Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

Abstract

In this paper we present LSJ, a contraction-free sequent calculus for Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent σ we can extract a Kripke countermodel for σ . Finally, we provide a procedure that given a sequent σ returns either a proof of σ in LSJ or a refutation in RJ such that the extracted counter-model is of minimal depth.

JTabWb: a Java Framework for Implementing Terminating Sequent and Tableau Calculi

“…E.g., the method getAppliedRule() of IterationInfo returns the rule applied by the engine in the last move; in many cases this is the only data needed to choose the next rule to apply to goal. For instance, this is an high-level description of the strategy for a terminating sequent calculus for intuitionistic propositional logic (as, e.g., the calculus LSJ described in [2]). …”

“…g3ibu is a prover based on the sequent calculi Gbu and Rbu [3]. lsj is a prover based on the sequent calculi LSJ and RJ [2]. Both these provers allow one to generate counter-models for unprovable sequents.…”

Intuitionistic Decision Procedures Since Gentzen

“…Sequents are [24] of the form Γ Θ = ⇒ ∆, the components Γ, Θ, ∆ being sets (of formulae) rather than multisets. Let us use < for ≤ without equality.…”

“…A syntactic proof of cut-admissibility for this calculus seems difficult; a semantic proof is in [24]. Using our own implementation of LSJ, with Prolog cuts to prune the search space wherever seemed appropriate, the first (indeed, only) proof we found of the formula that is the type of the S combinator is 87 lines long.…”

An Evaluation-Driven Decision Procedure for G3i

“…In the construction of a branch, the formulas decomposed by right rules are stored in the history; loops are avoided by preventing the application of some right rules to formulas already in the history. The second line of research, which in proof-theoretical terms concerns the identification of contraction-free calculi, generated several calculi where the reuse of implicative formulas is prevented by replacing A → B on the left with "simpler" formulas or adopting a nonstandard notion of sequent [Vorob'ev 1970;Dyckhoff 1992;Hudelmaier 1993;Miglioli et al 1997;Ferrari et al 2009Ferrari et al , 2013a.…”