Nonhomogeneous Stochastic Automata

In this note we consider a nonhomogeneous Markov chain type stochastic automaton which is a generalization of Bartoszyński’s stochastic automaton. The latter is a generalization of the Pawlak’s known machine in a stochastic direction. By nonhomogeneous stochastic automaton we mean a system ⟨ T, α, {A(n), n ⩾ 1}⟩, where T is a finite nonempty set, α, is an initial distribution on T, and {A(n), n ⩾ 1} is a matrix sequence whose every element is a stochastic matrix called a transition probability matrix. If A(n) = A for all n ⩾ 1, then we obtain Bartoszyński’s automaton. The sequence (ti0, ti1, …), tij ∈ T, j = 0, 1, 2, … is called a word of automata if α(ti0) > 0 and A(k)(tik-1, tik) > 0 for every k ⩾ 1. The goal of this note is to give necessary and sufficient conditions for the existence of an extension and a shrinkage of the automata under consideration. These problems for T, A were considered for the first time by Bartoszyński. The shrinkage problem deals with the existence of a stochastic automaton which generates only all sequences of states of T which are simultaneously generated by two given automata while the extension problem treats of the existence of a stochastic automaton which generates all sequences of states of which are generated by at least one of two given automata. Moreover, we introduce some new notions: attainable state, concordance of automata in a wide and a narrow sense, which help us to solve the problems mentioned above.