Let $ L $ to be a language defined on $ Sigma = left {a, b right } $ such as

$ L = left {x #y mid x, y in Sigma ^ *, # text {is a constant and} x neq y right $

Indicate if the language L is a CFL or not? Give valid reasons for the same thing.

Now, I think that the given language is not a CFL. I've used the lemma pumping test to show that L is not a CFL.

Specifically, I did the following-

Consider a string $ w = abb # aab $. obviously, $ w in L $.

Let, $ u = epsilon \ v = a \ w = bb #aa \ x = b \ y = epsilon $

Right here, $ | vx | geq 1 $

But, $ uv ^ 2wx ^ 2y = aabb #aabb notin L $

Therefore, the result of the pumping lemma test is negative. Therefore, we can conclude that the given language is not a CFL.

Now, I have a doubt about the method above.

I know that in the case of a CFL, if we want to do the pumping lemma test for the CFL, we must always use chains whose length is greater than or equal to the minimum pumping length. In fact, this also confirms the condition that the length of the chain $ w $ used for the pumping test of the lemma (noted $ | w | $) must be greater than or equal to n.

Therefore, when I use $ w = abb # aab $ to perform the pumping lemma test, I implicitly assume that 7 is greater than or equal to the minimum pumping length (if $ L $ had to be a CFL). *Am I correct or incorrect?*