On Rational Stochastic Languages

The goal of the present paper is to provide a study of rational stochastic languages over a semiring K is an element of {Q, R, Q(+), R(+)}. A rational stochastic language is a probability distribution over a free monoid Sigma*, which is rational over K, that is, which can be generated by a multiplicity automaton with parameters in K. We study the relations between the classes of rational stochastic languages S(K)(rat)(Sigma). We define the notion of residual language of a stochastic language and we use it to investigate properties of several subclasses of rational stochastic languages. Then, we study the representation of rational stochastic languages by means of multiplicity automata. Lastly, we show some connections between properties of rational stochastic languages and results obtained in the field of probabilistic grammatical inference.