Localising Stochasticity in Weighted Automata

TL;DR

This paper demonstrates that weighted automata over nonnegative reals can be normalized into probabilistic automata when their transition matrices have spectral radius less than one, revealing a fundamental connection between these models.

Contribution

It introduces a normalization technique for weighted automata to probabilistic automata using Perron-Frobenius theory, and characterizes probabilistic automata via stochastic regular expressions with weighted stars.

Findings

Weighted automata with spectral radius < 1 can be normalized to probabilistic automata.

Finite-mass weighted automata and probabilistic automata are equivalent up to normalization.

Behavior of arbitrary weighted automata decomposes into growth rate and stochastic component.

Abstract

Weighted automata over the nonnegative reals form a fundamental model for quantitative languages. We show that, up to scaling, this model collapses to probabilistic automata. Concretely, we prove that every weighted automaton whose transition matrix has spectral radius strictly less than one can be normalised, by a semantics-preserving rescaling of transition weights, into an equivalent locally stochastic probabilistic automaton. Thus, finite-mass weighted automata and probabilistic automata coincide up to normalisation. The construction is effective and relies on Perron-Frobenius theory. We further characterise probabilistic automata by stochastic regular expressions equipped with a geometrically weighted star. Beyond the finite-mass setting, we show that the behaviour of an arbitrary weighted automaton admits a decomposition into an exponential growth rate and a normalised probabilistic component, separating quantitative growth from stochastic structure.