Analytic Proofs for Tense Logic

Abstract

The first algorithm to transform a proof in Nishimura’s sequent calculus GKt for tense logic Kt into an analytic proof of the same sequent is presented. In an analytic proof, every rule instance is analytic i.e., each formula in every premise is a subformula of some formula in its conclusion. We call this algorithm analytic restriction to convey that it extends analytic cut-restriction where just the cut-rule instances are made analytic. This distinction is essential in tense logic since cut and modal rules can both cause non-analyticity. Analytic cut-restriction is itself an extension of cut-elimination so our work contributes to a broader program of transforming arbitrary sequent proofs into ones constructed from a designated set of formulas—not necessarily subformulas. As with cut-elimination, the aim is to limit the proof search space and support proof-theoretic and meta-logical investigations.