Independence, Relative Randomness, and PA Degrees

TL;DR

This paper explores the properties of pairs of reals that are mutually random with respect to non-computable measures, extending van Lambalgen's theorem, and investigates the independence spectrum of r.e. sets, with implications for PA degrees.

Contribution

It generalizes van Lambalgen's theorem to non-computable measures and characterizes the independence spectrum of r.e. sets, linking to PA degrees.

Findings

Generalized van Lambalgen's theorem for non-computable measures

No Δ^0_2 set in the independence spectrum of r.e. sets

If A is r.e. and P is PA degree with P not Turing above A, then A⊕P computes 0'

Abstract

We study pairs of reals that are mutually Martin-L\"{o}f random with respect to a common, not necessarily computable probability measure. We show that a generalized version of van Lambalgen's Theorem holds for non-computable probability measures, too. We study, for a given real $A$, the \emph{independence spectrum} of $A$, the set of all $B$ so that there exists a probability measure $\mu$ so that $\mu\{A,B\} = 0$ and $(A,B)$ is $\mu\times\mu$-random. We prove that if $A$ is r.e., then no $\Delta^0_2$ set is in the independence spectrum of $A$. We obtain applications of this fact to PA degrees. In particular, we show that if $A$ is r.e.\ and $P$ is of PA degree so that $P \not\geq_{T} A$, then $A \oplus P \geq_{T} 0'$.