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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)

Abstract

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.

Keywords

Cellular Automaton Turing Machine Infinite Chain Canonical Order Firing Squad 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AC87]
    Albert J., Culik II K.: A simple universal cellular automaton and its one-way and totalisting version. Complex Systems 1 (1987) 1–16.MATHMathSciNetGoogle Scholar
  2. [CPY89]
    Culik II K., Pachl J., Yu S.: On the limit sets of cellular automata. SIAM J. Computing 18 (1989) 831–842.MATHCrossRefMathSciNetGoogle Scholar
  3. [Kar92]
    Kari J.: The nilpotency problem of one-dimensional cellular automata. SIAM J. Computing 21 (1992) 571–586.MATHCrossRefMathSciNetGoogle Scholar
  4. [Mar94]
    Martin B.: A universal cellular automaton in quasi-linear time and its s-m-n form. Theoretical Computer Science 123(2) (1994) 199–237.MATHCrossRefMathSciNetGoogle Scholar
  5. [Moo64]
    Moore E.F.: Sequential machines, selected papers. Addison Wesley Reading Mass. (1964) 213–214.MATHGoogle Scholar
  6. [MR92]
    Mazoyer J., Reimen N.: A linear speed-up theorem for cellular automata. Theoretical Computer Science 101 (1992) 59–98.MATHCrossRefMathSciNetGoogle Scholar
  7. [Smi71]
    Smith III A.R.: Simple computation-universal cellular spaces. Journal ACM 18 (1971) 339–353.MATHCrossRefGoogle Scholar
  8. [Wol84]
    Wolfram S.: Universality and complexity in cellular automata. Physica D 10 (1984) 1–35.CrossRefMathSciNetGoogle Scholar

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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