- For replicators as a general concept, see Replicator.
Replicator is a Life-like cellular automaton where a cell survives or is born if there are an odd number of neighbors. It is one of two Life-like Fredkin replicator rules. Under this ruleset, every pattern self-replicates; furthermore, every pattern will eventually produce an arbitrary number of copies of itself, all arbitrarily far away from each other.
The replication property follows from a property of Fredkin replicator rules, in which patterns can be modelled as an infinite grid whose entries are elements of the cyclic group Zn, where n is the number of states. In this case, n=2, 0 is the off state, and 1 is the on state. The rule can be expressed equivalently as assigning a new value to a cell by summing all neighboring cells. Since Zn is an abelian group, addition is commutative and associative; hence applying the rule to a sum (XOR) of two patterns is the same as summing the two patterns after the rule is applied to each one.
Thus, to find the nth generation of a pattern, it suffices to XOR together the nth generation of each of the single cells which compose the pattern. A single cell is a replicator. More specifically, an on-cell at (0,0) at time 0 will produce, at time 2n, 8 on-cells at all positions (b,c) where b and c are any of -2n, 0, or 2n, and b and c are not both 0 (this can be proven using induction on n). When n is large enough, the 8 cells are arbitrarily far away, and thus, for a pattern, the XOR sum of the (2n)th generation of each of its cells forms the pattern's (2n)th generation, 8 copies of the original. Repeating this process produces an arbitrary number of copies, all at arbitrary distance.
The replication habit seen can be classified as (150,90), where the classification (E,C) refers to the one-dimensional rules simulated by the edge of the replicator and the diagonal of the square.
Replicator 2, also known as Fredkin, is a related totalistic Fredkin replicator rule, where a cell survives or is born if the number of neighbors, including itself, is odd. It has the totalistic rulestring 13579. Like Replicator, every pattern self-replicates, although according to 150,150 instead.
- muzik (November 1, 2018). New method of classifying two-dimensional replicators (discussion thread) at the ConwayLife.com forums
- Replicator (discussion thread) at the ConwayLife.com forums