cs373 fa10 hw5 - w R denote the “reverse” of the string...

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Problem Set 5 CS 373: Theory of Computation Assigned: October 5, 2010 Due on: October 12, 2010 at 10am Instructions: This homework has no practice problems. The homework can be solved in groups of size at most 3. Please strictly follow the homework guidelines given on the class website; submitions not following these guidelines will not be graded. Recommended Reading: Lectures 10, 11, and 12. Homework Problems Problem 1 . [Category: Comprehension+Design] Consider the following DFA M . Let L = L ( M ). q 0 q 1 q 2 q 3 q 4 q 5 1 1 0 0 0 1 0 1 0 , 1 1 0 Figure 1: DFA M for Problem 1 1. What are the following sets: suffix( L,± ), suffix( L, 0), suffix( L, 00), and suffix( L, 01)? [2 points] 2. What are the following sets: suffix( M,q 0 ), suffix( M,q 2 ), suffix( M,q 3 ), and suffix( M,q 4 )? [2 points] 3. Construct the minimum DFA that is equivalent to M , showing the steps of the construction clearly. [3 points] 4. For every pair of states in the minimal DFA constructed in the previous part, give a string that “distinguishes” the states. [3 points] Problem 2 . [Category: Comprehension+Proof] For a string w Σ * , let
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Unformatted text preview: w R denote the “reverse” of the string w . Consider Pal = { w ∈ Σ * | w = w R } ; thus, Pal is the collection of palindromes over Σ. Prove that Pal is not regular using the Myhill-Nerode Theorem, by demonstrating that C suf (Pal) is infinite, i.e., there is an infinite set W ⊆ Σ * such that for any x,y ∈ W , suffix(Pal , x) 6 = suffix(Pal , y). [10 points] Problem 3 . [Category: Proof] For a language L ⊆ Σ * and string x ∈ Σ * , define the prefix language of L with respect to x as prefix( L,x ) = { y | yx ∈ L } 1 Note, the difference between this and the way we defined suffix( L,x ) in class. Once again, the class of prefix languages (denoted as C pref ( L )) of L is defined as C pref ( L ) = { prefix( L,x ) | x ∈ Σ * } Prove that L is regular if and only if C pref ( L ) is finite. Hint: Can you see a connection between the prefix languages of L and the suffix languages of L R ? [10 points] 2...
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cs373 fa10 hw5 - w R denote the “reverse” of the string...

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