Non-deterministic semantics for dynamic topological logic

Abstract

Dynamic Topological Logic ( D T L ) is a combination of S 4 , under its topological interpretation, and the temporal logic L T L interpreted over the natural numbers. D T L is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models, which are tuples < X , T , f , V > , where < X , T > is a topological space, f a function on X and V a truth valuation assigning subsets of X to propositional variables. Our main result is that the set of valid formulas of D T L over spaces with continuous functions is recursively enumerable. We show this by defining alternative semantics for D T L . Under standard semantics, D T L is not complete for Kripke frames. However, we introduce the notion of a non-deterministic quasimodel, where the function f is replaced by a binary relation g assigning to each world multiple temporal successors. We place restrictions on the successors so that the logic remains unchanged; under these alternative semantics, D T L becomes Kripke-complete. We then apply model-search techniques to enumerate the set of all valid formulas.