Graded Semantics and Graded Logics for Eilenberg-Moore Coalgebras
Jonas Forster, Lutz Schröder, Paul Wild, Harsh Beohar, Sebastian Gurke, Karla Messing
TL;DR
The paper tackles aligning coalgebraic behavioural semantics with language and trace notions by formulating Eilenberg-Moore semantics as a special case of graded semantics. It develops a graded-logic framework, parametrized by a quantale, and shows that EM semantics admit expressive, invariant logics whose modalities arise from the language-type functor and a single branching-type modality. A key result is that expressivity of graded logics for EM semantics reduces to the branching-time expressivity of the corresponding language-type modalities, enabling principled construction of expressive logics for both two-valued and quantitative settings. The work unifies determinization-based semantics with graded modal logic, providing practical modal tools for probabilistic, weighted, and nondeterministic automata, and setting the stage for further integration with fixpoint logics and Kleisli-style trace semantics.
Abstract
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 termining 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.