Dynamic Topological Logic of Metric Spaces

Abstract

Dynamic Topological Logic (DT L) is a modal framework for reasoning about dynamical systems, that is, pairs hX; fi 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 rst show that any formula which can be satis ed on an arbitrary dynamic topological system can be satis ed 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 formula can be satis ed 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 satis able on a dynamical system based on a complete metric space is also satis ed on one based on the Cantor space