cs373 fa10 hw4sol

# cs373 fa10 hw4sol - Solutions for Problem Set 4 CS 373...

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Unformatted text preview: Solutions for Problem Set 4 CS 373: Theory of Computation Assigned: September 21, 2010 Due on: September 28, 2010 at 10am Homework Problems Problem 1 . [Category: Proof] Solve problem 1.49(b) using the pumping lemma. Solution: Let p be the pumping length for C . Consider w = 1 p 01 p ∈ C . Suppose x,y,z are such that w = xyz , with | xy | ≤ p and | y | > 0. Since, | xy | ≤ p , without loss of generality, x = 1 r , y = 1 s , and z = 1 t 01 p , with r + s + t = p and s > 0. Now, xy z = 1 r 1 t 01 p = 1 r + t 01 p . Since s > 0, and r + s + t = p , we have r + t < p . Hence, xy z 6∈ C and therefore, C does not satisfy the pumping lemma and is not regular. Problem 2 . [Category: Comprehension+Proof] Solve problem 1.54 Solution: Recall that the problem defines F = { a i b j c k | i,j,k ≥ 0 and if i = 1 then j = k } . a. Consider A = F ∩ L ( ab * c * ) = { ab n c n | n ≥ } . Define h : { a,b,c } * → { , 1 } * where h ( a ) = , h ( b ) = 0 and h ( c ) = 1. Then, h ( A ) = { n 1 n | n ≥ } = K , which is known to be not regular. Thus, F is not regular as K was obtained from F by applying a series of regularity preserving operations. b. Take the pumping length p = 3. Consider any w = a i b j c k ∈ F , such that | w | ≥ p ....
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cs373 fa10 hw4sol - Solutions for Problem Set 4 CS 373...

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