Abstract
Deterministic cellular automata (CA) with additive rules are studied by exploiting the properties of circulant matrices on finite fields. Complete state transition diagrams for higher-order and multidimensional CA on finite lattices are analyzed. Conditions on the rules which make them reversible are obtained. It is shown that all state transition diagrams of the CA have identical trees rooted on cycles. General formulae for cycle lengths and multiplicities are given.
Similar content being viewed by others
References
S. Wolfram,Rev. Mod. Phys. 55:601 (1983);Physica 10D:247 (1984).
O. Martin, A. M. Odlyzko, and S. Wolfram,Commun. Math. Phys. 93:219 (1984).
Philip J. Davis,Circulant Matrices (John Wiley & Sons, New York, 1979).
F. J. MacWilliams and N. J. A. Sloane,The Theory of Error-Correcting Codes (North-Holland, New York, 1977).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Guan, Ph., He, Y. Exact results for deterministic cellular automata with additive rules. J Stat Phys 43, 463–478 (1986). https://doi.org/10.1007/BF01020648
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01020648