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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 MyhillNerode 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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 Fall '08
 Viswanathan,M
 Set Theory, suffix languages

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