Using random sets as oracles

Abstract

Let R be a notion of algorithmic randomness for individual subsets of N. A set B is a base for R randomness if there is a Z T B such that Z is R random relative to B. We show that the bases for 1-randomness are exactly the K-trivial sets, and discuss several consequences of this result. On the other hand, the bases for computable randomness include every Δ 0 2 set that is not diagonally noncomputable, but no set of PA-degree. As a consequence, an n-c.e. set is a base for computable randomness if and only if it is Turing incomplete.

Joining non-low C.E. sets with diagonally non-computable functions

“…The same theorem with 1-random instead of PA or DNC was proved by Hirschfeldt, Nies and Stephan [5].…”

“…However, Z T C, because a c.e. set computable from an incomplete random set must be K-trivial (Hirschfeldt, Nies and Stephan [5]). Therefore, we see that in Theorem 2.1 we cannot replace PA with DNC.…”

Almost everywhere domination and superhighness

“…Subsequent work by Binns, Cholak, Greenberg, Kjos-Hanssen, Lerman, Miller, and Solomon [2,5,13,14] has greatly illuminated this concept and established its close relationship to the decisive results on K-triviality and low-for-randomness which are due to Downey, Hirschfeldt, Kučera, Nies, Stephan, and Terwijn [16,9,21,11]. The purpose of this paper is to update the Dobrinen/Simpson account of almost everywhere domination by expositing this subsequent research.…”

Benign cost functions and lowness properties

“…This property of a set A ∈ 2 ω expresses that A is as far from random is possible, in that its initial segments are as compressible as possible: for all n, K(A n ) ≤ + K(n). 1 The robustness of this class is expressed by its coincidence with several notions indicating that the set is computationally feeble (Nies; Hirschfeldt and Nies; Hirschfeldt, Nies and Stephan; see [16,9,15]):…”