csymbol cd="latexml" id="S3.Thmtheorem2.p6.7.7.m7.1.1.cmml" xref="S3.Thmtheorem2.p6.7.7.m7.1.1">direct-sum</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.7.7.m7.1c">\oplus</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.7.7.m7.1d">⊕</annotation></semantics></math>) to the entry matching its assignment to <math alttext="S" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.8.8.m8.1"><semantics id="S3.Thmtheorem2.p6.8.8.m8.1a"><mi id="S3.Thmtheorem2.p6.8.8.m8.1.1" xref="S3.Thmtheorem2.p6.8.8.m8.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.8.8.m8.1b"><ci id="S3.Thmtheorem2.p6.8.8.m8.1.1.cmml" xref="S3.Thmtheorem2.p6.8.8.m8.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.8.8.m8.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.8.8.m8.1d">italic_S</annotation></semantics></math>. Next, during step (<a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S3.I5.i2" title="Item 2 ‣ Proof 3.2. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">2</span></a>), we iterate over every fact in the relation matching the neighbor of <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.9.9.m9.1"><semantics id="S3.Thmtheorem2.p6.9.9.m9.1a"><mi id="S3.Thmtheorem2.p6.9.9.m9.1.1" xref="S3.Thmtheorem2.p6.9.9.m9.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.9.9.m9.1b"><ci id="S3.Thmtheorem2.p6.9.9.m9.1.1.cmml" xref="S3.Thmtheorem2.p6.9.9.m9.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.9.9.m9.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.9.9.m9.1d">italic_v</annotation></semantics></math> and multiply its existing annotation (using the semiring <math alttext="\otimes" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.10.10.m10.1"><semantics id="S3.Thmtheorem2.p6.10.10.m10.1a"><mo id="S3.Thmtheorem2.p6.10.10.m10.1.1" xref="S3.Thmtheorem2.p6.10.10.m10.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.10.10.m10.1b"><csymbol cd="latexml" id="S3.Thmtheorem2.p6.10.10.m10.1.1.cmml" xref="S3.Thmtheorem2.p6.10.10.m10.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.10.10.m10.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.10.10.m10.1d">⊗</annotation></semantics></math>) with the value stored for its assignment to <math alttext="S" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.11.11.m11.1"><semantics id="S3.Thmtheorem2.p6.11.11.m11.1a"><mi id="S3.Thmtheorem2.p6.11.11.m11.1.1" xref="S3.Thmtheorem2.p6.11.11.m11.1.1.cmml">S</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.11.11.m11.1b"><ci id="S3.Thmtheorem2.p6.11.11.m11.1.1.cmml" xref="S3.Thmtheorem2.p6.11.11.m11.1.1">𝑆</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.11.11.m11.1c">S</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.11.11.m11.1d">italic_S</annotation></semantics></math>. Once we are done, each fact in the relation matching the neighbor of <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.12.12.m12.1"><semantics id="S3.Thmtheorem2.p6.12.12.m12.1a"><mi id="S3.Thmtheorem2.p6.12.12.m12.1.1" xref="S3.Thmtheorem2.p6.12.12.m12.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.12.12.m12.1b"><ci id="S3.Thmtheorem2.p6.12.12.m12.1.1.cmml" xref="S3.Thmtheorem2.p6.12.12.m12.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.12.12.m12.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.12.12.m12.1d">italic_v</annotation></semantics></math> had its annotation multiplied by the sum of the annotations of the facts that agree with it in the relation of <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p6.13.13.m13.1"><semantics id="S3.Thmtheorem2.p6.13.13.m13.1a"><mi id="S3.Thmtheorem2.p6.13.13.m13.1.1" xref="S3.Thmtheorem2.p6.13.13.m13.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p6.13.13.m13.1b"><ci id="S3.Thmtheorem2.p6.13.13.m13.1.1.cmml" xref="S3.Thmtheorem2.p6.13.13.m13.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p6.13.13.m13.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p6.13.13.m13.1d">italic_v</annotation></semantics></math>. In terms of complexity, we perform the elimination of a single vertex in loglinear time. Since the number of elimination steps is bounded by the size of the query, the entire elimination takes loglinear time.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem2.p7"> <p class="ltx_p" id="S3.Thmtheorem2.p7.4"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p7.4.4">As for the correctness, consider the variables that appear in <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p7.1.1.m1.1"><semantics id="S3.Thmtheorem2.p7.1.1.m1.1a"><mi id="S3.Thmtheorem2.p7.1.1.m1.1.1" xref="S3.Thmtheorem2.p7.1.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p7.1.1.m1.1b"><ci id="S3.Thmtheorem2.p7.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p7.1.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p7.1.1.m1.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p7.1.1.m1.1d">italic_v</annotation></semantics></math> but not in its neighbor. Since <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p7.2.2.m2.1"><semantics id="S3.Thmtheorem2.p7.2.2.m2.1a"><mi id="S3.Thmtheorem2.p7.2.2.m2.1.1" xref="S3.Thmtheorem2.p7.2.2.m2.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p7.2.2.m2.1b"><ci id="S3.Thmtheorem2.p7.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p7.2.2.m2.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p7.2.2.m2.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p7.2.2.m2.1d">italic_v</annotation></semantics></math> is chosen to be a leaf, the running intersection property guarantees that these variables only appear in <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p7.3.3.m3.1"><semantics id="S3.Thmtheorem2.p7.3.3.m3.1a"><mi id="S3.Thmtheorem2.p7.3.3.m3.1.1" xref="S3.Thmtheorem2.p7.3.3.m3.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p7.3.3.m3.1b"><ci id="S3.Thmtheorem2.p7.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p7.3.3.m3.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p7.3.3.m3.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p7.3.3.m3.1d">italic_v</annotation></semantics></math>. This means that they do not appear in the free vertices, so they are existential and must be projected out. As joins and projections are commutative over annotated databases <cite class="ltx_cite ltx_citemacro_cite">[<span class="ltx_ref ltx_missing_citation ltx_ref_self">10.1145/1265530.1265535</span>]</cite>, we can do the projection first. As these variables appear only in <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p7.4.4.m4.1"><semantics id="S3.Thmtheorem2.p7.4.4.m4.1a"><mi id="S3.Thmtheorem2.p7.4.4.m4.1.1" xref="S3.Thmtheorem2.p7.4.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p7.4.4.m4.1b"><ci id="S3.Thmtheorem2.p7.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p7.4.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p7.4.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p7.4.4.m4.1d">italic_v</annotation></semantics></math>, the projection can be done locally. Thus, the results of the modified query over the modified database remain unchanged.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem2.p8"> <p class="ltx_p" id="S3.Thmtheorem2.p8.14"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p8.14.14">Finally, we adapt the relations matching the free vertices into a database for <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p8.1.1.m1.1"><semantics id="S3.Thmtheorem2.p8.1.1.m1.1a"><msub id="S3.Thmtheorem2.p8.1.1.m1.1.2"><mi id="S3.Thmtheorem2.p8.1.1.m1.1.2.2">Q</mi><mrow id="S3.Thmtheorem2.p8.1.1.m1.1.1.1"><mo fence="false" id="S3.Thmtheorem2.p8.1.1.m1.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p8.1.1.m1.1.1.1.3">free</mi><mrow id="S3.Thmtheorem2.p8.1.1.m1.1.1.1.4"><mo id="S3.Thmtheorem2.p8.1.1.m1.1.1.1.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem2.p8.1.1.m1.1.1.1.1">Q</mi><mo id="S3.Thmtheorem2.p8.1.1.m1.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.1.1.m1.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.1.1.m1.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math>. We first claim that for each atom <math alttext="\varphi" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.2.2.m2.1"><semantics id="S3.Thmtheorem2.p8.2.2.m2.1a"><mi id="S3.Thmtheorem2.p8.2.2.m2.1.1" xref="S3.Thmtheorem2.p8.2.2.m2.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.2.2.m2.1b"><ci id="S3.Thmtheorem2.p8.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p8.2.2.m2.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.2.2.m2.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.2.2.m2.1d">italic_φ</annotation></semantics></math> of <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.3.3.m3.1"><semantics id="S3.Thmtheorem2.p8.3.3.m3.1a"><mi id="S3.Thmtheorem2.p8.3.3.m3.1.1" xref="S3.Thmtheorem2.p8.3.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.3.3.m3.1b"><ci id="S3.Thmtheorem2.p8.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p8.3.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.3.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.3.3.m3.1d">italic_Q</annotation></semantics></math>, we get that <math alttext="\mathrm{free}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.4.4.m4.1"><semantics id="S3.Thmtheorem2.p8.4.4.m4.1a"><mrow id="S3.Thmtheorem2.p8.4.4.m4.1.2" xref="S3.Thmtheorem2.p8.4.4.m4.1.2.cmml"><mi id="S3.Thmtheorem2.p8.4.4.m4.1.2.2" xref="S3.Thmtheorem2.p8.4.4.m4.1.2.2.cmml">free</mi><mo id="S3.Thmtheorem2.p8.4.4.m4.1.2.1" xref="S3.Thmtheorem2.p8.4.4.m4.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p8.4.4.m4.1.2.3.2" xref="S3.Thmtheorem2.p8.4.4.m4.1.2.cmml"><mo id="S3.Thmtheorem2.p8.4.4.m4.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p8.4.4.m4.1.2.cmml">(</mo><mi id="S3.Thmtheorem2.p8.4.4.m4.1.1" xref="S3.Thmtheorem2.p8.4.4.m4.1.1.cmml">φ</mi><mo id="S3.Thmtheorem2.p8.4.4.m4.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p8.4.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.4.4.m4.1b"><apply id="S3.Thmtheorem2.p8.4.4.m4.1.2.cmml" xref="S3.Thmtheorem2.p8.4.4.m4.1.2"><times id="S3.Thmtheorem2.p8.4.4.m4.1.2.1.cmml" xref="S3.Thmtheorem2.p8.4.4.m4.1.2.1"></times><ci id="S3.Thmtheorem2.p8.4.4.m4.1.2.2.cmml" xref="S3.Thmtheorem2.p8.4.4.m4.1.2.2">free</ci><ci id="S3.Thmtheorem2.p8.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p8.4.4.m4.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.4.4.m4.1c">\mathrm{free}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.4.4.m4.1d">roman_free ( italic_φ )</annotation></semantics></math> is contained in some free vertex. As <math alttext="T" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.5.5.m5.1"><semantics id="S3.Thmtheorem2.p8.5.5.m5.1a"><mi id="S3.Thmtheorem2.p8.5.5.m5.1.1" xref="S3.Thmtheorem2.p8.5.5.m5.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.5.5.m5.1b"><ci id="S3.Thmtheorem2.p8.5.5.m5.1.1.cmml" xref="S3.Thmtheorem2.p8.5.5.m5.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.5.5.m5.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.5.5.m5.1d">italic_T</annotation></semantics></math> is a ext-free-connex tree, <math alttext="\mathrm{vars}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.6.6.m6.1"><semantics id="S3.Thmtheorem2.p8.6.6.m6.1a"><mrow id="S3.Thmtheorem2.p8.6.6.m6.1.2" xref="S3.Thmtheorem2.p8.6.6.m6.1.2.cmml"><mi id="S3.Thmtheorem2.p8.6.6.m6.1.2.2" xref="S3.Thmtheorem2.p8.6.6.m6.1.2.2.cmml">vars</mi><mo id="S3.Thmtheorem2.p8.6.6.m6.1.2.1" xref="S3.Thmtheorem2.p8.6.6.m6.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p8.6.6.m6.1.2.3.2" xref="S3.Thmtheorem2.p8.6.6.m6.1.2.cmml"><mo id="S3.Thmtheorem2.p8.6.6.m6.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p8.6.6.m6.1.2.cmml">(</mo><mi id="S3.Thmtheorem2.p8.6.6.m6.1.1" xref="S3.Thmtheorem2.p8.6.6.m6.1.1.cmml">φ</mi><mo id="S3.Thmtheorem2.p8.6.6.m6.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p8.6.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.6.6.m6.1b"><apply id="S3.Thmtheorem2.p8.6.6.m6.1.2.cmml" xref="S3.Thmtheorem2.p8.6.6.m6.1.2"><times id="S3.Thmtheorem2.p8.6.6.m6.1.2.1.cmml" xref="S3.Thmtheorem2.p8.6.6.m6.1.2.1"></times><ci id="S3.Thmtheorem2.p8.6.6.m6.1.2.2.cmml" xref="S3.Thmtheorem2.p8.6.6.m6.1.2.2">vars</ci><ci id="S3.Thmtheorem2.p8.6.6.m6.1.1.cmml" xref="S3.Thmtheorem2.p8.6.6.m6.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.6.6.m6.1c">\mathrm{vars}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.6.6.m6.1d">roman_vars ( italic_φ )</annotation></semantics></math> is a vertex of <math alttext="T" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.7.7.m7.1"><semantics id="S3.Thmtheorem2.p8.7.7.m7.1a"><mi id="S3.Thmtheorem2.p8.7.7.m7.1.1" xref="S3.Thmtheorem2.p8.7.7.m7.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.7.7.m7.1b"><ci id="S3.Thmtheorem2.p8.7.7.m7.1.1.cmml" xref="S3.Thmtheorem2.p8.7.7.m7.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.7.7.m7.1c">T</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.7.7.m7.1d">italic_T</annotation></semantics></math>. If <math alttext="\mathrm{vars}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.8.8.m8.1"><semantics id="S3.Thmtheorem2.p8.8.8.m8.1a"><mrow id="S3.Thmtheorem2.p8.8.8.m8.1.2" xref="S3.Thmtheorem2.p8.8.8.m8.1.2.cmml"><mi id="S3.Thmtheorem2.p8.8.8.m8.1.2.2" xref="S3.Thmtheorem2.p8.8.8.m8.1.2.2.cmml">vars</mi><mo id="S3.Thmtheorem2.p8.8.8.m8.1.2.1" xref="S3.Thmtheorem2.p8.8.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p8.8.8.m8.1.2.3.2" xref="S3.Thmtheorem2.p8.8.8.m8.1.2.cmml"><mo id="S3.Thmtheorem2.p8.8.8.m8.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p8.8.8.m8.1.2.cmml">(</mo><mi id="S3.Thmtheorem2.p8.8.8.m8.1.1" xref="S3.Thmtheorem2.p8.8.8.m8.1.1.cmml">φ</mi><mo id="S3.Thmtheorem2.p8.8.8.m8.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p8.8.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.8.8.m8.1b"><apply id="S3.Thmtheorem2.p8.8.8.m8.1.2.cmml" xref="S3.Thmtheorem2.p8.8.8.m8.1.2"><times id="S3.Thmtheorem2.p8.8.8.m8.1.2.1.cmml" xref="S3.Thmtheorem2.p8.8.8.m8.1.2.1"></times><ci id="S3.Thmtheorem2.p8.8.8.m8.1.2.2.cmml" xref="S3.Thmtheorem2.p8.8.8.m8.1.2.2">vars</ci><ci id="S3.Thmtheorem2.p8.8.8.m8.1.1.cmml" xref="S3.Thmtheorem2.p8.8.8.m8.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.8.8.m8.1c">\mathrm{vars}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.8.8.m8.1d">roman_vars ( italic_φ )</annotation></semantics></math> is a free vertex, then <math alttext="\mathrm{vars}(\varphi)=\mathrm{free}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.9.9.m9.2"><semantics id="S3.Thmtheorem2.p8.9.9.m9.2a"><mrow id="S3.Thmtheorem2.p8.9.9.m9.2.3" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.cmml"><mrow id="S3.Thmtheorem2.p8.9.9.m9.2.3.2" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2.cmml"><mi id="S3.Thmtheorem2.p8.9.9.m9.2.3.2.2" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2.2.cmml">vars</mi><mo id="S3.Thmtheorem2.p8.9.9.m9.2.3.2.1" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p8.9.9.m9.2.3.2.3.2" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2.cmml"><mo id="S3.Thmtheorem2.p8.9.9.m9.2.3.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2.cmml">(</mo><mi id="S3.Thmtheorem2.p8.9.9.m9.1.1" xref="S3.Thmtheorem2.p8.9.9.m9.1.1.cmml">φ</mi><mo id="S3.Thmtheorem2.p8.9.9.m9.2.3.2.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem2.p8.9.9.m9.2.3.1" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.1.cmml">=</mo><mrow id="S3.Thmtheorem2.p8.9.9.m9.2.3.3" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3.cmml"><mi id="S3.Thmtheorem2.p8.9.9.m9.2.3.3.2" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3.2.cmml">free</mi><mo id="S3.Thmtheorem2.p8.9.9.m9.2.3.3.1" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p8.9.9.m9.2.3.3.3.2" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3.cmml"><mo id="S3.Thmtheorem2.p8.9.9.m9.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3.cmml">(</mo><mi id="S3.Thmtheorem2.p8.9.9.m9.2.2" xref="S3.Thmtheorem2.p8.9.9.m9.2.2.cmml">φ</mi><mo id="S3.Thmtheorem2.p8.9.9.m9.2.3.3.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.9.9.m9.2b"><apply id="S3.Thmtheorem2.p8.9.9.m9.2.3.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.3"><eq id="S3.Thmtheorem2.p8.9.9.m9.2.3.1.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.1"></eq><apply id="S3.Thmtheorem2.p8.9.9.m9.2.3.2.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2"><times id="S3.Thmtheorem2.p8.9.9.m9.2.3.2.1.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2.1"></times><ci id="S3.Thmtheorem2.p8.9.9.m9.2.3.2.2.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.2.2">vars</ci><ci id="S3.Thmtheorem2.p8.9.9.m9.1.1.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.1.1">𝜑</ci></apply><apply id="S3.Thmtheorem2.p8.9.9.m9.2.3.3.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3"><times id="S3.Thmtheorem2.p8.9.9.m9.2.3.3.1.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3.1"></times><ci id="S3.Thmtheorem2.p8.9.9.m9.2.3.3.2.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.3.3.2">free</ci><ci id="S3.Thmtheorem2.p8.9.9.m9.2.2.cmml" xref="S3.Thmtheorem2.p8.9.9.m9.2.2">𝜑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.9.9.m9.2c">\mathrm{vars}(\varphi)=\mathrm{free}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.9.9.m9.2d">roman_vars ( italic_φ ) = roman_free ( italic_φ )</annotation></semantics></math> is a free vertex. Otherwise, consider the nearest free vertex to <math alttext="\mathrm{vars}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.10.10.m10.1"><semantics id="S3.Thmtheorem2.p8.10.10.m10.1a"><mrow id="S3.Thmtheorem2.p8.10.10.m10.1.2" xref="S3.Thmtheorem2.p8.10.10.m10.1.2.cmml"><mi id="S3.Thmtheorem2.p8.10.10.m10.1.2.2" xref="S3.Thmtheorem2.p8.10.10.m10.1.2.2.cmml">vars</mi><mo id="S3.Thmtheorem2.p8.10.10.m10.1.2.1" xref="S3.Thmtheorem2.p8.10.10.m10.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p8.10.10.m10.1.2.3.2" xref="S3.Thmtheorem2.p8.10.10.m10.1.2.cmml"><mo id="S3.Thmtheorem2.p8.10.10.m10.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p8.10.10.m10.1.2.cmml">(</mo><mi id="S3.Thmtheorem2.p8.10.10.m10.1.1" xref="S3.Thmtheorem2.p8.10.10.m10.1.1.cmml">φ</mi><mo id="S3.Thmtheorem2.p8.10.10.m10.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p8.10.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.10.10.m10.1b"><apply id="S3.Thmtheorem2.p8.10.10.m10.1.2.cmml" xref="S3.Thmtheorem2.p8.10.10.m10.1.2"><times id="S3.Thmtheorem2.p8.10.10.m10.1.2.1.cmml" xref="S3.Thmtheorem2.p8.10.10.m10.1.2.1"></times><ci id="S3.Thmtheorem2.p8.10.10.m10.1.2.2.cmml" xref="S3.Thmtheorem2.p8.10.10.m10.1.2.2">vars</ci><ci id="S3.Thmtheorem2.p8.10.10.m10.1.1.cmml" xref="S3.Thmtheorem2.p8.10.10.m10.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.10.10.m10.1c">\mathrm{vars}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.10.10.m10.1d">roman_vars ( italic_φ )</annotation></semantics></math>. Since <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.11.11.m11.1"><semantics id="S3.Thmtheorem2.p8.11.11.m11.1a"><msup id="S3.Thmtheorem2.p8.11.11.m11.1.1" xref="S3.Thmtheorem2.p8.11.11.m11.1.1.cmml"><mi id="S3.Thmtheorem2.p8.11.11.m11.1.1.2" xref="S3.Thmtheorem2.p8.11.11.m11.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p8.11.11.m11.1.1.3" xref="S3.Thmtheorem2.p8.11.11.m11.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.11.11.m11.1b"><apply id="S3.Thmtheorem2.p8.11.11.m11.1.1.cmml" xref="S3.Thmtheorem2.p8.11.11.m11.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p8.11.11.m11.1.1.1.cmml" xref="S3.Thmtheorem2.p8.11.11.m11.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p8.11.11.m11.1.1.2.cmml" xref="S3.Thmtheorem2.p8.11.11.m11.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p8.11.11.m11.1.1.3.cmml" xref="S3.Thmtheorem2.p8.11.11.m11.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.11.11.m11.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.11.11.m11.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> contains all free variables of <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.12.12.m12.1"><semantics id="S3.Thmtheorem2.p8.12.12.m12.1a"><mi id="S3.Thmtheorem2.p8.12.12.m12.1.1" xref="S3.Thmtheorem2.p8.12.12.m12.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.12.12.m12.1b"><ci id="S3.Thmtheorem2.p8.12.12.m12.1.1.cmml" xref="S3.Thmtheorem2.p8.12.12.m12.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.12.12.m12.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.12.12.m12.1d">italic_Q</annotation></semantics></math>, in particular <math alttext="\mathrm{free}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.13.13.m13.1"><semantics id="S3.Thmtheorem2.p8.13.13.m13.1a"><mrow id="S3.Thmtheorem2.p8.13.13.m13.1.2" xref="S3.Thmtheorem2.p8.13.13.m13.1.2.cmml"><mi id="S3.Thmtheorem2.p8.13.13.m13.1.2.2" xref="S3.Thmtheorem2.p8.13.13.m13.1.2.2.cmml">free</mi><mo id="S3.Thmtheorem2.p8.13.13.m13.1.2.1" xref="S3.Thmtheorem2.p8.13.13.m13.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p8.13.13.m13.1.2.3.2" xref="S3.Thmtheorem2.p8.13.13.m13.1.2.cmml"><mo id="S3.Thmtheorem2.p8.13.13.m13.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p8.13.13.m13.1.2.cmml">(</mo><mi id="S3.Thmtheorem2.p8.13.13.m13.1.1" xref="S3.Thmtheorem2.p8.13.13.m13.1.1.cmml">φ</mi><mo id="S3.Thmtheorem2.p8.13.13.m13.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p8.13.13.m13.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.13.13.m13.1b"><apply id="S3.Thmtheorem2.p8.13.13.m13.1.2.cmml" xref="S3.Thmtheorem2.p8.13.13.m13.1.2"><times id="S3.Thmtheorem2.p8.13.13.m13.1.2.1.cmml" xref="S3.Thmtheorem2.p8.13.13.m13.1.2.1"></times><ci id="S3.Thmtheorem2.p8.13.13.m13.1.2.2.cmml" xref="S3.Thmtheorem2.p8.13.13.m13.1.2.2">free</ci><ci id="S3.Thmtheorem2.p8.13.13.m13.1.1.cmml" xref="S3.Thmtheorem2.p8.13.13.m13.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.13.13.m13.1c">\mathrm{free}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.13.13.m13.1d">roman_free ( italic_φ )</annotation></semantics></math>, then from the running intersection property, we know that the nearest vertex must also contain <math alttext="\mathrm{free}(\varphi)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p8.14.14.m14.1"><semantics id="S3.Thmtheorem2.p8.14.14.m14.1a"><mrow id="S3.Thmtheorem2.p8.14.14.m14.1.2" xref="S3.Thmtheorem2.p8.14.14.m14.1.2.cmml"><mi id="S3.Thmtheorem2.p8.14.14.m14.1.2.2" xref="S3.Thmtheorem2.p8.14.14.m14.1.2.2.cmml">free</mi><mo id="S3.Thmtheorem2.p8.14.14.m14.1.2.1" xref="S3.Thmtheorem2.p8.14.14.m14.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p8.14.14.m14.1.2.3.2" xref="S3.Thmtheorem2.p8.14.14.m14.1.2.cmml"><mo id="S3.Thmtheorem2.p8.14.14.m14.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p8.14.14.m14.1.2.cmml">(</mo><mi id="S3.Thmtheorem2.p8.14.14.m14.1.1" xref="S3.Thmtheorem2.p8.14.14.m14.1.1.cmml">φ</mi><mo id="S3.Thmtheorem2.p8.14.14.m14.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem2.p8.14.14.m14.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p8.14.14.m14.1b"><apply id="S3.Thmtheorem2.p8.14.14.m14.1.2.cmml" xref="S3.Thmtheorem2.p8.14.14.m14.1.2"><times id="S3.Thmtheorem2.p8.14.14.m14.1.2.1.cmml" xref="S3.Thmtheorem2.p8.14.14.m14.1.2.1"></times><ci id="S3.Thmtheorem2.p8.14.14.m14.1.2.2.cmml" xref="S3.Thmtheorem2.p8.14.14.m14.1.2.2">free</ci><ci id="S3.Thmtheorem2.p8.14.14.m14.1.1.cmml" xref="S3.Thmtheorem2.p8.14.14.m14.1.1">𝜑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p8.14.14.m14.1c">\mathrm{free}(\varphi)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p8.14.14.m14.1d">roman_free ( italic_φ )</annotation></semantics></math>.</span></p> </div> <div class="ltx_para" id="S3.Thmtheorem2.p9"> <p class="ltx_p" id="S3.Thmtheorem2.p9.14"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p9.14.14">Consider a vertex of <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.1.1.m1.1"><semantics id="S3.Thmtheorem2.p9.1.1.m1.1a"><msup id="S3.Thmtheorem2.p9.1.1.m1.1.1" xref="S3.Thmtheorem2.p9.1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem2.p9.1.1.m1.1.1.2" xref="S3.Thmtheorem2.p9.1.1.m1.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p9.1.1.m1.1.1.3" xref="S3.Thmtheorem2.p9.1.1.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.1.1.m1.1b"><apply id="S3.Thmtheorem2.p9.1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p9.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p9.1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem2.p9.1.1.m1.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p9.1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem2.p9.1.1.m1.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p9.1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem2.p9.1.1.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.1.1.m1.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.1.1.m1.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> which does not match an atom of <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p9.2.2.m2.1"><semantics id="S3.Thmtheorem2.p9.2.2.m2.1a"><msub id="S3.Thmtheorem2.p9.2.2.m2.1.2"><mi id="S3.Thmtheorem2.p9.2.2.m2.1.2.2">Q</mi><mrow id="S3.Thmtheorem2.p9.2.2.m2.1.1.1"><mo fence="false" id="S3.Thmtheorem2.p9.2.2.m2.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p9.2.2.m2.1.1.1.3">free</mi><mrow id="S3.Thmtheorem2.p9.2.2.m2.1.1.1.4"><mo id="S3.Thmtheorem2.p9.2.2.m2.1.1.1.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem2.p9.2.2.m2.1.1.1.1">Q</mi><mo id="S3.Thmtheorem2.p9.2.2.m2.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.2.2.m2.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.2.2.m2.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math>. Since it is a vertex of <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.3.3.m3.1"><semantics id="S3.Thmtheorem2.p9.3.3.m3.1a"><msup id="S3.Thmtheorem2.p9.3.3.m3.1.1" xref="S3.Thmtheorem2.p9.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem2.p9.3.3.m3.1.1.2" xref="S3.Thmtheorem2.p9.3.3.m3.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p9.3.3.m3.1.1.3" xref="S3.Thmtheorem2.p9.3.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.3.3.m3.1b"><apply id="S3.Thmtheorem2.p9.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p9.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p9.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem2.p9.3.3.m3.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p9.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem2.p9.3.3.m3.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p9.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem2.p9.3.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.3.3.m3.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.3.3.m3.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>, we know that it only contains free variables, and that its variables are contained in some atom of <math alttext="Q" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.4.4.m4.1"><semantics id="S3.Thmtheorem2.p9.4.4.m4.1a"><mi id="S3.Thmtheorem2.p9.4.4.m4.1.1" xref="S3.Thmtheorem2.p9.4.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.4.4.m4.1b"><ci id="S3.Thmtheorem2.p9.4.4.m4.1.1.cmml" xref="S3.Thmtheorem2.p9.4.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.4.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.4.4.m4.1d">italic_Q</annotation></semantics></math>. This means that its variables are also contained in some atom of <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p9.5.5.m5.1"><semantics id="S3.Thmtheorem2.p9.5.5.m5.1a"><msub id="S3.Thmtheorem2.p9.5.5.m5.1.2"><mi id="S3.Thmtheorem2.p9.5.5.m5.1.2.2">Q</mi><mrow id="S3.Thmtheorem2.p9.5.5.m5.1.1.1"><mo fence="false" id="S3.Thmtheorem2.p9.5.5.m5.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p9.5.5.m5.1.1.1.3">free</mi><mrow id="S3.Thmtheorem2.p9.5.5.m5.1.1.1.4"><mo id="S3.Thmtheorem2.p9.5.5.m5.1.1.1.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem2.p9.5.5.m5.1.1.1.1">Q</mi><mo id="S3.Thmtheorem2.p9.5.5.m5.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.5.5.m5.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.5.5.m5.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> and thus some other vertex of <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.6.6.m6.1"><semantics id="S3.Thmtheorem2.p9.6.6.m6.1a"><msup id="S3.Thmtheorem2.p9.6.6.m6.1.1" xref="S3.Thmtheorem2.p9.6.6.m6.1.1.cmml"><mi id="S3.Thmtheorem2.p9.6.6.m6.1.1.2" xref="S3.Thmtheorem2.p9.6.6.m6.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p9.6.6.m6.1.1.3" xref="S3.Thmtheorem2.p9.6.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.6.6.m6.1b"><apply id="S3.Thmtheorem2.p9.6.6.m6.1.1.cmml" xref="S3.Thmtheorem2.p9.6.6.m6.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p9.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem2.p9.6.6.m6.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p9.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem2.p9.6.6.m6.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p9.6.6.m6.1.1.3.cmml" xref="S3.Thmtheorem2.p9.6.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.6.6.m6.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.6.6.m6.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. Therefore, we can eliminate <math alttext="v" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.7.7.m7.1"><semantics id="S3.Thmtheorem2.p9.7.7.m7.1a"><mi id="S3.Thmtheorem2.p9.7.7.m7.1.1" xref="S3.Thmtheorem2.p9.7.7.m7.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.7.7.m7.1b"><ci id="S3.Thmtheorem2.p9.7.7.m7.1.1.cmml" xref="S3.Thmtheorem2.p9.7.7.m7.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.7.7.m7.1c">v</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.7.7.m7.1d">italic_v</annotation></semantics></math> (and its relation counterpart) by joining its relation into the relation of the containing vertex in the same fashion as in step (<a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S3.I5.i2" title="Item 2 ‣ Proof 3.2. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">2</span></a>). Next, consider an atom of <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem2.p9.8.8.m8.1"><semantics id="S3.Thmtheorem2.p9.8.8.m8.1a"><msub id="S3.Thmtheorem2.p9.8.8.m8.1.2"><mi id="S3.Thmtheorem2.p9.8.8.m8.1.2.2">Q</mi><mrow id="S3.Thmtheorem2.p9.8.8.m8.1.1.1"><mo fence="false" id="S3.Thmtheorem2.p9.8.8.m8.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="S3.Thmtheorem2.p9.8.8.m8.1.1.1.3">free</mi><mrow id="S3.Thmtheorem2.p9.8.8.m8.1.1.1.4"><mo id="S3.Thmtheorem2.p9.8.8.m8.1.1.1.4.1" stretchy="false">(</mo><mi id="S3.Thmtheorem2.p9.8.8.m8.1.1.1.1">Q</mi><mo id="S3.Thmtheorem2.p9.8.8.m8.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.8.8.m8.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.8.8.m8.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> which does not have a matching vertex in <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.9.9.m9.1"><semantics id="S3.Thmtheorem2.p9.9.9.m9.1a"><msup id="S3.Thmtheorem2.p9.9.9.m9.1.1" xref="S3.Thmtheorem2.p9.9.9.m9.1.1.cmml"><mi id="S3.Thmtheorem2.p9.9.9.m9.1.1.2" xref="S3.Thmtheorem2.p9.9.9.m9.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p9.9.9.m9.1.1.3" xref="S3.Thmtheorem2.p9.9.9.m9.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.9.9.m9.1b"><apply id="S3.Thmtheorem2.p9.9.9.m9.1.1.cmml" xref="S3.Thmtheorem2.p9.9.9.m9.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p9.9.9.m9.1.1.1.cmml" xref="S3.Thmtheorem2.p9.9.9.m9.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p9.9.9.m9.1.1.2.cmml" xref="S3.Thmtheorem2.p9.9.9.m9.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p9.9.9.m9.1.1.3.cmml" xref="S3.Thmtheorem2.p9.9.9.m9.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.9.9.m9.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.9.9.m9.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. We know that <math alttext="T^{\prime}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.10.10.m10.1"><semantics id="S3.Thmtheorem2.p9.10.10.m10.1a"><msup id="S3.Thmtheorem2.p9.10.10.m10.1.1" xref="S3.Thmtheorem2.p9.10.10.m10.1.1.cmml"><mi id="S3.Thmtheorem2.p9.10.10.m10.1.1.2" xref="S3.Thmtheorem2.p9.10.10.m10.1.1.2.cmml">T</mi><mo id="S3.Thmtheorem2.p9.10.10.m10.1.1.3" xref="S3.Thmtheorem2.p9.10.10.m10.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.10.10.m10.1b"><apply id="S3.Thmtheorem2.p9.10.10.m10.1.1.cmml" xref="S3.Thmtheorem2.p9.10.10.m10.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p9.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem2.p9.10.10.m10.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p9.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem2.p9.10.10.m10.1.1.2">𝑇</ci><ci id="S3.Thmtheorem2.p9.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem2.p9.10.10.m10.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.10.10.m10.1c">T^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.10.10.m10.1d">italic_T start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> has a vertex <math alttext="u" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.11.11.m11.1"><semantics id="S3.Thmtheorem2.p9.11.11.m11.1a"><mi id="S3.Thmtheorem2.p9.11.11.m11.1.1" xref="S3.Thmtheorem2.p9.11.11.m11.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.11.11.m11.1b"><ci id="S3.Thmtheorem2.p9.11.11.m11.1.1.cmml" xref="S3.Thmtheorem2.p9.11.11.m11.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.11.11.m11.1c">u</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.11.11.m11.1d">italic_u</annotation></semantics></math> containing this atom. For this atom, we take the relation of <math alttext="u" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.12.12.m12.1"><semantics id="S3.Thmtheorem2.p9.12.12.m12.1a"><mi id="S3.Thmtheorem2.p9.12.12.m12.1.1" xref="S3.Thmtheorem2.p9.12.12.m12.1.1.cmml">u</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.12.12.m12.1b"><ci id="S3.Thmtheorem2.p9.12.12.m12.1.1.cmml" xref="S3.Thmtheorem2.p9.12.12.m12.1.1">𝑢</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.12.12.m12.1c">u</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.12.12.m12.1d">italic_u</annotation></semantics></math> (without the annotations), project it accordingly, and annotate the facts with <math alttext="\mathord{\bar{1}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.13.13.m13.1"><semantics id="S3.Thmtheorem2.p9.13.13.m13.1a"><mover accent="true" id="S3.Thmtheorem2.p9.13.13.m13.1.1" xref="S3.Thmtheorem2.p9.13.13.m13.1.1a.cmml"><mn id="S3.Thmtheorem2.p9.13.13.m13.1.1.2" xref="S3.Thmtheorem2.p9.13.13.m13.1.1.2.cmml">1</mn><mo id="S3.Thmtheorem2.p9.13.13.m13.1.1.1" xref="S3.Thmtheorem2.p9.13.13.m13.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.13.13.m13.1b"><ci id="S3.Thmtheorem2.p9.13.13.m13.1.1a.cmml" xref="S3.Thmtheorem2.p9.13.13.m13.1.1"><mover accent="true" id="S3.Thmtheorem2.p9.13.13.m13.1.1.cmml" xref="S3.Thmtheorem2.p9.13.13.m13.1.1"><mn id="S3.Thmtheorem2.p9.13.13.m13.1.1.2.cmml" xref="S3.Thmtheorem2.p9.13.13.m13.1.1.2">1</mn><mo id="S3.Thmtheorem2.p9.13.13.m13.1.1.1.cmml" xref="S3.Thmtheorem2.p9.13.13.m13.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.13.13.m13.1c">\mathord{\bar{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.13.13.m13.1d">start_ID over¯ start_ARG 1 end_ARG end_ID</annotation></semantics></math>. As we only added facts projected from existing relations and annotated by <math alttext="\mathord{\bar{1}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p9.14.14.m14.1"><semantics id="S3.Thmtheorem2.p9.14.14.m14.1a"><mover accent="true" id="S3.Thmtheorem2.p9.14.14.m14.1.1" xref="S3.Thmtheorem2.p9.14.14.m14.1.1a.cmml"><mn id="S3.Thmtheorem2.p9.14.14.m14.1.1.2" xref="S3.Thmtheorem2.p9.14.14.m14.1.1.2.cmml">1</mn><mo id="S3.Thmtheorem2.p9.14.14.m14.1.1.1" xref="S3.Thmtheorem2.p9.14.14.m14.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p9.14.14.m14.1b"><ci id="S3.Thmtheorem2.p9.14.14.m14.1.1a.cmml" xref="S3.Thmtheorem2.p9.14.14.m14.1.1"><mover accent="true" id="S3.Thmtheorem2.p9.14.14.m14.1.1.cmml" xref="S3.Thmtheorem2.p9.14.14.m14.1.1"><mn id="S3.Thmtheorem2.p9.14.14.m14.1.1.2.cmml" xref="S3.Thmtheorem2.p9.14.14.m14.1.1.2">1</mn><mo id="S3.Thmtheorem2.p9.14.14.m14.1.1.1.cmml" xref="S3.Thmtheorem2.p9.14.14.m14.1.1.1">¯</mo></mover></ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p9.14.14.m14.1c">\mathord{\bar{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p9.14.14.m14.1d">start_ID over¯ start_ARG 1 end_ARG end_ID</annotation></semantics></math>, the query answers and annotations remain unchanged.</span></p> </div> </div> <div class="ltx_para" id="Thmthm7.p3"> <p class="ltx_p" id="Thmthm7.p3.4"><span class="ltx_text ltx_font_italic" id="Thmthm7.p3.4.4">We note that in the case where the semiring operations require only constant time, or in the case that <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm6" title="Lemma 6. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a> is applied to CQs or AggCQs, the preprocessing overhead of <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm6" title="Lemma 6. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm8" title="Lemma 8. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">8</span></a> is linear time instead of loglinear time. In addition, when using <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm8" title="Lemma 8. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">8</span></a>, two free variables are neighbors in <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="Thmthm7.p3.1.1.m1.1"><semantics id="Thmthm7.p3.1.1.m1.1a"><msub id="Thmthm7.p3.1.1.m1.1.2"><mi id="Thmthm7.p3.1.1.m1.1.2.2">Q</mi><mrow id="Thmthm7.p3.1.1.m1.1.1.1"><mo fence="false" id="Thmthm7.p3.1.1.m1.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="Thmthm7.p3.1.1.m1.1.1.1.3">free</mi><mrow id="Thmthm7.p3.1.1.m1.1.1.1.4"><mo id="Thmthm7.p3.1.1.m1.1.1.1.4.1" stretchy="false">(</mo><mi id="Thmthm7.p3.1.1.m1.1.1.1.1">Q</mi><mo id="Thmthm7.p3.1.1.m1.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="Thmthm7.p3.1.1.m1.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p3.1.1.m1.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> if and only if they are neighbors in <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm7.p3.2.2.m2.1"><semantics id="Thmthm7.p3.2.2.m2.1a"><mi id="Thmthm7.p3.2.2.m2.1.1" xref="Thmthm7.p3.2.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p3.2.2.m2.1b"><ci id="Thmthm7.p3.2.2.m2.1.1.cmml" xref="Thmthm7.p3.2.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p3.2.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p3.2.2.m2.1d">italic_Q</annotation></semantics></math> and since the order remains the same, <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="Thmthm7.p3.3.3.m3.1"><semantics id="Thmthm7.p3.3.3.m3.1a"><msub id="Thmthm7.p3.3.3.m3.1.2"><mi id="Thmthm7.p3.3.3.m3.1.2.2">Q</mi><mrow id="Thmthm7.p3.3.3.m3.1.1.1"><mo fence="false" id="Thmthm7.p3.3.3.m3.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="Thmthm7.p3.3.3.m3.1.1.1.3">free</mi><mrow id="Thmthm7.p3.3.3.m3.1.1.1.4"><mo id="Thmthm7.p3.3.3.m3.1.1.1.4.1" stretchy="false">(</mo><mi id="Thmthm7.p3.3.3.m3.1.1.1.1">Q</mi><mo id="Thmthm7.p3.3.3.m3.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="Thmthm7.p3.3.3.m3.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p3.3.3.m3.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> will have a disruptive trio if and only if <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm7.p3.4.4.m4.1"><semantics id="Thmthm7.p3.4.4.m4.1a"><mi id="Thmthm7.p3.4.4.m4.1.1" xref="Thmthm7.p3.4.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p3.4.4.m4.1b"><ci id="Thmthm7.p3.4.4.m4.1.1.cmml" xref="Thmthm7.p3.4.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p3.4.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p3.4.4.m4.1d">italic_Q</annotation></semantics></math> has a disruptive trio.</span></p> </div> <div class="ltx_para" id="Thmthm7.p4"> <p class="ltx_p" id="Thmthm7.p4.3"><span class="ltx_text ltx_font_italic" id="Thmthm7.p4.3.3">By putting <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm6" title="Lemma 6. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">6</span></a> (with <math alttext="V=\mathrm{free}{(Q)}" class="ltx_Math" display="inline" id="Thmthm7.p4.1.1.m1.1"><semantics id="Thmthm7.p4.1.1.m1.1a"><mrow id="Thmthm7.p4.1.1.m1.1.2" xref="Thmthm7.p4.1.1.m1.1.2.cmml"><mi id="Thmthm7.p4.1.1.m1.1.2.2" xref="Thmthm7.p4.1.1.m1.1.2.2.cmml">V</mi><mo id="Thmthm7.p4.1.1.m1.1.2.1" xref="Thmthm7.p4.1.1.m1.1.2.1.cmml">=</mo><mrow id="Thmthm7.p4.1.1.m1.1.2.3" xref="Thmthm7.p4.1.1.m1.1.2.3.cmml"><mi id="Thmthm7.p4.1.1.m1.1.2.3.2" xref="Thmthm7.p4.1.1.m1.1.2.3.2.cmml">free</mi><mo id="Thmthm7.p4.1.1.m1.1.2.3.1" xref="Thmthm7.p4.1.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="Thmthm7.p4.1.1.m1.1.2.3.3.2" xref="Thmthm7.p4.1.1.m1.1.2.3.cmml"><mo id="Thmthm7.p4.1.1.m1.1.2.3.3.2.1" stretchy="false" xref="Thmthm7.p4.1.1.m1.1.2.3.cmml">(</mo><mi id="Thmthm7.p4.1.1.m1.1.1" xref="Thmthm7.p4.1.1.m1.1.1.cmml">Q</mi><mo id="Thmthm7.p4.1.1.m1.1.2.3.3.2.2" stretchy="false" xref="Thmthm7.p4.1.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm7.p4.1.1.m1.1b"><apply id="Thmthm7.p4.1.1.m1.1.2.cmml" xref="Thmthm7.p4.1.1.m1.1.2"><eq id="Thmthm7.p4.1.1.m1.1.2.1.cmml" xref="Thmthm7.p4.1.1.m1.1.2.1"></eq><ci id="Thmthm7.p4.1.1.m1.1.2.2.cmml" xref="Thmthm7.p4.1.1.m1.1.2.2">𝑉</ci><apply id="Thmthm7.p4.1.1.m1.1.2.3.cmml" xref="Thmthm7.p4.1.1.m1.1.2.3"><times id="Thmthm7.p4.1.1.m1.1.2.3.1.cmml" xref="Thmthm7.p4.1.1.m1.1.2.3.1"></times><ci id="Thmthm7.p4.1.1.m1.1.2.3.2.cmml" xref="Thmthm7.p4.1.1.m1.1.2.3.2">free</ci><ci id="Thmthm7.p4.1.1.m1.1.1.cmml" xref="Thmthm7.p4.1.1.m1.1.1">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p4.1.1.m1.1c">V=\mathrm{free}{(Q)}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p4.1.1.m1.1d">italic_V = roman_free ( italic_Q )</annotation></semantics></math>) and <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm8" title="Lemma 8. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Lemma</span> <span class="ltx_text ltx_ref_tag">8</span></a> together, we get that <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm7.p4.2.2.m2.1"><semantics id="Thmthm7.p4.2.2.m2.1a"><mi id="Thmthm7.p4.2.2.m2.1.1" xref="Thmthm7.p4.2.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p4.2.2.m2.1b"><ci id="Thmthm7.p4.2.2.m2.1.1.cmml" xref="Thmthm7.p4.2.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p4.2.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p4.2.2.m2.1d">italic_Q</annotation></semantics></math> and <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="Thmthm7.p4.3.3.m3.1"><semantics id="Thmthm7.p4.3.3.m3.1a"><msub id="Thmthm7.p4.3.3.m3.1.2"><mi id="Thmthm7.p4.3.3.m3.1.2.2">Q</mi><mrow id="Thmthm7.p4.3.3.m3.1.1.1"><mo fence="false" id="Thmthm7.p4.3.3.m3.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="Thmthm7.p4.3.3.m3.1.1.1.3">free</mi><mrow id="Thmthm7.p4.3.3.m3.1.1.1.4"><mo id="Thmthm7.p4.3.3.m3.1.1.1.4.1" stretchy="false">(</mo><mi id="Thmthm7.p4.3.3.m3.1.1.1.1">Q</mi><mo id="Thmthm7.p4.3.3.m3.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="Thmthm7.p4.3.3.m3.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p4.3.3.m3.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> are equally hard for direct access.</span></p> </div> <div class="ltx_theorem ltx_theorem_thm" id="Thmthm9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm9.1.1.1">Theorem 9</span></span><span class="ltx_text ltx_font_bold" id="Thmthm9.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm9.p1"> <p class="ltx_p" id="Thmthm9.p1.8"><span class="ltx_text ltx_font_italic" id="Thmthm9.p1.8.8">Let <math alttext="(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})" class="ltx_Math" display="inline" id="Thmthm9.p1.1.1.m1.5"><semantics id="Thmthm9.p1.1.1.m1.5a"><mrow id="Thmthm9.p1.1.1.m1.5.6.2" xref="Thmthm9.p1.1.1.m1.5.6.1.cmml"><mo id="Thmthm9.p1.1.1.m1.5.6.2.1" stretchy="false" xref="Thmthm9.p1.1.1.m1.5.6.1.cmml">(</mo><mi id="Thmthm9.p1.1.1.m1.1.1" xref="Thmthm9.p1.1.1.m1.1.1.cmml">𝕂</mi><mo id="Thmthm9.p1.1.1.m1.5.6.2.2" rspace="0em" xref="Thmthm9.p1.1.1.m1.5.6.1.cmml">,</mo><mo id="Thmthm9.p1.1.1.m1.2.2" lspace="0em" rspace="0em" xref="Thmthm9.p1.1.1.m1.2.2.cmml">⊕</mo><mo id="Thmthm9.p1.1.1.m1.5.6.2.3" rspace="0em" xref="Thmthm9.p1.1.1.m1.5.6.1.cmml">,</mo><mo id="Thmthm9.p1.1.1.m1.3.3" lspace="0em" rspace="0em" xref="Thmthm9.p1.1.1.m1.3.3.cmml">⊗</mo><mo id="Thmthm9.p1.1.1.m1.5.6.2.4" xref="Thmthm9.p1.1.1.m1.5.6.1.cmml">,</mo><mover accent="true" id="Thmthm9.p1.1.1.m1.4.4" xref="Thmthm9.p1.1.1.m1.4.4a.cmml"><mn id="Thmthm9.p1.1.1.m1.4.4.2" xref="Thmthm9.p1.1.1.m1.4.4.2.cmml">0</mn><mo id="Thmthm9.p1.1.1.m1.4.4.1" xref="Thmthm9.p1.1.1.m1.4.4.1.cmml">¯</mo></mover><mo id="Thmthm9.p1.1.1.m1.5.6.2.5" xref="Thmthm9.p1.1.1.m1.5.6.1.cmml">,</mo><mover accent="true" id="Thmthm9.p1.1.1.m1.5.5" xref="Thmthm9.p1.1.1.m1.5.5a.cmml"><mn id="Thmthm9.p1.1.1.m1.5.5.2" xref="Thmthm9.p1.1.1.m1.5.5.2.cmml">1</mn><mo id="Thmthm9.p1.1.1.m1.5.5.1" xref="Thmthm9.p1.1.1.m1.5.5.1.cmml">¯</mo></mover><mo id="Thmthm9.p1.1.1.m1.5.6.2.6" stretchy="false" xref="Thmthm9.p1.1.1.m1.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.1.1.m1.5b"><vector id="Thmthm9.p1.1.1.m1.5.6.1.cmml" xref="Thmthm9.p1.1.1.m1.5.6.2"><ci id="Thmthm9.p1.1.1.m1.1.1.cmml" xref="Thmthm9.p1.1.1.m1.1.1">𝕂</ci><csymbol cd="latexml" id="Thmthm9.p1.1.1.m1.2.2.cmml" xref="Thmthm9.p1.1.1.m1.2.2">direct-sum</csymbol><csymbol cd="latexml" id="Thmthm9.p1.1.1.m1.3.3.cmml" xref="Thmthm9.p1.1.1.m1.3.3">tensor-product</csymbol><ci id="Thmthm9.p1.1.1.m1.4.4a.cmml" xref="Thmthm9.p1.1.1.m1.4.4"><mover accent="true" id="Thmthm9.p1.1.1.m1.4.4.cmml" xref="Thmthm9.p1.1.1.m1.4.4"><mn id="Thmthm9.p1.1.1.m1.4.4.2.cmml" xref="Thmthm9.p1.1.1.m1.4.4.2">0</mn><mo id="Thmthm9.p1.1.1.m1.4.4.1.cmml" xref="Thmthm9.p1.1.1.m1.4.4.1">¯</mo></mover></ci><ci id="Thmthm9.p1.1.1.m1.5.5a.cmml" xref="Thmthm9.p1.1.1.m1.5.5"><mover accent="true" id="Thmthm9.p1.1.1.m1.5.5.cmml" xref="Thmthm9.p1.1.1.m1.5.5"><mn id="Thmthm9.p1.1.1.m1.5.5.2.cmml" xref="Thmthm9.p1.1.1.m1.5.5.2">1</mn><mo id="Thmthm9.p1.1.1.m1.5.5.1.cmml" xref="Thmthm9.p1.1.1.m1.5.5.1">¯</mo></mover></ci></vector></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.1.1.m1.5c">(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.1.1.m1.5d">( blackboard_K , ⊕ , ⊗ , start_ID over¯ start_ARG 0 end_ARG end_ID , start_ID over¯ start_ARG 1 end_ARG end_ID )</annotation></semantics></math> be a logarithmic-time commutative semiring, and let <math alttext="Q(\vec{x},\star,\vec{z})" class="ltx_Math" display="inline" id="Thmthm9.p1.2.2.m2.3"><semantics id="Thmthm9.p1.2.2.m2.3a"><mrow id="Thmthm9.p1.2.2.m2.3.4" xref="Thmthm9.p1.2.2.m2.3.4.cmml"><mi id="Thmthm9.p1.2.2.m2.3.4.2" xref="Thmthm9.p1.2.2.m2.3.4.2.cmml">Q</mi><mo id="Thmthm9.p1.2.2.m2.3.4.1" xref="Thmthm9.p1.2.2.m2.3.4.1.cmml">⁢</mo><mrow id="Thmthm9.p1.2.2.m2.3.4.3.2" xref="Thmthm9.p1.2.2.m2.3.4.3.1.cmml"><mo id="Thmthm9.p1.2.2.m2.3.4.3.2.1" stretchy="false" xref="Thmthm9.p1.2.2.m2.3.4.3.1.cmml">(</mo><mover accent="true" id="Thmthm9.p1.2.2.m2.1.1" xref="Thmthm9.p1.2.2.m2.1.1.cmml"><mi id="Thmthm9.p1.2.2.m2.1.1.2" xref="Thmthm9.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="Thmthm9.p1.2.2.m2.1.1.1" stretchy="false" xref="Thmthm9.p1.2.2.m2.1.1.1.cmml">→</mo></mover><mo id="Thmthm9.p1.2.2.m2.3.4.3.2.2" rspace="0em" xref="Thmthm9.p1.2.2.m2.3.4.3.1.cmml">,</mo><mo id="Thmthm9.p1.2.2.m2.2.2" lspace="0em" rspace="0em" xref="Thmthm9.p1.2.2.m2.2.2.cmml">⋆</mo><mo id="Thmthm9.p1.2.2.m2.3.4.3.2.3" xref="Thmthm9.p1.2.2.m2.3.4.3.1.cmml">,</mo><mover accent="true" id="Thmthm9.p1.2.2.m2.3.3" xref="Thmthm9.p1.2.2.m2.3.3.cmml"><mi id="Thmthm9.p1.2.2.m2.3.3.2" xref="Thmthm9.p1.2.2.m2.3.3.2.cmml">z</mi><mo id="Thmthm9.p1.2.2.m2.3.3.1" stretchy="false" xref="Thmthm9.p1.2.2.m2.3.3.1.cmml">→</mo></mover><mo id="Thmthm9.p1.2.2.m2.3.4.3.2.4" stretchy="false" xref="Thmthm9.p1.2.2.m2.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.2.2.m2.3b"><apply id="Thmthm9.p1.2.2.m2.3.4.cmml" xref="Thmthm9.p1.2.2.m2.3.4"><times id="Thmthm9.p1.2.2.m2.3.4.1.cmml" xref="Thmthm9.p1.2.2.m2.3.4.1"></times><ci id="Thmthm9.p1.2.2.m2.3.4.2.cmml" xref="Thmthm9.p1.2.2.m2.3.4.2">𝑄</ci><vector id="Thmthm9.p1.2.2.m2.3.4.3.1.cmml" xref="Thmthm9.p1.2.2.m2.3.4.3.2"><apply id="Thmthm9.p1.2.2.m2.1.1.cmml" xref="Thmthm9.p1.2.2.m2.1.1"><ci id="Thmthm9.p1.2.2.m2.1.1.1.cmml" xref="Thmthm9.p1.2.2.m2.1.1.1">→</ci><ci id="Thmthm9.p1.2.2.m2.1.1.2.cmml" xref="Thmthm9.p1.2.2.m2.1.1.2">𝑥</ci></apply><ci id="Thmthm9.p1.2.2.m2.2.2.cmml" xref="Thmthm9.p1.2.2.m2.2.2">⋆</ci><apply id="Thmthm9.p1.2.2.m2.3.3.cmml" xref="Thmthm9.p1.2.2.m2.3.3"><ci id="Thmthm9.p1.2.2.m2.3.3.1.cmml" xref="Thmthm9.p1.2.2.m2.3.3.1">→</ci><ci id="Thmthm9.p1.2.2.m2.3.3.2.cmml" xref="Thmthm9.p1.2.2.m2.3.3.2">𝑧</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.2.2.m2.3c">Q(\vec{x},\star,\vec{z})</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.2.2.m2.3d">italic_Q ( over→ start_ARG italic_x end_ARG , ⋆ , over→ start_ARG italic_z end_ARG )</annotation></semantics></math> be a self-join-free free-connex CQ<sup class="ltx_sup" id="Thmthm9.p1.8.8.1"><span class="ltx_text ltx_font_upright" id="Thmthm9.p1.8.8.1.1">⋆</span></sup>. Then, direct access for <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm9.p1.4.4.m4.1"><semantics id="Thmthm9.p1.4.4.m4.1a"><mi id="Thmthm9.p1.4.4.m4.1.1" xref="Thmthm9.p1.4.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.4.4.m4.1b"><ci id="Thmthm9.p1.4.4.m4.1.1.cmml" xref="Thmthm9.p1.4.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.4.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.4.4.m4.1d">italic_Q</annotation></semantics></math> is in <math alttext="\mathord{\langle T_{p},T_{a}\rangle}" class="ltx_Math" display="inline" id="Thmthm9.p1.5.5.m5.2"><semantics id="Thmthm9.p1.5.5.m5.2a"><mrow id="Thmthm9.p1.5.5.m5.2.2.2" xref="Thmthm9.p1.5.5.m5.2.2.3.cmml"><mo id="Thmthm9.p1.5.5.m5.2.2.2.3" stretchy="false" xref="Thmthm9.p1.5.5.m5.2.2.3.cmml">⟨</mo><msub id="Thmthm9.p1.5.5.m5.1.1.1.1" xref="Thmthm9.p1.5.5.m5.1.1.1.1.cmml"><mi id="Thmthm9.p1.5.5.m5.1.1.1.1.2" xref="Thmthm9.p1.5.5.m5.1.1.1.1.2.cmml">T</mi><mi id="Thmthm9.p1.5.5.m5.1.1.1.1.3" xref="Thmthm9.p1.5.5.m5.1.1.1.1.3.cmml">p</mi></msub><mo id="Thmthm9.p1.5.5.m5.2.2.2.4" xref="Thmthm9.p1.5.5.m5.2.2.3.cmml">,</mo><msub id="Thmthm9.p1.5.5.m5.2.2.2.2" xref="Thmthm9.p1.5.5.m5.2.2.2.2.cmml"><mi id="Thmthm9.p1.5.5.m5.2.2.2.2.2" xref="Thmthm9.p1.5.5.m5.2.2.2.2.2.cmml">T</mi><mi id="Thmthm9.p1.5.5.m5.2.2.2.2.3" xref="Thmthm9.p1.5.5.m5.2.2.2.2.3.cmml">a</mi></msub><mo id="Thmthm9.p1.5.5.m5.2.2.2.5" stretchy="false" xref="Thmthm9.p1.5.5.m5.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.5.5.m5.2b"><list id="Thmthm9.p1.5.5.m5.2.2.3.cmml" xref="Thmthm9.p1.5.5.m5.2.2.2"><apply id="Thmthm9.p1.5.5.m5.1.1.1.1.cmml" xref="Thmthm9.p1.5.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm9.p1.5.5.m5.1.1.1.1.1.cmml" xref="Thmthm9.p1.5.5.m5.1.1.1.1">subscript</csymbol><ci id="Thmthm9.p1.5.5.m5.1.1.1.1.2.cmml" xref="Thmthm9.p1.5.5.m5.1.1.1.1.2">𝑇</ci><ci id="Thmthm9.p1.5.5.m5.1.1.1.1.3.cmml" xref="Thmthm9.p1.5.5.m5.1.1.1.1.3">𝑝</ci></apply><apply id="Thmthm9.p1.5.5.m5.2.2.2.2.cmml" xref="Thmthm9.p1.5.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="Thmthm9.p1.5.5.m5.2.2.2.2.1.cmml" xref="Thmthm9.p1.5.5.m5.2.2.2.2">subscript</csymbol><ci id="Thmthm9.p1.5.5.m5.2.2.2.2.2.cmml" xref="Thmthm9.p1.5.5.m5.2.2.2.2.2">𝑇</ci><ci id="Thmthm9.p1.5.5.m5.2.2.2.2.3.cmml" xref="Thmthm9.p1.5.5.m5.2.2.2.2.3">𝑎</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.5.5.m5.2c">\mathord{\langle T_{p},T_{a}\rangle}</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.5.5.m5.2d">⟨ italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟩</annotation></semantics></math> if and only if direct access for <math alttext="Q_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="Thmthm9.p1.6.6.m6.1"><semantics id="Thmthm9.p1.6.6.m6.1a"><msub id="Thmthm9.p1.6.6.m6.1.2"><mi id="Thmthm9.p1.6.6.m6.1.2.2">Q</mi><mrow id="Thmthm9.p1.6.6.m6.1.1.1"><mo fence="false" id="Thmthm9.p1.6.6.m6.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="Thmthm9.p1.6.6.m6.1.1.1.3">free</mi><mrow id="Thmthm9.p1.6.6.m6.1.1.1.4"><mo id="Thmthm9.p1.6.6.m6.1.1.1.4.1" stretchy="false">(</mo><mi id="Thmthm9.p1.6.6.m6.1.1.1.1">Q</mi><mo id="Thmthm9.p1.6.6.m6.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow></msub><annotation encoding="application/x-tex" id="Thmthm9.p1.6.6.m6.1b">Q_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.6.6.m6.1c">italic_Q start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> is in <math alttext="\mathord{\langle T_{p},T_{a}\rangle}" class="ltx_Math" display="inline" id="Thmthm9.p1.7.7.m7.2"><semantics id="Thmthm9.p1.7.7.m7.2a"><mrow id="Thmthm9.p1.7.7.m7.2.2.2" xref="Thmthm9.p1.7.7.m7.2.2.3.cmml"><mo id="Thmthm9.p1.7.7.m7.2.2.2.3" stretchy="false" xref="Thmthm9.p1.7.7.m7.2.2.3.cmml">⟨</mo><msub id="Thmthm9.p1.7.7.m7.1.1.1.1" xref="Thmthm9.p1.7.7.m7.1.1.1.1.cmml"><mi id="Thmthm9.p1.7.7.m7.1.1.1.1.2" xref="Thmthm9.p1.7.7.m7.1.1.1.1.2.cmml">T</mi><mi id="Thmthm9.p1.7.7.m7.1.1.1.1.3" xref="Thmthm9.p1.7.7.m7.1.1.1.1.3.cmml">p</mi></msub><mo id="Thmthm9.p1.7.7.m7.2.2.2.4" xref="Thmthm9.p1.7.7.m7.2.2.3.cmml">,</mo><msub id="Thmthm9.p1.7.7.m7.2.2.2.2" xref="Thmthm9.p1.7.7.m7.2.2.2.2.cmml"><mi id="Thmthm9.p1.7.7.m7.2.2.2.2.2" xref="Thmthm9.p1.7.7.m7.2.2.2.2.2.cmml">T</mi><mi id="Thmthm9.p1.7.7.m7.2.2.2.2.3" xref="Thmthm9.p1.7.7.m7.2.2.2.2.3.cmml">a</mi></msub><mo id="Thmthm9.p1.7.7.m7.2.2.2.5" stretchy="false" xref="Thmthm9.p1.7.7.m7.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.7.7.m7.2b"><list id="Thmthm9.p1.7.7.m7.2.2.3.cmml" xref="Thmthm9.p1.7.7.m7.2.2.2"><apply id="Thmthm9.p1.7.7.m7.1.1.1.1.cmml" xref="Thmthm9.p1.7.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm9.p1.7.7.m7.1.1.1.1.1.cmml" xref="Thmthm9.p1.7.7.m7.1.1.1.1">subscript</csymbol><ci id="Thmthm9.p1.7.7.m7.1.1.1.1.2.cmml" xref="Thmthm9.p1.7.7.m7.1.1.1.1.2">𝑇</ci><ci id="Thmthm9.p1.7.7.m7.1.1.1.1.3.cmml" xref="Thmthm9.p1.7.7.m7.1.1.1.1.3">𝑝</ci></apply><apply id="Thmthm9.p1.7.7.m7.2.2.2.2.cmml" xref="Thmthm9.p1.7.7.m7.2.2.2.2"><csymbol cd="ambiguous" id="Thmthm9.p1.7.7.m7.2.2.2.2.1.cmml" xref="Thmthm9.p1.7.7.m7.2.2.2.2">subscript</csymbol><ci id="Thmthm9.p1.7.7.m7.2.2.2.2.2.cmml" xref="Thmthm9.p1.7.7.m7.2.2.2.2.2">𝑇</ci><ci id="Thmthm9.p1.7.7.m7.2.2.2.2.3.cmml" xref="Thmthm9.p1.7.7.m7.2.2.2.2.3">𝑎</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.7.7.m7.2c">\mathord{\langle T_{p},T_{a}\rangle}</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.7.7.m7.2d">⟨ italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_T start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ⟩</annotation></semantics></math>, assuming <math alttext="T_{p}=\Omega(\mathrm{loglinear})" class="ltx_Math" display="inline" id="Thmthm9.p1.8.8.m8.1"><semantics id="Thmthm9.p1.8.8.m8.1a"><mrow id="Thmthm9.p1.8.8.m8.1.2" xref="Thmthm9.p1.8.8.m8.1.2.cmml"><msub id="Thmthm9.p1.8.8.m8.1.2.2" xref="Thmthm9.p1.8.8.m8.1.2.2.cmml"><mi id="Thmthm9.p1.8.8.m8.1.2.2.2" xref="Thmthm9.p1.8.8.m8.1.2.2.2.cmml">T</mi><mi id="Thmthm9.p1.8.8.m8.1.2.2.3" xref="Thmthm9.p1.8.8.m8.1.2.2.3.cmml">p</mi></msub><mo id="Thmthm9.p1.8.8.m8.1.2.1" xref="Thmthm9.p1.8.8.m8.1.2.1.cmml">=</mo><mrow id="Thmthm9.p1.8.8.m8.1.2.3" xref="Thmthm9.p1.8.8.m8.1.2.3.cmml"><mi id="Thmthm9.p1.8.8.m8.1.2.3.2" mathvariant="normal" xref="Thmthm9.p1.8.8.m8.1.2.3.2.cmml">Ω</mi><mo id="Thmthm9.p1.8.8.m8.1.2.3.1" xref="Thmthm9.p1.8.8.m8.1.2.3.1.cmml">⁢</mo><mrow id="Thmthm9.p1.8.8.m8.1.2.3.3.2" xref="Thmthm9.p1.8.8.m8.1.2.3.cmml"><mo id="Thmthm9.p1.8.8.m8.1.2.3.3.2.1" stretchy="false" xref="Thmthm9.p1.8.8.m8.1.2.3.cmml">(</mo><mi id="Thmthm9.p1.8.8.m8.1.1" xref="Thmthm9.p1.8.8.m8.1.1.cmml">loglinear</mi><mo id="Thmthm9.p1.8.8.m8.1.2.3.3.2.2" stretchy="false" xref="Thmthm9.p1.8.8.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm9.p1.8.8.m8.1b"><apply id="Thmthm9.p1.8.8.m8.1.2.cmml" xref="Thmthm9.p1.8.8.m8.1.2"><eq id="Thmthm9.p1.8.8.m8.1.2.1.cmml" xref="Thmthm9.p1.8.8.m8.1.2.1"></eq><apply id="Thmthm9.p1.8.8.m8.1.2.2.cmml" xref="Thmthm9.p1.8.8.m8.1.2.2"><csymbol cd="ambiguous" id="Thmthm9.p1.8.8.m8.1.2.2.1.cmml" xref="Thmthm9.p1.8.8.m8.1.2.2">subscript</csymbol><ci id="Thmthm9.p1.8.8.m8.1.2.2.2.cmml" xref="Thmthm9.p1.8.8.m8.1.2.2.2">𝑇</ci><ci id="Thmthm9.p1.8.8.m8.1.2.2.3.cmml" xref="Thmthm9.p1.8.8.m8.1.2.2.3">𝑝</ci></apply><apply id="Thmthm9.p1.8.8.m8.1.2.3.cmml" xref="Thmthm9.p1.8.8.m8.1.2.3"><times id="Thmthm9.p1.8.8.m8.1.2.3.1.cmml" xref="Thmthm9.p1.8.8.m8.1.2.3.1"></times><ci id="Thmthm9.p1.8.8.m8.1.2.3.2.cmml" xref="Thmthm9.p1.8.8.m8.1.2.3.2">Ω</ci><ci id="Thmthm9.p1.8.8.m8.1.1.cmml" xref="Thmthm9.p1.8.8.m8.1.1">loglinear</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm9.p1.8.8.m8.1c">T_{p}=\Omega(\mathrm{loglinear})</annotation><annotation encoding="application/x-llamapun" id="Thmthm9.p1.8.8.m8.1d">italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = roman_Ω ( roman_loglinear )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="Thmthm7.p5"> <p class="ltx_p" id="Thmthm7.p5.5"><span class="ltx_text ltx_font_italic" id="Thmthm7.p5.5.5">The positive side of <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm9" title="Theorem 9. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">9</span></a> can easily be used also in case <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm7.p5.1.1.m1.1"><semantics id="Thmthm7.p5.1.1.m1.1a"><mi id="Thmthm7.p5.1.1.m1.1.1" xref="Thmthm7.p5.1.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p5.1.1.m1.1b"><ci id="Thmthm7.p5.1.1.m1.1.1.cmml" xref="Thmthm7.p5.1.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p5.1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p5.1.1.m1.1d">italic_Q</annotation></semantics></math> contains self-joins by removing the self-joins while duplicating the relevant relations. We say that a query <math alttext="Q^{\mathrm{sf}}" class="ltx_Math" display="inline" id="Thmthm7.p5.2.2.m2.1"><semantics id="Thmthm7.p5.2.2.m2.1a"><msup id="Thmthm7.p5.2.2.m2.1.1" xref="Thmthm7.p5.2.2.m2.1.1.cmml"><mi id="Thmthm7.p5.2.2.m2.1.1.2" xref="Thmthm7.p5.2.2.m2.1.1.2.cmml">Q</mi><mi id="Thmthm7.p5.2.2.m2.1.1.3" xref="Thmthm7.p5.2.2.m2.1.1.3.cmml">sf</mi></msup><annotation-xml encoding="MathML-Content" id="Thmthm7.p5.2.2.m2.1b"><apply id="Thmthm7.p5.2.2.m2.1.1.cmml" xref="Thmthm7.p5.2.2.m2.1.1"><csymbol cd="ambiguous" id="Thmthm7.p5.2.2.m2.1.1.1.cmml" xref="Thmthm7.p5.2.2.m2.1.1">superscript</csymbol><ci id="Thmthm7.p5.2.2.m2.1.1.2.cmml" xref="Thmthm7.p5.2.2.m2.1.1.2">𝑄</ci><ci id="Thmthm7.p5.2.2.m2.1.1.3.cmml" xref="Thmthm7.p5.2.2.m2.1.1.3">sf</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p5.2.2.m2.1c">Q^{\mathrm{sf}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p5.2.2.m2.1d">italic_Q start_POSTSUPERSCRIPT roman_sf end_POSTSUPERSCRIPT</annotation></semantics></math> is a self-join-free version of a query <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm7.p5.3.3.m3.1"><semantics id="Thmthm7.p5.3.3.m3.1a"><mi id="Thmthm7.p5.3.3.m3.1.1" xref="Thmthm7.p5.3.3.m3.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p5.3.3.m3.1b"><ci id="Thmthm7.p5.3.3.m3.1.1.cmml" xref="Thmthm7.p5.3.3.m3.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p5.3.3.m3.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p5.3.3.m3.1d">italic_Q</annotation></semantics></math> if it can be obtained from <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm7.p5.4.4.m4.1"><semantics id="Thmthm7.p5.4.4.m4.1a"><mi id="Thmthm7.p5.4.4.m4.1.1" xref="Thmthm7.p5.4.4.m4.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm7.p5.4.4.m4.1b"><ci id="Thmthm7.p5.4.4.m4.1.1.cmml" xref="Thmthm7.p5.4.4.m4.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p5.4.4.m4.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p5.4.4.m4.1d">italic_Q</annotation></semantics></math> by replacing the relation symbols such that each relation symbol appears at most once in <math alttext="Q^{\mathrm{sf}}" class="ltx_Math" display="inline" id="Thmthm7.p5.5.5.m5.1"><semantics id="Thmthm7.p5.5.5.m5.1a"><msup id="Thmthm7.p5.5.5.m5.1.1" xref="Thmthm7.p5.5.5.m5.1.1.cmml"><mi id="Thmthm7.p5.5.5.m5.1.1.2" xref="Thmthm7.p5.5.5.m5.1.1.2.cmml">Q</mi><mi id="Thmthm7.p5.5.5.m5.1.1.3" xref="Thmthm7.p5.5.5.m5.1.1.3.cmml">sf</mi></msup><annotation-xml encoding="MathML-Content" id="Thmthm7.p5.5.5.m5.1b"><apply id="Thmthm7.p5.5.5.m5.1.1.cmml" xref="Thmthm7.p5.5.5.m5.1.1"><csymbol cd="ambiguous" id="Thmthm7.p5.5.5.m5.1.1.1.cmml" xref="Thmthm7.p5.5.5.m5.1.1">superscript</csymbol><ci id="Thmthm7.p5.5.5.m5.1.1.2.cmml" xref="Thmthm7.p5.5.5.m5.1.1.2">𝑄</ci><ci id="Thmthm7.p5.5.5.m5.1.1.3.cmml" xref="Thmthm7.p5.5.5.m5.1.1.3">sf</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm7.p5.5.5.m5.1c">Q^{\mathrm{sf}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm7.p5.5.5.m5.1d">italic_Q start_POSTSUPERSCRIPT roman_sf end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> <div class="ltx_theorem ltx_theorem_cor" id="Thmthm10"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm10.1.1.1">Corollary 10</span></span><span class="ltx_text ltx_font_bold" id="Thmthm10.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm10.p1"> <p class="ltx_p" id="Thmthm10.p1.9"><span class="ltx_text ltx_font_italic" id="Thmthm10.p1.9.9">Let <math alttext="(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})" class="ltx_Math" display="inline" id="Thmthm10.p1.1.1.m1.5"><semantics id="Thmthm10.p1.1.1.m1.5a"><mrow id="Thmthm10.p1.1.1.m1.5.6.2" xref="Thmthm10.p1.1.1.m1.5.6.1.cmml"><mo id="Thmthm10.p1.1.1.m1.5.6.2.1" stretchy="false" xref="Thmthm10.p1.1.1.m1.5.6.1.cmml">(</mo><mi id="Thmthm10.p1.1.1.m1.1.1" xref="Thmthm10.p1.1.1.m1.1.1.cmml">𝕂</mi><mo id="Thmthm10.p1.1.1.m1.5.6.2.2" rspace="0em" xref="Thmthm10.p1.1.1.m1.5.6.1.cmml">,</mo><mo id="Thmthm10.p1.1.1.m1.2.2" lspace="0em" rspace="0em" xref="Thmthm10.p1.1.1.m1.2.2.cmml">⊕</mo><mo id="Thmthm10.p1.1.1.m1.5.6.2.3" rspace="0em" xref="Thmthm10.p1.1.1.m1.5.6.1.cmml">,</mo><mo id="Thmthm10.p1.1.1.m1.3.3" lspace="0em" rspace="0em" xref="Thmthm10.p1.1.1.m1.3.3.cmml">⊗</mo><mo id="Thmthm10.p1.1.1.m1.5.6.2.4" xref="Thmthm10.p1.1.1.m1.5.6.1.cmml">,</mo><mover accent="true" id="Thmthm10.p1.1.1.m1.4.4" xref="Thmthm10.p1.1.1.m1.4.4a.cmml"><mn id="Thmthm10.p1.1.1.m1.4.4.2" xref="Thmthm10.p1.1.1.m1.4.4.2.cmml">0</mn><mo id="Thmthm10.p1.1.1.m1.4.4.1" xref="Thmthm10.p1.1.1.m1.4.4.1.cmml">¯</mo></mover><mo id="Thmthm10.p1.1.1.m1.5.6.2.5" xref="Thmthm10.p1.1.1.m1.5.6.1.cmml">,</mo><mover accent="true" id="Thmthm10.p1.1.1.m1.5.5" xref="Thmthm10.p1.1.1.m1.5.5a.cmml"><mn id="Thmthm10.p1.1.1.m1.5.5.2" xref="Thmthm10.p1.1.1.m1.5.5.2.cmml">1</mn><mo id="Thmthm10.p1.1.1.m1.5.5.1" xref="Thmthm10.p1.1.1.m1.5.5.1.cmml">¯</mo></mover><mo id="Thmthm10.p1.1.1.m1.5.6.2.6" stretchy="false" xref="Thmthm10.p1.1.1.m1.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.1.1.m1.5b"><vector id="Thmthm10.p1.1.1.m1.5.6.1.cmml" xref="Thmthm10.p1.1.1.m1.5.6.2"><ci id="Thmthm10.p1.1.1.m1.1.1.cmml" xref="Thmthm10.p1.1.1.m1.1.1">𝕂</ci><csymbol cd="latexml" id="Thmthm10.p1.1.1.m1.2.2.cmml" xref="Thmthm10.p1.1.1.m1.2.2">direct-sum</csymbol><csymbol cd="latexml" id="Thmthm10.p1.1.1.m1.3.3.cmml" xref="Thmthm10.p1.1.1.m1.3.3">tensor-product</csymbol><ci id="Thmthm10.p1.1.1.m1.4.4a.cmml" xref="Thmthm10.p1.1.1.m1.4.4"><mover accent="true" id="Thmthm10.p1.1.1.m1.4.4.cmml" xref="Thmthm10.p1.1.1.m1.4.4"><mn id="Thmthm10.p1.1.1.m1.4.4.2.cmml" xref="Thmthm10.p1.1.1.m1.4.4.2">0</mn><mo id="Thmthm10.p1.1.1.m1.4.4.1.cmml" xref="Thmthm10.p1.1.1.m1.4.4.1">¯</mo></mover></ci><ci id="Thmthm10.p1.1.1.m1.5.5a.cmml" xref="Thmthm10.p1.1.1.m1.5.5"><mover accent="true" id="Thmthm10.p1.1.1.m1.5.5.cmml" xref="Thmthm10.p1.1.1.m1.5.5"><mn id="Thmthm10.p1.1.1.m1.5.5.2.cmml" xref="Thmthm10.p1.1.1.m1.5.5.2">1</mn><mo id="Thmthm10.p1.1.1.m1.5.5.1.cmml" xref="Thmthm10.p1.1.1.m1.5.5.1">¯</mo></mover></ci></vector></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.1.1.m1.5c">(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.1.1.m1.5d">( blackboard_K , ⊕ , ⊗ , start_ID over¯ start_ARG 0 end_ARG end_ID , start_ID over¯ start_ARG 1 end_ARG end_ID )</annotation></semantics></math> be a logarithmic-time commutative semiring, and let <math alttext="Q(\vec{x},\star,\vec{z})" class="ltx_Math" display="inline" id="Thmthm10.p1.2.2.m2.3"><semantics id="Thmthm10.p1.2.2.m2.3a"><mrow id="Thmthm10.p1.2.2.m2.3.4" xref="Thmthm10.p1.2.2.m2.3.4.cmml"><mi id="Thmthm10.p1.2.2.m2.3.4.2" xref="Thmthm10.p1.2.2.m2.3.4.2.cmml">Q</mi><mo id="Thmthm10.p1.2.2.m2.3.4.1" xref="Thmthm10.p1.2.2.m2.3.4.1.cmml">⁢</mo><mrow id="Thmthm10.p1.2.2.m2.3.4.3.2" xref="Thmthm10.p1.2.2.m2.3.4.3.1.cmml"><mo id="Thmthm10.p1.2.2.m2.3.4.3.2.1" stretchy="false" xref="Thmthm10.p1.2.2.m2.3.4.3.1.cmml">(</mo><mover accent="true" id="Thmthm10.p1.2.2.m2.1.1" xref="Thmthm10.p1.2.2.m2.1.1.cmml"><mi id="Thmthm10.p1.2.2.m2.1.1.2" xref="Thmthm10.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="Thmthm10.p1.2.2.m2.1.1.1" stretchy="false" xref="Thmthm10.p1.2.2.m2.1.1.1.cmml">→</mo></mover><mo id="Thmthm10.p1.2.2.m2.3.4.3.2.2" rspace="0em" xref="Thmthm10.p1.2.2.m2.3.4.3.1.cmml">,</mo><mo id="Thmthm10.p1.2.2.m2.2.2" lspace="0em" rspace="0em" xref="Thmthm10.p1.2.2.m2.2.2.cmml">⋆</mo><mo id="Thmthm10.p1.2.2.m2.3.4.3.2.3" xref="Thmthm10.p1.2.2.m2.3.4.3.1.cmml">,</mo><mover accent="true" id="Thmthm10.p1.2.2.m2.3.3" xref="Thmthm10.p1.2.2.m2.3.3.cmml"><mi id="Thmthm10.p1.2.2.m2.3.3.2" xref="Thmthm10.p1.2.2.m2.3.3.2.cmml">z</mi><mo id="Thmthm10.p1.2.2.m2.3.3.1" stretchy="false" xref="Thmthm10.p1.2.2.m2.3.3.1.cmml">→</mo></mover><mo id="Thmthm10.p1.2.2.m2.3.4.3.2.4" stretchy="false" xref="Thmthm10.p1.2.2.m2.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.2.2.m2.3b"><apply id="Thmthm10.p1.2.2.m2.3.4.cmml" xref="Thmthm10.p1.2.2.m2.3.4"><times id="Thmthm10.p1.2.2.m2.3.4.1.cmml" xref="Thmthm10.p1.2.2.m2.3.4.1"></times><ci id="Thmthm10.p1.2.2.m2.3.4.2.cmml" xref="Thmthm10.p1.2.2.m2.3.4.2">𝑄</ci><vector id="Thmthm10.p1.2.2.m2.3.4.3.1.cmml" xref="Thmthm10.p1.2.2.m2.3.4.3.2"><apply id="Thmthm10.p1.2.2.m2.1.1.cmml" xref="Thmthm10.p1.2.2.m2.1.1"><ci id="Thmthm10.p1.2.2.m2.1.1.1.cmml" xref="Thmthm10.p1.2.2.m2.1.1.1">→</ci><ci id="Thmthm10.p1.2.2.m2.1.1.2.cmml" xref="Thmthm10.p1.2.2.m2.1.1.2">𝑥</ci></apply><ci id="Thmthm10.p1.2.2.m2.2.2.cmml" xref="Thmthm10.p1.2.2.m2.2.2">⋆</ci><apply id="Thmthm10.p1.2.2.m2.3.3.cmml" xref="Thmthm10.p1.2.2.m2.3.3"><ci id="Thmthm10.p1.2.2.m2.3.3.1.cmml" xref="Thmthm10.p1.2.2.m2.3.3.1">→</ci><ci id="Thmthm10.p1.2.2.m2.3.3.2.cmml" xref="Thmthm10.p1.2.2.m2.3.3.2">𝑧</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.2.2.m2.3c">Q(\vec{x},\star,\vec{z})</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.2.2.m2.3d">italic_Q ( over→ start_ARG italic_x end_ARG , ⋆ , over→ start_ARG italic_z end_ARG )</annotation></semantics></math> be a free-connex CQ<sup class="ltx_sup" id="Thmthm10.p1.9.9.1"><span class="ltx_text ltx_font_upright" id="Thmthm10.p1.9.9.1.1">⋆</span></sup>. Let <math alttext="Q^{\mathrm{sf}}" class="ltx_Math" display="inline" id="Thmthm10.p1.4.4.m4.1"><semantics id="Thmthm10.p1.4.4.m4.1a"><msup id="Thmthm10.p1.4.4.m4.1.1" xref="Thmthm10.p1.4.4.m4.1.1.cmml"><mi id="Thmthm10.p1.4.4.m4.1.1.2" xref="Thmthm10.p1.4.4.m4.1.1.2.cmml">Q</mi><mi id="Thmthm10.p1.4.4.m4.1.1.3" xref="Thmthm10.p1.4.4.m4.1.1.3.cmml">sf</mi></msup><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.4.4.m4.1b"><apply id="Thmthm10.p1.4.4.m4.1.1.cmml" xref="Thmthm10.p1.4.4.m4.1.1"><csymbol cd="ambiguous" id="Thmthm10.p1.4.4.m4.1.1.1.cmml" xref="Thmthm10.p1.4.4.m4.1.1">superscript</csymbol><ci id="Thmthm10.p1.4.4.m4.1.1.2.cmml" xref="Thmthm10.p1.4.4.m4.1.1.2">𝑄</ci><ci id="Thmthm10.p1.4.4.m4.1.1.3.cmml" xref="Thmthm10.p1.4.4.m4.1.1.3">sf</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.4.4.m4.1c">Q^{\mathrm{sf}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.4.4.m4.1d">italic_Q start_POSTSUPERSCRIPT roman_sf end_POSTSUPERSCRIPT</annotation></semantics></math> be a self-join-free version of <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm10.p1.5.5.m5.1"><semantics id="Thmthm10.p1.5.5.m5.1a"><mi id="Thmthm10.p1.5.5.m5.1.1" xref="Thmthm10.p1.5.5.m5.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.5.5.m5.1b"><ci id="Thmthm10.p1.5.5.m5.1.1.cmml" xref="Thmthm10.p1.5.5.m5.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.5.5.m5.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.5.5.m5.1d">italic_Q</annotation></semantics></math>. If direct access for <math alttext="Q^{\mathrm{sf}}_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="Thmthm10.p1.6.6.m6.1"><semantics id="Thmthm10.p1.6.6.m6.1a"><msubsup id="Thmthm10.p1.6.6.m6.1.2"><mi id="Thmthm10.p1.6.6.m6.1.2.2.2">Q</mi><mrow id="Thmthm10.p1.6.6.m6.1.1.1"><mo fence="false" id="Thmthm10.p1.6.6.m6.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="Thmthm10.p1.6.6.m6.1.1.1.3">free</mi><mrow id="Thmthm10.p1.6.6.m6.1.1.1.4"><mo id="Thmthm10.p1.6.6.m6.1.1.1.4.1" stretchy="false">(</mo><mi id="Thmthm10.p1.6.6.m6.1.1.1.1">Q</mi><mo id="Thmthm10.p1.6.6.m6.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow><mi id="Thmthm10.p1.6.6.m6.1.2.2.3">sf</mi></msubsup><annotation encoding="application/x-tex" id="Thmthm10.p1.6.6.m6.1b">Q^{\mathrm{sf}}_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.6.6.m6.1c">italic_Q start_POSTSUPERSCRIPT roman_sf end_POSTSUPERSCRIPT start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math> is in <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="Thmthm10.p1.7.7.m7.2"><semantics id="Thmthm10.p1.7.7.m7.2a"><mrow id="Thmthm10.p1.7.7.m7.2.2.4" xref="Thmthm10.p1.7.7.m7.2.2.3.cmml"><mo id="Thmthm10.p1.7.7.m7.2.2.4.1" stretchy="false" xref="Thmthm10.p1.7.7.m7.2.2.3.cmml">⟨</mo><mi id="Thmthm10.p1.7.7.m7.1.1.1" xref="Thmthm10.p1.7.7.m7.1.1.1.cmml">loglinear</mi><mo id="Thmthm10.p1.7.7.m7.2.2.4.2" xref="Thmthm10.p1.7.7.m7.2.2.3.cmml">,</mo><mi id="Thmthm10.p1.7.7.m7.2.2.2" xref="Thmthm10.p1.7.7.m7.2.2.2.cmml">log</mi><mo id="Thmthm10.p1.7.7.m7.2.2.4.3" stretchy="false" xref="Thmthm10.p1.7.7.m7.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.7.7.m7.2b"><list id="Thmthm10.p1.7.7.m7.2.2.3.cmml" xref="Thmthm10.p1.7.7.m7.2.2.4"><ci id="Thmthm10.p1.7.7.m7.1.1.1.cmml" xref="Thmthm10.p1.7.7.m7.1.1.1">loglinear</ci><log id="Thmthm10.p1.7.7.m7.2.2.2.cmml" xref="Thmthm10.p1.7.7.m7.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.7.7.m7.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.7.7.m7.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math>, then direct access for <math alttext="Q" class="ltx_Math" display="inline" id="Thmthm10.p1.8.8.m8.1"><semantics id="Thmthm10.p1.8.8.m8.1a"><mi id="Thmthm10.p1.8.8.m8.1.1" xref="Thmthm10.p1.8.8.m8.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.8.8.m8.1b"><ci id="Thmthm10.p1.8.8.m8.1.1.cmml" xref="Thmthm10.p1.8.8.m8.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.8.8.m8.1c">Q</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.8.8.m8.1d">italic_Q</annotation></semantics></math> is in <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="Thmthm10.p1.9.9.m9.2"><semantics id="Thmthm10.p1.9.9.m9.2a"><mrow id="Thmthm10.p1.9.9.m9.2.2.4" xref="Thmthm10.p1.9.9.m9.2.2.3.cmml"><mo id="Thmthm10.p1.9.9.m9.2.2.4.1" stretchy="false" xref="Thmthm10.p1.9.9.m9.2.2.3.cmml">⟨</mo><mi id="Thmthm10.p1.9.9.m9.1.1.1" xref="Thmthm10.p1.9.9.m9.1.1.1.cmml">loglinear</mi><mo id="Thmthm10.p1.9.9.m9.2.2.4.2" xref="Thmthm10.p1.9.9.m9.2.2.3.cmml">,</mo><mi id="Thmthm10.p1.9.9.m9.2.2.2" xref="Thmthm10.p1.9.9.m9.2.2.2.cmml">log</mi><mo id="Thmthm10.p1.9.9.m9.2.2.4.3" stretchy="false" xref="Thmthm10.p1.9.9.m9.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm10.p1.9.9.m9.2b"><list id="Thmthm10.p1.9.9.m9.2.2.3.cmml" xref="Thmthm10.p1.9.9.m9.2.2.4"><ci id="Thmthm10.p1.9.9.m9.1.1.1.cmml" xref="Thmthm10.p1.9.9.m9.1.1.1">loglinear</ci><log id="Thmthm10.p1.9.9.m9.2.2.2.cmml" xref="Thmthm10.p1.9.9.m9.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="Thmthm10.p1.9.9.m9.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="Thmthm10.p1.9.9.m9.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math>.</span></p> </div> </div> </div> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_font_italic ltx_title_section"> <span class="ltx_tag ltx_tag_section">4. </span>Incorporating Annotation and Aggregation in the Answers</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.5"><span class="ltx_text ltx_font_italic" id="S4.p1.5.1">In this section, we discuss the existence of efficient direct access in the case where the order </span><em class="ltx_emph" id="S4.p1.5.2">does not</em><span class="ltx_text ltx_font_italic" id="S4.p1.5.3"> involve the computed value, that is, the annotation (for CQ</span><sup class="ltx_sup" id="S4.p1.5.4">⋆</sup><span class="ltx_text ltx_font_italic" id="S4.p1.5.5">s) or the aggregate values (for AggCQs). Equivalently, these are queries where the computed value is last in order, that is, the vector </span><math alttext="\vec{z}" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><mover accent="true" id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml"><mi id="S4.p1.2.m2.1.1.2" xref="S4.p1.2.m2.1.1.2.cmml">z</mi><mo id="S4.p1.2.m2.1.1.1" stretchy="false" xref="S4.p1.2.m2.1.1.1.cmml">→</mo></mover><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><apply id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1"><ci id="S4.p1.2.m2.1.1.1.cmml" xref="S4.p1.2.m2.1.1.1">→</ci><ci id="S4.p1.2.m2.1.1.2.cmml" xref="S4.p1.2.m2.1.1.2">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">\vec{z}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">over→ start_ARG italic_z end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.p1.5.6"> in the head is empty. Hence, we focus on CQ</span><sup class="ltx_sup" id="S4.p1.5.7">⋆</sup><span class="ltx_text ltx_font_italic" id="S4.p1.5.8">s of the form </span><math alttext="Q(\vec{x},\star)" class="ltx_Math" display="inline" id="S4.p1.4.m4.2"><semantics id="S4.p1.4.m4.2a"><mrow id="S4.p1.4.m4.2.3" xref="S4.p1.4.m4.2.3.cmml"><mi id="S4.p1.4.m4.2.3.2" xref="S4.p1.4.m4.2.3.2.cmml">Q</mi><mo id="S4.p1.4.m4.2.3.1" xref="S4.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S4.p1.4.m4.2.3.3.2" xref="S4.p1.4.m4.2.3.3.1.cmml"><mo id="S4.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S4.p1.4.m4.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.p1.4.m4.1.1" xref="S4.p1.4.m4.1.1.cmml"><mi id="S4.p1.4.m4.1.1.2" xref="S4.p1.4.m4.1.1.2.cmml">x</mi><mo id="S4.p1.4.m4.1.1.1" stretchy="false" xref="S4.p1.4.m4.1.1.1.cmml">→</mo></mover><mo id="S4.p1.4.m4.2.3.3.2.2" rspace="0em" xref="S4.p1.4.m4.2.3.3.1.cmml">,</mo><mo id="S4.p1.4.m4.2.2" lspace="0em" rspace="0em" xref="S4.p1.4.m4.2.2.cmml">⋆</mo><mo id="S4.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S4.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.4.m4.2b"><apply id="S4.p1.4.m4.2.3.cmml" xref="S4.p1.4.m4.2.3"><times id="S4.p1.4.m4.2.3.1.cmml" xref="S4.p1.4.m4.2.3.1"></times><ci id="S4.p1.4.m4.2.3.2.cmml" xref="S4.p1.4.m4.2.3.2">𝑄</ci><interval closure="open" id="S4.p1.4.m4.2.3.3.1.cmml" xref="S4.p1.4.m4.2.3.3.2"><apply id="S4.p1.4.m4.1.1.cmml" xref="S4.p1.4.m4.1.1"><ci id="S4.p1.4.m4.1.1.1.cmml" xref="S4.p1.4.m4.1.1.1">→</ci><ci id="S4.p1.4.m4.1.1.2.cmml" xref="S4.p1.4.m4.1.1.2">𝑥</ci></apply><ci id="S4.p1.4.m4.2.2.cmml" xref="S4.p1.4.m4.2.2">⋆</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m4.2c">Q(\vec{x},\star)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.2d">italic_Q ( over→ start_ARG italic_x end_ARG , ⋆ )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.p1.5.9"> and AggCQs of the form </span><math alttext="Q(\vec{x},\alpha(\vec{w}))" class="ltx_Math" display="inline" id="S4.p1.5.m5.3"><semantics id="S4.p1.5.m5.3a"><mrow id="S4.p1.5.m5.3.3" xref="S4.p1.5.m5.3.3.cmml"><mi id="S4.p1.5.m5.3.3.3" xref="S4.p1.5.m5.3.3.3.cmml">Q</mi><mo id="S4.p1.5.m5.3.3.2" xref="S4.p1.5.m5.3.3.2.cmml">⁢</mo><mrow id="S4.p1.5.m5.3.3.1.1" xref="S4.p1.5.m5.3.3.1.2.cmml"><mo id="S4.p1.5.m5.3.3.1.1.2" stretchy="false" xref="S4.p1.5.m5.3.3.1.2.cmml">(</mo><mover accent="true" id="S4.p1.5.m5.2.2" xref="S4.p1.5.m5.2.2.cmml"><mi id="S4.p1.5.m5.2.2.2" xref="S4.p1.5.m5.2.2.2.cmml">x</mi><mo id="S4.p1.5.m5.2.2.1" stretchy="false" xref="S4.p1.5.m5.2.2.1.cmml">→</mo></mover><mo id="S4.p1.5.m5.3.3.1.1.3" xref="S4.p1.5.m5.3.3.1.2.cmml">,</mo><mrow id="S4.p1.5.m5.3.3.1.1.1" xref="S4.p1.5.m5.3.3.1.1.1.cmml"><mi id="S4.p1.5.m5.3.3.1.1.1.2" xref="S4.p1.5.m5.3.3.1.1.1.2.cmml">α</mi><mo id="S4.p1.5.m5.3.3.1.1.1.1" xref="S4.p1.5.m5.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.p1.5.m5.3.3.1.1.1.3.2" xref="S4.p1.5.m5.1.1.cmml"><mo id="S4.p1.5.m5.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.p1.5.m5.1.1.cmml">(</mo><mover accent="true" id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml"><mi id="S4.p1.5.m5.1.1.2" xref="S4.p1.5.m5.1.1.2.cmml">w</mi><mo id="S4.p1.5.m5.1.1.1" stretchy="false" xref="S4.p1.5.m5.1.1.1.cmml">→</mo></mover><mo id="S4.p1.5.m5.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.p1.5.m5.1.1.cmml">)</mo></mrow></mrow><mo id="S4.p1.5.m5.3.3.1.1.4" stretchy="false" xref="S4.p1.5.m5.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.3b"><apply id="S4.p1.5.m5.3.3.cmml" xref="S4.p1.5.m5.3.3"><times id="S4.p1.5.m5.3.3.2.cmml" xref="S4.p1.5.m5.3.3.2"></times><ci id="S4.p1.5.m5.3.3.3.cmml" xref="S4.p1.5.m5.3.3.3">𝑄</ci><interval closure="open" id="S4.p1.5.m5.3.3.1.2.cmml" xref="S4.p1.5.m5.3.3.1.1"><apply id="S4.p1.5.m5.2.2.cmml" xref="S4.p1.5.m5.2.2"><ci id="S4.p1.5.m5.2.2.1.cmml" xref="S4.p1.5.m5.2.2.1">→</ci><ci id="S4.p1.5.m5.2.2.2.cmml" xref="S4.p1.5.m5.2.2.2">𝑥</ci></apply><apply id="S4.p1.5.m5.3.3.1.1.1.cmml" xref="S4.p1.5.m5.3.3.1.1.1"><times id="S4.p1.5.m5.3.3.1.1.1.1.cmml" xref="S4.p1.5.m5.3.3.1.1.1.1"></times><ci id="S4.p1.5.m5.3.3.1.1.1.2.cmml" xref="S4.p1.5.m5.3.3.1.1.1.2">𝛼</ci><apply id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.3.3.1.1.1.3.2"><ci id="S4.p1.5.m5.1.1.1.cmml" xref="S4.p1.5.m5.1.1.1">→</ci><ci id="S4.p1.5.m5.1.1.2.cmml" xref="S4.p1.5.m5.1.1.2">𝑤</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.3c">Q(\vec{x},\alpha(\vec{w}))</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.3d">italic_Q ( over→ start_ARG italic_x end_ARG , italic_α ( over→ start_ARG italic_w end_ARG ) )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.p1.5.10">. In other words, the problem is similar to the CQ case, except that the access algorithm should also retrieve the aggregated value from the data structure. In Section </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2303.05327v2#S4.SS1" title="4.1. Generalized Dichotomies ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">4.1</span></a><span class="ltx_text ltx_font_italic" id="S4.p1.5.11">, we will use annotated databases to identify the cases where this can be done efficiently for min, max, count, sum, and average. By contrast, in Section </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2303.05327v2#S4.SS2" title="4.2. Count Distinct ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">4.2</span></a><span class="ltx_text ltx_font_italic" id="S4.p1.5.12"> we will show that for count-distinct, the class of tractable cases is more restricted unless the domain of the elements we count is small (logarithmic size).</span></p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_font_italic ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1. </span>Generalized Dichotomies</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1"><span class="ltx_text ltx_font_italic" id="S4.SS1.p1.1.1">We now show that </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2303.05327v2#S3" title="3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3</span></a><span class="ltx_text ltx_font_italic" id="S4.SS1.p1.1.2"> extends to databases with annotations, and so, also to queries with aggregate functions that can be efficiently simulated by annotations.</span></p> </div> <div class="ltx_theorem ltx_theorem_thm" id="Thmthm11"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm11.1.1.1">Theorem 11</span></span><span class="ltx_text ltx_font_bold" id="Thmthm11.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm11.p1"> <p class="ltx_p" id="Thmthm11.p1.3"><span class="ltx_text ltx_font_italic" id="Thmthm11.p1.3.3">Let <math alttext="(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})" class="ltx_Math" display="inline" id="Thmthm11.p1.1.1.m1.5"><semantics id="Thmthm11.p1.1.1.m1.5a"><mrow id="Thmthm11.p1.1.1.m1.5.6.2" xref="Thmthm11.p1.1.1.m1.5.6.1.cmml"><mo id="Thmthm11.p1.1.1.m1.5.6.2.1" stretchy="false" xref="Thmthm11.p1.1.1.m1.5.6.1.cmml">(</mo><mi id="Thmthm11.p1.1.1.m1.1.1" xref="Thmthm11.p1.1.1.m1.1.1.cmml">𝕂</mi><mo id="Thmthm11.p1.1.1.m1.5.6.2.2" rspace="0em" xref="Thmthm11.p1.1.1.m1.5.6.1.cmml">,</mo><mo id="Thmthm11.p1.1.1.m1.2.2" lspace="0em" rspace="0em" xref="Thmthm11.p1.1.1.m1.2.2.cmml">⊕</mo><mo id="Thmthm11.p1.1.1.m1.5.6.2.3" rspace="0em" xref="Thmthm11.p1.1.1.m1.5.6.1.cmml">,</mo><mo id="Thmthm11.p1.1.1.m1.3.3" lspace="0em" rspace="0em" xref="Thmthm11.p1.1.1.m1.3.3.cmml">⊗</mo><mo id="Thmthm11.p1.1.1.m1.5.6.2.4" xref="Thmthm11.p1.1.1.m1.5.6.1.cmml">,</mo><mover accent="true" id="Thmthm11.p1.1.1.m1.4.4" xref="Thmthm11.p1.1.1.m1.4.4a.cmml"><mn id="Thmthm11.p1.1.1.m1.4.4.2" xref="Thmthm11.p1.1.1.m1.4.4.2.cmml">0</mn><mo id="Thmthm11.p1.1.1.m1.4.4.1" xref="Thmthm11.p1.1.1.m1.4.4.1.cmml">¯</mo></mover><mo id="Thmthm11.p1.1.1.m1.5.6.2.5" xref="Thmthm11.p1.1.1.m1.5.6.1.cmml">,</mo><mover accent="true" id="Thmthm11.p1.1.1.m1.5.5" xref="Thmthm11.p1.1.1.m1.5.5a.cmml"><mn id="Thmthm11.p1.1.1.m1.5.5.2" xref="Thmthm11.p1.1.1.m1.5.5.2.cmml">1</mn><mo id="Thmthm11.p1.1.1.m1.5.5.1" xref="Thmthm11.p1.1.1.m1.5.5.1.cmml">¯</mo></mover><mo id="Thmthm11.p1.1.1.m1.5.6.2.6" stretchy="false" xref="Thmthm11.p1.1.1.m1.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmthm11.p1.1.1.m1.5b"><vector id="Thmthm11.p1.1.1.m1.5.6.1.cmml" xref="Thmthm11.p1.1.1.m1.5.6.2"><ci id="Thmthm11.p1.1.1.m1.1.1.cmml" xref="Thmthm11.p1.1.1.m1.1.1">𝕂</ci><csymbol cd="latexml" id="Thmthm11.p1.1.1.m1.2.2.cmml" xref="Thmthm11.p1.1.1.m1.2.2">direct-sum</csymbol><csymbol cd="latexml" id="Thmthm11.p1.1.1.m1.3.3.cmml" xref="Thmthm11.p1.1.1.m1.3.3">tensor-product</csymbol><ci id="Thmthm11.p1.1.1.m1.4.4a.cmml" xref="Thmthm11.p1.1.1.m1.4.4"><mover accent="true" id="Thmthm11.p1.1.1.m1.4.4.cmml" xref="Thmthm11.p1.1.1.m1.4.4"><mn id="Thmthm11.p1.1.1.m1.4.4.2.cmml" xref="Thmthm11.p1.1.1.m1.4.4.2">0</mn><mo id="Thmthm11.p1.1.1.m1.4.4.1.cmml" xref="Thmthm11.p1.1.1.m1.4.4.1">¯</mo></mover></ci><ci id="Thmthm11.p1.1.1.m1.5.5a.cmml" xref="Thmthm11.p1.1.1.m1.5.5"><mover accent="true" id="Thmthm11.p1.1.1.m1.5.5.cmml" xref="Thmthm11.p1.1.1.m1.5.5"><mn id="Thmthm11.p1.1.1.m1.5.5.2.cmml" xref="Thmthm11.p1.1.1.m1.5.5.2">1</mn><mo id="Thmthm11.p1.1.1.m1.5.5.1.cmml" xref="Thmthm11.p1.1.1.m1.5.5.1">¯</mo></mover></ci></vector></annotation-xml><annotation encoding="application/x-tex" id="Thmthm11.p1.1.1.m1.5c">(\mathbb{K},\oplus,\otimes,\mathord{\bar{0}},\mathord{\bar{1}})</annotation><annotation encoding="application/x-llamapun" id="Thmthm11.p1.1.1.m1.5d">( blackboard_K , ⊕ , ⊗ , start_ID over¯ start_ARG 0 end_ARG end_ID , start_ID over¯ start_ARG 1 end_ARG end_ID )</annotation></semantics></math> be a logarithmic-time commutative semiring, and let <math alttext="Q(\vec{x},\star)" class="ltx_Math" display="inline" id="Thmthm11.p1.2.2.m2.2"><semantics id="Thmthm11.p1.2.2.m2.2a"><mrow id="Thmthm11.p1.2.2.m2.2.3" xref="Thmthm11.p1.2.2.m2.2.3.cmml"><mi id="Thmthm11.p1.2.2.m2.2.3.2" xref="Thmthm11.p1.2.2.m2.2.3.2.cmml">Q</mi><mo id="Thmthm11.p1.2.2.m2.2.3.1" xref="Thmthm11.p1.2.2.m2.2.3.1.cmml">⁢</mo><mrow id="Thmthm11.p1.2.2.m2.2.3.3.2" xref="Thmthm11.p1.2.2.m2.2.3.3.1.cmml"><mo id="Thmthm11.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="Thmthm11.p1.2.2.m2.2.3.3.1.cmml">(</mo><mover accent="true" id="Thmthm11.p1.2.2.m2.1.1" xref="Thmthm11.p1.2.2.m2.1.1.cmml"><mi id="Thmthm11.p1.2.2.m2.1.1.2" xref="Thmthm11.p1.2.2.m2.1.1.2.cmml">x</mi><mo id="Thmthm11.p1.2.2.m2.1.1.1" stretchy="false" xref="Thmthm11.p1.2.2.m2.1.1.1.cmml">→</mo></mover><mo id="Thmthm11.p1.2.2.m2.2.3.3.2.2" rspace="0em" xref="Thmthm11.p1.2.2.m2.2.3.3.1.cmml">,</mo><mo id="Thmthm11.p1.2.2.m2.2.2" lspace="0em" rspace="0em" xref="Thmthm11.p1.2.2.m2.2.2.cmml">⋆</mo><mo id="Thmthm11.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="Thmthm11.p1.2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm11.p1.2.2.m2.2b"><apply id="Thmthm11.p1.2.2.m2.2.3.cmml" xref="Thmthm11.p1.2.2.m2.2.3"><times id="Thmthm11.p1.2.2.m2.2.3.1.cmml" xref="Thmthm11.p1.2.2.m2.2.3.1"></times><ci id="Thmthm11.p1.2.2.m2.2.3.2.cmml" xref="Thmthm11.p1.2.2.m2.2.3.2">𝑄</ci><interval closure="open" id="Thmthm11.p1.2.2.m2.2.3.3.1.cmml" xref="Thmthm11.p1.2.2.m2.2.3.3.2"><apply id="Thmthm11.p1.2.2.m2.1.1.cmml" xref="Thmthm11.p1.2.2.m2.1.1"><ci id="Thmthm11.p1.2.2.m2.1.1.1.cmml" xref="Thmthm11.p1.2.2.m2.1.1.1">→</ci><ci id="Thmthm11.p1.2.2.m2.1.1.2.cmml" xref="Thmthm11.p1.2.2.m2.1.1.2">𝑥</ci></apply><ci id="Thmthm11.p1.2.2.m2.2.2.cmml" xref="Thmthm11.p1.2.2.m2.2.2">⋆</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm11.p1.2.2.m2.2c">Q(\vec{x},\star)</annotation><annotation encoding="application/x-llamapun" id="Thmthm11.p1.2.2.m2.2d">italic_Q ( over→ start_ARG italic_x end_ARG , ⋆ )</annotation></semantics></math> be a CQ<sup class="ltx_sup" id="Thmthm11.p1.3.3.1"><span class="ltx_text ltx_font_upright" id="Thmthm11.p1.3.3.1.1">⋆</span></sup>.</span></p> <ol class="ltx_enumerate" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.4.1">If </span><math alttext="Q" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.1"><semantics id="S4.I1.i1.p1.1.m1.1a"><mi id="S4.I1.i1.p1.1.m1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.1b"><ci id="S4.I1.i1.p1.1.m1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.1d">italic_Q</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.4.2"> is free-connex and with no disruptive trio, then direct access for </span><math alttext="Q" class="ltx_Math" display="inline" id="S4.I1.i1.p1.2.m2.1"><semantics id="S4.I1.i1.p1.2.m2.1a"><mi id="S4.I1.i1.p1.2.m2.1.1" xref="S4.I1.i1.p1.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.2.m2.1b"><ci id="S4.I1.i1.p1.2.m2.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.2.m2.1d">italic_Q</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.4.3"> is in </span><math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.3.m3.2"><semantics id="S4.I1.i1.p1.3.m3.2a"><mrow id="S4.I1.i1.p1.3.m3.2.2.4" xref="S4.I1.i1.p1.3.m3.2.2.3.cmml"><mo id="S4.I1.i1.p1.3.m3.2.2.4.1" stretchy="false" xref="S4.I1.i1.p1.3.m3.2.2.3.cmml">⟨</mo><mi id="S4.I1.i1.p1.3.m3.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.cmml">loglinear</mi><mo id="S4.I1.i1.p1.3.m3.2.2.4.2" xref="S4.I1.i1.p1.3.m3.2.2.3.cmml">,</mo><mi id="S4.I1.i1.p1.3.m3.2.2.2" xref="S4.I1.i1.p1.3.m3.2.2.2.cmml">log</mi><mo id="S4.I1.i1.p1.3.m3.2.2.4.3" stretchy="false" xref="S4.I1.i1.p1.3.m3.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.3.m3.2b"><list id="S4.I1.i1.p1.3.m3.2.2.3.cmml" xref="S4.I1.i1.p1.3.m3.2.2.4"><ci id="S4.I1.i1.p1.3.m3.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1">loglinear</ci><log id="S4.I1.i1.p1.3.m3.2.2.2.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.3.m3.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.3.m3.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.4.4"> on </span><math alttext="\mathbb{K}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.4.m4.1"><semantics id="S4.I1.i1.p1.4.m4.1a"><mi id="S4.I1.i1.p1.4.m4.1.1" xref="S4.I1.i1.p1.4.m4.1.1.cmml">𝕂</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.4.m4.1b"><ci id="S4.I1.i1.p1.4.m4.1.1.cmml" xref="S4.I1.i1.p1.4.m4.1.1">𝕂</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.4.m4.1c">\mathbb{K}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.4.m4.1d">blackboard_K</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i1.p1.4.5">-databases.</span></p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.3.1">Otherwise, if </span><math alttext="Q" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.1"><semantics id="S4.I1.i2.p1.1.m1.1a"><mi id="S4.I1.i2.p1.1.m1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.1.m1.1b"><ci id="S4.I1.i2.p1.1.m1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.1d">italic_Q</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.3.2"> is also self-join-free, then direct access for </span><math alttext="Q" class="ltx_Math" display="inline" id="S4.I1.i2.p1.2.m2.1"><semantics id="S4.I1.i2.p1.2.m2.1a"><mi id="S4.I1.i2.p1.2.m2.1.1" xref="S4.I1.i2.p1.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.2.m2.1b"><ci id="S4.I1.i2.p1.2.m2.1.1.cmml" xref="S4.I1.i2.p1.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.2.m2.1d">italic_Q</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.3.3"> is not in </span><math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S4.I1.i2.p1.3.m3.2"><semantics id="S4.I1.i2.p1.3.m3.2a"><mrow id="S4.I1.i2.p1.3.m3.2.2.4" xref="S4.I1.i2.p1.3.m3.2.2.3.cmml"><mo id="S4.I1.i2.p1.3.m3.2.2.4.1" stretchy="false" xref="S4.I1.i2.p1.3.m3.2.2.3.cmml">⟨</mo><mi id="S4.I1.i2.p1.3.m3.1.1.1" xref="S4.I1.i2.p1.3.m3.1.1.1.cmml">loglinear</mi><mo id="S4.I1.i2.p1.3.m3.2.2.4.2" xref="S4.I1.i2.p1.3.m3.2.2.3.cmml">,</mo><mi id="S4.I1.i2.p1.3.m3.2.2.2" xref="S4.I1.i2.p1.3.m3.2.2.2.cmml">log</mi><mo id="S4.I1.i2.p1.3.m3.2.2.4.3" stretchy="false" xref="S4.I1.i2.p1.3.m3.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.3.m3.2b"><list id="S4.I1.i2.p1.3.m3.2.2.3.cmml" xref="S4.I1.i2.p1.3.m3.2.2.4"><ci id="S4.I1.i2.p1.3.m3.1.1.1.cmml" xref="S4.I1.i2.p1.3.m3.1.1.1">loglinear</ci><log id="S4.I1.i2.p1.3.m3.2.2.2.cmml" xref="S4.I1.i2.p1.3.m3.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.3.m3.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.3.m3.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I1.i2.p1.3.4">, assuming the</span></p> </div> </li> </ol> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Proof 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.2"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.2.2">For the negative side of the dichotomy, we simply use the negative side of <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S3" title="3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3</span></a>. This can be done since each answer to a CQ<sup class="ltx_sup" id="S4.Thmtheorem1.p1.2.2.1"><span class="ltx_text ltx_font_upright" id="S4.Thmtheorem1.p1.2.2.1.1">⋆</span></sup> contains the ordinary (non-annotated) answer to the CQ obtained by removing <math alttext="\star" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.2.m2.1"><semantics id="S4.Thmtheorem1.p1.2.2.m2.1a"><mo id="S4.Thmtheorem1.p1.2.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml">⋆</mo><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.2.m2.1b"><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1">⋆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.2.m2.1c">\star</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.1d">⋆</annotation></semantics></math>, and the answers have the same order.</span></p> </div> <div class="ltx_para" id="S4.Thmtheorem1.p2"> <p class="ltx_p" id="S4.Thmtheorem1.p2.7"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p2.7.7">For the positive side of the dichotomy, <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm10" title="Corollary 10. ‣ Hypothesis 7. ‣ 3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">10</span></a> implies that it is enough to show an algorithm for <math alttext="Q^{\prime}(\vec{x},\star)=Q^{\mathrm{sf}}_{|\mathrm{free}{(Q)}}" class="ltx_math_unparsed" display="inline" id="S4.Thmtheorem1.p2.1.1.m1.3"><semantics id="S4.Thmtheorem1.p2.1.1.m1.3a"><mrow id="S4.Thmtheorem1.p2.1.1.m1.3.4"><mrow id="S4.Thmtheorem1.p2.1.1.m1.3.4.2"><msup id="S4.Thmtheorem1.p2.1.1.m1.3.4.2.2"><mi id="S4.Thmtheorem1.p2.1.1.m1.3.4.2.2.2">Q</mi><mo id="S4.Thmtheorem1.p2.1.1.m1.3.4.2.2.3">′</mo></msup><mo id="S4.Thmtheorem1.p2.1.1.m1.3.4.2.1">⁢</mo><mrow id="S4.Thmtheorem1.p2.1.1.m1.3.4.2.3.2"><mo id="S4.Thmtheorem1.p2.1.1.m1.3.4.2.3.2.1" stretchy="false">(</mo><mover accent="true" id="S4.Thmtheorem1.p2.1.1.m1.2.2"><mi id="S4.Thmtheorem1.p2.1.1.m1.2.2.2">x</mi><mo id="S4.Thmtheorem1.p2.1.1.m1.2.2.1" stretchy="false">→</mo></mover><mo id="S4.Thmtheorem1.p2.1.1.m1.3.4.2.3.2.2" rspace="0em">,</mo><mo id="S4.Thmtheorem1.p2.1.1.m1.3.3" lspace="0em" rspace="0em">⋆</mo><mo id="S4.Thmtheorem1.p2.1.1.m1.3.4.2.3.2.3" stretchy="false">)</mo></mrow></mrow><mo id="S4.Thmtheorem1.p2.1.1.m1.3.4.1">=</mo><msubsup id="S4.Thmtheorem1.p2.1.1.m1.3.4.3"><mi id="S4.Thmtheorem1.p2.1.1.m1.3.4.3.2.2">Q</mi><mrow id="S4.Thmtheorem1.p2.1.1.m1.1.1.1"><mo fence="false" id="S4.Thmtheorem1.p2.1.1.m1.1.1.1.2" rspace="0.167em" stretchy="false">|</mo><mi id="S4.Thmtheorem1.p2.1.1.m1.1.1.1.3">free</mi><mrow id="S4.Thmtheorem1.p2.1.1.m1.1.1.1.4"><mo id="S4.Thmtheorem1.p2.1.1.m1.1.1.1.4.1" stretchy="false">(</mo><mi id="S4.Thmtheorem1.p2.1.1.m1.1.1.1.1">Q</mi><mo id="S4.Thmtheorem1.p2.1.1.m1.1.1.1.4.2" stretchy="false">)</mo></mrow></mrow><mi id="S4.Thmtheorem1.p2.1.1.m1.3.4.3.2.3">sf</mi></msubsup></mrow><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p2.1.1.m1.3b">Q^{\prime}(\vec{x},\star)=Q^{\mathrm{sf}}_{|\mathrm{free}{(Q)}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p2.1.1.m1.3c">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( over→ start_ARG italic_x end_ARG , ⋆ ) = italic_Q start_POSTSUPERSCRIPT roman_sf end_POSTSUPERSCRIPT start_POSTSUBSCRIPT | roman_free ( italic_Q ) end_POSTSUBSCRIPT</annotation></semantics></math>, which is a full self-join-free acyclic CQ<sup class="ltx_sup" id="S4.Thmtheorem1.p2.7.7.1"><span class="ltx_text ltx_font_upright" id="S4.Thmtheorem1.p2.7.7.1.1">⋆</span></sup> with no disruptive trios. Using <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S3" title="3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3</span></a>, we have direct access for <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p2.3.3.m3.1"><semantics id="S4.Thmtheorem1.p2.3.3.m3.1a"><msup id="S4.Thmtheorem1.p2.3.3.m3.1.1" xref="S4.Thmtheorem1.p2.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem1.p2.3.3.m3.1.1.2" xref="S4.Thmtheorem1.p2.3.3.m3.1.1.2.cmml">Q</mi><mo id="S4.Thmtheorem1.p2.3.3.m3.1.1.3" xref="S4.Thmtheorem1.p2.3.3.m3.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p2.3.3.m3.1b"><apply id="S4.Thmtheorem1.p2.3.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p2.3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p2.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem1.p2.3.3.m3.1.1">superscript</csymbol><ci id="S4.Thmtheorem1.p2.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem1.p2.3.3.m3.1.1.2">𝑄</ci><ci id="S4.Thmtheorem1.p2.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem1.p2.3.3.m3.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p2.3.3.m3.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p2.3.3.m3.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p2.4.4.m4.2"><semantics id="S4.Thmtheorem1.p2.4.4.m4.2a"><mrow id="S4.Thmtheorem1.p2.4.4.m4.2.2.4" xref="S4.Thmtheorem1.p2.4.4.m4.2.2.3.cmml"><mo id="S4.Thmtheorem1.p2.4.4.m4.2.2.4.1" stretchy="false" xref="S4.Thmtheorem1.p2.4.4.m4.2.2.3.cmml">⟨</mo><mi id="S4.Thmtheorem1.p2.4.4.m4.1.1.1" xref="S4.Thmtheorem1.p2.4.4.m4.1.1.1.cmml">loglinear</mi><mo id="S4.Thmtheorem1.p2.4.4.m4.2.2.4.2" xref="S4.Thmtheorem1.p2.4.4.m4.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem1.p2.4.4.m4.2.2.2" xref="S4.Thmtheorem1.p2.4.4.m4.2.2.2.cmml">log</mi><mo id="S4.Thmtheorem1.p2.4.4.m4.2.2.4.3" stretchy="false" xref="S4.Thmtheorem1.p2.4.4.m4.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p2.4.4.m4.2b"><list id="S4.Thmtheorem1.p2.4.4.m4.2.2.3.cmml" xref="S4.Thmtheorem1.p2.4.4.m4.2.2.4"><ci id="S4.Thmtheorem1.p2.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem1.p2.4.4.m4.1.1.1">loglinear</ci><log id="S4.Thmtheorem1.p2.4.4.m4.2.2.2.cmml" xref="S4.Thmtheorem1.p2.4.4.m4.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p2.4.4.m4.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p2.4.4.m4.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math> if we ignore the annotations. Since <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p2.5.5.m5.1"><semantics id="S4.Thmtheorem1.p2.5.5.m5.1a"><msup id="S4.Thmtheorem1.p2.5.5.m5.1.1" xref="S4.Thmtheorem1.p2.5.5.m5.1.1.cmml"><mi id="S4.Thmtheorem1.p2.5.5.m5.1.1.2" xref="S4.Thmtheorem1.p2.5.5.m5.1.1.2.cmml">Q</mi><mo id="S4.Thmtheorem1.p2.5.5.m5.1.1.3" xref="S4.Thmtheorem1.p2.5.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p2.5.5.m5.1b"><apply id="S4.Thmtheorem1.p2.5.5.m5.1.1.cmml" xref="S4.Thmtheorem1.p2.5.5.m5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p2.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem1.p2.5.5.m5.1.1">superscript</csymbol><ci id="S4.Thmtheorem1.p2.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem1.p2.5.5.m5.1.1.2">𝑄</ci><ci id="S4.Thmtheorem1.p2.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem1.p2.5.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p2.5.5.m5.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p2.5.5.m5.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> is full, from an answer to <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p2.6.6.m6.1"><semantics id="S4.Thmtheorem1.p2.6.6.m6.1a"><msup id="S4.Thmtheorem1.p2.6.6.m6.1.1" xref="S4.Thmtheorem1.p2.6.6.m6.1.1.cmml"><mi id="S4.Thmtheorem1.p2.6.6.m6.1.1.2" xref="S4.Thmtheorem1.p2.6.6.m6.1.1.2.cmml">Q</mi><mo id="S4.Thmtheorem1.p2.6.6.m6.1.1.3" xref="S4.Thmtheorem1.p2.6.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p2.6.6.m6.1b"><apply id="S4.Thmtheorem1.p2.6.6.m6.1.1.cmml" xref="S4.Thmtheorem1.p2.6.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p2.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem1.p2.6.6.m6.1.1">superscript</csymbol><ci id="S4.Thmtheorem1.p2.6.6.m6.1.1.2.cmml" xref="S4.Thmtheorem1.p2.6.6.m6.1.1.2">𝑄</ci><ci id="S4.Thmtheorem1.p2.6.6.m6.1.1.3.cmml" xref="S4.Thmtheorem1.p2.6.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p2.6.6.m6.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p2.6.6.m6.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> we compute the annotation by finding the tuples that agree with the answer in every relation of <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p2.7.7.m7.1"><semantics id="S4.Thmtheorem1.p2.7.7.m7.1a"><msup id="S4.Thmtheorem1.p2.7.7.m7.1.1" xref="S4.Thmtheorem1.p2.7.7.m7.1.1.cmml"><mi id="S4.Thmtheorem1.p2.7.7.m7.1.1.2" xref="S4.Thmtheorem1.p2.7.7.m7.1.1.2.cmml">Q</mi><mo id="S4.Thmtheorem1.p2.7.7.m7.1.1.3" xref="S4.Thmtheorem1.p2.7.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p2.7.7.m7.1b"><apply id="S4.Thmtheorem1.p2.7.7.m7.1.1.cmml" xref="S4.Thmtheorem1.p2.7.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p2.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem1.p2.7.7.m7.1.1">superscript</csymbol><ci id="S4.Thmtheorem1.p2.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem1.p2.7.7.m7.1.1.2">𝑄</ci><ci id="S4.Thmtheorem1.p2.7.7.m7.1.1.3.cmml" xref="S4.Thmtheorem1.p2.7.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p2.7.7.m7.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p2.7.7.m7.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and multiply the annotations of these tuples; this can be done in logarithmic time.</span></p> </div> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.1"><span class="ltx_text ltx_font_italic" id="S4.SS1.p2.1.1">From the positive side of </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2303.05327v2#Thmthm11" title="Theorem 11. ‣ 4.1. Generalized Dichotomies ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">11</span></a><span class="ltx_text ltx_font_italic" id="S4.SS1.p2.1.2">, we conclude efficient direct access for AggCQs </span><math alttext="Q(\vec{x},\alpha(\vec{w}))" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.3"><semantics id="S4.SS1.p2.1.m1.3a"><mrow id="S4.SS1.p2.1.m1.3.3" xref="S4.SS1.p2.1.m1.3.3.cmml"><mi id="S4.SS1.p2.1.m1.3.3.3" xref="S4.SS1.p2.1.m1.3.3.3.cmml">Q</mi><mo id="S4.SS1.p2.1.m1.3.3.2" xref="S4.SS1.p2.1.m1.3.3.2.cmml">⁢</mo><mrow id="S4.SS1.p2.1.m1.3.3.1.1" xref="S4.SS1.p2.1.m1.3.3.1.2.cmml"><mo id="S4.SS1.p2.1.m1.3.3.1.1.2" stretchy="false" xref="S4.SS1.p2.1.m1.3.3.1.2.cmml">(</mo><mover accent="true" id="S4.SS1.p2.1.m1.2.2" xref="S4.SS1.p2.1.m1.2.2.cmml"><mi id="S4.SS1.p2.1.m1.2.2.2" xref="S4.SS1.p2.1.m1.2.2.2.cmml">x</mi><mo id="S4.SS1.p2.1.m1.2.2.1" stretchy="false" xref="S4.SS1.p2.1.m1.2.2.1.cmml">→</mo></mover><mo id="S4.SS1.p2.1.m1.3.3.1.1.3" xref="S4.SS1.p2.1.m1.3.3.1.2.cmml">,</mo><mrow id="S4.SS1.p2.1.m1.3.3.1.1.1" xref="S4.SS1.p2.1.m1.3.3.1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.3.3.1.1.1.2" xref="S4.SS1.p2.1.m1.3.3.1.1.1.2.cmml">α</mi><mo id="S4.SS1.p2.1.m1.3.3.1.1.1.1" xref="S4.SS1.p2.1.m1.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S4.SS1.p2.1.m1.3.3.1.1.1.3.2" xref="S4.SS1.p2.1.m1.1.1.cmml"><mo id="S4.SS1.p2.1.m1.3.3.1.1.1.3.2.1" stretchy="false" xref="S4.SS1.p2.1.m1.1.1.cmml">(</mo><mover accent="true" id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml"><mi id="S4.SS1.p2.1.m1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.2.cmml">w</mi><mo id="S4.SS1.p2.1.m1.1.1.1" stretchy="false" xref="S4.SS1.p2.1.m1.1.1.1.cmml">→</mo></mover><mo id="S4.SS1.p2.1.m1.3.3.1.1.1.3.2.2" stretchy="false" xref="S4.SS1.p2.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p2.1.m1.3.3.1.1.4" stretchy="false" xref="S4.SS1.p2.1.m1.3.3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.3b"><apply id="S4.SS1.p2.1.m1.3.3.cmml" xref="S4.SS1.p2.1.m1.3.3"><times id="S4.SS1.p2.1.m1.3.3.2.cmml" xref="S4.SS1.p2.1.m1.3.3.2"></times><ci id="S4.SS1.p2.1.m1.3.3.3.cmml" xref="S4.SS1.p2.1.m1.3.3.3">𝑄</ci><interval closure="open" id="S4.SS1.p2.1.m1.3.3.1.2.cmml" xref="S4.SS1.p2.1.m1.3.3.1.1"><apply id="S4.SS1.p2.1.m1.2.2.cmml" xref="S4.SS1.p2.1.m1.2.2"><ci id="S4.SS1.p2.1.m1.2.2.1.cmml" xref="S4.SS1.p2.1.m1.2.2.1">→</ci><ci id="S4.SS1.p2.1.m1.2.2.2.cmml" xref="S4.SS1.p2.1.m1.2.2.2">𝑥</ci></apply><apply id="S4.SS1.p2.1.m1.3.3.1.1.1.cmml" xref="S4.SS1.p2.1.m1.3.3.1.1.1"><times id="S4.SS1.p2.1.m1.3.3.1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.3.3.1.1.1.1"></times><ci id="S4.SS1.p2.1.m1.3.3.1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.3.3.1.1.1.2">𝛼</ci><apply id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.3.3.1.1.1.3.2"><ci id="S4.SS1.p2.1.m1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1.1">→</ci><ci id="S4.SS1.p2.1.m1.1.1.2.cmml" xref="S4.SS1.p2.1.m1.1.1.2">𝑤</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.3c">Q(\vec{x},\alpha(\vec{w}))</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.3d">italic_Q ( over→ start_ARG italic_x end_ARG , italic_α ( over→ start_ARG italic_w end_ARG ) )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.SS1.p2.1.3">, as long as we can efficiently formulate the aggregate function as an annotation over some logarithmic-time commutative semiring. This is stated in the following corollary of </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2303.05327v2#Thmthm11" title="Theorem 11. ‣ 4.1. Generalized Dichotomies ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">11</span></a><span class="ltx_text ltx_font_italic" id="S4.SS1.p2.1.4">.</span></p> </div> <div class="ltx_theorem ltx_theorem_cor" id="Thmthm14"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm14.1.1.1">Corollary 14</span></span><span class="ltx_text ltx_font_bold" id="Thmthm14.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm14.p1"> <p class="ltx_p" id="Thmthm14.p1.7"><span class="ltx_text ltx_font_italic" id="Thmthm14.p1.7.7">Consider an AggCQ <math alttext="Q(\vec{x},\alpha(\vec{w})){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,% \varphi_{\ell}(\vec{x},\vec{y})" class="ltx_Math" display="inline" id="Thmthm14.p1.1.1.m1.10"><semantics id="Thmthm14.p1.1.1.m1.10a"><mrow 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xref="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.2.cmml">α</mi><mo id="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.1" xref="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.1.cmml">⁢</mo><mrow id="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.3.2" xref="Thmthm14.p1.1.1.m1.1.1.cmml"><mo id="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.3.2.1" stretchy="false" xref="Thmthm14.p1.1.1.m1.1.1.cmml">(</mo><mover accent="true" id="Thmthm14.p1.1.1.m1.1.1" xref="Thmthm14.p1.1.1.m1.1.1.cmml"><mi id="Thmthm14.p1.1.1.m1.1.1.2" xref="Thmthm14.p1.1.1.m1.1.1.2.cmml">w</mi><mo id="Thmthm14.p1.1.1.m1.1.1.1" stretchy="false" xref="Thmthm14.p1.1.1.m1.1.1.1.cmml">→</mo></mover><mo id="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.3.2.2" stretchy="false" xref="Thmthm14.p1.1.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="Thmthm14.p1.1.1.m1.8.8.1.1.1.4" rspace="0.448em" stretchy="false" xref="Thmthm14.p1.1.1.m1.8.8.1.1.2.cmml">)</mo></mrow></mrow><mpadded width="0.226em"><mo id="Thmthm14.p1.1.1.m1.10.10.4" xref="Thmthm14.p1.1.1.m1.10.10.4.cmml">:</mo></mpadded><mrow id="Thmthm14.p1.1.1.m1.10.10.3.2" 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id="Thmthm14.p1.1.1.m1.3.3.2" xref="Thmthm14.p1.1.1.m1.3.3.2.cmml">x</mi><mo id="Thmthm14.p1.1.1.m1.3.3.1" stretchy="false" xref="Thmthm14.p1.1.1.m1.3.3.1.cmml">→</mo></mover><mo id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.3.2.2" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.3.1.cmml">,</mo><mover accent="true" id="Thmthm14.p1.1.1.m1.4.4" xref="Thmthm14.p1.1.1.m1.4.4.cmml"><mi id="Thmthm14.p1.1.1.m1.4.4.2" xref="Thmthm14.p1.1.1.m1.4.4.2.cmml">y</mi><mo id="Thmthm14.p1.1.1.m1.4.4.1" stretchy="false" xref="Thmthm14.p1.1.1.m1.4.4.1.cmml">→</mo></mover><mo id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.3.2.3" stretchy="false" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="Thmthm14.p1.1.1.m1.10.10.3.2.3" xref="Thmthm14.p1.1.1.m1.10.10.3.3.cmml">,</mo><mi id="Thmthm14.p1.1.1.m1.7.7" mathvariant="normal" xref="Thmthm14.p1.1.1.m1.7.7.cmml">…</mi><mo id="Thmthm14.p1.1.1.m1.10.10.3.2.4" xref="Thmthm14.p1.1.1.m1.10.10.3.3.cmml">,</mo><mrow id="Thmthm14.p1.1.1.m1.10.10.3.2.2" 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xref="Thmthm14.p1.1.1.m1.10.10.3.2.2.3.1.cmml">,</mo><mover accent="true" id="Thmthm14.p1.1.1.m1.6.6" xref="Thmthm14.p1.1.1.m1.6.6.cmml"><mi id="Thmthm14.p1.1.1.m1.6.6.2" xref="Thmthm14.p1.1.1.m1.6.6.2.cmml">y</mi><mo id="Thmthm14.p1.1.1.m1.6.6.1" stretchy="false" xref="Thmthm14.p1.1.1.m1.6.6.1.cmml">→</mo></mover><mo id="Thmthm14.p1.1.1.m1.10.10.3.2.2.3.2.3" stretchy="false" xref="Thmthm14.p1.1.1.m1.10.10.3.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm14.p1.1.1.m1.10b"><apply id="Thmthm14.p1.1.1.m1.10.10.cmml" xref="Thmthm14.p1.1.1.m1.10.10"><ci id="Thmthm14.p1.1.1.m1.10.10.4.cmml" xref="Thmthm14.p1.1.1.m1.10.10.4">:</ci><apply id="Thmthm14.p1.1.1.m1.8.8.1.cmml" xref="Thmthm14.p1.1.1.m1.8.8.1"><times id="Thmthm14.p1.1.1.m1.8.8.1.2.cmml" xref="Thmthm14.p1.1.1.m1.8.8.1.2"></times><ci id="Thmthm14.p1.1.1.m1.8.8.1.3.cmml" xref="Thmthm14.p1.1.1.m1.8.8.1.3">𝑄</ci><interval closure="open" id="Thmthm14.p1.1.1.m1.8.8.1.1.2.cmml" xref="Thmthm14.p1.1.1.m1.8.8.1.1.1"><apply id="Thmthm14.p1.1.1.m1.2.2.cmml" xref="Thmthm14.p1.1.1.m1.2.2"><ci id="Thmthm14.p1.1.1.m1.2.2.1.cmml" xref="Thmthm14.p1.1.1.m1.2.2.1">→</ci><ci id="Thmthm14.p1.1.1.m1.2.2.2.cmml" xref="Thmthm14.p1.1.1.m1.2.2.2">𝑥</ci></apply><apply id="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.cmml" xref="Thmthm14.p1.1.1.m1.8.8.1.1.1.1"><times id="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.1.cmml" xref="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.1"></times><ci id="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.2.cmml" xref="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.2">𝛼</ci><apply id="Thmthm14.p1.1.1.m1.1.1.cmml" xref="Thmthm14.p1.1.1.m1.8.8.1.1.1.1.3.2"><ci id="Thmthm14.p1.1.1.m1.1.1.1.cmml" xref="Thmthm14.p1.1.1.m1.1.1.1">→</ci><ci id="Thmthm14.p1.1.1.m1.1.1.2.cmml" xref="Thmthm14.p1.1.1.m1.1.1.2">𝑤</ci></apply></apply></interval></apply><list id="Thmthm14.p1.1.1.m1.10.10.3.3.cmml" xref="Thmthm14.p1.1.1.m1.10.10.3.2"><apply id="Thmthm14.p1.1.1.m1.9.9.2.1.1.cmml" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1"><minus id="Thmthm14.p1.1.1.m1.9.9.2.1.1.1.cmml" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1"></minus><apply id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.cmml" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2"><times id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.1.cmml" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.1"></times><apply id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.2.cmml" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.2"><csymbol cd="ambiguous" id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.2.1.cmml" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.2">subscript</csymbol><ci id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.2.2.cmml" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.2.2">𝜑</ci><cn id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.2.3.cmml" type="integer" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.2.3">1</cn></apply><interval closure="open" id="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.3.1.cmml" xref="Thmthm14.p1.1.1.m1.9.9.2.1.1.2.3.2"><apply id="Thmthm14.p1.1.1.m1.3.3.cmml" xref="Thmthm14.p1.1.1.m1.3.3"><ci id="Thmthm14.p1.1.1.m1.3.3.1.cmml" xref="Thmthm14.p1.1.1.m1.3.3.1">→</ci><ci id="Thmthm14.p1.1.1.m1.3.3.2.cmml" xref="Thmthm14.p1.1.1.m1.3.3.2">𝑥</ci></apply><apply id="Thmthm14.p1.1.1.m1.4.4.cmml" xref="Thmthm14.p1.1.1.m1.4.4"><ci id="Thmthm14.p1.1.1.m1.4.4.1.cmml" xref="Thmthm14.p1.1.1.m1.4.4.1">→</ci><ci id="Thmthm14.p1.1.1.m1.4.4.2.cmml" xref="Thmthm14.p1.1.1.m1.4.4.2">𝑦</ci></apply></interval></apply></apply><ci id="Thmthm14.p1.1.1.m1.7.7.cmml" xref="Thmthm14.p1.1.1.m1.7.7">…</ci><apply id="Thmthm14.p1.1.1.m1.10.10.3.2.2.cmml" xref="Thmthm14.p1.1.1.m1.10.10.3.2.2"><times id="Thmthm14.p1.1.1.m1.10.10.3.2.2.1.cmml" xref="Thmthm14.p1.1.1.m1.10.10.3.2.2.1"></times><apply id="Thmthm14.p1.1.1.m1.10.10.3.2.2.2.cmml" xref="Thmthm14.p1.1.1.m1.10.10.3.2.2.2"><csymbol cd="ambiguous" id="Thmthm14.p1.1.1.m1.10.10.3.2.2.2.1.cmml" xref="Thmthm14.p1.1.1.m1.10.10.3.2.2.2">subscript</csymbol><ci id="Thmthm14.p1.1.1.m1.10.10.3.2.2.2.2.cmml" xref="Thmthm14.p1.1.1.m1.10.10.3.2.2.2.2">𝜑</ci><ci id="Thmthm14.p1.1.1.m1.10.10.3.2.2.2.3.cmml" xref="Thmthm14.p1.1.1.m1.10.10.3.2.2.2.3">ℓ</ci></apply><interval closure="open" id="Thmthm14.p1.1.1.m1.10.10.3.2.2.3.1.cmml" xref="Thmthm14.p1.1.1.m1.10.10.3.2.2.3.2"><apply id="Thmthm14.p1.1.1.m1.5.5.cmml" xref="Thmthm14.p1.1.1.m1.5.5"><ci id="Thmthm14.p1.1.1.m1.5.5.1.cmml" xref="Thmthm14.p1.1.1.m1.5.5.1">→</ci><ci id="Thmthm14.p1.1.1.m1.5.5.2.cmml" xref="Thmthm14.p1.1.1.m1.5.5.2">𝑥</ci></apply><apply id="Thmthm14.p1.1.1.m1.6.6.cmml" xref="Thmthm14.p1.1.1.m1.6.6"><ci id="Thmthm14.p1.1.1.m1.6.6.1.cmml" xref="Thmthm14.p1.1.1.m1.6.6.1">→</ci><ci id="Thmthm14.p1.1.1.m1.6.6.2.cmml" xref="Thmthm14.p1.1.1.m1.6.6.2">𝑦</ci></apply></interval></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm14.p1.1.1.m1.10c">Q(\vec{x},\alpha(\vec{w})){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,% \varphi_{\ell}(\vec{x},\vec{y})</annotation><annotation encoding="application/x-llamapun" id="Thmthm14.p1.1.1.m1.10d">italic_Q ( over→ start_ARG italic_x end_ARG , italic_α ( over→ start_ARG italic_w end_ARG ) ) : - italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) , … , italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math> where <math alttext="\alpha" class="ltx_Math" display="inline" id="Thmthm14.p1.2.2.m2.1"><semantics id="Thmthm14.p1.2.2.m2.1a"><mi id="Thmthm14.p1.2.2.m2.1.1" xref="Thmthm14.p1.2.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="Thmthm14.p1.2.2.m2.1b"><ci id="Thmthm14.p1.2.2.m2.1.1.cmml" xref="Thmthm14.p1.2.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm14.p1.2.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="Thmthm14.p1.2.2.m2.1d">italic_α</annotation></semantics></math> is one of <math alttext="\mathsf{Min}" class="ltx_Math" display="inline" id="Thmthm14.p1.3.3.m3.1"><semantics id="Thmthm14.p1.3.3.m3.1a"><mi id="Thmthm14.p1.3.3.m3.1.1" xref="Thmthm14.p1.3.3.m3.1.1.cmml">𝖬𝗂𝗇</mi><annotation-xml encoding="MathML-Content" id="Thmthm14.p1.3.3.m3.1b"><ci id="Thmthm14.p1.3.3.m3.1.1.cmml" xref="Thmthm14.p1.3.3.m3.1.1">𝖬𝗂𝗇</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm14.p1.3.3.m3.1c">\mathsf{Min}</annotation><annotation encoding="application/x-llamapun" id="Thmthm14.p1.3.3.m3.1d">sansserif_Min</annotation></semantics></math>, <math alttext="\mathsf{Max}" class="ltx_Math" display="inline" id="Thmthm14.p1.4.4.m4.1"><semantics id="Thmthm14.p1.4.4.m4.1a"><mi id="Thmthm14.p1.4.4.m4.1.1" xref="Thmthm14.p1.4.4.m4.1.1.cmml">𝖬𝖺𝗑</mi><annotation-xml encoding="MathML-Content" id="Thmthm14.p1.4.4.m4.1b"><ci id="Thmthm14.p1.4.4.m4.1.1.cmml" xref="Thmthm14.p1.4.4.m4.1.1">𝖬𝖺𝗑</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm14.p1.4.4.m4.1c">\mathsf{Max}</annotation><annotation encoding="application/x-llamapun" id="Thmthm14.p1.4.4.m4.1d">sansserif_Max</annotation></semantics></math>, <math alttext="\mathsf{Count}" class="ltx_Math" display="inline" id="Thmthm14.p1.5.5.m5.1"><semantics id="Thmthm14.p1.5.5.m5.1a"><mi id="Thmthm14.p1.5.5.m5.1.1" xref="Thmthm14.p1.5.5.m5.1.1.cmml">𝖢𝗈𝗎𝗇𝗍</mi><annotation-xml encoding="MathML-Content" id="Thmthm14.p1.5.5.m5.1b"><ci id="Thmthm14.p1.5.5.m5.1.1.cmml" xref="Thmthm14.p1.5.5.m5.1.1">𝖢𝗈𝗎𝗇𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm14.p1.5.5.m5.1c">\mathsf{Count}</annotation><annotation encoding="application/x-llamapun" id="Thmthm14.p1.5.5.m5.1d">sansserif_Count</annotation></semantics></math>, <math alttext="\mathsf{Sum}" class="ltx_Math" display="inline" id="Thmthm14.p1.6.6.m6.1"><semantics id="Thmthm14.p1.6.6.m6.1a"><mi id="Thmthm14.p1.6.6.m6.1.1" xref="Thmthm14.p1.6.6.m6.1.1.cmml">𝖲𝗎𝗆</mi><annotation-xml encoding="MathML-Content" id="Thmthm14.p1.6.6.m6.1b"><ci id="Thmthm14.p1.6.6.m6.1.1.cmml" xref="Thmthm14.p1.6.6.m6.1.1">𝖲𝗎𝗆</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm14.p1.6.6.m6.1c">\mathsf{Sum}</annotation><annotation encoding="application/x-llamapun" id="Thmthm14.p1.6.6.m6.1d">sansserif_Sum</annotation></semantics></math>, and <math alttext="\mathsf{Avg}" class="ltx_Math" display="inline" id="Thmthm14.p1.7.7.m7.1"><semantics id="Thmthm14.p1.7.7.m7.1a"><mi id="Thmthm14.p1.7.7.m7.1.1" xref="Thmthm14.p1.7.7.m7.1.1.cmml">𝖠𝗏𝗀</mi><annotation-xml encoding="MathML-Content" id="Thmthm14.p1.7.7.m7.1b"><ci id="Thmthm14.p1.7.7.m7.1.1.cmml" xref="Thmthm14.p1.7.7.m7.1.1">𝖠𝗏𝗀</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm14.p1.7.7.m7.1c">\mathsf{Avg}</annotation><annotation encoding="application/x-llamapun" id="Thmthm14.p1.7.7.m7.1d">sansserif_Avg</annotation></semantics></math>.</span></p> <ol class="ltx_enumerate" id="S4.I2"> <li class="ltx_item" id="S4.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(1)</span> <div class="ltx_para" id="S4.I2.i1.p1"> <p class="ltx_p" id="S4.I2.i1.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.3.1">If the CQ </span><math alttext="Q^{\prime}(\vec{x}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{% \ell}(\vec{x},\vec{y})" class="ltx_Math" display="inline" id="S4.I2.i1.p1.1.m1.8"><semantics id="S4.I2.i1.p1.1.m1.8a"><mrow id="S4.I2.i1.p1.1.m1.8.8" xref="S4.I2.i1.p1.1.m1.8.8.cmml"><mrow id="S4.I2.i1.p1.1.m1.8.8.4" xref="S4.I2.i1.p1.1.m1.8.8.4.cmml"><msup id="S4.I2.i1.p1.1.m1.8.8.4.2" xref="S4.I2.i1.p1.1.m1.8.8.4.2.cmml"><mi id="S4.I2.i1.p1.1.m1.8.8.4.2.2" xref="S4.I2.i1.p1.1.m1.8.8.4.2.2.cmml">Q</mi><mo id="S4.I2.i1.p1.1.m1.8.8.4.2.3" xref="S4.I2.i1.p1.1.m1.8.8.4.2.3.cmml">′</mo></msup><mo id="S4.I2.i1.p1.1.m1.8.8.4.1" xref="S4.I2.i1.p1.1.m1.8.8.4.1.cmml">⁢</mo><mrow id="S4.I2.i1.p1.1.m1.8.8.4.3.2" xref="S4.I2.i1.p1.1.m1.1.1.cmml"><mo id="S4.I2.i1.p1.1.m1.8.8.4.3.2.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S4.I2.i1.p1.1.m1.1.1" xref="S4.I2.i1.p1.1.m1.1.1.cmml"><mi id="S4.I2.i1.p1.1.m1.1.1.2" xref="S4.I2.i1.p1.1.m1.1.1.2.cmml">x</mi><mo id="S4.I2.i1.p1.1.m1.1.1.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.1.1.1.cmml">→</mo></mover><mo id="S4.I2.i1.p1.1.m1.8.8.4.3.2.2" rspace="0.448em" stretchy="false" xref="S4.I2.i1.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><mpadded width="0.226em"><mo id="S4.I2.i1.p1.1.m1.8.8.3" xref="S4.I2.i1.p1.1.m1.8.8.3.cmml">:</mo></mpadded><mrow id="S4.I2.i1.p1.1.m1.8.8.2.2" xref="S4.I2.i1.p1.1.m1.8.8.2.3.cmml"><mrow id="S4.I2.i1.p1.1.m1.7.7.1.1.1" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.cmml"><mo id="S4.I2.i1.p1.1.m1.7.7.1.1.1a" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.cmml">−</mo><mrow id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.cmml"><msub id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.cmml"><mi id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.2" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.2.cmml">φ</mi><mn id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.3" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.1" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.2" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.1.cmml"><mo id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.2.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.1.cmml">(</mo><mover accent="true" id="S4.I2.i1.p1.1.m1.2.2" xref="S4.I2.i1.p1.1.m1.2.2.cmml"><mi id="S4.I2.i1.p1.1.m1.2.2.2" xref="S4.I2.i1.p1.1.m1.2.2.2.cmml">x</mi><mo id="S4.I2.i1.p1.1.m1.2.2.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.2.2.1.cmml">→</mo></mover><mo id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.2.2" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.1.cmml">,</mo><mover accent="true" id="S4.I2.i1.p1.1.m1.3.3" xref="S4.I2.i1.p1.1.m1.3.3.cmml"><mi id="S4.I2.i1.p1.1.m1.3.3.2" xref="S4.I2.i1.p1.1.m1.3.3.2.cmml">y</mi><mo id="S4.I2.i1.p1.1.m1.3.3.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.3.3.1.cmml">→</mo></mover><mo id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.2.3" stretchy="false" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S4.I2.i1.p1.1.m1.8.8.2.2.3" xref="S4.I2.i1.p1.1.m1.8.8.2.3.cmml">,</mo><mi id="S4.I2.i1.p1.1.m1.6.6" mathvariant="normal" xref="S4.I2.i1.p1.1.m1.6.6.cmml">…</mi><mo id="S4.I2.i1.p1.1.m1.8.8.2.2.4" xref="S4.I2.i1.p1.1.m1.8.8.2.3.cmml">,</mo><mrow id="S4.I2.i1.p1.1.m1.8.8.2.2.2" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.cmml"><msub id="S4.I2.i1.p1.1.m1.8.8.2.2.2.2" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.cmml"><mi id="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.2" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.2.cmml">φ</mi><mi id="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.3" mathvariant="normal" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.3.cmml">ℓ</mi></msub><mo id="S4.I2.i1.p1.1.m1.8.8.2.2.2.1" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.1.cmml">⁢</mo><mrow id="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.2" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.1.cmml"><mo id="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.2.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.1.cmml">(</mo><mover accent="true" id="S4.I2.i1.p1.1.m1.4.4" xref="S4.I2.i1.p1.1.m1.4.4.cmml"><mi id="S4.I2.i1.p1.1.m1.4.4.2" xref="S4.I2.i1.p1.1.m1.4.4.2.cmml">x</mi><mo id="S4.I2.i1.p1.1.m1.4.4.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.4.4.1.cmml">→</mo></mover><mo id="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.2.2" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.1.cmml">,</mo><mover accent="true" id="S4.I2.i1.p1.1.m1.5.5" xref="S4.I2.i1.p1.1.m1.5.5.cmml"><mi id="S4.I2.i1.p1.1.m1.5.5.2" xref="S4.I2.i1.p1.1.m1.5.5.2.cmml">y</mi><mo id="S4.I2.i1.p1.1.m1.5.5.1" stretchy="false" xref="S4.I2.i1.p1.1.m1.5.5.1.cmml">→</mo></mover><mo id="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.2.3" stretchy="false" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.1.m1.8b"><apply id="S4.I2.i1.p1.1.m1.8.8.cmml" xref="S4.I2.i1.p1.1.m1.8.8"><ci id="S4.I2.i1.p1.1.m1.8.8.3.cmml" xref="S4.I2.i1.p1.1.m1.8.8.3">:</ci><apply id="S4.I2.i1.p1.1.m1.8.8.4.cmml" xref="S4.I2.i1.p1.1.m1.8.8.4"><times id="S4.I2.i1.p1.1.m1.8.8.4.1.cmml" xref="S4.I2.i1.p1.1.m1.8.8.4.1"></times><apply id="S4.I2.i1.p1.1.m1.8.8.4.2.cmml" xref="S4.I2.i1.p1.1.m1.8.8.4.2"><csymbol cd="ambiguous" id="S4.I2.i1.p1.1.m1.8.8.4.2.1.cmml" xref="S4.I2.i1.p1.1.m1.8.8.4.2">superscript</csymbol><ci id="S4.I2.i1.p1.1.m1.8.8.4.2.2.cmml" xref="S4.I2.i1.p1.1.m1.8.8.4.2.2">𝑄</ci><ci id="S4.I2.i1.p1.1.m1.8.8.4.2.3.cmml" xref="S4.I2.i1.p1.1.m1.8.8.4.2.3">′</ci></apply><apply id="S4.I2.i1.p1.1.m1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.8.8.4.3.2"><ci id="S4.I2.i1.p1.1.m1.1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.1.1.1">→</ci><ci id="S4.I2.i1.p1.1.m1.1.1.2.cmml" xref="S4.I2.i1.p1.1.m1.1.1.2">𝑥</ci></apply></apply><list id="S4.I2.i1.p1.1.m1.8.8.2.3.cmml" xref="S4.I2.i1.p1.1.m1.8.8.2.2"><apply id="S4.I2.i1.p1.1.m1.7.7.1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1"><minus id="S4.I2.i1.p1.1.m1.7.7.1.1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1"></minus><apply id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.cmml" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2"><times id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.1.cmml" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.1"></times><apply id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.cmml" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2"><csymbol cd="ambiguous" id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.1.cmml" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2">subscript</csymbol><ci id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.2.cmml" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.2">𝜑</ci><cn id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.3.cmml" type="integer" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.2.3">1</cn></apply><interval closure="open" id="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.1.cmml" xref="S4.I2.i1.p1.1.m1.7.7.1.1.1.2.3.2"><apply id="S4.I2.i1.p1.1.m1.2.2.cmml" xref="S4.I2.i1.p1.1.m1.2.2"><ci id="S4.I2.i1.p1.1.m1.2.2.1.cmml" xref="S4.I2.i1.p1.1.m1.2.2.1">→</ci><ci id="S4.I2.i1.p1.1.m1.2.2.2.cmml" xref="S4.I2.i1.p1.1.m1.2.2.2">𝑥</ci></apply><apply id="S4.I2.i1.p1.1.m1.3.3.cmml" xref="S4.I2.i1.p1.1.m1.3.3"><ci id="S4.I2.i1.p1.1.m1.3.3.1.cmml" xref="S4.I2.i1.p1.1.m1.3.3.1">→</ci><ci id="S4.I2.i1.p1.1.m1.3.3.2.cmml" xref="S4.I2.i1.p1.1.m1.3.3.2">𝑦</ci></apply></interval></apply></apply><ci id="S4.I2.i1.p1.1.m1.6.6.cmml" xref="S4.I2.i1.p1.1.m1.6.6">…</ci><apply id="S4.I2.i1.p1.1.m1.8.8.2.2.2.cmml" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2"><times id="S4.I2.i1.p1.1.m1.8.8.2.2.2.1.cmml" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.1"></times><apply id="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.cmml" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.2"><csymbol cd="ambiguous" id="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.1.cmml" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.2">subscript</csymbol><ci id="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.2.cmml" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.2">𝜑</ci><ci id="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.3.cmml" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.2.3">ℓ</ci></apply><interval closure="open" id="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.1.cmml" xref="S4.I2.i1.p1.1.m1.8.8.2.2.2.3.2"><apply id="S4.I2.i1.p1.1.m1.4.4.cmml" xref="S4.I2.i1.p1.1.m1.4.4"><ci id="S4.I2.i1.p1.1.m1.4.4.1.cmml" xref="S4.I2.i1.p1.1.m1.4.4.1">→</ci><ci id="S4.I2.i1.p1.1.m1.4.4.2.cmml" xref="S4.I2.i1.p1.1.m1.4.4.2">𝑥</ci></apply><apply id="S4.I2.i1.p1.1.m1.5.5.cmml" xref="S4.I2.i1.p1.1.m1.5.5"><ci id="S4.I2.i1.p1.1.m1.5.5.1.cmml" xref="S4.I2.i1.p1.1.m1.5.5.1">→</ci><ci id="S4.I2.i1.p1.1.m1.5.5.2.cmml" xref="S4.I2.i1.p1.1.m1.5.5.2">𝑦</ci></apply></interval></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.1.m1.8c">Q^{\prime}(\vec{x}){\,:\!\!-\,}\varphi_{1}(\vec{x},\vec{y}),\dots,\varphi_{% \ell}(\vec{x},\vec{y})</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.1.m1.8d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( over→ start_ARG italic_x end_ARG ) : - italic_φ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ) , … , italic_φ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.3.2"> is free-connex with no disruptive trio, then direct access for </span><math alttext="Q" class="ltx_Math" display="inline" id="S4.I2.i1.p1.2.m2.1"><semantics id="S4.I2.i1.p1.2.m2.1a"><mi id="S4.I2.i1.p1.2.m2.1.1" xref="S4.I2.i1.p1.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.2.m2.1b"><ci id="S4.I2.i1.p1.2.m2.1.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.2.m2.1d">italic_Q</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.3.3"> is in </span><math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S4.I2.i1.p1.3.m3.2"><semantics id="S4.I2.i1.p1.3.m3.2a"><mrow id="S4.I2.i1.p1.3.m3.2.2.4" xref="S4.I2.i1.p1.3.m3.2.2.3.cmml"><mo id="S4.I2.i1.p1.3.m3.2.2.4.1" stretchy="false" xref="S4.I2.i1.p1.3.m3.2.2.3.cmml">⟨</mo><mi id="S4.I2.i1.p1.3.m3.1.1.1" xref="S4.I2.i1.p1.3.m3.1.1.1.cmml">loglinear</mi><mo id="S4.I2.i1.p1.3.m3.2.2.4.2" xref="S4.I2.i1.p1.3.m3.2.2.3.cmml">,</mo><mi id="S4.I2.i1.p1.3.m3.2.2.2" xref="S4.I2.i1.p1.3.m3.2.2.2.cmml">log</mi><mo id="S4.I2.i1.p1.3.m3.2.2.4.3" stretchy="false" xref="S4.I2.i1.p1.3.m3.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.3.m3.2b"><list id="S4.I2.i1.p1.3.m3.2.2.3.cmml" xref="S4.I2.i1.p1.3.m3.2.2.4"><ci id="S4.I2.i1.p1.3.m3.1.1.1.cmml" xref="S4.I2.i1.p1.3.m3.1.1.1">loglinear</ci><log id="S4.I2.i1.p1.3.m3.2.2.2.cmml" xref="S4.I2.i1.p1.3.m3.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.3.m3.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.3.m3.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.3.4">.</span></p> </div> </li> <li class="ltx_item" id="S4.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(2)</span> <div class="ltx_para" id="S4.I2.i2.p1"> <p class="ltx_p" id="S4.I2.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.3.1">Otherwise, if </span><math alttext="Q" class="ltx_Math" display="inline" id="S4.I2.i2.p1.1.m1.1"><semantics id="S4.I2.i2.p1.1.m1.1a"><mi id="S4.I2.i2.p1.1.m1.1.1" xref="S4.I2.i2.p1.1.m1.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.1.m1.1b"><ci id="S4.I2.i2.p1.1.m1.1.1.cmml" xref="S4.I2.i2.p1.1.m1.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.1.m1.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.1.m1.1d">italic_Q</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.3.2"> is also self-join-free, then direct access for </span><math alttext="Q" class="ltx_Math" display="inline" id="S4.I2.i2.p1.2.m2.1"><semantics id="S4.I2.i2.p1.2.m2.1a"><mi id="S4.I2.i2.p1.2.m2.1.1" xref="S4.I2.i2.p1.2.m2.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.2.m2.1b"><ci id="S4.I2.i2.p1.2.m2.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.2.m2.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.2.m2.1d">italic_Q</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.3.3"> is not in </span><math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S4.I2.i2.p1.3.m3.2"><semantics id="S4.I2.i2.p1.3.m3.2a"><mrow id="S4.I2.i2.p1.3.m3.2.2.4" xref="S4.I2.i2.p1.3.m3.2.2.3.cmml"><mo id="S4.I2.i2.p1.3.m3.2.2.4.1" stretchy="false" xref="S4.I2.i2.p1.3.m3.2.2.3.cmml">⟨</mo><mi id="S4.I2.i2.p1.3.m3.1.1.1" xref="S4.I2.i2.p1.3.m3.1.1.1.cmml">loglinear</mi><mo id="S4.I2.i2.p1.3.m3.2.2.4.2" xref="S4.I2.i2.p1.3.m3.2.2.3.cmml">,</mo><mi id="S4.I2.i2.p1.3.m3.2.2.2" xref="S4.I2.i2.p1.3.m3.2.2.2.cmml">log</mi><mo id="S4.I2.i2.p1.3.m3.2.2.4.3" stretchy="false" xref="S4.I2.i2.p1.3.m3.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.3.m3.2b"><list id="S4.I2.i2.p1.3.m3.2.2.3.cmml" xref="S4.I2.i2.p1.3.m3.2.2.4"><ci id="S4.I2.i2.p1.3.m3.1.1.1.cmml" xref="S4.I2.i2.p1.3.m3.1.1.1">loglinear</ci><log id="S4.I2.i2.p1.3.m3.2.2.2.cmml" xref="S4.I2.i2.p1.3.m3.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.3.m3.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.3.m3.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.3.4">, assuming the</span></p> </div> </li> </ol> </div> </div> <div class="ltx_theorem ltx_theorem_proof" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.1.1.1">Proof 4.2</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.8"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.p1.8.8">For the positive side, we simply apply <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#Thmthm11" title="Theorem 11. ‣ 4.1. Generalized Dichotomies ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Theorem</span> <span class="ltx_text ltx_ref_tag">11</span></a> with the corresponding semiring. In the case where <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.1.m1.1"><semantics id="S4.Thmtheorem2.p1.1.1.m1.1a"><mi id="S4.Thmtheorem2.p1.1.1.m1.1.1" xref="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.1.m1.1b"><ci id="S4.Thmtheorem2.p1.1.1.m1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.1.m1.1d">italic_α</annotation></semantics></math> is <math alttext="\mathsf{Avg}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.2.m2.1"><semantics id="S4.Thmtheorem2.p1.2.2.m2.1a"><mi id="S4.Thmtheorem2.p1.2.2.m2.1.1" xref="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml">𝖠𝗏𝗀</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.2.m2.1b"><ci id="S4.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.2.m2.1.1">𝖠𝗏𝗀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.2.m2.1c">\mathsf{Avg}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.2.m2.1d">sansserif_Avg</annotation></semantics></math>, we compute <math alttext="\mathsf{Sum}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.3.m3.1"><semantics id="S4.Thmtheorem2.p1.3.3.m3.1a"><mi id="S4.Thmtheorem2.p1.3.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml">𝖲𝗎𝗆</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.3.m3.1b"><ci id="S4.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.3.m3.1.1">𝖲𝗎𝗆</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.3.m3.1c">\mathsf{Sum}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.3.m3.1d">sansserif_Sum</annotation></semantics></math> and <math alttext="\mathsf{Count}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.4.m4.1"><semantics id="S4.Thmtheorem2.p1.4.4.m4.1a"><mi id="S4.Thmtheorem2.p1.4.4.m4.1.1" xref="S4.Thmtheorem2.p1.4.4.m4.1.1.cmml">𝖢𝗈𝗎𝗇𝗍</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.4.4.m4.1b"><ci id="S4.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem2.p1.4.4.m4.1.1">𝖢𝗈𝗎𝗇𝗍</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.4.m4.1c">\mathsf{Count}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.4.m4.1d">sansserif_Count</annotation></semantics></math> separately and divide the results. The negative side carries over from <a class="ltx_ref" href="https://arxiv.org/html/2303.05327v2#S3" title="3. The Direct-Access Problem ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Section</span> <span class="ltx_text ltx_ref_tag">3</span></a> since a direct access solution for <math alttext="Q" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.5.5.m5.1"><semantics id="S4.Thmtheorem2.p1.5.5.m5.1a"><mi id="S4.Thmtheorem2.p1.5.5.m5.1.1" xref="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.5.5.m5.1b"><ci id="S4.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p1.5.5.m5.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.5.5.m5.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.5.5.m5.1d">italic_Q</annotation></semantics></math> in <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.6.6.m6.2"><semantics id="S4.Thmtheorem2.p1.6.6.m6.2a"><mrow id="S4.Thmtheorem2.p1.6.6.m6.2.2.4" xref="S4.Thmtheorem2.p1.6.6.m6.2.2.3.cmml"><mo id="S4.Thmtheorem2.p1.6.6.m6.2.2.4.1" stretchy="false" xref="S4.Thmtheorem2.p1.6.6.m6.2.2.3.cmml">⟨</mo><mi id="S4.Thmtheorem2.p1.6.6.m6.1.1.1" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1.cmml">loglinear</mi><mo id="S4.Thmtheorem2.p1.6.6.m6.2.2.4.2" xref="S4.Thmtheorem2.p1.6.6.m6.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem2.p1.6.6.m6.2.2.2" xref="S4.Thmtheorem2.p1.6.6.m6.2.2.2.cmml">log</mi><mo id="S4.Thmtheorem2.p1.6.6.m6.2.2.4.3" stretchy="false" xref="S4.Thmtheorem2.p1.6.6.m6.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.6.6.m6.2b"><list id="S4.Thmtheorem2.p1.6.6.m6.2.2.3.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.2.2.4"><ci id="S4.Thmtheorem2.p1.6.6.m6.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.1.1.1">loglinear</ci><log id="S4.Thmtheorem2.p1.6.6.m6.2.2.2.cmml" xref="S4.Thmtheorem2.p1.6.6.m6.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.6.6.m6.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.6.6.m6.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math> is also a direct access solution for <math alttext="Q^{\prime}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.7.7.m7.1"><semantics id="S4.Thmtheorem2.p1.7.7.m7.1a"><msup id="S4.Thmtheorem2.p1.7.7.m7.1.1" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.cmml"><mi id="S4.Thmtheorem2.p1.7.7.m7.1.1.2" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.2.cmml">Q</mi><mo id="S4.Thmtheorem2.p1.7.7.m7.1.1.3" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.7.7.m7.1b"><apply id="S4.Thmtheorem2.p1.7.7.m7.1.1.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.7.7.m7.1.1.1.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.7.7.m7.1.1.2.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.2">𝑄</ci><ci id="S4.Thmtheorem2.p1.7.7.m7.1.1.3.cmml" xref="S4.Thmtheorem2.p1.7.7.m7.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.7.7.m7.1c">Q^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.7.7.m7.1d">italic_Q start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="\mathord{\langle\mathrm{loglinear},\log\rangle}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.8.8.m8.2"><semantics id="S4.Thmtheorem2.p1.8.8.m8.2a"><mrow id="S4.Thmtheorem2.p1.8.8.m8.2.2.4" xref="S4.Thmtheorem2.p1.8.8.m8.2.2.3.cmml"><mo id="S4.Thmtheorem2.p1.8.8.m8.2.2.4.1" stretchy="false" xref="S4.Thmtheorem2.p1.8.8.m8.2.2.3.cmml">⟨</mo><mi id="S4.Thmtheorem2.p1.8.8.m8.1.1.1" xref="S4.Thmtheorem2.p1.8.8.m8.1.1.1.cmml">loglinear</mi><mo id="S4.Thmtheorem2.p1.8.8.m8.2.2.4.2" xref="S4.Thmtheorem2.p1.8.8.m8.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem2.p1.8.8.m8.2.2.2" xref="S4.Thmtheorem2.p1.8.8.m8.2.2.2.cmml">log</mi><mo id="S4.Thmtheorem2.p1.8.8.m8.2.2.4.3" stretchy="false" xref="S4.Thmtheorem2.p1.8.8.m8.2.2.3.cmml">⟩</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.8.8.m8.2b"><list id="S4.Thmtheorem2.p1.8.8.m8.2.2.3.cmml" xref="S4.Thmtheorem2.p1.8.8.m8.2.2.4"><ci id="S4.Thmtheorem2.p1.8.8.m8.1.1.1.cmml" xref="S4.Thmtheorem2.p1.8.8.m8.1.1.1">loglinear</ci><log id="S4.Thmtheorem2.p1.8.8.m8.2.2.2.cmml" xref="S4.Thmtheorem2.p1.8.8.m8.2.2.2"></log></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.8.8.m8.2c">\mathord{\langle\mathrm{loglinear},\log\rangle}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.8.8.m8.2d">⟨ roman_loglinear , roman_log ⟩</annotation></semantics></math> if we ignore the aggregated values.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_rem" id="Thmthm17"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmthm17.1.1.1">Remark 17</span></span><span class="ltx_text ltx_font_bold" id="Thmthm17.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm17.p1"> <p class="ltx_p" id="Thmthm17.p1.4"><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2303.05327v2#Thmthm14" title="Corollary 14. ‣ 4.1. Generalized Dichotomies ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">14</span></a><span class="ltx_text ltx_font_italic" id="Thmthm17.p1.4.4"> can be easily extended to support multiple aggregate functions <math alttext="\alpha_{1}(\vec{w}_{1})" class="ltx_Math" display="inline" id="Thmthm17.p1.1.1.m1.1"><semantics id="Thmthm17.p1.1.1.m1.1a"><mrow id="Thmthm17.p1.1.1.m1.1.1" xref="Thmthm17.p1.1.1.m1.1.1.cmml"><msub id="Thmthm17.p1.1.1.m1.1.1.3" xref="Thmthm17.p1.1.1.m1.1.1.3.cmml"><mi id="Thmthm17.p1.1.1.m1.1.1.3.2" xref="Thmthm17.p1.1.1.m1.1.1.3.2.cmml">α</mi><mn id="Thmthm17.p1.1.1.m1.1.1.3.3" xref="Thmthm17.p1.1.1.m1.1.1.3.3.cmml">1</mn></msub><mo id="Thmthm17.p1.1.1.m1.1.1.2" xref="Thmthm17.p1.1.1.m1.1.1.2.cmml">⁢</mo><mrow id="Thmthm17.p1.1.1.m1.1.1.1.1" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.cmml"><mo id="Thmthm17.p1.1.1.m1.1.1.1.1.2" stretchy="false" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.cmml">(</mo><msub id="Thmthm17.p1.1.1.m1.1.1.1.1.1" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.cmml"><mover accent="true" id="Thmthm17.p1.1.1.m1.1.1.1.1.1.2" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.cmml"><mi id="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.2" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.2.cmml">w</mi><mo id="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.1" stretchy="false" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.1.cmml">→</mo></mover><mn id="Thmthm17.p1.1.1.m1.1.1.1.1.1.3" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="Thmthm17.p1.1.1.m1.1.1.1.1.3" stretchy="false" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm17.p1.1.1.m1.1b"><apply id="Thmthm17.p1.1.1.m1.1.1.cmml" xref="Thmthm17.p1.1.1.m1.1.1"><times id="Thmthm17.p1.1.1.m1.1.1.2.cmml" xref="Thmthm17.p1.1.1.m1.1.1.2"></times><apply id="Thmthm17.p1.1.1.m1.1.1.3.cmml" xref="Thmthm17.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmthm17.p1.1.1.m1.1.1.3.1.cmml" xref="Thmthm17.p1.1.1.m1.1.1.3">subscript</csymbol><ci id="Thmthm17.p1.1.1.m1.1.1.3.2.cmml" xref="Thmthm17.p1.1.1.m1.1.1.3.2">𝛼</ci><cn id="Thmthm17.p1.1.1.m1.1.1.3.3.cmml" type="integer" xref="Thmthm17.p1.1.1.m1.1.1.3.3">1</cn></apply><apply id="Thmthm17.p1.1.1.m1.1.1.1.1.1.cmml" xref="Thmthm17.p1.1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm17.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="Thmthm17.p1.1.1.m1.1.1.1.1">subscript</csymbol><apply id="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.cmml" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.2"><ci id="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.1.cmml" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.1">→</ci><ci id="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.2.cmml" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.2.2">𝑤</ci></apply><cn id="Thmthm17.p1.1.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="Thmthm17.p1.1.1.m1.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm17.p1.1.1.m1.1c">\alpha_{1}(\vec{w}_{1})</annotation><annotation encoding="application/x-llamapun" id="Thmthm17.p1.1.1.m1.1d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( over→ start_ARG italic_w end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, …, <math alttext="\alpha_{k}(\vec{w}_{k})" class="ltx_Math" display="inline" id="Thmthm17.p1.2.2.m2.1"><semantics id="Thmthm17.p1.2.2.m2.1a"><mrow id="Thmthm17.p1.2.2.m2.1.1" xref="Thmthm17.p1.2.2.m2.1.1.cmml"><msub id="Thmthm17.p1.2.2.m2.1.1.3" xref="Thmthm17.p1.2.2.m2.1.1.3.cmml"><mi id="Thmthm17.p1.2.2.m2.1.1.3.2" xref="Thmthm17.p1.2.2.m2.1.1.3.2.cmml">α</mi><mi id="Thmthm17.p1.2.2.m2.1.1.3.3" xref="Thmthm17.p1.2.2.m2.1.1.3.3.cmml">k</mi></msub><mo id="Thmthm17.p1.2.2.m2.1.1.2" xref="Thmthm17.p1.2.2.m2.1.1.2.cmml">⁢</mo><mrow id="Thmthm17.p1.2.2.m2.1.1.1.1" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.cmml"><mo id="Thmthm17.p1.2.2.m2.1.1.1.1.2" stretchy="false" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.cmml">(</mo><msub id="Thmthm17.p1.2.2.m2.1.1.1.1.1" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.cmml"><mover accent="true" id="Thmthm17.p1.2.2.m2.1.1.1.1.1.2" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.cmml"><mi id="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.2" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.2.cmml">w</mi><mo id="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.1" stretchy="false" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.1.cmml">→</mo></mover><mi id="Thmthm17.p1.2.2.m2.1.1.1.1.1.3" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="Thmthm17.p1.2.2.m2.1.1.1.1.3" stretchy="false" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm17.p1.2.2.m2.1b"><apply id="Thmthm17.p1.2.2.m2.1.1.cmml" xref="Thmthm17.p1.2.2.m2.1.1"><times id="Thmthm17.p1.2.2.m2.1.1.2.cmml" xref="Thmthm17.p1.2.2.m2.1.1.2"></times><apply id="Thmthm17.p1.2.2.m2.1.1.3.cmml" xref="Thmthm17.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmthm17.p1.2.2.m2.1.1.3.1.cmml" xref="Thmthm17.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="Thmthm17.p1.2.2.m2.1.1.3.2.cmml" xref="Thmthm17.p1.2.2.m2.1.1.3.2">𝛼</ci><ci id="Thmthm17.p1.2.2.m2.1.1.3.3.cmml" xref="Thmthm17.p1.2.2.m2.1.1.3.3">𝑘</ci></apply><apply id="Thmthm17.p1.2.2.m2.1.1.1.1.1.cmml" xref="Thmthm17.p1.2.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm17.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="Thmthm17.p1.2.2.m2.1.1.1.1">subscript</csymbol><apply id="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.cmml" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.2"><ci id="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.1.cmml" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.1">→</ci><ci id="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.2.cmml" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.2.2">𝑤</ci></apply><ci id="Thmthm17.p1.2.2.m2.1.1.1.1.1.3.cmml" xref="Thmthm17.p1.2.2.m2.1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm17.p1.2.2.m2.1c">\alpha_{k}(\vec{w}_{k})</annotation><annotation encoding="application/x-llamapun" id="Thmthm17.p1.2.2.m2.1d">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( over→ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. For that, we can simply solve the problem for each <math alttext="\alpha_{i}(\vec{w}_{i})" class="ltx_Math" display="inline" id="Thmthm17.p1.3.3.m3.1"><semantics id="Thmthm17.p1.3.3.m3.1a"><mrow id="Thmthm17.p1.3.3.m3.1.1" xref="Thmthm17.p1.3.3.m3.1.1.cmml"><msub id="Thmthm17.p1.3.3.m3.1.1.3" xref="Thmthm17.p1.3.3.m3.1.1.3.cmml"><mi id="Thmthm17.p1.3.3.m3.1.1.3.2" xref="Thmthm17.p1.3.3.m3.1.1.3.2.cmml">α</mi><mi id="Thmthm17.p1.3.3.m3.1.1.3.3" xref="Thmthm17.p1.3.3.m3.1.1.3.3.cmml">i</mi></msub><mo id="Thmthm17.p1.3.3.m3.1.1.2" xref="Thmthm17.p1.3.3.m3.1.1.2.cmml">⁢</mo><mrow id="Thmthm17.p1.3.3.m3.1.1.1.1" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.cmml"><mo id="Thmthm17.p1.3.3.m3.1.1.1.1.2" stretchy="false" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.cmml">(</mo><msub id="Thmthm17.p1.3.3.m3.1.1.1.1.1" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.cmml"><mover accent="true" id="Thmthm17.p1.3.3.m3.1.1.1.1.1.2" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.cmml"><mi id="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.2" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.2.cmml">w</mi><mo id="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.1" stretchy="false" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.1.cmml">→</mo></mover><mi id="Thmthm17.p1.3.3.m3.1.1.1.1.1.3" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="Thmthm17.p1.3.3.m3.1.1.1.1.3" stretchy="false" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmthm17.p1.3.3.m3.1b"><apply id="Thmthm17.p1.3.3.m3.1.1.cmml" xref="Thmthm17.p1.3.3.m3.1.1"><times id="Thmthm17.p1.3.3.m3.1.1.2.cmml" xref="Thmthm17.p1.3.3.m3.1.1.2"></times><apply id="Thmthm17.p1.3.3.m3.1.1.3.cmml" xref="Thmthm17.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="Thmthm17.p1.3.3.m3.1.1.3.1.cmml" xref="Thmthm17.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="Thmthm17.p1.3.3.m3.1.1.3.2.cmml" xref="Thmthm17.p1.3.3.m3.1.1.3.2">𝛼</ci><ci id="Thmthm17.p1.3.3.m3.1.1.3.3.cmml" xref="Thmthm17.p1.3.3.m3.1.1.3.3">𝑖</ci></apply><apply id="Thmthm17.p1.3.3.m3.1.1.1.1.1.cmml" xref="Thmthm17.p1.3.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="Thmthm17.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="Thmthm17.p1.3.3.m3.1.1.1.1">subscript</csymbol><apply id="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.2"><ci id="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.1.cmml" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.1">→</ci><ci id="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.2.cmml" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.2.2">𝑤</ci></apply><ci id="Thmthm17.p1.3.3.m3.1.1.1.1.1.3.cmml" xref="Thmthm17.p1.3.3.m3.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmthm17.p1.3.3.m3.1c">\alpha_{i}(\vec{w}_{i})</annotation><annotation encoding="application/x-llamapun" id="Thmthm17.p1.3.3.m3.1d">italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( over→ start_ARG italic_w end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> separately, and extract the aggregate values of an answer from the <math alttext="k" class="ltx_Math" display="inline" id="Thmthm17.p1.4.4.m4.1"><semantics id="Thmthm17.p1.4.4.m4.1a"><mi id="Thmthm17.p1.4.4.m4.1.1" xref="Thmthm17.p1.4.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="Thmthm17.p1.4.4.m4.1b"><ci id="Thmthm17.p1.4.4.m4.1.1.cmml" xref="Thmthm17.p1.4.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmthm17.p1.4.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="Thmthm17.p1.4.4.m4.1d">italic_k</annotation></semantics></math> data structures that we construct in the preprocessing phase. (Moreover, a practical implementation can handle all aggregate values in the same structure.) ∎</span></p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_font_italic ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2. </span>Count Distinct</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1"><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.1">Can we generalize </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2303.05327v2#Thmthm14" title="Corollary 14. ‣ 4.1. Generalized Dichotomies ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">14</span></a><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.2"> beyond the stated aggregate functions? The most notable missing aggregate function is </span><math alttext="\mathsf{CountD}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">𝖢𝗈𝗎𝗇𝗍𝖣</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝖢𝗈𝗎𝗇𝗍𝖣</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">\mathsf{CountD}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">sansserif_CountD</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.3"> (count-distinct). Next, we show that we </span><em class="ltx_emph" id="S4.SS2.p1.1.4">cannot</em><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.5"> have similar tractability for count-distinct, and we illustrate it with a specific query (</span><span class="ltx_ref ltx_missing_label ltx_font_italic ltx_ref_self">LABEL:lem:hsc</span><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.6">). After that, we show how precisely </span><a class="ltx_ref ltx_font_italic" href="https://arxiv.org/html/2303.05327v2#Thmthm14" title="Corollary 14. ‣ 4.1. Generalized Dichotomies ‣ 4. Incorporating Annotation and Aggregation in the Answers ‣ Direct Access for Answers to Conjunctive Queries with Aggregation"><span class="ltx_text ltx_ref_tag">Corollary</span> <span class="ltx_text ltx_ref_tag">14</span></a><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.7"> changes for count-distinct, and particularly which AggCQs move from the tractable to the intractable side (</span><span class="ltx_ref ltx_missing_label ltx_font_italic ltx_ref_self">LABEL:thm:countd-dichotomy</span><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.8">). We later explain that the tractable cases of </span><span class="ltx_ref ltx_missing_label ltx_font_italic ltx_ref_self">LABEL:cor:aggregate-order-tractable</span><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.9"> are fully restored if we assume that the domain size is logarithmic in that of the input (</span><span class="ltx_ref ltx_missing_label ltx_font_italic ltx_ref_self">LABEL:rem:countd-small-domain</span><span class="ltx_text ltx_font_italic" id="S4.SS2.p1.1.10">).</span></p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1"><span class="ltx_text ltx_font_italic" id="S4.SS2.p2.1.1">The negative results in this section require the </span><em class="ltx_emph" id="S4.SS2.p2.1.2">small-universe Hitting Set Conjecture (</em></p> </div> <div class="ltx_theorem ltx_theorem_hypo" id="Thmthm18"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><em class="ltx_emph ltx_font_bold" id="Thmthm18.1.1.1">Hypothesis 18</em></span><span class="ltx_text ltx_font_bold" id="Thmthm18.2.2">.</span> </h6> <div class="ltx_para" id="Thmthm18.p1"> <p class="ltx_p" id="Thmthm18.p1.1"><span class="ltx_text ltx_font_italic" id="Thmthm18.p1.1.1">HSC)</span></p> </div> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.1"><cite class="ltx_cite ltx_citemacro_cite"><span class="ltx_text ltx_font_italic" id="S4.SS2.p3.1.1.1">[</span><span class="ltx_ref ltx_missing_citation ltx_ref_self">vassilevska2015hardness</span><span class="ltx_text ltx_font_italic" id="S4.SS2.p3.1.2.2">]</span></cite><span class="ltx_text ltx_font_italic" id="S4.SS2.p3.1.3">. 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