[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84325-en":3,"doc-seo-84325-105":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},84325,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Finite Convergence of the Modal µ-Calculus on Almost-Periodic Words","Finite convergence in modal µ-calculus holds when some finite unfolding of a formula defines the same set, and a structure has finite convergence when this property holds for every µ-calculus formula. While examples exist of non-ultimately periodic words that still enjoy finite convergence, the notion is characterized here using almost-periodicity. A word is almost-periodic if each factor either occurs only finitely often or reappears in every sufficiently long factor window. The paper proves the converse, establishing an equivalence and recovering related decidability results.","Finite Convergence of the Modal µ-Calculus on Almost-Periodic Words  \nFabian Lehr TU Munich, Germany  \nFlorian Bruse TU Munich, Germany  \narXiv :2607 .08 18 1v 1 [ cs .LO] 9 Jul 2026  \nJuly 10, 2026  \nAbstract  \nA formula of the modal µ-calculus enjoys finite convergence on a structure if there is some finite unfolding of the formula that defines the same set. A structure enjoys finite convergence if all formulas of the µ-calculus enjoy finite convergence on said structure. It is known that there are words that are not ultimately periodic, but have finite convergence.  \nAn almost-periodic word w is one in which each finite word v either appears only finitely often, or within each factor of some length that only depends only on w and v. It is immediate that words that have finite convergence must be almost periodic. In this paper we show the converse, namely that all almost-periodic words have finite convergence. This characterizes finite convergence on infinite words, and also re-provesa decidability result due to Semenov (’84) .  \n1 Introduction  \nThe Modal µ-calclus (Lµ ) [21, 17] is a central logic for model-checking and verification, in particular due to its role as the yardstick for regular temporal logics such as LTL and CTL. Lµ extends basic modal logic by the ability to formulate recursive definitions via least and greatest fixpoints.  \nSince Lµ serves as the prototypical temporal logic, the exact nature of its recursive definitions has been studied intensively. For example, it is known that nesting more fixpoint definitions of opposite polarity (e.g., least fixpoints that are mutually recursive with greatest fixpoints, etc.) gives strictly more expressive power [5] . Hence, the so-called alternation hierarchy of the µ-calculus is strict. This is not true if one restricts oneself to words or word-like structures [13, 12], where the µ-calculus collapses into its alternation-free fragment. Finally, over structures without infinite paths, least fixpoints can be rewritten into greatest fixpoints and vice versa [18] .  \nAnother central question is how fast the fixpoint of such a recursive definition is reached. It follows from the Kleene Fixpoint Theorem [15] that both least and greatest fixpoints can be replaced by an ordinal-indexed sequence of approximations, such that, over any given structure, some approximation is equivalent to the actual fixpoint definition. Hence, the fixpoint definition converges at this ordinal. This ordinal is then called the structure’s closure ordinal of the fixpoint  \ndefinition, respectively that of the formula that expresses it. The exact nature of the closure ordinal has consequences for e.g., model checking [6] .  \nGiven a formula, the least ordinal that bounds the closure ordinal of the formula on all structures is simply called the closure ordinal of the formula, if it exists. For example, the formula µX. □x, which expresses that all paths in the structure are finite, has no closure ordinal. There has been a long chain of research establishing the exact nature of closure ordinals of formulas. For example, [9] establishes that for every ordinal less that ω 2 , there is a formula with this closure ordinal. [11] gives formulas with uncountable closure ordinals, and [2, 1] show for the so-called Σ-fragment that closure ordinals are either below ω 2 or at least ω 1 . Moreover, [20] presents a similar dichotomy for monadic second-order logic on trees. See [1] for a good overview over the literature.  \nA related line of research concerns the closure behavior of structures, i.e. , the closure ordinals of formulas on a fixed structure. On finite structures, all fixpoint definitions uniformly converge at the cardinality of the structure, and this extends to structures with finite bisimulation quotient, where convergence happens at the size of the quotient. A natural question is then whether finite convergence also requires a finite bisimulation quotient. This is of interest because e.g., on struct","cbCaislvIctIW8gI","https://ap.wps.com/l/cbCaislvIctIW8gI","pdf",471009,1,14,"English","en",105,"# Abstract\n# Introduction\n## Background: Modal µ-Calculus and Alternation\n## Closure Ordinals and Convergence Speed\n## Finite Convergence vs Bisimulation Quotients\n## Almost-Periodic Words Characterization\n## Decidability and Automata on Almost-Periodic Words","[{\"question\":\"What does “finite convergence” mean for a modal µ-calculus formula and a structure?\",\"answer\":\"A formula has finite convergence on a structure if a finite unfolding defines the same set as the original recursive definition. A structure has finite convergence when every µ-calculus formula does so on that structure.\"},{\"question\":\"Why do almost-periodic words matter for finite convergence on infinite words?\",\"answer\":\"Finite convergence implies almost-periodicity for the underlying word. This paper proves the converse: all almost-periodic words have finite convergence, yielding a complete characterization.\"},{\"question\":\"How does this work relate to closure ordinals and fixpoint convergence?\",\"answer\":\"The paper builds on closure ordinals, which measure when iterative approximations reach the least or greatest fixpoint. It connects the existence of finite closure behavior on a word with the almost-periodic property, and discusses related decidability via automata behavior.\"}]",1784194829,35,{"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},"finite-convergence-of-the-modal-calculus-on-almost-periodic-words","",{"@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/finite-convergence-of-the-modal-calculus-on-almost-periodic-words/84325/",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 “finite convergence” mean for a modal µ-calculus formula and a structure?","Question",{"text":75,"@type":76},"A formula has finite convergence on a structure if a finite unfolding defines the same set as the original recursive definition. A structure has finite convergence when every µ-calculus formula does so on that structure.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why do almost-periodic words matter for finite convergence on infinite words?",{"text":80,"@type":76},"Finite convergence implies almost-periodicity for the underlying word. This paper proves the converse: all almost-periodic words have finite convergence, yielding a complete characterization.",{"name":82,"@type":73,"acceptedAnswer":83},"How does this work relate to closure ordinals and fixpoint convergence?",{"text":84,"@type":76},"The paper builds on closure ordinals, which measure when iterative approximations reach the least or greatest fixpoint. It connects the existence of finite closure behavior on a word with the almost-periodic property, and discusses related decidability via automata behavior.","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"]