Bad oracles in higher computability and randomness

Laurent Bienvenu; Noam Greenberg; Benoit Monin

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arXiv · 1912.00807

Bad oracles in higher computability and randomness

Laurent Bienvenu·Noam Greenberg·Benoit Monin

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Abstract

Many constructions in computability theory rely on "time tricks". In the higher setting, relativising to some oracles shows the necessity of these. We construct an oracle~$A$ and a set~$X$, higher Turing reducible to~$X$, but for which $\Psi(A)\ne X$ for any higher functional~$\Psi$ which is consistent on all oracles. We construct an oracle~$A$ relative to which there is no universal higher ML-test. On the other hand, we show that badness has its limits: there are no higher self-PA oracles, and for no~$A$ can we construct a higher $A$-c.e.\ set which is also higher $A$-ML-random. We study various classes of bad oracles and differentiate between them using other familiar classes. For example, bad oracles for consistent reductions can be higher ML-random, whereas bad oracles for universal tests cannot.

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BibTeX RIS

Laurent Bienvenu, Noam Greenberg, Benoit Monin. 2019-12-02. Bad oracles in higher computability and randomness. https://arxiv.org/abs/1912.00807

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