For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

State-based systems and modal logics for reasoning about them often heterogeneously combine a number of features such as non-determinism and probabilities. Here, we show that the combination of features can be reflected algorithmically and develop modular decision procedures for heterogeneous modal logics. The modularity is achieved by formalising the underlying state-based systems as multi-sorted coalgebras and associating both a logical and an algorithmic description to a number of basic building blocks. Our main result is that logics arising as combinations of these building blocks can be decided in polynomial space provided that this is the case for the components. By instantiating the general framework to concrete cases, we obtain PSPACE decision procedures for a wide variety of structurally different logics, describing e.g. Segala systems and games with uncertain information.

Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatised in rank 1. Here we establish the converse, i.e. every rank 1 modal logic has a sound and strongly complete coalgebraic semantics. As a consequence, recent results on coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, become applicable to arbitrary rank 1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of these results. As an extended example, we apply our framework to recently defined deontic logics.

We study sequent calculi for propositional modal logics, interpreted over coalgebras, with admissibility of cut being the main result. As applications we present a new proof of the (already known) interpolation property for coalition logic and establish the interpolation property for the conditional logics CK and CK Id.

CoLoSS, the Coalgebraic Logic Satisfiability Solver, decides satisfiability of modal formulas in a generic and compositional way. It implements a uniform polynomial space algorithm to decide satisfiability for modal logics that are amenable to coalgebraic semantics. This includes e.g. the logics K, KD, Pauly’s coalition logic, graded modal logic, and probabilistic modal logic. Logics are easily integrated into CoLoSS by providing a complete axiomatisation of their coalgebraic semantics in a specific format. Moreover, CoLoSS is compositional: it synthesises decision procedures for modular combinations of logics that include the fusion of two modal logics as a special case. One thus automatically obtains reasoning support e.g. for logics interpreted over probabilistic automata that combine non-determinism and probabilities in different ways.

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.

Coalgebras provide a uniform framework for the semantics of a large class of (mostly non-normal) modal logics, including e.g. monotone modal logic, probabilistic and graded modal logic, and coalition logic, as well as the usual Kripke semantics of modal logic. In earlier work, the finite model property for coalgebraic logics has been established w.r.t. the class of all structures appropriate for a given logic at hand; the corresponding modal logics are characterised by being axiomatised in rank 1, i.e. without nested modalities. Here, we extend the range of coalgebraic techniques to cover logics that impose global properties on their models, formulated as frame conditions with possibly nested modalities on the logical side (in generalisation of frame conditions such as symmetry or transitivity in the context of Kripke frames). We show that the finite model property for such logics follows from the finite algebra property of the associated class of complex algebras, and then investigate sufficient conditions for the finite algebra property to hold. Example applications include extensions of coalition logic and logics of uncertainty and knowledge.

Modal logics see a wide variety of applications in artificial intelligence, e.g. in reasoning about knowledge, belief, uncertainty, agency, defaults, and relevance. From the perspective of applications, the attractivity of modal logics stems from a combination of expressive power and comparatively low computational complexity. Compared to the classical treatment of modal logics with relational semantics, the use of modal logics in AI has two characteristic traits: Firstly, a large and growing variety of logics is used, adapted to the concrete situation at hand, and secondly, these logics are often non-normal. Here, we present a shallow model construction that witnesses PSPACE bounds for a broad class of mostly non-normal modal logics. Our approach is uniform and generic: we present general criteria that uniformly apply to and are easily checked in large numbers of examples. Thus, we not only re-prove known complexity bounds for a wide variety of structurally different logics and obtain previously unknown PSPACE-bounds, e.g. for Elgesem’s logic of agency, but also lay the foundations upon which the complexity of newly emerging logics can be determined.

The modelling of parthood relations in description logics via transitive roles often leads to undecidability when combined with number restrictions and role hierarchies. Here, we introduce the description logic PHQ that explicitly supports reasoning about parthood in the presence of qualified number restrictions. Our main results are completeness and decidability in NEXPTIME. Conceptually, we argue that PHQ provides a better semantic fit for many applications: more often than not, parthoods occurring e.g. in biomedical ontologies are expected to be tree-like. In such cases, PHQ supports stronger inferences than standard description logics. Technically this is achieved by explicitly excluding the merging of descendants, which, at the same time, eliminates the prime source of undecidability. We work in the general setting of coalgebraic modal logic, a generic semantic framework for not-necessarily-normal modal logics. This added generality allows the re-use of many of our results for other logics of sometimes quite different flavour.

Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. The coalgebraic approach is generic and compositional: tools and techniques simultaneously apply to a large class of application areas and can moreover be combined in a modular way. In particular, this facilitates a pick-and-choose approach to domain specific formalisms, applicable across the entire scope of application areas, leading to generic software tools that are easier to design, to implement, and to maintain. This paper substantiates the authors ’ firm belief that the systematic exploitation of the coalgebraic nature of modal logic will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrency/mobility.