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A new construction for 8-Williamson matrices yields examples in every odd order up to 35, while matrices with orders divisible by 3 and less than 70 are also enumerated, including a new set of order 63.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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tools and paradigms does the paper use for the Williamson conjecture?","Question",{"text":63,"@type":64},"It combines computer algebra systems with SAT solvers under a “SAT+CAS” paradigm, and also uses SMT-inspired techniques by learning conflict clauses through a problem-tailored code.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What does the paper achieve for even-order Williamson matrices?",{"text":68,"@type":64},"It completely enumerates all Williamson matrices of even order up to and including 70 and finds they exist in every even order within that range.",{"name":70,"@type":61,"acceptedAnswer":71},"What additional results are reported for 8-Williamson matrices and other orders?",{"text":72,"@type":64},"A new construction for 8-Williamson matrices produces them in all odd orders up to and including 35, and the paper enumerates cases divisible by 3 below 70, identifying a previously unknown set of order 63.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},267271,1789416583,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & 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the Williamson Conjecture  \nCurtis Brighta , Ilias Kotsireasb , Vijay Ganesha  \na University of Waterloo  \nb Wilfrid Laurier University  \nAbstract  \nWe employ tools from the fields of symbolic computation and satisfiability checking—namely, computer algebra systems and SAT solvers—to study the Williamson conjecture from combinatorial design theory and increase the bounds to which Williamson matrices have been enumerated. In particular, we completely enumerate all Williamson matrices of even order up to and including 70 which gives us deeper insight into the behaviour and distribution of Williamson matrices. We find that, in contrast to the case when the order is odd, Williamson matrices of even order are quite plentiful and exist in every even order up to and including 70. As a consequence of this and a new construction for 8-Williamson matrices we construct 8-Williamson matrices in all odd orders up to and including 35. We additionally enumerate all Williamson matrices whose orders are divisible by 3 and less than 70, finding one previously unknown set of Williamson matrices of order 63.  \nKeywords: Williamson matrices; Boolean satisfiability; SAT solvers; Exhaustive search; Autocorrelation  \n1. Introduction  \nIn recent years SAT solvers have been used to solve or make progress on mathematical conjectures which have otherwise resisted solution from some of the world's best mathematicians. Some prominent problems which fit into this trend include the Erds discrepancy conjecture, which was open for 80 years and had a special case solved by Konev and Lisitsa (2014); the Ruskey–Savage conjecture, which has been open for 25 years and had a special case solved by Zulkoski et al. (2015); the Boolean Pythagorean triples problem, which was open for 30 years and solved by Heule et al. (2016); and the determination of the fifth Schur number, which was open for 100 years and solved by Heule (2018) . Although these are problems which arise in completely separate fieldsand have no obvious connection to propositional satisfiability checking, nevertheless SAT solvers were found to be extremely effective at pushing the state-of-the-art and sometimes absolutely crucial in the problem's ultimate solution.  \nIn this paper we apply a SAT solver to the Williamson conjecture from combinatorial design theory. Our work is similar in spirit to the aforementioned works but we would like to highlight two main differences. Firstly, we employ an approach inspired by SMT  \nTo appear in the Journal of Symbolic Computation March 15, 2019  \n(SAT modulo theories) solvers and use a SAT solver that is able to learn conflict clauses through a piece of code specifically tailored to the problem domain. This code encodes domain-specific knowledge that an off-the-shelf SAT solver would otherwise not be able to exploit. This framework is not limited to any specific domain; any external library or function can be used as long as it is callable by the SAT solver. As we will see in Section 3, the clauses that are learned in this fashion can enormously cut down the search space as well as the solver's runtime.  \nSecondly, similar in style to (Zulkoski et al., 2015) we incorporate functionality from computer algebra systems to increase the efficiency of the search in what we call the “SAT+CAS” paradigm. This approach of combining computer algebra systems with SAT or SMT solvers was also independently proposed at the conference ISSAC by Ábrahám (2015) . More recently, it has been argued by the SC2 project (Ábrahámet al., 2016) that the fields of satisfiability checking and symbolic computation are complementary and combining the tools of both fields (i.e., SAT solvers and computer algebra systems) in the right way can solve problems more efficiently than could be done by applying the tools of either field in isolation, and our work provides evidence for this view.  \nWe describe the Williamson c","cbCaijHyfZnluon7","https://ap.wps.com/l/cbCaijHyfZnluon7","pdf",354197,"English","# Introduction\n## SAT solvers in mathematical conjectures\n## SAT+CAS paradigm overview\n# The Williamson conjecture\n## Matrices and construction context","[{\"question\":\"What tools and paradigms does the paper use for the Williamson conjecture?\",\"answer\":\"It combines computer algebra systems with SAT solvers under a “SAT+CAS” paradigm, and also uses SMT-inspired techniques by learning conflict clauses through a problem-tailored code.\"},{\"question\":\"What does the paper achieve for even-order Williamson matrices?\",\"answer\":\"It completely enumerates all Williamson matrices of even order up to and including 70 and finds they exist in every even order within that range.\"},{\"question\":\"What additional results are reported for 8-Williamson matrices and other orders?\",\"answer\":\"A new construction for 8-Williamson matrices produces them in all odd orders up to and including 35, and the paper enumerates cases divisible by 3 below 70, identifying a previously unknown set of order 63.\"}]","Applying Computer Algebra Systems with SAT Solvers to the Williamson Conjecture - v2 | PDF",76]