Question

Consider the language L _ {w E {a,b)' : the longest run of a's in u, is longer than any run of b's in w} For example, abbbbaabbbaaaaa E L because the longest run of b's in it has length four, while the longest run of a's has length six. Prove that L is not context-free.

Answer 1

To Prove a language is not Context Free we use Pumping Lemma for CFL.

If L is any CFL and z $\small \epsilon$ L such that |z| $\tiny \geq$ n, if z = uvwxy $\small \epsilon$ L ( |uvx| $\tiny \leq$ |z| , |vx| != 0)

then uvⁱwxⁱy $\small \epsilon$ L , $\small \forall$ i $\tiny \geq$ 0.

Every CFL satisfies pumping lemma.

Now consider z = abbbbaabbbaaaaaa

take u = a, v = bbbb, w = aa, x = bbb, y = aaaaaa

Here in uvⁱwxⁱy take i = 2 now it will bw uv²wx²y

Hence uv²wx²y = a(bbbb)²aa(bbb)²aaaaaa = abbbbbbbbaabbbbbbaaaaaa

Now in this string it is saying that longest run of a's in w is shorter than any run of b's in w.

This is a contradiction, Hence this Language is not CFL.