hw3sol

hw3sol - CSE 105 Introduction to the Theory of Comptuation...

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Unformatted text preview: CSE 105: Introduction to the Theory of Comptuation Fall 2010 Problem Set 3 Instructor: Daniele Micciancio Due on: Wed. Oct. 20, 2010 Problem 1 Let L be the set of all strings (over the alphabet { , 1 , + , = } ) of the form ” x + y = z ” , where x,y and z are the binary representation of three numbers such that the sum of x and y equals z . Prove that the language L is not regular using the pumping lemma. Proof: Assume for contradiction L is regular. By the pumping lemma there is an integer p such that all strings in L of length at least p can be pumped. Consider the string w= “ 1 p + 0 = 1 p . Clearly w ∈ L and the length of w is 2 p + 3 > p . So, w = xyz where x,y,z are strings satisfying the three properties in the pumping lemma. Let a = | x | and b = | y | be the lengths of x and y . Using property | xy | ≤ p , we get that a + b = | xy | ≤ p . Since xy are the first a + b characters of w and the first p ≥ a + b characters of w are all 1’s, we get that x = 1 a and y = 1 b . From the pumping lemma we know that xz = xy z ∈ L . But xz = “ 1 p- b + 0 = 1 p ” is in L only if 1 p- b = 1 p , which is possible only when b = 0 . This contradicts the property....
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This note was uploaded on 08/31/2011 for the course CSE 105 taught by Professor Paturi during the Fall '99 term at UCSD.

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hw3sol - CSE 105 Introduction to the Theory of Comptuation...

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