[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85595-en":3,"doc-seo-85595-105":30,"detail-sidebar-cat-0-en-105":83},{"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},85595,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Continuous Algebras with Hypotheses","This paper unifies prior variants of Kleene algebra with hypotheses—including Kleene algebra with tests, commutative KA, bi-KA, and concurrent forms—within a single framework that also covers regular tree languages. The approach orders algebras by complete lattices and computes least fixpoints. A canonical model of closed languages is established and proved sound and complete for all continuous models. The work further develops quasi-equational axiomatisations using modular tool support to derive new completeness results under various hypotheses.","arXiv :2605 . 18450v2 [ cs .LO] 13 Jul 2026  \nContinuous Algebras with Hypotheses  \nLukas Mulder  \nRadboud University, The Netherlands Damien Pous 􀀚  \nCNRS, LIP, ENS de Lyon, France Jana Wagemaker 􀀚  \nRadboud University, The Netherlands  \n~~ Abstract ~~  \nIn the literature on Kleene algebra (KA), a number of variants have been proposed such as Kleene algebra with tests, commutative KA, bi-KA, and concurrent KA. The equational theories of some of these structures have then been studied in the presence of additional assumptions, called hypotheses. We propose a unifying framework encompassing all the previous structures, as well as regular tree languages. This is done by considering algebras ordered by complete lattices, where least fixpoints can be computed. We provide a canonical model consisting of closed languages, which we prove sound and complete with respect to all continuous models. Then we study quasi-equational axiomatisations. It is illusory to hope for a generic axiomatisation which would be sound and complete for all instances. Instead, we provide a generic axiomatisation which we prove sound and we setup tools that make it possible to get complete ones in a modular way, building on previous works from the literature. We showcase these tools by proving new completeness results for commutative KA, bi-KA, and regular tree languages, in each case extended with various hypotheses.  \n2012 ACM Subject Classification Theory of computation → Algebraic language theory Keywords and phrases Kleene algebra, complete lattices, languages, completeness  \nRelated Version This article appeared in Proc . CONCUR 2026 and the present version contains the appendices with all proofs: [https://doi.org/10.4230/LIPIcs.CONCUR.2026.28](https://doi.org/10.4230/LIPIcs.CONCUR.2026.28)  \nFunding This work was partially supported by the Dutch research council (NWO) under grant no.  \nVI.Veni.242.134 (VerHyp) and by the LABEX MILYON (ANR-10-LABX-0070)  \nAcknowledgements We thank Paul Brunet and Amina Doumane for early discussions on this work.  \n 1  Introduction  \nKleene algebras (KA) [27, 13] are structures with sequential composition, iteration and choice. They were first proposed as the algebra of regular expressions, and their axioms are sound and complete w.r.t. relational models and language models [5, 28 , 35] . Their equational theory is decidable via automata algorithms, which makes it possible to design automation tactics in proof assistants [7, 34 , 39] . In such applications, we often want to reason under assumptions, or hypotheses. Unfortunately, the Horn theory of Kleene algebras is undecidable [30, 36] and we may only automate reasoning under specific forms of hypotheses [11, 33, 21, 31] .  \nIn each case, proving completeness and extending decidability is a delicate issue. This has motivated the development of a general theory of Kleene algebra with hypotheses [31, 15 , 41] . There, models of closed languages are shown to characterise the Horn theory of star-continuous Kleene algebras, and a notion of reduction makes it possible to approach completeness and decidability proofs in a principled and modular way.  \nBeside hypotheses, Kleene algebras have also been extended to deal with common programming or logical constructs: conditionals in Kleene algebra with tests (KAT) [29], mutable tests [20], concurrency in bi-KA [37], concurrent KA [22] and synchronous KA (SKA) [44, 48], intersection [16], or converse [18, 3] . Here again, completeness proofs are  \n2 Continuous Algebras with Hypotheses  \nnotoriously hard, and while some of the above extensions do fit—with some effort—into the framework of KA with hypotheses (e.g., KAT, SKA and KA with converse), some of them do not, typically when operations such as parallel composition or intersection are added, or when the required closures do not preserve regularity. An extended framework was developed [25, 23 , 47] to deal with hypotheses in concurrent versions of KA, but a unifying theory is s","cbCaicc52Krk4Say","https://ap.wps.com/l/cbCaicc52Krk4Say","pdf",631893,4,1,38,"English","en",105,"# Introduction\n## Overview of Kleene algebra and hypotheses\n## Unifying framework using complete lattices and fixpoints\n## Parameterized syntax and language models","[{\"question\":\"Does the paper propose a universal axiomatisation for all hypotheses?\",\"answer\":\"No generic axiomatisation is expected to be both sound and complete for all cases; instead, the paper provides a generic axiomatisation and modular tools to obtain complete instances for specific variants.\"}]",1784204827,96,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"continuous-algebras-with-hypotheses","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/continuous-algebras-with-hypotheses/85595/",{"url":52,"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","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Does the paper propose a universal axiomatisation for all hypotheses?","Question",{"text":75,"@type":76},"No generic axiomatisation is expected to be both sound and complete for all cases; instead, the paper provides a generic axiomatisation and modular tools to obtain complete instances for specific variants.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]