8. (6 points) Here is a weaker version of the Pumping Lemma for context-free languages (some- times used in textbooks): Let CF be a context-free language. Then there exists p >= 1 such that any string w 2 CF of length greater than p can be rewritten w = uvxyz such that jvyj 6= 0 and uvixyiz 2 CF for every i >= 0. Explain why this weaker Pumping Lemma is insucient to prove that Lww = fww : w 2 fa; bg g is not context-free. Do this by starting a proof that the language is not context-free, then indicating where the proof breaks down.

8. (6 points) Here is a weaker version of the Pumping Lemma for context-free languages (some- times used in textbooks): Let CF be a context-free language. Then there exists p 21 such that any string w e CF of length greater than p can be rewritten w = uvxyz such that Ivy|0 and uv ry'z E CF for every i 20. (ww : uE Explain why this weaker Pumping Lemma is insufficient to prove that Lwu (a, b}* } is not context-free. Do this by starting a proof that the language is not context-free, then indicating where the proof breaks down