Maehara-style Modal Nested Calculi

Lutz Strassburger (1, 2) , Roman Kuznets (3)

1 Inria Saclay - Ile de France
2 PARSIFAL - Proof search and reasoning with logic specifications
3 TU Wien - Vienna University of Technology = Technische Universität Wien

We develop multi-conclusion nested sequent calculi for the fifteen logics of the intuitionistic modal cube between IK and IS5. The proof of cut-free completeness for all logics is provided both syntactically via a Maehara-style translation and semantically by constructing an infinite birelational countermodel from a failed proof search. Interestingly, the Maehara-style translation for proving soundness syntactically fails due to the hierarchical structure of nested sequents. Consequently, we only provide the semantic proof of soundness. The countermodel construction used to prove completeness required a completely novel approach to deal with two independent sources of non-termination in the proof search present in the case of transitive and Euclidean logics.

Domaines: Logique en informatique [cs.LO]

Lutz Strassburger, Roman Kuznets. Maehara-style Modal Nested Calculi. [Research Report] RR-9123, Inria Saclay. 2017. ⟨hal-01644750⟩