Weighted automata are a generalization of nondeterministic automata that associate a weight drawn from a semiring $K$ with every transition and every state. Their behaviours can be formalized either as weighted language equivalence or weighted bisimulation. In this paper we explore the properties of weighted automata in the framework of coalgebras over (i) the category $\mathsf{SMod}$ of semimodules over a semiring $K$ and $K$-linear maps, and (ii) the category $\mathsf{Set}$ of sets and maps. We show that the behavioural equivalences defined by the corresponding final coalgebras in these two cases characterize weighted language equivalence and weighted bisimulation, respectively. These results extend earlier work by Bonchi et al. using the category $\mathsf{Vect}$ of vector spaces and linear maps as the underlying model for weighted automata with weights drawn from a field $K$. The key step in our work is generalizing the notions of linear relation and linear bisimulation of Boreale from vector spaces to semimodules using the concept of the kernel of a $K$-linear map in the sense of universal algebra. We also provide an abstract procedure for forward partition refinement for computing weighted language equivalence. Since for weighted automata defined over semirings the problem is undecidable in general, it is guaranteed to halt only in special cases. We provide sufficient conditions for the termination of our procedure. Although the results are similar to those of Bonchi et al., many of our proofs are new, especially those about the coalgebra in $\mathsf{SMod}$ characterizing weighted language equivalence.
翻译:加权自动数据是非确定性自动数据的一种概括化 { 加权自动数据 { 加权自动数据 } 将从半美元中抽取的重量与每个过渡阶段和每个州联系起来。 它们的行为可以正式化为加权语言等值或加权刺激。 在本文中, 我们探索了在以下框架中加权自动数据在煤层框架范围内的属性 :(一) 在半美元和线性地图上 $mathsf{Smod} 的半模块类别 { K$- 线性自动数据, (二) 从每半美元中抽取的半K美元和地图的加权值 。 我们工作的关键步骤是将线性关系和双线性平流的等值 。 在这两起案件中, 相应的最终煤层对等值定义的行为等等同性分别是加权语言对等值和加权刺激 。 我们的平流电子平流层平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面, 。