day24 - Diagonalization and Reducibility Click to edit...

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Click to edit Master subtitle style 10/4/11 Diagonalization and Reducibility
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10/4/11 Dec 2, © UCF (Charles E. 22 Non-re Problems There are even “practical” problems that are worse than unsolvable -- they’re not even semi- decidable. The classic non-re problem is the Uniform Halting Problem, that is, the problem to decide of an arbitrary effective procedure P, whether or not P is an algorithm. Assume that the algorithms can be enumerated, and that F accomplishes this. Then F(x) = Fx where F0, F1, F2, … is a list of all the algorithms
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10/4/11 Dec 2, © UCF (Charles E. 33 The Contradiction But then G is itself an algorithm. Assume it is the g-th one F(g) = Fg = G Then, G(g) = Fg(g) + 1 = G(g) + 1 But then G contradicts its own existence since G would need to be an algorithm. This cannot be used to show that the effective procedures are non-enumerable, since the above is not a contradiction when G(g) is undefined. In fact, we already have shown how to enumerate the (partial) recursive functions.
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Dec 2, © UCF (Charles E. 44 The Set TOT The listing of all algorithms can be
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day24 - Diagonalization and Reducibility Click to edit...

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