Chapter 1: Regular Languages

For Your Enjoyment

• Regular languages play an important role in lexical analysis (the scanner) for a compiler.

Chapter 1.3, Regular Expressions

GNFAs

The book requires GNFAs to have the following three properties:

• The start state has transition arrows going to every other state but no arrows coming in from any other state.
• There is only a single accept state, and it has arrows coming in from every other state but no arrows going to any other state. Furthermore, the accept state is not the start state.
• Except for the start and accept states, one arrow goes from every state to every other state and also from each state to itself.

A generalized nondeterministic finite automaton (GNFA) is a 5-tuple where

• Q is the finite set of states,
• Σ is the alphabet,
• δ(Q - {q_accept}) × (Q - {q_start}) —> R is the transition function (R is a regular expression),
• q_start is the start state, and
• q_accept is the accept state.

A GNFA accepts a string w in Σ^* if w = w₁...w_k where each w_i is in Σ^* and a sequence of states q₀...q_k exists such that

• q₀ = q_start is the start state,
• q_k = q_accept is the accept state, and
• for each i, w_i ∈ L(R_i) where R_i = δ(q_i-1, q_i).

DFA to Regular Expression Conversion Process

1. Create an n+2 state GNFA from an n state DFA as follows:
   • Add a new start state with an ε transition to the DFA start state.
   • Add a new accept state with ε transitions from the DFA accept states to the new accept state. (Change the DFA accept states to non-accept states.)
   • A transition should contain the union of the DFA transition labels.
   • Add the ∅ transition to pairs of states in the DFA that had no transition between them.
2. Repeatedly rip out one of the former DFA states using the following procedure until only the start state and accept state are left
   • Call the state being removed q_rip
   • Consider a pair of states q_j and q_k
   • if q_j goes to q_rip with R₁, q_rip goes to q_rip with R₂, q_rip goes to q_k with R₃, and q_j goes to q_k with R₄, then the transition from q_j to q_k in the machine with q_rip removed is now R₁(R₂)^*R₃ ∪ R₄.
3. The regular expression equivalent appears on the transition from the start state to the accept state.

Lecture Problem

• Construct a DFA that recognizes any string that starts with an a over Σ = {a, b}.
• Using the procedure above, find the equivalent regular expression.

Active Learning Problem

• Construct a DFA that recognizes any string that starts with an a and ends with a b over Σ = {a, b}.
• Using the procedure above, find the equivalent regular expression.

Active Learning Problem

• Understand Example 1.68 on page 76.

In-Class Notes