[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-267335-105":59,"doc-detail-267335-en":131},{"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":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","the-fractal-dimension-of-sat-formulas-paper","The Fractal Dimension of SAT Formulas - Paper","","Modern SAT solvers have advanced substantially on industrial instances, largely through experimental techniques believed to exploit hidden instance structure, even though a precise definition of that structure is still lacking. This work studies the fractal dimension of SAT instances to extend the model describing industrial problem structure. The paper demonstrates that many industrial formula families are self-similar and possess a small fractal dimension, and that this dimension changes when learnt clauses are added during solver execution.",{"@graph":69,"@context":123},[70,84,106],{"@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/the-fractal-dimension-of-sat-formulas-paper/267335/",{"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/the-fractal-dimension-of-sat-formulas-paper/267335.png","ImageObject",300,407,{"name":92,"@type":93},"Rizky","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-18","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What concept does the paper use to study the structure of industrial SAT instances?","Question",{"text":113,"@type":114},"It studies the fractal dimension of SAT instances, treating formulas in a graph-based way to capture structural properties like self-similarity.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What structural property do the authors show for many industrial SAT formula families?",{"text":118,"@type":114},"They show that many industrial families are self-similar and have a small fractal dimension.",{"name":120,"@type":111,"acceptedAnswer":121},"How does SAT solver execution affect the measured fractal dimension?",{"text":122,"@type":114},"The paper shows the fractal dimension is affected by the addition of learnt clauses during execution.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},267335,1789417158,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},962085564807,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","The Fractal Dimension of SAT Formulas  \nCarlos Ans􀀓otegui 1 , Maria Luisa Bonet2 , Jes􀀓us Gir􀀓aldez-Cru3 , and Jordi Levy3  \n1 DIEI, Univ. de Lleida, [carlos@diei.udl.cat](carlos@diei.udl.cat)  \n2 LSI, UPC, [bonet@lsi.upc.edu](bonet@lsi.upc.edu)  \n3 IIIA-CSIC, jgiraldez,[levy@iiia.csic.es](levy@iiia.csic.es)  \nAbstract. Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there is not a precise de􀀌nition of the notion of structure.  \nRecently, there have been some attempts to analyze this structure in terms of complex networks, with the long-term aim of explaining the success of SAT solving techniques, and possibly improving them.  \nWe study the fractal dimension of SAT instances with the aim of complementing the model that describes the structure of industrial instances.  \nWe show that many industrial families of formulas are self-similar, with a small fractal dimension. We also show how this dimension is a􀀋ected by the addition of learnt clauses during the execution of SAT solvers.  \n1 Introduction  \nThe SAT community has been able to come up with successful SAT solvers for industrial applications. However, nowadays we can hardly explain why these solvers are so e􀀎cient working on industrial SAT instances with hundreds of thousands of variables and not on random instances with hundreds of variables. The common wisdom is that the success of modern SAT/CSP solvers is correlated to their ability to exploit the hidden structure of real-world instances [13] . Unfortunately, there is no precise de􀀌nition of the notion of structure.  \nParallelly, the community of complex networks has produced tools for describing and analyzing the structure of social, biological and communication networks [1] which can explain some interactions in the real-world. Preferential attachment (where the probability that a new edge is attached to a node is proportional to its degree) has been proposed as the responsible of scalefree structure in real-world graphs [5] . Thus, in the web, the probability of a webpage to get new connections is proportional to its popularity (the number of connections it already has) . In cite [9], it is proposed similarity (where nodes tend to get connected to similar nodes, according to some topological distance) as a mechanism that, together with preferential attachment or popularity, explains the structure of some real-world graphs. This explains the self-similarity property observed in many real-world graphs [11] .  \nRepresenting SAT instances as graphs, we can use some of the techniques from complex networks to characterize the structure of SAT instances. Recently,  \nsome progress has been made in this direction. It is known that many industrial instances have the small-world property [12], exhibit high modularity [4], and have a scale-free structure [2] . In this later work, it is shown that in many formulas the number of occurrences of a variable (i.e. the degree of graph nodes) follows a powerlaw distribution with hub variables having a huge number of occurrences. A method to generate scale-free random instances is proposed in [3] . They show that SAT solvers specialized on industrial formulas perform better than random-specialized solvers on these scale-free random instances. In [7], theeigenvector centrality of variables in industrial instances is analyzed. They show that it is correlated with some aspects of SAT solvers. For instance, decision variables selected by the SAT solvers are usually the most central variables in the formula. However, how these analyses may help to improve the performance of SAT solvers is not known at this stage.  \nThe contribution of this paper is to analyze the existence of self-similarity in industrial SAT instances. The existence of a self-similar structure woul","cbCairfMJbHgtbOa","https://ap.wps.com/l/cbCairfMJbHgtbOa","pdf",485156,15,"English","# Abstract\n# 1 Introduction\n## Motivation and background\n## Graph-based analysis using complex networks\n## Self-similarity and fractal dimension\n## Applications and experimental findings","[{\"question\":\"What concept does the paper use to study the structure of industrial SAT instances?\",\"answer\":\"It studies the fractal dimension of SAT instances, treating formulas in a graph-based way to capture structural properties like self-similarity.\"},{\"question\":\"What structural property do the authors show for many industrial SAT formula families?\",\"answer\":\"They show that many industrial families are self-similar and have a small fractal dimension.\"},{\"question\":\"How does SAT solver execution affect the measured fractal dimension?\",\"answer\":\"The paper shows the fractal dimension is affected by the addition of learnt clauses during execution.\"}]","The Fractal Dimension of SAT Formulas - Paper | PDF",38]