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# What is the pumping lemma for regular language?

There are two Pumping Lemmas (PL), which are defined for Regular Languages and Context - Free Languages.

## Pumping lemma for Regular languages

- It gives a method for pumping (generating) many substrings from a given string.
- In other words, we say it provides means to break a given long input string into several substrings.
- Lt gives necessary condition(s) to prove a set of strings is not regular.

## Theorem

For any regular language L, there exists an integer P, such that for all w in L

|w|>=P

We can break w into three strings, w=xyz such that.

(1)lxyl < P

(2)lyl > 1

(3)for all k>= 0: the string xy^{k}z is also in L

## Application of pumping lemma

Pumping lemma is to be applied to show that certain languages are not regular.

It should never be used to show a language is regular.

- If L is regular, it satisfies the Pumping lemma.
- If L does not satisfy the Pumping Lemma, it is not regular.

**Steps to prove that a language is not regular by using PL**are as follows−

- step 1 − We have to assume that L is regular
- step 2 − So, the pumping lemma should hold for L.
- step 3 − It has to have a pumping length (say P).
- step 4 − All strings longer that P can be pumped |w|>=p.
- step 5 − Now find a string 'w' in L such that |w|>=P
- step 6 − Divide w into xyz.
- step 7 − Show that xy
^{i}z ∉ L for some i. - step 8 − Then consider all ways that w can be divided into xyz.
- step 9 − Show that none of these can satisfy all the 3 pumping conditions at same time.
- step 10 − w cannot be pumped = CONTRADICTION.

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