A General Framework for the Derivation of Regular Expressions

TL;DR

This paper introduces a unified theoretical framework for deriving regular expressions using a space of generic structures, simplifying proofs of key properties and enabling automaton construction.

Contribution

It presents a general formalism that encompasses Brzozowski's and Antimirov's derivatives, allowing systematic automaton construction from regular expressions.

Findings

Unified framework for regular expression derivatives

Simplifies proof of finiteness of derivatives

Enables construction of DFA, NFA, and AFA from derivatives

Abstract

The aim of this paper is to design a theoretical framework that allows us to perform the computation of regular expression derivatives through a space of generic structures. Thanks to this formalism, the main properties of regular expression derivation, such as the finiteness of the set of derivatives, need only be stated and proved one time, at the top level. Moreover, it is shown how to construct an alternating automaton associated with the derivation of a regular expression in this general framework. Finally, Brzozowski's derivation and Antimirov's derivation turn out to be a particular case of this general scheme and it is shown how to construct a DFA, a NFA and an AFA for both of these derivations.