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

Tiziano Dalmonte (1, 2) , Björn Lellmann (3) , Nicola Olivetti (1, 2) , Elaine Pimentel (4)

1 LIRICA - Logique, Interaction, Raisonnement et Inférence, Complexité, Algèbre
2 AMU - Aix Marseille Université
3 TU Wien - Vienna University of Technology = Technische Universität Wien
4 UFRN - Universidade Federal do Rio Grande do Norte [Natal]

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms T , P , D, and, for every n ≥ 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 axiomatisation 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.

Mots clés: Non-normal modal logic; deontic logic; hypersequent calculus; neighbourhood semantics; countermodels; optimal complexity

Domaines: Logique en informatique [cs.LO]

Tiziano Dalmonte, Björn Lellmann, Nicola Olivetti, Elaine Pimentel. Hypersequent calculi for non-normal modal and deontic logics: countermodels and optimal complexity. Journal of Logic and Computation, 2021, 31 (1), pp.67-111. ⟨10.1093/logcom/exaa072⟩. ⟨hal-03159947⟩