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Journal of Algebraic Combinatorics

, Volume 33, Issue 1, pp 11–35 | Cite as

Dynamics groups of asynchronous cellular automata

  • Matthew Macauley
  • Jon McCammond
  • Henning S. Mortveit
Open Access
Article

Abstract

We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is π-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible \(2^{2^{3}}=256\) cellular automaton rules are π-independent. In the article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research.

Keywords

Sequential dynamical systems Cellular automata Update order Dynamics groups Coxeter groups Periodic points Fibonacci numbers Lucas numbers 

References

  1. 1.
    Hansson, A.Å., Mortveit, H.S., Reidys, C.M.: On asynchronous cellular automata. Adv. Complex Syst. 8, 521–538 (2005) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Macauley, M., McCammond, J., Mortveit, H.S.: Order independence in asynchronous cellular automata. J. Cell. Autom. 3, 37–56 (2008) MATHMathSciNetGoogle Scholar
  3. 3.
    Martin, O., Odlyzko, A.M., Wolfram, S.: Algebraic properties of cellular automata. Commun. Math. Phys. 93, 219–258 (1984) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Mortveit, H.S., Reidys, C.M.: An Introduction to Sequential Dynamical Systems. Springer, Berlin (2007) Google Scholar
  5. 5.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Published electronically at: http://www.research.att.com/~njas/sequences/
  6. 6.
    Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge University Press, Cambridge (1997) MATHGoogle Scholar
  7. 7.
    Weir, A.J.: Sylow p-subgroups of the general linear group over finite fields of characteristic p. Proc. Am. Math. Soc. 6, 454–464 (1955) MATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Matthew Macauley
    • 1
  • Jon McCammond
    • 2
  • Henning S. Mortveit
    • 3
  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  3. 3.Department of MathematicsVirginia TechBlacksburgUSA

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