Communications in Mathematical Physics

, Volume 93, Issue 2, pp 219–258

Algebraic properties of cellular automata


  • Olivier Martin
    • California Institute of Technology
  • Andrew M. Odlyzko
    • Bell Laboratories
  • Stephen Wolfram
    • Bell Laboratories
    • The Institute for Advanced Study

DOI: 10.1007/BF01223745

Cite this article as:
Martin, O., Odlyzko, A.M. & Wolfram, S. Commun.Math. Phys. (1984) 93: 219. doi:10.1007/BF01223745


Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata. The complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities. The systems are usually irreversible, and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations.

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© Springer-Verlag 1984