COT5310F07FinalSamplesKey

# COT5310F07FinalSamplesKey - COT 5310 Fall 2006 More Final...

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COT 5310 Fall 2006 More Final Exam Samples Name: KEY 12 1 . , (?) unknown , categorize each of the following decision problems. No proofs are required. Problem / Language Class Regular Context Free Context Sensitive Choosing from among (D) decidable , (U) undecidable L = Σ * ? D U U L = φ D D U ? L = L 2 ? D U U x L 2 , for arbitrary x ? D D D 8 2 . , (?) unknown , categorize each of the following closure properties. No proofs are required. Problem / Language Class Regular Context Free Choosing from among (Y) yes , (N) No Closed under intersection? Y N Closed under quotient? Y N Closed under quotient with Regular languages? Y Y Closed under complement? Y N Consider the two operations on languages max and min max(L) = { x | x L and, for no y does xy L } and min(L) = { x | x L and, for no prefix of x , y , does y L } D L 1 m 8 3. , where escribe the languages produced by max and min . for each of the following: = { a i b j c k | k i or k j } ax(L 1 ) = { a i b j c k | k =max(i, j) } in(L ) = { λ } (string of length 0) m 1 = { a i b j c k | k > i or k > j } ax(L 2 ) = { } (empty) L 2 m min(L 2 ) = { a i b j c k | k =min(i, j)+1 }

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COT 5310 Formal Languages and Automata Theory – 2 – Key for Final Samples – Hughes 12 4 . Prove that any class of languages, C , closed under union, concatenation, intersection with regular languages, homomorphism and substitution (e.g., the Context-Free Languages) is closed under MissingMiddle , where, assuming L is over the alphabet Σ , MissingMiddle(L) = { xz | y Σ * such that xyz L } You must be very explicit, describing what is produced by each transformation you apply. Define the alphabet Σ ’ = { a’ | a ∈Σ }, where, of course, a’ is a “new” symbol, i.e., one not in . Define homomorphisms g and h, and substitution f as follows: g(a) = a’ a h(a) = a ; h(a’) = λ a f(a) = {a, a’ } a Consider R = * g( *) * = { x y’ z | x, y, z * and y’=g(y) ’* } * is regular since it is the Kleene star closure of a finite set. g( *) is regular since it is the homomorphic image of a regular language. R is regular as it is the concatenation of regular languages.
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• Spring '08
• Staff
• Formal language, CFL, Formal Languages and Automata Theory, CONSTANT ≤m TOT, Rice’s theorem, CONSTANT ≡m TOT

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COT5310F07FinalSamplesKey - COT 5310 Fall 2006 More Final...

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