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Symbolic methods for solving such games need to be able to construct intricate arguments to establish the existence of winning strategies. Often, large problem instances require prohibitively complex arguments. Therefore, techniques that identify smaller and simpler sub-problems and exploit the respective results for the given game-solving task are highly desirable. In this paper, we propose the first such technique for infinite-state games. The main idea is to enhance symbolic game-solving with the results of localized attractor computations performed in sub-games. The crux of our approach lies in identifying useful sub-games by computing permissive winning strategy templates in finite abstractions of the infinite-state game. 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href="#Aff9">9</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Philippe-Heim" data-author-popup="auth-Philippe-Heim" data-corresp-id="c1">Philippe Heim<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-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-5433-8133"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-5433-8133</a></span><sup class="u-js-hide"><a href="#Aff10">10</a></sup>, </li><li class="c-article-author-list__item c-article-author-list__item--hide-small-screen"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Rayna-Dimitrova" data-author-popup="auth-Rayna-Dimitrova">Rayna Dimitrova</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0009-0006-2494-8690"><span class="u-visually-hidden">ORCID: </span>orcid.org/0009-0006-2494-8690</a></span><sup class="u-js-hide"><a href="#Aff10">10</a></sup> & </li><li class="c-article-author-list__show-more" aria-label="Show all 4 authors for this article" title="Show all 4 authors for this article">…</li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Satya_Prakash-Nayak" data-author-popup="auth-Satya_Prakash-Nayak">Satya Prakash Nayak</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-4407-8681"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-4407-8681</a></span><sup class="u-js-hide"><a href="#Aff9">9</a></sup> </li></ul><button aria-expanded="false" class="c-article-author-list__button"><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-down-medium"></use></svg><span>Show authors</span></button> <div class="app-overview-section__separator app-overview-section__book-series"> <div class="app-book-series-listing"> <div> <svg class="app-book-series-listing__icon" width="24" height="24" aria-hidden="true" focusable="false"><use href="#icon-eds-i-book-series-medium"></use></svg> </div> <div> <p data-test="series-link"> <span class="app-book-series-listing__description">Part of the book series:</span> <a href="https://www.springer.com/series/558" data-track="click" data-track-action="open book series" data-track-label="link">Lecture Notes in Computer Science</a> ((LNCS,volume 14683)) </p> </div> </div> </div> <div class="app-book-series-listing app-overview-section__conf-series"> <div> <svg class="app-book-series-listing__icon" width="24" height="24" aria-hidden="true" focusable="false"><use href="#icon-eds-i-conference-series-medium"></use></svg> </div> <div> <p class="app-book-series-listing__description" data-test="conference-series-link">Included in the following conference series:</p> <ul class="app-book-series-listing__list" data-component="data-book-show-more"> <li class="app-book-series-listing__item"><a href="https://link.springer.com/conference/cav" data-test="conference-series-link" data-track="click" data-track-action="open conference" data-track-label="link">International Conference on Computer Aided Verification</a></li> </ul> </div> </div> <div class="app-overview-section__separator" data-test="article-metrics"> <div id="altmetric-container"> <ul class="app-article-metrics-bar u-list-reset" data-test="article-metrics"> <li class="app-article-metrics-bar__item" data-test="access-count"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>1210 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> </ul> </div> </div> </div> </header> </div> <div data-article-body="true" 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>Infinite-state games are a commonly used model for the synthesis of reactive systems with unbounded data domains. Symbolic methods for solving such games need to be able to construct intricate arguments to establish the existence of winning strategies. Often, large problem instances require prohibitively complex arguments. Therefore, techniques that identify smaller and simpler sub-problems and exploit the respective results for the given game-solving task are highly desirable.</p><p>In this paper, we propose the first such technique for infinite-state games. The main idea is to enhance symbolic game-solving with the results of localized attractor computations performed in sub-games. The crux of our approach lies in identifying useful sub-games by computing permissive winning strategy templates in finite abstractions of the infinite-state game. The experimental evaluation of our method demonstrates that it outperforms existing techniques and is applicable to infinite-state games beyond the state of the art.</p></div></div></section><div class="c-article-section__content c-article-section__content--separator"><p>Authors are ordered randomly, denoted by P. Heim, R. Dimitrova and S. Prakash. The publicly verifiable record of the randomization is available at <a href="https://www.aeaweb.org/journals/policies/random-author-order/search?RandomAuthorsSearch%5Bsearch%5D=fKy1kA2NiEmL">www.aeaweb.org</a>.</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/10.1007/978-3-031-65633-0_7.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/978-3-031-57228-9?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/10.1007/978-3-031-57228-9_4?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1007/978-3-031-57228-9_4">Symbolic Solution of Emerson-Lei Games for Reactive Synthesis </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">© 2024</span> </div> </div> </article> </div> <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/978-3-030-85037-1?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/10.1007/978-3-030-85037-1_3?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/978-3-030-85037-1_3">Stubborn Set Reduction for Timed Reachability and Safety Games </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">© 2021</span> </div> </div> </article> </div> <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/978-3-319-08867-9?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/10.1007/978-3-319-08867-9_35?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/978-3-319-08867-9_35">Solving Games without Controllable Predecessor </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">© 2014</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1733921669, embedded_user: 'null' } }); </script> <div class="main-content"> <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><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figa_HTML.png?as=webp"><img aria-describedby="Figa" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figa_HTML.png" alt="figure a" loading="lazy" width="181" height="240"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-a-desc"></div></div></figure></div><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><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figb_HTML.png?as=webp"><img aria-describedby="Figb" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figb_HTML.png" alt="figure b" loading="lazy" width="181" height="240"></picture></div></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-b-desc"></div></div></figure></div><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><i>Games on graphs</i> provide an effective way to formalize the automatic synthesis of <i>correct-by-design</i> software in cyber-physical systems. The prime examples are algorithms that synthesize <i>control software</i> to ensure high-level logical specifications in response to external environmental behavior. These systems typically operate over unbounded data domains. For instance, in smart-home applications [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 35" title="Sylla, A.N., Louvel, M., Rutten, E., Delaval, G.: Modular and hierarchical discrete control for applications and middleware deployment in IoT and smart buildings. In: 2018 IEEE Conference on Control Technology and Applications (CCTA), pp. 1472–1479. IEEE (2018)" href="#ref-CR35" id="ref-link-section-d2892493e745">35</a>], they need to regulate real-valued quantities like room temperature and lighting in response to natural conditions, day-time, or energy costs. Also, unbounded data domains are valuable for over-approximating large countable numbers of products in a smart manufacturing line [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Gueye, S.M.K., Delaval, G., Rutten, E., Diguet, J.P.: Discrete and logico-numerical control for dynamic partial reconfigurable FPGA-based embedded systems: a case study. In: 2018 IEEE Conference on Control Technology and Applications (CCTA), pp. 1480–1487. IEEE (2018)" href="#ref-CR20" id="ref-link-section-d2892493e748">20</a>]. The tight integration of many specialized machines makes their efficient control challenging. Similar control synthesis problems occur in robotic warehouse systems [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Girija, P., Mareena, J., Fenny, J., Swapna, K., Kaewkhiaolueang, K.: Amazon robotic service (ARS) (2021)" href="#ref-CR18" id="ref-link-section-d2892493e751">18</a>], underwater robots for oil-pipe inspections [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Kelasidi, E., Liljebäck, P., Pettersen, K.Y., Gravdahl, J.T.: Innovation in underwater robots: biologically inspired swimming snake robots. IEEE Robotics Autom. Mag. 23(1), 44–62 (2016). 
 https://doi.org/10.1109/MRA.2015.2506121
 
 " href="#ref-CR25" id="ref-link-section-d2892493e755">25</a>], and electric smart-grid regulation [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Masselot, M., Patil, S., Zhabelova, G., Vyatkin, V.: Towards a formal model of protection functions for power distribution networks. In: IECON 2016-42nd Annual Conference of the IEEE Industrial Electronics Society, pp. 5302–5309. IEEE (2016)" href="#ref-CR29" id="ref-link-section-d2892493e758">29</a>].</p><p>Algorithmically, the outlined synthesis problems can be formalized via <i>infinite-state games</i> that model the ongoing interaction between the system (with its to-be-designed control software) and its environment over their <i>infinite</i> data domains. Due to their practical relevance and their challenging complexity, there has been an increasing interest in <i>automated techniques</i> for solving infinite-state games to obtain correct-by-design control implementations. As the game-solving problem is in general undecidable in the presence of infinite data domains, this problem is substantially more challenging than its finite-state counterpart.</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/10.1007/978-3-031-65633-0_7/figures/1" rel="nofollow"><picture><source type="image/webp" srcset="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Fig1_HTML.png?as=webp"><img aria-describedby="Fig1" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Fig1_HTML.png" alt="figure 1" loading="lazy" width="623" height="638"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-1-desc"><p>Schematic paper outline; contributions highlighted in blue. (Color figure online)</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/10.1007/978-3-031-65633-0_7/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>Within the literature<sup><a href="#Fn1"><span class="u-visually-hidden">Footnote </span>1</a></sup>, there are two prominent directions to attack this problem. One comprises <i>abstraction-based approaches</i>, where either the overall synthesis problem (e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Henzinger, T.A., Jhala, R., Majumdar, R.: Counterexample-guided control. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 886–902. Springer, Heidelberg (2003). 
 https://doi.org/10.1007/3-540-45061-0_69
 
 " href="#ref-CR23" id="ref-link-section-d2892493e806">23</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Walker, A., Ryzhyk, L.: Predicate abstraction for reactive synthesis. In: Formal Methods in Computer-Aided Design, FMCAD 2014, Lausanne, Switzerland, October 21–24, 2014. pp. 219–226. IEEE (2014). 
 https://doi.org/10.1109/FMCAD.2014.6987617
 
 " href="#ref-CR38" id="ref-link-section-d2892493e809">38</a>]) or the specification (e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Choi, W., Finkbeiner, B., Piskac, R., Santolucito, M.: Can reactive synthesis and syntax-guided synthesis be friends? In: Jhala, R., Dillig, I. (eds.) PLDI ’22: 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation, San Diego, CA, USA, 13–17 June, 2022, pp. 229–243. ACM (2022). 
 https://doi.org/10.1145/3519939.3523429
 
 " href="#ref-CR8" id="ref-link-section-d2892493e812">8</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Finkbeiner, B., Klein, F., Piskac, R., Santolucito, M.: Temporal stream logic: synthesis beyond the bools. In: Dillig, I., Tasiran, S. (eds.) CAV 2019. LNCS, vol. 11561, pp. 609–629. Springer, Cham (2019). 
 https://doi.org/10.1007/978-3-030-25540-4_35
 
 " href="#ref-CR14" id="ref-link-section-d2892493e816">14</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Maderbacher, B., Bloem, R.: Reactive synthesis modulo theories using abstraction refinement. In: Griggio, A., Rungta, N. (eds.) 22nd Formal Methods in Computer-Aided Design, FMCAD 2022, Trento, Italy, October 17-21, 2022, pp. 315–324. IEEE (2022). 
 https://doi.org/10.34727/2022/ISBN.978-3-85448-053-2_38
 
 " href="#ref-CR27" id="ref-link-section-d2892493e819">27</a>]) are abstracted, resulting in a finite-state game, to which classical techniques apply. The other one are <i>constraint-based techniques</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Faella, M., Parlato, G.: Reachability games modulo theories with a bounded safety player. In: Proceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence. AAAI’23/IAAI’23/EAAI’23. AAAI Press (2023). 
 https://doi.org/10.1609/aaai.v37i5.25779
 
 " href="#ref-CR9" id="ref-link-section-d2892493e825">9</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Farzan, A., Kincaid, Z.: Strategy synthesis for linear arithmetic games. Proc. ACM Program. Lang. 2(POPL), 61:1-61:30 (2018). 
 https://doi.org/10.1145/3158149
 
 " href="#ref-CR10" id="ref-link-section-d2892493e828">10</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Samuel, S., D’Souza, D., Komondoor, R.: Gensys: a scalable fixed-point engine for maximal controller synthesis over infinite state spaces. In: Spinellis, D., Gousios, G., Chechik, M., Penta, M.D. (eds.) ESEC/FSE ’21: 29th ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineering, Athens, Greece, August 23–28, 2021, pp. 1585–1589. ACM (2021). 
 https://doi.org/10.1145/3468264.3473126
 
 " href="#ref-CR32" id="ref-link-section-d2892493e831">32</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Samuel, S., D’Souza, D., Komondoor, R.: Symbolic fixpoint algorithms for logical LTL games. In: 38th IEEE/ACM International Conference on Automated Software Engineering, ASE 2023, Luxembourg, September 11–15, 2023, pp. 698–709. IEEE (2023). 
 https://doi.org/10.1109/ASE56229.2023.00212
 
 " href="#ref-CR33" id="ref-link-section-d2892493e835">33</a>], that work directly on a symbolic representation of the infinite-state game. Due to the undecidability of the overall synthesis problem, both categories are inherently constrained. While abstraction-based approaches are limited by the abstraction domain they employ, constraint-based techniques typically diverge due to non-terminating fixpoint computations.</p><p>To address these limitations, a recent constraint-based technique called <i>attractor acceleration</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e844">21</a>] employs <i>ranking arguments</i> to improve the convergence of symbolic game-solving algorithms. While this technique has shown superior performance over the state-of-the art, the utilized ranking arguments become complex, and thus difficult to find, as the size of the games increases. This makes the approach from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e850">21</a>] infeasible in such cases, often resulting in divergence in larger and more complex games.</p><p>In this paper, we propose an approach to overcoming the above limitation and thus extending the applicability of synthesis over infinite state games towards realistic applications. The key idea is to utilize efficient abstraction-based pre-computations that <i>localize</i> attractor computations to <i>small and useful sub-games</i>. In that way, acceleration can be applied locally to small sub-games, and the results utilized by the procedure for solving the global game. This often avoids the computationally inefficient attractor acceleration over the complete game. To <i>guide</i> the identification of useful sub-games, our approach computes <i>strategy templates</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Anand, A., Nayak, S.P., Schmuck, A.: Synthesizing permissive winning strategy templates for parity games. In: Enea, C., Lal, A. (eds.) CAV 2023, Part I. LNCS, vol. 13964, pp. 436–458. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-37706-8_22
 
 " href="#ref-CR2" id="ref-link-section-d2892493e869">2</a>] – a concise representation of a possibly infinite number of winning strategies – in finite abstractions of the infinite-state game. Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig1">1</a> shows an overview of our method which also serves as an outline of the paper.</p><p>Our experimental evaluation demonstrates the superior performance of our approach compared to the state of the art. Existing tools fail on almost all benchmarks, while our implementation terminates within minutes.</p><p>To build up more intuition, we illustrate the main idea of our approach with the following example, which will also serve as our running example.</p><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-2" data-title="Fig. 2."><figure><figcaption><b id="Fig2" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 2.</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/10.1007/978-3-031-65633-0_7/figures/2" rel="nofollow"><picture><img aria-describedby="Fig2" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Fig2_HTML.png" alt="figure 2" loading="lazy" width="685" height="179"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-2-desc"><p>A reactive program game for a sample-collecting robot with locations <span class="mathjax-tex">${ base , move , mine }$</span>, integer-type program variables <span class="mathjax-tex">${ pos , done , req , samp }$</span> and input variable <span class="mathjax-tex">${ inpReq }$</span>. We use the following abbreviations: <span class="mathjax-tex">$ enterBase ~\widehat{=}~( pos = 12 \wedge done = 1)$</span>, <span class="mathjax-tex">$ atMine ~\widehat{=}~( pos = 23)$</span>, <span class="mathjax-tex">$ haveSamples ~\widehat{=}~(a > 0 \vee b > 0)$</span>, <span class="mathjax-tex">$ enough ~\widehat{=}~ samp \ge req $</span>, <span class="mathjax-tex">$ sampleA ~\widehat{=}~( samp := samp + a)$</span>, <span class="mathjax-tex">$ sampleB ~\widehat{=}~( samp := samp + b)$</span>, and <span class="mathjax-tex">$ sampleS ~\widehat{=}~( samp := samp + 1)$</span>. In each round of the game, the environment chooses a value for the input <span class="mathjax-tex">${ inpReq }$</span>. Based on guards over program variables and inputs, the game transitions to a black square. The system then chooses one of the possible updates to the program variables, thus determining the next location.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-031-65633-0_7/figures/2" data-track-dest="link:Figure2 Full size image" aria-label="Full size image figure 2" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar1">Example 1</h3> <p>Figure <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig2">2</a> shows a reactive program game for a sample-collecting robot. The robot moves along tracks, and its position is determined by the integer program variable <span class="mathjax-tex">$ pos $</span>. The robot remains in location <span class="mathjax-tex">$ base $</span> until prompted by the environment to collect <span class="mathjax-tex">$ inpReq $</span> many samples. It cannot return to <span class="mathjax-tex">$ base $</span> until the required samples are collected, as enforced by the variable <span class="mathjax-tex">$ done $</span>. From the right position, it can enter the <span class="mathjax-tex">$ mine $</span>, where it must stay and collect samples from two sites, <i>a</i> and <i>b</i>. However, it has to choose the correct site in each iteration, as they might not have samples all the time (if both do not have samples, it can get one sample itself). Once enough samples are collected, the robot can return to <span class="mathjax-tex">$ base $</span>. The requirement on the robot’s strategy is to be at <i>base</i> infinitely often.</p> <p><i>Attractor acceleration</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e1744">21</a>] uses ranking arguments to establish that by iterating some strategy an unbounded number of times through some location, a player in the game can enforce reaching a set of target states. In this example, to reach <span class="mathjax-tex">$ samp \ge req $</span> in location <span class="mathjax-tex">$ mine $</span> (the target) the robot can iteratively increase the value of <span class="mathjax-tex">$ samp $</span> by choosing the right updates (the iterated strategy). This works, since if <span class="mathjax-tex">$ samp $</span> is increased repeatedly, eventually <span class="mathjax-tex">$ samp \ge req $</span> will hold (the ranking argument). Establishing the existence of the iterated strategy (i.e. the robot can increment <span class="mathjax-tex">$ samp $</span>) is a game-solving problem, since the behavior of the robot is influenced by the environment. This game-solving problem potentially considers the whole game, since the iterated strategy is not known a priori. In addition, identifying locations where acceleration can be applied and finding the right ranking arguments is challenging. This impacts the scalability and applicability of acceleration, making it infeasible for large games.</p> <p>Consequently, our method aims to identify <i>small and useful sub-games</i> and <i>cache the results obtained by solving these sub-games</i>. In Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar1">1</a>, a useful sub-game would be the game restricted to the <span class="mathjax-tex">$ mine $</span> location with the target state <span class="mathjax-tex">$ samp \ge req $</span>. Applying the acceleration technique to this sub-game, provides the ranking argument described earlier. These cached results are then utilized to enhance the symbolic game-solving procedure for the entire game.</p> <p>To identify these small and useful sub-games, we use <i>permissive strategy templates</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Anand, A., Nayak, S.P., Schmuck, A.: Synthesizing permissive winning strategy templates for parity games. In: Enea, C., Lal, A. (eds.) CAV 2023, Part I. LNCS, vol. 13964, pp. 436–458. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-37706-8_22
 
 " href="#ref-CR2" id="ref-link-section-d2892493e1976">2</a>] in finite-state abstracted games. They describe a potentially infinite set of winning strategies using local conditions on the transitions of the game. These local conditions (in the abstract game) provide guidance about local behavior in the solution of the infinite-state game without solving it. This local behavior (e.g. incrementing <span class="mathjax-tex">$ samp $</span> in <span class="mathjax-tex">$ mine $</span>) induces our sub-games.</p> </div></div></section><section data-title="Preliminaries"><div class="c-article-section" id="Sec2-section"><h2 id="Sec2" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">2 </span>Preliminaries</h2><div class="c-article-section__content" id="Sec2-content"><p><b>Sequences and First-Order Logic.</b> For a set <i>V</i>, <span class="mathjax-tex">$V^*$</span> and <span class="mathjax-tex">$V^\omega $</span> denote the sets of finite, respectively infinite, sequences of elements of <i>V</i>, and let <span class="mathjax-tex">$V^\infty = V^* \cup V^\omega $</span>. For <span class="mathjax-tex">$\pi \in V^\infty $</span>, we denote with <span class="mathjax-tex">$|\pi | \in \mathbb {N}\cup \{ \infty \}$</span> the length of <span class="mathjax-tex">$\pi $</span>, and define <span class="mathjax-tex">$ dom (\pi ):=\{0,\ldots , |\pi | - 1\}$</span>. For <span class="mathjax-tex">$\pi = v_0v_1\ldots \in V^\infty $</span> and <span class="mathjax-tex">$i, j\in dom (\pi )$</span> with <span class="mathjax-tex">$i<j$</span>, we define <span class="mathjax-tex">$\pi [i]:=v_i$</span> and <span class="mathjax-tex">$\pi [i,j]:=v_i\ldots v_j$</span>. <span class="mathjax-tex">$ last (\pi )$</span> is the last element of a finite sequence <span class="mathjax-tex">$\pi $</span>.</p><p>Let <span class="mathjax-tex">$\mathcal {V}$</span> be the set of all values of arbitrary types, <span class="mathjax-tex">$ Vars $</span> be the set of all variables, <span class="mathjax-tex">$\mathcal {F}$</span> be the set of all functions, and <span class="mathjax-tex">$\varSigma _F$</span> be the set of all function symbols. Let <span class="mathjax-tex">$\mathcal {T}_F$</span> be the set of all function terms defined by the grammar <span class="mathjax-tex">$\mathcal {T}_F\ni \tau _f :\,\!:= x \,|\, f(\tau _f^1, \dots \tau _f^n)$</span> for <span class="mathjax-tex">$f \in \varSigma _F$</span> and <span class="mathjax-tex">$x \in Vars $</span>. A function <span class="mathjax-tex">$\nu : Vars \rightarrow \mathcal {V}$</span> is called an <i>assignment</i>. The set of all assignments over variables <span class="mathjax-tex">$X \subseteq Vars $</span> is denoted as <span class="mathjax-tex">$ Assignments (X)$</span>. We denote the combination of two assignments <span class="mathjax-tex">$\nu ', \nu ''$</span> over disjoint sets of variables by <span class="mathjax-tex">$\nu ' \uplus \nu ''$</span>. A function <span class="mathjax-tex">$\mathcal {I}: \varSigma _F\rightarrow \mathcal {F}$</span> is called an <i>interpretation</i>. The set of all interpretations is denoted as <span class="mathjax-tex">$ Interpretations (\varSigma _F)$</span>. The evaluation of function terms <span class="mathjax-tex">$\chi _{\nu ,\mathcal {I}}: \mathcal {T}_F\rightarrow \mathcal {V}$</span> is defined by <span class="mathjax-tex">$\chi _{\nu ,\mathcal {I}}(x) := \nu (x)$</span> for <span class="mathjax-tex">$x \in Vars $</span>, <span class="mathjax-tex">$\chi _{\nu ,\mathcal {I}}(f(\tau _0, \dots \tau _n)) := \mathcal {I}(f)(\chi _{\nu ,\mathcal {I}}(\tau _0), \dots \chi _{\nu ,\mathcal {I}}(\tau _n))$</span> for <span class="mathjax-tex">$f \in \varSigma _F$</span> and <span class="mathjax-tex">$\tau _0, \dots \tau _n \in \mathcal {T}_F$</span>. We denote the set of all first-order formulas as <span class="mathjax-tex">$ FOL $</span> and by <span class="mathjax-tex">$ QF $</span> the set of all quantifier-free formulas in <span class="mathjax-tex">$ FOL $</span>. Let <span class="mathjax-tex">$\varphi $</span> be a formula and <span class="mathjax-tex">$X = \{x_1,\ldots ,x_n\} \subseteq Vars $</span> be a set of variables. We write <span class="mathjax-tex">$\varphi (X)$</span> to denote that the free variables of <span class="mathjax-tex">$\varphi $</span> are a subset of <i>X</i>. We also denote with <span class="mathjax-tex">$ FOL (X)$</span> and <span class="mathjax-tex">$ QF (X)$</span> the set of formulas (respectively quantifier-free formulas) whose free variables belong to <i>X</i>. For a quantifier <span class="mathjax-tex">$Q \in \{\exists , \forall \}$</span>, we write <span class="mathjax-tex">$Q X.\varphi $</span> as a shortcut for <span class="mathjax-tex">$Q x_1.\ldots Q x_n.\varphi $</span>. We denote with <span class="mathjax-tex">$\models : Assignments ( Vars ) \times Interpretations (\varSigma _F) \times FOL $</span> the entailment of first-order logic formulas. A <i>first-order theory</i> <span class="mathjax-tex">$T \subseteq Interpretations (\varSigma _F)$</span> with <span class="mathjax-tex">$T \ne \emptyset $</span> restricts the possible interpretations of function and predicate symbols. Given a theory <i>T</i>, for a formula <span class="mathjax-tex">$\varphi (X)$</span> and assignment <span class="mathjax-tex">$\nu \in Assignments (X)$</span> we define that <span class="mathjax-tex">$\nu \models _{T} \varphi $</span> if and only if <span class="mathjax-tex">$\nu , \mathcal {I}\models \varphi $</span> for all <span class="mathjax-tex">$\mathcal {I}\in T$</span>.</p><p>For exposition on first-order logic and first-order theories, see c.f. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Bradley, A.R., Manna, Z.: The Calculus of Computation - Decision Procedures with Applications to Verification. Springer, Heidelberg (2007). 
 https://doi.org/10.1007/978-3-540-74113-8
 
 " href="#ref-CR7" id="ref-link-section-d2892493e4244">7</a>].</p><p><b>Two-Player Graph Games.</b> A <i>game graph</i> is a tuple <span class="mathjax-tex">$G = (V,V_ Env ,V_ Sys ,\rho )$</span> where <span class="mathjax-tex">$V = V_ Env \uplus V_ Sys $</span> are the vertices, partitioned between the environment player (<i>player</i> <span class="mathjax-tex">$ Env $</span>) and the system player (<i>player</i> <span class="mathjax-tex">$ Sys $</span>), and <span class="mathjax-tex">$\rho \subseteq (V_ Env \times V_ Sys ) \cup (V_ Sys \times V_ Env )$</span> is the <i>transition relation</i>. A <i>play</i> in <i>G</i> is a sequence <span class="mathjax-tex">$\pi \in V^\infty $</span> where <span class="mathjax-tex">$(\pi [i],\pi [i+1])\in \rho $</span> for all <span class="mathjax-tex">$i \in dom (\pi )$</span>, and if <span class="mathjax-tex">$\pi $</span> is finite then <span class="mathjax-tex">$ last (\pi )$</span> is a dead-end.</p><p>For <span class="mathjax-tex">$p = Sys $</span> (or <span class="mathjax-tex">$ Env $</span>) we define <span class="mathjax-tex">$1 - p := Env $</span> (respectively <span class="mathjax-tex">$ Sys $</span>). A <i>strategy for player</i> <i>p</i> is a partial function <span class="mathjax-tex">$\sigma : V^*V_{p} \rightarrow V$</span> where <span class="mathjax-tex">$\sigma (\pi \cdot v) = v'$</span> implies <span class="mathjax-tex">$(v,v') \in \rho $</span> and <span class="mathjax-tex">$\sigma $</span> is defined for all <span class="mathjax-tex">$\pi \cdot v \in V^*V_{p}$</span> unless <i>v</i> is a dead-end. <span class="mathjax-tex">$ Strat _{p}(G)$</span> denotes the set of all strategies for player <i>p</i> in <i>G</i>. A play <span class="mathjax-tex">$\pi $</span> is <i>consistent with</i> <span class="mathjax-tex">$\sigma $</span> <i>for player</i> <i>p</i> if <span class="mathjax-tex">$\pi [i+1] = \sigma (\pi [0,i])$</span> for every <span class="mathjax-tex">$i \in dom (\pi )$</span> where <span class="mathjax-tex">$\pi [i] \in V_p$</span>. <span class="mathjax-tex">$ Plays _G(v,\sigma )$</span> is the set of all plays in <i>G</i> starting in <i>v</i> and consistent with strategy <span class="mathjax-tex">$\sigma $</span>.</p><p>An <i>objective</i> in <i>G</i> is a set <span class="mathjax-tex">$\varOmega \subseteq V^\infty $</span>. A <i>two-player turn-based game</i> is a pair <span class="mathjax-tex">$(G,\varOmega )$</span>, where <i>G</i> is a game graph and <span class="mathjax-tex">$\varOmega $</span> is an objective for player <span class="mathjax-tex">$ Sys $</span>. A sequence <span class="mathjax-tex">$\pi \in V^\infty $</span> is <i>winning for player</i> <span class="mathjax-tex">$ Sys $</span> if and only if <span class="mathjax-tex">$\pi \in \varOmega $</span>, and is <i>winning for player</i> <span class="mathjax-tex">$ Env $</span> otherwise. The definitions of different types of common objectives can be found in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Schmuck, A.K., Heim, P., Dimitrova, R., Nayak, S.P.: Localized attractor computations for infinite-state games (full version) (2024). 
 https://doi.org/10.48550/ARXIV.2405.09281
 
 " href="#ref-CR34" id="ref-link-section-d2892493e5511">34</a>]. The <i>winning region</i> <span class="mathjax-tex">$W_p(G,\varOmega )$</span> <i>of player</i> <i>p</i> <i>in</i> <span class="mathjax-tex">$(G,\varOmega )$</span> is the set of all vertices <i>v</i> from which player <i>p</i> has a strategy <span class="mathjax-tex">$\sigma $</span> such that every play in <span class="mathjax-tex">$ Plays _G(v,\sigma )$</span> is winning for player <i>p</i>. A strategy <span class="mathjax-tex">$\sigma $</span> of player <i>p</i> is <i>winning</i> if for every <span class="mathjax-tex">$v\in W_p(G,\varOmega )$</span>, every play in <span class="mathjax-tex">$ Plays _G(v,\sigma )$</span> is winning for player <i>p</i>.</p><p><b>Acceleration-Based Solving of Infinite-State Games.</b> We represent infinite-state games using the same formalism as [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e5798">21</a>], called reactive program games. Intuitively, reactive program games describe symbolically, using <span class="mathjax-tex">$ FOL $</span> formulas and terms, the possible interactions between the system player and the environment player in two-player games over infinite data domains.</p> <h3 class="c-article__sub-heading" id="FPar2">Definition 1</h3> <p><b>(Reactive Program Game Structure</b> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e5831">21</a>]<b>).</b> A <i>reactive program game structure</i> is a tuple <span class="mathjax-tex">$\mathcal G = (T,\mathbb {I}, \mathbb {X}, L, Inv ,\delta )$</span> with the following components. <i>T</i> is a first-order theory. <span class="mathjax-tex">$\mathbb {I}\subseteq Vars $</span> is a finite set of <i>input variables</i>. <span class="mathjax-tex">$\mathbb {X}\subseteq Vars $</span> is a finite set of <i>program variables</i> where <span class="mathjax-tex">$\mathbb {I}\cap \mathbb {X}= \emptyset $</span>. <i>L</i> is a finite set of <i>game locations</i>. <span class="mathjax-tex">$ Inv : L \rightarrow FOL (\mathbb {X})$</span> maps each location to a <i>location invariant</i>. <span class="mathjax-tex">$\delta \subseteq L \times QF (\mathbb {X}\cup \mathbb {I}) \times (\mathbb {X}\rightarrow \mathcal {T}_F) \times L$</span> is a finite <i>symbolic transition relation</i> where </p><ol class="u-list-style-none"> <li> <span class="u-custom-list-number">(1)</span> <p>for every <span class="mathjax-tex">$l \in L$</span> the set of <i>outgoing transition guards</i> <span class="mathjax-tex">$ Guards (l) := \{ g \mid \exists u, l'.~(l, g, u , l') \in \delta \}$</span> is such that <span class="mathjax-tex">$\bigvee _{g \in Guards (l)} g \equiv _{T} \top $</span>, and for all <span class="mathjax-tex">$g_1, g_2 \in Guards (l)$</span> with <span class="mathjax-tex">$g_1 \ne g_2$</span> it holds that <span class="mathjax-tex">$g_1 \wedge g_2 \equiv _{T} \bot $</span>,</p> </li> <li> <span class="u-custom-list-number">(2)</span> <p>for all <span class="mathjax-tex">$l,g, u, l_1, l_2$</span>, if <span class="mathjax-tex">$(l, g, u, l_1) \in \delta $</span> and <span class="mathjax-tex">$(l, g, u, l_2) \in \delta $</span>, then <span class="mathjax-tex">$l_1 = l_2$</span>, and</p> </li> <li> <span class="u-custom-list-number">(3)</span> <p>for every <span class="mathjax-tex">$l \in L$</span> and <span class="mathjax-tex">$\textbf{x} \in Assignments (\mathbb {X})$</span> such that <span class="mathjax-tex">$\textbf{x} \models _{T} Inv (l)$</span>, and <span class="mathjax-tex">$\textbf{i} \in Assignments (\mathbb {I})$</span>, there exist a transition <span class="mathjax-tex">$(l, g, u, l') \in \delta $</span> such that <span class="mathjax-tex">$\textbf{x} \uplus \textbf{i}\models _{T} g$</span> and <span class="mathjax-tex">$\mathbf {x'} \models _{T} Inv (l')$</span> where <span class="mathjax-tex">$\textbf{x}'(x) = \chi _{\textbf{x}\uplus \textbf{i},\mathcal {I}}(u(x))$</span> for all <span class="mathjax-tex">$x \in \mathbb {X}$</span> and <span class="mathjax-tex">$\mathcal {I}\in T$</span>, and</p> </li> <li> <span class="u-custom-list-number">(4)</span> <p>for every <span class="mathjax-tex">$(l, g, u, l') \in \delta $</span>, <span class="mathjax-tex">$f \in \varSigma _F(u)$</span>, <span class="mathjax-tex">$\mathcal {I}_1, \mathcal {I}_2 \in T$</span> it holds that <span class="mathjax-tex">$\mathcal {I}_1(f) = \mathcal {I}_2(f)$</span>.</p> </li> </ol> <p>The requirements on <span class="mathjax-tex">$\delta $</span> imply for each <span class="mathjax-tex">$l\in L$</span> that: (1) the guards in <span class="mathjax-tex">$ Guards (l)$</span> partition the set <span class="mathjax-tex">$ Assignments (\mathbb {X}\cup \mathbb {I})$</span>, (2) each pair of <span class="mathjax-tex">$g \in Guards (l)$</span> and update <i>u</i> can label at most one outgoing transition from <i>l</i>, (3) if there is an assignment satisfying the invariant at <i>l</i>, then for every input assignment there is a possible transition, and (4) the theory <i>T</i> determines the meaning of functions in updates uniquely. Given locations <span class="mathjax-tex">$l,l' \in L$</span>, we define <span class="mathjax-tex">$ Labels (l,l'):=\{(g,u) \mid (l, g, u, l') \in \delta \}$</span> as the set of <i>labels on transitions from</i> <i>l</i> <i>to</i> <span class="mathjax-tex">$l'$</span>. We define as <span class="mathjax-tex">$\textsf{RPGS}$</span> the set of all reactive program game structures. The semantics of the reactive program game structure <span class="mathjax-tex">$\mathcal G$</span> is a (possibly infinite) game graph defined as follows.</p> <h3 class="c-article__sub-heading" id="FPar3">Definition 2</h3> <p><b>(Semantics of Reactive Program Game Structures).</b> Let <span class="mathjax-tex">$\mathcal G = (T,\mathbb {I}, \mathbb {X}, L, Inv , \delta )$</span> be a reactive program game structure. The semantics of <span class="mathjax-tex">$\mathcal G$</span> is the game graph <span class="mathjax-tex">$\llbracket \mathcal \rrbracket {G} = (\mathcal {S}, \mathcal {S}_ Env ,\mathcal {S}_ Sys ,\rho )$</span> where <span class="mathjax-tex">$\mathcal {S}:= \mathcal {S}_ Env \uplus \mathcal {S}_ Sys $</span> and</p><ul class="u-list-style-dash"> <li> <p><span class="mathjax-tex">$\mathcal {S}_ Env := \{ (l,\textbf{x}) \in L \times Assignments (\mathbb {X}) \mid \textbf{x} \models _{T} Inv (l)\}$</span>;</p> </li> <li> <p><span class="mathjax-tex">$\mathcal {S}_ Sys :=\mathcal {S}_ Env \times Assignments (\mathbb {I})$</span>;</p> </li> <li> <p><span class="mathjax-tex">$\rho \subseteq (\mathcal {S}_ Env \times \mathcal {S}_ Sys ) \cup (\mathcal {S}_ Sys \times \mathcal {S}_ Env )$</span> is the smallest relation such that</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$(s,(s,\textbf{i})) \in \rho $</span> for every <span class="mathjax-tex">$s \in \mathcal {S}_ Env $</span> and <span class="mathjax-tex">$\textbf{i} \in Assignments (\mathbb {I})$</span>,</p> </li> <li> <p><span class="mathjax-tex">$(((l,\textbf{x}), \textbf{i}),(l',\textbf{x}')) \in \rho $</span> iff <span class="mathjax-tex">$\textbf{x}' \models _{T} Inv (l')$</span> and there exists <span class="mathjax-tex">$(g,u) \in Labels (l,l')$</span> such that <span class="mathjax-tex">$\textbf{x}\uplus \textbf{i} \models _{T} g$</span>, <span class="mathjax-tex">$\textbf{x}'(x) = \chi _{\textbf{x}\uplus \textbf{i}, \mathcal {I}}(u(x))$</span> for every <span class="mathjax-tex">$x \in \mathbb {X}$</span> and <span class="mathjax-tex">$\mathcal {I}\in T$</span>.</p> </li> </ul> </li> </ul> <p>Note that this semantics differs from the original one in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e8825">21</a>] where the semantic game structure is not split into environment and system states. We do that in order to consistently use the notion of a game graph. Both semantics are equivalent. We refer to the vertices of <span class="mathjax-tex">$\llbracket G\rrbracket $</span> as <i>states</i>. We define the function <span class="mathjax-tex">$ loc : \mathcal {S}\rightarrow L$</span> where <span class="mathjax-tex">$ loc (s) := l$</span> for any <span class="mathjax-tex">$s= (l,\textbf{x}) \in \mathcal {S}_ Env $</span> and any <span class="mathjax-tex">$s = ((l,\textbf{x}),\textbf{i}) \in \mathcal {S}_ Sys $</span>. By abusing notation, we extend the function <span class="mathjax-tex">$ loc $</span> to sequences of states, defining <span class="mathjax-tex">$ loc : \mathcal {S}^\infty \rightarrow L^\infty $</span> where <span class="mathjax-tex">$ loc (\pi ) = l_0l_1l_2\ldots $</span> iff <span class="mathjax-tex">$ loc (\pi [i]) = l_i$</span> for all <span class="mathjax-tex">$i \in dom (\pi )$</span>. For simplicity of the notation, we write <span class="mathjax-tex">$W_p(\mathcal {G},\varOmega )$</span> instead of <span class="mathjax-tex">$W_p(\llbracket \mathcal {G}\rrbracket ,\varOmega )$</span>. We represent and manipulate possibly infinite sets of states symbolically, using formulas in <span class="mathjax-tex">$ FOL $</span>(<span class="mathjax-tex">$\mathbb {X}$</span>) to describe sets of assignments to the variables in <span class="mathjax-tex">$\mathbb {X}$</span>. Our <i>symbolic domain</i> <span class="mathjax-tex">$\mathcal {D}:= L \rightarrow FOL (\mathbb {X})$</span> is the set of functions mapping locations to formulas in <span class="mathjax-tex">$ FOL $</span>(<span class="mathjax-tex">$\mathbb {X}$</span>). An element <span class="mathjax-tex">$d \in \mathcal {D}$</span> represents the states <span class="mathjax-tex">$\llbracket d\rrbracket : = \{((l,\textbf{x}) \in \mathcal {S}\mid \textbf{x} \models _{T} d(l)\}.$</span> With <span class="mathjax-tex">$\{ l_1 \mapsto \varphi _1, \dots , l_n \mapsto \varphi _n \}$</span> we denote <span class="mathjax-tex">$d \in \mathcal {D}$</span> s.t. <span class="mathjax-tex">$d(l_i) = \varphi _i$</span> and <span class="mathjax-tex">$d(l) = \bot $</span> for <span class="mathjax-tex">$l \not \in \{l_1, \dots , l_n\}$</span>. For brevity, we sometimes refer to elements of <span class="mathjax-tex">$\mathcal {D}$</span> as sets of states.</p><p>Note that the elements of the symbolic domain <span class="mathjax-tex">$\mathcal {D}$</span> represent subsets of <span class="mathjax-tex">$\mathcal {S}_ Env $</span>, i.e., sets of environment states. Environment states are pairs of location and valuation of the program variables. The system states, on the other hand, correspond to intermediate configurations that additionally store the current input from the environment. This input is not stored further on (unless assigned to program variables). Thus, we restrict the symbolic domain to environment states.</p><p><i>Solving Reactive Program Games.</i> We consider <i>objectives defined over the locations</i> of a reactive program game structure <span class="mathjax-tex">$\mathcal {G}$</span>. That is, we require that if <span class="mathjax-tex">$\pi ',\pi '' \in \mathcal {S}^\infty $</span> are such that <span class="mathjax-tex">$ loc (\pi ') = loc (\pi '')$</span>, then <span class="mathjax-tex">$\pi ' \in \varOmega $</span> iff <span class="mathjax-tex">$\pi '' \in \varOmega $</span>. We consider the problem of solving reactive program games. Given <span class="mathjax-tex">$\mathcal {G}$</span> and an objective <span class="mathjax-tex">$\varOmega $</span> for Player <span class="mathjax-tex">$ Sys $</span> defined over the locations of <span class="mathjax-tex">$\mathcal {G}$</span>, we want to compute <span class="mathjax-tex">$W_{ Sys }(\llbracket \mathcal \rrbracket {G},\varOmega )$</span>.</p><p><i>Attractor Computation and Acceleration.</i> A core building block of many algorithms for solving two-player games is the computation of <i>attractors</i>. Intuitively, an attractor is the set of states from which a given player <i>p</i> can enforce reaching a given set of target states no matter what the other player does. Formally, for a reactive program game structure <span class="mathjax-tex">$\mathcal {G}$</span>, and <span class="mathjax-tex">$R \subseteq \mathcal {S}$</span> the <i>player-</i><i>p</i> <i>attractor for</i> <i>R</i> is</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ Attr _{\llbracket \mathcal \rrbracket {G},p}(R) := \{s \in \mathcal {S}\mid \exists \sigma \in Strat _{p}(\llbracket \mathcal \rrbracket {G}).\forall \pi \in Plays _{\llbracket \mathcal {G}\rrbracket }(s,\sigma ).\exists n\in \mathbb {N}.\;\pi [n]\in R\}.$$</span></div></div><p>In this work, we are concerned with the symbolic computation of attractors in reactive program games. Attractors in reactive program games are computed using the so-called <i>enforceable predecessor operator</i> over the symbolic domain <span class="mathjax-tex">$\mathcal {D}$</span>. For <span class="mathjax-tex">$d \in \mathcal {D}$</span>, <span class="mathjax-tex">$ CPre _{\mathcal {G},p}(d) \in \mathcal {D}$</span> represents the states from which player <i>p</i> can enforce reaching <span class="mathjax-tex">$\llbracket d\rrbracket $</span> in one step in <span class="mathjax-tex">$\mathcal {G}$</span> (i.e. one move by each player). More precisely,</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{ll} \llbracket CPre _{\mathcal {G}, Sys }(d)\rrbracket &{}= \{ s \in \mathcal {S}_ Env \mid \forall s'. \; ((s,s') \in \rho ) \rightarrow \exists s''.\; (s',s'') \in \rho \wedge s ''\in \llbracket d\rrbracket \},~\text {and} \\ \llbracket CPre _{\mathcal {G}, Env }(d)\rrbracket &{}= \{s \in \mathcal {S}_ Env \mid \exists s'. \; ((s,s') \in \rho ) \wedge \forall s''.\; ((s',s'') \in \rho ) \rightarrow s'' \in \llbracket d\rrbracket \}. \end{array} $$</span></div></div><p>The player-<i>p</i> attractor for <span class="mathjax-tex">$\llbracket d\rrbracket $</span> can be computed as a fixpoint of the enforceable predecessor operator:</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{ll} Attr _{\llbracket \mathcal \rrbracket {G},p}(\llbracket d\rrbracket ) \cap \mathcal {S}_ Env &= \llbracket \mu X. \;d \vee CPre _{\mathcal {G},p}(X)\rrbracket , \end{array} $$</span></div></div><p>where <span class="mathjax-tex">$\mu $</span> denotes the least fixpoint. Note that since <span class="mathjax-tex">$\mathcal {S}$</span> is infinite, an iterative computation of the attractor is not guaranteed to terminate.</p><p>In Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar1">1</a>, consider the computation of the player-<span class="mathjax-tex">$ Sys $</span> attractor for <span class="mathjax-tex">$\llbracket d\rrbracket $</span> where <span class="mathjax-tex">$d = \{ base \mapsto \top , move \mapsto \top , mine \mapsto \bot \}$</span>. Applying <span class="mathjax-tex">$ CPre _{\mathcal {G}, Sys }(d)$</span> will produce <span class="mathjax-tex">$\{ base \mapsto \top , move \mapsto \top , mine \mapsto samp \ge req \}$</span> as in one step player-<span class="mathjax-tex">$ Sys $</span> can enforce reaching <span class="mathjax-tex">$ move $</span> if <span class="mathjax-tex">$ samp \ge req $</span> in <span class="mathjax-tex">$ mine $</span>. Since in <span class="mathjax-tex">$ mine $</span> the system player can enforce to increment <span class="mathjax-tex">$ samp $</span> by at least one, a second iteration of <span class="mathjax-tex">$ CPre _{\mathcal {G}, Sys }(\cdot )$</span> gives <span class="mathjax-tex">$\{\dots , mine \mapsto samp \ge req - 1\}$</span>, a third <span class="mathjax-tex">$\{\dots , mine \mapsto samp \ge req - 2\}$</span>, and so on. Thus, a naive iterative fixpoint computation does no terminate here. To avoid this non-termination, [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e11901">21</a>] introduced <i>attractor acceleration</i>. It will compute that, as explained in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec1">1</a>, the fixpoint is indeed <span class="mathjax-tex">$\{\dots , mine \mapsto \top \}$</span>.</p><p><b>Permissive Strategy Templates.</b> The main objective of this work is to identify small and useful sub-games, for which the results can enhance the symbolic game-solving process. To achieve this, we use a technique called <i>permissive strategy templates</i> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Anand, A., Nayak, S.P., Schmuck, A.: Synthesizing permissive winning strategy templates for parity games. In: Enea, C., Lal, A. (eds.) CAV 2023, Part I. LNCS, vol. 13964, pp. 436–458. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-37706-8_22
 
 " href="#ref-CR2" id="ref-link-section-d2892493e11960">2</a>], designed for finite game graphs. These templates can represent (potentially infinite) sets of winning strategies through local edge conditions. This motivates our construction of sub-games based on templates in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec6">4.2</a>.</p><p>These strategy templates are structured using three local edge conditions: <i>safety</i>, <i>co-live</i>, and <i>live-group</i> templates. Formally, given a game <span class="mathjax-tex">$(G,\varOmega )$</span> with <span class="mathjax-tex">$G=(V,V_ Env , V_ Sys ,\rho )$</span> and <span class="mathjax-tex">$E_p=\rho \cap (V_p\times V_{p-1})$</span>, a <i>strategy template for player</i> <i>p</i> is a tuple <span class="mathjax-tex">$(U,D,\mathcal {H})$</span> consisting of a set of <i>unsafe</i> edges <span class="mathjax-tex">$U\subseteq E_p$</span>, a set of <i>co-live</i> edges <span class="mathjax-tex">$D\subseteq E_p$</span>, and a set of live-groups <span class="mathjax-tex">$\mathcal {H}\subseteq 2^{E_p}$</span>. A strategy template <span class="mathjax-tex">$(U,D,\mathcal {H})$</span> represents the set of plays <span class="mathjax-tex">$\varPsi = \varPsi _U\cap \varPsi _D\cap \varPsi _\mathcal {H}\subseteq Plays (G)$</span>, where</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \varPsi _U&:= \{ \pi \mid \forall i.~(\pi [i], \pi [i+1]) \not \in U\}, \ \ \varPsi _D:= \{ \pi \mid \exists k.~\forall i > k.~(\pi [i], \pi [i+1]) \not \in D\}, \\ \varPsi _\mathcal {H}&:= \bigcap _{H\in \mathcal {H}} \{ \pi \mid (\forall i.~\exists j > i.~ \pi [j] \in \textsc {src}(H)) \rightarrow (\forall i.~\exists j > i.~ (\pi [j], \pi [j+1]) \in H) \}, \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\textsc {src}(H)$</span> contains the sources <span class="mathjax-tex">$\{u \mid (u,v)\in H\}$</span> of the edges in <span class="mathjax-tex">$H$</span>. A strategy <span class="mathjax-tex">$\sigma $</span> for player <i>p</i> <i>satisfies</i> a strategy template <span class="mathjax-tex">$\varPsi $</span> if it is winning in the game <span class="mathjax-tex">$(G,\varPsi )$</span> for player <i>p</i>. Intuitively, <span class="mathjax-tex">$\sigma $</span> satisfies a strategy template if every play <span class="mathjax-tex">$\pi $</span> consistent with <span class="mathjax-tex">$\sigma $</span> for player <i>p</i> is contained in <span class="mathjax-tex">$\varPsi $</span>, that is, (i) <span class="mathjax-tex">$\pi $</span> never uses the unsafe edges in <span class="mathjax-tex">$U$</span> (i.e., <span class="mathjax-tex">$\pi \in \varPsi _U$</span>), (ii) <span class="mathjax-tex">$\pi $</span> stops using the co-live edges in <span class="mathjax-tex">$D$</span> eventually (i.e., <span class="mathjax-tex">$\pi \in \varPsi _D$</span>), and (iii) for every live-group <span class="mathjax-tex">$H\in \mathcal {H}$</span>, if <span class="mathjax-tex">$\rho $</span> visits <span class="mathjax-tex">$\textsc {src}(H)$</span> infinitely often, then it also uses the edges in <span class="mathjax-tex">$H$</span> infinitely often (i.e., <span class="mathjax-tex">$\pi \in \varPsi _\mathcal {H}$</span>). Strategy templates can be used as a concise representation of winning strategies as formalized next.</p> <h3 class="c-article__sub-heading" id="FPar4">Definition 3</h3> <p><b>(Winning Strategy Template</b> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Anand, A., Nayak, S.P., Schmuck, A.: Synthesizing permissive winning strategy templates for parity games. In: Enea, C., Lal, A. (eds.) CAV 2023, Part I. LNCS, vol. 13964, pp. 436–458. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-37706-8_22
 
 " href="#ref-CR2" id="ref-link-section-d2892493e13319">2</a>]<b>).</b> A strategy template <span class="mathjax-tex">$\varPsi $</span> for player <i>p</i> is <i>winning</i> if every strategy satisfying <span class="mathjax-tex">$\varPsi $</span> is winning for <i>p</i> in <span class="mathjax-tex">$(G,\varOmega )$</span>.</p> <p>We note that the algorithms for computing winning strategy templates in safety, Büchi, co-Büchi, and parity games, presented in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Anand, A., Nayak, S.P., Schmuck, A.: Synthesizing permissive winning strategy templates for parity games. In: Enea, C., Lal, A. (eds.) CAV 2023, Part I. LNCS, vol. 13964, pp. 436–458. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-37706-8_22
 
 " href="#ref-CR2" id="ref-link-section-d2892493e13403">2</a>], exhibit the same worst-case computation time as standard methods for solving such (finite-state) games.</p></div></div></section><section data-title="Attractor Computation with Caching"><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>Attractor Computation with Caching</h2><div class="c-article-section__content" id="Sec3-content"><p>As outlined in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec1">1</a>, the core of our method consists of the pre-computation of attractor sets for local sub-games and the utilization of the results in the attractor computations performed when solving the complete reactive program game. We call the pre-computed results <i>attractor cache</i>. We use the cache during attractor computations to <i>directly add</i> to the computed attractor sets of states from which, <i>based on the pre-computed information</i>, the respective player can enforce reaching the current attractor subset. In that way, if the local attractor computation requires acceleration, we can avoid performing the acceleration during the attractor computation for the overall game. This section presents the formal definition of an attractor cache and shows how it is used.</p><p>Intuitively, an attractor cache is a finite set of tuples or <i>cache entries</i> of the form <span class="mathjax-tex">$(\mathcal {G}, p, src , targ , \mathbb {X}_ ind )$</span>. <span class="mathjax-tex">$\mathcal {G}$</span> is a reactive program game structure and <i>p</i> the player the cache entry applies to. The sets of states <span class="mathjax-tex">$ src , targ \in \mathcal {D}$</span> are related via enforceable reachability: player <i>p</i> can enforce reaching <span class="mathjax-tex">$\llbracket targ \rrbracket $</span> from <span class="mathjax-tex">$\llbracket src \rrbracket $</span> in <span class="mathjax-tex">$\mathcal {G}$</span>. <span class="mathjax-tex">$\mathbb {X}_ ind $</span> are the so-called <i>independent variables</i> – the enforcement relation must hold independently of and preserve the values of <span class="mathjax-tex">$\mathbb {X}_ ind $</span>. Independent variables are useful when a cache entry only concerns a part of the game structure where these variables are irrelevant. This allows the utilization of the cache entry under different conditions on those variables. We formalize this intuition in the next definition.</p> <h3 class="c-article__sub-heading" id="FPar5">Definition 4</h3> <p><b>(Attractor Cache).</b> A finite set <span class="mathjax-tex">$C \subseteq \textsf{RPGS}\times \{ Sys , Env \} \times \mathcal {D}\times \mathcal {D}\times 2^{\mathbb {X}}$</span> is called an <i>attractor cache</i> if and only if for all <span class="mathjax-tex">$(\mathcal {G}, p, src , targ , \mathbb {X}_ ind ) \in C$</span> and all <span class="mathjax-tex">$\varphi \in FOL (\mathbb {X}_ ind )$</span> it holds that <span class="mathjax-tex">$ \llbracket src \wedge \lambda l. ~\varphi \rrbracket \subseteq Attr _{\llbracket \mathcal \rrbracket {G},p}(\llbracket targ \wedge \lambda l. ~\varphi \rrbracket ). $</span></p> <p>We use the <i>lambda abstraction</i> <span class="mathjax-tex">$\lambda l. ~\varphi $</span> to denote the anonymous function that maps each location in <i>L</i> to the formula <span class="mathjax-tex">$\varphi $</span>.</p> <h3 class="c-article__sub-heading" id="FPar6">Example 2</h3> <p>Recall the game from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar1">1</a>. From every state with location <span class="mathjax-tex">$ mine $</span>, player <span class="mathjax-tex">$ Sys $</span> can enforce eventually reaching <span class="mathjax-tex">$ samp \ge req $</span> by choosing at every step the update that increases variable <span class="mathjax-tex">$ samp $</span>. As this argument only concerns location <span class="mathjax-tex">$ mine $</span>, the program variables <span class="mathjax-tex">$ done $</span> and <span class="mathjax-tex">$ pos $</span> are independent. Since it is not updated, <span class="mathjax-tex">$ req $</span> is also independent (we prove this in the next section). Hence, <span class="mathjax-tex">$C_ ex = \{ (\mathcal {G}_ ex , Sys , src , targ , \mathbb {X}_ ind ) \}$</span> where <span class="mathjax-tex">$\mathcal {G}_ ex $</span> is from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig2">2</a>, <span class="mathjax-tex">$ src = \{ mine \mapsto \top \}$</span>, <span class="mathjax-tex">$ targ = \{ mine \mapsto samp \ge req \}$</span>, and <span class="mathjax-tex">$\mathbb {X}_ ind = \{ done , pos , req \}$</span> is an attractor cache.</p> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-c" data-title="Algorithm 1"><figure><figcaption><b id="Figc" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 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/10.1007/978-3-031-65633-0_7/figures/c" rel="nofollow"><picture><img aria-describedby="Figc" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figc_HTML.png" alt="figure c" loading="lazy" width="685" height="254"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-c-desc"><p>Attractor computation using an attractor cache.</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/10.1007/978-3-031-65633-0_7/figures/c" data-track-dest="link:Figurec Full size image" aria-label="Full size image figure c" 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>Algorithm 1 shows how we use an attractor cache to enhance accelerated attractor computations. <span class="u-small-caps">AttractorAccCache</span> extends the procedure <span class="u-small-caps">AttractorAcc</span> for accelerated symbolic attractor computation presented in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e14597">21</a>]. <span class="u-small-caps">AttractorAccCache</span> takes a cache as an additional argument and at each iteration of the attractor computation checks if some cache entry is applicable. For each such cache entry, if <span class="mathjax-tex">$\llbracket targ \rrbracket $</span> is a subset of <span class="mathjax-tex">$\llbracket a^n\rrbracket $</span>, we can add <span class="mathjax-tex">$ src $</span> to <span class="mathjax-tex">$a^n$</span> since we know that <span class="mathjax-tex">$ targ $</span> is enforceable from <span class="mathjax-tex">$ src $</span>. However, <span class="mathjax-tex">$a^n$</span> may constrain the values of <span class="mathjax-tex">$\mathbb {X}_ ind $</span> making this subset check fail unnecessarily. Therefore, <span class="u-small-caps">StrengthenTarget</span> computes a formula <span class="mathjax-tex">$\varphi \in FOL (\mathbb {X}_ ind )$</span> such that <span class="mathjax-tex">$ targ $</span> strengthened with <span class="mathjax-tex">$\varphi $</span> is a subset of <span class="mathjax-tex">$a^n$</span>. Intuitively, <span class="mathjax-tex">$\varphi $</span> describes the values of the independent variables that remain unchanged in the cached attractor. Note that <span class="mathjax-tex">$\varphi $</span> always exists as we could pick <span class="mathjax-tex">$\bot $</span>, which we have to do if <span class="mathjax-tex">$ targ $</span> is truly <i>not</i> a subset of <span class="mathjax-tex">$a^n$</span>.</p><p>The next lemma formalizes this intuition and the correctness of <span class="u-small-caps">AttractorAccCache</span> under the above condition on <span class="u-small-caps">StrengthenTarget</span>. Note that since the cache is used in the context of attractor computation, the objective <span class="mathjax-tex">$\varOmega $</span> of the reactive program game is not relevant here.</p> <h3 class="c-article__sub-heading" id="FPar7">Lemma 1</h3> <p><b>(Correctness of Cache Utilization).</b> Let <span class="mathjax-tex">$\mathcal {G}$</span> be a reactive program game structure, <span class="mathjax-tex">$p \in \{ Sys , Env \}$</span>, <span class="mathjax-tex">$d \in \mathcal {D}$</span> and <i>C</i> be an attractor cache. Furthermore, suppose that for every <span class="mathjax-tex">$ targ \in \mathcal {D}$</span>, <span class="mathjax-tex">$a \in \mathcal {D}$</span> and every <span class="mathjax-tex">$\mathbb {X}_ ind \subseteq \mathbb {X}$</span> it holds that if <span class="mathjax-tex">$\textsc {StrengthenTarget}( targ , \mathbb {X}_ ind , a) = \varphi $</span>, then <span class="mathjax-tex">$\varphi \in FOL (\mathbb {X}_ ind )$</span> and <span class="mathjax-tex">$\llbracket targ \wedge \lambda l. ~\varphi \rrbracket \subseteq \llbracket a\rrbracket $</span>. Then, if the procedure <span class="mathjax-tex">$\textsc {AttractorAccCache}(\mathcal G, p, d, C)$</span> terminates returning <span class="mathjax-tex">$ attr \in \mathcal {D}$</span>, then it holds that <span class="mathjax-tex">$\llbracket attr \rrbracket = Attr _{\llbracket \mathcal \rrbracket {G},p}(\llbracket d\rrbracket ) \cap \mathcal {S}_ Env $</span>.</p> <p>We realize <span class="mathjax-tex">$\textsc {StrengthenTarget}( targ ,\mathbb {X}_ ind ,a)$</span> such that it returns the formula <span class="mathjax-tex">$\bigwedge _{l \in L} \big (\forall (\mathbb {X}\backslash \mathbb {X}_ ind ).~ targ (l) \rightarrow a(l)\big )$</span> which satisfies the condition in Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar7">1</a>.</p> <h3 class="c-article__sub-heading" id="FPar8">Example 3</h3> <p>Recall the game from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar1">1</a> and the cache <span class="mathjax-tex">$C_ ex $</span> from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar6">2</a>. Suppose that we are computing the attractor for player <span class="mathjax-tex">$ Sys $</span> to <span class="mathjax-tex">$d = \{ base \mapsto \top \}$</span>, i.e. <span class="mathjax-tex">$\textsc {AttractorAccCache}(\mathcal {G}_ ex , Sys , d, C_ ex )$</span> without acceleration, i.e., <span class="mathjax-tex">$\textsf{Accelerate}$</span> returns <span class="mathjax-tex">$\bot $</span> in line 11 in Algorithm 1. Initially, <span class="mathjax-tex">$a^1 = \{ base \mapsto \top \}$</span>. After one iteration of applying <span class="mathjax-tex">$ Cpre $</span>, we get <span class="mathjax-tex">$a^2 = \{ base \mapsto \top , move \mapsto pos = 12 \wedge done = 1\}$</span>. Then we get <span class="mathjax-tex">$a^3 = \{ \dots , mine \mapsto pos = 12 \wedge samp \ge req \}$</span>. In the only entry of <span class="mathjax-tex">$C_ ex $</span>, the target set <span class="mathjax-tex">$ targ = \{ mine \mapsto samp \ge req \}$</span> contains more states in <span class="mathjax-tex">$ mine $</span> (i.e., all possible positions of the robot) then <span class="mathjax-tex">$a^3$</span> (which asserts <span class="mathjax-tex">$ pos = 12$</span>). However, <span class="mathjax-tex">$\textsc {StrengthenTarget}( targ , \mathbb {X}_ ind , a^3)$</span> as implemented above, will return the strengthening <span class="mathjax-tex">$ pos = 12$</span> (after simplifying the formula), which makes the cache entry with <span class="mathjax-tex">$ targ $</span> applicable. Since <span class="mathjax-tex">$ src = \{ mine \mapsto \top \}$</span>, we update <span class="mathjax-tex">$a^3$</span> to <span class="mathjax-tex">$\{ \dots , mine \mapsto pos = 12\}$</span> in line 10 of the algorithm.</p> </div></div></section><section data-title="Abstract Template-Based Cache Generation"><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>Abstract Template-Based Cache Generation</h2><div class="c-article-section__content" id="Sec4-content"><p>Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec3">3</a> defined attractor caches and showed their utilization for attractor computations via Algorithm 1. We motivated this approach by the observation that there often exist <i>small local sub-games</i> that entail essential attractors, and pre-computing these attractors within the sub-games, caching them and then using them via Algorithm 1 is more efficient then only applying acceleration over the entire game (as in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e16827">21</a>]). To formalize this workflow, Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec5">4.1</a> explains the generation of cache entries from sub-game structures of the given reactive program game, and Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec6">4.2</a> discusses the identification of helpful sub-game structures via permissive strategy templates in finite-state abstractions of the given game.</p><h3 class="c-article__sub-heading" id="Sec5"><span class="c-article-section__title-number">4.1 </span>Generating Attractor Caches from Sub-Games</h3><p>Within this subsection, we consider a sub-game structure <span class="mathjax-tex">$\mathcal {G}'$</span> which is induced by a subset of locations <span class="mathjax-tex">$L_ sub \subseteq L$</span> of the original reactive program game structure <span class="mathjax-tex">$\mathcal {G}$</span>, as formalized next. Intuitively, we remove all locations from <span class="mathjax-tex">$\mathcal {G}$</span> not in <span class="mathjax-tex">$L_ sub $</span> and redirect their incoming transitions to a new sink location <span class="mathjax-tex">$\textsf{sink}_ sub $</span>.</p> <h3 class="c-article__sub-heading" id="FPar9">Definition 5</h3> <p><b>(Induced Sub-Game Structure).</b> Let <span class="mathjax-tex">$\mathcal {G}= (T,\mathbb {I}, \mathbb {X}, L, Inv ,\delta )$</span> be a reactive program game structure and let <span class="mathjax-tex">$L_ sub \subseteq L$</span> be a set of locations. The <i>sub-game structure induced by</i> <span class="mathjax-tex">$L_ sub $</span> is the reactive program game structure</p> <p><span class="mathjax-tex">$\textsf{SubGame}(\mathcal {G},L_ sub ) := (T,\mathbb {I},\mathbb {X}',L', Inv ',\delta ')$</span> where <span class="mathjax-tex">$L' := L_ sub \cup \{\textsf{sink}_{ sub }\}$</span>,</p> <p><span class="mathjax-tex">$\mathbb {X}' : = \{ x \in \mathbb {X}\mid x~\text {appears in transitions from or invariants of } L_{ sub } \text { in } \mathcal {G}' \}$</span>,</p> <p><span class="mathjax-tex">$ Inv '(l) := Inv (l)$</span> for all <span class="mathjax-tex">$l \in L' \setminus \{\textsf{sink}_{ sub }\} $</span> and <span class="mathjax-tex">$ Inv '(\textsf{sink}_{ sub }) := \top $</span>, and</p> <p><span class="mathjax-tex">$\begin{array}{lll} \delta ' &{} := &{} \{(l,g,u,l') \in \delta \mid l,l' \in L' \} \cup \{(\textsf{sink}_{ sub },\top ,\lambda x.\;x,\textsf{sink}_{ sub })\} \cup \\ {} &{}&{} \{(l,g,\lambda x.x,\textsf{sink}_{ sub }) \mid \exists l' \in L.\; (l,g,u,l') \in \delta \wedge l\in L' \wedge l' \not \in L'\}. \end{array}$</span></p> <p>Recall that <span class="mathjax-tex">$\mathcal {D}= L \rightarrow FOL (\mathbb {X})$</span>. Let <span class="mathjax-tex">$\mathcal {D}' := L' \rightarrow FOL (\mathbb {X}')$</span> be the symbolic domain for a sub-game structure with locations <span class="mathjax-tex">$L'$</span>. As <span class="mathjax-tex">$\mathbb {X}' \subseteq \mathbb {X}$</span>, <span class="mathjax-tex">$ FOL (\mathbb {X}') \subseteq FOL (\mathbb {X})$</span> which allows us to extend each element of <span class="mathjax-tex">$\mathcal {D}'$</span> to an element of <span class="mathjax-tex">$\mathcal {D}$</span> that agrees on <span class="mathjax-tex">$L'$</span>. Formally, we define <span class="mathjax-tex">$\textsf{extend}_{L} : \mathcal {D}' \rightarrow \mathcal {D}$</span> such that for <span class="mathjax-tex">$d'\in \mathcal {D}'$</span> and <span class="mathjax-tex">$l \in L$</span> we have <span class="mathjax-tex">$\textsf{extend}_{L} (d')(l) := \texttt {if } l \in L' \texttt { then }d'(l) \texttt { else } \bot $</span>.</p><p>The computation of an attractor cache from an induced sub-game is detailed in Algorithm 2. Given a reactive program game structure <span class="mathjax-tex">$\mathcal {G}$</span>, a player <i>p</i>, and a subset of locations <span class="mathjax-tex">$L_ sub $</span>, Algorithm 2 first computes the induced sub-game (line 2). The quantifier elimination ([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Bradley, A.R., Manna, Z.: The Calculus of Computation - Decision Procedures with Applications to Verification. Springer, Heidelberg (2007). 
 https://doi.org/10.1007/978-3-540-74113-8
 
 " href="#ref-CR7" id="ref-link-section-d2892493e18489">7</a>, Ch. 7]) <span class="mathjax-tex">$\textsf{QElim}$</span> in line 3 projects the given <span class="mathjax-tex">$d \in \mathcal {D}$</span> to an element <span class="mathjax-tex">$d'$</span> of the symbolic domain <span class="mathjax-tex">$\mathcal {D}'$</span> of the sub-game structure. Then, in line 4, we perform the accelerated attractor computation from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e18584">21</a>] with target set <span class="mathjax-tex">$d'$</span> to obtain the set of states <i>a</i> from which player <i>p</i> can <i>enforce</i> reaching <span class="mathjax-tex">$d'$</span> in <span class="mathjax-tex">$\mathcal {G}'$</span>. The independent variables are those variables in <span class="mathjax-tex">$\mathbb {X}$</span> that are not updated in any of the transitions in <span class="mathjax-tex">$\mathcal {G}'$</span>. Formally, we define those as <span class="mathjax-tex">$\textsf{IndependentVars}(\mathcal {G},\mathcal {G}'):=\{x \in \mathbb {X}\mid \forall (l, g, u, l') \in \delta '.~u(x)=x\}$</span>. In order to output an attractor cache for the original game <span class="mathjax-tex">$\mathcal {G}$</span>, we extend the computed source and target sets <i>a</i> and <span class="mathjax-tex">$d'$</span> via the previously defined function <span class="mathjax-tex">$\textsf{extend}_{L}$</span> (line 6). Intuitively, the attractor computed over a sub-game <span class="mathjax-tex">$\mathcal {G}'$</span> is also an attractor for the overall game <span class="mathjax-tex">$\mathcal {G}$</span> as sub-games are only restricted by location (not by variables). Hence, player <i>p</i> can also enforce reaching the target set in the original game <span class="mathjax-tex">$\mathcal {G}$</span>, if he can do so in <span class="mathjax-tex">$\mathcal {G}'$</span>. This is formalized by the next lemma.</p> <h3 class="c-article__sub-heading" id="FPar10">Lemma 2</h3> <p>Let <span class="mathjax-tex">$\mathcal {G}= (T,\mathbb {I}, \mathbb {X}, L, Inv ,\delta )$</span> be a reactive program game structure, and let <span class="mathjax-tex">$\mathcal {G}'= (T,\mathbb {I},\mathbb {X}',L', Inv ',\delta ')$</span> be an induced sub-game structure with sink location <span class="mathjax-tex">$\textsf{sink}_{ sub }$</span> constructed as above. Let <span class="mathjax-tex">$ src ', targ ' \in \mathcal {D}'$</span> be such that <span class="mathjax-tex">$ targ '(\textsf{sink}_{ sub }) = \bot $</span> and <span class="mathjax-tex">$\llbracket src '\rrbracket \subseteq Attr _{\llbracket \mathcal \rrbracket {G'},p}(\llbracket targ '\rrbracket )$</span> for some player <span class="mathjax-tex">$p\in \{ Sys , Env \}$</span>. Furthermore, let <span class="mathjax-tex">$\mathbb {Y} \subseteq \textsf{IndependentVars}(\mathcal {G},\mathcal {G}')$</span>. Then, for every <span class="mathjax-tex">$\varphi \in FOL (\mathbb {Y} )$</span> it holds that</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\llbracket \textsf{extend}_L( src ') \wedge \lambda l.\varphi \rrbracket \subseteq Attr _{\llbracket \mathcal \rrbracket {G},p}(\llbracket \textsf{extend}_L( targ ') \wedge \lambda l.\varphi \rrbracket ).$$</span></div></div> <p>This results in the following correctness statement.</p> <h3 class="c-article__sub-heading" id="FPar11">Lemma 3</h3> <p><span class="mathjax-tex">$\textsc {SubgameCache}(\mathcal {G}, p, L_ sub , d)$</span> returns an attractor cache over <span class="mathjax-tex">$\mathcal {G}$</span>.</p> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-d" data-title="Algorithm 2"><figure><figcaption><b id="Figd" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 2</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/10.1007/978-3-031-65633-0_7/figures/d" rel="nofollow"><picture><img aria-describedby="Figd" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figd_HTML.png" alt="figure d" loading="lazy" width="685" height="151"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-d-desc"><p>Cache generation based on an induced sub-game.</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/10.1007/978-3-031-65633-0_7/figures/d" data-track-dest="link:Figured Full size image" aria-label="Full size image figure d" 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> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-3" data-title="Fig. 3."><figure><figcaption><b id="Fig3" class="c-article-section__figure-caption" data-test="figure-caption-text">Fig. 3.</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/10.1007/978-3-031-65633-0_7/figures/3" rel="nofollow"><picture><img aria-describedby="Fig3" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Fig3_HTML.png" alt="figure 3" loading="lazy" width="685" height="112"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-3-desc"><p>Induced sub-game structure <span class="mathjax-tex">$\textsf{SubGame}(\mathcal {G}_ ex ,\{ mine \})$</span> of the reactive program game structure <span class="mathjax-tex">$\mathcal {G}_ ex $</span> from Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig2">2</a>, with the same abbreviations as in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig2">2</a>.</p></div></div><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="chapter-link" data-track="click" data-track-label="button" data-track-action="view figure" href="/chapter/10.1007/978-3-031-65633-0_7/figures/3" data-track-dest="link:Figure3 Full size image" aria-label="Full size image figure 3" rel="nofollow"><span>Full size image</span><svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-chevron-right-small"></use></svg></a></div></figure></div> <h3 class="c-article__sub-heading" id="FPar12">Example 4</h3> <p>Consider the reactive program game structure <span class="mathjax-tex">$\mathcal {G}_ ex $</span> from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar1">1</a>. We apply <span class="mathjax-tex">$\textsc {SubgameCache}(\mathcal {G}_ ex , Sys , \{ mine \}, d)$</span> with <span class="mathjax-tex">$d =\{ mine \mapsto samp \ge req \wedge pos = 12 \wedge done \ne 1 \}$</span>. First, we construct the induced sub-game structure in Figrue <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig3">3</a>. Quantifier elimination produces the target set <span class="mathjax-tex">$d' = \{ mine \mapsto samp \ge req \}$</span>. If we compute the attractor in this sub-game to set <span class="mathjax-tex">$d'$</span>, we get <span class="mathjax-tex">$\{ mine \mapsto \top \}$</span>. Note that since the number of steps needed to reach <span class="mathjax-tex">$d'$</span> depends on the initial value of <span class="mathjax-tex">$ samp $</span> and is hence unbounded, a technique like acceleration [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e20322">21</a>] is necessary to compute this attractor. As in this sub-game structure only the variable <span class="mathjax-tex">$ samp $</span> is updated, the independent variables are <span class="mathjax-tex">$\mathbb {X}_ ind = \{ done , pos , req \}$</span>. With this we get the cache entry from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar6">2</a>.</p> <h3 class="c-article__sub-heading" id="Sec6"><span class="c-article-section__title-number">4.2 </span>Constructing Sub-games from Abstract Strategy Templates</h3><p>The procedure from the previous subsection yields attractor caches regardless of how the sub-games are chosen. In this section we describe our approach to identifying “useful” sub-game structures. These sub-game structures are induced by so-called <i>helpful edges</i> determined by permissive strategy templates. Since the game graph described by a reactive program game structure is in general infinite, we first construct finite abstract games in which we compute permissive strategy templates for the two players. We start by describing the abstract games.</p><p><b>Finite Abstractions of Reactive Program Games.</b> Here we describe the construction of a game graph <span class="mathjax-tex">$\widehat{G}=(V, V_ Env , V_ Sys , \widehat{\rho })$</span> from a reactive program game structure <span class="mathjax-tex">$\mathcal {G}= (T,\mathbb {I}, \mathbb {X}, L, Inv ,\delta )$</span> with semantics <span class="mathjax-tex">$\llbracket \mathcal {G}\rrbracket = (\mathcal {S}, \mathcal {S}_ Env ,\mathcal {S}_ Sys , \rho )$</span>. While <span class="mathjax-tex">$\llbracket \mathcal {G}\rrbracket $</span> is also a game graph, its vertex set is typically infinite. The game graph <span class="mathjax-tex">$\widehat{G}$</span>, which is an abstraction of <span class="mathjax-tex">$\llbracket \mathcal {G}\rrbracket $</span>, has a finite vertex set instead.</p><p>We construct the game graph <span class="mathjax-tex">$\widehat{G}$</span> from <span class="mathjax-tex">$\mathcal {G}$</span> by performing abstraction with respect to a given abstract domain. The abstract domain consists of two finite sets of quantifier-free first-order formulas which are used to define the vertex sets of the game graph <span class="mathjax-tex">$\widehat{G}$</span>. The conditions that we impose in the definition of abstraction domain given below ensure that it can partition the state space of <span class="mathjax-tex">$\mathcal {G}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar13">Definition 6</h3> <p><b>(Game Abstraction Domain).</b> A <i>game abstraction domain</i> for a reactive program game structure <span class="mathjax-tex">$\mathcal {G}= (T,\mathbb {I}, \mathbb {X}, L, Inv ,\delta )$</span> is a pair of finite sets of quantifier-free first-order formulas <span class="mathjax-tex">$(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}}) \in QF (\mathbb {X}) \times QF (\mathbb {X}\cup \mathbb {I})$</span> such that for <span class="mathjax-tex">$\mathcal {P} = \mathcal P_{\mathbb {X}}$</span> (resp. <span class="mathjax-tex">$\mathcal {P}= \mathcal {P}_{\mathbb {X}\cup \mathbb {I}}$</span>) and <span class="mathjax-tex">$V = \mathbb {X}$</span> (resp. <span class="mathjax-tex">$V = \mathbb {X}\cup \mathbb {I}$</span>), <span class="mathjax-tex">$\mathcal {P}$</span> partitions <span class="mathjax-tex">$ Assignments (V)$</span>, i.e. <span class="mathjax-tex">$ Assignments (V) = \bigcup _{\varphi \in \mathcal {P}}\{ \textbf{v} \mid \textbf{v} \models _{T} \varphi \}$</span> and for every <span class="mathjax-tex">$\varphi _1,\varphi _2 \in \mathcal {P}$</span> with <span class="mathjax-tex">$\varphi _1\wedge \varphi _2$</span> satisfiable it holds that <span class="mathjax-tex">$\varphi _1 = \varphi _2$</span>.</p> <p>The abstraction domain we use consists of all conjunctions of atomic predicates (and their negations) that appear in the guards of the reactive program game structure <span class="mathjax-tex">$\mathcal {G}$</span>. Let <span class="mathjax-tex">$ GA $</span> be the set of atomic formulas appearing in the guards of <span class="mathjax-tex">$\mathcal {G}$</span>. We use the abstraction domain <span class="mathjax-tex">$\textsf{AbstractDomain}(\mathcal {G}):=(\mathcal P_{\mathbb {X}}^{ GA },\mathcal {P}_{\mathbb {X}\cup \mathbb {I}}^{ GA })$</span> where</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{ll} \mathcal P_{\mathbb {X}}^{ GA } &{} := \{ \bigwedge \nolimits _{\varphi \in J}\varphi \wedge \bigwedge \nolimits _{\varphi \not \in J} \lnot \varphi \mid J \subseteq GA \cap FOL (\mathbb {X})\}, \\ \mathcal {P}_{\mathbb {X}\cup \mathbb {I}}^{ GA } &{} := \{ \bigwedge \nolimits _{\varphi \in J}\varphi \wedge \bigwedge \nolimits _{\varphi \not \in J} \lnot \varphi \mid J \subseteq GA \cap ( FOL (\mathbb {X}\cup \mathbb {I})\setminus FOL (\mathbb {X}))\}. \end{array} $$</span></div></div> <h3 class="c-article__sub-heading" id="FPar14">Example 5</h3> <p>In the game structure <span class="mathjax-tex">$\mathcal {G}_ ex $</span> from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar1">1</a>, we get for <span class="mathjax-tex">$\mathcal P_{\mathbb {X}}^{ GA }$</span> all combinations of <span class="mathjax-tex">$\varphi _1 \wedge \varphi _2 \wedge \varphi _3$</span>, where <span class="mathjax-tex">$\varphi _1 \in \{ req < samp , req \ge samp \}$</span>, <span class="mathjax-tex">$\varphi _2 \in \{ pos = 12, pos = 23, pos \ne 12 \wedge pos \ne 23\}$</span>, and <span class="mathjax-tex">$\varphi _3 \in \{ task = 1, task \ne 1 \}$</span>. For <span class="mathjax-tex">$\mathcal {P}_{\mathbb {X}\cup \mathbb {I}}^{ GA }$</span> we get all combinations of <span class="mathjax-tex">$\psi _1 \wedge \psi _2 \wedge \psi _3$</span>, where <span class="mathjax-tex">$\psi _1 \in \{a \le 0, a > 0\}$</span>, <span class="mathjax-tex">$\psi _2 \in \{ b \le 0, b > 0\}$</span>, and <span class="mathjax-tex">$\psi _3 \in \{ inpReq \le 0 , inpReq > 0 \}$</span>.</p> <p>We choose this abstraction domain as a baseline since the predicates appearing in the guards are natural delimiters in the program variable state space. However, the abstraction we define now is independent of this specific domain.</p><p>Given a game abstraction domain <span class="mathjax-tex">$(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}})$</span>, we construct two abstract game graphs, <span class="mathjax-tex">$\widehat{G}^\uparrow $</span> and <span class="mathjax-tex">$\widehat{G}^\downarrow $</span>. They have the same sets of vertices but differ in the transition relations. The transition relation in <span class="mathjax-tex">$\widehat{G}^\uparrow $</span> overapproximates the transitions originating from states of player <span class="mathjax-tex">$ Sys $</span> and underapproximates the transitions from states of player <span class="mathjax-tex">$ Env $</span>. In <span class="mathjax-tex">$\widehat{G}^\downarrow $</span> the approximation of the two players is reversed.</p> <h3 class="c-article__sub-heading" id="FPar15">Definition 7</h3> <p><b>(Abstract Game Graphs).</b> Let <span class="mathjax-tex">$\mathcal {G}= (T,\mathbb {I}, \mathbb {X}, L, Inv ,\delta )$</span> be a reactive program game structure and <span class="mathjax-tex">$(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}})$</span> be an abstraction domain. The game graphs <span class="mathjax-tex">$\widehat{G}^\circ =(V, V_ Env , V_ Sys , \widehat{\rho }^\circ )$</span> with <span class="mathjax-tex">$\circ \in \{\uparrow ,\downarrow \}$</span> are the <span class="mathjax-tex">$(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}})$</span>-induced abstractions of <span class="mathjax-tex">$\mathcal {G}$</span> if <span class="mathjax-tex">$V := V_ Sys \cup V_ Env $</span>, <span class="mathjax-tex">$V_ Env := L \times \mathcal P_{\mathbb {X}}$</span>, and <span class="mathjax-tex">$V_ Sys := L \times \mathcal P_{\mathbb {X}}\times \mathcal {P}_{\mathbb {X}\cup \mathbb {I}}$</span>; and <span class="mathjax-tex">$\widehat{\rho }^\circ \subseteq (V_ Env \times V_ Sys ) \cup (V_ Sys \times V_ Env )$</span> is the smallest relation such that</p><ul class="u-list-style-dash"> <li> <p><span class="mathjax-tex">$((l, \varphi ), (l, \varphi , \varphi _I)) \in \widehat{\rho }^\circ \cap (V_ Env \times V_ Sys )$</span> iff the following formula is valid </p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ move ^\bullet _\mathbb {X}( Inv (l) \wedge \varphi (\mathbb {X}), \exists \mathbb {I}.~ \varphi _I(\mathbb {X}, \mathbb {I}))$$</span></div></div><p> for <span class="mathjax-tex">$\bullet = \downarrow $</span> if <span class="mathjax-tex">$\circ = \uparrow $</span> and <span class="mathjax-tex">$\bullet = \uparrow $</span> if <span class="mathjax-tex">$\circ = \downarrow $</span>,</p> </li> <li> <p><span class="mathjax-tex">$((l, \varphi , \varphi _I), (l', \varphi ')) \in \widehat{\rho }^\circ \cap (V_ Sys \times V_ Env )$</span> iff the following formula is valid </p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ move ^\circ _{\mathbb {X}\cup \mathbb {I}}\left( Inv (l) \wedge \varphi (\mathbb {X}) \wedge \varphi _I(\mathbb {X}, \mathbb {I}), \exists (g , u) \in Labels (l, l').~ trans (g,u,l',\varphi ')\right) $$</span></div></div><p> for <span class="mathjax-tex">$ trans (g,u,l',\varphi '):=g(\mathbb {X}, \mathbb {I}) \wedge \big (\varphi '\wedge Inv (l')\big )(u(\mathbb {X}, \mathbb {I}))$</span>,</p> </li> </ul><p>where <span class="mathjax-tex">$ move ^\uparrow _{V}(\varphi , \varphi ') := \exists V. \varphi (V) \wedge \varphi '(V)$</span> and <span class="mathjax-tex">$ move ^\downarrow _{V}(\varphi , \varphi ') := \forall V. \varphi (V) \rightarrow \varphi '(V)$</span>.</p> <p>Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar15">7</a> provides us with a procedure <span class="u-small-caps">AbstractRPG</span> for constructing the pair of abstractions <span class="mathjax-tex">$(\widehat{G}^\uparrow ,\widehat{G}^\downarrow ) := \textsc {AbstractRPG}(\mathcal {G},(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}}))$</span>.</p><p>We refer to the vertices in the abstract game graphs as abstract states. By slightly overloading notation, we define the projection from abstract states <span class="mathjax-tex">$v\in V$</span> to the respective location by <span class="mathjax-tex">$ loc : V \rightarrow L$</span> s.t. <span class="mathjax-tex">$ loc ((l,\varphi )) =l$</span> and <span class="mathjax-tex">$ loc ((l,\varphi ,\varphi _I)) =l$</span>. This definition naturally extends to sequences of abstract states <span class="mathjax-tex">$\pi \in V^\infty $</span> s.t. <span class="mathjax-tex">$ loc (\widehat{\pi })[i] = loc (\widehat{\pi }[i])$</span> for all <span class="mathjax-tex">$i \in dom(\pi )$</span>, and to sets of vertices: <span class="mathjax-tex">$ loc : 2^V \rightarrow 2^L$</span>. Given <span class="mathjax-tex">$\mathcal {G}$</span> with semantics <span class="mathjax-tex">$\llbracket \mathcal {G}\rrbracket = (\mathcal {S}, \mathcal {S}_ Env ,\mathcal {S}_ Sys , \rho )$</span> and an abstract game graph <span class="mathjax-tex">$\widehat{G}^\circ =(V, V_ Env , V_ Sys , \widehat{\rho }^\circ )$</span>, we define the following functions between their respective state spaces. The <i>concretization function</i> <span class="mathjax-tex">$\gamma : V \rightarrow 2^\mathcal {S}$</span> is defined s.t.</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{ll} \gamma ((l,\varphi )) &{}:= \{ (l, \textbf{x}) \in \mathcal {S}_ Env \mid \textbf{x} \models _{T} \varphi \}~\text {and}\\ \gamma ((l,\varphi ,\varphi _I)) &{}:= \{ ((l, \textbf{x}),\textbf{i}) \in \mathcal {S}_ Sys \mid \textbf{x} \uplus \textbf{i} \models _{T} \varphi \wedge \varphi _I \}. \end{array} $$</span></div></div><p>The <i>abstraction function</i> <span class="mathjax-tex">$\alpha : \mathcal {S}\rightarrow 2^V$</span> is defined s.t. <span class="mathjax-tex">$v\in \alpha (s)$</span> iff <span class="mathjax-tex">$s\in \gamma (v)$</span>. We extend both function from states to (finite or infinite) state sequences <span class="mathjax-tex">$\pi \in \mathcal {S}^\infty $</span> and <span class="mathjax-tex">$\widehat{\pi }\in V^\infty $</span> s.t.</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$ \begin{array}{ll} \gamma (\widehat{\pi }) &{}:=\{ \pi \in \mathcal {S}^\infty \mid |\pi |=|\widehat{\pi }| \wedge \forall i\in dom(\pi ).~\pi [i] \in \gamma (\widehat{\pi }[ i])\},~\text {and}\\ \alpha ( \pi ) &{}:=\{ \widehat{\pi }\in V^\infty \mid |\pi |=|\widehat{\pi }| \wedge \forall i\in dom(\pi ).~\widehat{\pi }[i] \in \alpha (\pi [ i])\}. \end{array} $$</span></div></div><p>Both functions naturally extend to <i>sets</i> of states or infinite sequences of states by letting <span class="mathjax-tex">$\gamma (A) : = \bigcup _{a\in A} \gamma (a)$</span> for <span class="mathjax-tex">$A \subseteq V$</span> and <span class="mathjax-tex">$A\subseteq V^\omega $</span> and <span class="mathjax-tex">$\alpha (C) : = \bigcup _{c\in C} \alpha (c)$</span> for <span class="mathjax-tex">$C \subseteq \mathcal {S}$</span> and <span class="mathjax-tex">$C\subseteq \mathcal {S}^\omega $</span>. Note that it follows from the partitioning conditions imposed on <span class="mathjax-tex">$(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}})$</span> in Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar13">6</a> that <span class="mathjax-tex">$\alpha $</span> is a total function and always maps states and state sequences to a singleton set. We abuse notation and write <span class="mathjax-tex">$\alpha (s) = v$</span> (resp. <span class="mathjax-tex">$\alpha (\pi ) = \widehat{\pi }$</span>) instead of <span class="mathjax-tex">$\alpha (s) = \{v\}$</span> (resp. <span class="mathjax-tex">$\alpha (\pi ) = \{\widehat{\pi }\}$</span>).</p><p>Let <span class="mathjax-tex">$\varOmega \subseteq \mathcal {S}^\infty $</span> be an objective for the semantic game <span class="mathjax-tex">$\llbracket \mathcal {G}\rrbracket $</span>. With the relational functions <span class="mathjax-tex">$\langle \alpha ,\gamma \rangle $</span> defined before, <span class="mathjax-tex">$\varOmega $</span> naturally induces an abstract objective <span class="mathjax-tex">$\widehat{\varOmega } := \alpha (\varOmega )\subseteq V^\infty $</span> over the abstract state space <i>V</i>.</p><p>Recall that we consider winning conditions <span class="mathjax-tex">$\varOmega \subseteq \mathcal {S}^\infty $</span> for <span class="mathjax-tex">$\llbracket \mathcal {G}\rrbracket $</span> defined over the set <i>L</i> of locations of <span class="mathjax-tex">$\mathcal {G}$</span>. As <span class="mathjax-tex">$\alpha $</span> preserves the location part of the states, <span class="mathjax-tex">$\widehat{\pi }\in \widehat{\varOmega }$</span> iff <span class="mathjax-tex">$\gamma (\widehat{\pi }) \subseteq \varOmega $</span>. That is, a sequence of abstract states is winning according to <span class="mathjax-tex">$\widehat{\varOmega }$</span> iff all the corresponding concrete state sequences are winning according to <span class="mathjax-tex">$\varOmega $</span>.</p><p>The next lemma states the correctness property that the abstraction satisfies. More concretely, <span class="mathjax-tex">$\widehat{G}^\uparrow $</span> overapproximates the winning region of player <span class="mathjax-tex">$ Sys $</span> in the concrete game, and <span class="mathjax-tex">$\widehat{G}^\downarrow $</span> underapproximates it.</p> <h3 class="c-article__sub-heading" id="FPar16">Lemma 4</h3> <p><b>(Correctness of the Abstraction).</b> Given a reactive program game structure <span class="mathjax-tex">$\mathcal {G}$</span> with semantics <span class="mathjax-tex">$\llbracket \mathcal {G}\rrbracket = (\mathcal {S}, \mathcal {S}_ Env ,\mathcal {S}_ Sys , \rho )$</span> and location-based objective <span class="mathjax-tex">$\varOmega $</span>, let <span class="mathjax-tex">$\widehat{G}^\circ =(V, V_ Env , V_ Sys , \widehat{\rho }^\circ )$</span> with <span class="mathjax-tex">$\circ \in \{\uparrow , \downarrow \}$</span> be its <span class="mathjax-tex">$(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}})$</span>-induced abstractions with relational functions <span class="mathjax-tex">$\langle \alpha ,\gamma \rangle $</span>. Then it holds that (1) <span class="mathjax-tex">$W_{ Sys }(\llbracket \mathcal {G}\rrbracket ,\varOmega ) \subseteq \gamma (W_{ Sys }(\widehat{G}^\uparrow ,\widehat{\varOmega }))$</span>, and (2) <span class="mathjax-tex">$\gamma (W_{ Sys }(\widehat{G}^\downarrow ,\widehat{\varOmega })) \subseteq W_{ Sys }(\llbracket \mathcal {G}\rrbracket ,\varOmega )$</span>.</p> <p>By duality, <span class="mathjax-tex">$\widehat{G}^\downarrow $</span> results in an overapproximation of the winning region of player <span class="mathjax-tex">$ Env $</span> in the concrete game. Given an abstraction <span class="mathjax-tex">$\widehat{G}^\circ $</span>, we denote with <span class="mathjax-tex">$ OverapproxP (\widehat{G}^\circ )$</span> the player whose winning region is overapproximated in <span class="mathjax-tex">$\widehat{G}^\circ $</span>: <span class="mathjax-tex">$ OverapproxP (\widehat{G}^\circ ):= Sys $</span> if <span class="mathjax-tex">$\circ =\uparrow $</span> and <span class="mathjax-tex">$ OverapproxP (\widehat{G}^\circ ):= Env $</span> if <span class="mathjax-tex">$\circ =\downarrow $</span>.</p><p><b>Abstract Strategy Templates and Their Induced Sub-games.</b> We now describe how we use a permissive strategy template for a player <i>p</i> in an abstract game to identify sub-game structures of the given reactive program game from which to generate attractor caches for player <i>p</i>.</p><p>We determine the sub-game structures and local target sets based on so-called <i>helpful edges</i> for player <i>p</i> in the abstract game where <i>p</i> is over-approximated. A helpful edge is a live-edge or an alternative choice to a co-live edge of a permissive strategy template. Intuitively, a helpful edge is an edge that player <i>p</i> might have to take eventually in order to win the abstract game. As our chosen abstraction domain is based on the guards, a helpful edge often corresponds to the change of conditions necessary to enable a guard in the reactive program game. Since reaching this change might require an unbounded number of steps, our method attempts a local attractor computation and potentially acceleration. Identifying helpful edges based on permissive strategy templates rather than on winning strategies has the following advantages. First, templates reflect multiple abstract winning strategies for player <i>p</i>, capturing multiple possibilities to make progress towards the objective. Moreover, they describe local conditions, facilitating the localization our method aims for. Helpful edges are defined as follows.</p> <h3 class="c-article__sub-heading" id="FPar17">Definition 8</h3> <p><b>(Helpful Edge).</b> Given a strategy template <span class="mathjax-tex">$(U,D,\mathcal {H})$</span> for player <i>p</i> in a game <span class="mathjax-tex">$(G,\varOmega )$</span> with <span class="mathjax-tex">$G=(V,V_ Env , V_ Sys ,\rho )$</span>, we call an edge <span class="mathjax-tex">$e \in \rho $</span> <i>helpful for player</i> <i>p</i> <i>w.r.t. the template</i> <span class="mathjax-tex">$(U,D,\mathcal {H})$</span> if and only if the following holds: There exists a live-group <span class="mathjax-tex">$H \in \mathcal {H}$</span> such that <span class="mathjax-tex">$e \in H$</span>, or <span class="mathjax-tex">$e \not \in U\cup D$</span> and there exists a co-live edge <span class="mathjax-tex">$(v_s,v_t) \in D$</span> with <span class="mathjax-tex">$v_s = \textsc {src}(e)$</span>. We define <span class="mathjax-tex">$\textsf{Helpful}_{G,p}(U,D,\mathcal {H})$</span> to be the set of helpful edges for player <i>p</i> in <i>G</i> w.r.t. <span class="mathjax-tex">$(U,D,\mathcal {H})$</span>.</p> <p>For each helpful edge, we define <i>pre-</i> and <i>post-sets</i> which are the abstract environment states before and after that edge. This is formalized as follows.</p> <h3 class="c-article__sub-heading" id="FPar18">Definition 9</h3> <p><b>(Pre- and Post-Sets).</b> Let <span class="mathjax-tex">$\widehat{G}^\circ =(V, V_ Env , V_ Sys , \widehat{\rho }^\circ )$</span> for some <span class="mathjax-tex">$\circ \in \{\uparrow ,\downarrow \}$</span> be a <span class="mathjax-tex">$(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}})$</span>-induced abstraction of <span class="mathjax-tex">$\mathcal {G}$</span>, let <span class="mathjax-tex">$p_{ over }:= OverapproxP (\widehat{G}^\circ )$</span>, and <span class="mathjax-tex">$e = (v_s,v_t) \in \textsf{Helpful}_{\widehat{G}^\circ ,p_{ over }}(U,D,\mathcal {H})$</span> for some template <span class="mathjax-tex">$(U,D,\mathcal {H})$</span>. If <span class="mathjax-tex">$p_{ over } = Env $</span>, we have <span class="mathjax-tex">$e \in V_ Env \times V_ Sys $</span> and define <span class="mathjax-tex">$\textsf{Pre}(e,p_{ over }) := \{ v_s \}$</span> and <span class="mathjax-tex">$\textsf{Post}(e,p_{ over }) := \{ v \in V \mid (v_t,v) \in \widehat{\rho }^\circ \}$</span>. If <span class="mathjax-tex">$p_{ over } = Sys $</span> we have that <span class="mathjax-tex">$e \in V_ Sys \times V_ Env $</span> and define <span class="mathjax-tex">$\textsf{Pre}(e,p_{ over }) := \{ v \in V \mid (v,v_s) \in \widehat{\rho }^\circ \}$</span> and <span class="mathjax-tex">$\textsf{Post}(e,p_{ over }) := \{ v_t \}$</span>. Note that in both cases it holds that <span class="mathjax-tex">$\textsf{Pre}(e,p_{ over }), \textsf{Post}(e,p_{ over }) \subseteq V_ Env \subseteq L \times \mathcal P_{\mathbb {X}}$</span>.</p> <div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-e" data-title="Algorithm 3"><figure><figcaption><b id="Fige" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 3</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/10.1007/978-3-031-65633-0_7/figures/e" rel="nofollow"><picture><img aria-describedby="Fige" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Fige_HTML.png" alt="figure e" loading="lazy" width="685" height="326"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-e-desc"><p>Generation of a cache based on a strategy template.</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/10.1007/978-3-031-65633-0_7/figures/e" data-track-dest="link:Figuree Full size image" aria-label="Full size image figure e" 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>As a helpful edge represents potential “progress” for player <i>p</i>, we consider the question of whether player <i>p</i> has a strategy in the concrete game to reach the <i>post-set</i> from the <i>pre-set</i>. This motivates the construction of sub-game structures induced by the locations connecting those two sets in the reactive program game.</p><p>Procedure <span class="u-small-caps">GenerateCache</span> in Algorithm 3 formalizes this idea. It takes an abstract game and a strategy template for the over-approximated player <span class="mathjax-tex">$p_ over $</span> in this game. For each helpful edge <i>e</i>, it constructs the sub-game structure induced by the set of locations that lie on a simple path in the location graph from the locations of the <i>pre-set</i> to the <i>post-set</i> of <i>e</i>. The optional parameter <i>b</i> allows for heuristically tuning the locality of the sub-games by bounding the paths’ length.</p><p>For each sub-game structure, the target sets for the local attractor computations are determined by the post-sets of the helpful edges that induced this sub-game structure (it might be more than one). They are computed by</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textsc {ConstructTargets}(L_ sub , PostSet ) = T_1 \cup T_2 \cup T_3 \end{aligned}$$</span></div><div class="c-article-equation__number"> (1) </div></div><p>where the sets <span class="mathjax-tex">$T_1,T_2 \text { and }T_3$</span> of elements of <span class="mathjax-tex">$\mathcal {D}$</span> are defined as follows.</p><ul class="u-list-style-dash"> <li> <p><span class="mathjax-tex">$T_1:=\{d\in \mathcal {D}\mid \exists P. \;(L_ sub ,P) \in PostSet \wedge \forall l \in L.\;d(l) = \bigvee _{(l,\varphi ) \in P} \varphi \}$</span> consists of targets that are determined by a single post-set.</p> </li> <li> <p><span class="mathjax-tex">$T_2:=\{d_\cup \}$</span>, where for every <span class="mathjax-tex">$l \in L$</span>, <span class="mathjax-tex">$d_\cup (l) = \bigvee _{P \text { s.t. } (L_ sub ,P) \in PostSet } \bigvee _{(l,\varphi ) \in P} \varphi $</span> is the singleton containing the union of the targets of all post-sets.</p> </li> <li> <p><span class="mathjax-tex">$T_3:=\{d_\top \}$</span>, where for <span class="mathjax-tex">$l \in L$</span>, <span class="mathjax-tex">$d_\top (l)= \exists P, \varphi . (L_ sub ,P) \in PostSet \wedge (\varphi , l) \in P$</span> contains the target that is <span class="mathjax-tex">$\top $</span> iff the location appears in some post-set.</p> </li> </ul> <p>Once the targets are constructed, <span class="u-small-caps">GenerateCache</span> uses <span class="u-small-caps">SubgameCache</span> from Algorithm 2 to compute the attractor caches for those targets and respective sub-game structures. By Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar11">3</a>, <span class="u-small-caps">SubgameCache</span> returns attractor caches. As attractor caches are closed under set union, we conclude the following.</p> <h3 class="c-article__sub-heading" id="FPar19">Corollary 1</h3> <p>The set <i>C</i> returned by <span class="mathjax-tex">$\textsc {GenerateCache}$</span> is an attractor cache.</p> <h3 class="c-article__sub-heading" id="FPar20">Example 6</h3> <p>The abstractions of <span class="mathjax-tex">$\mathcal {G}_ ex $</span> from Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar1">1</a> and respective templates are too large to depict. One helpful edge for <span class="mathjax-tex">$ Sys $</span> is <span class="mathjax-tex">$e = (( mine , \varphi , \varphi _I), ( mine , \varphi '))$</span> with <img src="//media.springernature.com/lw288/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figf_HTML.gif" alt="">, <img src="//media.springernature.com/lw296/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figg_HTML.gif" alt="">, and <span class="mathjax-tex">$\varphi _I = a > 0 \wedge b \le 0 \wedge inpReq \le 0$</span>. This edge <i>e</i> is in a live group where the other edges are similar with different <span class="mathjax-tex">$\varphi _I$</span>. They correspond to the situation where the value of <span class="mathjax-tex">$ samp $</span> finally becomes greater or equal to <span class="mathjax-tex">$ req $</span>. For <i>e</i>, <span class="mathjax-tex">$\textsf{Pre}(e, Sys ) = \{( mine , \varphi )\}$</span> and <span class="mathjax-tex">$\textsf{Post}(e, Sys ) = \{( mine , \varphi ')\}$</span> result in <span class="mathjax-tex">$L_ sub = \{ mine \}$</span> and the target <span class="mathjax-tex">$\{ mine \mapsto \varphi ' \}$</span>. With this, we generate a cache as in Example <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar12">4</a>.</p> </div></div></section><section data-title="Game Solving with Abstract Template-Based Caching"><div class="c-article-section" id="Sec7-section"><h2 id="Sec7" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">5 </span>Game Solving with Abstract Template-Based Caching</h2><div class="c-article-section__content" id="Sec7-content"><div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-h" data-title="Algorithm 4"><figure><figcaption><b id="Figh" class="c-article-section__figure-caption" data-test="figure-caption-text">Algorithm 4</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/10.1007/978-3-031-65633-0_7/figures/h" rel="nofollow"><picture><img aria-describedby="Figh" src="//media.springernature.com/lw685/springer-static/image/chp%3A10.1007%2F978-3-031-65633-0_7/MediaObjects/563039_1_En_7_Figh_HTML.png" alt="figure h" loading="lazy" width="685" height="235"></picture></a></div><div class="c-article-section__figure-description" data-test="bottom-caption" id="figure-h-desc"><p>Game solving with abstract template-based caching.</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/10.1007/978-3-031-65633-0_7/figures/h" data-track-dest="link:Figureh Full size image" aria-label="Full size image figure h" 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>This section summarizes our approach for reactive progam game solving via Algorithm 4, which combines the procedures introduced in Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec3">3</a> and Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec4">4</a> as schematically illustrated in Fig. <a data-track="click" data-track-label="link" data-track-action="figure anchor" href="#Fig1">1</a> of Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec1">1</a>. Algorithm 4 starts by computing the abstract domain and both abstractions. For each abstract game, <span class="mathjax-tex">$\textsc {SolveAbstract}$</span> computes a strategy template [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Anand, A., Nayak, S.P., Schmuck, A.: Synthesizing permissive winning strategy templates for parity games. In: Enea, C., Lal, A. (eds.) CAV 2023, Part I. LNCS, vol. 13964, pp. 436–458. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-37706-8_22
 
 " href="#ref-CR2" id="ref-link-section-d2892493e30765">2</a>]. Then, <span class="u-small-caps">GenerateCache</span> is invoked to construct the respective attractor cache. <span class="u-small-caps">RPGSolveWithCache</span> solves reactive program games in direct analogy to <span class="u-small-caps">RPGSolve</span> from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e30777">21</a>], but instead of using <span class="u-small-caps">AttractorAcc</span>, it uses the new algorithm <span class="u-small-caps">AttractorAccCache</span> which utilizes the attractor cache <i>C</i>. The overall correctness of <span class="u-small-caps">RPGCacheSolve</span> follows from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar7">1</a>, Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar19">1</a>, and the correctness of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e30799">21</a>].</p> <h3 class="c-article__sub-heading" id="FPar21">Theorem 1</h3> <p><b>(Correctness).</b> Given a reactive program game structure <span class="mathjax-tex">$\mathcal {G}$</span> and a location-based objective <span class="mathjax-tex">$\varOmega $</span>, for any <span class="mathjax-tex">$b \in \mathbb {N}$</span>, if <span class="u-small-caps">RPGCacheSolve</span> terminates, then it returns <span class="mathjax-tex">$W_{ Sys }(\llbracket \mathcal {G}\rrbracket ,\varOmega ).$</span></p> <h3 class="c-article__sub-heading" id="FPar22">Remark 1</h3> <p>In addition to using the strategy templates from the abstract games for caching, we can make use of the winning regions in the abstract games, which are computed together with the templates. Thanks to Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="#FPar16">4</a>, we know that outside of its winning region in the abstract game the over-approximated player loses for sure. Thus, we can <i>prune</i> parts of the reactive program game that correspond to the abstract states where the over-approximated player loses. As our experiments show that the main performance advantage is gained by caching rather than pruning, we give the formal details for pruning in the extended version [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Schmuck, A.K., Heim, P., Dimitrova, R., Nayak, S.P.: Localized attractor computations for infinite-state games (full version) (2024). 
 https://doi.org/10.48550/ARXIV.2405.09281
 
 " href="#ref-CR34" id="ref-link-section-d2892493e30939">34</a>].</p> <p><i>Discussion.</i> The procedure <span class="u-small-caps">RPGCacheSolve</span> depends on the choice of game abstraction domain <span class="mathjax-tex">$(\mathcal P_{\mathbb {X}},\mathcal {P}_{\mathbb {X}\cup \mathbb {I}})$</span> and on the construction of the local games performed in <span class="u-small-caps">GenerateCache</span>. The abstraction based on guards is natural, as it is obtained from the predicates appearing in the game. Acceleration [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e31007">21</a>] is often needed to establish that some guards can eventually be enabled. Therefore, we choose an abstraction domain that represents precisely the guards in the game.</p><p>Helpful edges capture transitions that a player might need to take, hence the game solving procedure has to establish that the player can eventually enable their guards. This might require acceleration, and hence motivates our use of helpful edges to construct the local games. Investigating alternatives to these design choices and their further refinement is a subject of future work.</p></div></div></section><section data-title="Experimental Evaluation"><div class="c-article-section" id="Sec8-section"><h2 id="Sec8" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">6 </span>Experimental Evaluation</h2><div class="c-article-section__content" id="Sec8-content"><div class="c-article-table" data-test="inline-table" data-container-section="table" id="table-1"><figure><figcaption class="c-article-table__figcaption"><b id="Tab1" data-test="table-caption">Table 1. Evaluation Results. ST is the variable domain type (additional to <span class="mathjax-tex">$\mathbb {B}$</span>). |<i>L</i>|, <span class="mathjax-tex">$|\mathbb {X}|$</span>, <span class="mathjax-tex">$|\mathbb {I}|$</span> are the number of respective game elements. We show the wall-clock running time in seconds for our prototype <span class="u-sans-serif">rpg-STeLA</span> in three settings (one with normal caching, one with additional pruning, one that only prunes), <span class="u-sans-serif">rpgsolve</span>, and <span class="u-sans-serif">MuVal</span> (with clause exchange). TO means timeout after 30 min, MO means out of memory (8GB). We highlight in bold the fastest solving runtime result. The evaluation was performed on a computer equipped with an Intel(R) Core(TM) i5-10600T CPU @ 2.40 GHz.</b></figcaption><div class="u-text-right u-hide-print"><a class="c-article__pill-button" data-test="table-link" data-track="click" data-track-action="view table" data-track-label="button" rel="nofollow" href="/chapter/10.1007/978-3-031-65633-0_7/tables/1" aria-label="Full size table 1"><span>Full size table</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>We implemented Algorithm 4 for solving reactive program games in a prototype tool<sup><a href="#Fn2"><span class="u-visually-hidden">Footnote </span>2</a></sup> <span class="u-sans-serif">rpg-STeLA</span> (Strategy Template-based Localized Acceleration). Our implementation is based on the open-source reactive program game solver <span class="u-sans-serif">rpgsolve</span> from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e33702">21</a>]. Specifically, we use <span class="u-sans-serif">rpgsolve</span> for the <span class="u-small-caps">AttractorAcc</span> and <span class="u-small-caps">RPGSolveWithCache</span> methods to compute attractors via acceleration and to solve reactive program games utilizing the precomputed cache, respectively. We realize <span class="u-small-caps">SolveAbstract</span> by using <span class="u-sans-serif">PeSTel</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Anand, A., Nayak, S.P., Schmuck, A.: Synthesizing permissive winning strategy templates for parity games. In: Enea, C., Lal, A. (eds.) CAV 2023, Part I. LNCS, vol. 13964, pp. 436–458. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-37706-8_22
 
 " href="#ref-CR2" id="ref-link-section-d2892493e33721">2</a>], which computes strategy templates in finite games. We do not use the bound <i>b</i> in Algorithm 4.</p><p>We compare our tool <span class="u-sans-serif">rpg-STeLA</span> to the solver <span class="u-sans-serif">rpgsolve</span> and the <span class="mathjax-tex">$\mu $</span>CLP solver <span class="u-sans-serif">MuVal</span> [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Unno, H., Satake, Y., Terauchi, T., Koskinen, E.: Program verification via predicate constraint satisfiability modulo theories. CoRR abs/2007.03656 (2020). 
 https://arxiv.org/abs/2007.03656
 
 " href="#ref-CR36" id="ref-link-section-d2892493e33756">36</a>]. Those are the only available techniques that can handle unbounded strategy loops, as stated in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e33760">21</a>]. Other tools from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="Beyene, T.A., Chaudhuri, S., Popeea, C., Rybalchenko, A.: A constraint-based approach to solving games on infinite graphs. In: Jagannathan, S., Sewell, P. (eds.) The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ’14, San Diego, CA, USA, January 20–21, 2014, pp. 221–234. ACM (2014). 
 https://doi.org/10.1145/2535838.2535860
 
 " href="#ref-CR5" id="ref-link-section-d2892493e33763">5</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Choi, W., Finkbeiner, B., Piskac, R., Santolucito, M.: Can reactive synthesis and syntax-guided synthesis be friends? In: Jhala, R., Dillig, I. (eds.) PLDI ’22: 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation, San Diego, CA, USA, 13–17 June, 2022, pp. 229–243. ACM (2022). 
 https://doi.org/10.1145/3519939.3523429
 
 " href="#ref-CR8" id="ref-link-section-d2892493e33766">8</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Faella, M., Parlato, G.: Reachability games modulo theories with a bounded safety player. In: Proceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence. AAAI’23/IAAI’23/EAAI’23. AAAI Press (2023). 
 https://doi.org/10.1609/aaai.v37i5.25779
 
 " href="#ref-CR9" id="ref-link-section-d2892493e33766_1">9</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Farzan, A., Kincaid, Z.: Strategy synthesis for linear arithmetic games. Proc. ACM Program. Lang. 2(POPL), 61:1-61:30 (2018). 
 https://doi.org/10.1145/3158149
 
 " href="#ref-CR10" id="ref-link-section-d2892493e33769">10</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Maderbacher, B., Bloem, R.: Reactive synthesis modulo theories using abstraction refinement. In: Griggio, A., Rungta, N. (eds.) 22nd Formal Methods in Computer-Aided Design, FMCAD 2022, Trento, Italy, October 17-21, 2022, pp. 315–324. IEEE (2022). 
 https://doi.org/10.34727/2022/ISBN.978-3-85448-053-2_38
 
 " href="#ref-CR27" id="ref-link-section-d2892493e33772">27</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Markgraf, O., Hong, C.-D., Lin, A.W., Najib, M., Neider, D.: Parameterized synthesis with safety properties. In: Oliveira, B.C.S. (ed.) APLAS 2020. LNCS, vol. 12470, pp. 273–292. Springer, Cham (2020). 
 https://doi.org/10.1007/978-3-030-64437-6_14
 
 " href="#ref-CR28" id="ref-link-section-d2892493e33775">28</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Neider, D., Topcu, U.: An automaton learning approach to solving safety games over infinite graphs. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 204–221. Springer, Heidelberg (2016). 
 https://doi.org/10.1007/978-3-662-49674-9_12
 
 " href="#ref-CR31" id="ref-link-section-d2892493e33779">31</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Samuel, S., D’Souza, D., Komondoor, R.: Gensys: a scalable fixed-point engine for maximal controller synthesis over infinite state spaces. In: Spinellis, D., Gousios, G., Chechik, M., Penta, M.D. (eds.) ESEC/FSE ’21: 29th ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineering, Athens, Greece, August 23–28, 2021, pp. 1585–1589. ACM (2021). 
 https://doi.org/10.1145/3468264.3473126
 
 " href="#ref-CR32" id="ref-link-section-d2892493e33779_1">32</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Samuel, S., D’Souza, D., Komondoor, R.: Symbolic fixpoint algorithms for logical LTL games. In: 38th IEEE/ACM International Conference on Automated Software Engineering, ASE 2023, Luxembourg, September 11–15, 2023, pp. 698–709. IEEE (2023). 
 https://doi.org/10.1109/ASE56229.2023.00212
 
 " href="#ref-CR33" id="ref-link-section-d2892493e33782">33</a>] cannot handle those, are outperformed by <span class="u-sans-serif">rpgsolve</span>, or are not available. For <span class="u-sans-serif">MuVal</span>, we encoded the games into <span class="mathjax-tex">$\mu $</span>CLP as outlined in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Unno, H., Satake, Y., Terauchi, T., Koskinen, E.: Program verification via predicate constraint satisfiability modulo theories. CoRR abs/2007.03656 (2020). 
 https://arxiv.org/abs/2007.03656
 
 " href="#ref-CR36" id="ref-link-section-d2892493e33808">36</a>] and done in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e33812">21</a>].</p><p><i>Benchmarks.</i> We performed the evaluation on three newly introduced sets of benchmarks (described in detail in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Schmuck, A.K., Heim, P., Dimitrova, R., Nayak, S.P.: Localized attractor computations for infinite-state games (full version) (2024). 
 https://doi.org/10.48550/ARXIV.2405.09281
 
 " href="#ref-CR34" id="ref-link-section-d2892493e33820">34</a>]). They all have unbounded variable ranges, contain unbounded strategy loops, and have Büchi winning conditions. On the literature benchmarks from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Bodlaender, M.H.L., Hurkens, C.A.J., Kusters, V.J.J., Staals, F., Woeginger, G.J., Zantema, H.: Cinderella versus the wicked stepmother. In: Baeten, J.C.M., Ball, T., de Boer, F.S. (eds.) TCS 2012. LNCS, vol. 7604, pp. 57–71. Springer, Heidelberg (2012). 
 https://doi.org/10.1007/978-3-642-33475-7_5
 
 " href="#ref-CR6" id="ref-link-section-d2892493e33823">6</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e33826">21</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Maderbacher, B., Bloem, R.: Reactive synthesis modulo theories using abstraction refinement. In: Griggio, A., Rungta, N. (eds.) 22nd Formal Methods in Computer-Aided Design, FMCAD 2022, Trento, Italy, October 17-21, 2022, pp. 315–324. IEEE (2022). 
 https://doi.org/10.34727/2022/ISBN.978-3-85448-053-2_38
 
 " href="#ref-CR27" id="ref-link-section-d2892493e33829">27</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Neider, D., Topcu, U.: An automaton learning approach to solving safety games over infinite graphs. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 204–221. Springer, Heidelberg (2016). 
 https://doi.org/10.1007/978-3-662-49674-9_12
 
 " href="#ref-CR31" id="ref-link-section-d2892493e33832">31</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="Woeginger: Combinatorics problem c5 (2009)" href="#ref-CR39" id="ref-link-section-d2892493e33836">39</a>] <span class="u-sans-serif">rpgsolve</span> performs well as [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e33842">21</a>] shows. Hence, we did not use them as local attractor caches are unnecessary, and they are smaller than our new ones. Our new benchmark categories are:</p><p><i>(1) Complex Global Strategy (Scheduler and Item Processing).</i> These benchmarks consist of a scheduler and an item processing unit. The core feature of these benchmarks is that the system needs to perform tasks that require complex global strategic decisions and local strategic decisions requiring acceleration.</p><p><i>(2) Parametric Benchmarks (Chains).</i> These benchmarks each consist of two parametric chains of local sub-tasks requiring acceleration and local strategic reasoning and more lightweight global strategic reasoning. The number of variables scales differently in both chains, showcasing differences in scalability.</p><p><i>(3) Simple Global Strategy (Robot and Smart Home).</i> These benchmarks represent different tasks for a robot and a smart home. The robot moves along tracks (with one-dimensional discrete position) and must perform tasks like collecting several products. The smart home must, e.g., maintain temperature levels and adjust blinds depending on whether the house is empty or on the current time of day. These benchmarks need acceleration and local strategic reasoning, but their global reasoning is usually simpler and more deterministic.</p><p><i>Analysis.</i> The experimental results in Table <a data-track="click" data-track-label="link" data-track-action="table anchor" href="#Tab1">1</a> demonstrate that local attractor pre-computation and caching have a significant impact on solving complex games. This is evidenced by the performance of <span class="u-sans-serif">rpg-STeLA</span> that is superior to the other two tools. We further see that pruning (without caching) is not sufficient, which underscores the need to use more elaborate local strategic information in the form of an attractor cache. This necessitates the computation of strategy templates, and simply solving an abstract game is insufficient. However, as pruning does not cause significant overhead, it offers an additional optimization.</p></div></div></section><section data-title="Related Work"><div class="c-article-section" id="Sec9-section"><h2 id="Sec9" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number">7 </span>Related Work</h2><div class="c-article-section__content" id="Sec9-content"><p>A body of methods for solving infinite-state games and synthesizing reactive systems operating over unbounded data domains exists. Abstraction-based approaches reduce the synthesis problem to the finite-state case. Those include abstraction of two-player games [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Finkbeiner, B., Mallik, K., Passing, N., Schledjewski, M., Schmuck, A.: BOCoSy: small but powerful symbolic output-feedback control. In: Bartocci, E., Putot, S. (eds.) HSCC ’22: 25th ACM International Conference on Hybrid Systems: Computation and Control, Milan, Italy, May 4–6, 2022, pp. 24:1–24:11. ACM (2022). 
 https://doi.org/10.1145/3501710.3519535
 
 " href="#ref-CR15" id="ref-link-section-d2892493e33880">15</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Grumberg, O., Lange, M., Leucker, M., Shoham, S.: When not losing is better than winning: Abstraction and refinement for the full mu-calculus. Inf. Comput. 205(8), 1130–1148 (2007). 
 https://doi.org/10.1016/j.ic.2006.10.009
 
 " href="#ref-CR19" id="ref-link-section-d2892493e33883">19</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Henzinger, T.A., Jhala, R., Majumdar, R.: Counterexample-guided control. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 886–902. Springer, Heidelberg (2003). 
 https://doi.org/10.1007/3-540-45061-0_69
 
 " href="#ref-CR23" id="ref-link-section-d2892493e33886">23</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 37" title="Vechev, M.T., Yahav, E., Yorsh, G.: Abstraction-guided synthesis of synchronization. Int. J. Softw. Tools Technol. Transf. 15(5–6), 413–431 (2013). 
 https://doi.org/10.1007/S10009-012-0232-3
 
 " href="#ref-CR37" id="ref-link-section-d2892493e33889">37</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Walker, A., Ryzhyk, L.: Predicate abstraction for reactive synthesis. In: Formal Methods in Computer-Aided Design, FMCAD 2014, Lausanne, Switzerland, October 21–24, 2014. pp. 219–226. IEEE (2014). 
 https://doi.org/10.1109/FMCAD.2014.6987617
 
 " href="#ref-CR38" id="ref-link-section-d2892493e33892">38</a>], which extends ideas from verification, such as abstract interpretation and counterexample-guided abstraction refinement, to games. The temporal logic LTL has recently been extended with data properties, resulting in TSL [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Finkbeiner, B., Klein, F., Piskac, R., Santolucito, M.: Temporal stream logic: synthesis beyond the bools. In: Dillig, I., Tasiran, S. (eds.) CAV 2019. LNCS, vol. 11561, pp. 609–629. Springer, Cham (2019). 
 https://doi.org/10.1007/978-3-030-25540-4_35
 
 " href="#ref-CR14" id="ref-link-section-d2892493e33896">14</a>] and its extension with logical theories [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 13" title="Finkbeiner, B., Heim, P., Passing, N.: Temporal stream logic modulo theories. In: FoSSaCS 2022. LNCS, vol. 13242, pp. 325–346. Springer, Cham (2022). 
 https://doi.org/10.1007/978-3-030-99253-8_17
 
 " href="#ref-CR13" id="ref-link-section-d2892493e33899">13</a>]. Synthesis techniques for those [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Choi, W., Finkbeiner, B., Piskac, R., Santolucito, M.: Can reactive synthesis and syntax-guided synthesis be friends? In: Jhala, R., Dillig, I. (eds.) PLDI ’22: 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation, San Diego, CA, USA, 13–17 June, 2022, pp. 229–243. ACM (2022). 
 https://doi.org/10.1145/3519939.3523429
 
 " href="#ref-CR8" id="ref-link-section-d2892493e33902">8</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Finkbeiner, B., Klein, F., Piskac, R., Santolucito, M.: Temporal stream logic: synthesis beyond the bools. In: Dillig, I., Tasiran, S. (eds.) CAV 2019. LNCS, vol. 11561, pp. 609–629. Springer, Cham (2019). 
 https://doi.org/10.1007/978-3-030-25540-4_35
 
 " href="#ref-CR14" id="ref-link-section-d2892493e33905">14</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Maderbacher, B., Bloem, R.: Reactive synthesis modulo theories using abstraction refinement. In: Griggio, A., Rungta, N. (eds.) 22nd Formal Methods in Computer-Aided Design, FMCAD 2022, Trento, Italy, October 17-21, 2022, pp. 315–324. IEEE (2022). 
 https://doi.org/10.34727/2022/ISBN.978-3-85448-053-2_38
 
 " href="#ref-CR27" id="ref-link-section-d2892493e33908">27</a>] are based on propositional abstraction of the temporal specification and iterative refinement by introducing assumptions. The synthesis task’s main burden in abstraction-based methods falls on the finite-state synthesis procedure. In contrast, we use abstraction not as the core solving mechanism but as a means to derive helpful sub-games. Another class of techniques reason directly over the infinite-state space. Several constraint-based approaches [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Faella, M., Parlato, G.: Reachability games modulo theories with a bounded safety player. In: Proceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence. AAAI’23/IAAI’23/EAAI’23. AAAI Press (2023). 
 https://doi.org/10.1609/aaai.v37i5.25779
 
 " href="#ref-CR9" id="ref-link-section-d2892493e33911">9</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Farzan, A., Kincaid, Z.: Strategy synthesis for linear arithmetic games. Proc. ACM Program. Lang. 2(POPL), 61:1-61:30 (2018). 
 https://doi.org/10.1145/3158149
 
 " href="#ref-CR10" id="ref-link-section-d2892493e33915">10</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="Katis, A., et al.: Validity-guided synthesis of reactive systems from assume-guarantee contracts. In: Beyer, D., Huisman, M. (eds.) TACAS 2018. LNCS, vol. 10806, pp. 176–193. Springer, Cham (2018). 
 https://doi.org/10.1007/978-3-319-89963-3_10
 
 " href="#ref-CR24" id="ref-link-section-d2892493e33918">24</a>] have been proposed for specific types of objectives. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Samuel, S., D’Souza, D., Komondoor, R.: Gensys: a scalable fixed-point engine for maximal controller synthesis over infinite state spaces. In: Spinellis, D., Gousios, G., Chechik, M., Penta, M.D. (eds.) ESEC/FSE ’21: 29th ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineering, Athens, Greece, August 23–28, 2021, pp. 1585–1589. ACM (2021). 
 https://doi.org/10.1145/3468264.3473126
 
 " href="#ref-CR32" id="ref-link-section-d2892493e33921">32</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Samuel, S., D’Souza, D., Komondoor, R.: Symbolic fixpoint algorithms for logical LTL games. In: 38th IEEE/ACM International Conference on Automated Software Engineering, ASE 2023, Luxembourg, September 11–15, 2023, pp. 698–709. IEEE (2023). 
 https://doi.org/10.1109/ASE56229.2023.00212
 
 " href="#ref-CR33" id="ref-link-section-d2892493e33924">33</a>] lift fixpoint-based methods for finite-state game solving to a symbolic representation of infinite state sets. However, a naive iterative fixpoint computation can be successful on a relatively limited class of games. Recently, [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Heim, P., Dimitrova, R.: Solving infinite-state games via acceleration. Proc. ACM Program. Lang. 8(POPL) (2024). 
 https://doi.org/10.1145/3632899
 
 " href="#ref-CR21" id="ref-link-section-d2892493e33927">21</a>] proposed a technique that addresses this limitation by accelerating symbolic attractor computations. However, as we demonstrate, their approach has limited scalability when the size of the game structure grows. Our method mitigates this by identifying small helpful sub-games and composing their solutions to solve the game.</p><p>There are many approaches for compositional synthesis from LTL specifications [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Filiot, E., Jin, N., Raskin, J.: Antichains and compositional algorithms for LTL synthesis. Formal Methods Syst. Des. 39(3), 261–296 (2011). 
 https://doi.org/10.1007/S10703-011-0115-3
 
 " href="#ref-CR11" id="ref-link-section-d2892493e33933">11</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Finkbeiner, B., Geier, G., Passing, N.: Specification decomposition for reactive synthesis. Innov. Syst. Softw. Eng. 19(4), 339–357 (2023). 
 https://doi.org/10.1007/S11334-022-00462-6
 
 " href="#ref-CR12" id="ref-link-section-d2892493e33936">12</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Finkbeiner, B., Passing, N.: Dependency-based compositional synthesis. In: Hung, D.V., Sokolsky, O. (eds.) ATVA 2020. LNCS, vol. 12302, pp. 447–463. Springer, Cham (2020). 
 https://doi.org/10.1007/978-3-030-59152-6_25
 
 " href="#ref-CR16" id="ref-link-section-d2892493e33939">16</a>]. To the best of our knowledge, no techniques for decomposing infinite-state games exist prior to our work.</p><p>In verification, acceleration [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 3" title="Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: FAST: fast acceleration of symbolic transition systems. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 118–121. Springer, Heidelberg (2003). 
 https://doi.org/10.1007/978-3-540-45069-6_12
 
 " href="#ref-CR3" id="ref-link-section-d2892493e33945">3</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 4" title="Bardin, S., Finkel, A., Leroux, J., Schnoebelen, P.: Flat acceleration in symbolic model checking. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 474–488. Springer, Heidelberg (2005). 
 https://doi.org/10.1007/11562948_35
 
 " href="#ref-CR4" id="ref-link-section-d2892493e33948">4</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Finkel, A., Leroux, J.: How to compose presburger-accelerations: applications to broadcast protocols. In: Agrawal, M., Seth, A. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 145–156. Springer, Heidelberg (2002). 
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 " href="#ref-CR17" id="ref-link-section-d2892493e33951">17</a>] and loop summarization [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Kroening, D., Sharygina, N., Tonetta, S., Tsitovich, A., Wintersteiger, C.M.: Loop summarization using state and transition invariants. Formal Methods Syst. Des. 42(3), 221–261 (2013). 
 https://doi.org/10.1007/s10703-012-0176-y
 
 " href="#ref-CR26" id="ref-link-section-d2892493e33954">26</a>] are applied to the loops in <i>given program</i> and can thus be easily combined with subsequent analysis. In contrast, in the setting of games, acceleration relies on <i>establishing the existence of a strategy</i> which needs more guidance.</p><p>Permissive strategy templates were introduced in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Anand, A., Mallik, K., Nayak, S.P., Schmuck, A.K.: Computing adequately permissive assumptions for synthesis. In: Sankaranarayanan, S., Sharygina, N. (eds.) Tools and Algorithms for the Construction and Analysis of Systems, pp. 211–228. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-30820-8_15
 
 " href="#ref-CR1" id="ref-link-section-d2892493e33967">1</a>] and used in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Anand, A., Nayak, S.P., Schmuck, A.: Synthesizing permissive winning strategy templates for parity games. In: Enea, C., Lal, A. (eds.) CAV 2023, Part I. LNCS, vol. 13964, pp. 436–458. Springer, Cham (2023). 
 https://doi.org/10.1007/978-3-031-37706-8_22
 
 " href="#ref-CR2" id="ref-link-section-d2892493e33970">2</a>] to represent sets of winning strategies for the system player in two-player games. They were used to synthesize hybrid controllers for non-linear dynamical systems [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Nayak, S.P., Egidio, L.N., Della Rossa, M., Schmuck, A.K., Jungers, R.M.: Context-triggered abstraction-based control design. IEEE Open J. Control Syst. 2, 277–296 (2023). 
 https://doi.org/10.1109/OJCSYS.2023.3305835
 
 " href="#ref-CR30" id="ref-link-section-d2892493e33973">30</a>]. Similar to our work, [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Nayak, S.P., Egidio, L.N., Della Rossa, M., Schmuck, A.K., Jungers, R.M.: Context-triggered abstraction-based control design. IEEE Open J. Control Syst. 2, 277–296 (2023). 
 https://doi.org/10.1109/OJCSYS.2023.3305835
 
 " href="#ref-CR30" id="ref-link-section-d2892493e33976">30</a>] uses templates over abstractions to localize the compuation of continuous feedback controllers. While this inspired the solution methodology for infinite-state systems developed in this paper, the abstraction methodology and the semantics of the underlying system and its controllers are very different in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Nayak, S.P., Egidio, L.N., Della Rossa, M., Schmuck, A.K., Jungers, R.M.: Context-triggered abstraction-based control design. IEEE Open J. Control Syst. 2, 277–296 (2023). 
 https://doi.org/10.1109/OJCSYS.2023.3305835
 
 " href="#ref-CR30" id="ref-link-section-d2892493e33979">30</a>]. Our work is the first which uses permissive strategy templates as a guide for localizing the computation of fixpoints in infinite-state games.</p></div></div></section><section data-title="Conclusion"><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">8 </span>Conclusion</h2><div class="c-article-section__content" id="Sec10-content"><p>We presented a method that extends the applicability of synthesis over infinite-state games towards realistic applications. The key idea is to reduce the game solving problem to smaller and simpler sub-problems by utilizing winning strategy templates computed in finite abstractions of the infinite-state game. The resulting sub-problems are solved using a symbolic method based on attractor acceleration. Thus, in our approach abstraction and symbolic game solving work in concert, using strategy templates as the interface between them. This opens up multiple avenues for future work, such as exploring different abstraction techniques, as well as developing data-flow analysis techniques for reactive program games that can be employed in the context of symbolic game-solving procedures.</p></div></div></section> </div> <section data-title="Data Availability Statement"><div class="c-article-section" id="data-availability-statement-section"><h2 id="data-availability-statement" class="c-article-section__title js-section-title js-c-reading-companion-sections-item"><span class="c-article-section__title-number"> </span>Data Availability Statement</h2><div class="c-article-section__content" id="data-availability-statement-content"> <p>The software generated during and/or analysed during the current study is available in the Zenodo repository [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Heim, P., Nayak, S.P., Dimitrova, R., Schmuck, A.K.: Artifact of “Localized Attractor Computations for Infinite-State Games” (2024). 
 https://doi.org/10.5281/zenodo.10939871
 
 " href="#ref-CR22" id="ref-link-section-d2892493e34003">22</a>]. A full version of this paper including proofs is available through arXiv [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Schmuck, A.K., Heim, P., Dimitrova, R., Nayak, S.P.: Localized attractor computations for infinite-state games (full version) (2024). 
 https://doi.org/10.48550/ARXIV.2405.09281
 
 " href="#ref-CR34" id="ref-link-section-d2892493e34006">34</a>].</p> </div></div></section><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>See Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="#Sec9">7</a> for a detailed discussion of related work.</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>Available at <a href="https://doi.org/10.5281/zenodo.10939871">https://doi.org/10.5281/zenodo.10939871</a>.</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">Anand, A., Mallik, K., Nayak, S.P., Schmuck, A.K.: Computing adequately permissive assumptions for synthesis. 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(2024). Localized Attractor Computations for Infinite-State Games. In: Gurfinkel, A., Ganesh, V. (eds) Computer Aided Verification. CAV 2024. Lecture Notes in Computer Science, vol 14683. 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