We develop a new bisimulation (pseudo)metric for weighted finite automata (WFA) that generalizes Boreale's linear bisimulation relation. Our metrics are induced by seminorms on the state space of WFA. Our development is based on spectral properties of sets of linear operators. In particular, the joint spectral radius of the transition matrices of WFA plays a central role. We also study continuity properties of the bisimulation pseudometric, establish an undecidability result for computing the metric, and give a preliminary account of applications to spectral learning of weighted automata.

中文翻译：

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实加权自动机的互仿真度量和规范

我们为加权有限自动机 (WFA) 开发了一种新的互仿真（伪）度量，它概括了 Boreale 的线性互仿真关系。我们的指标是由 WFA 状态空间上的半范数引起的。我们的开发基于线性算子集的频谱特性。特别是，WFA 过渡矩阵的联合光谱半径起着核心作用。我们还研究了互模拟伪度量的连续性，建立了计算度量的不可判定性结果，并初步说明了在加权自动机谱学习中的应用。