# hw4 - say why this does not contradict the Pumping Lemma....

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ECS 120: Intoduction to the Theory of Computation Problem Set 4 Practice only, do not submit! Problem 1. 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 } (b) { 1 2 n | n = 0 , 1 , 2 ,... } (c) { w | w is not a palindrome } (d) { 0 n 1 m 2 n - m | n m 0 } Problem 2. Consider languages over a ﬁxed alphabet Σ, with | Σ | = 2. Prove or disprove the following. (a) If L 1 is nonregular and L 1 L 2 then L 2 is nonregular. (b) If L 1 L 2 and L 2 is nonregular, then L 1 is nonregular. (c) If L 1 is nonregular, then its complement L 1 is nonregular. (d) If L 1 is regular, then L 1 L 2 is regular for any language L 2 . (e) If L 1 and L 2 are nonregular, then L 1 L 2 is nonregular. Problem 3. Sipser, Problem 1.54. Note that you have 3 things to do here: show that F is not regular, show that it satisﬁes the 3 conditions of the Pumping Lemma, and

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Unformatted text preview: say why this does not contradict the Pumping Lemma. Problem 4. Sipser, Exercise 2.4. 1 Problem 5. Exhibit a context free grammar for the language L = { a n b m : n 6 = 2 m } . Then describe a PDA for the same language. You do not need to be formal about the PDA; explaining how it behaves in English is ﬁne. Problem 6. Consider the grammar G deﬁned by S → AA , A → AAA | bA | Ab | a . (a) Carefully and precisely describe the L ( G ) in an easy-to-recognize form. (b) Is L ( G ) regular? Prove your answer either way. (c) Is G ambiguous? Prove your answer either way. (d) Is L ( G ) inherently ambiguous? Give a convincing argument either way. Problem 7 Sipser, Page 129, Exercise 2.6, b), d). 2...
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## This note was uploaded on 04/29/2010 for the course ECS 222 taught by Professor Mr. during the Spring '10 term at UC Davis.

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hw4 - say why this does not contradict the Pumping Lemma....

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