First, we closely investigated the proof-theoretic framework of hypersequents and its relation to other proof-theoretic frameworks. This yielded, e.g. limitative results, stating that certain logics are not captured in the hypersequent framework, as well as positive results such as the first hypersequent-based decision procedure of optimal complexity for the important modal logic S5. Moreover, these investigations led to the introduction of the proof-theoretic framework of grafted hypersequents, which we used to provide novel complexity-optimal calculi for modal logics K5 and KD5.
A further investigation into the relations between other sequent-style calculi led to the introduction of the linear nested sequent framework, an intermediate framework between ordinary sequents and nested sequents. The introduction of this framework and the development of a new method for the construction of so-called standard calculi from ordinary sequent calculi was pivotal for the project. In particular, we used the developed method to construct novel modular calculi for an infinite class of multimodal non-normal and normal modal logics, based on classical or substructural propositional logic, and containing most of the standard logics. Investigating the connections between linear nested and labelled sequents yielded encodings of these calculi in linear logic. This included the surprising result that subexponential linear logic can be encoded in linear logic, and hence the two logics are equally expressive. Due to the modularity of the introduced calculi they naturally lend themselves to implementations. We heeded this call and implemented both the encoding of the calculi in a linear logic theorem prover as well as two theorem provers based directly on the calculi for the logics based on classical and substructural logic, respectively.
As an important application of the developed methods, in a series of publications we constructed novel standard calculi for a large class of conditional logics introduced by D. Lewis. Apart from giving rise to a practical decision procedure for the considered logics, the constructed calculi have a strong connection to the semantics of these logics, and thus can be used for counter model generation. A first implementation is available online.
The obtained results were disseminated in a number of scientific publications, both in international journals such as Theoretical Computer Science or the Logic Journal of the IGPL, and in the proceedings of renowned international conferences, such as TABLEAUX, JELIA, or LPAR.
