Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.

That is, makes the diagram C R 1 oo 2 // D TC TR T1 oo T2 // TD 1 commute. We call a pair (c; d) 2 C D bisimilar, if 9R C D. R bisimulation and (c; d) 2 R. If c and d are bisimilar, this is denoted by c - d. 2 On Bisimulation If f : A ! B is a function, denote its Graph by G(f) = f(a; f(a)) j a 2<F12.2

For the remainder of this exposition assume that T: Set! Set is an endofunctor. Unless otherwise stated, the results and proofs of the material presented is taken from Rutten [10]. 1 Preliminaries Deo/nition 1.1. (i) A T-coalgebra (T-system) is pair (C; fl) where C is a set and fl: C! T C. (ii) If (C; fl) and (D; ffi) are T-systems, then f: C! D is a T-coalgebra homomorphism (T-morphism), if ffi ffi f = T f ffi fl, that is, if the diagram C fl fflffl f // D

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Coalgebras for a functor on the category of sets subsume many formulations of the notion of transition system, including labelled transition systems, Kripke models, Kripke frames and many types of automata. This paper presents a multimodal language which is bisimulation invariant and (under a natural completeness condition) expressive enough to characterise elements of the underlying state space up to bisimulation. Like Moss' coalgebraic logic, the theory can be applied to an arbitrary signature functor on the category of sets. Also, an upper bound for the size of conjunctions and disjunctions needed to obtain characteristic formulas is given.

It has by now been recognised that coalgebras of various different endofunctors can be used to give semantics to a wide range of modal logics [5] and the last decade has seen an intensive study of modal