Two Complete Axiom Systems for the Algebra of Regular Events

The theory of finite automata is closely linked with the theory of Kleene's regular expressions. In this paper, two formal systems for the algebraic transformation of regular expressions are developed. Both systems are consistent and complete; i.e., the set of equations derivable within the system equals the set of equations between two regular expressions denoting the same event. One of the systems is based upon the uniqueness of the solution of certain regular expression equations, whereas some facts concerning the representation theory of regular events are used in connection with the other.