Deciding regular grammar logics with converse through first-order logic

Stéphane Demri (1, 2) , Hans de Nivelle (3)

1 CNRS - Centre National de la Recherche Scientifique
2 LSV - Laboratoire Spécification et Vérification [Cachan]
3 Max Planck Institute for Informatics [Saarbrücken]

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF 2 , which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. The class of regular grammar logics includes numerous logics from quite different application domains. The notion of regularity is directly related to the frame condition of the logics. A consequence of the translation is that the general satisfiability problem for every regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show that some other modal logics can be naturally translated into GF2 , including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF 2 without extra machinery such as fixed point-operators.

Mots clés: modal logic; relational translation; guarded fragment; 2-variable fragment; temporal logic; finite-state automaton; regular language; nominal tense logic; intuitionnistic logic; information logic; closure

Domaines: Informatique [cs]; Logique en informatique [cs.LO]

Stéphane Demri, Hans de Nivelle. Deciding regular grammar logics with converse through first-order logic. Journal of Logic, Language and Information, 2005, 14 (3), pp.289-329. ⟨10.1007/s10849-005-5788-9⟩. ⟨hal-03187843⟩