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Discrete Baker Transformation for Binary Valued Cylindrical Cellular Automata

  • Burton Voorhees
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4173)

Abstract

Recently, the discrete baker transformation has been defined for linear cellular automata acting on multi-dimensional tori with alphabet of prime cardinality. Here we specialize to binary valued cylindrical cellular automata and generalizing the discrete baker transformation to non-linear rules. We show that for a cellular automaton, defined on a cylinder of size n = 2 k m with m odd, the equivalence classes of rules that map to the same rule under the discrete baker transformation fall into equivalence classes labeled by the set of 2 m cellular automata defined on a cylinder of size m. We also derive the relation between the state transition diagram of a cellular automata rule and that of its baker transformation and discuss cycle periods of the baker transformation for odd n.

Keywords

Cellular Automaton Additive Rule State Transition Diagram Bruijn Sequence Rule Component 
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References

  1. 1.
    Bulitko, V., Voorhees, B., Bulitko, V.: Discrete baker transformation for linear cellular automata analysis. Journal of Cellular Automata 1(1) (to appear, 2006)Google Scholar
  2. 2.
    Jen, E.: Cylindrical cellular automata. Communications in Mathematical Physics 118, 569–590 (1988)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Voorhees, B.: Computational Analysis of One Dimensional Cellular Automata. World Scientific, Singapore (1995)CrossRefGoogle Scholar
  4. 4.
    Golomb, S.W.: Shift Register Sequences. Holden-Day, San Francisco (1967)MATHGoogle Scholar
  5. 5.
    Ralston, A.: De Bruijn sequences—a model example of the interaction of discrete mathematics and computer science. Mathematics Magazine 55, 131–143 (1982)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bulitko, V.: Discrete baker transformation and estimation of heights and cycle lengths for additive cellular automata (unpublished manuscript) (2004)Google Scholar
  7. 7.
    Martin, O., Odlyzko, A.M., Wolfram, S.: Algebraic properties of cellular automata. Communications in Mathematical Physics 93, 219–258 (1984)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Burton Voorhees
    • 1
  1. 1.Center for Science – Athabasca UniversityAtahbascaCanada

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