{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T21:57:37Z","timestamp":1747173457507,"version":"3.40.5"},"reference-count":22,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2023,12,13]],"date-time":"2023-12-13T00:00:00Z","timestamp":1702425600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2024,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Modal logic enjoys topological semantics that may be traced back to McKinsey and Tarski, and the classification of topological spaces via modal axioms is a lively area of research. In the past two decades, there has been interest in extending topological modal logic to the language of the mu-calculus, but previously no class of topological spaces was known to be mu-calculus definable that was not already modally definable. In this paper, we show that the full mu-calculus is indeed more expressive than standard modal logic, in the sense that there are classes of topological spaces (and weakly transitive Kripke frames), which are mu-definable but not modally definable. The classes we exhibit satisfy a modally definable property outside of their perfect core, and thus we dub them <jats:italic>imperfect spaces.<\/jats:italic> We show that the mu-calculus is sound and complete for these classes. Our examples are minimal in the sense that they use a single instance of a greatest fixed point, and we show that least fixed points alone do not suffice to define any class of spaces that is not already modally definable.<\/jats:p>","DOI":"10.1017\/s0960129523000385","type":"journal-article","created":{"date-parts":[[2023,12,13]],"date-time":"2023-12-13T07:07:20Z","timestamp":1702451240000},"page":"81-97","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Fixed point logics and definable topological properties"],"prefix":"10.1017","volume":"34","author":[{"given":"David","family":"Fern\u00e1ndez-Duque","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3886-1143","authenticated-orcid":false,"given":"Quentin","family":"Gougeon","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,12,13]]},"reference":[{"key":"S0960129523000385_ref3","doi-asserted-by":"crossref","unstructured":"Baltag, A. , Bezhanishvili, N. and Fern\u00e1ndez-Duque, D. (2021). The topological mu-calculus: completeness and decidability. In: 36th Annual ACM\/IEEE Symposium on Logic in Computer Science, LICS 2021, Rome, Italy, June 29\u2013July 2, 2021, IEEE, 1\u201313.","DOI":"10.1109\/LICS52264.2021.9470560"},{"key":"S0960129523000385_ref20","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s11225-006-8301-9","article-title":"Modal frame correspondences and fixed-points","volume":"83","author":"van Benthem","year":"2006","journal-title":"Studia Logica"},{"key":"S0960129523000385_ref11","first-page":"827","article-title":"Succinctness in subsystems of the spatial m-calculus","volume":"5","author":"Fern\u00e1ndez-Duque","year":"2018","journal-title":"FLAP"},{"key":"S0960129523000385_ref18","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.apal.2010.07.003","article-title":"Completeness for flat modal fixpoint logics","volume":"162","author":"Santocanale","year":"2010","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0960129523000385_ref22","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1006\/inco.1999.2836","article-title":"Completeness of Kozen\u2019s axiomatisation of the propositional m-calculus","volume":"157","author":"Walukiewicz","year":"2000","journal-title":"Information and Computation"},{"key":"S0960129523000385_ref14","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1016\/0304-3975(82)90125-6","article-title":"Results on the propositional","volume":"27","author":"Kozen","year":"1983","journal-title":"Theoretical Computer Science"},{"key":"S0960129523000385_ref9","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/s11225-011-9339-x","article-title":"On the modal definability of simulability by finite transitive models","volume":"98","author":"Fern\u00e1ndez-Duque","year":"2011","journal-title":"Studia Logica"},{"key":"S0960129523000385_ref2","first-page":"501","article-title":"Diskrete R\u00e4ume","volume":"2","author":"Aleksandroff","year":"1937","journal-title":"Matematicheskii Sbornik"},{"key":"S0960129523000385_ref8","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.apal.2003.11.013","article-title":"Intuitionistic logic and modality via topology","volume":"127","author":"Esakia","year":"2004","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0960129523000385_ref7","first-page":"244","article-title":"Weak transitivity\u2013a restitution","volume":"8","author":"Esakia","year":"2001","journal-title":"Logical Investigations"},{"key":"S0960129523000385_ref19","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1080\/11663081.1999.10510972","article-title":"Everywhere\u201d and \u201chere","volume":"9","author":"Shehtman","year":"1999","journal-title":"Journal of Applied Non-Classical Logics"},{"key":"S0960129523000385_ref6","volume-title":"Oxford Logic Guides","volume":"35","author":"Chagrov","year":"1997"},{"key":"S0960129523000385_ref21","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/978-1-4020-5587-4_5","volume-title":"Handbook of Spatial Logics","author":"van Benthem","year":"2007"},{"volume-title":"Cambridge Tracts in Theoretical Computer Science","year":"2001","author":"Blackburn","key":"S0960129523000385_ref5"},{"key":"S0960129523000385_ref13","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s11225-017-9732-1","article-title":"The finite model property for logics with the tangle modality","volume":"106","author":"Goldblatt","year":"2018","journal-title":"Studia Logica"},{"key":"S0960129523000385_ref10","unstructured":"Fern\u00e1ndez-Duque, D. (2011b). Tangled modal logic for spatial reasoning. In: Twenty-Second International Joint Conference on Artificial Intelligence."},{"key":"S0960129523000385_ref17","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.apal.2007.11.001","article-title":"Completions of m-algebras","volume":"154","author":"Santocanale","year":"2008","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0960129523000385_ref15","doi-asserted-by":"crossref","unstructured":"Kudinov, A. and Shehtman, V. (2014). Derivational modal logics with the difference modality. In: Leo Esakia on Duality in Modal and Intuitionistic Logics, Springer, 291\u2013334.","DOI":"10.1007\/978-94-017-8860-1_11"},{"key":"S0960129523000385_ref16","doi-asserted-by":"crossref","first-page":"141","DOI":"10.2307\/1969080","article-title":"The algebra of topology","volume":"45","author":"McKinsey","year":"1944","journal-title":"Annals of Mathematics"},{"key":"S0960129523000385_ref1","doi-asserted-by":"crossref","unstructured":"Afshari, B. and Leigh, G. (2017). Cut-free completeness for modal mu-calculus. In: 32nd Annual ACM\/IEEE Symposium on Logic in Computer Science LICS, IEEE Press, 1\u201312.","DOI":"10.1109\/LICS.2017.8005088"},{"key":"S0960129523000385_ref12","doi-asserted-by":"crossref","first-page":"1032","DOI":"10.1016\/j.apal.2016.11.006","article-title":"Spatial logic of tangled closure operators and modal mu-calculus","volume":"168","author":"Goldblatt","year":"2017","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0960129523000385_ref4","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1215\/00294527-1306190","article-title":"An algebraic approach to subframe logics. Modal case","volume":"52","author":"Bezhanishvili","year":"2011","journal-title":"Notre Dame Journal of Formal Logic"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129523000385","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,5]],"date-time":"2024-02-05T11:24:29Z","timestamp":1707132269000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129523000385\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,13]]},"references-count":22,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,2]]}},"alternative-id":["S0960129523000385"],"URL":"https:\/\/doi.org\/10.1017\/s0960129523000385","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"type":"print","value":"0960-1295"},{"type":"electronic","value":"1469-8072"}],"subject":[],"published":{"date-parts":[[2023,12,13]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}