From Finite Automata to Regular Expressions and Back — A Summary on Descriptional Complexity

The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. As an interesting special case also one-unambiguous regular expressions, a sort of a deterministic version of a regular expression, are considered. We also briefly recall the known bounds for the removal of spontaneous transitions (ε-transitions) on non-ε-free nondeter-ministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and new developments on the state elimination algorithm that converts finite automata to regular expressions.