Bad oracles in higher computability and randomness

TL;DR

This paper explores the limitations and properties of oracles in higher computability and randomness, revealing the existence of bad oracles and their characteristics, and establishing boundaries for their behavior in higher Turing reducibility and randomness tests.

Contribution

It constructs specific bad oracles in higher computability, analyzes their properties, and differentiates between classes of bad oracles using various randomness and reducibility notions.

Findings

Existence of an oracle A and set X with higher Turing reducibility but no consistent higher functional

Construction of an oracle A with no universal higher ML-test

Demonstration that bad oracles for consistent reductions can be higher ML-random

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.