Coalgebraic Logic over Measurable Spaces: Behavioral and Logical Equivalence

We study the relationship between logical and behavioral equivalence for coalgebras on general measurable spaces. Modal logics are interpreted in these coalgebras using predicate liftings. Prominent examples include stochastic relations and labelled Markov transition systems and corresponding Hennessy–Milner type logics. Local versions of logical and behavioral equivalence are introduced and it is shown that these notions coincide for a wide class of functors. We relate these notions to the corresponding global ones common in model checking. Throughout, we work in general measurable spaces. In contrast to previous work, no topological assumptions on the state spaces are needed.