We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that innnitary modal logic contains characterizing formulas. That is, every model-world pair is characterized up to bisimulation by an innnitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a frangm...

We show that certain fixpoint expressions used to describe finite Kripke polynomial coalgebras can be seen as coalgebraic modal fixpoint formulae. Both the synthesis of a coalgebra from its expression and the ability to check behavioural equivalence follow from the same tableau construction. There is an associated complete equational logic, analogous to Kleene algebra, which may now be seen as ...

This papers reviews the classical theory of deterministic automata and regular languages from a categorical perspective. The basis is formed by Rutten's description of the Brzozowski automaton structure in a coalgebraic framework. We enlarge the framework to a so-called bialgebraic one, by including algebras together with suitable distributive laws connecting the algebraic and coalgebraic struc...

The coalgebraic perspective on objects and classes in object-oriented programming is elaborated: objects consist of a (unique) identiier, a local state, and a collection of methods described as a coalgebra; classes are coalgebraic (behavioural) speciications of objects. The creation of a \new" object of a class is described in terms of the terminal coalgebra satisfying the speciication. We pres...

This paper examines three important topics in computer science: behavioural equivalence, bisimulation, and minimal realisation of automata, and discusses the relationships that hold between them. Central to all three topics is a notion of equivalence of behaviour, and by taking a coalgebraic approach to algebraic speciications we show that bisimulation is`the same as' behavioural equivalence. M...

The microcosm principle, advocated by Baez and Dolan and formalized for Lawvere theories lately by three of the authors, has been applied to coalgebras in order to describe compositional behavior systematically. Here we further illustrate the usefulness of the approach by extending it to a many-sorted setting. Then we can show that the coalgebraic component calculi of Barbosa are examples, with...

The main aim of this paper is an extension of the theory of algebraic graph transformation systems by a loose semantics. For this purpose, graph transitions are introduced as a loose interpretation of graph productions. They are deened using a double pullback construction in contrast to classical graph derivations based on double-pushouts. Two characterisation results relate graph transitions t...