Functions associated to a partition on state set of a probabilistic automaton

In this paper we consider a finite probabilistic automaton, a partition P on the state set of this automaton and we characterize the probability that auto maton having the initial state in a block p \gE P and receiving an input word x1x2 … xk to move into successive states s1 \gE p1,…, sk \gE {itpk}, where p1,…, pk are blocks of P.This problem has been suggested by Paz (1971) where it is studied for the case with one letter input alphabet and is left as an open problem for the case of input alphabet with more than one letter. In this paper we study this case