Bisimulation Metrics for Weighted Automata

TL;DR

This paper introduces a new bisimulation pseudometric for weighted finite automata based on spectral properties, extending previous relations and exploring its theoretical properties and applications.

Contribution

It develops a spectral-based bisimulation metric for WFA, generalizing existing relations and analyzing its properties and computational challenges.

Findings

The bisimulation metric is induced by seminorms on WFA state space.

Spectral properties, especially joint spectral radius, are central to the metric.

Computing the metric is shown to be undecidable.

Abstract

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.