Two-State Two-Symbol Probabilistic Automata

In this paper, we discuss the algebraic treatments of probabilistic automata with two states. We derive the result that the matrix product corresponding to a given input tape can be decomposed into the sum of a finite number of fundamental matrices which are determined by the matrices corresponding to the input symbols. We introduced the new concept of probabilistic automata completely isolated by the L-th approximation. With respect to these automata, all tapes of length greater than or equal to L + 1 can be classified into 2 T M sets by means of their (L + 1)-suffixes. By using this concept, we demonstrate that a two-input symbol actual automaton with two states can realize any definite event.