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# hw2_sol - Problem Set 2 Solutions CS373 Spring 2011 Due...

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Unformatted text preview: Problem Set 2 Solutions CS373 - Spring 2011 Due: Thursday Feb 24 at 2:00 PM in class (151 Everitt Lab) Please follow the homework format guidelines posted on the class web page: http://www.cs.uiuc.edu/class/sp11/cs373/ 1. Is it regular? [ Category : Proof, Points : 12] Prove that each of the following languages is regular by building a nite automaton or regular expression, or prove non-regularity by using the pumping lemma. i) A = { w ∈ { , 1 } * | | w | is even and the number of zeros in w 's rst half is even } (4 Points) [SOLUTION] Given pumping length p , let w = 1 p 01 p- 1 . This string has an even number ( 2 p ) of characters, and an even number of 's (zero) in the rst half. Since | xy | ≤ p , y = 1 k for some 1 ≤ k ≤ p . If k is odd, then | xy 2 z | is odd, so xy 2 z 6∈ A . If k is even, then k ≥ 2 . Therefore, | xy z | ≤ 2 n- 2 . Since there are p- 1 1 's after the zero in w , the in xy z will be in the rst half of the string. Since there is only one in xy z , and it is in the rst half of the string, xy z 6∈ A . ii) A = { xy | x,y ∈ { , 1 } * , the number of 0s in x is equal to the number of 1s in y } (4 Points) [SOLUTION] This language includes every string and has the regular expression (0 ∪ 1) * . We will prove that all strings w = a 1 a 2 a 3 ...a n are in A . Let Δ( i ) be the number of zeros in all a k for k ≤ i minus the number of ones in all a j for j > i . Notice that if Δ( i ) = 0 for some i , then w is in A , since the `border' between x and y is between a i and a ( i + 1) . The number Δ(0) ≤ , as there are no zeros to the left of position zero. The number Δ( n ) ≥ , as there are no ones to the right of position n . Note that if the i th character is a zero, then Δ( i ) = Δ( i- 1) + 1 , as the number of zeros to the left of character i has increased by one, and the number of ones to the right is unchanged. Also, if the i th character is a one, then Δ( i ) = Δ( i- 1) + 1 , as the number of zeros to the left of i has not changed, but the number of ones to the right of i has decreased by one. Since Δ( i ) can only change by one at each character, and Δ(0) ≤ , and Δ( n ) ≥ , there must be an i such that Δ( i ) = 0 , so there is a way to decompose any string w into xy = w such that the number of zeros in x is equal to the number of ones in y . iii) A = { xy | x,y ∈ { , 1 } * , the number of 0s in x is equal to the number of 0s in y } (4 Points) [SOLUTION] This language is regular, and is equal to (1 * 01 * 01 * ) * . This is the regular expression that requires the number of zeros to be even. Any string w with an even number of zeros can be decomposed into xy = w such that the number of zeros in x is the same as the number of zeros in y . If w contains 2 n zeros, then split the string right after the n th zero....
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hw2_sol - Problem Set 2 Solutions CS373 Spring 2011 Due...

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