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Unformatted text preview: phism.) b). [20 points] The language L 2 = { n 1 m 2 nm  n ≥ m ≥ } over Σ = { , 1 , 2 } . Problem 3. [15 points] (This is exercise 4.1.2(b) in both the second and third editions of the textbook.) Prove that { n  n is a perfect cube } is not a regular language. ( Hint: See the solution for Exercise 4.1.2(a).) Problem 4. [15 points] (Exercise 4.2.6(c) in both editions of the textbook.) Show that the regular languages are closed under the following operation: init ( L ) = { w  for some string x, wx is in L } . Hint : It is easiest to start with a DFA for L and perform a construction to get the desired language. Problem 5. [15 points] (Exercise 4.2.1(f) in both editions of the textbook.) Suppose h is the homomorphism from the alphabet { , 1 , 2 } to the alphabet { a, b } de±ned by: h (0) = a ; h (1) = ab , and h (2) = ba . If L is the language L ( a ( ba ) * ), what is h1 ( L )? Provide a proof to justify your claim. 2...
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 '08
 Motwani,R
 Formal language, Formal languages, Regular expression, Regular language, 1m, Complexity Theory Handout

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