Using the Pumping Lemma

Abstract

Let’s use the pumping lemma in the form of the demon game to show that the set

$$ A = \left\{ {{a^n}{b^n}\left| {n \ge m} \right.} \right\}$$

is not regular. The set A is the set of strings in a * b * with no more b’s than a’s. The demon, who is betting that A is regular, picks some number k. A good response for you is to pick x = a ^k, y = b ^k, and z = ɛ. Then xyz = a ^k b ^k ∈ A and |y| = k; so far you have followed the rules. The demon must now pick u, v, w such that y = uvw and v ≠ ɛ. Say the demon picks u, v, w of length j, m, n, respectively, with k = j + m + n and m > 0. No matter what the demon picks, you can take i = 2 and you win:

$$ \begin{array}{l} xu{v^2}wz = {a^k}{b^j}{b^m}{b^m}{b^n} \\ = {a^k}{b^{j + 2m + n}} \\ = {a^k}{b^{k + m}} \\ \end{array}$$

, which is not in A, because the number of b’s is strictly larger than the number of a’s.