is this statement correct pumping lemma for regular languages cannot be used to prove that certa

Name: is this statement correct pumping lemma for regular languages cannot be used to prove that certain languages are not regular 19513
Uploaded: 2023-05-04T14:56:50-08:00
Duration: 3 min 9 s
Channel: Madhur L
Description: is this statement correct pumping lemma for regular languages cannot be used to prove that certain languages are not regular 19513

Is this statement correct pumping lemma for regular...

John Doe claims that the language L, of all strings over the alphabet Σ = {a, b} that contain an even number of occurrences of the letter 'a', is not a regular language. He offers the following "pumping lemma proof". Explain what is wrong with the "proof" given below. "Pumping Lemma Proof" We assume that L is regular. Then, according to the pumping lemma, every long string in L (of length m or more) must be "pumpable". We choose the specific string w = a^2m. Clearly, w is in L, and |w| >= m, so this w must be "pumpable". That means that the string w consists of 3 substrings, x, y, and z. In other words, w = xyz. We choose y = a. Then if we "pump up" to produce the string xy^2z, this string should also be in the language L, according to the Pumping Lemma. But xy^2z = a^(2m+1), and clearly xy^2z is NOT in L, since this new string has an odd number of a's. Therefore, our original assumption that L is regular was incorrect. Q.E.D.

Madhur L.