Learning Probabilistic Residual Finite State Automata

We introduce a new class of probabilistic automata: Probabilistic Residual Finite State Automata. We show that this class can be characterized by a simple intrinsic property of the stochastic languages they generate (the set of residual languages is finitely generated) and that it admits canonical minimal forms. We prove that there are more languages generated by PRFA than by Probabilistic Deterministic Finite Automata (PDFA). We present a first inference algorithm using this representation and we show that stochastic languages represented by PRFA can be identified from a characteristic sample if words are provided with their probabilities of appearance in the target language. Introduction In the field of machine learning, most realistic situations deal with data provided by a stochastic source and probabilistic models, such as Hidden Markov Models (HMMs) or probabilistic automata (PA), become increasingly important. For example, speech recognition, computational biology and more generally, every field where statistical sequence analysis is needed, may use this kind of models. In this paper, we focus on Probabilistic Automata. A probabilistic automata can be described by its structure (a Finite State Automata) and by a set of continuous parameters (probability to emit a given letter from a given state or to end the generation process). There exist several fairly good methods to adjust the continuous parameters of a given structure to a training set of examples. However the efficient building of the structure from given data is still an open problem. Hence most applications of HMMs or PA assume a fixed model structure, which is either chosen as general as possible (i.e. a complete graph) or a priori selected using domain knowledge. Several learning algorithms, based on previous works in the field of grammatical inference, have been designed to output a deterministic structure (Probabilistic Deterministic Finite State Automata: PDFA) from training data ([1], [2], [3]; see also [4], [5] for early works) and several interesting theoretical and experimental results have been obtained. However, unlike to the case of non stochastic