{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T07:48:24Z","timestamp":1748418504231},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":741,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Dynamic Topological Logic <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200001055_inline1\" \/> is a modal framework for reasoning about <jats:italic>dynamical systems<\/jats:italic>, that is, pairs \u3008<jats:italic>X, f<\/jats:italic>\u3009 where <jats:italic>X<\/jats:italic> is a topological space and <jats:italic>f<\/jats:italic>: <jats:italic>X<\/jats:italic> \u2192 <jats:italic>X<\/jats:italic> a continuous function.<\/jats:p><jats:p>In this paper we consider the case where <jats:italic>X<\/jats:italic> is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on \u211a.<\/jats:p><jats:p>We then show that the situation changes when considering <jats:italic>complete<\/jats:italic> metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space.<\/jats:p>","DOI":"10.2178\/jsl\/1327068705","type":"journal-article","created":{"date-parts":[[2012,1,20]],"date-time":"2012-01-20T09:16:13Z","timestamp":1327050973000},"page":"308-328","source":"Crossref","is-referenced-by-count":6,"title":["Dynamic topological logic of metric spaces"],"prefix":"10.1017","volume":"77","author":[{"given":"David","family":"Fern\u00e1ndez-Duque","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200001055_ref014","doi-asserted-by":"crossref","first-page":"477","DOI":"10.17323\/1609-4514-2005-5-2-477-492","article-title":"On completeness of dynamic topological logic","volume":"5","author":"Slavnov","year":"2005","journal-title":"Moscow Mathematics Journal"},{"key":"S0022481200001055_ref013","unstructured":"Slavnov S. , Two counterexamples in the logic of dynamic topological systems, Technical Report TR-2003015, Cornell University, 2003."},{"key":"S0022481200001055_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-005-0285-z"},{"key":"S0022481200001055_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-010-0185-8"},{"key":"S0022481200001055_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-010-9160-4"},{"key":"S0022481200001055_ref006","volume-title":"Real analysis: Modern techniques and their applications","author":"Folland","year":"1999"},{"key":"S0022481200001055_ref003","doi-asserted-by":"publisher","DOI":"10.1093\/jigpal\/jzl036"},{"key":"S0022481200001055_ref001","volume-title":"The general topology of dynamical systems","author":"Akin","year":"1993"},{"key":"S0022481200001055_ref002","doi-asserted-by":"crossref","unstructured":"Artemov S. 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M. , and Nerode A. , Modal logics and topological semantics for hybrid systems, Technical Report MSI 97-05, Cornell University, 1997.","DOI":"10.21236\/ADA344355"},{"key":"S0022481200001055_ref009","article-title":"Propositional temporal logics: Decidability and completeness","volume":"8","author":"Lichtenstein","journal-title":"Logic Jounal of the IGPL"},{"key":"S0022481200001055_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2004.06.004"},{"key":"S0022481200001055_ref012","doi-asserted-by":"publisher","DOI":"10.4064\/fm-1-1-11-16"},{"key":"S0022481200001055_ref011","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/exn034"},{"key":"S0022481200001055_ref016","volume-title":"The infinite-dimensional topology of function spaces","author":"van Mill","year":"2001"},{"key":"S0022481200001055_ref015","doi-asserted-by":"publisher","DOI":"10.4064\/fm-31-1-103-134"},{"key":"S0022481200001055_ref004","first-page":"110","volume-title":"Annals of Pure and Applied Logic","volume":"157","author":"Fern\u00e1ndez-Duque","year":"2009"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200001055","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T20:57:03Z","timestamp":1556225823000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200001055\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,3]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,3]]}},"alternative-id":["S0022481200001055"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1327068705","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,3]]}}}