Chapter 1: Regular Languages

For Your Enjoyment

Chapter 1.4, Nonregular Languages

Pumping Lemma

If A is a regular language, then there is a number p (the pumping length) where if s is any string in A of length at least p, then s may be divided into three pieces s = xyz, satisfying the following:

  1. for each i ≥ 0, xyiz ∈ A,
  2. |y| > 0, and
  3. |xy| ≤ p.


The book shows how to prove the above lemma. The proof involves setting p equal to the number of states in the DFA.

Lecture Problem

Demonstrate that the pumping lemma holds for a few strings in L = {w | the number of 0s in w is a multiple of 3} where Σ = {0, 1].

Lecture Problem

Use the pumping lemma to show that{0n1n | n ≥ 0} is not regular.

Active Learning Problem

Use the pumping lemma to show that {0i1j | i > j} is not regular.

Practice for the Midterm

  1. Problem 1.29 (b) on page 88.
  2. Problem 1.30 on page 88.
  3. Prove that the language, {w | w is a palindrome}, over the binary alphabet is not regular.
  4. Problem 1.48 on page 90.

In-Class Notes

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