Bisimulation for Weakly Expressive Coalgebraic Modal Logics.

Abstract

Research on the expressiveness of coalgebraic modal logics with respect to semantic equivalence notions has so far focused mainly on finding logics that are able to distinguish states that are not behaviourally equivalent (such logics are said to be expressive). In other words, the notion of behavioural equivalence is taken as the starting point, and the expressiveness of the logic is evaluated against it. However, for some applications, modal logics that are not expressive are of independent interest. Such an example is given by contingency logic. We can now turn the question of expressiveness around and ask, given a modal logic, what is a suitable notion of semantic equivalence? In this paper, we propose a notion of \Lambda-bisimulation which is parametric in a collection \Lambda of predicate liftings. We study the basic properties of \Lambda-bisimilarity, and prove as our main result a Hennessy-Milner style theorem, which shows that (for finitary functors) \Lambda-bisimilarity exactly matches the expressiveness of the coalgebraic modal logic arising from \Lambda.