Highness properties close to PA-completeness

TL;DR

This paper investigates the highness properties of oracles related to computably enumerable objects, showing they are often close to or equivalent to PA-completeness, with some properties being distinct from it.

Contribution

It identifies and analyzes various highness properties near PA-completeness, including a separation result for the continuous covering property.

Findings

Highness properties are often close to PA-completeness.

Separation of the continuous covering property from PA-completeness.

Analysis of oracle computations for c.e. objects and their approximations.

Abstract

Suppose we are given a computably enumerable object arise from algorithmic randomness or computable analysis. We are interested in the strength of oracles which can compute an object that approximates this c.e. object. It turns out that, depending on the type of object, the resulting highness property is either close to, or equivalent to being PA-complete. We examine, for example, dominating a c.e. martingale by an oracle-computable martingale, computing compressions functions for two variants of Kolmogorov complexity, and computing subtrees of positive measure of a given $\Pi^0_1$ tree of positive measure without dead ends. We prove a separation result from PA-completeness for the latter property, called the \emph{continuous covering property}. We also separate the corresponding principles in reverse mathematics.