Dense computability, upper cones, and minimal pairs

Abstract

This paper concerns algorithms that give correct answers with (asymptotic) density 1. A dense description of a function g : ω → ω is a partial function f on ω such that {n : f (n) = g(n)} has density 1. We define g to be densely computable if it has a partial computable dense description f . Several previous authors have studied the stronger notions of generic computability and coarse computability, which correspond respectively to requiring in addition that g and f agree on the domain of f , and to requiring that f be total. Strengthening these two notions, call a function g effectively densely computable if it has a partial computable dense description f such that the domain of f is a computable set and f and g agree on the domain of f . We compare these notions as well as asymptotic approximations to them that require for each ε > 0 the existence of an appropriate description that is correct on a set of lower density of at least 1−ε. We determine which implications hold among these various notions of approximate computability and show that any Boolean combination of these notions is satisfied by a c.e. set unless it is ruled out by these implications. We define reducibilities corresponding to dense and effectively dense reducibility and show that their uniform and nonuniform versions are different. We show that there are natural embeddings of the Turing degrees into the corresponding degree structures, and that these embeddings are not surjective and indeed that sufficiently random sets have quasiminimal degree. We show that nontrivial upper cones in the generic, dense, and effective dense degrees are of measure 0 and use this fact to show that there are minimal pairs in the dense degrees.

A Minimal Pair in the Generic Degrees

“…On the other hand, Theorem 6 does show that the methods of [6], and the related ones of [1], are not available here. Nevertheless, in this paper we give a positive answer to Question 4 in both cases.…”

“…Thus generic computability captures the idea of computing a set while allowing for a small number of errors of omission, while coarse computability captures the idea of computing a set while allowing for a small number of errors of commission. We can also consider notions that allow both kinds of errors, as was done by Astor, Hirschfeldt, and Jockusch [1].…”

“…Astor, Hirschfeldt, and Jockusch [1] also studied the following notion, which had been briefly considered much earlier by Meyer [13] and Lynch [12]. Definition 10.…”

“…We say that A is uniformly effectively densely reducible to B if there is a Turing functional Φ such that if f is an effective dense description of B, then Φ f is an effective dense description of A. Open Question 11 (Astor, Hirschfeldt, and Jockusch [1]). Are there minimal pairs in the (nonuniform or uniform) effective dense degrees?…”

Density-1-Bounding and Quasiminimality in the Generic Degrees

“…Fix a condition p = , that forces that φ G is a computation of B. Fix k such that 1 k < . By Lemma 6.2, fix i < k such that B T A =i .…”