2 12 5 Points A B L S T S T A B Andsis A Substring Oft As 2769727

2. (12.5 points).?= {a,b,#} L = {s#t:s,t?{a,b}+, andsis a substring oft}. As examples,ba#ba?L;abab??L;bb#aaaabba?L;bb#aaa??L. Use the CFL pumping theorem to prove that L??CFLs, following these guidelines carefully.

• Start by expressingwin terms of k, the pumping length, and the symbols inS. The other requirements are thatw?L, and |w|=k.

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• For all possible values ofvandy, generatew’fromwby pumping in or out, and convince me that w’??L. If you pump in, state how many times.

• Never propose literal values forw,v,y, |w|, |v|, |y|, or k. For example, statements like “let k =3” are not allowed, and you cannot assume that |v| = 1.

• Formulas foru,x, andz, the non-pumpable parts ofw, are rarely useful. Do not use them in this proof.

• End with a conclusion convincing me that your proof shows that L??CFLs.