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Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines

Author

Listed:
  • Fernando Soler-Toscano
  • Hector Zenil
  • Jean-Paul Delahaye
  • Nicolas Gauvrit

Abstract

Drawing on various notions from theoretical computer science, we present a novel numerical approach, motivated by the notion of algorithmic probability, to the problem of approximating the Kolmogorov-Chaitin complexity of short strings. The method is an alternative to the traditional lossless compression algorithms, which it may complement, the two being serviceable for different string lengths. We provide a thorough analysis for all binary strings of length and for most strings of length by running all Turing machines with 5 states and 2 symbols ( with reduction techniques) using the most standard formalism of Turing machines, used in for example the Busy Beaver problem. We address the question of stability and error estimation, the sensitivity of the continued application of the method for wider coverage and better accuracy, and provide statistical evidence suggesting robustness. As with compression algorithms, this work promises to deliver a range of applications, and to provide insight into the question of complexity calculation of finite (and short) strings.Additional material can be found at the Algorithmic Nature Group website at http://www.algorithmicnature.org. An Online Algorithmic Complexity Calculator implementing this technique and making the data available to the research community is accessible at http://www.complexitycalculator.com.

Suggested Citation

  • 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.
    • Handle: RePEc:plo:pone00:0096223
    DOI: 10.1371/journal.pone.0096223
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    References listed on IDEAS

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    1. Jean-Paul Delahaye & Hector Zenil, 2007. "On the Kolmogorov Complexity for Short Sequences," Post-Print hal-00731944, HAL.
    2. 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.
    3. 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.
    4. 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.
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    Citations

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    Cited by:

    1. 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).
    2. Ferman Hasan, 2021. "Contribution of eye-tracking to the study on perception of the complexity," Technium Social Sciences Journal, Technium Science, vol. 20(1), pages 612-626, June.
    3. 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.
    4. repec:thr:techub:10020:y:2021:i:1:p:612-626 is not listed on IDEAS
    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.

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