Inducing an order on cellular automata by a grouping operation

  • Jacques Mazoyer
  • Ivan Rapaport
Automata and Formal Languages I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1373)


A grouped instance of a cellular automaton (CA) is another one obtained by grouping several states into blocks and by letting interact neighbor blocks. Based on this operation a preorder ≤ on the set of one dimensional CA is introduced. It is shown that (CA,≤) admits a global minimum and that on the bottom of (CA,≤) very natural equivalence classes are located. These classes remind us the first two well-known Wolfram ones because they capture global (or dynamical) properties as nilpotency or periodicity. Non trivial properties as the undecidability of ≤ and the existence of bounded infinite chains are also proved. Finally, it is shown that (CA,≤) admits no maximum. This result allows us to conclude that, in a “grouping sense”, there is no universal CA.


Cellular Automaton Turing Machine Infinite Chain Canonical Order Firing Squad 
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Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Jacques Mazoyer
    • 1
  • Ivan Rapaport
    • 1
  1. 1.LIP-École Normale Supérieure Supérieure de LyonLyon Cedex 07France

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