cot5310F07Midterm1Key

# cot5310F07Midterm1Key - COT 5310 Fall 2007 Midterm#1 Name...

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Unformatted text preview: COT 5310 Fall 2007 Midterm#1 Name: KEY 12 1 . Choosing from among (REC) recursive , (RE) re non-recursive, (coRE) co-re non-recursive , (NR) 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.) { f | whenever x>y and f(x) ↓ and f(y) ↓ then f(x)>f(y) } coRE Justification: ∀ x ∀ y ∀ t [(STP(x,f,t) & STP(y,f,t) & (x>y)) ⇒ (VALUE(x,f,t)>VALUE(y,f,t))] b.) { f | size of range(f) is at most 1 } coRE Justification: ∀ x ∀ y ∀ t [(STP(x, f, t) & STP(y, f, t)) ⇒ (VALUE(x, f, t) = VALUE(y, f, t))] I allowed ∃ K ∀ x ∀ t [STP(x, f, t) ⇒ (VALUE(x, f, t) = K)], which is NR c.) { <f,x> | f(x) converges in at most x 2 steps } REC Justification: STP(x, f, x 2 ) d.) { f | domain(f) contains the value 0 } RE Justification: ∃ t STP(0, f, t) 12 2 . Let set A be recursive, and both B and C be re non-recursive. 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. You may find it useful to know that, if possible categories....
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cot5310F07Midterm1Key - COT 5310 Fall 2007 Midterm#1 Name...

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