# Chapter 2: Context-Free Languages

## For Your Enjoyment

## Chapter 2.3, Non-Context-Free Languages

### CFL Pumping Lemma

If A is a context-free 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 five pieces s = uvxyz satisfying
the conditions

- for each i ≥ 0, uv
^{i}xy^{i}z ∈ A,
- |vy| > 0, and
- |vxy| ≤ p.

### Lecture Problem

- Use the pumping lemma to show that A =
{a
^{n}b^{n}c^{n} | n ≥ 0} is not context-free.

### Active Learning Problem

- Use the pumping lemma to show that B = {ww
^{R}w | w ∈ {0,1}*}
is not context-free.
- Show that the string 0
^{p}0^{p}0^{p}
can be pumped.
- Show that the string 0
^{p}110^{p}0^{p}1
can be pumped.