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Unformatted text preview: COT 6410 Fall 2010 Exam#1 Name: KEY 12 1 . Choosing from among (REC) recursive , (RE) re nonrecursive, (coRE) core nonrecursive , (NRNC) nonre/noncore , categorize each of the sets in a) through d). Justify your answer by showing some minimal quantification of some known recursive predicate. a) A = { f  f(x) for all x } 2200 <x,t> ~STP(f,x,t) coRE b.) B = { f  domain(f) is a proper subset of ; that is f diverges at some points } 5 x 2200 t ~STP(f,x,t) NRNC c.) C = { f  f(x) > x for at least one value x } 5 <x,t> [STP(f,x,t)&VALUE(f,x,t)>x] RE d.) D = { <f,x>  f(x) converges in at most x steps } STP(f,x,x) REC 6 2. Prove that the Uniform Halting Problem (the set TOTAL ) is nonre within any formal model of computation. (Hint: A diagonalization proof is required here.) Look at Notes 6 3 . Let set A be recursive, B be re nonrecursive and C be nonre. Choosing from among (REC) recursive , (RE) re nonrecursive , (NR) nonre , categorize the set D in each of a) through d) by listing all possible categories. No justification is required. a.) D = ~B NR b.) D A REC, RE, NR c.) D = A B REC, RE d.) D = C  A REC, RE, NR 10 4 . Let set A be nonempty recursive, B be re nonrecursive and C be nonre. Using the terminology (REC) recursive , ( RE) re nonrecursive , (NR) nonre , categorize each set by dealing with the cases I present, saying whether or not the set can be of the given category and. briefly, but convincingly, justifying each answer. You may assume, for any set S , the existence of comparably hard sets S E = {2xx S} and...
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 Fall '10
 Dutton

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