Relating Sequent Calculi for Bi-intuitionistic Propositional Logic

TL;DR

This paper compares three different sequent calculi for bi-intuitionistic propositional logic, analyzing their completeness, cut-elimination properties, and inter-translation capabilities to understand their relative strengths and limitations.

Contribution

It provides a detailed comparison and translation methods among three sequent calculi for bi-intuitionistic logic, highlighting their completeness and cut-elimination features.

Findings

Standard sequent calculus is incomplete without cut

Nested sequents encapsulate cut rules into unnest rules

Labelled sequent calculus is cut-free and based on Kripke semantics

Abstract

Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for bi-intuitionistic propositional logic: (1) a basic standard-style sequent calculus that restricts the premises of implication-right and exclusion-left inferences to be single-conclusion resp. single-assumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Gore et al., where a complete class of cuts is encapsulated into special "unnest" rules and (3) a cut-free labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standard-style sequent calculus.