Rational Numbers and Regular Events

In this paper the relation between a sequential circuit and its regular expression is investigated. The circuits are without special starting units. One method of analysis of a circuit leads to a set of equations whose solutions are regular expressions related to the state diagram of the circuit. In another approach, a set of regular equations, identical in form to the next state equations, is obtained directly from the circuit. By reversing the regular equations and using derivatives, the regular equations are transformed to a form related to the reverse state diagram. The discussion clarifies the relationship among circuits, regular expressions and state diagrams. Moreover, further insight is obtained into the solution of equations with as unknowns.