[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82437-en":3,"doc-seo-82437-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82437,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Kleene Algebra with Transitive Commutativity Conditions","Kleene algebra provides an algebraic basis for reasoning about program structure and control flow, while commutativity conditions extend it to capture reorderings and partial independence between atomic actions. The document addresses when the equational theory of Kleene algebra with commutativity conditions is decidable. It establishes a complete characterization: decidability holds exactly when the commutativity relation is transitive, and it refines both sides by relating the star-free and star variants via equality of their equational theories. It also proves undecidability of universality when transitivity fails.","arXiv :2607 .09635v 1 [ cs .PL] 10 Jul 2026  \nKleene Algebra with Transitive Commutativity Conditions  \nHAN XU, Princeton University, United States  \nCHENYU ZHOU, University of Southern California, United States DAVID WALKER, Princeton University, United States ZACHARY KINCAID, Princeton University, United States  \nKleene algebra (􀀠􀀖) provides a foundational algebraic framework for reasoning about program structure and control flow. To capture equivalences arising from reordering or independence of actions, Kozen [1996] purposed that 􀀠􀀖 can be extended with commutativity conditions, that is, equations of the form { 􀀰􀀱 = 􀀱􀀰 |(􀀰, 􀀱) ∈ 􀀘 }, where 􀀘 is a binary relation on constant symbols. This paper studies the following question: for which relations 􀀘 is the equational theory of 􀀠􀀖 + 􀀘 decidable?  \nEarly related work [Bertoni et al. 1982; Ibarra 1978] showed that regular languages modulo commutativity conditions 􀀘 are decidable if and only if 􀀘 is transitive. For Kleene algebra 􀀠􀀖 and commutativity conditions 􀀘, however, the situation is substantially more difficult. Only very recently, Kuznetsov [2023] showed that the equational theory of Kleene algebra 􀀠􀀖 + 􀀘 is undecidable under certain specific commutativity conditions, settling the first nontrivial cases more than 25 years after the corresponding problem for 􀀠􀀖∗ + 􀀘 was resolved by Kozen [1996] . Nevertheless, the decidability problem of 􀀠􀀖 + 􀀘 remained open.  \nIn this work, we resolve this question completely by showing that the equational theory of 􀀠􀀖 + 􀀘 is decidable if and only if 􀀘 is transitive. Moreover, we strengthen the result in both directions. On the negative side, we show that when 􀀘 is not transitive, the universality problem for 􀀠􀀖 + 􀀘 is already undecidable. On the positive side, we show that for transitive 􀀘, the equational theories of 􀀠􀀖∗ + 􀀘 and 􀀠􀀖 + 􀀘 coincide.  \nCCS Concepts: • Software and its engineering → General programming languages.  \nAdditional Key Words and Phrases: Kleene Algebra; Decision Procedure  \nACM Reference Format:  \nHan Xu, Chenyu Zhou, David Walker, and Zachary Kincaid. 2018. Kleene Algebra with Transitive Commutativity Conditions. In Proceedings of Make sure to enter the correct conference title from your rights confirmation email (Conference acronym ’XX). ACM, New York, NY, USA, 42 pages. [https://doi.org/XXXXXXX.XXXXXXX](https://doi.org/XXXXXXX.XXXXXXX)  \n1 Introduction  \nKleene algebra (􀀠􀀖) provides an algebraic foundation for reasoning about the control structure of programs. Its operators—addition, multiplication, and Kleene star—correspond naturally to nondeterministic choice, sequential composition, and iteration. This correspondence makes 􀀠􀀖 a powerful framework for expressing and verifying program equivalences through algebraic manipulation, and it has found applications in program verification, compiler optimization, network analysis and the study of regular languages [Anderson et al. 2014; Conway 1971; Kozen 1994, 1997] .  \nAuthors’ Contact Information: Han Xu, [hx3501@princeton.edu](hx3501@princeton.edu), Princeton University, United States; Chenyu Zhou, [czhou691@usc.edu](czhou691@usc.edu), University of Southern California, United States; David Walker, [dpw@princeton.edu](dpw@princeton.edu), Princeton University, United States; Zachary Kincaid, [zkincaid@cs.princeton.edu](zkincaid@cs.princeton.edu), Princeton University, United States.  \nPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires [prior specific permission and/or a fee. Request permissions from permissions@acm.org](prior","cbCaibSHXzeHz0JQ","https://ap.wps.com/l/cbCaibSHXzeHz0JQ","pdf",779690,1,42,"English","en",105,"# Introduction\n# Kleene algebra and commutativity conditions\n# Decidability problem and main results\n# Related work and prior characterizations","[{\"question\":\"What does the paper study about Kleene algebra with commutativity conditions?\",\"answer\":\"It studies for which commutativity relations the equational theory of Kleene algebra extended with those conditions is decidable.\"},{\"question\":\"When is the equational theory of Kleene algebra with commutativity conditions decidable?\",\"answer\":\"It is decidable if and only if the commutativity relation is transitive.\"},{\"question\":\"What happens when the commutativity relation is not transitive?\",\"answer\":\"When it is not transitive, the universality problem for the extended Kleene algebra is already undecidable.\"}]",1784180375,106,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"kleene-algebra-with-transitive-commutativity-conditions","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/kleene-algebra-with-transitive-commutativity-conditions/82437/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 the paper study about Kleene algebra with commutativity conditions?","Question",{"text":75,"@type":76},"It studies for which commutativity relations the equational theory of Kleene algebra extended with those conditions is decidable.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"When is the equational theory of Kleene algebra with commutativity conditions decidable?",{"text":80,"@type":76},"It is decidable if and only if the commutativity relation is transitive.",{"name":82,"@type":73,"acceptedAnswer":83},"What happens when the commutativity relation is not transitive?",{"text":84,"@type":76},"When it is not transitive, the universality problem for the extended Kleene algebra is already undecidable.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]