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A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences

Author

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  • Fernando Soler-Toscano
  • Hector Zenil

Abstract

Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity and that, usually, generic lossless compression algorithms fall short at characterizing features other than statistical ones not different from entropy evaluations, here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure calculated from the output distribution of small Turing machines. We introduce and justify finite approximations that have been used in some applications as an alternative to lossless compression algorithms for approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of the relevant properties of both and and compare them to Levin’s Universal Distribution. We provide error estimations of with respect to . Finally, we present an application to integer sequences from the On-Line Encyclopedia of Integer Sequences, which suggests that our AP-based measures may characterize nonstatistical patterns, and we report interesting correlations with textual, function, and program description lengths of the said sequences.

Suggested Citation

  • Fernando Soler-Toscano & Hector Zenil, 2017. "A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences," Complexity, Hindawi, vol. 2017, pages 1-10, December.
    • Handle: RePEc:hin:complx:7208216
    DOI: 10.1155/2017/7208216
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    References listed on IDEAS

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    1. Hector Zenil & Jean‐Paul Delahaye, 2011. "An Algorithmic Information Theoretic Approach To The Behaviour Of Financial Markets," Journal of Economic Surveys, Wiley Blackwell, vol. 25(3), pages 431-463, July.
    2. Jean-Paul Delahaye & Hector Zenil, 2011. "An algorithmic information-theoretic approach to the behaviour of financial markets," Post-Print hal-00825528, HAL.
    3. Fernando Soler-Toscano & Hector Zenil & Jean-Paul Delahaye & Nicolas Gauvrit, 2014. "Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines," PLOS ONE, Public Library of Science, vol. 9(5), pages 1-18, May.
    4. Zenil, Hector & Soler-Toscano, Fernando & Dingle, Kamaludin & Louis, Ard A., 2014. "Correlation of automorphism group size and topological properties with program-size complexity evaluations of graphs and complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 404(C), pages 341-358.
    5. Jean-Paul Delahaye & Hector Zenil, 2012. "Numerical Evaluation of Algorithmic Complexity for Short Strings: A Glance into the Innermost Structure of Randomness," Post-Print hal-00825530, HAL.
    6. Jean-Paul Delahaye & Hector Zenil, 2011. "On the Algorithmic Nature of the World," Post-Print hal-00826620, HAL.
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    Cited by:

    1. Maxwell Murialdo & Arturo Cifuentes, 2022. "Quantifying Value with Effective Complexity," Journal of Interdisciplinary Economics, , vol. 34(1), pages 69-85, January.

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