hwsol9 - Theory of Algorithms Spring 2000 Homework 9...

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Theory of Algorithms. Spring 2000. Homework 9 Solutions. Section 8.1 (5a) Let L = a n ww R a n : n 0 , w ∈ { a, b } * . Then L is context-free. In fact, L is just ww R : w ∈ { a, b } * . (5b) Let L = { a n b j a n b j : n 0 , j 0 } . Then L is not context-free. Proof. Assume towards a contradiction that L is context-free. Let m > 0 be given by the Context- Free Pumping Lemma. Then let w = a m b m a m b m . Notice that w L and | w | ≥ m . So let w = uvxyz be the decomposition of w given by the Pumping Lemma. So | vxy | ≤ m and | vy | ≥ 1. Here are some of the possibilities for what v and y look like: (Case 1) vxy is a substring of the first block of a ’s. In this case let i = 2. We have w 2 = uv 2 xy 2 z = a m + k b m a m b m where k = | vy | . Since k 6 = 0, w 2 / L . (Case 2) vx is a substring of the first block of a ’s, and y = a k b s for some k, s 1. In this case let i = 2. Notice that w 2 / L ( a * b * a * b * ), so w 2 / L . (Case 3) v is a substring of the first block of a ’s and y is a substring of the first block of b ’s. In this case let i = 2. We have w 2 = uv 2 xy 2 z = a m + k b m + s a m b m where k = | v | and s = | y | . Since k and s cannot both be 0, we have that w 2 / L .
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