Explain the concept of pumping lemma for regular languages.

The pumping lemma for regular languages is a fundamental tool used to prove that a language is not regular. It provides a way to demonstrate that certain properties of regular languages cannot hold for a given language.

The pumping lemma states that for any regular language L, there exists a constant p (the pumping length) such that any string s in L with length greater than or equal to p can be divided into five parts: s = xyzuv, satisfying the following conditions:

1. The length of xyuv is less than or equal to p.
2. The length of y and v combined is greater than zero.
3. For any non-negative integer n, the string xy^nzu^nv is also in L.

In simpler terms, the pumping lemma states that if a language is regular, then any sufficiently long string in that language can be "pumped" by repeating a portion of it any number of times and still remain in the language.

To prove that a language is not regular using the pumping lemma, one must show that for any choice of the string s in the language, there exists a division of s that violates at least one of the conditions mentioned above. This demonstrates that the language does not satisfy the pumping lemma and therefore cannot be regular.

In summary, the pumping lemma for regular languages is a powerful tool used to prove the non-regularity of languages by showing that they do not satisfy the conditions imposed by the lemma.