An Algebraic Approach to Proof Search in Sequent Calculi

Inference rules in the sequent calculus can be interpreted as both proof construction rules and proof search rules. However, the kind of information used in each case is somewhat different. In this paper we explore these differences by using a multiple-conclusioned sequent calculus for intuitionistic logic (LM) as a search calculus for proofs in the single-conclusioned intuitionistic sequent calculus LJ. We also show how the classical sequent calculus LK can be considered as both a search calculus and a proof calculus. The key technical issue is to determine the appropriate Boolean constraints to be attached to each formula, reflecting the search choices made. We also briefly discuss the possibilities for extending these techniques to intermediate logics via Avron’s hypersequents, and to an additive form of proof-nets for linear logic.