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Conditional Measure and the Violation of Van Lambalgen’s Theorem for Martin-Löf Randomness
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Conditional Measure and the Violation of Van Lambalgen’s Theorem for Martin-Löf Randomness
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Journal Article
Conditional Measure and the Violation of Van Lambalgen’s Theorem for Martin-Löf Randomness
2017
Overview
Van Lambalgen’s theorem states that a pair (
α
,
β
) of bit sequences is Martin-Löf random if and only if
α
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
]).
Publisher
Springer US,Springer Nature B.V
