Journal of Symbolic Logic

Jump inversions inside effectively closed sets and applications to randomness

George Barmpalias, Rod Downey, and Keng Meng Ng

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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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J. Symbolic Logic, Volume 76, Issue 2 (2011), 491-518.

First available in Project Euclid: 19 May 2011

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Barmpalias, George; Downey, Rod; Ng, Keng Meng. Jump inversions inside effectively closed sets and applications to randomness. J. Symbolic Logic 76 (2011), no. 2, 491--518. doi:10.2178/jsl/1305810761.

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