Converting Regular Expression to NFA

Regular Expressions

Regular expressions are a concise way to describe patterns in strings. They are built using a set of basic operations:

1. Basic Symbols: Individual characters from an alphabet
2. Concatenation: Writing expressions next to each other (e.g., "ab")
3. Union (|): Choice between alternatives (e.g., "a|b")
4. Kleene Star (*): Zero or more repetitions (e.g., "a*")
5. Plus (+): One or more repetitions (e.g., "a+")

Non-deterministic Finite Automata (NFA)

An NFA is a mathematical model of computation that consists of:

• A finite set of states
• A set of input symbols (alphabet)
• A transition function that maps (state, symbol) pairs to sets of states
• A start state
• A set of accept states

NFAs can have multiple transitions for the same input symbol from a state and can include ε-transitions (transitions that don't consume any input).

Thompson's Construction

Thompson's construction is an algorithm that converts regular expressions to NFAs. The algorithm works recursively by building NFAs for each subexpression and combining them according to the operators:

1. Basic Symbol: Creates a simple NFA with two states and a transition labeled with the symbol
2. Concatenation: Connects the accept state of the first NFA to the start state of the second NFA with an ε-transition
3. Union: Creates a new start state with ε-transitions to both NFAs and a new accept state with ε-transitions from both NFAs
4. Kleene Star: Creates a new start state and accept state, with ε-transitions to handle zero or more repetitions
5. Plus: Similar to Kleene star but requires at least one repetition

Equivalence of Regular Expressions and NFAs

The fundamental theorem states that:

• Every regular expression can be converted to an NFA (Thompson's construction)
• Every NFA can be converted to a regular expression
• Both represent the same class of languages: regular languages

Applications

Understanding the conversion between regular expressions and NFAs is crucial for:

1. Compiler design and lexical analysis
2. Pattern matching in text processing
3. String searching algorithms
4. Network protocol validation
5. Input validation in programming languages