Computing K-Trivial Sets by Incomplete Random Sets

Every K-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem. A major objective in algorithmic randomness is to understand how random sets and computably enumerable (c.e.) sets interact within the Turing degrees. At some level of randomness all interesting interactions cease. The lower and upper cones of noncomputable c.e. sets are definable null sets, and thus if a set is “sufficiently” random, it cannot compute, nor be computed by, a noncomputable c.e. set. However, the most studied notion of algorithmic randomness, Martin-Löf randomness, is not strong enough to support this argument, and in fact, significant interactions between Martin-Löf random sets and c.e. sets occur. The study of these interactions has lead to a number of surprising results that show a remarkably robust relationship between Martin-Löf random sets and the class of K-trivial sets. Interestingly, the significant interaction occurs “at the boundaries”: the Martin-Löf random sets in question are close to being non-random (in that they fail fairly simple statistical tests), and K-trivial c.e. sets are close to being computable. The following theorem resolves one of the main open questions in algorithmic randomness, and further strengthens the relationship between the Martin-Löf random sets and the K-trivial sets. Theorem 1. There is an incomplete Martin-Löf random set that computes every K-trivial set. This theorem is essentially a corollary of two recent results, both proved in 2012: the first by Bienvenu, Greenberg, Kučera, Nies and Turetsky [3]; and the second by Day and Miller [10]. In the remainder of this announcement, we will explain the background to the problem behind this theorem, and indicate the main ideas used in the proof. In this announcement, by “random” we will henceforth mean Martin-Löf random. We will give the full definition shortly, but essentially, an element X of Cantor space is Martin-Löf random if it is not an element of a particular kind of effectively Day was supported by a Miller Research Fellowship in the Department of Mathematics at the University of California, Berkeley. Greenberg was supported by the Marsden Fund of New Zealand, and by a Rutherford Discovery Fellowship. Miller was supported by the National Science Foundation under grant DMS-1001847. Nies was supported by the Marsden Fund. Turetsky was supported by the Marsden Fund, via a postdoctoral scholarship. A major part of the work by Bienvenu, Greenberg, Kučera, Nies and Turetsky was performed during a research-in-pairs stay at the Mathematisches Forschungsinstitut Oberwolfach.