An Institution of Modal Logics for Coalgebras

This paper presents a modular framework for the specification of certain inductively-defined coalgebraic types. Modal logics for coalgebras of polynomial endofunctors on the category of sets have been studied in [M. Rößiger, Coalgebras and modal logic, in: H. Reichel (Ed.), Coalgebraic Methods in Computer Science, Electronic Notes in Theoretical Computer Science, vol. 33, Elsevier Science, 2000, pp. 299–320; B. Jacobs, Many-sorted coalgebraic modal logic: a model-theoretic study, Theoretical Informatics and Applications 35(1) (2001) 31–59]. These logics are here generalised to endofunctors on categories of sorted sets, in order to allow collections of inter-related types to be specified simultaneously. The inductive nature of the coalgebraic types considered is then used to formalise semantic relationships between different types, and to define translations between the associated logics. The resulting logical framework is shown to be an institution, whose specifications and specification morphisms admit final and respectively cofree models

Coalgebraic Semantics for Timed Processes

We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad ” generated by the time domain. All our examples of time domains satisfy a partial closure property, yielding a distributive law of a monad for total monoid actions over the evolution comonad, and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions. We show that the induced coalgebras are exactly timed transition systems with delay operators. We then integrate our coalgebraic formulation of time qua timed transition systems into Turi and Plotkin’s formulation of structural operational semantics in terms of distributive laws. We combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact. We give a modular account of the operational semantics for a combination induced by that of each of its components. Our study necessitates the investigation of products of comonads. In particular, we characterise when a monad lifts to the category of coalgebras for a product comonad, providing constructions with which one can readily calculate. Key words: time domains, timed transition systems, evolution comonads, delay operators, structural operational semantics, modularity, distributive laws

On the final sequence of a finitary set functor

AbstractA standard construction of the final coalgebra of an endofunctor involves defining a chain of iterates, starting at the final object of the underlying category and successively applying the functor. In this paper we show that, for a finitary set functor, this construction always yields a final coalgebra in ω2=ω+ω steps

A modular approach to defining and characterising notions of simulation

We propose a modular approach to defining notions of simulation, and modal logics which characterise them. We use coalgebras to model state-based systems, relators to define notions of simulation for such systems, and inductive techniques to define the syntax and semantics of modal logics for coalgebras. We show that the expressiveness of an inductively defined logic for coalgebras w.r.t. a notion of simulation follows from an expressivity condition involving one step in the definition of the logic, and the relator inducing that notion of simulation. Moreover, we show that notions of simulation and associated characterising logics for increasingly complex system types can be derived by lifting the operations used to combine system types, to a relational level as well as to a logical level. We use these results to obtain Baltag’s logic for coalgebraic simulation, as well as notions of simulation and associated logics for a large class of non-deterministic and probabilistic systems

Coinductive Control of Inductive Data Types

We combine the theory of inductive data types with the theory of universal measurings. By doing so, we find that many categories of algebras of endofunctors are actually enriched in the corresponding category of coalgebras of the same endofunctor. The enrichment captures all possible partial algebra homomorphisms, defined by measuring coalgebras. Thus this enriched category carries more information than the usual category of algebras which captures only total algebra homomorphisms. We specify new algebras besides the initial one using a generalization of the notion of initial algebra

Relatively terminal coalgebras

AbstractDana Scott’s model of λ-calculus was based on a limit construction which started from an algebra of a suitable endofunctor F and continued by iterating F. We demonstrate that this is a special case of the concept we call coalgebra relatively terminal w.r.t. the given algebra A. This means a coalgebra together with a universal coalgebra-to-algebra morphism into A.We prove that by iterating F countably many times we obtain the relatively terminal coalgebras whenever F preserves limits of ωop-chains. If F is finitary, we need in general ω+ω steps. And for arbitrary accessible (=bounded) set functors we need an ordinal number of steps in general. Scott’s result is captured by the fact that in a CPO-enriched category, assuming that F is locally continuous, ω steps are sufficient for algebras given by projections