Let L = {w#w | w ∈ {a, b}*} be a language. Use the pumping lemma for context-free languages to prove that L is not context-free. Hint: What is the alphabet of L?

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Let L = {w#w | w ∈ {a, b}*} be a language. Use the pumping lemma for context-free languages to prove that L is not context-free. Hint: What is the alphabet of L?

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Step 1

L generates string like:

a#a, aa#aa, ab#ab, aabb#aabb, b#b,bb#bb, aabaabb#aabaabb . . . . 

This language is not context-free. 

let us take a string: aaabb#aaabb

Here up to #, In PDA, substring aaabb will be pushed to the stack. But when # comes, after that stack top element will be 'b' and the element that comes next will be 'a'. So there is no way to compare 'a' and 'b', so we won't be able to pop b with respect to 'a'. Therefore machine will not work for this language.

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