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# hw02 - phism b[20 points The language L 2 = n 1 m 2 n-m | n...

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CS 154 Intro. to Automata and Complexity Theory Handout 6 Autumn 2008 David Dill October 6, 2009 Problem Set 2 Due: October 13, 2009 Homework: (Total 100 points) Do the following exercises. Problem 1. [15 points] Consider the DFA: 1 2 3 4 b a, c b a, b, c c a, b, c a Using the state elimination procedure described in lecture, find a regular expression for the language of this automaton. Draw the GNFAs resulting from each step: (1) a GNFA equivalent to this DFA, (2) eliminate state 3, (3) elimination state 2, (4) eliminate state 1, (5) eliminate state 4. Problem 2. [40 points] Show that the two languages below are not regular. For each language, give two separate proofs , one using the Pumping Lemma, and (2) one using closure properties of the regular languages discussed in the textbook and lecture, and the fact that the language { 0 i 1 i | i 0 } is not regular, and not the Pumping Lemma. a). [20 points] The language L 1 over Σ = { 0 , 1 } consisting of all strings where the number of 1’s is exactly two times the number of 0’s. ( Hint: For the closure properties proof, using set intersection and inverse homomor-

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Unformatted text preview: phism.) b). [20 points] The language L 2 = { n 1 m 2 n-m | 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 h-1 ( L )? Provide a proof to justify your claim. 2...
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