Constructive equivalence relations on computable probability measures

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probabil...

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Bibliographic Details
Published inAnnals of pure and applied logic Vol. 160; no. 3; pp. 238 - 254
Main Authors Bienvenu, Laurent, Merkle, Wolfgang
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.09.2009
Subjects
03D55
03D80
03D99
Algorithmic randomness
Computable analysis
Kolmogorov complexity
Probability measures
03D80
Probability measures
Kolmogorov complexity
Computable analysis
03D55
03D99
Algorithmic randomness
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Summary:A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences. In what follows, we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general.
ISSN:0168-0072
DOI:10.1016/j.apal.2009.01.002