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Like notions of process equivalence, behavioural preorders on processes comein many flavours, ranging from fine-grained comparisons such as readysimulation to coarse-grained ones such as trace inclusion. Often, suchbehavioural preorders are characterized in terms of theory inclusion indedicated characteristic logics; e.g. simulation is characterized by theoryinclusion in the positive fragment of Hennessy-Milner logic. We introduce aunified semantic framework for behavioural preorders and their characteristiclogics in which we parametrize the system type as a functor on the category$\mathsf{Pos}$ of partially ordered sets following the paradigm of universalcoalgebra, while behavioural preorders are captured as graded monads on$\mathsf{Pos}$, in generalization of a previous approach to notions of processequivalence. We show that graded monads on $\mathsf{Pos}$ are induced by a formof graded inequational theories that we introduce here. Moreover, we provide ageneral notion of modal logic compatible with a given graded behaviouralpreorder, along with a criterion for expressiveness, in the indicated sense ofcharacterization of the behavioural preorder by theory inclusion. We illustrateour main result on various behavioural preorders on labelled transition systemsand probabilistic transition systems.

Behavioural Preorders via Graded Monads