Two Counterexamples in the Logic of Dynamic Topological Systems

The classical Tarski theorem on topological semantics of modal logic states that the logic S4 is complete in R for each n. Recently several authors have considered logics of dynamic topological systems, which is a topological space and a function on it. In [1] a bimodal logic S4C was introduced and proven to be complete with respect to the class of all continuous dynamic systems. A number of polymodal logics for dynamic topological systems were considered in [3, 4, 5]. In [5] a modal logic of dynamic systems with homeomorphisms was axiomatized and proven to enjoy the analogue of the Tarski theorem. In this note it is shown that the analogue of the Tarski theorem does not hold for S4C, a question posed by Artemov and Nerode. In the language with an iteration of the dynamic system function, we also construct an R-valid formula that does not hold in the logic of dynamic systems with homeomorphisms. This proves that the analogue of the Tarski theorem does not hold for the logic of homeomorphisms with iterations.