Dynamic topological logics over spaces with continuous functions

Dynamic topological logics are combinations of topological and temporal modal logics that are used for reasoning about dynamical systems consisting of a topological space and a continuous function on it. Here we partially solve a major open problem in the field by showing (by reduction of the ω-reachability problem for lossy channel systems) that the dynamic topological logic over arbitrary topological spaces as well as those over Rn, for each n ≥ 1, are undecidable. Actually, we prove this result for the natural and expressive fragment of the full dynamic topological language where the topological operators cannot be applied to formulas containing the temporal eventuality. Using Kruskal’s tree theorem we also show that the formulas of this fragment that are valid in arbitrary topological spaces with continuous functions are recursively enumerable, which is not the case for spaces with homeomorphisms.