A compositional approach to defining logics for coalgebras

We present a compositional approach to defining expressive logics for coalgebras of endofunctors on Set. This approach uses a notion of language constructor and an associated notion of semantics to capture one inductive step in the definition of a language for coalgebras and of its semantics. We show that suitable choices for the language constructors and for their associated semantics yield logics which are both adequate and expressive w.r.t, behavioural equivalence. Moreover, we show that type-building operations give rise to corresponding operations both on language constructors and on their associated semantics, thus allowing the derivation of expressive logics for increasingly complex coalgebraic types. Our framework subsumes several existing approaches to defining logics for coalgebras, and at the same time allows the derivation of new logics, with logics for probabilistic systems being the prime example.