Explain about pumping lemma for context free language?

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