cot6410F10MidtermSamplesKey

cot6410F10MidtermSamplesKey - COT 6410 Fall 2010 Sample...

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COT 6410 Fall 2010 Sample Midterm#1 Name: KEY 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 | domain(f) is finite } NR Justification: x y x t ~STP(y, f, t) b.) { f | domain(f) is empty } CO Justification: x t ~STP(x, f, t) c.) { <f,x> | f(x) converges in at most 20 steps } REC Justification: STP(x, f, 20) d.) { f | domain(f) converges in at most 20 steps for some input x } RE Justification: x t STP(x, f, t) 2 . 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 = ~C RE, NR b.) D A C REC, RE, NR c.) D = ~B NR d.) D = B A REC, RE 3. Prove that the Halting Problem (the set HALT = K 0 = L u ) is not recursive (decidable) within any formal model of computation. (Hint: A diagonalization proof is required here.) Look at notes. 4. Using reduction from the known undecidable HasZero, HZ = { f | x f(x) = 0 } , show the non- recursiveness (undecidability) of the problem to decide if an arbitrary primitive recursive function g has the property IsZero, Z = { f | x f(x) = 0 } ,. Hint: there is a very simple construction that uses STP to do this. Just giving that construction is not sufficient; you must also explain why it
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cot6410F10MidtermSamplesKey - COT 6410 Fall 2010 Sample...

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