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Page1ofMidterm1 - COT 5310 Spring 2006 Midterm#1 Name...

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COT 5310 Spring 2006 Midterm#1 Name: PAGE#1 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) ).
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