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.
Similar content being viewed by others
References
S. Wolfram, Statistical mechanics of cellular automata,Rev. Mod. Phys. 55:601 (1983).
P. Guan and Y. He, Exact results for deterministic cellular automata with additive rules,J. Stat. Phys. 43:463 (1986).
O. Martin, A. Odlyzko, and S. Wolfram, Algebraic properties of cellular automata,Commun. Math. Phys. 93:219 (1984).
R. Lidl and G. Pilz,Applied Abstract Algebra (Springer-Verlag, New York, 1984), p. 172.
S. Wolfram, Mathematica®, A System for Doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, California, 1991), p. 602.
R. W. Marsh, Table of Irreducible Polynomials over GF(2) Through Degree 19, U. S. Department of Commerce, PB161693 (1957).
N. Jacobsen,Lectures in Abstract Algebra, Vol. II-Linear Algebra (Van Nostrand, New York, 1953), p. 73.
Author information
Authors and Affiliations
Additional information
Deceased.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01052755