Relating Sequent Calculi for Bi-intuitionistic Propositional Logic

Luís Pinto
Tarmo Uustalu

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.

In Steffen van Bakel, Stefano Berardi and Ulrich Berger: Proceedings Third International Workshop on Classical Logic and Computation (CL&C 2010), Brno, Czech Republic, 21-22 August 2010, Electronic Proceedings in Theoretical Computer Science 47, pp. 57–72.
Published: 27th January 2011.

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