Communications in Mathematical Physics

, Volume 93, Issue 2, pp 219-258

First online:

Algebraic properties of cellular automata

  • Olivier MartinAffiliated withCalifornia Institute of Technology
  • , Andrew M. OdlyzkoAffiliated withBell Laboratories
  • , Stephen WolframAffiliated withBell LaboratoriesThe Institute for Advanced Study

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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.