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cot5310F07SampleMidterm1 - COT 5310 Fall 2006 Midterm#1...

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COT 5310 Fall 2006 Midterm#1 Sample Name:_____________________ 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. S is the domain of a partial recursive function g S . If I say a function g is partially computable, then there is an index g (I know that’s overloading, but that’s okay as long as we understand each other), such that Φ g (x) = Φ (x, g) = g(x) . Here Φ is a universal partially recursive function.
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