Explain about pumping lemma for context free language?


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.


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


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

Let take i=2


  • xxxxxxyyyyzzzzz

  • x6y4z5 ≠L

Because, it is not in the form of xnynzn