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Transient and cyclic behavior of cellular automata with null boundary conditions

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Abstract

One-dimensional cellular automata (CA) over finite fields are studied in which each interior cell is updated to contain the sum of the previous values of its two nearest neighbors. Boundary cells are updated according to null boundary conditions. For a given initial configuration, the CA evolves through transient configurations to an attracting cycle. The dependence of the maximal transient length and maximal cycle length on the number of cells is investigated. Both can be determined from the minimal polynomial of the update matrix, which in this case satisfies a useful recurrence relation. With cell values from a field of characteristic two, the explicit dependence of the maximal transient length on the number of cells is determined. Extensions and directions for future work are presented.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Montclair State College, 07043, Upper Montclair, New Jersey

    John G. Stevens

  2. Corporate Research, Exxon Research and Engineering Company, 08801, Annandale, New Jersey

    Ronald E. Rosensweig

  3. Department of Mechanical Engineering, New Jersey Institute of Technology, 07102, Newark, New Jersey

    A. E. Cerkanowicz

Authors
  1. John G. Stevens
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  2. Ronald E. Rosensweig
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  3. A. E. Cerkanowicz
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Stevens, J.G., Rosensweig, R.E. & Cerkanowicz, A.E. Transient and cyclic behavior of cellular automata with null boundary conditions. J Stat Phys 73, 159–174 (1993). https://doi.org/10.1007/BF01052755

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  • DOI: https://doi.org/10.1007/BF01052755

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