complexity_ans

complexity_ans - CS 535 - Fall 2008 - MIDTERM with Answers...

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CS 535 - Fall 2008 - MIDTERM with Answers DIRECTIONS: Do any 4 of the following 5 problems. Each problem is worth 10 points. Write all of your answers in your blue book. The test is open book and you can use one page of notes. 1. a. Let f be a total computable function, f: N N . For any integer y , f 1 ( y ) is de±ned to be the (possibly in±nite) set of all x s such that f ( x ) = y . Show that for any ±xed integer m , f 1 ( m ) is a c.e. set. Answer. If f 1 ( y ) is ±nite then it is c.e by de±nition. If f 1 ( y ) is in±nite we de±ne a computable function h which enumerates f 1 ( y ) by, h(0) = smallest number z 0 with f( z 0 ) = m For j > 0, h(j) = smallest number z larger than h(j-1) with f( z ) =m. h is a total computable function which enumerates f 1 ( m ). b. Given two disjoint computably enumerable sets, A and B , prove that if A is undecidable then A B is also undecidable. Ans. Assume A B is decidable. , We will get a contradiction to this as- sumption by showing that in this case that A is decidable. Given any input x, to decide if x A , ±rst check if x A B , If no, then x / A. If yes, then enumerate A and B until x appears in one (and only one) of these sets. Then
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complexity_ans - CS 535 - Fall 2008 - MIDTERM with Answers...

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