PSPACE Tableau Algorithms for Acyclic Modalized $\boldsymbol{\mathcal{ALC}}$

Abstract

We study $\mathcal{ALCK}_m$ and $\mathcal{ALCS}4_m$, which extend the description logic $\mathcal{ALC}$ by adding modal operators of the basic multi-modal logics K _m and S4 _m . We develop a sound and complete tableau algorithm $\Lambda_\mathcal{K}$ for answering $\mathcal{ALCK}_m$ queries w.r.t. an $\mathcal{ALCK}_m$ knowledge base with an acyclic TBox. Defining tableau expansion rules in the presence of acyclic definitions by considering only the concept names on the left-hand side of TBox definitions or their negations, allows us to give a PSPACE implementation for $\Lambda_\mathcal{K}$. We then consider answering $\mathcal{ALCS}4_m$ queries w.r.t. an $\mathcal{ALCS}4_m$ knowledge base (with an acyclic TBox) in which the epistemic operators correspond to those of classical multi-modal logic S4 _m . The expansion rules in the tableau algorithm Λ _S4 are designed to syntactically incorporate the epistemic properties. Blocking is corporated into the tableau expansion rules to ensure termination. We also provide a PSPACE implementation for Λ _S4. In light of the fact that the satisfiability problem for $\mathcal{ALCK}_m$ with general TBox and no epistemic properties (i.e., $\mathbf{K}_{\mathcal{ALC}}$) is NEXPTIME-complete, we conclude that both $\mathcal{ALCK}_m$ and $\mathcal{ALCS}4_m$ offer computationally manageable and practically useful fragments of $\mathbf{K}_{\mathcal{ALC}}$.