cot6410F10MidtermSamples

cot6410F10MidtermSamples - COT 6410 Fall 2010 Generally...

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COT 6410 Fall 2010 Generally useful information. The notation z = <x,y> denotes the pairing function with inverses x = <z> 1 and y = <z> 2 . The minimization notation μ y [P(…,y)] means the least y (starting at 0 ) such that P(…,y) is true. The bounded minimization (acceptable in primitive recursive functions) notation μ y (u y v) [P(…,y)] means the least y (starting at u and ending at v ) such that P(…,y) is true. Unlike the text, I find it convenient to define μ y (u y v) [P(…,y)] to be v+1 , when no y satisfies this bounded minimization. The tilde symbol, ~, means the complement. Thus, set ~S is the set complement of set S , and predicate ~P(x) is the logical complement of predicate P(x). A function P is a predicate if it is a logical function that returns either 1 ( true ) or 0 ( false ). Thus, P(x) means P evaluates to true on x , but we can also take advantage of the fact that true is 1 and false is 0 in formulas like y × P(x) , which would evaluate to either y (if P(x) ) or 0 (if ~P(x) ). A set S is recursive if S has a total recursive characteristic function χ S , such that x S χ S (x) . Note χ S is a predicate. Thus, it evaluates to 0 ( false ), if x S . When I say a set S is re, unless I explicitly say otherwise, you may assume any of the following equivalent characterizations: 1. S is either empty or the range of a total recursive function f S . 2.
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cot6410F10MidtermSamples - COT 6410 Fall 2010 Generally...

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