On extension of stochastic k-automata

There are a great many research works concerning the well-known stochastic automata of Moore, Mealy, Rabin, Turing and others. Recently an automaton of Markov’s chain type has been introduced by Bartoszyński. This automaton is obtained by a generalization of Pawlak’s deterministic machine. The aim of this note is to give a concept of a stochastic automaton of Markov’s generalized chain type. The introduced automaton called a stochastic k-automaton (s.k-a.) is a common generalization of Bartoszyński’s automaton and Grodzki’s deterministic k-machine. By a stochastic k-automaton we mean an ordered triple M k = ⟨ U , a , π ⟩, k ⩾ 1, where U denotes a finite non-empty set, a is a function from Uk to [0, 1] with ∑ v ∈ U k a ( v ) = 1, and π is a function from Uk+1 to [0,1] with ∑ u ∈ U π ( v , u ) = 1 for every v ∈ U k . For all N ⩾ k we can define a probability measure PN on U N = U × U × … × U as follows: P N ( u 1 , u 2 , … , u N ) = a ( u 1 , u 2 , … , u k ) π ( u 1 , u 2 , … , u k + 1 ) π ( u 2 , u 3 , … , u k + 2 ) … π ( u N − k , u N − k + 1 , … , u N ). We deal with the problems of the shrinkage and the extension of a system of s.k-a.’s M k ( i ) = ⟨ U , a ( i ) , π ( i ) ⟩, i = 1 , 2 , … , m , m ⩾ 2. In this note there are given conditions under which an s.k-a. M k = ⟨ U , a , π ⟩ exists and the language of this automaton defined as L M = { ( u 1 , u 2 , u 3 , … ) : ∧ N ⩾ 1 P N ( u l , u 2 , … u N ) > 0 } either contains the languages of all the automata M k ( i ) , i = 1 , 2 , … , m, or this language equals the intersection of all those languages.