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 introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of 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 LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) back into the more restrictive system LK(T). We then prove completeness of LKp(T) (and therefore of LK+(T)) with respect to first-order reasoning modulo the ground propositional lemmas of the background theory T .

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-00779199v4⟩