cot6410F10Midterm1Key

cot6410F10Midterm1Key - COT 6410 Fall 2010 Exam#1 Name: KEY...

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Unformatted text preview: COT 6410 Fall 2010 Exam#1 Name: KEY 12 1 . Choosing from among (REC) recursive , (RE) re non-recursive, (coRE) co-re non-recursive , (NRNC) non-re/non-co-re , 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 non-re 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 non-recursive and C be non-re. Choosing from among (REC) recursive , (RE) re non-recursive , (NR) non-re , 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 non-empty recursive, B be re non-recursive and C be non-re. Using the terminology (REC) recursive , ( RE) re non-recursive , (NR) non-re , 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 = {2x|x S} and...
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cot6410F10Midterm1Key - COT 6410 Fall 2010 Exam#1 Name: KEY...

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