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

Jia Tao, Giora Slutzki, Vasant G. Honavar

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 \(\mathbf{K}_{\mathcal{ALC}}\).