Dynamic topological logic of metric spaces

Abstract

Dynamic Topological Logic (DT L) 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 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.

Convergence, Continuity and Recurrence in Dynamic Epistemic Logic

“…With Theorem 1, a connection arises between dynamic epistemic logic and dynamic topological logic (see e.g. [23,24,34,35]): Each system (X d , f ) may be considered a dynamic topological model with atom set L Λ and the 'next' operator's semantics given by an application of f , equivalent to a f dynamic modality of DEL. The topological 'interior' operator has yet no DEL parallel.…”

The intuitionistic temporal logic of dynamical systems

“…Note that [18] considers a language without a universal modality, but adding it does not affect computable enumerability. Moreover, we proved in [19] that any satisfiable L C formula was also satisfiable on a model based on Q; as before, the universal modality can readily be incorporated, giving us the following: Theorem 4.6. Let ITL Q be the set of L I -formulas that are valid over the class of dynamical systems based on the rational numbers with the usual topology.…”

Complete Intuitionistic Temporal Logics in Topological Dynamics

“…Remark 11.4. Fernández-Duque [16] states Theorem 11.1 for L C , but the proof provided yields the result for all of L C . Roughly speaking, this is due to the fact that the simulations constructed in the proof are total and surjective.…”