Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness

TL;DR

This paper corrects and clarifies the proof of van Lambalgen's Theorem for various forms of Schnorr and computable randomness, highlighting errors and providing weaker versions of the theorem.

Contribution

It corrects Miyabe's proof for uniformly relative Schnorr and computable randomness and establishes a weaker form of van Lambalgen's Theorem.

Findings

Corrected proof of van Lambalgen's Theorem for uniformly relative Schnorr randomness.

Identified errors in Miyabe's proof for uniformly relative computable randomness.

Proved a weaker version of van Lambalgen's Theorem with mutual computable randomness.

Abstract

We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness. It has been claimed in the literature that this corollary (and the analogous result for computable randomness) is a "straightforward modification of the proof of van Lambalgen's Theorem." This is not so, and we point out why. We also point out an error in Miyabe's proof of van Lambalgen's Theorem for truth-table reducible randomness (which we will call uniformly relative computable randomness). While we do not fix the error, we do prove a weaker version of van Lambalgen's Theorem where each half is computably random uniformly relative to the other.