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Abstract. We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search.

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Abstract. The description logic SHI extends the basic description logic ALC with transitive roles, role hierarchies and inverse roles. The known tableau-based decision procedure [9] for SHI exhibit (at least) NEXP-TIME behaviour even though SHI is known to be EXPTIME-complete. The automata-based algorithms for SHI often yield optimal worst-case complexity results, but do not behave well in practice since good optimisations for them have yet to be found. We extend our method for global caching in ALC to SHI by adding analytic cut rules, thereby giving the first EXPTIME tableau-based decision procedure for SHI, and showing one way to incorporate global caching and inverse roles.