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June 2011 Jump inversions inside effectively closed sets and applications to randomness
George Barmpalias, Rod Downey, Keng Meng Ng
J. Symbolic Logic 76(2): 491-518 (June 2011). DOI: 10.2178/jsl/1305810761

Abstract

We study inversions of the jump operator on Π⁰₁ classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0' sets which are not 2-random. Both of the classes coincide with the degrees above 0' which are not 0'-dominated. A further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails.

Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable.

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George Barmpalias. Rod Downey. Keng Meng Ng. "Jump inversions inside effectively closed sets and applications to randomness." J. Symbolic Logic 76 (2) 491 - 518, June 2011. https://doi.org/10.2178/jsl/1305810761

Information

Published: June 2011
First available in Project Euclid: 19 May 2011

zbMATH: 1248.03065
MathSciNet: MR2830414
Digital Object Identifier: 10.2178/jsl/1305810761

Rights: Copyright © 2011 Association for Symbolic Logic

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Vol.76 • No. 2 • June 2011
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