{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:53:18Z","timestamp":1753894398151,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2023,12,18]],"date-time":"2023-12-18T00:00:00Z","timestamp":1702857600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>Topological semantics for modal logic based on the Cantor derivative operator\ngives rise to derivative logics, also referred to as $d$-logics. Unlike logics\nbased on the topological closure operator, $d$-logics have not previously been\nstudied in the framework of dynamical systems, which are pairs $(X,f)$\nconsisting of a topological space $X$ equipped with a continuous function\n$f\\colon X\\to X$. We introduce the logics $\\bf{wK4C}$, $\\bf{K4C}$ and\n$\\bf{GLC}$ and show that they all have the finite Kripke model property and are\nsound and complete with respect to the $d$-semantics in this dynamical setting.\nIn particular, we prove that $\\bf{wK4C}$ is the $d$-logic of all dynamic\ntopological systems, $\\bf{K4C}$ is the $d$-logic of all $T_D$ dynamic\ntopological systems, and $\\bf{GLC}$ is the $d$-logic of all dynamic topological\nsystems based on a scattered space. We also prove a general result for the case\nwhere $f$ is a homeomorphism, which in particular yields soundness and\ncompleteness for the corresponding systems $\\bf{wK4H}$, $\\bf{K4H}$ and\n$\\bf{GLH}$. The main contribution of this work is the foundation of a general\nproof method for finite model property and completeness of dynamic topological\n$d$-logics. Furthermore, our result for $\\bf{GLC}$ constitutes the first step\ntowards a proof of completeness for the trimodal topo-temporal language with\nrespect to a finite axiomatisation -- something known to be impossible over the\nclass of all spaces.<\/jats:p>","DOI":"10.46298\/lmcs-19(4:26)2023","type":"journal-article","created":{"date-parts":[[2023,12,19]],"date-time":"2023-12-19T17:55:06Z","timestamp":1703008506000},"source":"Crossref","is-referenced-by-count":0,"title":["Dynamic Cantor Derivative Logic"],"prefix":"10.46298","volume":"Volume 19, Issue 4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8604-4183","authenticated-orcid":false,"given":"David","family":"Fern\u00e1ndez-Duque","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9814-7323","authenticated-orcid":false,"given":"Yo\u00e0v","family":"Montacute","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2023,12,18]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/12717\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/12717\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,15]],"date-time":"2024-04-15T20:20:06Z","timestamp":1713212406000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/10042"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,18]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-19(4:26)2023","relation":{"has-preprint":[{"id-type":"arxiv","id":"2107.10349v4","asserted-by":"subject"},{"id-type":"arxiv","id":"2107.10349v3","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2107.10349","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2107.10349","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2023,12,18]]},"article-number":"10042"}}