Regular languages, derivatives and finite automata

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.