Coherent randomness tests and computing the $K$-trivial sets

Abstract

We show that a Martin-Löf random set for which the effective version of the Lebesgue density theorem fails computes every K-trivial set. Combined with a recent result by Day and Miller, this gives a positive solution to the ML-covering problem (Question 4.6 in Randomness and computability: Open questions. Bull. Symbolic Logic, 12 (3): 2006). On the other hand, we settle stronger variants of the covering problem in the negative. We show that any witness for the solution of the covering problem, namely an incomplete random set which computes all K-trivial sets, must be very close to being Turing complete. For example, such a random set must be LR-hard. Similarly, not every K-trivial set is computed by the two halves of a random set.The work passes through a notion of randomness which characterises computing K-trivial sets by random sets. This gives a "smart" K-trivial set, all randoms from whom this set is computed have to compute all K-trivial sets.

Density, forcing, and the covering problem

“…However, the Cantor space version follows immediately from the proof given in [3,Theorem 20]. Theorem 1.2 (Bienvenu, Greenberg, Kučera, Nies and Turetsky [2]). If X ∈ 2 ω is Martin-Löf random and not a density-one point, then X computes every K-trivial set.…”

“…The present paper, combined with theorems of Bienvenu, Greenberg, Kučera, Nies and Turetsky [2], and Bienvenu, Hölzl, Miller and Nies [3,4], gives a strong affirmative answer to the covering problem:…”

“…The original proof of Theorem 1.2, given in [2], involves several steps. A direct proof, though one relying on more of the theory of K-triviality, is given by Bienvenu, Hölzl, Miller and Nies [4].…”

Lebesgue Density and Classes

“…Theorem 6.2 has an interesting consequence. Bienvenu, et al [2] introduce Oberwolfach randomness and show that every Oberwolfach random real is a full density-one point. Based on earlier work by Figueira, Hirschfeldt, Miller, Ng, and Nies [7], they observe that one "half " of every Martin-Löf random real is always Oberwolfach random, hence full density-one 3 :…”

“…Thus there are arbitrarily long strings α extending satisfying condition (1). To satisfy (2) and (3), we simply choose an α long enough and designate it F s+1 ( i). Proof.…”