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Sipser 1.54 - 1.54 Consider the language F ={aibjck |i,j,k...

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1.54) Consider the language F = {a i b j c k | i , j , k ≥ 0 and if i = 1 then j = k } (A) Show that F is not regular. Assume that F is regular and conforms appropriately. If F is regular, then the reverse of F is regular; Use the pumping lemma, and let p be its hypothetical pumping number. Consider w = c p b p a , w є F R . Attempting to pump w will yield some c p + i b p a, which is not in the reverse of F , as p + i > p . Thus F R is not regular as it violates the pumping lemma, so F is not regular. (B) Show that F acts like a regular language in the pumping lemma For any string w in F , there are four possible forms: aaa*b*c* (two or more as, followed by any number of as, followed by any number of bs, followed by any number of cs), ab i c i (exactly one a, followed by some number of bs, possibly zero, followed by an equal number of cs), bb*c* (no as, followed by at least one b, followed by any number of cs, possibly zero), or c*(no as or bs, followed by any number of cs, possibly zero). We must now show that for each of these cases, w is suitable for
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