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

News
Feb 2013, accepted
26 Sep 2012, uploaded to arXiv
Aug 2012, submitted

Title
Van Lambalgen’s Theorem for uniformly relative Schnorr and computable randomness
(with Jason Rute)

Type
Full paper

Journal
Proceedings of the Twelfth Asian Logic Conference, World Scientific, 251-270

Abstract
We correct Miyabe’s proof of van Lambalgen’s Theorem for truth-table Schnorr randomness (which we will call uniformly rela- tive 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 com- putable 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 rel- ative 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.