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Kolmogorov-Loveland stochasticity and Kolmogorov complexity. (English) Zbl 1204.68110
Summary: W. Merkle, J. S. Miller, A. Nies, J. Reimann and F. Stephan [Ann. Pure Appl. Logic 138, No. 1–3, 183–210 (2006; Zbl 1097.03041)] showed that all Kolmogorov-Loveland stochastic infinite binary sequences have constructive Hausdorff dimension 1. In this paper, we go even further, showing that from an infinite sequence of dimension less than \(\mathcal {H}(\frac {1}{2}+\delta)\) (\(\mathcal H\) being the Shannon entropy function) one can extract by an effective selection rule a biased subsequence with bias at least \(\delta \). We also prove an analogous result for finite strings.

MSC:
68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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