day21 - Undecidability We Cant Do It All Click to edit...

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Click to edit Master subtitle style 10/4/11 Undecidability We Can’t Do It All
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10/4/11 Dec 2, © UCF (Charles E. 22 Undecidability Precursor We can see that there are undecidable functions merely by noting that there are an uncountable number of mappings from the natural numbers into the natural numbers. Since effective procedures are always over a language with a finite number of primitives, and since we restrict programs to finite length, there can be only a countable number of effective procedures. Thus no formalism can get us all mappings -- some must be non-computable. The above is a great existence proof, but is unappealing since it doesn’t help us to understand what kinds of problems are uncomputable. The classic unsolvable problem is called the Halting Problem. It is the problem to decide of an arbitrary effective procedure f: & + & , and an arbitrary n & & , whether or not f(n) is defined.
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10/4/11 Dec 2, © UCF (Charles E. 33 Halting Problem Assume we can decide the halting problem. Then there exists
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day21 - Undecidability We Cant Do It All Click to edit...

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