• A
  • A
  • A
  • АБB
  • АБB
  • АБB
  • А
  • А
  • А
  • А
  • А
Обычная версия сайта

Статья

Conditional Measure and the Violation of Van Lambalgen’s Theorem for Martin-Löf Randomness

Theory of Computing Systems. 2017. Vol. 60. No. 2. P. 314-323.

Van Lambalgen’s theorem states that a pair (αβ) of bit sequences is Martin-Löf random if and only ifα is Martin-Löf random and β is Martin-Löf random relative to α. 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 β for which the conditional probability P(⋅|β) 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 computable measure for which conditional measure is not computable. Such measures were first constructed by Ackerman et al. ([1]).