[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81859-en":3,"doc-seo-81859-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81859,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Toward Satisfiability Modulo Realizability","Problems complete for the existential theory of the reals (∃R) appear across discrete geometry. The work presents satisfiability modulo realizability, a SAT-based method for solving satisfiable ∃R instances by ensuring solutions correspond to realizable geometric configurations. It encodes an underapproximation as a SAT instance over abstract order types. Because almost all abstract order types are unrealizable, the approach uses diversity-driven sampling, partial realizability feedback, and a flippability heuristic that shares limited information among components. Applied to discrete geometry, it resolves an open problem by proving the maximum size is 23.","arXiv :2607 .02958v 1 [ cs .CG] 3 Jul 2026  \nToward Satisfiability Modulo Realizability  \nAndrew Krapivin , Benjamin Przybocki , and Marijn J. H. Heule   \nCarnegie Mellon University, Pittsburgh, USA {akrapivi,bprzyboc,[mheule}@andrew.cmu.edu](mheule}@andrew.cmu.edu)  \nAbstract. Problems complete for the existential theory of the reals (∃R) arise throughout discrete geometry. We introduce satisfiability modulo realizability, a SAT-based approach for solving satisfiable instances of ∃R whose solutions correspond to realizable geometric configurations. Our method encodes an underapproximation of a geometric problem as a SAT instance over abstract order types. Since almost all abstract order types are unrealizable, naive search is infeasible. We guide the search toward realizable order types using diversity-driven sampling, partial realizability feedback, and a novel flippability heuristic that passes only limited information between components. We apply our method to discrete geometry problems and resolve an open problem by showing that the largest set of points avoiding empty convex hexagons and convex heptagons is of size 23 .  \nKeywords: Discrete geometry · SAT solving · Realizability.  \n1 Introduction  \nThe success of SAT solving has inspired the motto that NP is the new P, but the same cannot yet be said for classes beyond NP. In this paper, we focus on problems that are complete for the class ∃R (pronounced “existential theory of the reals”), which is a continuous analogue of NP that often arises in areas including discrete geometry, game theory, and computer algebra [33] .  \nWe develop a SAT-based program, which we call PointSAT, for finding solutions to problems in discrete geometry, where solutions consist of sets of points in R2 satisfying combinatorial constraints.1 We evaluate PointSAT by applying it to variants of the happy ending problem posed by Esther Klein and studied by Erdős and Szekeres [9], a problem so named because it led to Klein and Szekeres getting married. Given a set of points in R2 in general position (i.e. , no three points are collinear), we say that a k-gon is a subset of k points in convex position, and a k-hole is a k-gon with no points in its interior. Using PointSAT, we obtained the following answer to a question of Heule and Scheucher [22], who proved the upper bound, with Figure 1 showing a witnessing point set: Theorem 1 . The largest point-set in R2 with no 6-hole or 7-gon is of size 23 . In addition to proving Theorem 1, we demonstrate the versatility of PointSAT by using it to find constructions for two related problems.  \n1 PointSAT is available at [https://github.com/andrewkrapivin/PointSAT](https://github.com/andrewkrapivin/PointSAT).  \n2 A. Krapivin et al.  \n( 0, 501)( 94, 479)(143, 409)  \n(174, 408)  \n(234, 412)  \n(263, 338)  \n(268, 325)  \n(317, 192)  \n(325, 451)  \n(335, 364)  \n(344, 227)  \n(348, 289)  \n(358, 107)  \n(359, 93)  \n(367, 120)  \n(379, 68)  \n(379, 257)  \n(389, 0)  \n(396, 409)  \n(476, 398)  \n(579, 426)  \n(648, 541)  \n(672, 566)  \nFig. 1: A set of 23 points with no 6-hole or 7-gon  \nAt a high level, our approach involves encoding an underapproximation of a given geometric problem into SAT, which means that every solution to the underlying geometric problem corresponds to a satisfying assignment but not vice versa. A satisfying assignment is called an abstract order type, and we say that an assignment corresponding to a geometric solution is realizable. There are two challenges to overcome:  \n1. Testing whether an abstract order type is realizable is computationally difficult (specifically, ∃R-complete) .  \n2. Almost all abstract order types are not realizable.  \nSubercaseaux, Mackey, Qian, and Heule [36] recently made significant progress on the first challenge by developing Localizer, a local-search solver for testing therealizability of abstract order types. Given an abstract order type as input, Localizer initializes a point configuration and iteratively perturbs the poin","cbCairpC888bBQgh","https://ap.wps.com/l/cbCairpC888bBQgh","pdf",597654,5,1,19,"English","en",105,"# Introduction\n## PointSAT and the SAT-based paradigm\n## Abstract order types and realizability challenges\n## Related work: Localizer\n## Contributions and heuristics","[{\"question\":\"What problem does satisfiability modulo realizability address?\",\"answer\":\"It targets satisfiable instances of the existential theory of the reals (∃R) where solutions must correspond to realizable geometric configurations in discrete geometry.\"},{\"question\":\"How does PointSAT connect SAT solving with realizability testing?\",\"answer\":\"PointSAT interfaces a SAT solver with Localizer, treating Localizer as a “theory solver” that tests realizability and guides the search toward geometric configurations that work.\"},{\"question\":\"What result does the paper achieve in discrete geometry?\",\"answer\":\"It resolves an open problem by showing that the largest point set avoiding empty convex hexagons and convex heptagons has size 23.\"}]","Toward Satisfiability Modulo Realizability | 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problem does satisfiability modulo realizability address?","Question",{"text":77,"@type":78},"It targets satisfiable instances of the existential theory of the reals (∃R) where solutions must correspond to realizable geometric configurations in discrete geometry.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does PointSAT connect SAT solving with realizability testing?",{"text":82,"@type":78},"PointSAT interfaces a SAT solver with Localizer, treating Localizer as a “theory solver” that tests realizability and guides the search toward geometric configurations that work.",{"name":84,"@type":75,"acceptedAnswer":85},"What result does the paper achieve in discrete geometry?",{"text":86,"@type":78},"It resolves an open problem by showing that the largest point set avoiding empty convex hexagons and convex heptagons has size 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