[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85449-en":3,"doc-seo-85449-105":29,"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":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},85449,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","The Complexity of Defining and Separating Fixpoint Formulae in Modal Logic","Modal separability for modal fixpoint formulae determines whether two given modal µ-calculus formulas φ and φ′ can be separated by a modal formula ψ such that φ ⊨ ψ and ψ ⊨ ¬φ′. The work studies separability and modal definability across model classes including arbitrary models, finite models, trees, and bounded outdegree structures. Results show PSpace-completeness on words, ExpTime-completeness for unrestricted and binary models, and 2-ExpTime-completeness when outdegree is bounded by d≥3, including a failure of Craig interpolation there.","arXiv :2509 .24583v4 [ cs .LO] 13 Jul 2026  \nTHE COMPLEXITY OF DEFINING AND SEPARATING FIXPOINT FORMULAE IN MODAL LOGIC  \nJEAN CHRISTOPH JUNG  a AND JĘDRZEJ KOŁODZIEJSKI  b  \na Department of Computer Science, TU Dortmund University, Germany e-mail address: [jean.jung@cs.tu-dortmund.de](jean.jung@cs.tu-dortmund.de)  \nb Faculty of Mathematics, Informatics and Mechanics, University of Warsaw e-mail address: [j.kolodziejski@uw.edu.pl](j.kolodziejski@uw.edu.pl)  \nAbstract. Modal separability for modal fixpoint formulae is the problem to decide for two given modal fixpoint formulae φ, φ′ whether there is a modal formula ψ that separates them, in the sense that φ |= ψ and ψ |= ¬φ′. We study modal separability and its special case modal definability over various classes of models, such as arbitrary models, finite models, trees, and models of bounded outdegree. Our main results are that modal separability is PSpace-complete over words, that is, models of outdegree ≤ 1 , ExpTime-complete over unrestricted and over binary models, and 2-ExpTime-complete over models of outdegree bounded by some d ≥ 3. Interestingly, this latter case behaves fundamentally different from the other cases also in that modal logic does not enjoy the Craig interpolation property over this class. Motivated by this we study also the induced interpolant existence problem as a special case of modal separability, and show that it is coNExpTime-complete and thus harder than validity in the logic. Besides deciding separability, we also provide algorithms for the effective construction of separators. Finally, we consider in a case study the extension of modal fixpoint formulae by graded modalities and investigate separability by modal formulae and graded modal formulae.  \n1. Introduction  \nFor given logics L, L+, the L-separability problem for L+ is to decide, given two L+-formulae φ, φ′, whether there is an L-formula ψ that separates φ and φ′ in the sense that φ |= ψ and ψ |= ¬φ′. Obviously, a separator can only exist when φ and φ′ are inconsistent: if φ ∧ φ′ is satisfiable in some model M, then M satisfies both any separator ψ (since φ |= ψ) and its negation ¬ψ (since ψ |= ¬φ′ and thus φ′ |= ¬ψ) . Moreover, when L ⊇ L+ and φ, φ′ are inconsistent, we can take φ or φ′ as trivial separators, so the separability problem is nontrivial only when L cannot express the entire L+ . Finally note that, for logics L+ closed under negation, L-separability generalizes the L-definability problem for L+: decide whether a given L+-formula is equivalent to an L-formula. Indeed, φ ∈ L+ is equivalent to an L-formula iff φ and ¬φ are L-separable. Since separability is more general than definability, solving it requires an even better understanding of the logics under consideration.  \nBoth definability and separability are central problems in logics, and both have a wide range of applications in computer science. On the one hand, L-definability in L+ is the essential question in understanding the expressive power of L within L+ . On the other  \nKey words and phrases: Modal Logic, Fixpoint Logic, Separability, Interpolation, Definability.  \nPreprint submitted to  \nLogical Methods in Computer Science  \n© J.Ch. Jung and J. Kołodziejski ⃝CC Creative Commons  \nhand, some applications require understanding the more general notion of separability, asthe following examples illustrate:  \n(1) In verification, a separator can be viewed as a property that separates good behavior, given by φ, from bad behavior, given by φ′. In this case φ, φ′ are not necessarily logical formulae but could be other symbolic representations of sets of reachable states. The separator can sometimes act as an easy abstraction of the reachable states, for instance, in the construction of invariants. This view of verification in terms of separability has induced the study of separability in formal languages. As seminal work in this area let us only mention definability and separability of regular word languages by languages re","cbCainrmc6imTdNg","https://ap.wps.com/l/cbCainrmc6imTdNg","pdf",716507,1,38,"English","en",105,"# Abstract\n# Introduction\n## L-separability and L-definability\n## Applications of separability","[{\"question\":\"What complexity results are obtained for modal separability?\",\"answer\":\"Modal separability is PSpace-complete over words, ExpTime-complete over unrestricted and binary models, and 2-ExpTime-complete when the outdegree is bounded by some d≥3. The d≥3 case also lacks Craig interpolation.\"}]",1784203626,96,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":27},"the-complexity-of-defining-and-separating-fixpoint-formulae-in-modal-logic","",{"@graph":35,"@context":77},[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/the-complexity-of-defining-and-separating-fixpoint-formulae-in-modal-logic/85449/",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],{"name":72,"@type":73,"acceptedAnswer":74},"What complexity results are obtained for modal separability?","Question",{"text":75,"@type":76},"Modal separability is PSpace-complete over words, ExpTime-complete over unrestricted and binary models, and 2-ExpTime-complete when the outdegree is bounded by some d≥3. The d≥3 case also lacks Craig interpolation.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]