Behavioural Preorders via Graded Monads

Like notions of process equivalence, behavioural preorders on processes come in many flavours, ranging from fine-grained comparisons such as ready simulation to coarse-grained ones such as trace inclusion. Often, such behavioural preorders are characterized in terms of theory inclusion in dedicated characteristic logics; e.g. simulation is characterized by theory inclusion in the positive fragment of Hennessy-Milner logic. We introduce a unified semantic framework for behavioural preorders and their characteristic logics in which we parametrize the system type as a functor on the category Pos of partially ordered sets following the paradigm of universal coalgebra, while behavioural preorders are captured as graded monads on Pos, in generalization of a previous approach to notions of process equivalence. We show that graded monads on Pos are induced by a form of graded inequational theories that we introduce here. Moreover, we provide a general notion of modal logic compatible with a given graded behavioural preorder, along with a criterion for expressiveness, in the indicated sense of characterization of the behavioural preorder by theory inclusion. We illustrate our main result on various behavioural preorders on labelled transition systems and probabilistic transition systems.