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Predictability in cellular automata.

Alexandru Agapie1, Anca Andreica2, Camelia Chira2

  • 1Bucharest University of Economic Studies, Department of Applied Mathematics, Bucharest, Romania, and Institute for Mathematical Statistics and Applied Mathematics, Bucharest, Romania.

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Probabilistic cellular automata always reach a stable state. A formula linking this state to zero-one borders, previously proven for asynchronous models, shows a perfect fit in synchronous models, suggesting a new proof is possible.

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Area of Science:

  • Computational physics
  • Statistical mechanics
  • Complex systems

Background:

  • Probabilistic cellular automata (PCA) are modeled as finite homogeneous Markov chains.
  • PCAs with local transition probabilities always possess a stationary distribution.
  • Predicting the final configuration requires linking stationary distribution probabilities to lattice configuration features.

Purpose of the Study:

  • To investigate if an exponential formula, previously established for asynchronous cellular automata, applies to synchronous cellular automata.
  • To determine if the number of zero-one borders is an intrinsic feature predicting stationary distributions in synchronous PCA.
  • To provide empirical evidence for a potential rigorous proof of the formula in the synchronous case.

Main Methods:

  • Computer experiments were conducted on a synchronous cellular automaton.
  • The empirical distribution of configurations was analyzed.
  • The observed distribution was compared against a theoretical formula based on the number of zero-one borders.

Main Results:

  • Numerical results indicate a perfect fit between the empirical distribution and the theoretical formula for neighborhood sizes of three and five.
  • The study validates the relevance of the zero-one border count as an intrinsic feature in synchronous PCA.
  • The findings suggest the formula's applicability extends beyond asynchronous models.

Conclusions:

  • The number of zero-one borders is a key feature for predicting stationary distributions in synchronous PCA.
  • Empirical evidence strongly supports the applicability of the exponential formula to synchronous cellular automata.
  • These results pave the way for a rigorous mathematical proof of the formula in the synchronous context.