OCA:Replicator
| Replicator | ||
|
||
| Rulestring | 1357/1357 B1357/S1357 |
|
|---|---|---|
| Rule integer | 87210 | |
| Character | Explosive | |
| Black/white reversal | B02468/S02468 | |
- 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.
Replication Property
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.[1]
Replicator 2
| Fredkin | |
| Rulestring | 02468/1357 B1357/S02468 |
|---|---|
| Rule integer | 174762 |
| Character | Explosive |
| Black/white reversal | B1357/S02468 |
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.[1]
References
- ↑ 1.0 1.1 muzik (November 1, 2018). New method of classifying two-dimensional replicators (discussion thread) at the ConwayLife.com forums
External links
- Replicator (discussion thread) at the ConwayLife.com forums
