A Sequent Calculus for Dynamic Topological Logic

TL;DR

This paper develops a sequent calculus for a fragment of dynamic topological logic, establishing its soundness and completeness, and sets the stage for future work on hypersequent calculus for related modal logics.

Contribution

It introduces a novel sequent calculus for a specific fragment of dynamic topological logic, combining propositional, modal, and temporal rules, with proven soundness and completeness.

Findings

Soundness of the calculus established

Completeness proved via existing axiomatization

A cut-free calculus constructed from known components

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 $\textbf{S5H}$.