Incompleteness, Approximation and Relative Randomness

INCOMPLETENESS, APPROXIMATION AND RELATIVE RANDOMNESS ANTHONY MORPHETT Abstract. We present some results about the structure of c.e. and ∆02 LR-degrees. First we give a technique for lower cone avoidance in the c.e. and ∆02 LR-degrees, and combine this with upper cone avoidance via Sacks restraints to construct a c.e. LR-degree which is incomparable with a given intermediate ∆02 LR-degree. Next we combine measure-guessing with an LR-incompleteness strategy to construct an incomplete c.e. LR-degree which is above a given low ∆02 LR-degree. This is in contrast to the Turing degrees, in which there is a low ∆02 Turing degree which is incomparable with all intermediate c.e. Turing degrees.

1. Introduction A basic task of computability theory is to study the information content of a set X by studying classes of computations obtained by relativising to X. The most fundamental class of relativised computations is the partial recursive functions relative to X, Rec(X). We can compare the information content of two sets A, B ⊆ N by comparing the classes Rec(A) and Rec(B). This approach gives rise to the familiar Turing reducibility, which may be defined as (1)

A ≤T B iff Rec(A) ⊆ Rec(B).

By replacing the class Rec(X) with another class of computations, we can obtain other reducibilities. One class of computations that has been of particular interest recently due to the study of algorithmic randomness is the class of effective approximations of nullsets (measure 0 subsets) of Cantor space. Effective approximations of nullsets are called Martin-L¨of tests, and give rise to the notion of Martin-L¨ of randomness (defined formally below). One may study the information content of a set A by examining the class of nullsets that are effectively approximable relative to the oracle A, or equivalently, the notion of Martin-L¨of randomness relativised to A. We obtain a reducibility, known as LR-reducibility, by comparing the contents of effectively approximable nullsets: A ≤LR B iff every nullset that is effectively approximable relative to A is contained in a nullset that is effectively approximable relative to B. LR-reducibility was first considered in [16], and has been further studied, for instance, in [3] and [4]. LR-reducibility has been shown to be equivalent to another reducibility arising from algorithmic randomness, known as LK-reducibility. For A, B ⊆ N, define A ≤LK B iff ∃c ∈ N ∀σ ∈ 2