Hypersequent calculi for non-normal modal and deontic logics: countermodels and optimal complexity

Abstract

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms ${T}$, ${P}$ and ${D}$ and for every $n \geq 1$, rule ${RD}_n^+$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatization by a syntactic proof of cut elimination. Then, we define a terminating proof search strategy in the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we show that from every failed proof of a formula or hypersequent it is possible to directly extract a countermodel of it in the bi-neighbourhood semantics of polynomial size for coNP logics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube.

Reasoning in Non-normal Modal Description Logics

“…In this section, we provide terminating, sound and complete tableau algorithms to check satisfiability of formulas in varying domain neighbourhood models. The notation partly adheres to that of Gabbay et al [27], while the model construction in the soundness proof is based on the strategy of Dalmonte et al [28].…”