day12 - Bounded Minimization 1 f(x = z(z x P(z if such a z...

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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 11 Bounded Minimization 1 f(x) = μ z (z ≤ x) [ P(z) ] if ° such a z, = x+1, otherwise where P(z) is primitive recursive. Can show f is primitive recursive by f(0) = 1-P(0) f(x+1) = f(x) if f(x) ≤ x = x+2-P(x+1) otherwise
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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 22 Bounded Minimization 2 f(x) = μ z (z < x) [ P(z) ] if ° such a z, = x, otherwise where P(z) is primitive recursive. Can show f is primitive recursive by f(0) = 0 f(x+1) = μ z (z ≤ x) [ P(z) ]
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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 33 Intermediate Arithmetic x // y: x//0 = 0 : silly, but want a value x//(y+1) = μ z (z<x) [ (z+1)*(y+1) > x ] x | y: x is a divisor of y x|y = ((y//x) * x) == y
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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 44 Primality firstFactor(x): first non-zero, non-one factor of x. firstfactor(x) = μ z (2 ≤ z ≤ x) [ z|x ] , 0 if none isPrime(x): isPrime(x) = firstFactor(x) == x && (x>1) prime(i) = i-th prime:
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