Sequent Calculi with procedure calls

Mahfuza Farooque (1, 2) , Stéphane Graham-Lengrand (1, 2)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 PARSIFAL - Proof search and reasoning with logic specifications

In this paper, we study 3 focussed sequent calculi that are based on Miller-Liang's LKF system for polarised classical logic, and integrate the possibility to call a decision procedure for some background theory T . The main sequent calculus out of the three is LK(T), in which we prove cut-elimination. Changing the polarities of literals and connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T ) that extends LK(T) with a relaxed focussing discipline. We then prove completeness of LK(T) (and therefore of LK+(T)) with respect to first-order reasoning modulo the ground propositional lemmas of the background theory T . A third sequent calculus is introduced, LKp(T), that extends LK(T) with the possibility to polarise literals "on the fly" during proof-search. This is used in other works to simulate the DPLL(T) procedure, and we show here its completeness. Encodings of LK+(T) and LKp(T) into the most restrictive system LK(T) are presented along the way.

Mots clés: Sequent calculus; polarities; focusing; cut-elimination; completeness

Domaines: Logique en informatique [cs.LO]

Mahfuza Farooque, Stéphane Graham-Lengrand. Sequent Calculi with procedure calls. 2013. ⟨hal-00779199v1⟩