[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-505015-105":59,"doc-detail-505015-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","optimizing-sat-encodings-for-arithmetic-constraints-equivalence-reasoning","Optimizing SAT Encodings for Arithmetic - Constraints - Equivalence Reasoning","","The paper addresses the challenge of encoding arithmetic constraints into SAT efficiently, focusing on the log encoding whose propagation limitations have historically reduced its effectiveness. It analyzes PicatSAT, an eager SAT compiler using sign-and-magnitude log encoding, and details optimizations that generate compact, efficient CNF. A central contribution is equivalence reasoning, which deduces Boolean relationships to reduce variables and clauses, lowering code size and improving solving time.",{"@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/optimizing-sat-encodings-for-arithmetic-constraints-equivalence-reasoning/505015/",{"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/optimizing-sat-encodings-for-arithmetic-constraints-equivalence-reasoning/505015.png","ImageObject",300,407,{"name":92,"@type":93},"Riley West","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-14","2026-10-04",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"Why is log encoding traditionally considered unsuitable for arithmetic constraints in SAT?","Question",{"text":112,"@type":113},"Log encoding was perceived as hindering propagation, making it hard to efficiently handle arithmetic constraints compared with other encodings.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What encoding scheme does PicatSAT use for integer domains?",{"text":117,"@type":113},"PicatSAT uses sign-and-magnitude log encoding. It employs log2(n) Boolean variables for magnitudes and adds another Boolean variable when both negative and positive values are present.",{"name":119,"@type":110,"acceptedAnswer":120},"What is equivalence reasoning and what benefits does it bring?",{"text":121,"@type":113},"Equivalence reasoning identifies whether Booleans are 0/1, equivalent to others, or negations. This reduces both Boolean variables and CNF clause counts, decreasing code size and sometimes solving time.","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},505015,1791289507,{"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":34,"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":144,"read_time":36},1099523885074,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Optimizing SAT Encodings for Arithmetic  \nConstraints  \nNeng-Fa Zhou 1 and H􀀗akan Kjellerstrand2  \n1 CUNY Brooklyn College & Graduate Center  \n[2](2 hakank.org)[ hakank.org](2 hakank.org)  \nAbstract. The log encoding has been perceived to be unsuited to arithmetic constraints due to its hindrance to propagation. The surprising performance of PicatSAT, which is a pure eager SAT compiler based on the log encoding, in the MiniZinc Challenge 2016 has revived interest in the log encoding. This paper details the optimizations used in PicatSAT for encoding arithmetic constraints. PicatSAT adopts some well-known optimizations from CP systems, language compilers, and hardware design systems for encoding constraints into compact and e􀀎cient SAT code. PicatSAT is also empowered by a novel optimization, called equivalence reasoning, for arithmetic constraints, which leads to reduction of code size and execution time. In a nutshell, this paper demonstrates that the optimized log encoding is competitive for encoding arithmetic constraints.  \n1 Introduction  \nThe drastic enhancement of SAT solvers' performance has made SAT a viable backbone for general CSP (Constraint Satisfaction Problem) solvers [8, 21, 30, 33, 34] . Many real-world combinatorial problems involve arithmetic constraints, and it remains a challenge to e􀀎ciently encode arithmetic constraints into SAT. The sparse encoding [20, 36] and order encoding [13, 25, 34] can easily blow up the code size, and the log encoding [22, 18] is perceived to be a poor choice, despite its compactness, due to its failure to maintain arc consistency, even for binary constraints. This dilemma of the eager approach has led to the emergence of the lazy approach, as represented by SMT solvers that use integer arithmetic as a theory [6, 14, 26] and the lazy clause generation (LCG) solver that combines SAT and constraint propagation [17, 29] . Both the eager and lazy approaches have strengths and weaknesses [27] . For problems that require frequent checking of arithmetic constraints the lazy approach may not be competitive due to the overhead, even when checking is done incrementally and in a priori manner. From an engineering perspective, the eager approach also has its merit, just like the separation of computer hardware and language compilers is bene􀀌cial.  \nThe surprising performance of the PicatSAT compiler in the MiniZinc Challenge 2016 is thought-provoking.3 PicatSAT is a pure SAT compiler that trans-  \n3 PicatSAT with the Lingeling SAT solver won two silver medals and one bronze medal in the competition ([http://www.minizinc.org/challenge2016/results2016.html](http://www.minizinc.org/challenge2016/results2016.html))  \nlates CSPs into log-encoded SAT code. PicatSAT adopts the sign-and-magnitude log encoding for domain variables. For a domain with the maximum absolute value n, it uses log2 (n) Boolean variables to encode the domain. If the domain contains both negative and positive values, then another Boolean variable is employed to encode the sign. Each combination of values of the Boolean variables represents a valuation for the domain variable. The addition constraint is encoded as logic adders, and the multiplication constraint is encoded as logic adders using the shift-and-add algorithm.  \nPicatSAT adopts some well-known optimizations from CP systems, language compilers, and hardware design systems for encoding constraints into compact and e􀀎cient SAT code: it preprocesses constraints before compilation in order to remove no-good values from the domains of variables whenever possible; it eliminates common subexpressions so that no primitive constraint is duplicated; it uses a logic optimizer to generate optimized code for adders. These optimizationssigni􀀌cantly improve the quality of the generated code.  \nThis paper proposes a new optimization, named equivalence reasoning, for log-encoded arithmetic constraints. Equivalence reasoning identi􀀌es information about if a Boolean variable is 0 or ","cbCaim3nUKREP7Qg","https://ap.wps.com/l/cbCaim3nUKREP7Qg","pdf",323771,16,"English","# Abstract\n# 1 Introduction\n## Eager vs. lazy approaches\n## PicatSAT overview\n# 2 The PicatSAT Compiler\n## 2.1 Preprocessing and Decomposition","[{\"question\":\"Why is log encoding traditionally considered unsuitable for arithmetic constraints in SAT?\",\"answer\":\"Log encoding was perceived as hindering propagation, making it hard to efficiently handle arithmetic constraints compared with other encodings.\"},{\"question\":\"What encoding scheme does PicatSAT use for integer domains?\",\"answer\":\"PicatSAT uses sign-and-magnitude log encoding. It employs log2(n) Boolean variables for magnitudes and adds another Boolean variable when both negative and positive values are present.\"},{\"question\":\"What is equivalence reasoning and what benefits does it bring?\",\"answer\":\"Equivalence reasoning identifies whether Booleans are 0/1, equivalent to others, or negations. This reduces both Boolean variables and CNF clause counts, decreasing code size and sometimes solving time.\"}]","Optimizing SAT Encodings for Arithmetic - Constraints - Equivalence Reasoning | PDF",1791086097]