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New Constructions in Cellular Automata$
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David Griffeath and Cristopher Moore

Print publication date: 2003

Print ISBN-13: 9780195137170

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195137170.001.0001

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Replicators and Larger-than-Life Examples

Replicators and Larger-than-Life Examples

(p.119) Replicators and Larger-than-Life Examples
New Constructions in Cellular Automata

Kellie Michele Evans

Oxford University Press

After watching a substantial number of cellular automaton dynamics generated by rules containing suitable ingredients, eventually a particular time-dependent pattern catches the eye. A configuration of occupied sites makes copies of itself, then the copies make copies of themselves, and these copies move toward one another and also toward the boundaries of the evolution. This continues as long as there is room for the evolution. When the innermost copies collide, they annihilate one another. Meanwhile, the outermost copies continue to reproduce, provided that no occupied sites from the outside impede. The pattern repeats, ad infinitum. We first saw this kind of evolution, which we call a replicator, in our studies of the Larger-than-Life (LtL) family of cellular automaton (CA) rules. The first replicators we found were all in the same region of LtL space. We thought an intrinsic property of this specific region was necessary for the existence of a replicator. However, we began seeing similar configurations, with slight variations, in many different subregions of LtL space. Then we learned of the range 1 HighLife cator. However, we began seeing similar configurations, with slight variations, in become quite famous. We saw more examples on Christopher Langton’s computer at the Santa Fe Institute in 1995; this convinced us that the behavior was not exclusive to LtL-like rules. Since then, new replicators have been discovered for a variety of CA rules. In this chapter, we define a replicator using an axiomatic approach and prove various theorems that follow from the axioms. We also present a collection of Larger-than-Life replicator examples, HighLife’s famous example, and propositions that generalize several of the LtL examples. We will begin by presenting a collection of Larger-than-Life replicator examples, but first let us define the family of Larger-than-Life update rules. Larger-than-Life (LtL) is a four-parameter family of two-state cellular automaton rules. The four parameters are the upper and lower bounds of the birth and survival intervals. At each time t, each site x∊ Zd is either live or dead. We think of a live site as being in state 1 and a dead site as being in state 0.

Keywords:   arrow replicator, bijections for replicators, centroids of replicator tiles, dimensions of replicators, global dynamics of replicators, nucleation models, one-dimensional replicators, projection maps, range

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