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In the current work, we show how infinite-duration games with Emerson-Lei objectives can be analyzed in two different ways. First, we show that the Zielonka tree of the Emerson-Lei condition naturally gives rise to a new reduction to parity games. This reduction, however, does not result in optimal analysis. Second, we show based on the first reduction (and the Zielonka tree) how to provide a direct fixpoint-based characterization of the winning region. The fixpoint-based characterization allows for symbolic analysis. It generalizes the solutions of games with known winning conditions such as Büchi, GR[1], parity, Streett, Rabin and Muller objectives, and in the case of these conditions reproduces previously known symbolic algorithms and complexity results. We also show how the capabilities of the proposed algorithm can be exploited in reactive synthesis, suggesting a new expressive fragment of LTL that can be handled symbolically. Our fragment combines a safety specification and a liveness part. The safety part is unrestricted and the liveness part allows to define Emerson-Lei conditions on occurrences of letters. The symbolic treatment is enabled due to the simplicity of determinization in the case of safety languages and by using our new algorithm for game solving. This approach maximizes the number of steps solved symbolically in order to maximize the potential for efficient symbolic implementations.","datePublished":"2024","isAccessibleForFree":true,"@type":"ScholarlyArticle","@context":"https://schema.org"}</script> </head> <body class="shared-article-renderer">  <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-vh-full"> <a class="c-skip-link" href="#main-content">Skip to main content</a> <div class="u-hide u-show-following-ad"></div> <aside class="c-ad c-ad--728x90" data-test="springer-doubleclick-ad"> <div class="c-ad__inner"> <p class="c-ad__label">Advertisement</p> <div id="div-gpt-ad-LB1" data-pa11y-ignore data-gpt data-test="LB1-ad" data-gpt-unitpath="/270604982/springerlink/book/chapter" data-gpt-sizes="728x90" style="min-width:728px;min-height:90px" data-gpt-targeting="pos=LB1;"></div> </div> </aside> <div class="app-elements"> <header class="eds-c-header" data-eds-c-header> <div class="eds-c-header__container" data-eds-c-header-expander-anchor> <div class="eds-c-header__brand"> <a href="https://link.springer.com" data-test=springerlink-logo data-track="click_imprint_logo" data-track-context="unified header" data-track-action="click logo link" data-track-category="unified header" data-track-label="link" > <img src="/oscar-static/images/darwin/header/img/logo-springer-nature-link-3149409f62.svg" alt="Springer Nature Link"> </a> </div> <a class="c-header__link eds-c-header__link" id="identity-account-widget" href='https://idp.springer.com/auth/personal/springernature?redirect_uri=https://link.springer.com/chapter/10.1007/978-3-031-57228-9_4?fromPaywallRec=false'><span class="eds-c-header__widget-fragment-title">Log in</span></a> </div> <nav class="eds-c-header__nav" aria-label="header navigation"> <div class="eds-c-header__nav-container"> <div class="eds-c-header__item eds-c-header__item--menu"> <a href="#eds-c-header-nav" class="eds-c-header__link" data-eds-c-header-expander> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-menu-medium"></use> </svg><span>Menu</span> </a> </div> <div class="eds-c-header__item eds-c-header__item--inline-links"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </div> <div class="eds-c-header__link-container"> <div class="eds-c-header__item eds-c-header__item--divider"> <a href="#eds-c-header-popup-search" class="eds-c-header__link" data-eds-c-header-expander data-eds-c-header-test-search-btn> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg><span>Search</span> </a> </div> <div id="ecommerce-header-cart-icon-link" class="eds-c-header__item ecommerce-cart" style="display:inline-block"> <a class="eds-c-header__link" href="https://order.springer.com/public/cart" style="appearance:none;border:none;background:none;color:inherit;position:relative"> <svg id="eds-i-cart" class="eds-c-header__icon" xmlns="http://www.w3.org/2000/svg" height="24" width="24" viewBox="0 0 24 24" aria-hidden="true" focusable="false"> <path fill="currentColor" fill-rule="nonzero" d="M2 1a1 1 0 0 0 0 2l1.659.001 2.257 12.808a2.599 2.599 0 0 0 2.435 2.185l.167.004 9.976-.001a2.613 2.613 0 0 0 2.61-1.748l.03-.106 1.755-7.82.032-.107a2.546 2.546 0 0 0-.311-1.986l-.108-.157a2.604 2.604 0 0 0-2.197-1.076L6.042 5l-.56-3.17a1 1 0 0 0-.864-.82l-.12-.007L2.001 1ZM20.35 6.996a.63.63 0 0 1 .54.26.55.55 0 0 1 .082.505l-.028.1L19.2 15.63l-.022.05c-.094.177-.282.299-.526.317l-10.145.002a.61.61 0 0 1-.618-.515L6.394 6.999l13.955-.003ZM18 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4ZM8 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"></path> </svg><span>Cart</span><span class="cart-info" style="display:none;position:absolute;top:10px;right:45px;background-color:#C65301;color:#fff;width:18px;height:18px;font-size:11px;border-radius:50%;line-height:17.5px;text-align:center"></span></a> <script>(function () { var exports = {}; 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aria-hidden="true"> <div class="c-context-bar__container u-container" data-track-context="sticky banner"> <div class="c-context-bar__title"> Symbolic Solution of Emerson-Lei Games for Reactive Synthesis </div> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both"> <a href="/content/pdf/10.1007/978-3-031-57228-9.pdf" rel="noopener" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-book-pdf="true" data-test="pdf-link" data-track="content_download" data-track-type="book pdf download" data-track-label="link" data-track-action="Book download - pdf" download> <span class="c-pdf-download__text"><span class="u-sticky-visually-hidden">Download</span> book PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"> <use xlink:href="#icon-eds-i-download-medium"/> </svg> </a> </div> <div class="c-pdf-download u-clear-both"> <a href="/download/epub/10.1007/978-3-031-57228-9.epub" 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data-track-component="chapter body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 id="Abs1" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number"> </span>Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>Emerson-Lei conditions have recently attracted attention due to both their succinctness and their favorable closure properties. In the current work, we show how infinite-duration games with Emerson-Lei objectives can be analyzed in two different ways. First, we show that the Zielonka tree of the Emerson-Lei condition naturally gives rise to a new reduction to parity games. This reduction, however, does not result in optimal analysis. Second, we show based on the first reduction (and the Zielonka tree) how to provide a direct fixpoint-based characterization of the winning region. The fixpoint-based characterization allows for symbolic analysis. It generalizes the solutions of games with known winning conditions such as Büchi, GR[1], parity, Streett, Rabin and Muller objectives, and in the case of these conditions reproduces previously known symbolic algorithms and complexity results.</p><p>We also show how the capabilities of the proposed algorithm can be exploited in reactive synthesis, suggesting a new expressive fragment of LTL that can be handled symbolically. Our fragment combines a safety specification and a liveness part. The safety part is unrestricted and the liveness part allows to define Emerson-Lei conditions on occurrences of letters. The symbolic treatment is enabled due to the simplicity of determinization in the case of safety languages and by using our new algorithm for game solving. This approach maximizes the number of steps solved symbolically in order to maximize the potential for efficient symbolic implementations.</p></div></div></section><div class="c-article-section__content c-article-section__content--separator"><p>This work is supported by the ERC Consolidator grant D-SynMA (No. 772459).</p></div> <div data-test="cobranding-download"> <div class="note test-pdf-link" id="cobranding-and-download-availability-text"> <div class="c-article-access-provider" data-component="provided-by-box"> <p class="c-article-access-provider__text c-article-access-provider__text--chapter"> You have full access to this open access chapter,  <a href="/content/pdf/10.1007/978-3-031-57228-9_4.pdf?pdf=inline%20link" class="c-pdf-download__link" id="js-body-chapter-download" style="display: inline; padding:0px!important;" target="_blank" rel="noopener" data-track="content_download" data-track-context="article body" data-track-type="conference paper PDF download" data-track-action="Pdf download" data-track-label="inline link" download>Download conference paper PDF</a> <svg width="24" height="24" focusable="false" role="img" aria-hidden="true" class="c-download-pdf-icon-large"> <use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use> </svg> </p> </div> </div> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w92h120/springer-static/cover-hires/book/978-3-319-10575-8?as=webp" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/978-3-319-10575-8_27?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 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3" data-track-label="10.1007/978-3-030-53291-8_21">Stochastic Games with Lexicographic Reachability-Safety Objectives </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Chapter</span> <span class="c-article-meta-recommendations__date">© 2020</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1733976288, embedded_user: 'null' } }); </script> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 id="Sec1" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>Infinite-duration two-player games are a strong tool that has been used, notably, for reactive synthesis from temporal specifications [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Sixteenth ACM Symposium on Principles of Programming Languages. pp. 179–190. ACM Press (1989). 
 https://doi.org/10.1145/75277.75293
 
 " href="#ref-CR38" id="ref-link-section-d15224108e661">38</a>]. Many different winning conditions are considered in the literature.</p><p>Emerson-Lei (EL) conditions [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Emerson, E.A., Lei, C.: Modalities for model checking: Branching time logic strikes back. Sci. Comput. Program. 8(3), 275–306 (1987). 
 https://doi.org/10.1016/0167-6423(87)90036-0
 
 " href="#ref-CR21" id="ref-link-section-d15224108e667">21</a>] were initially suggested in the context of automata but are among the most general (regular) winning conditions considered for such games. They succinctly express general liveness properties by encoding Boolean combinations of events that should occur infinitely or finitely often. Automata and games in which acceptance or winning is defined by Emerson-Lei conditions have garnered attention in recent years [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Hunter, P., Dawar, A.: Complexity bounds for regular games. In: 30th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3618, pp. 495–506. Springer (2005). 
 https://doi.org/10.1007/11549345_43
 
 " href="#ref-CR25" id="ref-link-section-d15224108e670">25</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="John, T., Jantsch, S., Baier, C., Klüppelholz, S.: From emerson-lei automata to deterministic, limit-deterministic or good-for-mdp automata. Innov. Syst. Softw. Eng. 18(3), 385–403 (2022). 
 https://doi.org/10.1007/s11334-022-00445-7
 
 " href="#ref-CR27" id="ref-link-section-d15224108e673">27</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Müller, D., Sickert, S.: LTL to deterministic emerson-lei automata. In: Bouyer, P., Orlandini, A., Pietro, P.S. (eds.) Proceedings Eighth International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2017, Roma, Italy, 20-22 September 2017. EPTCS, vol. 256, pp. 180–194 (2017). 
 https://doi.org/10.4204/EPTCS.256.13
 
 " href="#ref-CR35" id="ref-link-section-d15224108e676">35</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title="Renkin, F., Duret-Lutz, A., Pommellet, A.: Practical "paritizing" of emerson-lei automata. In: Hung, D.V., Sokolsky, O. (eds.) Automated Technology for Verification and Analysis - 18th International Symposium, ATVA 2020, Hanoi, Vietnam, October 19-23, 2020, Proceedings. Lecture Notes in Computer Science, vol. 12302, pp. 127–143. Springer (2020). 
 https://doi.org/10.1007/978-3-030-59152-6_7
 
 " href="#ref-CR40" id="ref-link-section-d15224108e679">40</a>], in particular because of their succinctness and good compositionality properties (Emerson-Lei objectives are closed under conjunction, disjunction, and negation). In this work, we show how infinite-duration two-player games with Emerson-Lei winning conditions can be solved symbolically.</p><p>It has been established that solving Emerson-Lei games is <span class="u-small-caps">PSpace</span>-complete and that an exponential amount of memory may be required by winning strategies [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Hunter, P., Dawar, A.: Complexity bounds for regular games. In: 30th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3618, pp. 495–506. Springer (2005). 
 https://doi.org/10.1007/11549345_43
 
 " href="#ref-CR25" id="ref-link-section-d15224108e688">25</a>]. Zielonka trees are succinct tree-representations of Muller objectives [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci. 200(1-2), 135–183 (1998). 
 https://doi.org/10.1016/S0304-3975(98)00009-7
 
 " href="#ref-CR47" id="ref-link-section-d15224108e691">47</a>]. They have been used to obtain tight bounds on the amount of memory needed for winning in Muller games [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Dziembowski, S., Jurdzinski, M., Walukiewicz, I.: How much memory is needed to win infinite games? In: 12th Annual IEEE Symposium on Logic in Computer Science. pp. 99–110. IEEE Computer Society (1997). 
 https://doi.org/10.1109/LICS.1997.614939
 
 " href="#ref-CR18" id="ref-link-section-d15224108e694">18</a>], and can also be applied to analyze Emerson-Lei objectives and games. One indirect way to solve Emerson-Lei games is by transformation to equivalent parity games using later-appearance-records [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Hunter, P., Dawar, A.: Complexity bounds for regular games. In: 30th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3618, pp. 495–506. Springer (2005). 
 https://doi.org/10.1007/11549345_43
 
 " href="#ref-CR25" id="ref-link-section-d15224108e697">25</a>], and then solving the resulting parity games. Another, more recent, indirect approach goes through Rabin games by first extracting history-deterministic Rabin automata from Zielonka trees and then solving the resulting Rabin games [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Casares, A., Colcombet, T., Lehtinen, K.: On the size of good-for-games rabin automata and its link with the memory in muller games. In: Bojanczyk, M., Merelli, E., Woodruff, D.P. (eds.) International Colloquium on Automata, Languages, and Programming, ICALP 2022. LIPIcs, vol. 229, pp. 117:1–117:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022). 
 https://doi.org/10.4230/LIPIcs.ICALP.2022.117
 
 " href="#ref-CR12" id="ref-link-section-d15224108e701">12</a>]. Both these indirect solution methods are enumerative by nature. Here, we give a direct symbolic algorithmic solution for Emerson-Lei games. We show how the Zielonka tree allows to directly encode the game as a parity game. Furthermore, building on this reduction, we show how to construct a fixpoint equation system that captures winning in the game. As usual, fixpoint equation systems are recipes for game solving algorithms that manipulate sets of states symbolically. To the best of our knowledge, we thereby give the first description of a fully symbolic algorithm for the solution of Emerson-Lei games.</p><p>The algorithm that we obtain in this way is adaptive in the sense that the nesting structure of recursive calls is obtained directly from the Zielonka tree of the given winning objective. As the Zielonka tree is specific to the objective, this means that the algorithm performs just the fixpoint computations that are required for that specific objective. In particular, our algorithm instantiates to previously known fixpoint iteration algorithms in the case that the objective is a (generalized) Büchi, GR[1], parity, Streett, Rabin or Muller condition, reproducing previously known algorithms and complexity results. As we use fixpoint iteration, the instantiation of our algorithm to parity game solving is not directly a quasipolynomial algorithm. In the general setting, the algorithm solves unrestricted Emerson-Lei games with <i>k</i> colors, <i>m</i> edges and <i>n</i> nodes in time <span class="mathjax-tex">$\mathcal {O}(k!\cdot m\cdot n^k)$</span> and yields winning strategies with memory <span class="mathjax-tex">$\mathcal {O}(k!)$</span>.</p><p>We apply our symbolic solution of Emerson-Lei games to the automated construction of safe systems. The ideas of synthesis of reactive systems from temporal specifications go back to the early days of computer science [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Church, A.: Logic, arithmetic, and automata. In: International Congress of Mathematicians. Institut Mittag-Leffler, Sweden (1963)" href="#ref-CR14" id="ref-link-section-d15224108e796">14</a>]. These concepts were modernized and connected to linear temporal logic (LTL) and finite-state automata by Pnueli and Rosner [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Sixteenth ACM Symposium on Principles of Programming Languages. pp. 179–190. ACM Press (1989). 
 https://doi.org/10.1145/75277.75293
 
 " href="#ref-CR38" id="ref-link-section-d15224108e799">38</a>]. In recent years, practical applications in robotics are using this form of synthesis as part of a framework producing correct-by-design controllers [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Bhatia, A., Maly, M.R., Kavraki, L.E., Vardi, M.Y.: Motion planning with complex goals. IEEE Robotics Autom. Mag. 18(3), 55–64 (2011). 
 https://doi.org/10.1109/MRA.2011.942115
 
 " href="#ref-CR6" id="ref-link-section-d15224108e802">6</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Kress-Gazit, H., Fainekos, G.E., Pappas, G.J.: Temporal-logic-based reactive mission and motion planning. IEEE Trans. Robotics 25(6), 1370–1381 (2009). 
 https://doi.org/10.1109/TRO.2009.2030225
 
 " href="#ref-CR28" id="ref-link-section-d15224108e805">28</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Liu, J., Ozay, N., Topcu, U., Murray, R.M.: Synthesis of reactive switching protocols from temporal logic specifications. IEEE Trans. Autom. Control. 58(7), 1771–1785 (2013). 
 https://doi.org/10.1109/TAC.2013.2246095
 
 " href="#ref-CR32" id="ref-link-section-d15224108e808">32</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Moarref, S., Kress-Gazit, H.: Automated synthesis of decentralized controllers for robot swarms from high-level temporal logic specifications. Auton. Robots 44(3-4), 585–600 (2020). 
 https://doi.org/10.1007/s10514-019-09861-4
 
 " href="#ref-CR34" id="ref-link-section-d15224108e812">34</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title="Wongpiromsarn, T., Topcu, U., Murray, R.M.: Receding horizon temporal logic planning. IEEE Trans. Autom. Control. 57(11), 2817–2830 (2012). 
 https://doi.org/10.1109/TAC.2012.2195811
 
 " href="#ref-CR44" id="ref-link-section-d15224108e815">44</a>].</p><p>A prominent way to extend the capacity of reasoning about state spaces is by reasoning <i>symbolically</i> about sets of states/paths. In order to apply this approach to reactive synthesis, different fragments of LTL that allow symbolic game analysis have been considered. Notably, the GR[1] fragment has been widely used for the applications in robotics mentioned above [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Bloem, R., Jobstmann, B., Piterman, N., Pnueli, A., Sa’ar, Y.: Synthesis of reactive(1) designs. J. Comput. Syst. Sci. 78(3), 911–938 (2012). 
 https://doi.org/10.1016/j.jcss.2011.08.007
 
 " href="#ref-CR7" id="ref-link-section-d15224108e824">7</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 37" title="Piterman, N., Pnueli, A., Sa’ar, Y.: Synthesis of reactive(1) designs. In: 7th International Conference on Verification, Model Checking, and Abstract Interpretation. Lecture Notes in Computer Science, vol. 3855, pp. 364–380. Springer (2006). 
 https://doi.org/10.1007/11609773_24
 
 " href="#ref-CR37" id="ref-link-section-d15224108e827">37</a>]. But also larger fragments are being considered and experimented with [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Ehlers, R.: Generalized rabin(1) synthesis with applications to robust system synthesis. In: Third International Symposium on NASA Formal Methods. Lecture Notes in Computer Science, vol. 6617, pp. 101–115. Springer (2011). 
 https://doi.org/10.1007/978-3-642-20398-5_9
 
 " href="#ref-CR19" id="ref-link-section-d15224108e830">19</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Ehlers, R.: Unbeast: Symbolic bounded synthesis. In: 17th International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Lecture Notes in Computer Science, vol. 6605, pp. 272–275. Springer (2011). 
 https://doi.org/10.1007/978-3-642-19835-9_25
 
 " href="#ref-CR20" id="ref-link-section-d15224108e833">20</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Sohail, S., Somenzi, F.: Safety first: a two-stage algorithm for the synthesis of reactive systems. Int. J. Softw. Tools Technol. Transf. 15(5-6), 433–454 (2013). 
 https://doi.org/10.1007/s10009-012-0224-3
 
 " href="#ref-CR41" id="ref-link-section-d15224108e837">41</a>]. Recently, De Giacomo and Vardi suggested that similar advantages can be had by changing the usual semantics of LTL from considering infinite models to finite models (LTL<span class="mathjax-tex">$_f$</span>) [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Giacomo, G.D., Vardi, M.Y.: Synthesis for LTL and LDL on finite traces. In: Yang, Q., Wooldridge, M.J. (eds.) Twenty-Fourth International Joint Conference on Artificial Intelligence. pp. 1558–1564. AAAI Press (2015)" href="#ref-CR22" id="ref-link-section-d15224108e862">22</a>]. The complexity of the problem remains doubly-exponential, however, symbolic techniques can be applied. As models are finite, it is possible to use the classical subset construction (in contrast to Büchi determinization), which can be reasoned about symbolically. Furthermore, the resulting games have simple reachability objectives. This approach with finite models is used for applications in planning [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Camacho, A., McIlraith, S.A.: Learning interpretable models expressed in linear temporal logic. In: Twenty-Ninth International Conference on Automated Planning and Scheduling. pp. 621–630. AAAI Press (2019). 
 https://doi.org/10.1609/icaps.v29i1.3529
 
 " href="#ref-CR10" id="ref-link-section-d15224108e865">10</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Camacho, A., Triantafillou, E., Muise, C.J., Baier, J.A., McIlraith, S.A.: Non-deterministic planning with temporally extended goals: LTL over finite and infinite traces. In: Thirty-First AAAI Conference on Artificial Intelligence. pp. 3716–3724. AAAI Press (2017). 
 https://doi.org/10.1609/aaai.v31i1.11058
 
 " href="#ref-CR11" id="ref-link-section-d15224108e868">11</a>] and robotics [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Bhatia, A., Maly, M.R., Kavraki, L.E., Vardi, M.Y.: Motion planning with complex goals. IEEE Robotics Autom. Mag. 18(3), 55–64 (2011). 
 https://doi.org/10.1109/MRA.2011.942115
 
 " href="#ref-CR6" id="ref-link-section-d15224108e871">6</a>].</p><p>Here, we harness our symbolic solution to Emerson-Lei games to suggest a large fragment of LTL that can be reasoned about symbolically. We introduce the <i>Safety and Emerson-Lei</i> fragment whose formulas are conjunctions <span class="mathjax-tex">$\varphi _{\textrm{safety}}\wedge \varphi _{\textrm{EL}}$</span> between an (unrestricted) safety condition and an (unrestricted) Emerson-Lei condition defined in terms of game states. This fragment generalizes GR[1] and the previously mentioned works in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Ehlers, R.: Generalized rabin(1) synthesis with applications to robust system synthesis. In: Third International Symposium on NASA Formal Methods. Lecture Notes in Computer Science, vol. 6617, pp. 101–115. Springer (2011). 
 https://doi.org/10.1007/978-3-642-20398-5_9
 
 " href="#ref-CR19" id="ref-link-section-d15224108e914">19</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Ehlers, R.: Unbeast: Symbolic bounded synthesis. In: 17th International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Lecture Notes in Computer Science, vol. 6605, pp. 272–275. Springer (2011). 
 https://doi.org/10.1007/978-3-642-19835-9_25
 
 " href="#ref-CR20" id="ref-link-section-d15224108e917">20</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Sohail, S., Somenzi, F.: Safety first: a two-stage algorithm for the synthesis of reactive systems. Int. J. Softw. Tools Technol. Transf. 15(5-6), 433–454 (2013). 
 https://doi.org/10.1007/s10009-012-0224-3
 
 " href="#ref-CR41" id="ref-link-section-d15224108e920">41</a>]. We approach safety and Emerson-Lei LTL synthesis in two steps: first, consider only the safety part and convert it to a symbolic safety automaton; second, reason symbolically on this automaton by solving Emerson-Lei games using our novel symbolic algorithm.</p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-a"><figure><div class="c-article-section__figure-content" id="Figa"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><img aria-describedby="Figa" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-57228-9_4/MediaObjects/560586_1_En_4_Figa_HTML.png" alt="figure a" loading="lazy" width="685" height="128"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-a-desc"></div></div></figure></div> <p>We show that realizability of a safety and Emerson-Lei formula <span class="mathjax-tex">$\varphi _{\textrm{safety}}\wedge \varphi _{\textrm{EL}}$</span> can be checked in time <span class="mathjax-tex">$2^{\mathcal {O}(m\cdot \log m\cdot 2^n)}$</span>, where <span class="mathjax-tex">$n=|\varphi _{\textrm{safety}}|$</span> and <span class="mathjax-tex">$m=|\varphi _{\textrm{EL}}|$</span>. The overall procedure therefore is doubly-exponential in the size of the safety part but only single-exponential in the size of the liveness part; notably, both the automaton determinization and game solving parts can be implemented symbolically.</p><p>We begin by recalling Emerson-Lei games and Zielonka trees in Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec2">2</a>, and also prove an upper bound on the size of Zielonka trees. Next we show how to solve Emerson-Lei games by fixpoint computation in Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec3">3</a>. In Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec4">4</a> we formally introduce the safety and Emerson-Lei fragment of LTL and show how to construct symbolic games with Emerson-Lei objectives that characterize realizability and that can be solved using the algorithm proposed in Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec3">3</a>. Omitted proofs and further details can be found in the full version of this paper [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Hausmann, D., Lehaut, M., Piterman, N.: Symbolic solution of Emerson-Lei games for reactive synthesis. CoRR abs/2305.02793 (2023), 
 https://arxiv.org/abs/2305.02793
 
 " href="#ref-CR23" id="ref-link-section-d15224108e1116">23</a>].</p></div></div></section><section data-title="Emerson-Lei Games and Zielonka Trees"><div class="c-article-section" id="Sec2-section"><h2 id="Sec2" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">2 </span>Emerson-Lei Games and Zielonka Trees</h2><div class="c-article-section__content" id="Sec2-content"><p>We recall the basics of Emerson-Lei games [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Hunter, P., Dawar, A.: Complexity bounds for regular games. In: 30th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3618, pp. 495–506. Springer (2005). 
 https://doi.org/10.1007/11549345_43
 
 " href="#ref-CR25" id="ref-link-section-d15224108e1127">25</a>] and Zielonka trees [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci. 200(1-2), 135–183 (1998). 
 https://doi.org/10.1016/S0304-3975(98)00009-7
 
 " href="#ref-CR47" id="ref-link-section-d15224108e1130">47</a>], and also show an apparently novel bound on the size of Zielonka trees; previously, the main interest was on the size of winning strategies induced by Zielonka trees, which is smaller [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Dziembowski, S., Jurdzinski, M., Walukiewicz, I.: How much memory is needed to win infinite games? In: 12th Annual IEEE Symposium on Logic in Computer Science. pp. 99–110. IEEE Computer Society (1997). 
 https://doi.org/10.1109/LICS.1997.614939
 
 " href="#ref-CR18" id="ref-link-section-d15224108e1133">18</a>].</p><p><i>Emerson-Lei games.</i> We consider two-player games played between the <i>existential player</i> <span class="mathjax-tex">$\exists $</span> and its opponent, the <i>universal player</i> <span class="mathjax-tex">$\forall $</span>. A <i>game arena</i> <span class="mathjax-tex">$A=(V, V_\exists ,V_\forall ,E)$</span> consists of a set <span class="mathjax-tex">$V=V_\exists \uplus V_\forall $</span> of nodes, partitioned into sets of <i>existential nodes</i> <span class="mathjax-tex">$V_\exists $</span> and <i>universal nodes</i> <span class="mathjax-tex">$V_\forall $</span>, and a set <span class="mathjax-tex">$E\subseteq V\times V$</span> of <i>moves</i>; we put <span class="mathjax-tex">$E(v)=\{v'\in V\mid (v,v')\in E\}$</span> for <span class="mathjax-tex">$v\in V$</span>. A <i>play</i> <span class="mathjax-tex">$\pi =v_0 v_1\ldots $</span> then is a sequence of nodes such that for all <span class="mathjax-tex">$i\ge 0$</span>, <span class="mathjax-tex">$(v_i,v_{i+1})\in E$</span>; we denote the set of plays in <i>A</i> by <span class="mathjax-tex">$\textsf{plays}(A)$</span>. A <i>game</i> <span class="mathjax-tex">$G=(A,\alpha )$</span> consists of a game arena <i>A</i> together with an objective <span class="mathjax-tex">$\alpha \subseteq \textsf{plays}(A)$</span>.</p><p>A <i>strategy</i> for the existential player is a function <span class="mathjax-tex">$\sigma :\, V^*\cdot V_\exists \rightarrow V$</span> such that for all <span class="mathjax-tex">$\pi \in V^*$</span> and <span class="mathjax-tex">$v\in V_\exists $</span> we have <span class="mathjax-tex">$(v,\sigma (\pi v))\in E$</span>. A play <span class="mathjax-tex">$v_0 v_1\ldots $</span> is said to be <i>compliant</i> with strategy <i>f</i> if for all <span class="mathjax-tex">$i\ge 0$</span> such that <span class="mathjax-tex">$v_i\in V_\exists $</span> we have <span class="mathjax-tex">$v_{i+1}=\sigma (v_0\ldots v_i)$</span>. Strategy <span class="mathjax-tex">$\sigma $</span> is <i>winning</i> for the existential player from node <span class="mathjax-tex">$v\in V$</span> if all plays starting in <i>v</i> that are compliant with <span class="mathjax-tex">$\sigma $</span> are contained in <span class="mathjax-tex">$\alpha $</span>; then we say that the existential player <i>wins</i> <i>v</i>. We denote by <span class="mathjax-tex">$W_\exists $</span> the <i>winning region</i> for the existential player (that is, the set of nodes that the existential player wins).</p><p>In <i>Emerson-Lei games</i>, each node is colored by a set of colors, and the objective <span class="mathjax-tex">$\alpha $</span> is induced by a formula that specifies combinations of colors that have to be visited infinitely often, or are allowed to be visited only finitely often. Formally, we fix a set <i>C</i> of colors and use <i>Emerson-Lei formulas</i>, that is, finite positive Boolean formulas <span class="mathjax-tex">$\varphi \in \mathbb {B}_+(\{\textsf{Inf}\,c, \textsf{Fin}\,c\}_{c\in C})$</span> over atoms of the shape <span class="mathjax-tex">$\textsf{Inf}\,c$</span> or <span class="mathjax-tex">$\textsf{Fin}\,c$</span>, to define sets of plays. The satisfaction relation <span class="mathjax-tex">$\models $</span> for a set <span class="mathjax-tex">$D\subseteq C$</span> of colors and an Emerson-Lei formula <span class="mathjax-tex">$\varphi $</span> (written <span class="mathjax-tex">$D\models \varphi $</span>) is defined in the usual inductive way; <i>D</i> will represent the set of colors that are visited infinitely often by plays. E.g. the clauses for atoms <span class="mathjax-tex">$\textsf{Inf}\,c$</span> and <span class="mathjax-tex">$\textsf{Fin}\,c$</span> are</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} D\models \textsf{Inf}\, c & \Leftrightarrow c\in D & D\models \textsf{Fin}\, c & \Leftrightarrow c\notin D \end{aligned}$$</span></div></div><p>Consider a game arena <span class="mathjax-tex">$A=(V,V_\exists , V_\forall ,E)$</span>. An <i>Emerson-Lei condition</i> is given by an Emerson-Lei formula <span class="mathjax-tex">$\varphi $</span> together with a coloring function <span class="mathjax-tex">$\gamma : V\rightarrow 2^C$</span> that assigns a (possibly empty) set <span class="mathjax-tex">$\gamma (v)$</span> of colors to each node <span class="mathjax-tex">$v\in V$</span>. The formula <span class="mathjax-tex">$\varphi $</span> and the coloring function <span class="mathjax-tex">$\gamma $</span> together specify the objective</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \alpha _{\gamma ,\varphi }=\Big \{v_0v_1\ldots \in \textsf{plays}(A) \Big | \{c\in C\mid \forall i.~\exists j \ge i.~c\in \gamma (v_j)\}\models \varphi \Big \} \end{aligned}$$</span></div></div><p>Thus a play <span class="mathjax-tex">$\pi =v_0v_1\ldots $</span> is winning for the existential player (formally: <span class="mathjax-tex">$\pi \in \alpha _{\gamma ,\varphi }$</span>) if and only if the set of colors that are visited infinitely often by <span class="mathjax-tex">$\pi $</span> satisfies <span class="mathjax-tex">$\varphi $</span>. Below, we will also make use of <i>parity games</i>, denoted by <span class="mathjax-tex">$(V,V_\exists , V_\forall ,E,\varOmega )$</span> where <span class="mathjax-tex">$\varOmega :V\rightarrow \{1,\ldots , {2k}\}$</span> (for <span class="mathjax-tex">$k\in \mathbb {N}$</span>) is a priority function, assigning priorities to game nodes. The objective of the existential player then is that the maximal priority that is visited infinitely often is an even number. Parity games are an instance of Emerson-Lei games, obtained with set <span class="mathjax-tex">$C=\{p_1,\ldots , p_{2k}\}$</span> of colors, a coloring function that assigns exactly one color to each node and with objective</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textsf{Parity}(p_1,\ldots , p_{2k})=\textstyle \bigvee _{i\text { even}} \left( \textsf{Inf}\, p_{i} \wedge \textstyle \bigwedge _{i<j\le 2k} \textsf{Fin}\, p_{j} \right) . \end{aligned}$$</span></div></div><p>Similarly, Emerson-Lei objectives directly encode (combinations of) other standard objectives, such as Büchi, Rabin, Streett or Muller conditions: </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-b"><figure><div class="c-article-section__figure-content" id="Figb"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><img aria-describedby="Figb" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-57228-9_4/MediaObjects/560586_1_En_4_Figb_HTML.png" alt="figure b" loading="lazy" width="685" height="102"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-b-desc"></div></div></figure></div> <p><i>Zielonka Trees.</i> We introduce a succinct encoding of the algorithmic essence of Emerson-Lei objectives in the form of so-called Zielonka trees [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Dziembowski, S., Jurdzinski, M., Walukiewicz, I.: How much memory is needed to win infinite games? In: 12th Annual IEEE Symposium on Logic in Computer Science. pp. 99–110. IEEE Computer Society (1997). 
 https://doi.org/10.1109/LICS.1997.614939
 
 " href="#ref-CR18" id="ref-link-section-d15224108e3306">18</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci. 200(1-2), 135–183 (1998). 
 https://doi.org/10.1016/S0304-3975(98)00009-7
 
 " href="#ref-CR47" id="ref-link-section-d15224108e3309">47</a>].</p> <h3 class="c-article__sub-heading" id="FPar1">Definition 1</h3> <p>The <i>Zielonka tree</i> for an Emerson-Lei formula <span class="mathjax-tex">$\varphi $</span> over set <i>C</i> of colors is a tuple <span class="mathjax-tex">$\mathcal {Z}_\varphi =(T,R,l)$</span> where <span class="mathjax-tex">$(T,R\subseteq T\times T)$</span> is a tree and <span class="mathjax-tex">$l:T\rightarrow 2^C$</span> is a labeling function that assigns sets <i>l</i>(<i>t</i>) of colors to vertices <span class="mathjax-tex">$t\in T$</span>. We denote the root of (<i>T</i>, <i>R</i>) by <i>r</i>. Then <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span> is defined to be the unique tree (up to reordering of child vertices) that satisfies the following constraints.</p> <ul class="u-list-style-dash"> <li> <p>The root vertex is labeled with <i>C</i>, that is, <span class="mathjax-tex">$l(r)=C$</span>.</p> </li> <li> <p>Each vertex <i>t</i> has exactly one child vertex <span class="mathjax-tex">$t_D$</span> (labeled with <span class="mathjax-tex">$l(t_D)=D$</span>) for each set <i>D</i> of colors that is maximal in <span class="mathjax-tex">$\{D'\subsetneq l(t)\mid D'\models \varphi \Leftrightarrow l(t)\not \models \varphi \}$</span>.</p> </li> </ul> <p>For <span class="mathjax-tex">$s,t\in T$</span> such that <i>s</i> is an ancestor of <i>t</i>, we write <span class="mathjax-tex">$s\le t$</span>. Given a vertex <span class="mathjax-tex">$s\in T$</span>, we denote its set of direct successors by <span class="mathjax-tex">$R(s)=\{t\in T\mid (s,t)\in R\}$</span> and the set of leafs below it by <span class="mathjax-tex">$L(s) = \{t\in T \mid s\le t \text { and } R(t)=\emptyset \}$</span>; we write <i>L</i> for the set of all leafs. We assume some fixed total order <span class="mathjax-tex">$\preceq $</span> on <i>T</i> that respects <span class="mathjax-tex">$\le $</span>; this order induces a numbering of <i>T</i>. A vertex <i>t</i> in the Zielonka tree is said to be <i>winning</i> if <span class="mathjax-tex">$l(t)\models \varphi $</span>, and <i>losing</i> otherwise. We let <span class="mathjax-tex">$T_\square $</span> and <span class="mathjax-tex">$T_\bigcirc $</span> denote the sets of winning and losing vertices in <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span>, respectively. Finally, we assign a <i>level</i> <span class="mathjax-tex">$\textsf{lev}(t)$</span> to each vertex <span class="mathjax-tex">$t\in T$</span> so that <span class="mathjax-tex">$\textsf{lev}(r)=|C|$</span>, and <span class="mathjax-tex">$\textsf{lev}(s')=\textsf{lev}(s)-1$</span> for all <span class="mathjax-tex">$(s,s')\in R$</span>.</p> <h3 class="c-article__sub-heading" id="FPar2">Example 2</h3> <p>As mentioned above, Emerson-Lei games and Zielonka trees instantiate naturally to games with, e.g., Büchi, generalized Büchi, GR[1], parity, Rabin, Streett and Muller objectives; for brevity, we illustrate this for selected examples here (more instances can be found in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Hausmann, D., Lehaut, M., Piterman, N.: Symbolic solution of Emerson-Lei games for reactive synthesis. CoRR abs/2305.02793 (2023), 
 https://arxiv.org/abs/2305.02793
 
 " href="#ref-CR23" id="ref-link-section-d15224108e4307">23</a>]).</p> <ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p><i>Generalized Büchi condition</i>: Given <i>k</i> colors <span class="mathjax-tex">$f_1,\ldots ,f_k$</span>, the winning objective <span class="mathjax-tex">$\varphi =\bigwedge _{1\le i\le k}\textsf{Inf}~f_i$</span> expresses that all colors are visited infinitely often (not necessarily simultaneously); the induced Zielonka tree is depicted below with boxes and circles representing winning and losing vertices, respectively. </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-c"><figure><div class="c-article-section__figure-content" id="Figc"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><img aria-describedby="Figc" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-57228-9_4/MediaObjects/560586_1_En_4_Figc_HTML.png" alt="figure c" loading="lazy" width="685" height="143"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-c-desc"></div></div></figure></div> </li> <li> <span class="u-custom-list-number">2.</span> <p><i>Streett condition</i>: The vertices in the Zielonka tree for Streett condition given by <span class="mathjax-tex">$\varphi =\bigwedge _{1\le i\le k}\left( \textsf{Fin}~r_i \vee \textsf{Inf}~g_i \right) $</span> are identified by duplicate-free lists <span class="mathjax-tex">$\textsf{L}$</span> of colors (each entry being <span class="mathjax-tex">$r_i$</span> or <span class="mathjax-tex">$g_i$</span> for some <span class="mathjax-tex">$1\le i\le k$</span>) that encode the vertex position in the tree. Vertex <span class="mathjax-tex">$\textsf{L}$</span> has label <span class="mathjax-tex">$l(\textsf{L})=C\setminus \textsf{L}$</span> and is winning if and only if <span class="mathjax-tex">$|\textsf{L}|$</span> is even. Winning vertices <span class="mathjax-tex">$\textsf{L}$</span> have one child vertex <span class="mathjax-tex">${\textsf{L}:g_j}$</span> for each <span class="mathjax-tex">$g_j\in C\setminus \textsf{L}$</span> resulting in <span class="mathjax-tex">$|C\setminus \textsf{L}|/2$</span> many child vertices. Losing vertices <span class="mathjax-tex">$\textsf{L}$</span> have the single child vertex <span class="mathjax-tex">$\textsf{L}:r_j$</span> where the last entry <span class="mathjax-tex">$\textsf{last}(\textsf{L})$</span> in <span class="mathjax-tex">$\textsf{L}$</span> is <span class="mathjax-tex">$g_j$</span>. All leafs are winning and are labeled with <span class="mathjax-tex">$\emptyset $</span>. The tree has height 2<i>k</i> and 2(<i>k</i>!) vertices.</p> </li> <li> <span class="u-custom-list-number">3.</span> <p>To obtain a Zielonka tree that has branching at both winning and losing vertices, we consider the objective <span class="mathjax-tex">$\varphi _{EL}=(\textsf{Fin}~a\vee \textsf{Inf}~b)\wedge ((\textsf{Fin}~a\vee \textsf{Fin}~d)\wedge \textsf{Inf}~c)$</span>. This property can be seen as the conjunction of a Streett pair (<i>a</i>, <i>b</i>) with two disjunctive Rabin pairs (<i>c</i>, <i>a</i>) and (<i>c</i>, <i>d</i>), altogether stating that <i>c</i> occurs infinitely often and <i>a</i> occurs finitely often or <i>b</i> occurs infinitely often and <i>d</i> occurs finitely often. Below we depict the induced Zielonka tree. </p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-d"><figure><div class="c-article-section__figure-content" id="Figd"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><img aria-describedby="Figd" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-57228-9_4/MediaObjects/560586_1_En_4_Figd_HTML.png" alt="figure d" loading="lazy" width="456" height="448"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-d-desc"></div></div></figure></div> </li> </ol> <h3 class="c-article__sub-heading" id="FPar3">Lemma 3</h3> <p>The height and the branching width of <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span> are bounded by |<i>C</i>| and <span class="mathjax-tex">$2^{|C|}$</span> respectively; the number of vertices in <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span> is bounded by <i>e</i>|<i>C</i>|! (where <i>e</i> is Euler’s number).</p> </div></div></section><section data-title="Solving Emerson-Lei Games"><div class="c-article-section" id="Sec3-section"><h2 id="Sec3" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">3 </span>Solving Emerson-Lei Games</h2><div class="c-article-section__content" id="Sec3-content"><p>We now show how to extract from the Zielonka tree of an Emerson-Lei objective a fixpoint characterization of the winning regions of an Emerson-Lei game. Solving the game then reduces to computing the fixpoint, yielding a game solving algorithm that works by fixpoint iteration and hence is directly open to symbolic implementation. The algorithm is adaptive in the sense that the structure of its recursive calls is extracted from the Zielonka tree and hence tailored to the objective. As a stepping stone towards obtaining our fixpoint characterization, we first show how Zielonka trees can be used to reduce Emerson-Lei games to parity games that are structured into tree-like subgames.</p><p>Recall that <span class="mathjax-tex">$G=(V,V_\exists ,V_\forall ,E,\alpha _{\gamma ,\varphi })$</span> is an Emerson-Lei game and that the associated Zielonka tree is <span class="mathjax-tex">$\mathcal {Z}_\varphi =(T,R,l)$</span> with set <i>L</i> of leaves, sets <span class="mathjax-tex">$T_\bigcirc $</span> and <span class="mathjax-tex">$T_\square $</span> of winning and losing vertices, respectively, and with root <i>r</i>. Following [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Dziembowski, S., Jurdzinski, M., Walukiewicz, I.: How much memory is needed to win infinite games? In: 12th Annual IEEE Symposium on Logic in Computer Science. pp. 99–110. IEEE Computer Society (1997). 
 https://doi.org/10.1109/LICS.1997.614939
 
 " href="#ref-CR18" id="ref-link-section-d15224108e5388">18</a>], we define the <i>anchor vertex</i> of <span class="mathjax-tex">$v\in V$</span> and <span class="mathjax-tex">$t\in T$</span> by</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textsf{anchor}(v,t)= \max \textstyle _{\le }\{s\in T \mid s \le t \wedge \gamma (v)\subseteq l(s)\}; \end{aligned}$$</span></div></div><p>it is the lower-most ancestor of <i>t</i> that contains <span class="mathjax-tex">$\gamma (v)$</span> in its label.</p><p><i>A novel reduction to parity games.</i> Intuitively, our reduction annotates nodes in <i>G</i> with leaves of <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span> that act as a memory, holding information about the order in which colors have been visited. In the reduced game, the memory value <span class="mathjax-tex">$t\in L$</span> is updated according to a move from <i>v</i> to <i>w</i> in <i>G</i> by playing a subgame along the Zielonka tree. This subgame starts at the anchor vertex of <i>v</i> and <i>t</i> and the players in turn pick child vertices, with the existential player choosing the branch that is taken at vertices from <span class="mathjax-tex">$T_\bigcirc $</span> and the universal player choosing at vertices from <span class="mathjax-tex">$T_\square $</span>.<sup><a href="#Fn1"><span class="u-visually-hidden">Footnote </span>1</a></sup> Once this subgame reaches a leaf <span class="mathjax-tex">$t'\in L$</span>, the memory value is updated to <span class="mathjax-tex">$t'$</span> and another step of <i>G</i> is played. Due to the tree structure of <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span> every play in the reduced game (walking through the Zielonka tree in the described way, repeatedly jumping from a leaf to an anchor vertex and then descending to a leaf again) has a unique topmost vertex from <i>T</i> that it visits infinitely often; by the definition of anchor vertices, the label of this vertex corresponds to the set of colors that is visited infinitely often by the according play of <i>G</i>. A parity condition can be used to decide whether this vertex is winning or losing.</p><p>Formally, we define the parity game <span class="mathjax-tex">$P_G=(V',V'_\exists ,V'_\forall ,E',\varOmega )$</span>, played over <span class="mathjax-tex">$V'=V\times T$</span>, as follows. Nodes <span class="mathjax-tex">$(v,t)\in V'$</span> are owned by the existential player if either <i>t</i> is not a leaf, and it is not a winning vertex (<span class="mathjax-tex">$t\notin L$</span> and <span class="mathjax-tex">$t\in T_\bigcirc $</span>), or if <i>t</i> is a leaf and, in <i>G</i>, <i>v</i> is owned by the existential player (<span class="mathjax-tex">$t\in L$</span> and <span class="mathjax-tex">$v\in V_\exists $</span>); all other nodes are owned by the universal player. Moves and priorities are defined by</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} E'(v,t) &= {\left\{ \begin{array}{ll} \{v\}\times R(t)&{} t\notin L \\ E(v)\times \{\textsf{anchor}(v,t)\}&{} t\in L \end{array}\right. } & \,\,\,\varOmega (v,t)&={\left\{ \begin{array}{ll} 2\cdot \textsf{lev}(t) &{} t\in T_\square \\ 2\cdot \textsf{lev}(t)+1&{} t\in T_\bigcirc \end{array}\right. } \end{aligned}$$</span></div></div><p>for <span class="mathjax-tex">$(v,t)\in V'$</span>. Thus from (<i>v</i>, <i>t</i>) such that <i>t</i> is a leaf (<span class="mathjax-tex">$t\in L$</span>), the owner of <i>v</i> picks a move <span class="mathjax-tex">$(v,w)\in E$</span> and the game continues with <span class="mathjax-tex">$(w,\textsf{anchor}(v,t))$</span>. From (<i>v</i>, <i>t</i>) such that <i>t</i> is not a leaf (<span class="mathjax-tex">$t\notin L$</span>), the owner of <i>t</i> picks a child <span class="mathjax-tex">$t'\in R(t)$</span> of <i>t</i> in the Zielonka tree and the game continues with <span class="mathjax-tex">$(v,t')$</span>, leaving the game node component <i>v</i> unchanged. Therefore, plays in <span class="mathjax-tex">$P_G$</span> correspond to plays from <i>G</i> that are annotated with memory values <span class="mathjax-tex">$t\in T$</span> that are updated according to the colors that are visited (by moving to the anchor vertex); in addition to that, the owners of vertices in the Zielonka Tree are allowed to decide (by selecting one of the child vertices) with which colors they intend to satisfy the sub-objectives that are encoded by vertex labels. The priority function <span class="mathjax-tex">$\varOmega $</span> then is used to identify the top-most anchor vertex <i>s</i> that is visited infinitely often in a play of <span class="mathjax-tex">$P_G$</span>, deciding a play to be winning if and only if <i>s</i> is a winning vertex (<span class="mathjax-tex">$t\in T_\square $</span>). We note that <span class="mathjax-tex">$|V'|=|V|\cdot |T|\le |V|\cdot e|C|!$</span> by Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar3">3</a>.</p> <h3 class="c-article__sub-heading" id="FPar4">Theorem 4</h3> <p>For all <span class="mathjax-tex">$v\in V$</span>, the existential player wins <i>v</i> in the Emerson-Lei game <i>G</i> if and only if the existential player wins (<i>v</i>, <i>r</i>) in the parity game <span class="mathjax-tex">$P_G$</span>.</p> <p>This reduction yields a novel indirect method to solve Emerson-Lei games with <i>n</i> nodes and <i>k</i> colors by solving parity games with <span class="mathjax-tex">$n\cdot ek!$</span> nodes and 2<i>k</i> priorities; by itself, this reduction does not improve upon using later appearance records [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Hunter, P., Dawar, A.: Complexity bounds for regular games. In: 30th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3618, pp. 495–506. Springer (2005). 
 https://doi.org/10.1007/11549345_43
 
 " href="#ref-CR25" id="ref-link-section-d15224108e7026">25</a>]. However, the game <span class="mathjax-tex">$P_G$</span> consists of subgames of particular tree-like shapes. The remainder of this section is dedicated to showing how the special structure of <span class="mathjax-tex">$P_G$</span> allows for direct symbolic solution by solving equivalent systems of fixpoint equations over <i>V</i> (rather than over the exponential-sized set <span class="mathjax-tex">$V'$</span>).</p><p><i>Fixpoint equation systems.</i> Recall (from e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Baldan, P., König, B., Mika-Michalski, C., Padoan, T.: Fixpoint games on continuous lattices. Proc. ACM Program. Lang. 3(POPL), 26:1–26:29 (2019). 
 https://doi.org/10.1145/3290339
 
 " href="#ref-CR4" id="ref-link-section-d15224108e7104">4</a>]) that a hierarchical system of fixpoint equations is given by equations</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ X_i =_{\eta _i} f_i(X_1,\ldots ,X_k) $$</span></div></div><p>for <span class="mathjax-tex">$1\le i\le k$</span>, where <span class="mathjax-tex">$\eta _i\in \{\textsf{GFP},\textsf{LFP}\}$</span> and the <span class="mathjax-tex">$f_i:\mathcal {P}(V)^k\rightarrow \mathcal {P}(V)$</span> are <i>monotone</i> functions, that is, <span class="mathjax-tex">$f_i(A_1,\ldots ,A_k)\subseteq f_i(B_1,\ldots ,B_k)$</span> whenever <span class="mathjax-tex">$A_j\subseteq B_j$</span> for all <span class="mathjax-tex">$1\le j\le k$</span>. As we aim to use fixpoint equation systems to characterize winning regions of games, it is convenient to define the semantics of equation systems also in terms of games, as proposed in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Baldan, P., König, B., Mika-Michalski, C., Padoan, T.: Fixpoint games on continuous lattices. Proc. ACM Program. Lang. 3(POPL), 26:1–26:29 (2019). 
 https://doi.org/10.1145/3290339
 
 " href="#ref-CR4" id="ref-link-section-d15224108e7477">4</a>]. For a system <i>S</i> of <i>k</i> fixpoint equations, the <i>fixpoint game</i> <span class="mathjax-tex">$G_S=(V,V_\exists ,V_\forall ,E,\varOmega )$</span> is a parity game with sets of nodes <span class="mathjax-tex">$V_\exists =V\times \{1,\ldots ,k\}$</span> and <span class="mathjax-tex">$V_\forall = \mathcal {P}(V)^k$</span>. The set of edges <i>E</i> and the priority function <span class="mathjax-tex">$\varOmega :V\rightarrow \{0,\ldots ,2k-1\}$</span> are defined, for <span class="mathjax-tex">$(v,i)\in V_\exists $</span> and <span class="mathjax-tex">$\bar{A}=(A_1,\ldots ,A_k)\in V_\forall $</span>, by</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} E(v,i)&=\{\bar{A}\in V_\forall \mid v\in f_i(\bar{A})\} & E(\bar{A})&=\{(v,i)\in V_\exists \mid v\in A_i\} \end{aligned}$$</span></div></div><p>and by <span class="mathjax-tex">$\varOmega (v,i)=2(k-i)+\iota _i$</span> and <span class="mathjax-tex">$\varOmega (\bar{A})=0$</span>, where <span class="mathjax-tex">$\iota _i=1$</span> if <span class="mathjax-tex">$\eta _i=\textsf{LFP}$</span> and <span class="mathjax-tex">$\iota _i=0$</span> if <span class="mathjax-tex">$\eta _i=\textsf{GFP}$</span>. We say that <i>v</i> is contained in the <i>solution</i> of variable <span class="mathjax-tex">$X_i$</span> (denoted by <span class="mathjax-tex">$v\in \llbracket X_i \rrbracket $</span>) if and only if the existential player wins the node (<i>v</i>, <i>i</i>) in <span class="mathjax-tex">$G_S$</span>. In order to show containment of a node <i>v</i> in the solution of <span class="mathjax-tex">$X_i$</span>, the existential player thus has to provide a solution <span class="mathjax-tex">$(A_1,\ldots ,A_k)\in V_\forall $</span> for all variables such that <span class="mathjax-tex">$v\in f_i(A_1,\ldots ,A_k)$</span>; the universal player in turn can challenge a claimed solution <span class="mathjax-tex">$(A_1,\ldots ,A_k)$</span> by picking some <span class="mathjax-tex">$1\le i\le k$</span> and <span class="mathjax-tex">$v\in A_i$</span> and moving to (<i>v</i>, <i>i</i>). The game objective checks whether the dominating equation in a play (that is, the equation with minimal index among the equations that are evaluated infinitely often in the play) is a least or a greatest fixpoint equation.</p><p>Baldan et al. have shown in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Baldan, P., König, B., Mika-Michalski, C., Padoan, T.: Fixpoint games on continuous lattices. Proc. ACM Program. Lang. 3(POPL), 26:1–26:29 (2019). 
 https://doi.org/10.1145/3290339
 
 " href="#ref-CR4" id="ref-link-section-d15224108e8593">4</a>] that this game characterization is equivalent to the more traditional Knaster-Tarski-style definition of the semantics of fixpoint equation systems in terms of nested fixpoints of the involved functions <span class="mathjax-tex">$f_i$</span>.</p><p>To give a flavor of the close connection between fixpoint equation systems and winning regions in games, we recall that for a given set <i>V</i> of nodes, the <i>controllable predecessor function</i> <span class="mathjax-tex">$\textsf{CPre}:2^V\rightarrow 2^V$</span> is defined, for <span class="mathjax-tex">$X\subseteq V$</span>, by</p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \textsf{CPre}(X)=\{v\in V_\exists \mid E(v)\cap X\ne \emptyset \}\cup \{v\in V_\forall \mid E(v)\subseteq X\}. $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar5">Example 5</h3> <p>Given a Büchi game <span class="mathjax-tex">$(V,V_\exists ,V_\forall ,E,\textsf{Inf}~f)$</span> with coloring function <span class="mathjax-tex">$\gamma :V\rightarrow 2^{\{f\}}$</span>, the winning region of the existential player is the solution of the equation system</p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_1 & =_{\textsf{GFP}} X_2 & X_2 & =_{\textsf{LFP}} (f\cap \textsf{CPre}(X_1))\cup (\overline{f}\cap \textsf{CPre}(X_2)) \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$f=\{v\in V\mid \gamma (v)=\{f\}\}$</span> and <span class="mathjax-tex">$\overline{f}=V\setminus f$</span>.</p> <p>Our upcoming fixpoint characterization of winning regions in Emerson-Lei games uses the following notation that relates game nodes with anchor vertices in the Zielonka tree.</p> <h3 class="c-article__sub-heading" id="FPar6">Definition 6</h3> <p>For a set <span class="mathjax-tex">$D\subseteq C$</span> of colors, and <span class="mathjax-tex">${\bowtie }\in \{\subseteq , \not \subseteq \}$</span> we put <span class="mathjax-tex">$\gamma ^{-1}_{\bowtie D}=\{v\in V\mid \gamma (v)\, \bowtie \,D\}$</span>. For <span class="mathjax-tex">$s,t\in T$</span> such that <span class="mathjax-tex">$s< t$</span> (that is, <i>s</i> is an ancestor of <i>t</i> in <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span>), we define</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textsf{anc}^s_{t}=\gamma ^{-1}_{\subseteq l(s)}\cap \gamma ^{-1}_{\not \subseteq l(s_t)} \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$s_t$</span> is <i>the</i> child vertex of <i>s</i> that leads to <i>t</i>; we also put <span class="mathjax-tex">$\textsf{anc}^t_{t}=\gamma ^{-1}_{\subseteq l(t)}$</span>.</p> <p>Note that for fixed <span class="mathjax-tex">$t\in T$</span> and <span class="mathjax-tex">$v\in V$</span>, there is a unique <span class="mathjax-tex">$s\in T$</span> such that <span class="mathjax-tex">$s\le t$</span> and <span class="mathjax-tex">$v\in \textsf{anc}^s_{t}$</span> (possibly, <span class="mathjax-tex">$s=t$</span>); this <i>s</i> is the anchor vertex of <i>t</i> at <i>v</i>.</p><p>Next, we present our fixpoint characterization of winning in Emerson-Lei games, noting that it closely follows the definition of <span class="mathjax-tex">$P_G$</span>.</p> <h3 class="c-article__sub-heading" id="FPar7">Definition 7</h3> <p><b>(Emerson-Lei equation system).</b> We define the system <span class="mathjax-tex">$S_\varphi $</span> of fixpoint equations for the objective <span class="mathjax-tex">$\varphi $</span> by putting</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_s &=_{\eta _s} {\left\{ \begin{array}{ll} \textstyle \bigcup _{t\in R(s)}X_{t} &{} R(s)\ne \emptyset ,s\in T_\bigcirc \\ \textstyle \bigcap _{t\in R(s)}X_{t} &{} R(s)\ne \emptyset ,s\in T_\square \\ \textstyle \bigcup _{s'\le s} \left( \textsf{anc}^{s'}_{s} \cap \textsf{CPre}(X_{s'})\right) &{} R(s)=\emptyset \end{array}\right. } \end{aligned}$$</span></div></div><p>for <span class="mathjax-tex">$s\in T$</span>. For every <span class="mathjax-tex">$t\in T$</span>, we use <span class="mathjax-tex">$X_t$</span> to refer to the variable <span class="mathjax-tex">$X_i$</span> where <i>i</i> is the index of <i>t</i> according to <span class="mathjax-tex">$\preceq $</span> and similarly for <span class="mathjax-tex">$\eta _t$</span>. Furthermore, <span class="mathjax-tex">$\eta _s=\textsf{GFP}$</span> if <span class="mathjax-tex">$s\in T_\square $</span> and <span class="mathjax-tex">$\eta _s=\textsf{LFP}$</span> if <span class="mathjax-tex">$s\in T_\bigcirc $</span>.</p> <h3 class="c-article__sub-heading" id="FPar8">Example 8</h3> <p>Instantiating Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar7">7</a> to the Büchi objective <span class="mathjax-tex">$\varphi =\textsf{Inf}~f$</span> yields exactly the equation system given in Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar5">5</a>. Revisiting the objectives from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar2">2</a>, we obtain the following fixpoint characterizations (further examples can be found in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Hausmann, D., Lehaut, M., Piterman, N.: Symbolic solution of Emerson-Lei games for reactive synthesis. CoRR abs/2305.02793 (2023), 
 https://arxiv.org/abs/2305.02793
 
 " href="#ref-CR23" id="ref-link-section-d15224108e10463">23</a>]).</p> <ol class="u-list-style-none"> <li> <span class="u-custom-list-number">1.</span> <p><i>Generalized Büchi condition:</i></p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_{s_0} &=_\textsf{GFP}\textstyle \bigcap _{1\le i\le k}X_{s_i} & X_{s_i} &=_\textsf{LFP} (\textsf{anc}^{s_0}_{s_i}\cap \textsf{CPre}(X_{s_0})) \cup (\textsf{anc}^{s_i}_{s_i}\cap \textsf{CPre}(X_{s_i})) \end{aligned}$$</span></div></div><p> where <span class="mathjax-tex">$\textsf{anc}^{s_0}_{s_i}=\gamma ^{-1}_{\subseteq C} \cap \gamma ^{-1}_{\not \subseteq C\setminus \{f_i\}}=\{v\in V\mid f_i\in \gamma (v)\}$</span> and <span class="mathjax-tex">$\textsf{anc}^{s_i}_{s_i}=\gamma ^{-1}_{\subseteq C\setminus \{f_i\}}$</span>.</p> </li> <li> <span class="u-custom-list-number">2.</span> <p><i>Streett condition</i>: </p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_{\textsf{L}}&=_{\eta _{\textsf{L}}} {\left\{ \begin{array}{ll} \textstyle \bigcap _{g_j\notin {\textsf{L}}}X_{{\textsf{L}}:g_j} &{} |{\textsf{L}}| \text { even}, |{\textsf{L}}|<2k\\ X_{{\textsf{L}}:r_j}&{} |{\textsf{L}}| \text { odd},\textsf{last}({\textsf{L}})=g_j\\ (\textsf{anc}^{[]}_{{\textsf{L}}}\cap \textsf{CPre}(X_{[]})) \cup \ldots \cup (\textsf{anc}^{{\textsf{L}}}_{{\textsf{L}}}\cap \textsf{CPre}(X_{{\textsf{L}}})) &{} |{\textsf{L}}|=2k \end{array}\right. } \end{aligned}$$</span></div></div><p> where <span class="mathjax-tex">$\eta _{\textsf{L}}=\textsf{GFP}$</span> if <span class="mathjax-tex">$|{\textsf{L}}|$</span> is even and <span class="mathjax-tex">$\eta _{\textsf{L}}=\textsf{LFP}$</span> if <span class="mathjax-tex">$|{\textsf{L}}|$</span> is odd. Here, <span class="mathjax-tex">$\textsf{anc}^{\textsf{K}}_{{\textsf{L}}}=\gamma ^{-1}_{\subseteq C\setminus \textsf{K}}\cap \gamma ^{-1}_{\not \subseteq C\setminus I}$</span> for <span class="mathjax-tex">$\textsf{K}\ne {\textsf{L}}$</span> and <span class="mathjax-tex">$I=\textsf{K}_{\textsf{L}}$</span>, and <span class="mathjax-tex">$\textsf{anc}^{{\textsf{L}}}_{{\textsf{L}}}=\gamma ^{-1}_{\subseteq \emptyset }$</span>, both for <span class="mathjax-tex">${\textsf{L}}$</span> such that <span class="mathjax-tex">$|{\textsf{L}}|=2k$</span>.</p> </li> <li> <span class="u-custom-list-number">3.</span> <p>The equation system associated to the Zielonka tree for the complex objective <span class="mathjax-tex">$\varphi _{EL}$</span> from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar2">2</a>.3 is as follows, where we use a formula over the colors to denote the set of vertices whose label satisfies the formula. For example, <span class="mathjax-tex">$b \wedge \lnot d$</span> corresponds to vertices whose set of colors contains <i>b</i> but does not contain <i>d</i>. </p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} X_{1}&=_{\textsf{LFP}} X_{2}\cup X_{3} \qquad X_{2} =_{\textsf{GFP}} X_{4}\cap X_{5}\qquad X_{3}=_{\textsf{GFP}} X_{6}\qquad \,\, X_{5} =_{\textsf{LFP}} X_{7}\qquad X_{7} =_{\textsf{GFP}} X_{8}\\ X_{4} &=_{\textsf{LFP}} (\lnot c\wedge \lnot d \cap \textsf{Cpre}(X_4))\cup (c\wedge \lnot d\cap \textsf{Cpre}(X_{2}))\cup (d\cap \textsf{Cpre}(X_{1}))\\ X_{6} &=_{\textsf{LFP}} (\lnot a\wedge \lnot c\cap \textsf{Cpre}(X_6))\cup (\lnot a\wedge c\cap \textsf{Cpre}(X_3))\cup (a\cap \textsf{Cpre}(X_{1}))\\ X_{8} &=_{\textsf{LFP}} (\lnot a\wedge \lnot b \wedge \lnot c\wedge \lnot d\cap \textsf{Cpre}(X_{8}))\cup (\lnot a\wedge \lnot b \wedge c\wedge \lnot d \cap \textsf{Cpre}(X_{7}))\,\cup \\ &\quad \quad \,\,\,(a \wedge \lnot b \wedge \lnot d \cap \textsf{Cpre}(X_{5}))\cup (b \wedge \lnot d\cap \textsf{Cpre}(X_{2}))\cup (d\cap \textsf{Cpre}(X_{1})), \end{aligned}$$</span></div></div> </li> </ol> <h3 class="c-article__sub-heading" id="FPar9">Theorem 9</h3> <p>Referring to the equation system from Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar7">7</a> and recalling that <i>r</i> is the root of the Zielonka tree <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span>, the solution of the variable <span class="mathjax-tex">$X_r$</span> is the winning region of the existential player in the Emerson-Lei game <i>G</i>.</p> <p>By Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar4">4</a>, it suffices to mutually transform winning strategies in <span class="mathjax-tex">$P_G$</span> and the fixpoint game <span class="mathjax-tex">$G_{S_\varphi }$</span> for the equation system <span class="mathjax-tex">$S_\varphi $</span> from Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar7">7</a>.</p><p>Given the fixpoint characterization of winning regions in Emerson-Lei games with objective <span class="mathjax-tex">$\varphi $</span> in Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar7">7</a>, we obtain a fixpoint iteration algorithm that computes the solution of Emerson-Lei games. The algorithm is by nature open to symbolic implementation. The main function is recursive, taking as input one vertex <span class="mathjax-tex">$s\in T$</span> of the Zielonka tree <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span> and a list <i>l</i> of subsets of the set <i>V</i> of nodes, and returns a subset of <i>V</i> as result. For calls <span class="mathjax-tex">$\textsc {Solve}(s,ls)$</span>, we require that the argument list <i>ls</i> contains exactly one subset <span class="mathjax-tex">$X_{s'}$</span> of <i>V</i> for each ancestor <span class="mathjax-tex">$s'$</span> of <i>s</i> in the Zielonka tree (with <span class="mathjax-tex">$s'<s$</span>).</p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-e"><figure><div class="c-article-section__figure-content" id="Fige"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><img aria-describedby="Fige" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-57228-9_4/MediaObjects/560586_1_En_4_Fige_HTML.png" alt="figure e" loading="lazy" width="685" height="469"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-e-desc"></div></div></figure></div> <h3 class="c-article__sub-heading" id="FPar10">Lemma 10</h3> <p>For all <span class="mathjax-tex">$v\in V$</span>, we have <span class="mathjax-tex">$v\in \llbracket X_r \rrbracket $</span> if and only if <span class="mathjax-tex">$v\in \textsc {Solve}(r,[])$</span>.</p> <h3 class="c-article__sub-heading" id="FPar11">Proof (Sketch)</h3> <p>[Sketch] The algorithm computes the solution of the equation system by standard Kleene-approximation for nested least and greatest fixpoints.</p> <h3 class="c-article__sub-heading" id="FPar12">Lemma 11</h3> <p>Given an Emerson-Lei game <span class="mathjax-tex">$(V,V_\exists ,V_\forall ,E,\alpha _{\gamma ,\varphi })$</span> with set of colors <i>C</i> and induced Zielonka tree <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span>, the solution <span class="mathjax-tex">$\llbracket X_r \rrbracket $</span> of the equation system <span class="mathjax-tex">$S_\varphi $</span> from Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar7">7</a> can be computed in time <span class="mathjax-tex">$\mathcal {O}(|\mathcal {Z}_\varphi |\cdot |E|\cdot |V|^k)$</span>, where <span class="mathjax-tex">$k\le |C|$</span> denotes the height of <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span>.</p> <p>Combining Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar9">9</a> with Lemmas <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar3">3</a>, <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar10">10</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar12">11</a> we obtain</p> <h3 class="c-article__sub-heading" id="FPar13">Corollary 12</h3> <p>Solving Emerson-Lei games with <i>n</i> nodes, <i>m</i> edges and <i>k</i> colors can be implemented symbolically to run in time <span class="mathjax-tex">$\mathcal {O}(k!\cdot m\cdot n^k)$</span>; the resulting strategies require memory at most <span class="mathjax-tex">$e\cdot k!$</span>.</p> <h3 class="c-article__sub-heading" id="FPar14">Remark 13</h3> <p> Strategy extraction works as follows. The algorithm computes a set <span class="mathjax-tex">$\llbracket X_t \rrbracket $</span> for each Zielonka tree vertex <span class="mathjax-tex">$t\in \mathcal {Z}_\varphi $</span>. Furthermore it yields, for each non-leaf vertex <span class="mathjax-tex">$s\in T_\bigcirc $</span> and each <span class="mathjax-tex">$v\in \llbracket X_s \rrbracket $</span>, a single child vertex <span class="mathjax-tex">$\textsf{choice}(v,s)\in R(s)$</span> of <i>s</i> such that <span class="mathjax-tex">$v\in \llbracket X_{\textsf{choice}(v,s)} \rrbracket $</span>. The algorithm also yields, for each leaf vertex <i>t</i> and each <span class="mathjax-tex">$v\in V_\exists \cap \llbracket X_t \rrbracket $</span>, a single game move <span class="mathjax-tex">$\textsf{move}(v,t)$</span>. All these choices together constitute a winning strategy for existential player in the parity game <span class="mathjax-tex">$P_G$</span>. We define a strategy for the Emerson-Lei game that uses leaves of the Zielonka tree as memory values, following the ideas used in the construction of <span class="mathjax-tex">$P_G$</span>; the strategy moves, from a node <span class="mathjax-tex">$v\in V_\exists $</span> and having memory content <i>m</i>, to the node <span class="mathjax-tex">$\textsf{move}(v,m)$</span>. As initial memory value we pick some leaf of <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span> that <span class="mathjax-tex">$\textsf{choice}$</span> associates with the initial node in <i>G</i>. To update memory value <i>m</i> according to visiting game node <i>v</i>, we first take the anchor vertex <i>s</i> of <i>m</i> and <i>v</i>. Then we pick the next memory value <i>m</i> to be some leaf below <i>s</i> that can be reached by talking the choices <span class="mathjax-tex">$\textsf{choice}(v,s')$</span> for every vertex <span class="mathjax-tex">$s'\in T_\bigcirc $</span> passed along the way from <i>s</i> to the leaf; if <span class="mathjax-tex">$s\in T_\square $</span>, then we additionally require the following: let <span class="mathjax-tex">$q=|R(s)|$</span>, let <i>o</i> be the number such that <i>m</i> is a leaf below the <i>o</i>-th child of <i>s</i>, and put <span class="mathjax-tex">$j=o+1\mod q$</span>; then we require that <span class="mathjax-tex">$m'$</span> is a leaf below the <i>j</i>-th child of <i>s</i>. By the correctness of the algorithm, the constructed strategy is a winning strategy.</p> <p>Dziembowski et al. have shown that winning strategies can be extracted by using a walk through the Zielonka tree that requires memory only for the branching at winning vertices [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Dziembowski, S., Jurdzinski, M., Walukiewicz, I.: How much memory is needed to win infinite games? In: 12th Annual IEEE Symposium on Logic in Computer Science. pp. 99–110. IEEE Computer Society (1997). 
 https://doi.org/10.1109/LICS.1997.614939
 
 " href="#ref-CR18" id="ref-link-section-d15224108e14117">18</a>]. This yields, for instance, memoryless strategies for games with Rabin objectives, for which branching in the associated Zielonka trees takes place at losing vertices. Adapting the strategy extraction in our setting to this more economic method is straight-forward but notation-heavy, so we omit a more precise analysis of strategy size here.</p> <p>Our algorithm hence can be implemented to run in time <span class="mathjax-tex">$2^{\mathcal {O}(k \log n)}$</span> for games with <i>n</i> nodes and <span class="mathjax-tex">$k\le n$</span> colors, improving upon the bound <span class="mathjax-tex">$2^{\mathcal {O}(n^2)}$</span> stated in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Hunter, P., Dawar, A.: Complexity bounds for regular games. In: 30th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3618, pp. 495–506. Springer (2005). 
 https://doi.org/10.1007/11549345_43
 
 " href="#ref-CR25" id="ref-link-section-d15224108e14229">25</a>], where the authors only consider the case that every game node has a distinct color, implying <span class="mathjax-tex">$n=k$</span>. We note that the later appearance record construction used in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Hunter, P., Dawar, A.: Complexity bounds for regular games. In: 30th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 3618, pp. 495–506. Springer (2005). 
 https://doi.org/10.1007/11549345_43
 
 " href="#ref-CR25" id="ref-link-section-d15224108e14257">25</a>] is known to be hard to represent symbolically. Our fixpoint characterization generalizes previously known algorithms for e.g. parity games [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Bruse, F., Falk, M., Lange, M.: The fixpoint-iteration algorithm for parity games. In: International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2014. EPTCS, vol. 161, pp. 116–130 (2014). 
 https://doi.org/10.4204/EPTCS.161.12
 
 " href="#ref-CR8" id="ref-link-section-d15224108e14260">8</a>], and Streett and Rabin games [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Piterman, N., Pnueli, A.: Faster solutions of rabin and streett games. In: 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12-15 August 2006, Seattle, WA, USA, Proceedings. pp. 275–284. IEEE Computer Society (2006). 
 https://doi.org/10.1109/LICS.2006.23
 
 " href="#ref-CR36" id="ref-link-section-d15224108e14263">36</a>], recovering previously known bounds on worst-case running time of fixpoint iteration algorithms for these types of games.</p><p>While it has recently been shown that parity games can be solved in quasipolynomial time [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Calude, C., Jain, S., Khoussainov, B., Li, W., Stephan, F.: Deciding parity games in quasipolynomial time. In: Theory of Computing, STOC 2017. pp. 252–263. ACM (2017)" href="#ref-CR9" id="ref-link-section-d15224108e14269">9</a>], we note that in the case of parity objectives, our algorithm is not immediately quasipolynomial. However, there are quasipolynomial methods for solving nested fixpoints [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Arnold, A., Niwinski, D., Parys, P.: A quasi-polynomial black-box algorithm for fixed point evaluation. In: Baier, C., Goubault-Larrecq, J. (eds.) 29th EACSL Annual Conference on Computer Science Logic, CSL 2021, January 25-28, 2021, Ljubljana, Slovenia (Virtual Conference). LIPIcs, vol. 183, pp. 9:1–9:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021). 
 https://doi.org/10.4230/LIPIcs.CSL.2021.9
 
 " href="#ref-CR2" id="ref-link-section-d15224108e14272">2</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="Hausmann, D., Schröder, L.: Quasipolynomial computation of nested fixpoints. In: Groote, J.F., Larsen, K.G. (eds.) Tools and Algorithms for the Construction and Analysis of Systems - 27th International Conference, TACAS 2021, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021, Luxembourg City, Luxembourg, March 27 - April 1, 2021, Proceedings, Part I. Lecture Notes in Computer Science, vol. 12651, pp. 38–56. Springer (2021). 
 https://doi.org/10.1007/978-3-030-72016-2_3
 
 " href="#ref-CR24" id="ref-link-section-d15224108e14275">24</a>] (with the latter being open to symbolic implementation); in the case of parity objectives, these more involved algorithms can be used in place of fixpoint iteration to solve our equation system and recover the quasipolynomial bound. The precise complexity of using quasipolynomial methods for solving fixpoint equation systems beyond parity conditions is subject to ongoing research.</p></div></div></section><section data-title="Synthesis for Safety and Emerson-Lei LTL"><div class="c-article-section" id="Sec4-section"><h2 id="Sec4" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">4 </span>Synthesis for Safety and Emerson-Lei LTL</h2><div class="c-article-section__content" id="Sec4-content"><p>In this section we present an application of the results from Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec3">3</a>. We introduce the safety and Emerson-Lei fragment of LTL and show that synthesis for this fragment can be reasoned about symbolically. The idea for safety and Emerson-Lei LTL synthesis is twofold: first, consider only the safety part and create a symbolic arena capturing its satisfaction. Second, play a game on this arena by adding the Emerson-Lei part as a winning condition. Finally we use the results from the previous sections to solve the game symbolically.</p><h3 class="c-article__sub-heading" id="Sec5"><span class="c-article-section__title-number">4.1 </span>Safety LTL and Symbolic Safety Automata</h3><p>We start by defining safety LTL, symbolic safety automata, and recalling known results about those.</p> <h3 class="c-article__sub-heading" id="FPar15">Definition 14</h3> <p><b>(LTL and Safety LTL</b> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="Zhu, S., Tabajara, L.M., Li, J., Pu, G., Vardi, M.Y.: A symbolic approach to safety LTL synthesis. In: 13th International Haifa Verification Conference: Hardware and Software - Verification and Testing. Lecture Notes in Computer Science, vol. 10629, pp. 147–162. Springer (2017). 
 https://doi.org/10.1007/978-3-319-70389-3_10
 
 " href="#ref-CR45" id="ref-link-section-d15224108e14305">45</a>]<b>).</b> Given a non-empty set <span class="mathjax-tex">$\textsf{AP}$</span> of atomic propositions, the general syntax for LTL formulas is as follows:</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \varphi := \top \mid \bot \mid p \mid \lnot \varphi \mid \varphi _1 \wedge \varphi _2 \mid \varphi _1 \vee \varphi _2 \mid X \varphi \mid \varphi _1 U \varphi _2 \qquad \ p\in \textsf{AP}. \end{aligned}$$</span></div></div><p>Standard abbreviations are defined as follows: <span class="mathjax-tex">$\varphi _1 R \varphi _2 := \lnot (\lnot \varphi _1 U \lnot \varphi _2)$</span>, <span class="mathjax-tex">$F \varphi := \top U \varphi $</span>, and <span class="mathjax-tex">$G \varphi := \lnot F \lnot \varphi $</span>. We define the satisfaction relation <span class="mathjax-tex">$\models $</span> for a formula <span class="mathjax-tex">$\varphi $</span> and its language <span class="mathjax-tex">$\mathcal {L}(\varphi )$</span> as usual.</p> <p>An LTL formula is said to be a <i>safety formula</i> if it is in negative normal form (i.e. all negations are pushed to atomic propositions) and only uses <i>X</i>, <i>R</i>, <i>G</i> as temporal operators (i.e. no <i>U</i> or <i>F</i> are allowed ).</p> <p>It is a safety formula in the sense that every word that does not satisfy the formula has a finite prefix that already falsifies the formula. In other words, such a formula is satisfied as long as “bad states” are avoided forever.</p> <h3 class="c-article__sub-heading" id="FPar16">Definition 15</h3> <p><b>(Symbolic Safety Automata).</b> A symbolic safety automaton is a tuple <span class="mathjax-tex">$\mathcal {A}= (2^{\textsf{AP}}, V, T, \theta _0)$</span> where <i>V</i> is a set of variables, <span class="mathjax-tex">$T(V,V',\textsf{AP})$</span> is the transition assertion, and <span class="mathjax-tex">$\theta _0(V)$</span> is the initialization assertion. A run of <span class="mathjax-tex">$\mathcal {A}$</span> on the word <span class="mathjax-tex">$w \in (2^{\textsf{AP}})^\omega $</span> is a sequence <span class="mathjax-tex">$\rho = s_0 s_1 \dots $</span> where the <span class="mathjax-tex">$s_i \in 2^V$</span> are variable assignments such that 1. <span class="mathjax-tex">$s_0 \models \theta _0$</span>, and 2. for all <span class="mathjax-tex">$i \ge 0$</span>, <span class="mathjax-tex">$(s_i, s_{i+1}, w(i)) \models T$</span>. A word <i>w</i> is in <span class="mathjax-tex">$\mathcal {L}(\mathcal {A})$</span> if and only if there is an infinite run of <span class="mathjax-tex">$\mathcal {A}$</span> on <i>w</i>. <span class="mathjax-tex">$\mathcal {A}$</span> is deterministic if for all words <span class="mathjax-tex">$w \in (2^{\textsf{AP}})^\omega $</span> there is at most one run of <span class="mathjax-tex">$\mathcal {A}$</span> on <i>w</i>.</p> <p>Kupferman and Vardi show how to convert a safety LTL formula into an equivalent deterministic symbolic safety automaton [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Kupferman, O., Vardi, M.Y.: Model checking of safety properties. Formal methods in system design 19(3), 291–314 (2001). 
 https://doi.org/10.1023/A:1011254632723
 
 " href="#ref-CR30" id="ref-link-section-d15224108e15238">30</a>].</p> <h3 class="c-article__sub-heading" id="FPar17">Lemma 16</h3> <p>A safety LTL formula <span class="mathjax-tex">$\varphi $</span> can be translated to a deterministic symbolic safety automaton <span class="mathjax-tex">$\mathcal {D}_\textsf{symb}$</span> accepting the same language, with <span class="mathjax-tex">$|\mathcal {D}_\textsf{symb}| = 2^{|\varphi |}$</span>.</p> <p>The idea is to first convert <span class="mathjax-tex">$\varphi $</span> to a (non-symbolic) non-deterministic safety automaton <span class="mathjax-tex">$\mathcal {N}_\varphi $</span>, which is of size exponential of the size of the formula, and then symbolically determinize <span class="mathjax-tex">$\mathcal {N}_\varphi $</span> by a standard subset construction to obtain <span class="mathjax-tex">$\mathcal {D}_\textsf{symb}$</span>. Note that while the size of <span class="mathjax-tex">$\mathcal {D}_\textsf{symb}$</span> is only exponential in the size of the formula, its state space would be double exponential when fully expanded.</p> <h3 class="c-article__sub-heading" id="FPar18">Example 17</h3> <p>Let <span class="mathjax-tex">$\varphi = G(b \vee c) \wedge G(a \rightarrow b \vee XXb)$</span> be a safety LTL formula over <span class="mathjax-tex">$\textsf{AP}= \{a,b,c\}$</span>. An execution satisfying <span class="mathjax-tex">$\varphi $</span> must have at least one of <i>b</i> or <i>c</i> at every step, moreover every <i>a</i> sees a <i>b</i> present at the same step or two steps afterwards.</p> <p>As an intermediate step towards building the equivalent <span class="mathjax-tex">$\mathcal {D}_\textsf{symb}$</span>, we first present below a corresponding non-deterministic safety automaton <span class="mathjax-tex">$\mathcal {N}_\varphi $</span>.</p><div class="c-article-section__figure c-article-section__figure--no-border" data-test="figure" data-container-section="figure" id="figure-f"><figure><div class="c-article-section__figure-content" id="Figf"><div class="c-article-section__figure-item"><div class="c-article-section__figure-content"><picture><img aria-describedby="Figf" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-57228-9_4/MediaObjects/560586_1_En_4_Figf_HTML.png" alt="figure f" loading="lazy" width="685" height="180"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-f-desc"></div></div></figure></div> <p>For the sake of presentation, we use Boolean combinations of <span class="mathjax-tex">$\textsf{AP}$</span> in transitions instead of labeling them with elements of <span class="mathjax-tex">$2^\textsf{AP}$</span>, with the intended meaning that <span class="mathjax-tex">$s \xrightarrow {\psi } s' = \{s \xrightarrow {C} s' \mid C \in 2^\textsf{AP},~C \models \psi \}$</span>. We also omit the <span class="mathjax-tex">$G(b \vee c)$</span> part of the formula in the construction. One can simply append <span class="mathjax-tex">$\dots \wedge (b \vee c)$</span> to every transition of <span class="mathjax-tex">$\mathcal {N}_\varphi $</span> to get back the original formula. Intuitively state 1 correspond to not seeing an <i>a</i>, state 2 means that a <i>b</i> must be seen at the next step, state 3 means that there must be a <i>b</i> now, and state 4 that <i>b</i> is needed now and next as well.</p> <p>Then the symbolic safety automaton is <span class="mathjax-tex">$\mathcal {D}_\textsf{symb}= (2^\textsf{AP}, V, T, \theta _0)$</span> with:</p><ul class="u-list-style-dash"> <li> <p><span class="mathjax-tex">$V = \{v_1,v_2,v_3,v_4\}$</span> are the variables corresponding to the four states of <span class="mathjax-tex">$\mathcal {N}_\varphi $</span>,</p> </li> <li> <p><span class="mathjax-tex">$\theta _0 = v_1 \wedge \lnot v_2 \wedge \lnot v_3 \wedge \lnot v_4$</span> asserts that only the state <span class="mathjax-tex">$v_1$</span> is initial,</p> </li> <li> <p>The transition assertion is <span class="mathjax-tex">$T = \,(v'_1 \leftrightarrow (v_1 \wedge (\lnot a \vee b)) \vee (v_3 \wedge b))\,\wedge $</span></p> <p><span class="mathjax-tex">$(v'_2 \leftrightarrow (v_1 \wedge a) \vee (v_3 \wedge (a \wedge b)))\,\wedge (v'_3 \leftrightarrow (v_2 \wedge (\lnot a \vee b)) \vee (v_4 \wedge b))\,\wedge $</span></p> <p><span class="mathjax-tex">$(v'_4 \leftrightarrow (v_2 \wedge a) \vee (v_4 \wedge (a \wedge b)))\,\wedge (v_1 \vee v_2 \vee v_3 \vee v_4)$</span>.</p> </li> </ul> <p>Determinizing <span class="mathjax-tex">$\mathcal {N}_\varphi $</span> enumeratively would give an automaton with 9 states (see Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar24">23</a>).</p> <h3 class="c-article__sub-heading" id="FPar19">Remark 18</h3> <p>Restricting attention to safety LTL enables the two advantages mentioned above with respect to determinization. First, subset construction suffices (as observed also in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Zhu, S., Tabajara, L.M., Pu, G., Vardi, M.Y.: On the power of automata minimization in temporal synthesis. In: Proceedings 12th International Symposium on Games, Automata, Logics, and Formal Verification. EPTCS, vol. 346, pp. 117–134 (2021). 
 https://doi.org/10.4204/EPTCS.346.8
 
 " href="#ref-CR46" id="ref-link-section-d15224108e16633">46</a>]), avoiding the more complex Büchi determinization. Second, this construction, due to its simplicity, can be implemented symbolically. Interestingly, recent implementations of the synthesis from LTL<span class="mathjax-tex">$_f$</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Zhu, S., Tabajara, L.M., Pu, G., Vardi, M.Y.: On the power of automata minimization in temporal synthesis. In: Proceedings 12th International Symposium on Games, Automata, Logics, and Formal Verification. EPTCS, vol. 346, pp. 117–134 (2021). 
 https://doi.org/10.4204/EPTCS.346.8
 
 " href="#ref-CR46" id="ref-link-section-d15224108e16658">46</a>] or from safety LTL [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="Zhu, S., Tabajara, L.M., Li, J., Pu, G., Vardi, M.Y.: A symbolic approach to safety LTL synthesis. In: 13th International Haifa Verification Conference: Hardware and Software - Verification and Testing. Lecture Notes in Computer Science, vol. 10629, pp. 147–162. Springer (2017). 
 https://doi.org/10.1007/978-3-319-70389-3_10
 
 " href="#ref-CR45" id="ref-link-section-d15224108e16661">45</a>] have used indirect approaches for obtaining deterministic automata. For example, by translating LTL to first order logic and applying the tool MONA to the results [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="Zhu, S., Tabajara, L.M., Li, J., Pu, G., Vardi, M.Y.: A symbolic approach to safety LTL synthesis. In: 13th International Haifa Verification Conference: Hardware and Software - Verification and Testing. Lecture Notes in Computer Science, vol. 10629, pp. 147–162. Springer (2017). 
 https://doi.org/10.1007/978-3-319-70389-3_10
 
 " href="#ref-CR45" id="ref-link-section-d15224108e16664">45</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Zhu, S., Tabajara, L.M., Pu, G., Vardi, M.Y.: On the power of automata minimization in temporal synthesis. In: Proceedings 12th International Symposium on Games, Automata, Logics, and Formal Verification. EPTCS, vol. 346, pp. 117–134 (2021). 
 https://doi.org/10.4204/EPTCS.346.8
 
 " href="#ref-CR46" id="ref-link-section-d15224108e16668">46</a>], or by concentrating on minimization of deterministic automata [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Tabakov, D., Vardi, M.Y.: Experimental evaluation of classical automata constructions. In: 12th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning. Lecture Notes in Computer Science, vol. 3835, pp. 396–411. Springer (2005). 
 https://doi.org/10.1007/11591191_28
 
 " href="#ref-CR42" id="ref-link-section-d15224108e16671">42</a>]. The direct construction is similar to approaches used for checking universality of nondeterministic finite automata [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Tabakov, D., Vardi, M.Y.: Experimental evaluation of classical automata constructions. In: 12th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning. Lecture Notes in Computer Science, vol. 3835, pp. 396–411. Springer (2005). 
 https://doi.org/10.1007/11591191_28
 
 " href="#ref-CR42" id="ref-link-section-d15224108e16674">42</a>] or SAT-based bounded model checking [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Armoni, R., Egorov, S., Fraer, R., Korchemny, D., Vardi, M.Y.: Efficient LTL compilation for sat-based model checking. In: International Conference on Computer-Aided Design. pp. 877–884. IEEE Computer Society (2005). 
 https://doi.org/10.1109/ICCAD.2005.1560185
 
 " href="#ref-CR1" id="ref-link-section-d15224108e16677">1</a>]. We are not aware of uses of this direct implementation of the subset construction in reactive synthesis. The worst case complexity of this part is doubly-exponential, which, just like for LTL and LTL<span class="mathjax-tex">$_f$</span>, cannot be avoided [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Artale, A., Geatti, L., Gigante, N., Mazzullo, A., Montanari, A.: Complexity of safety and cosafety fragments of linear temporal logic. In: Proceedings of the AAAI Conference on Artificial Intelligence. vol. 37, pp. 6236–6244 (2023)" href="#ref-CR3" id="ref-link-section-d15224108e16702">3</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 43" title="Vardi, M.Y., Stockmeyer, L.J.: Improved upper and lower bounds for modal logics of programs: Preliminary report. In: Proceedings of the 17th Annual ACM Symposium on Theory of Computing. pp. 240–251. ACM (1985)" href="#ref-CR43" id="ref-link-section-d15224108e16706">43</a>].</p> <h3 class="c-article__sub-heading" id="Sec6"><span class="c-article-section__title-number">4.2 </span>Symbolic Games</h3><p>We use <i>symbolic game structures</i> to represent a certain class of games. Formally, a <i>symbolic game structure</i> <span class="mathjax-tex">$\mathcal{G} = \langle \mathcal{V}, \mathcal{X}, \mathcal{Y}, \theta _\exists , \rho _\exists , \varphi \rangle $</span> consists of:</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$\mathcal{V} = \{v_1,\ldots , v_n\}$</span> : A finite set of typed <i>variables</i> over finite domains. Without loss of generality, we assume they are all Boolean. A node <i>s</i> is an valuation of <span class="mathjax-tex">$\mathcal{V}$</span>, assigning to each variable <span class="mathjax-tex">$v_i\in \mathcal{V}$</span> a value <span class="mathjax-tex">$s[v_i]\in \{0,1\}$</span>. Let <span class="mathjax-tex">$\varSigma $</span> be the set of nodes.</p> <p>We extend the evaluation function <span class="mathjax-tex">$s[\cdot ]$</span> to Boolean expressions over <span class="mathjax-tex">$\mathcal{V}$</span> in the usual way. An <i>assertion</i> is a Boolean formula over <span class="mathjax-tex">$\mathcal{V}$</span>. A node <i>s</i> satisfies an assertion <span class="mathjax-tex">$\varphi $</span> denoted <span class="mathjax-tex">$s\models \varphi $</span>, if <span class="mathjax-tex">$s[\varphi ]=\textbf{true}$</span>. We say that <i>s</i> is a <span class="mathjax-tex">$\varphi $</span>-node if <span class="mathjax-tex">$s\models \varphi $</span>.</p> </li> <li> <p><span class="mathjax-tex">$\mathcal{X} \subseteq \mathcal{V}$</span> is a set of <i>input variables</i>. These are variables controlled by the universal player. Let <span class="mathjax-tex">$\varSigma _\mathcal{X}$</span> denote the possible valuations to variables in <span class="mathjax-tex">$\mathcal{X}$</span>.</p> </li> <li> <p><span class="mathjax-tex">$\mathcal{Y} = \mathcal{V} \setminus \mathcal{X}$</span> is a set of <i>output variables</i>. These are variables controlled by the existential player. Let <span class="mathjax-tex">$\varSigma _\mathcal{Y}$</span> denote the possible valuations to variables in <span class="mathjax-tex">$\mathcal{Y}$</span>.</p> </li> <li> <p><span class="mathjax-tex">$\theta _\exists (\mathcal{X},\mathcal{Y})$</span> is an assertion characterizing the initial condition.</p> </li> <li> <p><span class="mathjax-tex">$\rho _\exists (\mathcal{V},\mathcal{X}',\mathcal{Y}')$</span> is the transition relation. This is an assertion relating a node <span class="mathjax-tex">$s\in \varSigma $</span> and an input value <span class="mathjax-tex">$s_\mathcal{X} \in \varSigma _\mathcal{X}$</span> to an output value <span class="mathjax-tex">$s_\mathcal{Y} \in \varSigma _\mathcal{Y}$</span> by referring to primed and unprimed copies of <span class="mathjax-tex">$\mathcal{V}$</span>. The transition relation <span class="mathjax-tex">$\rho _\exists $</span> identifies a valuation <span class="mathjax-tex">$s_\mathcal{Y}\in \varSigma _\mathcal{Y}$</span> as a <i>possible output</i> in node <i>s</i> reading input <span class="mathjax-tex">$s_\mathcal{X}$</span> if <span class="mathjax-tex">$(s,(s_\mathcal{X},s_\mathcal{Y})) \models \rho _\exists $</span>, where <i>s</i> is the assignment to variables in <span class="mathjax-tex">$\mathcal V$</span> and <span class="mathjax-tex">$s_\mathcal{X}$</span> and <span class="mathjax-tex">$s_\mathcal{Y}$</span> are the assignment to variables in <span class="mathjax-tex">$\mathcal V'$</span> induced by <span class="mathjax-tex">$(s_\mathcal{X},s_\mathcal{Y})\in \varSigma $</span>.</p> </li> <li> <p><span class="mathjax-tex">$\varphi $</span> is the winning condition, given by an LTL formula.</p> </li> </ul> <p>For two nodes <i>s</i> and <span class="mathjax-tex">$s'$</span> of <span class="mathjax-tex">$\mathcal{G}$</span>, <span class="mathjax-tex">$s'$</span> is a <i>successor</i> of <i>s</i> if <span class="mathjax-tex">$(s,s') \models \rho _\exists $</span>.</p><p>A symbolic game structure <span class="mathjax-tex">$\mathcal{G}$</span> defines an arena <span class="mathjax-tex">$A_\mathcal{G}$</span>, where <span class="mathjax-tex">$V_\forall =\varSigma $</span>, <span class="mathjax-tex">$V_\exists = \varSigma \times \varSigma _\mathcal{X}$</span>, and <i>E</i> is defined as follows:</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ E= \{(s,(s,s_\mathcal{X})) ~|~ s\in \varSigma \text{ and } s_\mathcal{X}\in \varSigma _\mathcal{X} \} \cup \{((s,s_\mathcal{X}),(s_\mathcal{X},s_\mathcal{Y})) ~|~ (s,(s_\mathcal{X},s_\mathcal{Y}))\models \rho _\exists \}.$$</span></div></div><p>When reasoning about symbolic game structures we ignore the intermediate visits to <span class="mathjax-tex">$V_\exists $</span>. Indeed, they add no information as they can be deduced from the nodes in <span class="mathjax-tex">$V_\forall $</span> preceding and following them. Thus, a play <span class="mathjax-tex">$\pi =\,s_0s_1\ldots $</span> is <i>winning for the existential player</i> if <span class="mathjax-tex">$\sigma $</span> is infinite and satisfies <span class="mathjax-tex">$\varphi $</span>. Otherwise, <span class="mathjax-tex">$\sigma $</span> is <i>winning for the universal player</i>.</p><p>The notion of strategy and winning region is trivially generalized from games to symbolic game structures. When needed, we treat <span class="mathjax-tex">$W_\exists $</span> (the set of nodes winning for the existential player) as an assertion. We define winning in the <i>entire</i> game structure by incorporating the initial assertion: a game structure <span class="mathjax-tex">$\mathcal{G}$</span> is said to be <i>won</i> by the existential player, if for all <span class="mathjax-tex">$s_\mathcal{X} \in \varSigma _\mathcal{X}$</span> there exists <span class="mathjax-tex">$s_\mathcal{Y} \in \varSigma _\mathcal{Y}$</span> such that <span class="mathjax-tex">$(s_\mathcal{X},s_\mathcal{Y})\models \theta _\exists \wedge W_\exists $</span>.</p><h3 class="c-article__sub-heading" id="Sec7"><span class="c-article-section__title-number">4.3 </span>Realizability and Synthesis</h3><p>Let <span class="mathjax-tex">$\varphi $</span> be an LTL formula over input and output variables <i>I</i> and <i>O</i>, controlled by <i>the environment</i> and <i>the system</i>, respectively (the universal and the existential player, respectively).</p><p>The reactive synthesis problem asks whether there is a strategy for the system of the form <span class="mathjax-tex">$\sigma : (2^I)^+ \rightarrow 2^O$</span> such that for all sequences <span class="mathjax-tex">$x_0 x_1 \dots \in (2^I)^\omega $</span> we have:</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$(x_0 \cup \sigma (x_0)) (x_1 \cup \sigma (x_0 x_1)) \dots \models \varphi $$</span></div></div><p>If there is such a strategy we say that <span class="mathjax-tex">$\varphi $</span> is <i>realizable</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Sixteenth ACM Symposium on Principles of Programming Languages. pp. 179–190. ACM Press (1989). 
 https://doi.org/10.1145/75277.75293
 
 " href="#ref-CR38" id="ref-link-section-d15224108e18944">38</a>].</p><p>Equivalently, <span class="mathjax-tex">$\varphi $</span> is <i>realizable</i> if the system is winning in the symbolic game <span class="mathjax-tex">$\mathcal{G}_\varphi =\langle I \cup O, I, O, \top ,\top ,\varphi \rangle $</span> with <i>I</i> for input variables <span class="mathjax-tex">$\mathcal{X}$</span> and <i>O</i> for output <span class="mathjax-tex">$\mathcal{Y}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar20">Theorem 19</h3> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Sixteenth ACM Symposium on Principles of Programming Languages. pp. 179–190. ACM Press (1989). 
 https://doi.org/10.1145/75277.75293
 
 " href="#ref-CR38" id="ref-link-section-d15224108e19085">38</a>] Given an LTL formula <span class="mathjax-tex">$\varphi $</span>, the realizability of <span class="mathjax-tex">$\varphi $</span> can be determined in doubly exponential time. The problem is 2EXPTIME-complete.</p> <p>The game <span class="mathjax-tex">$\mathcal{G}_\varphi $</span> above uses neither the initial condition nor the system transition. Conversely, consider a symbolic game <span class="mathjax-tex">$\mathcal{G}=\langle \mathcal{V},\mathcal{X},\mathcal{Y},\theta _\exists ,\rho _\exists ,\varphi \rangle $</span>:</p> <h3 class="c-article__sub-heading" id="FPar21">Theorem 20</h3> <p>[<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Bloem, R., Jobstmann, B., Piterman, N., Pnueli, A., Sa’ar, Y.: Synthesis of reactive(1) designs. J. Comput. Syst. Sci. 78(3), 911–938 (2012). 
 https://doi.org/10.1016/j.jcss.2011.08.007
 
 " href="#ref-CR7" id="ref-link-section-d15224108e19223">7</a>] The system wins in <span class="mathjax-tex">$\mathcal{G}$</span> iff <span class="mathjax-tex">$\varphi _{_\mathcal{G}}= \theta _\exists \wedge G \rho _\exists \wedge \varphi $</span> is realizable.<sup><a href="#Fn2"><span class="u-visually-hidden">Footnote </span>2</a></sup> <sup><a href="#Fn3"><span class="u-visually-hidden">Footnote </span>3</a></sup> </p> <h3 class="c-article__sub-heading" id="Sec8"><span class="c-article-section__title-number">4.4 </span>Safety and Emerson-Lei Synthesis</h3><p>We now define the class of LTL formulas that are supported by our technique and show how to construct appropriate games capturing their realizability problem.</p><p>For <span class="mathjax-tex">$\psi \in \mathbb {B}(\textsf{AP})$</span>, let <span class="mathjax-tex">$\textsf{Inf}\,\psi := GF\psi $</span> and <span class="mathjax-tex">$\textsf{Fin}\,\psi := FG\lnot \psi = \lnot \textsf{Inf}\,\psi $</span>. The <i>Emerson-Lei fragment</i> of LTL consists of all formulas that are positive Boolean combinations of <span class="mathjax-tex">$\textsf{Inf}\,\psi $</span> and <span class="mathjax-tex">$\textsf{Fin}\,\psi $</span> for all Boolean formulas <span class="mathjax-tex">$\psi $</span> over atomic propositions. The satisfaction of such formulas depends only on the set of letters (truth assignments to propositions) appearing infinitely often in a word.</p> <h3 class="c-article__sub-heading" id="FPar22">Remark 21</h3> <p>The Emerson-Lei fragment easily accommodates various liveness properties that cannot be encoded in smaller fragments such as GR[1]. One prominent example for this is the property of <i>stability</i> (as encoded by LTL formulas of the shape <i>FG</i> <i>p</i>), which appears frequently as a guarantee in usage of synthesis for robotics and control (see, e.g., the work of Ehlers [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Ehlers, R.: Generalized rabin(1) synthesis with applications to robust system synthesis. In: Third International Symposium on NASA Formal Methods. Lecture Notes in Computer Science, vol. 6617, pp. 101–115. Springer (2011). 
 https://doi.org/10.1007/978-3-642-20398-5_9
 
 " href="#ref-CR19" id="ref-link-section-d15224108e19558">19</a>] and Ozay [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Liu, J., Ozay, N., Topcu, U., Murray, R.M.: Synthesis of reactive switching protocols from temporal logic specifications. IEEE Trans. Autom. Control. 58(7), 1771–1785 (2013). 
 https://doi.org/10.1109/TAC.2013.2246095
 
 " href="#ref-CR32" id="ref-link-section-d15224108e19561">32</a>]), and commonly is approximated in GR[1] but, as a guarantee or as part of a specification, cannot be captured exactly in the game context. Another important example is <i>strong fairness</i> (as encoded by LTL formulas of the shape <span class="mathjax-tex">$\bigwedge _{i} (GF~r_i\rightarrow GF~g_i)$</span>) which allows to capture the exact relation between cause and effect. Particularly, in GR[1] only if <i>all</i> “resources” are available infinitely often there is an obligation on the system to supply <i>all</i> its “guarantees”. In contrast, strong fairness allows to connect particular resources to particular supplied guarantees. Ongoing studies on fairness assumptions that arise from the abstraction of continuous state spaces to discrete state spaces [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Liu, J., Ozay, N., Topcu, U., Murray, R.M.: Synthesis of reactive switching protocols from temporal logic specifications. IEEE Trans. Autom. Control. 58(7), 1771–1785 (2013). 
 https://doi.org/10.1109/TAC.2013.2246095
 
 " href="#ref-CR32" id="ref-link-section-d15224108e19638">32</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Majumdar, R., Schmuck, A.: Supervisory controller synthesis for nonterminating processes is an obliging game. IEEE Trans. Autom. Control. 68(1), 385–392 (2023). 
 https://doi.org/10.1109/TAC.2022.3143108
 
 " href="#ref-CR33" id="ref-link-section-d15224108e19641">33</a>] provide further examples of fairness assumptions that can be expressed in EL but not in GR[1]. Emerson-Lei liveness allows free combination of all properties mentioned above and more.</p> <h3 class="c-article__sub-heading" id="FPar23">Definition 22</h3> <p>The <i>Safety and Emerson-Lei fragment</i> is the set of formulas of the form <span class="mathjax-tex">$\varphi = \varphi _{\textrm{safety}} \wedge \varphi _{\textrm{EL}}$</span>, where <span class="mathjax-tex">$\varphi _{\textrm{safety}}$</span> is a safety formula and <span class="mathjax-tex">$\varphi _{\textrm{EL}}$</span> is in the Emerson-Lei fragment.</p> <p>We assume a partition <span class="mathjax-tex">$\textsf{AP}= I \uplus O$</span> where <i>I</i> is a set of <i>input propositions</i> and <i>O</i> a set of <i>output propositions</i>, both non-empty. Let <span class="mathjax-tex">$\varphi = \varphi _{\textrm{safety}} \wedge \varphi _{\textrm{EL}}$</span> be a safety and Emerson-Lei formula over <span class="mathjax-tex">$\textsf{AP}$</span>, and let <span class="mathjax-tex">$\mathcal {D}_\textsf{symb}= (2^\textsf{AP}, V, T, \theta _0)$</span> be the symbolic deterministic safety automaton associated to <span class="mathjax-tex">$\varphi _{\textrm{safety}}$</span>. We construct <span class="mathjax-tex">$G_\varphi = \langle V \uplus \textsf{AP}, I, O \uplus V, \theta _0, T, \varphi _{\textrm{EL}} \rangle $</span>, thus <span class="mathjax-tex">$\mathcal{X}=I$</span> and <span class="mathjax-tex">$\mathcal{Y}=O \uplus V$</span>.</p> <h3 class="c-article__sub-heading" id="FPar24">Example 23</h3> <p>Let <span class="mathjax-tex">$\varphi _{\textrm{safety}} = G(b \vee c) \wedge G(a \rightarrow b \vee XXb)$</span>, our running safety example from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar18">17</a> with its associated symbolic deterministic automaton. Partition <span class="mathjax-tex">$\textsf{AP}$</span> into <span class="mathjax-tex">$I = \{a\}$</span> and <span class="mathjax-tex">$O = \{b,c\}$</span>. We depict the arena of the game <span class="mathjax-tex">$G_\varphi $</span> (independent of the formula <span class="mathjax-tex">$\varphi _{\textrm{EL}}$</span> that is yet to be defined) in Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig1">1</a>.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-1" data-title="Fig. 1."><figure><figcaption><b id="Fig1" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 1.</b></figcaption><div class="c-article-section__figure-content"><div class="c-article-section__figure-item"><a class="c-article-section__figure-link" data-test="img-link" data-track="click" data-track-label="image" data-track-action="view figure" href="/chapter/10.1007/978-3-031-57228-9_4/figures/1" rel="nofollow"><picture><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-57228-9_4/MediaObjects/560586_1_En_4_Fig1_HTML.png" alt="figure 1" loading="lazy" width="685" height="373"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>Game arena for <span class="mathjax-tex">$G_\varphi $</span></p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-031-57228-9_4/figures/1" data-track-dest="link:Figure1 Full size image" aria-label="Full size image figure 1" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <p>To keep the illustration readable and keep it from getting too large, a few modifications to the formal arena definition have been made. First, <i>c</i> labels on edges have been omitted: every transition labeled with <i>b</i> represent two transitions with sets <span class="mathjax-tex">$\{b\}$</span> and <span class="mathjax-tex">$\{b,c\}$</span>, while transitions labeled with <span class="mathjax-tex">$\lnot b$</span> stand for a single transition with set <span class="mathjax-tex">$\{c\}$</span> (due to the <span class="mathjax-tex">$G(b \vee c)$</span> requirement forbidding <span class="mathjax-tex">$\emptyset $</span>). Similarly, existential nodes have been omitted when all choices for the existential player lead to the same destination. Instead, the universal and existential moves have been combined in one transition: <span class="mathjax-tex">$a;*$</span> for an <i>a</i> followed by some existential move, and <i>a</i>; <i>b</i> for when an <i>a</i> requires the existential player to play <i>b</i> (with or without <i>c</i>, as above). Finally, states are only labeled with variables from <i>V</i> and not <span class="mathjax-tex">$\textsf{AP}$</span>, the latter is used to label edges instead. For a fully state-based labeling arena, states would have to store the last move, leading to various duplicate states.</p> <p>Note that this game arena is given only for illustration purposes, as we want to solve the symbolic game without explicitly enumerating all its states and transitions like here.</p> <h3 class="c-article__sub-heading" id="FPar25">Lemma 24</h3> <p>The system wins <span class="mathjax-tex">$G_\varphi $</span> if and only if <span class="mathjax-tex">$\varphi $</span> is realizable.</p> <p>Next we detail how to solve the symbolic game <span class="mathjax-tex">$G_\varphi $</span> by using the results from Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec3">3</a>.</p> <h3 class="c-article__sub-heading" id="FPar26">Lemma 25</h3> <p>Given a symbolic game <span class="mathjax-tex">$G = \langle \mathcal{V}, \mathcal{X}, \mathcal{Y}, \theta _\exists , \rho _\exists , \varphi \rangle $</span> such that <span class="mathjax-tex">$\varphi $</span> is an Emerson-Lei formula with set of colors</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} C=\{\psi \in \mathbb {B}(\textsf{AP})\mid \psi \text { is a subformula of }\varphi \}, \end{aligned}$$</span></div></div><p>the winning region <span class="mathjax-tex">$W_\exists $</span> of <i>G</i> is characterized by the equation system from Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar7">7</a>, using the assertion</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textsf{CPre}(S)=\forall s_{\mathcal {X}} \in \varSigma _{\mathcal {X}} .\,\exists s_{\mathcal {Y}} \in \varSigma _{\mathcal {Y}}.\, S'\wedge (v,s_{\mathcal {X}},s_{\mathcal {Y}})\models \rho _\exists . \end{aligned}$$</span></div></div> <p>The proof of this lemma is by straightforward adaptation of the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar9">9</a> to the symbolic setting, following the relation between symbolic game structures and game arenas described above.</p><p>Finally, this gives us a procedure to solve the synthesis problem for safety and Emerson-Lei LTL.</p> <h3 class="c-article__sub-heading" id="FPar27">Theorem 26</h3> <p>The realizability of a formula <span class="mathjax-tex">$\varphi = \varphi _\textsf{safety}\wedge \varphi _{EL}$</span> of the Safety and Emerson-Lei fragment of LTL can be checked in time <span class="mathjax-tex">$2^{\mathcal {O}({m\cdot \log m\cdot 2^{n}})}$</span>, where <span class="mathjax-tex">$n=|\varphi _\textsf{safety}|$</span> and <span class="mathjax-tex">$m=|\varphi _{EL}|$</span>. Realizable formulas can be realized by systems of size at most <span class="mathjax-tex">$2^{2^n}\cdot e\cdot m!$</span>.</p> <h3 class="c-article__sub-heading" id="FPar28">Proof</h3> <p>Using the construction described in this section, we obtain the symbolic game <span class="mathjax-tex">$G_{\varphi }$</span> of size <span class="mathjax-tex">$q=2^{2^n}$</span> with winning condition <span class="mathjax-tex">$\varphi _{EL}$</span>, using at most <i>m</i> colors; by Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar25">24</a>, this game characterizes realizibility of the formula. Using the results from the previous section, <span class="mathjax-tex">$G_{\varphi }$</span> can be solved in time <span class="mathjax-tex">$\mathcal {O}(m!\cdot q^2 \cdot q^m)\in \mathcal {O}(2^{m \log m}\cdot 2^{(m+2)2^n})\in 2^{\mathcal {O}({m\cdot \log m\cdot 2^{n}})}$</span>, resulting in winning strategies with memory at most <span class="mathjax-tex">$e\cdot m!$</span>.</p> <p>Both the automata determinization and the game solving can be implemented symbolically.</p> <h3 class="c-article__sub-heading" id="FPar29">Example 27</h3> <p>To illustrate the overall synthesis method, we consider the game that is obtained by combining the game arena <span class="mathjax-tex">$G_{\varphi _{\textsf{safety}}}$</span> from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar24">23</a> with the winning objective <span class="mathjax-tex">$\varphi _{EL}= (\textsf{Fin}~a\vee \textsf{Inf}~b)\wedge (\textsf{Fin}~a\vee \textsf{Fin}d)\wedge \textsf{Inf}~c$</span> from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar2">2</a>.3, where we instantiate the label <i>d</i> to nodes satisfying <span class="mathjax-tex">$b\wedge c$</span> thus creating a game-specific dependency between the colors. Solving this game amounts to solving the equation system shown in Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar8">8</a>.3. However, with the interpretation of <span class="mathjax-tex">$d=b\wedge c$</span>, some of the conditions become simpler. For example, <span class="mathjax-tex">$\lnot a \wedge \lnot b \wedge \lnot c \wedge \lnot d$</span> becomes <span class="mathjax-tex">$\lnot a \wedge \lnot b \wedge \lnot c$</span> and <span class="mathjax-tex">$b \wedge \lnot d$</span> becomes <span class="mathjax-tex">$b \wedge \lnot c$</span>. It turns out that the system player wins the node <span class="mathjax-tex">$v_1$</span>. Intuitively, the system can play <span class="mathjax-tex">$\{c\}$</span> whenever possible and thereby guarantee satisfaction of <span class="mathjax-tex">$\varphi _{EL}$</span>. We extract this strategy from the computed solution of the equation system in Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar2">2</a>.3 as described in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar14">13</a>. E.g. for partial runs <span class="mathjax-tex">$\pi $</span> that end in <span class="mathjax-tex">$v_1$</span> and for which the last leaf vertex in the induced walk <span class="mathjax-tex">$\rho _\pi $</span> through <span class="mathjax-tex">$\mathcal {Z}_\varphi $</span> is the vertex 8, the system can react by playing <span class="mathjax-tex">$\{b\}$</span>, <span class="mathjax-tex">$\{c\}$</span>, or even <span class="mathjax-tex">$\{b,c\}$</span> whenever the environment plays <span class="mathjax-tex">$\emptyset $</span>. The move <span class="mathjax-tex">$\{b\}$</span> continues the induced walk <span class="mathjax-tex">$\rho _\pi $</span> through vertex 2 to the leaf vertex 5; similarly, the move <span class="mathjax-tex">$\{b,c\}$</span> continues <span class="mathjax-tex">$\rho _\pi $</span> through the vertex 1 to the leaf vertex 6. The strategy construction gives precedence to the choice that leads through the lowest vertex in the Zielonka tree, which in this case means picking the move <span class="mathjax-tex">$\{c\}$</span> that continues <span class="mathjax-tex">$\rho _\pi $</span> through the vertex 7 to the leaf 8. Proceeding similarly for all other combinations of game nodes and vertices in the Zielonka tree, one obtains a strategy <span class="mathjax-tex">$\sigma $</span> for the system that always outputs singleton letters, giving precedence to <span class="mathjax-tex">$\{c\}$</span> whenever possible. To see that <span class="mathjax-tex">$\sigma $</span> is a winning strategy, let <span class="mathjax-tex">$\pi $</span> be a play that is compatible with <span class="mathjax-tex">$\sigma $</span>. If <span class="mathjax-tex">$\pi $</span> eventually loops at <span class="mathjax-tex">$v_1$</span> forever, then <span class="mathjax-tex">$s_\pi $</span> is the existential vertex 7 and the existential player wins the play since it satisfies both <span class="mathjax-tex">$\textsf{Fin}~a$</span> and <span class="mathjax-tex">$\textsf{Inf}~c$</span>. Any other play <span class="mathjax-tex">$\pi $</span> satisfies <span class="mathjax-tex">$\textsf{Inf}~a$</span>, <span class="mathjax-tex">$\textsf{Inf}~b$</span> and <span class="mathjax-tex">$\textsf{Inf}~c$</span> since all cycles that are compatible with <span class="mathjax-tex">$\sigma $</span> (excluding the loop at <span class="mathjax-tex">$v_1$</span>) contain at least one <i>a</i>-edge, at least one <i>b</i>-edge and also at least one <i>c</i>-edge that is prescribed by the strategy <span class="mathjax-tex">$\sigma $</span>. For these plays, <span class="mathjax-tex">$\rho _\pi $</span> eventually reaches the vertex 2. Since the system always plays singleton letters (so that <span class="mathjax-tex">$\pi $</span> in particular satisfies <span class="mathjax-tex">$\textsf{Fin}(b\wedge c)$</span>), the vertex 1 is not visited again by <span class="mathjax-tex">$\rho _\pi $</span>, once vertex 2 has been reached. Hence the dominating vertex for such plays is <span class="mathjax-tex">$s_\pi =2$</span>, an existential vertex.</p> <h3 class="c-article__sub-heading" id="Sec9"><span class="c-article-section__title-number">4.5 </span>Synthesis Extensions and Optimizations</h3><p>We have chosen to use safety-LTL as the safety part of the Safety-EL fragment to showcase the options opened by having symbolic algorithms for the analysis of very expressive liveness conditions. The crucial feature of the safety fragment is the ability to convert that part of the specification to a symbolic deterministic automaton. It is important to note that <i>every</i> fragment of LTL (or <span class="mathjax-tex">$\omega $</span>-regular in general) that can be easily converted to a symbolic deterministic automaton can be incorporated and handled with the same machinery. For example, it was suggested to extend the expressiveness of GR[1] by including deterministic automata in the safety part of the game and referring to their states in the liveness part [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Bloem, R., Jobstmann, B., Piterman, N., Pnueli, A., Sa’ar, Y.: Synthesis of reactive(1) designs. J. Comput. Syst. Sci. 78(3), 911–938 (2012). 
 https://doi.org/10.1016/j.jcss.2011.08.007
 
 " href="#ref-CR7" id="ref-link-section-d15224108e22765">7</a>]. Past LTL [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Lichtenstein, O., Pnueli, A., Zuck, L.D.: The glory of the past. In: Conference on Logics of Programs. Lecture Notes in Computer Science, vol. 193, pp. 196–218. Springer (1985). 
 https://doi.org/10.1007/3-540-15648-8_16
 
 " href="#ref-CR31" id="ref-link-section-d15224108e22768">31</a>] can be handled in the same way in that it is incorporated for GR[1] [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Bloem, R., Jobstmann, B., Piterman, N., Pnueli, A., Sa’ar, Y.: Synthesis of reactive(1) designs. J. Comput. Syst. Sci. 78(3), 911–938 (2012). 
 https://doi.org/10.1016/j.jcss.2011.08.007
 
 " href="#ref-CR7" id="ref-link-section-d15224108e22771">7</a>]. An extreme example is GR-EBR, where safety parts are allowed to use bounded future and pure past, which still allows the symbolic treatment [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Cimatti, A., Geatti, L., Gigante, N., Montanari, A., Tonetta, S.: Fairness, assumptions, and guarantees for extended bounded response ltl+p synthesis. Software and System Modeling (2023). 
 https://doi.org/10.1007/s10270-023-01122-4
 
 " href="#ref-CR15" id="ref-link-section-d15224108e22775">15</a>]. All of these alternatives can be incorporated in the safety part with no changes to our overall methodology. Unlike previous cases, if there is an easy translation to deterministic symbolic automata <i>with a non-trivial winning condition</i>, these can be incorporated as well with the EL part extended to handle their winning condition as well. We could consider also extensions to the liveness parts. For example, by using past LTL or reference to states of additional symbolic deterministic automata. The Boolean state formulas appearing as part of the EL condition can be replaced by formulas allowing one usage of the next operator, as in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Ehlers, R.: Generalized rabin(1) synthesis with applications to robust system synthesis. In: Third International Symposium on NASA Formal Methods. Lecture Notes in Computer Science, vol. 6617, pp. 101–115. Springer (2011). 
 https://doi.org/10.1007/978-3-642-20398-5_9
 
 " href="#ref-CR19" id="ref-link-section-d15224108e22781">19</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="Raman, V., Piterman, N., Finucane, C., Kress-Gazit, H.: Timing semantics for abstraction and execution of synthesized high-level robot control. IEEE Trans. Robotics 31(3), 591–604 (2015). 
 https://doi.org/10.1109/TRO.2015.2414134
 
 " href="#ref-CR39" id="ref-link-section-d15224108e22784">39</a>]. The generalization to handle transition-based EL games, which would be required in that case, rather than state-based EL games is straight-forward.</p><p>As the formulas we consider are conjunctions, optimizations can be applied to both conjuncts independently. This subsumes, for example, analyzing the winning region in a safety game prior to the full analysis [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Bansal, S., Giacomo, G.D., Stasio, A.D., Li, Y., Vardi, M.Y., Zhu, S.: Compositional safety LTL synthesis. In: 14th International Conference on Verified Software, Theories, Tools and Experiments. Lecture Notes in Computer Science, vol. 13800, pp. 1–19. Springer (2022). 
 https://doi.org/10.1007/978-3-031-25803-9_1
 
 " href="#ref-CR5" id="ref-link-section-d15224108e22790">5</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Bloem, R., Jobstmann, B., Piterman, N., Pnueli, A., Sa’ar, Y.: Synthesis of reactive(1) designs. J. Comput. Syst. Sci. 78(3), 911–938 (2012). 
 https://doi.org/10.1016/j.jcss.2011.08.007
 
 " href="#ref-CR7" id="ref-link-section-d15224108e22793">7</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Kugler, H., Segall, I.: Compositional synthesis of reactive systems from live sequence chart specifications. In: 15th International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Lecture Notes in Computer Science, vol. 5505, pp. 77–91. Springer (2009). 
 https://doi.org/10.1007/978-3-642-00768-2_9
 
 " href="#ref-CR29" id="ref-link-section-d15224108e22796">29</a>], reductions in the size of nondeterministic automata [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Duret-Lutz, A., Renault, E., Colange, M., Renkin, F., Aisse, A.G., Schlehuber-Caissier, P., Medioni, T., Martin, A., Dubois, J., Gillard, C., Lauko, H.: From spot 2.0 to spot 2.10: What’s new? In: 34th International Conference on Computer Aided Verification. Lecture Notes in Computer Science, vol. 13372, pp. 174–187. Springer (2022). 
 https://doi.org/10.1007/978-3-031-13188-2_9
 
 " href="#ref-CR17" id="ref-link-section-d15224108e22799">17</a>], or symbolic minimization of deterministic automata [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="D’Antoni, L., Veanes, M.: Minimization of symbolic automata. In: Symposium on Principles of Programming Languages (POPL). pp. 541–554. ACM (2014). 
 https://doi.org/10.1145/2535838.2535849
 
 " href="#ref-CR16" id="ref-link-section-d15224108e22802">16</a>].<sup><a href="#Fn4"><span class="u-visually-hidden">Footnote </span>4</a></sup> </p></div></div></section><section data-title="Conclusions and Future Work"><div class="c-article-section" id="Sec10-section"><h2 id="Sec10" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">5 </span>Conclusions and Future Work</h2><div class="c-article-section__content" id="Sec10-content"><p>We provide a symbolic algorithm to solve games with Emerson-Lei winning conditions. Our solution is based on an encoding of the Zielonka tree of the winning condition in a system of fixpoint equations. In case of known winning conditions, our algorithm recovers known algorithms and complexity results. As an application of this algorithm, we suggest an expressive fragment of LTL for which realizability can be reasoned about symbolically. Formulas in our fragment are conjunctions between an LTL safety formula and an Emerson-Lei liveness condition. This fragment is more general than, e.g., GR[1].</p><p>In the future, we believe that analysis of the Emerson-Lei part can reduce the size of Zielonka trees (and thus the symbolic algorithm). This can be done either through analysis and simplification of the LTL formula, e.g., [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="John, T., Jantsch, S., Baier, C., Klüppelholz, S.: Determinization and limit-determinization of Emerson-Lei automata. In: 19th International Symposium on Automated Technology for Verification and Analysis. Lecture Notes in Computer Science, vol. 12971, pp. 15–31. Springer (2021). 
 https://doi.org/10.1007/978-3-030-88885-5_2
 
 " href="#ref-CR26" id="ref-link-section-d15224108e22827">26</a>], by means of alternating-cycle decomposition [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Casares, A., Colcombet, T., Lehtinen, K.: On the size of good-for-games rabin automata and its link with the memory in muller games. In: Bojanczyk, M., Merelli, E., Woodruff, D.P. (eds.) International Colloquium on Automata, Languages, and Programming, ICALP 2022. LIPIcs, vol. 229, pp. 117:1–117:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022). 
 https://doi.org/10.4230/LIPIcs.ICALP.2022.117
 
 " href="#ref-CR12" id="ref-link-section-d15224108e22830">12</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Casares, A., Duret-Lutz, A., Meyer, K.J., Renkin, F., Sickert, S.: Practical applications of the alternating cycle decomposition. In: Fisman, D., Rosu, G. (eds.) Tools and Algorithms for the Construction and Analysis of Systems - 28th International Conference, TACAS 2022, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022, Munich, Germany, April 2-7, 2022, Proceedings, Part II. Lecture Notes in Computer Science, vol. 13244, pp. 99–117. Springer (2022). 
 https://doi.org/10.1007/978-3-030-99527-0_6
 
 " href="#ref-CR13" id="ref-link-section-d15224108e22833">13</a>], or by analyzing the semantic meaning of colors. We would also like to implement the proposed overall synthesis method.</p></div></div></section> </div> <section data-title="Notes" lang="en"><div class="c-article-section" id="notes-section"><h2 id="notes" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number"> </span>Notes</h2><div class="c-article-section__content" id="notes-content"><ol class="c-article-footnote c-article-footnote--listed"><li class="c-article-footnote--listed__item" id="Fn1"><span class="c-article-footnote--listed__index">1.</span><div class="c-article-footnote--listed__content"><p>Players choose from vertices where they lose, which explains the notation <span class="mathjax-tex">$T_\square $</span> and <span class="mathjax-tex">$T_\bigcirc $</span>.</p></div></li><li class="c-article-footnote--listed__item" id="Fn2"><span class="c-article-footnote--listed__index">2.</span><div class="c-article-footnote--listed__content"><p>Technically, <span class="mathjax-tex">$\rho _\exists $</span> contains primed variables and is not an LTL formula. This can be easily handled by using the next operator <i>X</i>. We thus ignore this issue.</p></div></li><li class="c-article-footnote--listed__item" id="Fn3"><span class="c-article-footnote--listed__index">3.</span><div class="c-article-footnote--listed__content"><p>We note that Bloem et al. consider more general games, where the environment also has an initial assertion and a transition relation. Our games are obtained from theirs by setting the initial assertion and the transition relation of the environment to true.</p></div></li><li class="c-article-footnote--listed__item" id="Fn4"><span class="c-article-footnote--listed__index">4.</span><div class="c-article-footnote--listed__content"><p>Notice that explicit minimization as done, e.g., in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Kupferman, O., Vardi, M.Y.: Model checking of safety properties. Formal methods in system design 19(3), 291–314 (2001). 
 https://doi.org/10.1023/A:1011254632723
 
 " href="#ref-CR30" id="ref-link-section-d15224108e22810">30</a>] would require to explicitly construct the potentially doubly exponential deterministic automaton, nullifying the entire effort to keep all analysis symbolic.</p></div></li></ol></div></div></section><div id="MagazineFulltextChapterBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 id="Bib1" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number"> </span>References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Armoni, R., Egorov, S., Fraer, R., Korchemny, D., Vardi, M.Y.: Efficient LTL compilation for sat-based model checking. 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