Pumping Lemma for CFG

Question

Consider the language, L = {1^k 0ⁱ 1ⁱ 0^j 1^j 0^k | i,j,k>0}. Is this language context free?

I tried to find this using pumping lemma,

By intution,

If we consider , by the definition of pumping lemma for CFG, u and y to be 1^k and 0^k respectively and vwx to be string in between,

if we consider v to be in 1ⁱ and x to be in 0^j and if the pump up the variables, surely string generated after pumping up wont be in the given language right?

Am I missing something?

Comments

Pumping lemma is to prove certain language is non cfl.

but given language is cfl.

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Why is this language context free?

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$1^{k}$ --push into the stack.

$0^{i}$ --push into the stack.

$1^{i}$--pop one '1' and one '0'.

$0^{j}$ --push into the stack.

$1^{j}$--pop one '1' and one '0'.

$0^{k}$---pop '0' for one 1 one.

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Cool bro! Got it!

Answer 1

I think it's not CFL as more than one comparison is needed