[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-267022-105":59,"doc-detail-267022-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","thinking-out-of-the-box-hybrid-sat-solving-by-unconstrained-continuous-optimization","Thinking Out of the Box - Hybrid SAT Solving by Unconstrained Continuous Optimization","","The Boolean satisfiability (SAT) problem underpins combinatorial optimization, software verification, cryptography, and machine learning. Many application settings need hybrid constraints beyond conjunctive normal form, including XOR, cardinality, and Not-All-Equal relations. The work develops unconstrained continuous optimization formulations using penalty terms for hybrid SAT solving, identifies when such penalties are theoretically required, and empirically shows that unconstrained optimizers like Adam can improve performance on hybrid benchmarks. Results connect continuous optimization with machine-learning-style methods for effective solving.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/thinking-out-of-the-box-hybrid-sat-solving-by-unconstrained-continuous-optimization/267022/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/thinking-out-of-the-box-hybrid-sat-solving-by-unconstrained-continuous-optimization/267022.png","ImageObject",300,407,{"name":92,"@type":93},"Noah","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-20","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does this paper address in SAT solving?","Question",{"text":112,"@type":113},"It addresses solving hybrid SAT instances that include constraints beyond CNF, such as XOR, cardinality, and Not-All-Equal constraints.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the proposed method handle hybrid SAT constraints?",{"text":117,"@type":113},"It formulates hybrid SAT solving as unconstrained continuous optimization using penalty terms.",{"name":119,"@type":110,"acceptedAnswer":120},"What evidence is provided that unconstrained optimizers help?",{"text":121,"@type":113},"The paper provides theoretical insights about penalty necessity and empirical results showing that unconstrained optimizers like Adam enhance solving on hybrid benchmarks.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},267022,1789413996,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":36},8796095462418,"https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780","arXiv :2506 .00674v1 [ cs .LO] 31 May 2025  \nThinking Out of the Box: Hybrid SAT Solving by Unconstrained Continuous Optimization  \nZhiwei Zhang \\#  \nDepartment of Computer Science, Rice University Samy Wu Fung \\#  \nDepartment of Applied Mathematics and Statistics, Colorado School of Mines Anastasios Kyrillidis \\#  \nDepartment of Computer Science, Rice University Stanley Osher \\#  \nDepartment of Mathematics, University of California, Los Angeles Moshe Y. Vardi \\#  \nDepartment of Computer Science, Rice University  \n~~ Abstract ~~  \nThe Boolean satisfiability (SAT) problem lies at the core of many applications in combinatorial optimization, software verification, cryptography, and machine learning. While state-of-the-art solvers have demonstrated high efficiency in handling conjunctive normal form (CNF) formulas, numerous applications require non-CNF (hybrid) constraints, such as XOR, cardinality, and Not-All-Equal constraints. Recent work leverages polynomial representations to represent such hybrid constraints, but it relies on box constraints that can limit the use of powerful unconstrained optimizers. In this paper, we propose unconstrained continuous optimization formulations for hybrid SAT solving by penalty terms. We provide theoretical insights into when these penalty terms are necessary and demonstrate empirically that unconstrained optimizers (e.g., Adam) can enhance SAT solving on hybrid benchmarks. Our results highlight the potential of combining continuous optimization and machine-learning-based methods for effective hybrid SAT solving.  \n2012 ACM Subject Classification Theory of computation → Constraint and logic programming  \nKeywords and phrases SAT solving, Optimization, Fourier analysis on Boolean functions, Unconstrained Penalty  \nDigital Object Identifier 10.4230/LIPIcs.CVIT.2016.23  \n 1  Introduction  \nThe Boolean satisfiability (SAT) problem [55] asks whether there exists a solution that satisfies all constraints in a given set of Boolean constraints. This fundamental problem holds immense significance in computer science with applications spanning combinatorial optimization [25], software verification [57], probabilistic inference [10], mathematical conjecture proving [23], machine learning [28], and quantum computing [52, 56] . While SAT is known to be NPcomplete, recent decades have witnessed remarkable advances in SAT solver technology [55] for both CDCL-based complete solvers [51, 4] and heuristic-search incomplete solvers [50] .  \nThe landscape of complete SAT solvers is dominated by Conflict-Driven Clause Learning (CDCL) [51, 4] approaches, which evolved from the seminal GRASP algorithm [35] . CDCL represents a significant advancement over the earlier backtracking Davis-Putnam-LogemannLoveland (DPLL) algorithm [14] . This paradigm has spawned numerous high-performance implementations, including groundbreaking tools like Chaff [38], MiniSat [16], and Glucose [1], alongside more contemporary developments such as MapleSAT [34] and Kissat [17] . The success of CDCL-based methods has established them as the predominant approach in SAT-solving research. Incomplete SAT solvers primarily employ discrete local search  \n© Jane Open Access and Joan R. Public;  \nlicensed under Creative Commons License CC-BY 4.0 42nd Conference on Very Important Topics (CVIT 2016) .  \nEditors: John Q. Open and Joan R. Access; Article No. 23; pp. 23:1–23:16  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n23:2 Thinking Out of the Box: Hybrid SAT Solving by Unconstrained Continuous Optimization  \n(DLS) [24] and message passing (MP) [7], both optimizing to minimize unsatisfied constraints.  \nImportant incomplete solvers include GSAT [50], WSAT [49], and newer innovations like TaSSAT [12], WalkSATlm [8], probSAT [3], Dimetheus [26], and Sparrow [2] . Despite lacking comprehensive guarantees, incomplete solvers are highly efficient for random and cr","cbCaiuto1O58cUmJ","https://ap.wps.com/l/cbCaiuto1O58cUmJ","pdf",1051813,16,"English","# Introduction\n## Complete SAT solvers (CDCL-based)\n## Incomplete SAT solvers (local search and message passing)\n## Non-CNF and hybrid constraints\n## Continuous optimization approaches (FourierSAT)\n## Paper contributions","[{\"question\":\"What problem does this paper address in SAT solving?\",\"answer\":\"It addresses solving hybrid SAT instances that include constraints beyond CNF, such as XOR, cardinality, and Not-All-Equal constraints.\"},{\"question\":\"How does the proposed method handle hybrid SAT constraints?\",\"answer\":\"It formulates hybrid SAT solving as unconstrained continuous optimization using penalty terms.\"},{\"question\":\"What evidence is provided that unconstrained optimizers help?\",\"answer\":\"The paper provides theoretical insights about penalty necessity and empirical results showing that unconstrained optimizers like Adam enhance solving on hybrid benchmarks.\"}]","Thinking Out of the Box - Hybrid SAT Solving by Unconstrained Continuous Optimization | PDF"]