Computability, Enumerability, Unsolvability: Directions in by S. B. Cooper, T. A. Slaman, S. S. Wainer PDF

By S. B. Cooper, T. A. Slaman, S. S. Wainer

ISBN-10: 0521557364

ISBN-13: 9780521557368

The basic principles pertaining to computation and recursion certainly locate their position on the interface among common sense and theoretical desktop technology. The contributions during this booklet offer an image of present principles and techniques within the ongoing investigations into the constitution of the computable and noncomputable universe. the various articles include introductory and heritage fabric that might make the amount a useful source for mathematicians and computing device scientists.

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Additional resources for Computability, Enumerability, Unsolvability: Directions in Recursion Theory

Example text

P. v. classes with diagonals in E. The latter was proved first by Fenner [Fe95]. 19 (Fenner [Fe95]). Let C be an r. p. and c. f. v. class such that E is not contained in C. There is a recursive function s and a k > 1 such that no (Et, s)-nt`-generic set is in C. So C is (Et, s)-p-meager, hence (Et, s) -meager in E. Proof. By assumption, there is a k > 1 and a set B E DTIME(2kn) such that B V C. 2 , there is a nondecreasing, time constructible function g such that every set which is g-similar to B does not belong to C.

His concept is defined in terms of (a new type of) extension functions, not on condition sets, but it can be easily reformulated in terms of the latter (as, conversely, our concept can be rephrased in the terms of Fenner's new extension function concept). The main difference between both concepts is, that Fenner admits only totally defined extension functions, whence his concept relates to E-category like F-category relates to G-category. 7 Extended s-bounded total category (or (Et, s) -category for short) is the category concept D = < D(t) : t recursive> where D(t) = {Dc : C is a total s-bounded t(n)-condition}.

A is [G, S] -p-generic 2. A is [G, k] -p-generic (k > 1) 3. A is [G, w] -p-generic If, moreover, A is recursive then the following are equivalent to the above too: 4. A is [G,,,, S]-p-generic 5. A is [G,,,, k]-p-generic (k > 1) 6. A is [G,,,,W]-p-generic. 13 fails if we consider nonrecursive sets. 10, [G,,,,* ]-p-genericity and [G,* ]-p-genericity (*E IS, k, w}) differ on the nonrecursive sets. 8 (b), we can prove the following weak hierarchy theorem for constant-size G genericity: For k > 1, there is a (nonrecursive) [G,,,, k]-p-generic set A which is not [G,,,, k + 1]-p-generic.

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Computability, Enumerability, Unsolvability: Directions in Recursion Theory by S. B. Cooper, T. A. Slaman, S. S. Wainer

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