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Mutual Dimension and Random Sequences

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Mathematical Foundations of Computer Science 2015 (MFCS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9235))

Abstract

If S and T are infinite sequences over a finite alphabet, then the lower and upper mutual dimensions mdim(S : T) and Mdim(S : T) are the upper and lower densities of the algorithmic information that is shared by S and T. In this paper we investigate the relationships between mutual dimension and coupled randomness, which is the algorithmic randomness of two sequences \(R_1\) and \(R_2\) with respect to probability measures that may be dependent on one another. For a restricted but interesting class of coupled probability measures we prove an explicit formula for the mutual dimensions \(mdim(R_1:R_2)\) and \(Mdim(R_1:R_2)\), and we show that the condition \(Mdim(R_1:R_2) = 0\) is necessary but not sufficient for \(R_1\) and \(R_2\) to be independently random.

We also identify conditions under which Billingsley generalizations of the mutual dimensions mdim(S : T) and Mdim(S : T) can be meaningfully defined; we show that under these conditions these generalized mutual dimensions have the “correct" relationships with the Billingsley generalizations of dim(S), Dim(S), dim(T), and Dim(T) that were developed and applied by Lutz and Mayordomo; and we prove a divergence formula for the values of these generalized mutual dimensions.

This research was supported in part by National Science Foundation Grants 0652519, 1143830, and 124705. Part of the second author’s work was done during a sabbatical at Caltech and the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge.

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Acknowledgment

We thank an anonymous reviewer of [4] for posing the question answered by Corollary 3.11. We also thank anonymous reviewers of this paper for useful comments, especially including Observation 3.2.

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Authors and Affiliations

  1. Department of Computer Science, Iowa State University, Ames, IA, 50011, USA

    Adam Case & Jack H. Lutz

Authors
  1. Adam Case
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  2. Jack H. Lutz
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Corresponding author

Correspondence to Adam Case .

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Editors and Affiliations

  1. University of Rome "Tor Vergata", Rome, Italy

    Giuseppe F. Italiano

  2. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  3. University of Edinburgh, Edinburgh, United Kingdom

    Donald T. Sannella

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Case, A., Lutz, J.H. (2015). Mutual Dimension and Random Sequences. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_17

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  • DOI: https://doi.org/10.1007/978-3-662-48054-0_17

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  • Print ISBN: 978-3-662-48053-3

  • Online ISBN: 978-3-662-48054-0

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