What is the pumping lemma for recursively enumerable languages?

The pumping lemma for recursively enumerable languages is a theorem that provides a necessary condition for a language to be recursively enumerable. It states that if a language L is recursively enumerable, then 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 = uvxyz, satisfying the following conditions:

1. For each i ≥ 0, the string u(v^i)x(y^i)z is also in L.
2. The length of the string vxy is less than or equal to p.
3. The length of the string vy is greater than 0.

In simpler terms, the pumping lemma states that for any recursively enumerable language, there exists a specific length after which any longer string in the language can be "pumped" by repeating or removing a portion of the string while still remaining in the language.

The pumping lemma is a useful tool in proving that certain languages are not recursively enumerable. If any of the conditions of the pumping lemma cannot be satisfied for a language, then the language is not recursively enumerable. However, it is important to note that satisfying the pumping lemma conditions does not guarantee that a language is recursively enumerable, as there may be other conditions or constraints that need to be considered.