A Bi-nested Calculus for Intuitionistic K: Proofs and Countermodels

TL;DR

This paper introduces a novel bi-nested sequent calculus for intuitionistic modal logic IK, enabling direct countermodel extraction and providing a decision procedure, thus advancing proof theory for this logic.

Contribution

It presents the first labelled-free bi-nested calculus for IK that supports countermodel extraction and simulates existing calculi, improving proof search and analysis.

Findings

Provides a decision procedure for IK

Enables direct countermodel extraction from failed proofs

Simulates existing nested and labelled calculi for IK

Abstract

The logic IK is the intuitionistic variant of modal logic introduced by Fischer Servi, Plotkin and Stirling, and studied by Simpson. This logic is considered a fundamental intuitionstic modal system as it corresponds, modulo the standard translation, to a fragment of intuitionstic first-order logic. In this paper we present a labelled-free bi-nested sequent calculus for IK. This proof system comprises two kinds of nesting, corresponding to the two relations of bi-relational models for IK: a pre-order relation, from intuitionistic models, and a binary relation, akin to the accessibility relation of Kripke models. The calculus provides a decision procedure for IK by means of a suitable proof-search strategy. This is the first labelled-free calculus for IK which allows direct counter-model extraction: from a single failed derivation, it is possible to construct a finite countermodel for the formula at the root. We further show the bi-nested calculus can simulate both the (standard) nested calculus and labelled sequent calculus, which are two best known calculi proposed in the literature for IK.