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Communications in Mathematical Physics

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

Cylindrical cellular automata

  • Erica Jen
Article

Abstract

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.

Keywords

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.

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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

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

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