Van Lambalgen's theorem fails for some computable measure

TL;DR

This paper demonstrates that Van Lambalgen's theorem does not hold universally for all computable measures by providing a counterexample, thus showing the necessity of the computability condition in the theorem.

Contribution

The authors construct a specific computable measure where Van Lambalgen's theorem fails, answering an open question about the necessity of the computability condition.

Findings

Counterexample measure where Van Lambalgen's theorem fails

Conditional measure is not computable in the constructed example

Supports the necessity of the computability condition in the theorem

Abstract

Van Lambalgen's theorem states that a pair $(\alpha,\beta)$ of bitsequences is Martin-L\"of random if and only if $\alpha$ is Martin-L\"of random and $\beta$ is Martin-L\"of random relative to $\alpha$. In [Information and Computation 209.2 (2011): 183-197, Theorem 3.3], Hayato Takahashi generalized van Lambalgen's theorem for computable measures $P$ on a product of two Cantor spaces; he showed that the equivalence holds for each $\beta$ for which the conditional probability $P(\cdot | \beta)$ is computable. He asked whether this computability condition is necessary. We give a positive answer by providing a computable measure for which van Lambalgen's theorem fails. We also present a simple construction of a measure for which conditional measure is not computable. Such measures were first constructed by N. Ackerman, C. Freer and D. Roy in [Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 107-116. IEEE (2011)].