A Characterization of Constructive Dimension

Authors:
Satyadev Nandakumar
Affiliations:
Department of Computer Science, Iowa State University, Ames, Iowa 50011, U.S.A.
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2008

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Abstract

In the context of Kolmogorov's algorithmic approach to the foundations of probability, Martin-Lof defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa, such that their success sets, for a suitably defined class of computable probability measures, are equal. The direct conversion is then generalized to give a new characterization of constructive dimensions, in particular, the constructive Hausdorff dimension and the constructive packing dimension.