Coalgebras for Bisimulation of Weighted Automata over Semirings

Weighted automata are a generalization of nondeterministic that associate weight drawn from semiring $K$ with every transition and state. Their behaviours can be formalized either as weighted language equivalence or bisimulation. In this paper we explore the properties in framework coalgebras over (i) category $\mathsf{SMod}$ semimodules $K$-linear maps, (ii) $\mathsf{Set}$ sets maps. We show behavioural equivalences defined by corresponding final these two cases characterize bisimulation, respectively. These results extend earlier work Bonchi et al. using $\mathsf{Vect}$ vector spaces linear maps underlying model for weights field $K$. The key step our is generalizing notions relation bisimulation Boreale to concept kernel map sense universal algebra. also provide an abstract procedure forward partition refinement computing equivalence. Since semirings problem undecidable general, it guaranteed halt only special cases. sufficient conditions termination procedure. Although similar those al., many proofs new, especially about coalgebra characterizing