Graded Monads for the Linear Time - Branching Time Spectrum
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State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the particular case of labelled transition 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 monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. In particular, it yields a generic notion of trace logic, which is maybe surprisingly based on using graded monad algebras as formulas. In the present paper, we focus on substantiating the genericity over process equivalences by elaborating concrete graded monads for a range of equivalences from the linear time - branching time spectrum. Moreover, we complete the theory of trace logics by adding an explicit propositional layer and providing a generic expressiveness criterion that generalizes the classical Hennessy-Milner theorem to coarser notions of process equivalence. We extract trace logics for our leading examples, and give exemplaric proofs of their trace invariance and expressiveness based on our generic criterion.
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