cot5310F07Midterm2Key - COT 5310 Fall 2007 Midterm#2...

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COT 5310 Fall 2007 Midterm#2 Name:_______KEY___ 12 1 . Choosing from among (REC) recursive , (RE) re non-recursive , (CO) co-re non-recursive , (NR) non-re , categorize each of the sets in a) through d). Justify your answer by showing some minimal quantification of some known recursive predicate. a.) { <f,x> | f(x) takes at least x 2 steps to converge } REC Justification: ~STP(x,f,x 2 -1) b.) { f | range(f) contains only even numbers } CO Justification: <x,t>[STP(x,f,t) isEven(x)] c.) { f | range(f) is not the set of natural numbers } NR Justification: x <y,t>[STP(y,f,t) Value(y,f,t) x] d.) { f | f converges on some pair of input, x, 2x } RE Justification: <x,t> [STP(x,f,t) && STP(2x,f,t)] 9 2 . Let A be re, possibly recursive, and B be re non-recursive. Let C = (A ~B) (B ~A). For each part, either show sets A and B with the specified property and justify in detail how these meet the required property, or present a demonstration that this property cannot hold. a.) Can C be re non-recursive? YES. Let A = φ . A is clearly re, even recursive since it is trivially decided by χ A (x) = 0. Then C = ( φ ~B) (B ) = B. B is given to be re, non-recursive. b.) Can C be co-re non-recursive? YES. Let A = . A is clearly re, even recursive since it is trivially decided by χ A (x) = 1. Then C = ( ~B) (B φ ) = ~ B. Since B is given to be re, non-recursive, its complement must be co-re non-recursive, as desired. 12 3 . Let set A and B be sets, such that A m B by the total m-1 recursive function f AB . For each of the following, be complete by addressing whether or not the specified set can be recursive, re non- recursive and/or non-re. a.) Assume A is re, non-recursive and semi-decided by the partial recursive functions g A . What can we say about the complexity (recursive, re, non-re) of B ? Address all three cases.
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