A Sequent Calculus for Dynamic Topological Logic

Abstract

We introduce a sequent calculus for the temporal-over-topological fragment $\textbf{DTL}_{0}^{\circ * \slash \Box}$ of dynamic topological logic $\textbf{DTL}$, prove soundness semantically, and prove completeness syntactically using the axiomatization of $\textbf{DTL}_{0}^{\circ * \slash \Box}$ given in \cite{paper3}. A cut-free sequent calculus for $\textbf{DTL}_{0}^{\circ * \slash \Box}$ is obtained as the union of the propositional fragment of Gentzen's classical sequent calculus, two $\Box$ structural rules for the modal extension, and nine $\circ$ (next) and $*$ (henceforth) structural rules for the temporal extension. Future research will focus on the construction of a hypersequent calculus for dynamic topological $\textbf{S5}$ logic in order to prove Kremer's Next Removal Conjecture for the logic of homeomorphisms on almost discrete spaces $\textbf{S5H}$.