🤖 AI Summary
The semantic relationship between nonnegative real-weighted automata and probabilistic automata remains poorly understood, particularly due to the lack of a systematic characterization of their stochastic structure. This work addresses this gap by introducing a semantics-preserving weight rescaling technique that normalizes finitely weighted automata into locally stochastic probabilistic automata, thereby disentangling their exponential growth component from their stochastic behavior. Leveraging Perron–Frobenius theory and spectral radius analysis, we establish an equivalence between the two models under normalization. We further propose a characterization via stochastic regular expressions based on geometrically weighted Kleene stars and develop a general behavioral decomposition framework that effectively constructs a locally stochastic probabilistic automaton from any given weighted automaton with finite total mass.
📝 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.