[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84593-en":3,"doc-seo-84593-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84593,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Algorithms and Fine-Grained Complexity for Nondeterministic and Symmetric Difference Automata","Symmetric difference automata (XNFA) extend finite automata by accepting an input word exactly when the number of accepting runs is odd, equivalently modeling computations as weighted automata over the two-element field. The work analyzes fine-grained complexity for core decision problems—acceptance, emptiness, and equivalence—optimizing the polynomial degree in running-time bounds. Under polynomial ambiguity, it gives a randomized reduction from NFA acceptance to XNFA acceptance. For bounded ambiguity it yields faster decision procedures, while certificate verification admits speedups even without ambiguity assumptions.","arXiv :2607 .00742v 1 [ cs .FL] 1 Jul 2026  \nAlgorithms and fine-grained complexity for nondeterministic and symmetric difference automata  \nDmitry Chistikov  \nUniversity of Warwick  \nRadosław Piórkowski  \nUniversity of Warwick  \nNeha Rino  \nUniversity of Warwick  \nBrink van der Merwe  \nStellenbosch University  \n~~ Abstract ~~  \nSymmetric difference automata (XNFA) are a variant of standard finite automata in which an input word is accepted iff the number of accepting runs is odd. Equivalently, these are weighted automata over the two-element field. We study the fine-grained complexity of the basic decision problems for XNFA: acceptance, emptiness, and equivalence, aiming to optimise the degree of the polynomial in their running-time bounds.  \nUnder the assumption of polynomial ambiguity, we provide a randomised reduction of NFA acceptance to XNFA acceptance. For automata of bounded ambiguity (e.g. , unambiguous automata), we show that acceptance for both NFA and XNFA can be decided faster than in the general case. Without ambiguity assumptions, we give faster algorithms for the verification of suitable certificates for (non)emptiness and (non)equivalence of XNFA. Several of our results extend to weighted automata over other semirings and fields.  \n2012 ACM Subject Classification Theory of computation → Quantitative automata  \nKeywords and phrases Automata theory, Fine-grained Complexity, Symmetric Difference Automata  \nFunding This research was supported by the Royal Society (IEC\\R2\\242349) . We would also like to thank the Centre for Discrete Mathematics and its Applications (DIMAP) and the Department of Computer Science, at the University of Warwick.  \nDmitry Chistikov: Supported in part by the Engineering and Physical Sciences Research Council [EP/X03027X/1] .  \nRadosław Piórkowski: Supported by the Engineering and Physical Sciences Research Council [EP/X03027X/1] .  \nNeha Rino: Supported by the Feuer International Scholarship in Artificial Intelligence by University of Warwick alumnus Jonathan Feuer.  \nAcknowledgements For Corollary 11 we are indebted to Stefan Kiefer and Andrew Ryzhikov, who brought the open problem from their work [26] to our attention and discussed the application of our Theorem 7 with us. We are also grateful to Michael Wehar for inspiring discussions and to Sarah J. Selkirk for making this project possible.  \n2 Algorithms and fine-grained complexity for NFA and XNFA  \n 1  Introduction  \nConnecting and comparing algorithmic problems over different algebraic structures is a well-known challenge. Consider the complexity classes NL and ⊕L corresponding to language recognition by logarithmic-space Turing machines. NL machines accept a word if there is a positive number of accepting runs while ⊕L machines accept if there is an odd number of accepting runs. It is still not known whether the inclusion NL ⊆ ⊕L (or its converse) holds; see [13, 19] . A complete problem for NL is the acceptance problem for nondeterministic finite automata, that is, given an NFA A and a word w, decide whether A accepts w. Similarly, a ⊕L-complete problem is the acceptance problem for weighted automata over the two-element field [34] .  \nWeighted automata over the two-element field are also known in the literature as symmetric difference automata (⊕-NFA) [39], or XOR-nondeterministic finite automata (XNFA) [42], or Mod 2 Multiplicity Automata (M2MA, for regular languages over infinite words) [3] . In this paper we use the term ‘XNFA’. A logspace reduction from NFA Acceptance to XNFA Acceptance would show that NL ⊆ ⊕L. In a similar spirit, it is not known how the time complexities of these two problems compare (as we discuss below in sections 1.1 and 2) .  \nIn this work, we study the possibility of fine-grained reductions between NFA and XNFA Acceptance, focusing on the time complexity.  \nXNFA are a variant of standard finite automata in which an input word is accepted if and only if the number of accepting runs is odd [40, 41] . ","cbCaiaxmatzN1rOI","https://ap.wps.com/l/cbCaiaxmatzN1rOI","pdf",718247,2,1,22,"English","en",105,"# Introduction\n## The acceptance problem\n# Algorithms and fine-grained complexity for NFA and XNFA","[{\"question\":\"What does it mean for a symmetric difference automaton (XNFA) to accept an input word?\",\"answer\":\"An XNFA accepts an input word exactly when the number of accepting runs for that word is odd.\"},{\"question\":\"Which decision problems are analyzed in terms of fine-grained complexity?\",\"answer\":\"The document studies fine-grained complexity for acceptance, non-emptiness, and equivalence of XNFA (and related results for NFA).\"},{\"question\":\"What kind of speedups or reductions are provided under ambiguity assumptions?\",\"answer\":\"Assuming polynomial ambiguity, it provides a randomized reduction from NFA acceptance to XNFA acceptance, and for bounded ambiguity it shows faster algorithms for acceptance for both NFA and XNFA.\"}]",1784196986,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"algorithms-and-fine-grained-complexity-for-nondeterministic-and-symmetric-difference-automata","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/algorithms-and-fine-grained-complexity-for-nondeterministic-and-symmetric-difference-automata/84593/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-22","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does it mean for a symmetric difference automaton (XNFA) to accept an input word?","Question",{"text":75,"@type":76},"An XNFA accepts an input word exactly when the number of accepting runs for that word is odd.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which decision problems are analyzed in terms of fine-grained complexity?",{"text":80,"@type":76},"The document studies fine-grained complexity for acceptance, non-emptiness, and equivalence of XNFA (and related results for NFA).",{"name":82,"@type":73,"acceptedAnswer":83},"What kind of speedups or reductions are provided under ambiguity assumptions?",{"text":84,"@type":76},"Assuming polynomial ambiguity, it provides a randomized reduction from NFA acceptance to XNFA acceptance, and for bounded ambiguity it shows faster algorithms for acceptance for both NFA and XNFA.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]