On Logics for Coalgebraic Simulation

We investigate logics for coalgebraic simulation from a compositional perspective. Specifically, we show that the expressiveness of an inductively-defined language for coalgebras w.r.t. a given notion of simulation comes as a consequence of an expressivity condition between the language constructor used to define the language for coalgebras, and the relator used to define the notion of simulation. This result can be instantiated to obtain Baltag's logics for coalgebraic simulation, as well as a logic which captures simulation on unlabelled probabilistic transition systems. Moreover, our approach is compositional w.r.t. coalgebraic types. This allows us to derive logics which capture other notions of simulation, including trace inclusion on labelled transition systems, and simulation on discrete Markov processes