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Numerical Evaluation of Algorithmic Complexity for Short Strings: A Glance into the Innermost Structure of Randomness

Author

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  • Jean-Paul Delahaye

    (LIFL - Laboratoire d'Informatique Fondamentale de Lille - Université de Lille, Sciences et Technologies - Inria - Institut National de Recherche en Informatique et en Automatique - Université de Lille, Sciences Humaines et Sociales - CNRS - Centre National de la Recherche Scientifique, SMAC - Systèmes Multi-Agents et Comportements - CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 - Centrale Lille - Université de Lille - CNRS - Centre National de la Recherche Scientifique)

  • Hector Zenil

    (LIFL - Laboratoire d'Informatique Fondamentale de Lille - Université de Lille, Sciences et Technologies - Inria - Institut National de Recherche en Informatique et en Automatique - Université de Lille, Sciences Humaines et Sociales - CNRS - Centre National de la Recherche Scientifique, SMAC - Systèmes Multi-Agents et Comportements - CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 - Centrale Lille - Université de Lille - CNRS - Centre National de la Recherche Scientifique)

Abstract

We describe an alternative method (to compression) that combines several theoretical and experimental results to numerically approximate the algorithmic Kolmogorov-Chaitin complexity of all Sigma(8)(n-1)2(n) bit strings up to 8 bits long, and for some between 9 and 16 bits long. This is done by an exhaustive execution of all deterministic 2-symbol Turing machines with up to four states for which the halting times are known thanks to the Busy Beaver problem, that is 11019960576 machines. An output frequency distribution is then computed, from which the algorithmic probability is calculated and the algorithmic complexity evaluated by way of the Levin-Chaitin coding theorem. (

Suggested Citation

  • 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.
    • Handle: RePEc:hal:journl:hal-00825530
    DOI: 10.1016/j.amc.2011.10.006
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    Cited by:

    1. 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.
    2. 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.
    3. Brandouy, Olivier & Delahaye, Jean-Paul & Ma, Lin & Zenil, Hector, 2014. "Algorithmic complexity of financial motions," Research in International Business and Finance, Elsevier, vol. 30(C), pages 336-347.
    4. Dingle, Kamaludin & Kamal, Rafiq & Hamzi, Boumediene, 2023. "A note on a priori forecasting and simplicity bias in time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    5. Maxwell Murialdo & Arturo Cifuentes, 2022. "Quantifying Value with Effective Complexity," Journal of Interdisciplinary Economics, , vol. 34(1), pages 69-85, January.
    6. 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.
    7. Mikołaj Morzy & Tomasz Kajdanowicz & Przemysław Kazienko, 2017. "On Measuring the Complexity of Networks: Kolmogorov Complexity versus Entropy," Complexity, Hindawi, vol. 2017, pages 1-12, November.
    8. Daniel Wilson-Nunn & Hector Zenil, 2014. "On the Complexity and Behaviour of Cryptocurrencies Compared to Other Markets," Papers 1411.1924, arXiv.org.

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