Cut Free Labelled Sequent Calculus for Dynamic Logic of Relation Changers

Abstract

Dynamic epistemic logic ($\mathbf {DEL}$) is known as a large family of logics that extend standard epistemic logic with dynamic operators. Such dynamic operators can be regarded as epistemic actions over Kripke semantics (or its variant). Therefore, $\mathbf {DEL}$ is often used to model changes of agents’ knowledge, belief or preference over Kripke semantics in terms of dynamic operators in many literatures. As a variant of $\mathbf {DEL}$, (van Benthem and Liu, J Appl Non-Classical Logics 17(2):157–18 2007; Liu, Reasoning about preference dynamics, Springer Science & Business Media, Berlin 2011) proposed dynamic logic of relation changers ($\mathbf {DLRC}$). They provided a general framework to capture many dynamic operators in terms of relation changing operation written in programs of propositional dynamic logic, and they also provided a sound and complete Hilbert-style axiomatization for $\mathbf {DLRC}$. While $\mathbf {DLRC}$ can cover many dynamic operators in a uniform manner, proof theory for $\mathbf {DLRC}$ is not well-studied except the Hilbert-style axiomatization. Therefore, we propose a cut-free labelled sequent calculus for $\mathbf {DLRC}$. We show that our sequent calculus is equipollent with the Hilbert-style axiomatization.