Learning Rational Stochastic Languages

Abstract

Given a finite set of words w ₁, ..., w _n independently drawn according to a fixed unknown distribution law P called a stochastic language, a usual goal in Grammatical Inference is to infer an estimate of P in some class of probabilistic models, such as Probabilistic Automata (PA). Here, we study the class ${{\mathcal S}_{\mathbb R}^{rat}(\Sigma)}$ of rational stochastic languages, which consists in stochastic languages that can be generated by Multiplicity Automata (MA) and which strictly includes the class of stochastic languages generated by PA. Rational stochastic languages have minimal normal representation which may be very concise, and whose parameters can be efficiently estimated from stochastic samples. We design an efficient inference algorithm DEES which aims at building a minimal normal representation of the target. Despite the fact that no recursively enumerable class of MA computes exactly ${{\mathcal S}_{\mathbb Q}^{rat}(\Sigma)}$, we show that DEES strongly identifies ${{\mathcal S}_{\mathbb Q}^{rat}(\Sigma)}$ in the limit. We study the intermediary MA output by DEES and show that they compute rational series which converge absolutely and which can be used to provide stochastic languages which closely estimate the target.