# 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:

- for each
*i* ≥ 0, *xy*^{i}z ∈ A,
- |
*y*| > 0, and
- |
*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{0^{n}1^{n} | n ≥ 0}
is not regular.

### Active Learning Problem

Use the pumping lemma to show that {0^{i}1^{j} | i > j}
is not regular.

### Practice for the Midterm

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