Sonia Marin (University College London), Focused nested calculi applied to the logic of bunched implications



Focusing is a general technique for syntactically compartmentalising the non-deterministic choices in a proof system, which not only improves proof search but also has the representational benefit of distilling sequent proofs into synthetic normal forms. However, since focusing was traditionally specified as a restriction of the sequent calculus, the technique had not been transferred to logics that lack a (shallow) sequent presentation, as is the case for some modal or substructural logics.

With K. Chaudhuri and L. Straßburger, we extended the focusing technique to nested sequents, a generalisation of ordinary sequents which allows us to capture all the logics of the classical and intuitionistic S5 cube in a modular fashion. This relied, following the method introduced by O. Laurent, on an adequate polarisation of the syntax and an internal cut-elimination procedure for the focused system which in turn is used to show its completeness.

Recently, with A. Gheorghiu, we applied a similar method to the logic of Bunched Implications (BI), a logic that freely combines intuitionistic logic and multiplicative linear logic. For this we had first to reformulate the traditional bunched calculus for BI using nested sequents, followed by a polarised and focused variant that (again) we show is sound and complete via a cut-elimination argument.