Sequent Calculi with procedure calls

TL;DR

This paper introduces two focused sequent calculi, LKp(T) and LK+(T), that incorporate decision procedures and dynamic polarization for background theories, enabling proof-search simulations of DPLL(T) and analyzing proof shapes.

Contribution

The paper presents novel sequent calculi integrating theory decision procedures and on-the-fly polarization, with proofs of cut-elimination and completeness in the context of background theories.

Findings

Cut-elimination is proven for LKp(T).

Polarities influence provability in the presence of a theory.

LK+(T) can be encoded into LKp(T), preserving completeness.

Abstract

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 .