Non-deterministic Semantics for Dynamic Topological Logic

Dynamic Topological Logic (DT L) is a combination of S4, under its topological interpretation, and the temporal logic LT L interpreted over the natural numbers. DT 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 DT L over spaces with continuous functions is recursively enumerable. We do this by defining alternative semantics for DT L. Under the standard semantics, DT 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, DT L becomes Kripke-complete. We then apply model-search techniques to enumerate the set of all valid formulas.