cs373 fa10 hw5

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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