Regular languages, derivatives and finite automata

TL;DR

This educational report introduces regular languages, derivatives, and finite automata, emphasizing derivatives for regex matching, submatching, and theoretical proofs, providing a comprehensive self-contained overview.

Contribution

It presents a detailed approach to regex matching using derivatives, including submatching techniques inspired by Laurikari, and connects theory with practical algorithms.

Findings

Derivatives can be used to prove key properties of regular languages.

A regex matching algorithm based on derivatives is summarized.

Submatching with derivatives is effectively implemented, inspired by Laurikari.

Abstract

This report is mostly written for educational purposes. It is meant as a self contained introduction to regular languages, regular expressions, and regular expression matching by using Brzozowski derivatives. As such it is mostly based on the work by Brzozowski[4] and Owens et al.[12] The language basics material have been inspired by books[2] and web material[16]. Chapter 1 introduces the fundamental concepts of formal languages, as well as the idea of string derivatives. In chapter 2 we define the class of regular languages, and further develops the theory of derivatives for that class. We use derivatives to prove the Myhill-Nerod theorem, the Pumping lemma, and the closure of regular languages under all Boolean connectives. In chapter 3 we introduce regular expressions and regular expression matching. Chapter 4 connects the theory of regular languages and derivatives with that of finite automata. Chapter 5 looks at the concept of anchors, and how this can be incorporated into a matcher based on derivatives. Chapter 6 discusses submatching using derivatives with an approach inspired by Laurikari and his work on tagged transitions[11]. This is the part we consider as our main contribution to the field. In the last chapter, chapter 7, we summarize by giving a regular expression matching algorithm using the previously discussed techniques. We also discuss related work by others.