Generalized Automata and Stochastic Languages

1. Introduction. In a probabilistic automaton, as defined e.g. in [2] and [l], the initial vector as well as the transition matrices are stochastic, and the final vector consists of O's and l's only. This paper is concerned with the so-called generalized automata where the elements of the initial vector, of the final vector and of the matrices are allowed to be arbitrary real numbers. Our purpose is to prove that a language is accepted by a generalized automaton if and only if it is accepted by a probabilistic automaton, i.e., if and only if it is a sto-chastic language. As an application, we show that the mirror image of a stochastic language is a stochastic language.