A state space approach to the finite automata

This paper proposes a state space approach for analyzing the finite automata. A Ψ-representation transforms a set of words into a formal power series for establishing the state equation of a finite automaton. We investigate the structure of the automaton via its corresponding state equation. It is shown that the solution of the state equation always exists and is unique. Furthermore, we prove that the solution field is a separable algebraic extension of the coefficient field. Finally, the concept of the substitution property of a partition is shown to be equivalent to that of invariant subspaces of the associated state space.