COMS W4115
Programming Languages and Translators
Lecture 6: Implementing a Lexical Analyzer
September 28, 2009

Lecture Outline

1. Finite automata
2. Equivalence of regular expressions and finite automata
3. Converting a regular expression to an NFA
4. Converting an NFA to a DFA
5. Simulating an NFA

1. Finite Automata

• Variants of finite automata are commonly used to recognize regular expressions.
• A nondeterministic finite automaton (NFA) consists of
• A finite set of states S.
• An input alphabet consisting of a finite set of symbols Σ.
• A transition function δ that maps S × (Σ ∪ {ε}) to subsets of S. This transition function can be represented by a transition graph in which the nodes are labeled by states and there is a directed edge labeled a from node w to node v if δ(w, a) contains v.
• An initial state s₀ in S.
• F, a subset of S, called the final (or accepting) states.
• An NFA accepts an input string x if there is a path in the transition graph from the initial state to a final state that spells out x.
• The language defined by an NFA is the set of strings accepted by the NFA.
• A deterministic finite automaton (DFA) is an NFA in which
1. There are no ε moves, and
2. For each state s and input symbol a there is exactly one transition out of s labeled a.

2. Equivalence of Regular Expressions and Finite Automata

• Regular expressions and finite automata define the same class of languages, namely the regular sets.
• From automata and language theory we know that
1. every regular expression can be converted into an equivalent NFA using the McNaughton-Yamada-Thompson algorithm.
2. every NFA can be converted into an equivalent DFA using the subset construction.
3. every finite automaton can be converted into a regular expression using Kleene's algorithm.

3. Converting a Regular Expression to an NFA

• The McNaughton-Yamada-Thompson algorithm: Algorithm 3.23, pp. 159-163.

4. Converting an NFA to a DFA

• The subset construction: Algorithm 3.20, ALSU, p. 153.

5. Simulating an NFA

• Two-stack simulation of an NFA: Algorithm 3.22, pp. 156-159.

6. Reading Assignment

• Read Chapter 3, all sections except 3.9.