Replicator

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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.

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).

Replicator in HighLife
RLE: here .
HighLife replicator animation.

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.

Two-dimensional replicators will either be a square (if the fastest travelling corner of the mass of replicators is moving either orthogonally or diagonally), or a rhombus (if one of the corners is moving in an oblique 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.[2] 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.[3]

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.[4]

A universal computer and constructor is known to exist also for B35/S236.[5] Therefore, replicators exist in that rule.

See Also

References

  1. David Eppstein. "Cellular Automata: Replicators". Retrieved on February 17, 2016.
  2. Gardner, Martin (1983), Wheels, Life, and Other Mathematical Amusements, W.H. Freeman, pp. 226-227
  3. Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (2004), Winning Ways for Your Mathematical Plays, 4 (2nd ed.), pp. 927-961
  4. Dave Greene (November 23, 2013). "Re: Geminoid Challenge". Retrieved on November 24, 2013.
  5. David Eppstein. "B35/S236". Retrieved on February 9, 2016.

External Links

  • Replicator at Eric Weisstein's Treasure Trove of Life