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Deterministic methods are given for quantified 3-CNFs with two quantifier blocks, and zero-error randomized algorithms extend tractability to multiple quantifier blocks. A second randomized approach addresses QBF on circuits, while complementary lower-bound arguments show that further improvements would imply new circuit complexity lower bounds via tight relationships between CNF and general Boolean formula settings.",{"@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/beating-exhaustive-search-for-quantied-boolean-formulas-and-connections-to-circuit-complexity/267025/",{"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/beating-exhaustive-search-for-quantied-boolean-formulas-and-connections-to-circuit-complexity/267025.png","ImageObject",300,407,{"name":92,"@type":93},"Aria","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-18","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 the paper study?","Question",{"text":112,"@type":113},"The paper studies algorithms for satisfifiability of quantified Boolean formulas (QBFs) and examines how faster QBF algorithms affect circuit complexity results.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What are the main algorithmic contributions for two quantifier blocks?",{"text":117,"@type":113},"It provides deterministic algorithms solving satisfiability of quantified 3-CNFs with one existential and one universal quantifier block in time improved over brute force search.",{"name":119,"@type":110,"acceptedAnswer":120},"How do the results relate to circuit complexity lower bounds?",{"text":121,"@type":113},"The paper complements the algorithms with hardness reductions showing that significant further speedups would imply new circuit complexity lower bounds.","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},267025,1789414002,{"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":144},2336464648322,"https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488","Beating Exhaustive Search for Quantified Boolean Formulas and Connections to  \nCircuit Complexity  \nRahul Santhanam􀀃 Ryan Williamsy  \nAbstract  \nWe study algorithms for the satisfiability problem for quantified Boolean formulas (QBFs), and consequences of faster algorithms for circuit complexity.  \n􀀏 We show that satisfiability of quantified 3-CNFs with m clauses, n variables, and two quantifier blocks (one existential block and one universal) can be solved deterministically in time 2n􀀀􀀊( pn) 􀀁 poly (m) . For the case of multiple quantifier blocks (alternations), we show that satisfiability of quantified CNFs of size poly (n) on n variables with qquantifier blocks can be solved in 2n􀀀n1=(q+1) 􀀁 poly(n) time by a zero-error randomized algorithm. These are the first provable improvements over brute force search in the general case, even for quantified polynomial-sized CNFs with two quantifier blocks.  \nA second zero-error randomized algorithm solves QBF on circuits of size s in 2n􀀀􀀊(q) 􀀁 poly(s) time when the number of quantifier blocks is q.  \n􀀏 We complement these algorithms by showing that improvements on them would imply new circuit complexity lower bounds. For example, if satisfiability of quantified CNF formulas with n variables, poly(n) size and at most q quantifier blocks can be solved in time 2n􀀀n! q(1=q), then the complexity class NEXP does not have O(log n) depth circuits of polynomial size. Furthermore, solving satisfiability of quantified CNF formulas with n variables, poly (n) size and O(log n) quantifier blocks in time 2n􀀀!(log(n)) time would imply the same circuit complexity lower bound. The proofs of these results proceed by establishing strong relationships between the time complexity of QBF satisfiability over CNF formulasand the time complexity of QBF satisfiability over arbitrary Boolean formulas.  \n1 Introduction  \nThe satisfiability (SAT) problem for Boolean formulas is the canonical NP-complete problem. Despite its apparent worstcase intractability, nowadays the ability to solve most SAT instances arising in practice is well-known. Over the past two decades, there have been many substantial advances in SAT solvers that led to the current state of affairs that “SATis generally easy in practice” [MZ09] . On the theoretical  \n􀀃 University of Edinburgh. Supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007- 2013) / ERC grant agreement no. 615075.  \ny Stanford University. Supported in part by a David Morgenthaler II Faculty Fellowship, and NSF CCF-1212372 . Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.  \nside, there are many known SAT algorithms which provably solve the problem faster than brute force search for formulasin conjunctive normal form (CNF) [MS85, PPZ97, Pud98, PPSZ98, Sch99, Sch05, DW06, CIP09, IMP12] .  \nThe quantified Boolean formula (QBF) problem is the analogue of SAT for the larger complexity class PSPACE. Variables can have arbitrary quantification (existential or universal), and the problem is to determine whether a given quantified formula is true or false. QBF would potentially have a much wider range of applications than SAT, if only we understood more about how to solve it. The best known QBF solvers can only tackle a very limited range of the problem space [Zha06, GIB09] .  \nMoreover, in theory, comparatively very little is known concerning general worst-case algorithms for QBF. Williams [Wil02] showed that QBFs over general CNF formulas with m clauses are solvable in O (1:71m ) time, and demonstrated that 3-CNF QBFs with two quantifier blocks can be solved in O (2n􀀀\"n) time (where n is the number of variables) for a constant \" > 0 depending on the clausevariable ratio m=n (however, \" ! 0 as m=n ! 2) . Santhanam [San10] showed that for every c 􀀕 1, there is a 􀀎 \u003C 1 such that Formula-SAT on m clauses and n variables can be ","cbCaivQi7EuxGGlp","https://ap.wps.com/l/cbCaivQi7EuxGGlp","pdf",260524,11,"English","# Abstract\n# Introduction","[{\"question\":\"What problem does the paper study?\",\"answer\":\"The paper studies algorithms for satisfifiability of quantified Boolean formulas (QBFs) and examines how faster QBF algorithms affect circuit complexity results.\"},{\"question\":\"What are the main algorithmic contributions for two quantifier blocks?\",\"answer\":\"It provides deterministic algorithms solving satisfiability of quantified 3-CNFs with one existential and one universal quantifier block in time improved over brute force search.\"},{\"question\":\"How do the results relate to circuit complexity lower bounds?\",\"answer\":\"The paper complements the algorithms with hardness reductions showing that significant further speedups would imply new circuit complexity lower bounds.\"}]","Beating Exhaustive Search for Quantified Boolean Formulas and Connections to Circuit Complexity | PDF",28]