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Unformatted text preview: COT 6410 Fall 2010 Exam#2 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(f) ↓ } 5 t STP(f,f,t) RE b.) B = { f  range(f) is a proper subset of ℵ } 5 x 2200 <y,t> [STP(f,x,t) ⇒ Value(f,y,t) ≠ x] NRNC c.) C = { f  f(0) take at least 100 steps to converge, if at all } ~STP(f,0,99) REC d.) D = { f  f diverges everywhere } 2200 <x,t> ~STP(f,x,t) coRE 6 2 . Define, compare and contrast the notions of Countable and Recursively Enumerable for some set S . Question 7 actually gives one of these definitions. A set S is countable iff it can be placed in a 11 correspondence with a subset of the Natural numbers. That is, S is countable iff there is an injective mapping (not necessarily computable) that associates each element of S with a unique element of ℵ . Alternatively, S is countable iff S is empty or there is a surjective mapping (not necessarily computable) that associates each element of ℵ with a unique element of S. A set S is recursively enumerable (re) iff it is either empty or there exists a total computable function that effectively maps the Natural numbers onto the set. That is, S is re, nonempty iff there is an algorithm (a total computable function), f, whose domain is ℵ and whose range is S. Every re set is countable, but not every countable set is re. The issue is that re requires a mapping that is computable, whereas countable just requires the existence of a mapping; whether or not that mapping is computable is not relevant. Thus, all subsets of that mapping is computable is not relevant....
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This note was uploaded on 07/14/2011 for the course COT 4610 taught by Professor Dutton during the Fall '10 term at University of Central Florida.
 Fall '10
 Dutton

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