Graded Semantics and Graded Logics for Eilenberg-Moore Coalgebras
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Coalgebra, as the abstract study of state-based systems, comes naturally equipped with a notion of behavioural equivalence that identifies states exhibiting the same behaviour. In many cases, however, this equivalence is finer than the intended semantics. Particularly in automata theory, behavioural equivalence of nondeterministic automata is essentially bisimilarity, and thus does not coincide with language equivalence. Language equivalence can be captured as behavioural equivalence on the determinization, which is obtained via the standard powerset construction. This construction can be lifted to coalgebraic generality, assuming a so-called Eilenberg-Moore distributive law between the functor determining the type of accepted structure (e.g. word languages) and a monad capturing the branching type (e.g. nondeterministic, weighted, probabilistic). Eilenberg-Moore-style coalgebraic semantics in this sense has been shown to be essentially subsumed by the more general framework of graded semantics, which is centrally based on graded monads. Graded semantics comes with a range of generic results, in particular regarding invariance and, under suitable conditions, expressiveness of dedicated modal logics for a given semantics; notably, these logics are evaluated on the original state space. We show that the instantiation of such graded logics to the case of Eilenberg-Moore-style semantics works extremely smoothly, and yields expressive modal logics in essentially all cases of interest. We additionally parametrize the framework over a quantale of truth values, thus in particular covering both the two-valued notions of equivalence and quantitative ones, i.e. behavioural distances.
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