Pumping Lemma(PL)

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The Pumping Lemma states that every regular language has a pumping length such that sufficiently long strings can be decomposed and repeated while remaining in the language.

The Pumping Lemma is a powerful theoretical tool that helps identify the limitations of finite automata and distinguish regular languages from non-regular ones.
The pumping lemma is a concept in formal language theory that states that for any regular language, sufficiently long strings can be divided into parts such that one part can be repeated any number of times to produce new strings that also belong to the language. It is often used to prove that certain languages are not regular.
Mathematically, the Pumping Lemma for every regular language L, there exists a constant p (called the pumping length) such that for any string w ∈ L where |w| ≥ p, we can divide w into three substrings w = xyz satisfying:
1. |xy| ≤ p (the pumpable part is within the first p characters)
2. |y| > 0 (y is non-empty – the “pumpable” portion)
3. For all i ≥ 0, xy^i z ∈ L (pumping y any number of times keeps the string in L)

It is mainly a proof technique.
It cannot be used to prove a language is regular.
The pumping lemma is a necessary property of regular and context-free languages.
It is applied to regular languages to prove that a given language is not regular.
The concept of PL is based on Finite Automata loops.
Since a finite automaton has a finite number of states, if a string is long enough, the automaton must visit some state more than once. This may create a loop. The looped part can be repeated (“pumped”). This may lead to PL.
If even one pumped string fails, the language is not regular.

To prove a language L is NOT regular:

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