# Replicator

*For the cellular automaton of the same name, see Replicator (CA).*

A **replicator** is any pattern that produces an arbitrary number of copies of itself. There is currently no precise definition.

## Contents

## Natural replicators

Replicators occur naturally in some cellular automata.^{[1]} Possibly the most well-known example in a Life-like cellular automaton is the simple replicator in HighLife (B36/S23), which repeatedly copies itself along a diagonal line every 12 generations according to a one-dimensional parity rule (Wolfram Rule 90).

Other rules with replicators include tHighLife.

In Life, the pre-pulsar produces an exact copy of itself after 15 generations. However, these duplicated copies then react with each other to form the pulsar, instead of replicating again. The pre-pulsar is therefore generally not considered a true replicator. The skewed variant of the pre-pulsar, and other pre-pulsar-like patterns of consistent spacing, also copy themselves after 15 generations, and also cannot replicate infinitely.

Parity-rule replicators are common in B1 rules. For example, the pattern consisting of a single alive cell is a replicator in many B1 rules such as Gnarl (B1/S1). In the parity-rule Life-like cellular automata Replicator rule (B1357/S1357) and Fredkin rule (B1357/S02468), every pattern is a replicator.

Replicators can alternatively be used to create spaceships by using objects to delete one replicated part, such as HighLife's bomber and the pre-pulsar spaceship in normal Life. Shuttles can also be created by subduing replicators, as seen in Pre-pulsar shuttle 28 and Pre-pulsar shuttle 29.

While the vast majority of one-dimensional replicators follow Rule 90, ones bound by other rules are known, such as Rule 150^{[2]}^{[3]} and Rule 1208925819614629174771760^{[4]}.

Two-dimensional replicators will either be a square (if the fastest travelling corner of the mass of replicators is moving either orthogonally or diagonally), a rectangle (if some of the replicators are moving in an oblique direction), or a rhombus (if some of the replicators are moving faster than the others in a certain direction).

## Construction-based replicators

John von Neumann proved the existence of a pattern of about 200,000 cells that self-replicates in a 29-state von Neumann neighbourhood cellular automaton.^{[5]} In particular, the cellular automaton supports both universal computation (by simulating a Turing machine) and universal construction and so a universal computer, connected to a universal constructor, would self-replicate when given a blueprint of itself.

In 1982, Berlekamp, Conway, and Guy proved that Life supports universal computation and universal construction, and thus that there exist self-replicating machines in Life.^{[6]}

Prior to 2013, no explicit examples of construction-based replicators in Life were known. However, on November 23, 2013, Dave Greene constructed an explicit example by feeding a universal slow-salvo constructor (without any underlying universal computer) a tape of gliders that functions as a recipe for the constructor's own construction.^{[7]}

A universal computer and constructor is likely to exist also for B35/S236, but no specific examples have been constructed.^{[8]} Therefore, replicators presumably exist in that rule, as in many other rules that appear to meet the requirements for construction universality.

## Classifying replicators

### Okanishi's classifications

A classification scheme for two-dimensional replicators was proposed by Luka Okanishi in December 2016:^{[9]}

- Successful replicators:
- Class S (strict); the replicated patterns appears 1, 2, 4, 4, 4, 8, 16, 8, 4, 8, ... times ( A173531).
- Class R (rectangular); same as above, although all copies of the replicator must be present.
- Class Q (quadruple); the replicated pattern appears 1, 4, 4, 16, 4, 16, 16, 64, 4, 16, ... times ( A102376).
- Class X (fourfold XOR): 1, 4, 8, 16, 16, 32, 32, 64, 32, 64, ... times ( A189007).
- Class U (unusual): all others.

- Failed replicators:
- Class A (almost)
- Class D (dirty)
- Class F (spacefilling)

### XOR-based classifications

An alternative classification for replicators was proposed in November 2018, based on underlying XOR rules.^{[10]}

## See also

## References

- ↑ David Eppstein. "Cellular Automata: Replicators". Retrieved on February 17, 2016.
- ↑ http://www.conwaylife.com/forums/viewtopic.php?f=11&t=2238&p=51593#p51593
- ↑ http://www.conwaylife.com/forums/viewtopic.php?f=11&t=3453#p61459
- ↑ http://www.conwaylife.com/forums/viewtopic.php?f=11&t=3453#p61463
- ↑ Gardner, Martin (1983),
*Wheels, Life, and Other Mathematical Amusements*, W.H. Freeman, pp. 226-227 - ↑ Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (2004),
*Winning Ways for Your Mathematical Plays*,**4**(2nd ed.), pp. 927-961 - ↑ Dave Greene (November 23, 2013). "Re: Geminoid Challenge". Retrieved on November 24, 2013.
- ↑ David Eppstein. "B35/S236". Retrieved on February 9, 2016.
- ↑ Luka Okanishi (AbhpzTa). 2D Replicator Classes (discussion thread) at the ConwayLife.com forums
- ↑ http://www.conwaylife.com/forums/viewtopic.php?f=11&t=3580

## External links

- Replicator at Wikipedia

- Replicator at the Life Lexicon