Communications in Mathematical Physics

, Volume 118, Issue 4, pp 569–590 | Cite as

Cylindrical cellular automata

  • Erica Jen


A one-dimensional cellular automaton with periodic boundary conditions may be viewed as a lattice of sites on a cylinder evolving according to a local interaction rule. A technique is described for finding analytically the set of attractors for such an automaton. Given any one-dimensional automaton rule, a matrixA is defined such that the number of fixed points on an arbitrary cylinder size is given by the trace ofA n , where the powern depends linearly on the cylinder size. More generally, the number of strings of arbitrary length that appear in limit cycles of any fixed period is found as the solution of a linear recurrence relation derived from the characteristic equation of an associated matrix. The technique thus makes it possible, for any rule, to compute the number of limit cycles of any period on any cylinder size. To illustrate the technique, closed-form expressions are provided for the complete attractor structure of all two-neighbor rules. The analysis of attractors also identifies shifts as a basic mechanism underlying periodic behavior. Every limit cycle can be equivalently defined as a set of strings on which the action of the rule is a shift of sizes/h; i.e., each string cyclically shifts bys sites inh iterations of the rule. The study of shifts provides detailed information on the structure and number of limit cycles for one-dimensional automata.


Periodic Boundary Periodic Boundary Condition Characteristic Equation Cellular Automaton Quantum Computing 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Martin, O., Odlyzko, A., Wolfram, S.: Algebraic properties of cellular automata. Commun. Math. Phys.93, 219–259 (1984)Google Scholar
  2. 2.
    Guan, P., He, Y.: Exact results for deterministic cellular automata. J. Stat. Phys.43, 463 (1986)Google Scholar
  3. 3.
    Guan, P., He, Y.: Upper bound on the number of cycles in border-decisive cellular automata. Complex Systems1, 181 (1987)Google Scholar
  4. 4.
    Wolfram, S.: Theory and applications of cellular automata. Singapore: World Scientific 1986Google Scholar
  5. 5.
    Jen, E.: Global properties of cellular automata. J. Stat. Phys.43, 219 (1986)Google Scholar
  6. 6.
    Jen, E.: Linear cellular automata and recurring sequences in finite fields. Los Alamos National Laboratory Report LA-UR-87-2026. Commun. Math. Phys. (to appear)Google Scholar
  7. 7.
    Jen, E.: Invariant strings and pattern-recognizing properties of one-dimensional cellular automata. J. Stat. Phys.43, 243 (1986)Google Scholar
  8. 8.
    Hedlund, G.: Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory3, 320 (1969)Google Scholar
  9. 9.
    Lind, D. A.: Applications of ergodic theory and sofic systems to cellular automata in reference [1]Google Scholar
  10. 10.
    Boyle, M.: Constraints on the degree of a sofic homomorphism and the induced multiplication of measures on unstable sets. Israel J. Math.53, 52 (1986)Google Scholar
  11. 11.
    Jen, E.: Scaling of preimages in cellular automata. Complex Systems6, 1045–1062 (1987)Google Scholar
  12. 12.
    Wolfram, S.: Random sequence generation by cellular automata. Adv. Appl. Math.7, 123 (1986)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Erica Jen
    • 1
  1. 1.Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA

Personalised recommendations