Dynamic topological logic of metric spaces.(English)Zbl 1241.03020

Summary: Dynamic Topological Logic ($$\mathcal {D T L}$$) is a modal framework for reasoning about dynamical systems, that is, pairs $$\langle X,f\rangle$$ where $$X$$ is a topological space and $$f: X\rightarrow X$$ 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 $$\mathbb Q$$.
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.

MSC:

 03B45 Modal logic (including the logic of norms) 03B80 Other applications of logic 54E35 Metric spaces, metrizability 54E50 Complete metric spaces 54H20 Topological dynamics (MSC2010)
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