We introduce a generic framework for hybrid logics, i.e. modal logics additionally featuring nominals and satisfaction operators, thus providing the necessary facilities for reasoning about individual states in a model. This framework, coalgebraic hybrid logic, works at the same level of generality as coalgebraic modal logic, and in particular subsumes, besides normal hybrid logics such as hybrid K, a wide variety of logics with non-normal modal operators such as probabilistic, graded, or coalitional modalities and non-monotonic conditionals. We prove a generic finite model property and an ensuing weak completeness result, and we give a semantic criterion for decidability in PSPACE. Moreover, we present a fully internalised PSPACE tableau calculus. These generic results are easily instantiated to particular hybrid logics and thus yield a wide range of new results, including e.g. decidability in PSPACE of probabilistic and graded hybrid logics.

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity are explicitly stated as open in the literature.

Conditional logics capture default entailment in a modal framework in which non-monotonic implication is a first-class citizen, and in particular can be negated and nested. There is a wide range of axiomatizations of conditionals in the literature, from weak systems such as the basic conditional logic CK, which allows only for equivalent exchange of conditional antecedents, to strong systems such as Burgess ’ system S, which imposes the full Kraus-Lehmann-Magidor properties of preferential logic. While tableaux systems implementing the actual complexity of the logic at hand have recently been developed for several weak systems, strong systems including in particular disjunction elimination or cautious monotonicity have so far eluded such efforts; previous results for strong systems are limited to semantics-based decision procedures and completeness proofs for Hilbert-style axiomatizations. Here, we present tableaux systems of optimal complexity PSPACE for several strong axiom systems in conditional logic, including system S; the arising decision procedure for system S is implemented in the generic reasoning tool CoLoSS.

Abstract. Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µ-calculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the µ-calculus. The family of such flat fixed point logics includes, e.g., CTL, the ∗-nesting-free fragment of PDL, and the logic of common knowledge. Here, we extend this notion to the generic semantic framework of coalgebraic logic, thus covering a wide range of logics beyond the standard µ-calculus including, e.g., flat fragments of the graded µ-calculus and the alternating-time µ-calculus (such as ATL), as well as probabilistic and monotone fixed point logics. Our main results are completeness of the Kozen-Park axiomatization and a timed-out tableaux method that matches EXPTIME upper bounds inherited from the coalgebraic µ-calculus but avoids using automata. 1