Uniform van Lambalgen's theorem fails for computable randomness

TL;DR

This paper demonstrates that the uniform version of van Lambalgen's theorem does not hold for computable randomness by constructing a specific counterexample involving bit sequences with particular randomness properties.

Contribution

It provides the first explicit counterexample showing the failure of the uniform van Lambalgen's theorem for computable randomness.

Findings

Existence of a non-computably random sequence with computably random odd bits

Odd bits are computably random independently of the even bits

Uniform van Lambalgen's theorem fails in the context of computable randomness

Abstract

We show that there exists a bitsequence that is not computably random for which its odd bits are computably random and its even bits are computably random relative to the odd bits. This implies that the uniform variant of van Lambalgen's theorem fails for computable randomness. (The other direction of van Lambalgen's theorem is known to hold in this case, and known to fail for the non-uniform variant.)