A parity rule replicator is a replicator whose evolution can be modeled via a parity rule. These are the patterns which are most commonly referred to as "replicators" when discussing cellular automata.
B3/S23 is known to support parity-rule replicators via constructs such as the 0E0P metacell, however no explicit finite-period elementary examples have been discovered so far. However, Life does support two types of parity rule replicator on a bounded grid with one finite axis and one infinite axis.
The first case is a simple superstring, operating on an infinite cylinder and travelling at the speed of light. While such a string emulates rule 22, rather than rule 90, it can still be proven to be strong if only every fourth cell is considered for a starting pattern, with each of the other three remaining dead; any configuration satisfying this constraint will replicate indefinitely.
The second case is the traffic light sequence, as run on an infinite bounded plane of height 5 or infinite cylinder of height 6. The pre-pulsar is a known example of a failed replicator; provided an infinite universe, it can replicate infinitely, at a speed of 35c/200.
By iterating their evolution three times and deleting one of the inner two copies, we can demonstrate both replicate strongly, as this specific arrangement of three replicators will reappear twice eight more iterations in, then sixteen, thirty-two, and so on:
Many examples of parity rule replicators exist in other rules, notably the 2c/12 diagonal case from HighLife, and the 4c/11 orthogonal replicator from tHighLife.
Classification
There are many parity rule replicators which have been documented. For the case of cellular automata running on a two-dimensional square tiling, they are often classed as either one-dimensional or two-dimensional, depending on the space they fill out as they expand.
One-dimensional
The HighLife and tHighLife replicators are notable examples of one-dimensional or linear replicators, in which a line is filled out.
tHighLife's replicator[1] (click above to open LifeViewer)
In these two cases, and indeed in most cases, the replicators will remain on the same, stationary line at all times, however this need not be the case; it is possible for the line on which all of the replicators lie to itself be in motion. Such replicators are classified as "migratory replicators".
It is also possible for replicators to move at different speeds in different directions, resulting in a directionally-biased, though still valid, replicator.
This pattern only replicates to the right (click above to open LifeViewer)
It is even possible for both ends to replicate at different periods; while there will never be a generation in which the entire universe will be composed solely of copies of the replicator after T=0, they still replicate strongly. Instead of annihilation happening at once, the reaction will instead burn from one end to the other, often faster than lightspeed. These are referred to as "Lorentz-shifted" replicators.[2]
Replicates at 5c/10 to the left and 8c/16 to the right[3] (click above to open LifeViewer)
Notation
All known one-dimensional, two-state, two-copy replicators in square-grid rules can be notated in full using two three-digit displacements, one for each of the two copies of the replicator:
(x1,y1)c/p1,(x2,y2)c/p2
The following table describes the possible displacement types, provided that one copy of the replicator is moving down and to the right (for orthogonal displacement, only to the right).
Class
Shorthand
Full notation
orthogonal replicator
xc/p
(x,0)c/p,(-x,0)c/p
diagonal replicator
xc/p
(x,x)c/p,(-x,-x)c/p
oblique replicator
(x,y)c/p
(x,y)c/p,(-x,-y)c/p
orthogonal biased replicator
x1c/p,x2c/p
(x1,0)c/p,(x2,0)c/p
diagonal biased replicator
x1c/p,x2c/p
(x1,x1)c/p,(x2,x2)c/p
orthogonal migratory replicator
(x,y)c/p,(-x,y)c/p
diagonal migratory replicator
(x,y)c/p,(y,x)c/p
general case
(x1,y1)c/p1,(x2,y2)c/p2
Two-dimensional
A two-dimensional or quadratic replicator expands in four directions, tracing out a planar figure (most often a square, less commonly a rectangle or rhombus).
Most commonly a square is traced out. For replicators that travel orthogonally, this square will have its four vertices oriented towards the cardinal directions (this orientation of a square is often called a "diamond" by non-mathematicians). For replicators that travel diagonally, the faces are oriented towards the cardinal directions. Oblique quadratic replicators exist that trace out square shapes of intermediate orientation.
An (8,4)c/20 replicator[6] (click above to open LifeViewer)
Two-dimensional replicators which create rectangular or rhombic fields are considerably rarer:
Rectangular cases arise where the individual units move in an oblique direction, but the overall arrangement has a D4 static symmetry rather than the rotational symmetry which yields a partially rotated square.
Rhombic cases arise when the units move faster in one direction than another, either both orthogonally or both diagonally.
A rhombic replicator: 15c/126 and 7c/126 diagonal[8] (click above to open LifeViewer)
Quadratic replicators can be notated using four displacements, one for each copy produced. However, there has been no need to use all four to describe a single replicator so far.
Other cases
Replicators that produce numbers of copies other than two or four also exist. A handful of cases are known of replicators producing three copies: if all of these are collinear, it emulates Rule 150, otherwise a Sierpiński triangle is traced out.[9]
The rules Replicator and Fredkin cause all configurations to produce eight or nine copies of themselves, respectively. The von Neumann neighbourhood equivalent of Replicator, B13/S13V, is the rule followed by most known two-dimensional replicators, whereas Fredkin's variant, B13/S024V, does not have any other known rules containing patterns emulating it in a strong way.
Replicators which use a modulo higher than 2 also exist. The first finite, strong example of a modulo-3 linear replicator was discovered on 29 March 2024.[10]