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State-based models of concurrent systems are traditionally considered under avariety of notions of process equivalence. In the particular case of labelledtransition systems, these equivalences range from trace equivalence to (strong)bisimilarity, and are organized in what is known as the linear time --branching time spectrum. A combination of universal coalgebra and graded monadsprovides a generic framework in which the semantics of concurrency can beparametrized both over the branching type of the underlying transition systemsand over the granularity of process equivalence. We show in the present paperthat this framework of graded semantics does subsume the most importantequivalences from the linear time -- branching time spectrum. An importantfeature of graded semantics is that it allows for the principled extraction ofcharacteristic modal logics. We have established invariance of these gradedlogics under the given graded semantics in earlier work; in the present paper,we extend the logical framework with an explicit propositional layer andprovide a generic expressiveness criterion that generalizes the classicalHennessy-Milner theorem to coarser notions of process equivalence. We extractgraded logics for a range of graded semantics on labelled transition systemsand probabilistic systems, and give exemplaric proofs of their expressivenessbased on our generic criterion.

Graded Monads and Graded Logics for the Linear Time -- Branching Time Spectrum