On Dynamic Topological and Metric Logics

The first result of this paper shows that some dynamic topological logics interpreted in various topological spaces with homeomorphisms are not recursively enumerable (and so are not recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints [11]. Second, we prove the non-elementary decidability of the dynamic metric logic with distance operators of the form ‘somewhere in the ball of radius a,’ for a ∈Q+, interpreted in arbitrary metric spaces with distance preserving automorphisms.