Axiomatizing the lexicographic products of modal logics with linear temporal logic

Philippe Balbiani (1, 2) , David Fernández-Duque (1, 3)

1 IRIT-LILaC - Logique, Interaction, Langue et Calcul
2 CNRS - Centre National de la Recherche Scientifique
3 ITAM - Instituto Tecnológico Autónomo de México

Given modal logics L1 and L2, their lexicographic product L1 x L2 is a new logic whose frames are the Cartesian products of an L1-frame and an L2-frame, but with the new accessibility relations reminiscent of a lexicographic ordering. This article considers the lexicographic products of several modal logics with linear temporal logic (LTL) based on ``next'' and ``always in the future''. We provide axiomatizations for logics of the form L x LTL and define cover-simple classes of frames; we then prove that, under fairly general conditions, our axiomatizations are sound and complete whenever the class of L-frames is cover-simple. Finally, we prove completeness for several concrete logics of the form L x LTL.

Mots clés: Modal logic; Linear temporal logic; Lexicographic product; Axiomatization

Domaines: Intelligence artificielle [cs.AI]; Informatique et langage [cs.CL]; Apprentissage [cs.LG]; Logique en informatique [cs.LO]

Philippe Balbiani, David Fernández-Duque. Axiomatizing the lexicographic products of modal logics with linear temporal logic. 11th conference on Advances in Modal logic (AiML 2016), Aug 2016, Budapest, Hungary. pp.78-96. ⟨hal-03658877⟩