An Evaluation-Driven Decision Procedure for G3i

Abstract

It is well known that G3i, the sequent calculus for intuitionistic propositional logic where weakening and contraction are absorbed into the rules, is not terminating. Indeed, due to the contraction in the rule for left implication, the naïve goal-oriented proof-search strategy, consisting in applying the rules of the calculus bottom up until possible, can generate branches of infinite length. The usual solution to this problem is to support the proof-search procedure with a loop checking mechanism that prevents the generation of infinite branches by storing and analyzing some information regarding the branch under development.In this article, we propose a new technique based on evaluation functions. An evaluation function is a lightweight computational mechanism that, analyzing only the current goal of the proof search, allows one to drive the application of rules to guarantee termination and to avoid useless backtracking. We describe an evaluation-driven proof-search procedure that given a sequent σ returns either a G3i-derivation of σ or a countermodel for σ . We prove that such a procedure is terminating and correct, and that the depth of the G3i-trees generated during proof search is quadratic in the size of σ . Finally, we discuss the overhead time introduced by evaluation functions in the proof-search procedure.

Linear Depth Deduction with Subformula Property for Intuitionistic Epistemic Logic

“…A feature of the calculi in [6] is that to get proofs in linear depth it can be required to introduce new propositional variables. Finally, a further line of work is to apply the results in [4] where it is showed a terminating strategy for the sequent calculus G3i.…”

Duality between Unprovability and Provability in Forward Refutation-search for Intuitionistic Propositional Logic

“…3, whenever we search for an FRJ(G)-derivation of G, we are also trying to build a countermodel for G in a backward style, starting from the final worlds down to the root. Thus, our countermodel construction technique is dual to standard proof-search procedures such as [1,8,9,10,13,19,20], where proofs and model are searched bottom-up, starting from the goal and backward applying the rules of the calculus. One of the advantages of forward vs. backward reasoning is that, provided one implements suitable redundancy checks, derivations are more concise since sequents are reused and not duplicated.…”

“…Thus, our forward proof-search procedure can be understood as a top-down method to build a countermodel for G, starting from the final worlds down to the root. This original approach is dual to the standard one, where countermodels are built bottom-up, mimicking the backward application of rules (see.e.g., [1,8,9,10,13,19,20]). This different viewpoint has a significant impact in the outcome.…”

A Semantical Analysis of Focusing and Contraction in Intuitionistic Logic

“…Given our track record [4,[13][14][15], we are particularly interested in decision procedures for intuitionistic propositional logic; from that standpoint, one of Gentzen's original structural rules is particularly worrisome: contraction (or duplication, seen from the bottom up), which permits the reuse of a formula in the antecedent or succedent of a sequent:…”