day44 - are required if P is re – If L is in(has property...

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Click to edit Master subtitle style 10/4/11 Recursion Theorem Self Reproducibility
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10/4/11 Dec 2, © UCF (Charles E. 22 Simple Form Theorem: There is an index e, such that Gx “ e(x) = e This means that we have a function that always produces its own description (index) no matter what input you give it. People used to have fun trying to find the smallest self-reproducing Lisp
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Click to edit Master subtitle style 10/4/11 Fixed Point Theorem A property of all indexing schemes
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10/4/11 Dec 2, © UCF (Charles E. 44 The Fixed Point Theorem Theorem: Let f(z) be any computable function. Then there is an index e such that &x & f(e)(x) = & e(x) There are many forms of computation that seek a fixed point. Correctness proofs are often of this sort.
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Click to edit Master subtitle style 10/4/11 Classifying Unsolvable Problems Rice-Shapiro Theorem Minimum Quantification
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10/4/11 Dec 2, © UCF (Charles E. 66 Rice-Shapiro Theorem Properties of a set of indices P that
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Unformatted text preview: are required if P is re: – If L is in (has property) P and L ’ some re set L’, then L’ is in P. – If L is an infinite set in P, then there is some finite subset L’ of L that is in P. – The set of finite languages in P is enumerable. 10/4/11 Dec 2, © UCF (Charles E. 77 Using Extended Rice • Violate Condition # 1 – L = “ – L is recursive – L is a singleton set – L is a regular set • Violates Condition # 2 – L = “ * • Violates Condition # 3 10/4/11 Dec 2, © UCF (Charles E. 88 RE Sets • L ” “ • L contains at least 3 numbers • W is in L, for a fixed W • L & Lu & ± 10/4/11 Dec 2, © UCF (Charles E. 99 Minimum Quantification • Recursive: unquantified total predicate • RE: existentially quantified • ~ RE: universally quantified • Recursive: Can express as RE & ~RE • TOT: universal/existential • ~TOT: existential/universal...
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day44 - are required if P is re – If L is in(has property...

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