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# day22 - Recursively Enumerable Properties of re Sets Click...

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Click to edit Master subtitle style 10/4/11 Recursively Enumerable Properties of re Sets

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10/4/11 Dec 2, © UCF (Charles E. 22 Definition of re Some texts define re in the same way as I have defined semi-decidable. S k “ is semi-decidable iff there exists a partially computable function g where I prefer the definition of re that says S “ & is re iff S = & or there exists a totally computable function f where S = { y | N x f(x) == y } We will prove these equivalent. ±ctually, f can be a primitive recursive function.
10/4/11 Dec 2, © UCF (Charles E. 33 Semi-Decidable Implies re Theorem: Let S be semi-decided by GS. Assume GS is the gS function in our enumeration of effective procedures. If S = Ø then S is re by definition, so we will assume wlog that there is some a ø S. Define the enumerating algorithm FS by FS(<x,t>) = x * STP(x, gs, t ) + a * (1-STP(x, gs, t )) Note: FS is primitive recursive and it enumerates every value in S infinitely often.

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day22 - Recursively Enumerable Properties of re Sets Click...

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