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
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