Dirk Pattinson

Coalgebraic modal logic: soundness, completeness and decidability of local consequence

This paper studies finitary modal logics, interpreted over coalgebras for an endofunctor, and establishes soundness, completeness and decidability results. The logics are studied within the abstract framework of coalgebraic modal logic, which can be instantiated with arbitrary endofunctors on the category of sets. This is achieved through the use of predicate liftings, which generalise atomic propositions and modal operators from Kripke models to arbitrary coalgebras. Predicate liftings also allow us to use induction along the terminal sequence of the underlying endofunctor as a proof principle. This induction principle is systematically exploited to establish soundness, completeness and decidability of the logics. We believe that this induction principle also opens new ways for reasoning about modal logics: Our proof of completeness does not rely on a canonical model construction, and the proof of the finite model property does not use filtrations.

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 axiomatization, in PSPACE. This leads not only to a unified derivation of (known) tight PSPACE-bounds for a number of logics including K, coalition logic, and graded modal logic (and to a new algorithm in the latter case), but also to a previously unknown tight PSPACE-bound for probabilistic modal logic, with rational probabilities coded in binary. 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

Modal Logics are Coalgebraic1

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.

Representations of stream processors using nested fixed points

We define representations of continuous functions on infinite streams of dis- crete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discrete-valued functions, the representatives are well-founded (finite- path) trees of a certain kind. The underlying idea can be traced back to Brouwers justi- fication of bar-induction, or to Kreisel and Troelstras elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an al- ternating fixpoint construction of some ubiquity.

Algebraic Semantics for Coalgebraic Logics

With coalgebras usually being defined in terms of an endofunctor T on sets, this paper shows that modal logics for T-coalgebras can be naturally described as functors L on boolean algebras. Building on this idea, we study soundness, completeness and expressiveness of coalgebraic logics from the perspective of duality theory. That is, given a logic L for coalgebras of an endofunctor T, we construct an endofunctor L such that L-algebras provide a sound and complete (algebraic) semantics of the logic. We show that if L is dual to T, then soundness and completeness of the algebraic semantics immediately yield the corresponding property of the coalgebraic semantics. We conclude by characterising duality between L and T in terms of the axioms of L. This provides a criterion for proving concretely given logics to be sound, complete and expressive.

Modular construction of complete coalgebraic logics

We present a modular approach to defining logics for a wide variety of state-based systems. The systems are modelled as coalgebras, and we use modal logics to specify their observable properties. We show that the syntax, semantics and proof systems associated with such logics can all be derived in a modular fashion. Moreover, we show that the logics thus obtained inherit soundness, completeness and expressiveness properties from their building blocks. We apply these techniques to derive sound, complete and expressive logics for a wide variety of probabilistic systems, for which no complete axiomatisation has been obtained so far.

Modular Construction of Modal Logics

We present a modular approach to defining logics for a wide variety of state-based systems. We use coalgebras to model the behaviour of systems, and modal logics to specify behavioural properties of systems. We show that the syntax, semantics and proof systems associated to such logics can all be derived in a modular way. Moreover, we show that the logics thus obtained inherit soundness, completeness and expressiveness properties from their building blocks. We apply these techniques to derive sound, complete and expressive logics for a wide variety of probabilistic systems.

Modular algorithms for heterogeneous modal logics

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.

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 axiomatization, in PSPACE. This leads not only to a unified derivation of (known) tight PSPACE-bounds for a number of logics including K, coalition logic, and graded modal logic (and to a new algorithm in the latter case), but also to a previously unknown tight PSPACE-bound for probabilistic modal logic, with rational probabilities coded in binary. 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.

Ultrafilter extensions for coalgebras

This paper studies finitary modal logics as specification languages for Set-coalgebras (coalgebras on the category of sets) using Stone duality. It is well-known that Set-coalgebras are not semantically adequate for finitary modal logics in the sense that bisimilarity does not in general coincide with logical equivalence. Stone-coalgebras (coalgebras over the category of Stone spaces), on the other hand, do provide an adequate semantics for finitary modal logics. This leads us to study the relationship of finitary modal logics and Set-coalgebras by uncovering the relationship between Set-coalgebras and Stone-coalgebras. This builds on a long tradition in modal logic, where one studies canonical extensions of modal algebras and ultrafilter extensions of Kripke frames to account for finitary logics. Our main contributions are the generalisations of two classical theorems in modal logic to coalgebras, namely the Jonsson-Tarski theorem giving a set-theoretic representation for each modal algebra and the bisimulation-somewhere-else theorem stating that two states of a coalgebra have the same (finitary modal) theory iff they are bisimilar (or behaviourally equivalent) in the ultrafilter extension of the coalgebra.