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sol02 - CS 154 Intro to Automata and Complexity Theory...

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CS 154 Intro. to Automata and Complexity Theory Handout 12 Autumn 2008 David Dill October 13, 2009 Solution Set 2 1
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Problem 1 L ( M ) = ( ab ) c Problem 2 a. L 1 = { w | w contains twice as many 1’s as 0’s } is not regular. Pumping Lemma: Proof Assume L 1 is regular and apply the Pumping Lemma. By P.L. there exists some pumping constant n > 0. We choose w = 0 n 1 2 n , which is in L 2 and satisfies | w | ≥ n . By P.L. w = xyz such that | xy | ≤ n and y negationslash = ǫ . Note that the first condition implies that both x and y contain only 0’s. Let | x | = a and | y | = b , the second condition implies b > 0. We choose k = 0: xz L 1 by P.L., but xz = 0 n b 1 2 n which is not in L 1 since b negationslash = 0. Thus we get a contradiction, which implies our initial assumption was wrong. We conclude that L 1 is not regular. ⊔⊓ Closure Properties: Proof Suppose L 1 were regular. Define h : { 0 , 1 } → { 0 , 1 } so that h (0) = 0 and h (1) = 11. Then { 0 i 1 i | i 0 } = h 1 [ L 1 L (0 1 )]. To see this, we show that w ∈ { 0 i 1 i | i 0 } iff h ( w ) h 1 [ L 1 L (0 1 )]. If w ∈ { 0 i 1 i | i 0 } , it is obvious that h ( w ) L 1 L (0 1 ). If w negationslash∈ { 0 i 1 i | i 0 } , then it must either have different numbers of 0’s and 1’s, in which case h ( w ) will not have twice as many 1’s and 0’s as required to be in L 1 , or w has a 1 before a 0, in which case h ( w ) will have the same property and will not be a member of L (0 1 ).) Since the regular languages are closed under inverse homomorphisms, { 0 i 1 i | i 0 } would be regular, but it is known not to be. Therefore, L 1 must not be regular. ⊔⊓ b. The language L 2 = { 0 n 1 m 2 n m | n m 0 } over Σ = { 0 , 1 , 2 } .
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