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# hmwrk3 - L 1 ∪ L 2 is nonregular 8 If L 1 is regular L 2...

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CSCI 250 GOLDWEBER Spring 05 2/2/03 Homework 3 Due Date: Monday February 7, 2005. There is a possible 52 points for this homework assignment. Problem 1 . (18 pts.) Use the pumping lemma to show that each of the following languages is not regular. 1. L = { 0 n 10 2 n | n 0 } 2. L = { 0 i 1 j 0 k | k > i + j } 3. L = { 0 i 1 j | j = i or j = 2 i } 4. L = { 0 i 1 j | j is a multiple of i } 5. L = { x ∈ { 0 , 1 } | n 0 ( x ) < 2 n 1 ( x ) } 6. L = { x ∈ { 0 , 1 } | no prefix of x has more 1 ’s than 0 ’s } Problem 2 . (16 pts.) For each statement below, decide whether it is true or false. If it is true, prove it. If not, give a counterexample. All parts refer to languages over the alphabet { 0 , 1 } . 1. If L 1 L 2 and L 1 is not regular, then L 2 is not regular. 2. If L 1 L 2 and L 2 is not regular, then L 1 is not regular. 3. If L 1 and L 2 are nonregular, then L 1 L 2 is nonregular. 4. If L 1 and L 2 are nonregular, then L 1 L 2 is nonregular. 5. If L is nonregular, then L is nonregular. 6. If L 1 is regular and L 2 is nonregular, then L 1 L 2 is nonregular. 7. If L 1 is regular, L 2 is nonregular, and L 1 L 2 is regular, then

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Unformatted text preview: L 1 ∪ L 2 is nonregular. 8. If L 1 is regular, L 2 is nonregular, and L 1 ∩ L 2 is nonregular, then L 1 ∪ L 2 is nonregular. Problem 3 . (18 pts.) A number of languages over { , 1 } are described below. In each case, decide whether the language is regular or not, and prove your answer is correct. 1. The set of all strings x beginning with a nonnull string of the form ww . 1 2. The set of all strings x containing some nonnull substring of the form ww . 3. The set of odd-length strings over { , 1 } with the middle symbol 0. 4. The set of even-length strings over { , 1 } with the two middle symbols equal. 5. The set of string over { , 1 } of the form xyx for some x with | x | ≥ 1 . 6. The set of nonpalindromes. 2...
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hmwrk3 - L 1 ∪ L 2 is nonregular 8 If L 1 is regular L 2...

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