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Dynamic topological logic of metric spaces

Published online by Cambridge University Press:  12 March 2014

David Fernández-Duque
David Fernández-Duque*
Affiliation:
Group for Logic, Language and Information, Universidad de Sevilla, Calle Camilo José Cela S/N, 41013 Seville, Spain, E-mail: [email protected]
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Abstract

Dynamic Topological Logic is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function.

In this paper we consider the case where X 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 ℚ.

We then show that the situation changes when considering complete 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 308 - 328
Copyright
Copyright © Association for Symbolic Logic 2012

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