## Section: New Results

### Focusing for Nested Sequents

Participants : Kaustuv Chaudhuri, Sonia Marin, Lutz Straßburger.

Focusing is a general technique for transforming a sequent proof system into one with a syntactic separation of non-deterministic choices without sacrificing completeness. This not only improves proof search, but also has the representational benefit of distilling sequent proofs into synthetic normal forms. We have shown how to apply the focusing technique to nested sequent calculi, a generalization of ordinary sequent calculi to tree-like instead of list-like structures. We thus improve the reach of focusing to the most commonly studied modal logics, the logics of the modal S5 cube. Among our key contributions is a focused cut-elimination theorem for focused nested sequents. This is published in [25].

Then we further extend our results to intuitionistic nested sequents, which can capture all the logics of the intuitionistic S5 cube in a modular fashion. We obtained an internal cut-elimination procedure for the focused system which in turn is used to show its completeness. This is published in [26]