Algebraic Models and Complete Proof Calculi for Classical BI

We consider the classical (propositional) version, CBI, of O’Hearn and Pym’s logic of bunched implications (BI) from a modeland proof-theoretic perspective. We make two main contributions in this paper. Firstly, we present a class of algebraic models for CBI which permit the full range of classical multiplicative connectives to be modelled. Our models can be seen as generalisations of Abelian groups, and include several computationally interesting models as concrete instances. Secondly, we give a display calculus proof system for CBI that is an instance of Belnap’s general display logic — hence cut-eliminating — and demonstrate this system to be sound and complete with respect to validity in our models. To achieve the latter, we first define a simple extension of the usual sequent calculus for BI by axioms that directly capture properties of our models, and show this extension to be sound and complete (though not cut-eliminating). Soundness and completeness of our display calculus then follows by establishing faithful translations between the display calculus and this extended sequent calculus.