Canonical coalgebraic linear time logics

We extend earlier work on linear time fixpoint logics for coalgebras with branching, by showing how propositional operators arising from the choice of branching monad can be canonically added to these logics. We then consider two semantics for the uniform modal fragments of such logics: the previously-proposed, step-wise semantics and a new semantics akin to those of path-based logics. We prove that the two semantics are equivalent, and show that the canonical choice made for resolving branching in these logics is crucial for this property. We also state conditions under which similar, non-canonical logics enjoy the same property – this applies both to the choice of a branching modality and to the choice of linear time modalities. Our logics allow reasoning about linear time behaviour in systems with non-deterministic, probabilistic or weighted branching. In all these cases, the logics enhanced with propositional operators gain in expressiveness. Another contribution of our work is a reformulation of fixpoint semantics, which applies to any coalgebraic modal logic whose semantics arises from a one-step semantics

Linear Time Logics - A Coalgebraic Perspective

We describe a general approach to deriving linear time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching behaviour and linear behaviour. Concretely, we define logics whose syntax is determined by the choice of linear behaviour and whose domain of truth values is determined by the choice of branching, and we provide two equivalent semantics for them: a step-wise semantics amenable to automata-based verification, and a path-based semantics akin to those of standard linear time logics. We also provide a semantic characterisation of the associated notion of logical equivalence, and relate it to previously-defined maximal trace semantics for such systems. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear time property. We conclude with a generalisation of the logics, dual in spirit to logics with discounting, which increases their practical appeal in the context of resource-aware computation by incorporating a notion of offsetting.Comment: Major revision of previous version: Sections 4 and 5 generalise the results in the previous version, with new proofs; Section 6 contains new result

Coalgebraic completeness-via-canonicity for distributive substructural logics

We prove strong completeness of a range of substructural logics with respect to a natural poset-based relational semantics using a coalgebraic version of completeness-via-canonicity. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions. Moreover, we believe that the coalgebraic framework provides a systematic and principled way to study the relationship between resource models on the semantics side, and substructural logics on the syntactic side.Comment: 36 page

PSPACE Bounds for Rank-1 Modal Logics

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 proof-theoretic 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

Generic Trace Semantics via Coinduction

Trace semantics has been defined for various kinds of state-based systems, notably with different forms of branching such as non-determinism vs. probability. In this paper we claim to identify one underlying mathematical structure behind these "trace semantics," namely coinduction in a Kleisli category. This claim is based on our technical result that, under a suitably order-enriched setting, a final coalgebra in a Kleisli category is given by an initial algebra in the category Sets. Formerly the theory of coalgebras has been employed mostly in Sets where coinduction yields a finer process semantics of bisimilarity. Therefore this paper extends the application field of coalgebras, providing a new instance of the principle "process semantics via coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page

Expressive Logics for Coinductive Predicates

The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata

Modal logics are coalgebraic

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