Smallest Linear Replicators Supporting Specific Speeds and Habits

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muzik
Posts: 3774
Joined: January 28th, 2016, 2:47 pm
Location: Scotland

Smallest Linear Replicators Supporting Specific Speeds and Habits

Post by muzik » January 23rd, 2020, 12:31 pm

Over the history of OCA research there's been a multitude of interesting linear and quadratic replicators discovered. This project is to serve as a means to consolidate all of the discovered replicators.

Currently this project has the same constraints as the 5s project - only outer-totalistic or isotropic non-totalistic range-1 2-state rules, with alternating and B0 also excluded, however this may be expanded in the future.

In addition to a displacement vector and period, replication habit is also something that must be taken into account for linear replicators - they preferably should act as a 0E0P-like unit cell of the rule they replicate, so that any amount of correctly spaced copies still function like an isolated one would. The notation is as follows:

(x1,y1,x2,y2)c/p kK Ff U
where
x1 = horizontal displacement of a replicator section
y1 = vertical displacement of a replicator section
x2 = overall horizontal displacement of the replicator
y2 = overall vertical displacement of the replicator
p = period
K = number of states in 1D parity rule
F = number of copies present at generation p
U = present if the replicator functions as a unit cell of a 1D parity rule - removing copies of the replicator will still result in it replicating in the same way in future
(this notation may need some refining though)

Here are some examples:

(1,0,0,0)c/1 k2 2f U

Code: Select all

x = 1, y = 2, rule = B2a/S
o$o!

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x = 15, y = 2, rule = B2a/S
obobobobobo3bo$obobobobobo3bo!
(4,0,0,0)c/4 k2 3f U

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x = 1, y = 5, rule = B2a/S1e3eiy5i
o$o2$o$o!

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x = 57, y = 5, rule = B2a/S1e3eiy5i
o3bo7bo3bo7bo3bo3bo7bo3bo11bo$o3bo7bo3bo7bo3bo3bo7bo3bo11bo2$o3bo7bo3b
o7bo3bo3bo7bo3bo11bo$o3bo7bo3bo7bo3bo3bo7bo3bo11bo!
(4,0,0,0)c/11 k2 2f U

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x = 5, y = 3, rule = B36/S2-i34q
5o$5o$5o!

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x = 61, y = 3, rule = B36/S2-i34q
5o3b5o3b5o3b5o3b5o3b5o11b5o$5o3b5o3b5o3b5o3b5o3b5o11b5o$5o3b5o3b5o3b5o
3b5o3b5o11b5o!
(2,2,0,0)c/12 k2 2f U

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x = 5, y = 5, rule = B36/S23
2b3o$bo2bo$o3bo$o2bo$3o!

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x = 33, y = 33, rule = B36/S23
2b3o$bo2bo$o3bo$o2bo$3o3b3o$5bo2bo$4bo3bo$4bo2bo$4b3o3b3o$9bo2bo$8bo3b
o$8bo2bo$8b3o3b3o$13bo2bo$12bo3bo$12bo2bo$12b3o3b3o$17bo2bo$16bo3bo$
16bo2bo$16b3o3b3o$21bo2bo$20bo3bo$20bo2bo$20b3o4$30b3o$29bo2bo$28bo3bo
$28bo2bo$28b3o!
(5,0,0,0)c/10 k3 2f

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x = 3, y = 4, rule = B3-e4ny56i7e/S2-ck3r4-nt5jq6
bo$3o$3o$bo!
(7,0,0,0)c/14 k4 2f

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x = 4, y = 3, rule = B2ei3-akny4acnqrw5aqy6aen/S1c2-ik3ack4cjknrw5acjk6c7
b2o$o2bo$b2o!
(7,0,0,0)c/14 k5 2f

Code: Select all

x = 4, y = 3, rule = B2e3-cnqy4rwyz5y6ace7c/S1c2ac3aiq4eiknrwy5cn6ekn7c8
b2o$o2bo$b2o!
(2,0,2,2)c/2 k2 2f U

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 x = 2, y = 2, rule = B2a4w/S2e3j
bo$2o!
There's a lot of these floating around, so post any more you come across below!
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!

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GUYTU6J
Posts: 906
Joined: August 5th, 2016, 10:27 am
Location: 中国

Re: Smallest Linear Replicators Supporting Specific Speeds and Habits

Post by GUYTU6J » January 23rd, 2020, 8:39 pm

Code: Select all

x = 3, y = 8, rule = B3-er4city/S23-a4city
2bo$obo$b2o3$b2o$obo$2bo!
More is in the thread "Rules with interesting replicators", including this one by 2718281828:

Code: Select all

x = 3, y = 2, rule = B2-a3ckq4ektwyz5-in6-i7e8/S1c3cijnr4ejrw5eq6ai
bo$obo!
I'd like to know how these three will be classified

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x = 3, y = 3, rule = B2k3-qy4c/S2-n3ijkqr4ir6ae
bo$b2o$o!

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x = 4, y = 5, rule = B3aijn4i6n/S2-n3ijqr4ikr
3o$o2bo$3bo$2b2o$bo!

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x = 5, y = 4, rule = B2kn3-kr4iz5cr/S2-cn3ijkqr4eiqr5ckr
4bo$2bo$3o$bo!
An oblique one:

Code: Select all

x = 5, y = 4, rule = B3aijqr4kr6n7e/S2-in3ijnqr4ir5i
3o$obo$o$b2obo!
Sorry but I prefer to contribute to cellular automata anonymously, so I made up a random string for my username.
-GUYTU6J

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