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New method of classifying two-dimensional replicators

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New method of classifying two-dimensional replicators

Postby muzik » November 1st, 2018, 7:01 pm

We first set the requirements that the replicator be a sawtooth, hereby excluding U-class, X-class, and failed replicators, and that it must be a square, rectangle or rhombus.

These replicators can be classified using the notation (E,C), where:

E - Edge rule; the one-dimensional rule which is simulated by the edge of the replicator

C - Core rule; the one-dimensional rule found on the diagonal of the square.
x = 46, y = 17, rule = bs012345678History
24.A$23.3A$22.A3.A$21.2A.A.2A$20.A.A.A.A.A$3C9.3C4.3A5.3A$C11.C.C3.A
5.A5.A$C3.3C.3C.3C2.2C.2C.3C.2C.CE$C3.C.C.C3.C5.A5.A5.E$3C.3C.C3.3C4.
3A5.2AE$20.A.A.A.A.E$21.2A.A.AE$22.A3.E4.3E3.E5.3E$23.2AE5.E5.E5.E.E$
24.E6.3E.3E.3E.3E$31.E3.E.E3.E.E$31.3E.3E.3E.3E!



Through these, we can currently identify four classes of replicators:

90,90 (includes all S, R and Q-class replicators):
x = 1, y = 1, rule = B1e/S
o!


90,150
x = 1, y = 1, rule = B1e/S04e
o!


150,90
x = 1, y = 1, rule = B13ci/S
o!


150,150
x = 1, y = 1, rule = B13i/S08
o!


0-fold and 1-fold replication rules have been excluded.

No XOR rules with higher than range 1 have been included (such as rules 1721342310 and 2523490710) as no replicators have been found that support these.
waiting for apgsearch to support one-dimensional rules
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Re: New method of classifying two-dimensional replicators

Postby muzik » November 1st, 2018, 7:15 pm

Examples of replicators:


90,90:

step2
x = 2, y = 2, rule = B2a/S3a4a
2o$2o!


step18
x = 3, y = 5, rule = B2e3aijkq4ac7c/S2-an3-a
bo$obo$3o$obo$bo!


step24
x = 7, y = 7, rule = B3578/S23
3o$obo$3o2$4b3o$bo2bobo$4b3o!


90,150:

step2
x = 2, y = 2, rule = B2a3i4w/S3a4a7c
2o$2o!
waiting for apgsearch to support one-dimensional rules
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Re: New method of classifying two-dimensional replicators

Postby Layz Boi » November 1st, 2018, 11:14 pm

Why exclude so many of them?

Here are some replicators that have an "Edge rule" and "Core rule." However, they are not saw-tooths:

x = 512, y = 512, rule = B12c4e/S1e2c4c
ooo$obo$ooo$!


x = 512, y = 512, rule = B1e2i4te/S
bo$obo$bo$!
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Re: New method of classifying two-dimensional replicators

Postby muzik » November 2nd, 2018, 5:49 am

These classes also match up with "Replicator"-type rules:

x = 16, y = 16, rule = B1e2ak3einqy4jnry5einqy6ak7e/S1e2ak3einqy4jnry5einqy6ak7e
3b2ob2obo3bobo$bo3b2o4bobo$o3b3o2bob2obo$3bo6b2o2b2o$bob3ob2obob2obo$o
b2ob2o2bob2ob2o$obobo2b2o2b2obo$b2o2b2ob5obo$bo4b2obobo2bo$2o2b2o2b2ob
obo$3bo2bo3b3o2bo$3o2bobo2b2obobo$2b2obob2o3b2o$bo2b3o2bo2b4o$2o4b2obo
bob3o$8b4obobo!


x = 16, y = 16, rule = B1e2ak3einqy4jnry5einqy6ak7e/S01c2-ak3acjkr4-jnry5acjkr6-ak7c8
3b2ob2obo3bobo$bo3b2o4bobo$o3b3o2bob2obo$3bo6b2o2b2o$bob3ob2obob2obo$o
b2ob2o2bob2ob2o$obobo2b2o2b2obo$b2o2b2ob5obo$bo4b2obobo2bo$2o2b2o2b2ob
obo$3bo2bo3b3o2bo$3o2bobo2b2obobo$2b2obob2o3b2o$bo2b3o2bo2b4o$2o4b2obo
bob3o$8b4obobo!


x = 16, y = 16, rule = B1357/S1357
3b2ob2obo3bobo$bo3b2o4bobo$o3b3o2bob2obo$3bo6b2o2b2o$bob3ob2obob2obo$o
b2ob2o2bob2ob2o$obobo2b2o2b2obo$b2o2b2ob5obo$bo4b2obobo2bo$2o2b2o2b2ob
obo$3bo2bo3b3o2bo$3o2bobo2b2obobo$2b2obob2o3b2o$bo2b3o2bo2b4o$2o4b2obo
bob3o$8b4obobo!


x = 16, y = 16, rule = B1357/S02468
3b2ob2obo3bobo$bo3b2o4bobo$o3b3o2bob2obo$3bo6b2o2b2o$bob3ob2obob2obo$o
b2ob2o2bob2ob2o$obobo2b2o2b2obo$b2o2b2ob5obo$bo4b2obobo2bo$2o2b2o2b2ob
obo$3bo2bo3b3o2bo$3o2bobo2b2obobo$2b2obob2o3b2o$bo2b3o2bo2b4o$2o4b2obo
bob3o$8b4obobo!


It is possible to see such behaviour in some S-class replicators:
x = 5, y = 6, rule = B1e2ak3einqy4jnry5einqy6ak7e/S1e2ak3einqy4jnry5einqy6ak7e
2bo$3bo$obobo$bo$2bo$3bo!


x = 47, y = 55, rule = B3578/S23
24b3o$24bobo$24b3o2$28b3o$25bo2bobo$28b3o2$32b3o$32bobo$32b3o2$36b3o$
33bo2bobo$36b3o2$24b3o13b3o$24bobo2bo10bobo$24b3o13b3o2$28b3o13b3o$28b
obo10bo2bobo$28b3o13b3o2$3o13b3o$obo2bo10bobo$3o13b3o2$4b3o13b3o$4bobo
10bo2bobo$4b3o13b3o2$8b3o$8bobo2bo$8b3o2$12b3o$12bobo$12b3o2$16b3o$16b
obo2bo$16b3o2$20b3o$20bobo$20b3o2$24b3o$24bobo2bo$24b3o2$28b3o$28bobo$
28b3o!
waiting for apgsearch to support one-dimensional rules
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Re: New method of classifying two-dimensional replicators

Postby muzik » November 5th, 2018, 9:18 am

What i believe to be the first five iterations of the 2523490710,1721342310 replication habit, as seen in R2B(1,3,5,7,9,11,13,15,17,19,21,23)/S(1,3,5,7,9,11,13,15,17,19,21,23):

x = 17, y = 129, rule = W0
8bo28$6b5o$6b5o$6b2ob2o$6b5o$6b5o24$4bobobobobo2$4bobobobobo2$4bobo3bo
bo2$4bobobobobo2$4bobobobobo20$2b2o2bobobo2b2o$2b2o2bobobo2b2o$4bobobo
bobo2$2b3ob5ob3o$6b5o$2b3ob2ob2ob3o$6b5o$2b3ob5ob3o2$4bobobobobo$2b2o
2bobobo2b2o$2b2o2bobobo2b2o16$o3bo3bo3bo3bo4$o3bo3bo3bo3bo4$o3bo7bo3bo
4$o3bo3bo3bo3bo4$o3bo3bo3bo3bo!


Is there anything that can simulate such rules?
waiting for apgsearch to support one-dimensional rules
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Re: New method of classifying two-dimensional replicators

Postby wwei23 » November 10th, 2018, 5:31 pm

From what I can tell:
E90 basically does this:
000
A00
0B0

Or this:
000
000
A0B

E150 does this:
000
000
ABC

C90 does this:
000
A0B
000

Or this:
A00
000
00B

C150 does this:
A00
0B0
00C

Or this:
000
ABC
000

So I think XOR 4 edge would do the following:
00000
00000
00000
00000
AB0CD

Center:
00000
00000
AB0CD
00000
00000

A0000
0B000
00000
000C0
0000D

I also think that the results of all the inputs are XORd.
And no, I am not going to go through all 2D range 2 neighborhoods.
UPDATE:
It's a bit more complicated, you OR the neighborhoods, then take the XOR of that.
x = 23, y = 10, rule = B1357S02468History
4.A3.3A3.A5.3A$.D2.AD3.D3.DA5.ACA$3A.A9.A5.3A5$A5.A$.C3.C$2.A.A!

So...this is probably what muzik is looking for.
x = 39, y = 13, rule = BS012345678History
13B.7B$B4.AB5.B.BA4.B$B4.AB5.B.B.A3.B5.D$B2.D.AB2.D2.B.B2.D2.B6.D4.7B
$B4.AB5.B.B3.A.B7.D3.B5AB$B4.AB5AB.B4.AB8.D2.B2A.2AB$13B.7B.9D.BA.D.A
B$B5ABA4.B.B4.AB8.D2.B2A.2AB$B5.BA4.B.B3.A.B7.D3.B5AB$B2.D2.BA.D2.B.B
2.D2.B6.D4.7B$B5.BA4.B.B.A3.B5.D$B5.BA4.B.BA4.B$13B.7B!
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Re: New method of classifying two-dimensional replicators

Postby muzik » November 14th, 2018, 7:15 pm

While I've yet to modify the classification system that integrates these, here are the rule tales for modulo-n replicators generated by FredkinModN-gen.py:

@RULE Fredkin_mod3_vonNeumann

@TREE

num_states=3
num_neighbors=4
num_nodes=13
1 0 0 0
1 1 1 1
1 2 2 2
2 0 1 2
2 1 2 0
2 2 0 1
3 3 4 5
3 4 5 3
3 5 3 4
4 6 7 8
4 7 8 6
4 8 6 7
5 9 10 11


@RULE Fredkin_mod4_vonNeumann

@TREE

num_states=4
num_neighbors=4
num_nodes=17
1 0 0 0 0
1 1 1 1 1
1 2 2 2 2
1 3 3 3 3
2 0 1 2 3
2 1 2 3 0
2 2 3 0 1
2 3 0 1 2
3 4 5 6 7
3 5 6 7 4
3 6 7 4 5
3 7 4 5 6
4 8 9 10 11
4 9 10 11 8
4 10 11 8 9
4 11 8 9 10
5 12 13 14 15


@RULE Fredkin_mod5_vonNeumann

@TREE

num_states=5
num_neighbors=4
num_nodes=21
1 0 0 0 0 0
1 1 1 1 1 1
1 2 2 2 2 2
1 3 3 3 3 3
1 4 4 4 4 4
2 0 1 2 3 4
2 1 2 3 4 0
2 2 3 4 0 1
2 3 4 0 1 2
2 4 0 1 2 3
3 5 6 7 8 9
3 6 7 8 9 5
3 7 8 9 5 6
3 8 9 5 6 7
3 9 5 6 7 8
4 10 11 12 13 14
4 11 12 13 14 10
4 12 13 14 10 11
4 13 14 10 11 12
4 14 10 11 12 13
5 15 16 17 18 19


@RULE Fredkin_mod6_vonNeumann

@TREE

num_states=6
num_neighbors=4
num_nodes=25
1 0 0 0 0 0 0
1 1 1 1 1 1 1
1 2 2 2 2 2 2
1 3 3 3 3 3 3
1 4 4 4 4 4 4
1 5 5 5 5 5 5
2 0 1 2 3 4 5
2 1 2 3 4 5 0
2 2 3 4 5 0 1
2 3 4 5 0 1 2
2 4 5 0 1 2 3
2 5 0 1 2 3 4
3 6 7 8 9 10 11
3 7 8 9 10 11 6
3 8 9 10 11 6 7
3 9 10 11 6 7 8
3 10 11 6 7 8 9
3 11 6 7 8 9 10
4 12 13 14 15 16 17
4 13 14 15 16 17 12
4 14 15 16 17 12 13
4 15 16 17 12 13 14
4 16 17 12 13 14 15
4 17 12 13 14 15 16
5 18 19 20 21 22 23


@RULE Fredkin_mod7_vonNeumann

@TREE

num_states=7
num_neighbors=4
num_nodes=29
1 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1
1 2 2 2 2 2 2 2
1 3 3 3 3 3 3 3
1 4 4 4 4 4 4 4
1 5 5 5 5 5 5 5
1 6 6 6 6 6 6 6
2 0 1 2 3 4 5 6
2 1 2 3 4 5 6 0
2 2 3 4 5 6 0 1
2 3 4 5 6 0 1 2
2 4 5 6 0 1 2 3
2 5 6 0 1 2 3 4
2 6 0 1 2 3 4 5
3 7 8 9 10 11 12 13
3 8 9 10 11 12 13 7
3 9 10 11 12 13 7 8
3 10 11 12 13 7 8 9
3 11 12 13 7 8 9 10
3 12 13 7 8 9 10 11
3 13 7 8 9 10 11 12
4 14 15 16 17 18 19 20
4 15 16 17 18 19 20 14
4 16 17 18 19 20 14 15
4 17 18 19 20 14 15 16
4 18 19 20 14 15 16 17
4 19 20 14 15 16 17 18
4 20 14 15 16 17 18 19
5 21 22 23 24 25 26 27


@RULE Fredkin_mod8_vonNeumann

@TREE

num_states=8
num_neighbors=4
num_nodes=33
1 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1
1 2 2 2 2 2 2 2 2
1 3 3 3 3 3 3 3 3
1 4 4 4 4 4 4 4 4
1 5 5 5 5 5 5 5 5
1 6 6 6 6 6 6 6 6
1 7 7 7 7 7 7 7 7
2 0 1 2 3 4 5 6 7
2 1 2 3 4 5 6 7 0
2 2 3 4 5 6 7 0 1
2 3 4 5 6 7 0 1 2
2 4 5 6 7 0 1 2 3
2 5 6 7 0 1 2 3 4
2 6 7 0 1 2 3 4 5
2 7 0 1 2 3 4 5 6
3 8 9 10 11 12 13 14 15
3 9 10 11 12 13 14 15 8
3 10 11 12 13 14 15 8 9
3 11 12 13 14 15 8 9 10
3 12 13 14 15 8 9 10 11
3 13 14 15 8 9 10 11 12
3 14 15 8 9 10 11 12 13
3 15 8 9 10 11 12 13 14
4 16 17 18 19 20 21 22 23
4 17 18 19 20 21 22 23 16
4 18 19 20 21 22 23 16 17
4 19 20 21 22 23 16 17 18
4 20 21 22 23 16 17 18 19
4 21 22 23 16 17 18 19 20
4 22 23 16 17 18 19 20 21
4 23 16 17 18 19 20 21 22
5 24 25 26 27 28 29 30 31


@RULE Fredkin_mod9_vonNeumann

@TREE

num_states=9
num_neighbors=4
num_nodes=37
1 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1
1 2 2 2 2 2 2 2 2 2
1 3 3 3 3 3 3 3 3 3
1 4 4 4 4 4 4 4 4 4
1 5 5 5 5 5 5 5 5 5
1 6 6 6 6 6 6 6 6 6
1 7 7 7 7 7 7 7 7 7
1 8 8 8 8 8 8 8 8 8
2 0 1 2 3 4 5 6 7 8
2 1 2 3 4 5 6 7 8 0
2 2 3 4 5 6 7 8 0 1
2 3 4 5 6 7 8 0 1 2
2 4 5 6 7 8 0 1 2 3
2 5 6 7 8 0 1 2 3 4
2 6 7 8 0 1 2 3 4 5
2 7 8 0 1 2 3 4 5 6
2 8 0 1 2 3 4 5 6 7
3 9 10 11 12 13 14 15 16 17
3 10 11 12 13 14 15 16 17 9
3 11 12 13 14 15 16 17 9 10
3 12 13 14 15 16 17 9 10 11
3 13 14 15 16 17 9 10 11 12
3 14 15 16 17 9 10 11 12 13
3 15 16 17 9 10 11 12 13 14
3 16 17 9 10 11 12 13 14 15
3 17 9 10 11 12 13 14 15 16
4 18 19 20 21 22 23 24 25 26
4 19 20 21 22 23 24 25 26 18
4 20 21 22 23 24 25 26 18 19
4 21 22 23 24 25 26 18 19 20
4 22 23 24 25 26 18 19 20 21
4 23 24 25 26 18 19 20 21 22
4 24 25 26 18 19 20 21 22 23
4 25 26 18 19 20 21 22 23 24
4 26 18 19 20 21 22 23 24 25
5 27 28 29 30 31 32 33 34 35


To my knowledge, no replicators have been found in any 2-state CA that are bound by these rules (indeed, no one-dimensional replicators that truly follow a composite Pascal triangle, either).
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Re: New method of classifying two-dimensional replicators

Postby muzik » November 15th, 2018, 7:23 am

"Range-3 Moore" seems to be bound by the following edge rule:
x = 55, y = 28, rule = Fredkin_mod3_Moore
27.A$26.3A$25.AB.BA$24.A2.A2.A$23.9A$22.AB7.BA$21.A2.B5.B2.A$20.3A3B
3.3B3A$19.AB.AB.AB.BA.BA.BA$18.A8.A8.A$17.3A6.3A6.3A$16.AB.BA4.AB.BA
4.AB.BA$15.A2.A2.A2.A2.A2.A2.A2.A2.A$14.27A$13.AB25.BA$12.A2.B23.B2.A
$11.3A3B21.3B3A$10.AB.AB.AB19.BA.BA.BA$9.A8.B17.B8.A$8.3A6.3B15.3B6.
3A$7.AB.BA4.BA.AB13.BA.AB4.AB.BA$6.A2.A2.A2.B2.B2.B11.B2.B2.B2.A2.A2.
A$5.9A9B9.9B9A$4.AB7.AB7.AB7.BA7.BA7.BA$3.A2.B5.A2.B5.A2.B5.B2.A5.B2.
A5.B2.A$2.3A3B3.3A3B3.3A3B3.3B3A3.3B3A3.3B3A$.AB.AB.AB.AB.AB.AB.AB.AB
.AB.BA.BA.BA.BA.BA.BA.BA.BA.BA$A26.A26.A!


and this core rule:
x = 55, y = 28, rule = Fredkin_mod3_Moore
27.A$26.A.A$25.A3BA$24.A5.A$23.A.A3.A.A$22.A2B.B.B.2BA$21.A2.B2.B2.B
2.A$20.A.AB.2B.2B.BA.A$19.A2B.9A.2BA$18.A17.A$17.A.A15.A.A$16.A3BA13.
A3BA$15.A5.A11.A5.A$14.A.A3.A.A9.A.A3.A.A$13.A2B.B.B.B.B7.B.B.B.B.2BA
$12.A2.B2.B5.B5.B5.B2.B2.A$11.A.AB.2B.B3.B.B3.B.B3.B.2B.BA.A$10.A2B.
5AB2.A.2AB.B2A.A2.B5A.2BA$9.A8.B8.B8.B8.A$8.A.A6.B.B6.B.B6.B.B6.A.A$
7.A3BA4.B3AB4.B3AB4.B3AB4.A3BA$6.A5.A2.B5.B2.B5.B2.B5.B2.A5.A$5.A.A3.
A.AB.B3.B.2B.B3.B.2B.B3.B.BA.A3.A.A$4.A2B.B.B.B.2A.A.A.4A.A.A.4A.A.A.
2A.B.B.B.2BA$3.A2.B2.B5.A2.A2.A2.A2.A2.A2.A2.A2.A5.B2.B2.A$2.A.AB.2B.
B3.A.2A.2A.2A.2A.2A.2A.2A.2A.A3.B.2B.BA.A$.A2B.5AB2.A27BA2.B5A.2BA$A
53.A!


What Wolfram 3-state rule integers do these rules have?
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