Explain about pumping lemma for context free language?

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Problem

Explain the pumping lemma for context free language by showing that the language of strings in the form xⁿyⁿzⁿ is not a context free language.

Solution

Pumping lemma (Context free grammar)

We can prove that a particular language is not context free grammar using pumping lemma.
Let’s take the concept of proof by contradiction
Here we assume that language is CFG

Conditions of pumping lemma

First of all consider a string and split into 5 parts those are pqrst it must satisfy the following conditions −

|qs|>=1
|qrs|=n (“ n” is pumping length)
pqⁱrsⁱt € L for different values of i

Let the L be the CF language.

Now we can take a string such that S={xⁿyⁿzⁿ}

We divide S in five parts.

Case 1 − let n=4 so S=x⁴y⁴z⁴

q and s each contain only one type of symbols

xxxxyyyyzzzz

p=x , q=xx, r=xyyyyz, s=z, t=zz

Let take i=2

Pq²rs²t

xxxxxxyyyyzzzzz
x⁶y⁴z⁵ ≠L

Because, it is not in the form of xⁿyⁿzⁿ

Updated on 16-Jun-2021 13:41:23

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