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Continuous higher randomness

Authors
  • Bienvenu, Laurent
  • Greenberg, Noam
  • Monin, Benoit
Type
Preprint
Publication Date
Mar 16, 2015
Submission Date
Mar 16, 2015
Identifiers
arXiv ID: 1503.04884
Source
arXiv
License
Yellow
External links

Abstract

We investigate the role of continuous reductions and continuous relativisation in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van-Lambalgen's theorem and the Miller-Yu / Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterisations of lowness for Martin-L\"of randomness. We also characterise computing higher $K$-trivial sets by higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak-2-randomness and $\Pi^1_1$-randomness. To do so we investigate classes of functions computable from Kleene's~$O$ based on strong forms of the higher limit lemma.

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