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This study introduces metric entropy to classify asynchronous elementary cellular automata (AECAs) robustness, overcoming limitations of previous density-based methods. The new approach enables complete classification and reveals faster convergence for AECAs with lower complexity.

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Area of Science:

  • Complex Systems
  • Computational Science
  • Information Theory

Background:

  • Asynchronous elementary cellular automata (AECAs) robustness was previously assessed using asymptotic cell density.
  • Asymptotic density struggles to differentiate the robustness of all AECAs.
  • A need exists for a more refined method to classify AECA robustness.

Purpose of the Study:

  • To introduce a novel method for classifying AECA robustness using metric entropy.
  • To analyze the asymptotic mean entropy of local pattern distribution in AECAs.
  • To explore Kolmogorov-Sinai entropy for classifying AECA uncertainty and evolution complexity.

Main Methods:

  • Adopted metric entropy (Martin, 2000) to measure asymptotic mean entropy of local pattern distribution.
  • Conducted numerical experiments on AECA models, including those with local patterns of length 1.
  • Applied Kolmogorov-Sinai entropy to assess AECA uncertainty and forward evolution complexity.

Main Results:

  • The entropy-based measure successfully classifies the robustness of all AECA models.
  • The method is effective even when restricted to local patterns of length 1.
  • AECAs with lower uncertainty, measured by Kolmogorov-Sinai entropy, exhibit faster convergence.

Conclusions:

  • Metric entropy provides a superior method for classifying AECA robustness compared to asymptotic density.
  • The proposed entropy-based approach offers a complete classification of AECA robustness.
  • AECA uncertainty, quantified by entropy, correlates with convergence speed, offering insights into their computational complexity.