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

TL;DR

This paper reviews historical and recent results on the complexity of converting between finite automata and regular expressions, including bounds, algorithms, and effects of epsilon-transitions.

Contribution

It summarizes known bounds, recent advances, and new developments in the descriptional complexity of automata and regular expressions conversions.

Findings

Bounds on automata to regular expressions conversion

Bounds on regular expressions to automata conversion

New algorithms for state elimination

Abstract

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. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.