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Motivates computational discovery of large structured objects and illustrates how Hadamard matrices and Williamson matrices can be found by computational methods. Explains Williamson submatrices for orders n=2^k and discusses whether the observed pattern continues beyond small k. Concludes that computational searches can establish that Williamson’s method works for powers of two and can uncover all cases up to a stated limit.",{"@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/computer-algebra-and-sat-for-mathematical-search-theoretical-foundations-of-satsmt-solving/267274/",{"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/computer-algebra-and-sat-for-mathematical-search-theoretical-foundations-of-satsmt-solving/267274.png","ImageObject",300,407,{"name":92,"@type":93},"Theodore","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-05","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 problem does the presentation focus on?","Question",{"text":113,"@type":114},"It focuses on using SAT/SMT solving and computer algebra techniques for mathematical search, exemplified through the discovery of Williamson matrices and related Hadamard matrix structures.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are Hadamard matrices connected to Williamson’s work in the slides?",{"text":118,"@type":114},"Hadamard matrices are defined as square matrices with ±1 entries whose rows are mutually orthogonal, and Williamson’s method is used to construct examples and Williamson submatrices of specific orders.",{"name":120,"@type":111,"acceptedAnswer":121},"Does the observed pattern of Williamson matrices continue for larger sizes?",{"text":122,"@type":114},"Yes. The slides state that computational searches found all Williamson matrices in even orders up to a given bound and that the uncovered patterns indicate Williamson’s method works for all orders that are powers of two.","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},267274,1789416591,{"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},7971461740886,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Computer Algebra and SAT for Mathematical Search  \nCurtis Bright  \nUniversity of Windsor  \nJoint work with  \nKevin Cheung, Brett Stevens, Ilias Kotsireas, Vijay Ganesh  \nTheoretical Foundations of SAT/SMT Solving Simons Institute  \nApril 21, 2021  \nMotivation  \nMathematicians have long been fascinated with searching for structures that satisfy elegant properties.  \nMotivation  \nHadamard matrices are square matrices with 􀀆1 entries whose rows are mutually orthogonal.  \nIn 1944, Williamson found a number of such matrices like this  \n8 􀀂 8 example:  \n\n| 1 | 1 | 1 | 1 | 1 | 􀀀1 | 1 | 􀀀1 |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| 1 | 1 | 1 | 1 | 􀀀1 | 1 | 􀀀1 | 1 |\n| 􀀀1 | 􀀀1 | 1 | 1 | 􀀀1 | 1 | 1 | 􀀀1 |\n| 􀀀1 | 􀀀1 | 1 | 1 | 1 | 􀀀1 | 􀀀1 | 1 |\n| 􀀀1 | 1 | 1 | 􀀀1 | 1 | 1 | 􀀀1 | 􀀀1 |\n| 1 | 􀀀1 | 􀀀1 | 1 | 1 | 1 | 􀀀1 | 􀀀1 |\n| 􀀀1 | 1 | 􀀀1 | 1 | 1 | 1 | 1 | 1 |\n| 1 | 􀀀1 | 1 | 􀀀1 | 1 | 1 | 1 | 1 |\n\nHadamard matrix found by Williamson  \nUsing Williamson submatrices of order n = 22 = 4.  \nHadamard matrix found by Williamson  \nUsing Williamson submatrices of order n = 23 = 8.  \nHadamard matrix found by Williamson  \nUsing Williamson submatrices of order n = 24 = 16.  \nHadamard matrix found by Williamson  \nUsing Williamson submatrices of order n = 25 = 32.  \nDoes this continue?  \nIt's unclear if this pattern extends to orders n = 2k for k > 5. . .  \nDoes this continue?  \nIt's unclear if this pattern extends to orders n = 2k for k > 5. . .  \nIt does!  \nIn 2019, we found all Williamson matrices in even orders n 􀀔 70.1 The patterns uncovered by these searches show that Williamson's method works for all orders that are powers of two.2  \n1 C. Bright, I. Kotsireas, V. Ganesh. Applying computer algebra systems with SAT solvers to the Williamson conjecture. Journal of Symbolic Computation, 2019 .  \n2——. New Infinite Families of Perfect Quaternion Sequences and Williamson Sequences. IEEE Transactions on Information Theory, 2020 .  \nMoral of the story  \nSometimes the only feasible way of discovering “large” examples of mathematical structures is through computational search.  \nPrevious search methods for Williamson matrices came from fields such as satisfiability solving and computer algebra .","cbCaihbRkYiWaSN1","https://ap.wps.com/l/cbCaihbRkYiWaSN1","pdf",2252707,42,"English","# Motivation\n## Hadamard matrices and Williamson matrices\n## Computational search results","[{\"question\":\"What problem does the presentation focus on?\",\"answer\":\"It focuses on using SAT/SMT solving and computer algebra techniques for mathematical search, exemplified through the discovery of Williamson matrices and related Hadamard matrix structures.\"},{\"question\":\"How are Hadamard matrices connected to Williamson’s work in the slides?\",\"answer\":\"Hadamard matrices are defined as square matrices with ±1 entries whose rows are mutually orthogonal, and Williamson’s method is used to construct examples and Williamson submatrices of specific orders.\"},{\"question\":\"Does the observed pattern of Williamson matrices continue for larger sizes?\",\"answer\":\"Yes. The slides state that computational searches found all Williamson matrices in even orders up to a given bound and that the uncovered patterns indicate Williamson’s method works for all orders that are powers of two.\"}]","Computer Algebra and SAT for Mathematical Search - Theoretical Foundations of SAT/SMT Solving | PDF",106]