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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations