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>1and y =<z>2. •The minimization notation µy [P(…,y)]means the least y(starting at 0) such that P(…,y)is true. The bounded minimization (acceptable in primitive recursive functions) notation µy (u≤y≤v) [P(…,y)]means the least y(starting at uand ending at v) such that P(…,y)is true. Unlike the text, I find it convenient to define µy (u≤y≤v) [P(…,y)] to be v+1, when no ysatisfies this bounded minimization. •The tilde symbol, ~,means the complement. Thus, set ~S is the set complement of setS, and predicate ~P(x) is the logical complement of predicateP(x).•A function Pis a predicate if it is a logical function that returns either 1(true) or 0(false). Thus, P(x)means Pevaluates to trueon x, but we can also take advantage of the fact that trueis 1and falseis 0in formulas like y ×P(x), which would evaluate to either y(if P(x)) or 0(if ~P(x)).
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