Continuous higher randomness

TL;DR

This paper explores the concept of higher randomness using continuous reductions, establishing new analogues of classical theorems, characterizations, and separations within the framework of higher computability and randomness.

Contribution

It introduces a higher analogue of Turing reducibility, studies lowness and characterizations of higher randomness, and differentiates between various higher randomness notions.

Findings

Higher Turing reducibility interacts well with higher randomness.

Characterizations of lowness for higher Martin-Löf randomness are equivalent.

Separation established between higher weak-2-randomness and -randomness.

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.