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hw4-sol - ECS 120 Intoduction to the Theory of Computation...

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ECS 120: Intoduction to the Theory of Computation Homework 4 Due Oct 26, at the beginning of the lecture or by 1pm in Kemper 2131 Problem 1. (counts as 2) Prove that the following languages are not regular (using the pumping lemma or closure properties). You can use the fact that L = { 0 n 1 n | n 0 } is non-regular. (a) { 0 n 1 m | n m } See Sipser, p. 82, Example 1.77 (b) { 1 2 n | n = 0 , 1 , 2 , . . . } Suppose for the purpose of contradiction that this language, L , is regular. Let p be the number in the pumping lemma and consider the string w = 1 2 p . Since | w | = 2 p > p this string satisfies the requirements of the lemma, and thus can be split three-ways, w = xyz , such that p ≥ | y | > 0 and | xy | ≤ p . Then, according to the pumping lemma, the string w = xy 2 z L . The length of this string is 2 p + | y | , and 2 p < 2 p + | y | ≤ 2 p + p < 2 p + 2 p = 2 p +1 . Thus w L . Contradiction. (c) { w | w is not a palindrome } Let L = { w | w is not a palindrome } , and L = { w | w is a palindrome } . Since L and L are complements of each other, and L is nonregular (shown in class), it follows from problem 2(c) below that L is nonregular too.
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