The Decision Problems of Definite Stochastic Automata

Abstract

A k-definite stochastic automaton is defined as a finite stochastic automaton in which the internal state probability distribution depends only on the last k input symbols for some definite integer k. Whether or not a stochastic automaton is definite depends on its state transition matrices. A set of stochastic matrices is called definite of order k by Paz if, for a fixed integer k, any product of k or more of the matrices is a matrix with all rows equal. A study is made of conditions in which linearly independent row vectors in the component matrices will result in identical row vectors in the product matrix. It is established that a definite stochastic automaton with n internal states and with highest rank of its transition matrices $n - m$ is at least ${{(n - m)} / m}$-definite. A time saving decision procedure for the definiteness of stochastic automata is presented and illustrated.