# Chapter 1: Regular Languages

## 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.

### Proof

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.