Experimental Mathematics

Tight bounds on periodic cell configurations in Life

David J. Buckingham and Paul B. Callahan

Abstract

Periodic configurations, or oscillators, occur in many cellular automata. In an oscillator, repeated applications of the automaton rules eventually restore the configuration to its initial state. This paper considers oscillators in Conway's Life; analogous techniques should apply to other rules. Three explicit methods are presented to construct oscillators in Life while guaranteeing certain complexity bounds, leading to the existence of \begin{itemize} \item an infinite sequence $K_n$ of oscillators of periods $n=58$, 59, 60, \dots \ and uniformly bounded population, and \item an infinite sequence $D_n$ of oscillators of periods $n=58$, 59, 60, \dots \ and diameter bounded by $b \sqrt{\log n}$, where $b$ is a uniform constant. \end{itemize} The proofs make use of the first explicit example of a stable glider reflector in Life, solving a longstanding open question about this cellular automaton.

Article information

Source
Experiment. Math., Volume 7, Issue 3 (1998), 221-241.

Dates
First available in Project Euclid: 14 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1047674205

Mathematical Reviews number (MathSciNet)
MR1676762

Zentralblatt MATH identifier
1002.92501

Subjects
Primary: 92B20: Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx]
Secondary: 68Q80: Cellular automata [See also 37B15]

Citation

Buckingham, David J.; Callahan, Paul B. Tight bounds on periodic cell configurations in Life. Experiment. Math. 7 (1998), no. 3, 221--241. https://projecteuclid.org/euclid.em/1047674205


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