Gliders and replicators in Neumann neigborhood {4,4} CA

[Naszvadi]

Some thoughts about 2D CA and neighborhood.

Moore neighborhood (and its generalizations on hyperbolic mosaics) are not "beautiful". They have a neighborhood graph, which is not edge-transitive.

Conjecture that is to be proven later: on every 2D mosaic 2-stage cellular automata, gliders cannot exist withOUT B0 in a rule when the mosaic's Schläfli-symbol is {n,k} and k>=4, and the neighborhood graph is derived from the edge-connections of the mosaic's polygons.

Golly is a powerful tool to investigate such Neumann-NH. rules with B0. Found a glider and several replicators with soup search. Here you are:

A glider:

B02S0Vglider.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 5, y = 3, rule = B02/S0V
bobo2$2ob2o!

One-dimensional replicators:

B0123S023V1drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 5, y = 4, rule = B0123/S023V
bobo$5o$5o$bobo!

B013S013V1drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B013/S013V
o$bo!

B013S03V1drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B013/S03V
o$bo!

B023S012V1drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B023/S012V
o$bo!

B023S02V1drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B023/S02V
o$bo!

B03S0V1drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 3, rule = B03/S0V
2o2$2o!

Two-dimensional replicators:

B0123S13V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B0123/S13V
o$bo!

B0123S3V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B0123/S3V
o$bo!

B0234S01V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B0234/S01V
o$bo!

B0234S0V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B0234/S0V
o$bo!

B023S013V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B023/S013V
o$bo!

B023S03V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B023/S03V
o$bo!

B024S01V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B024/S01V
o$bo!

B024S0V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B024/S0V
o$bo!

B024S3V2drep2.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 3, y = 3, rule = B024/S3V
bo$obo$bo!

B024S3V2drep.rle:

Code: Select all

#CXRLE Pos=0,0 Gen=0
x = 2, y = 2, rule = B024/S3V
o$bo!

[drc]

B13/S13V is equivalent to B1357/S1357:

Code: Select all

x = 15, y = 11, rule = B13/S13V
3ob3ob3obo$o3bobobobobo$o3bobobobobo$o3bobobobobo$3ob3ob3ob3o2$2o2bobo
b2o2b3o$obobobobobobo$obobobobobob3o$obobobobobobo$2o2b3ob2o2b3o!

Although this is a pattern for other CAs with all odd number transitions:

Code: Select all

x = 15, y = 11, rule = B135/S135H
3ob3ob3obo$o3bobobobobo$o3bobobobobo$o3bobobobobo$3ob3ob3ob3o2$2o2bobo
b2o2b3o$obobobobobobo$obobobobobob3o$obobobobobobo$2o2b3ob2o2b3o!

[Naszvadi]

...

Although this is a pattern for other CAs with all odd number transitions:

Code: Select all

x = 15, y = 11, rule = B135/S135H
3ob3ob3obo$o3bobobobobo$o3bobobobobo$o3bobobobobo$3ob3ob3ob3o2$2o2bobo
b2o2b3o$obobobobobobo$obobobobobob3o$obobobobobobo$2o2b3ob2o2b3o!

Okay, even on {7,3} mosaic? Maybe true, but this is not trivial.

[Naszvadi]

...
Conjecture that is to be proven later: on every 2D mosaic 2-stage cellular automata, gliders cannot exist withOUT B0 in a rule when the mosaic's Schläfli-symbol is {n,k} and k>=4, and the neighborhood graph is derived from the edge-connections of the mosaic's polygons.

Assumed that {n,k} should be planar or hyperbolical mosaic. (Only finite number of cases were excluded by this statement)

[Naszvadi]

It seems that no one recognized or cared about that even 2D Moore neighbourhood binary rules are asymmetrical, even the B\d*/S\d* rules, especially game of life. The connection graph of the cells is NOT edge-transitive!

Well, the (Moore) neighourhoods can be introduced as a hyperedge (each h-edge is a subset with 9 elements including a promoted one), so in that case, no asymmetries will occur. But it is cheating

So on this Euclidean lattice, I am still hunting for patterns, like gliders, guns, puffers, replicators, signal gates etc.

[muzik]

Those last two in the OP make for some interesting fractals in the ghost dead blocks.

I wonder, could such a pattern exist in the Moore neighbourhood, without B0 or B1?

[Naszvadi]

Those last two in the OP make for some interesting fractals in the ghost dead blocks.

I wonder, could such a pattern exist in the Moore neighbourhood, without B0 or B1?

#define OP
// pls

THX

p.s. conjectured Opening Post?

[Naszvadi]

Now, adding some information about the necessary properties of any {4,4} (Neumann-nbh) binary totalistic cellular automata.

Which automata could have gliders?

We use Golly's notation about the rulestrings, they must match with regexp /B0?1?2?3?4?/S0?1?2?3?4?V/.

• So if a rulestring does not contain B0 or B1, then any finite pattern is unable to leave its bounding box, so no gliders
• If there is B1 in the rulestring without B0, then any pattern fill grow infinitely in all orthogonal directions, which excludes the existence of a glider
• So B0 is a must. In that case trivial: S4 shouldn't be in the rulestring

At this "milestone", we got 256 rules. We need a lemma for eliminating 64 further.

Let's see the rulestring that matches this regexp: /B01{a}2{b}3${c}4{d}/S0{e}1{f}2{g}3{h}V/
where a, b, c, d, e, f, g, h are all 1-bit integers.

Rules could be decomposed to two alternating effective rules on generations' parity:

• Assume without the loss of generosity: gen. 0 = starting pattern
• to odd generation, the transition rule is: B1{1-a}2{1-b}3${1-c}4{1-d}/S0{1-e}1{1-f}2{1-g}3{1-h}V
• to even generation, the transition rule is: B1{h}2{g}3${f}4{e}/S0{d}1{c}2{b}3{a}4V

So if an above rule has a=1 and h=0 together (which means its original rulestring is B01[234]*/S[012]*V), no pattern could escape from its bounding box, so no gliders too.

Number of remaining interesting rules that may have gliders: 192.

Conjecture: It is worth focusing on rules that "a>=h"

I wonder is there is such a universal automaton or at least one that can handles and solves any kind of embedded 3-SAT problem. I put a c/2-p8 glider in the OP

[BlinkerSpawn]

B2 without B1 or S1 explodes from dominoes like B2 in Moore.
B3 without B1 or S1 explodes from monominoes like B1e in Moore.
S2 without B1 explodes from corners like B1c in Moore.
S3 without B1 explodes along any edge.
So our prerequisites so far are:
B0 and no S4
Not both B1 and S3
If B2 and/or B3 then B1 and/or S1
No S2 or S3 without B1
And what remains is:
B01[234]/S[012] (64 rules)
B0(2||3)[4]/S[0]1 (12 rules)
76 "interesting" rules pass this test for non-explosivity.
(A||B is common notation for logical OR)
EDIT: I've manually checked the group of 12 and they're all explosive and chaotic.

[Naszvadi]

...
And what remains is:
B01[234]/S[012] (64 rules)
B0(2||3)[4]/S[0]1 (12 rules)
76 "interesting" rules pass this test for non-explosivity.
(A||B is common notation for logical OR)
EDIT: I've manually checked the group of 12 and they're all explosive and chaotic.

Cool!

The remaining 64 rules should be checked using this tool here:
http://conwaylife.com/forums/viewtopic. ... 502#p38491

and what is needed: an appropriate glider searching tool, like a tweaked zfind.

[Naszvadi]

Well, the tools are available from now, especially here: ../forums/viewtopic.php?f=9&t=2761

[Naszvadi]

...
And what remains is:
B01[234]/S[012] (64 rules)
...

B01234/SV is trivially frozen, so 63 rules are left.

[Naszvadi]

...
And what remains is:
B01[234]/S[012] (64 rules)
B0(2||3)[4]/S[0]1 (12 rules)
76 "interesting" rules pass this test for non-explosivity.
(A||B is common notation for logical OR)
EDIT: I've manually checked the group of 12 and they're all explosive and chaotic.

It is completely wrong. Please check it, no glider could exist:

Code: Select all

1:B01/SV ('B234/S01234V', 'B/S34V')
2:B01/S2V ('B234/S0134V', 'B2/S34V')
3:B01/S1V ('B234/S0234V', 'B3/S34V')
4:B01/S12V ('B234/S034V', 'B23/S34V')
5:B01/S0V ('B234/S1234V', 'B4/S34V')
6:B01/S02V ('B234/S134V', 'B24/S34V')
7:B01/S01V ('B234/S234V', 'B34/S34V')
8:B01/S012V ('B234/S34V', 'B234/S34V')
9:B014/SV ('B23/S01234V', 'B/S034V')
10:B014/S2V ('B23/S0134V', 'B2/S034V')
11:B014/S1V ('B23/S0234V', 'B3/S034V')
12:B014/S12V ('B23/S034V', 'B23/S034V')
13:B014/S0V ('B23/S1234V', 'B4/S034V')
14:B014/S02V ('B23/S134V', 'B24/S034V')
15:B014/S01V ('B23/S234V', 'B34/S034V')
16:B014/S012V ('B23/S34V', 'B234/S034V')
17:B013/SV ('B24/S01234V', 'B/S134V')
18:B013/S2V ('B24/S0134V', 'B2/S134V')
19:B013/S1V ('B24/S0234V', 'B3/S134V')
20:B013/S12V ('B24/S034V', 'B23/S134V')
21:B013/S0V ('B24/S1234V', 'B4/S134V')
22:B013/S02V ('B24/S134V', 'B24/S134V')
23:B013/S01V ('B24/S234V', 'B34/S134V')
24:B013/S012V ('B24/S34V', 'B234/S134V')
25:B0134/SV ('B2/S01234V', 'B/S0134V')
26:B0134/S2V ('B2/S0134V', 'B2/S0134V')
27:B0134/S1V ('B2/S0234V', 'B3/S0134V')
28:B0134/S12V ('B2/S034V', 'B23/S0134V')
29:B0134/S0V ('B2/S1234V', 'B4/S0134V')
30:B0134/S02V ('B2/S134V', 'B24/S0134V')
31:B0134/S01V ('B2/S234V', 'B34/S0134V')
32:B0134/S012V ('B2/S34V', 'B234/S0134V')
33:B012/SV ('B34/S01234V', 'B/S234V')
34:B012/S2V ('B34/S0134V', 'B2/S234V')
35:B012/S1V ('B34/S0234V', 'B3/S234V')
36:B012/S12V ('B34/S034V', 'B23/S234V')
37:B012/S0V ('B34/S1234V', 'B4/S234V')
38:B012/S02V ('B34/S134V', 'B24/S234V')
39:B012/S01V ('B34/S234V', 'B34/S234V')
40:B012/S012V ('B34/S34V', 'B234/S234V')
41:B0124/SV ('B3/S01234V', 'B/S0234V')
42:B0124/S2V ('B3/S0134V', 'B2/S0234V')
43:B0124/S1V ('B3/S0234V', 'B3/S0234V')
44:B0124/S12V ('B3/S034V', 'B23/S0234V')
45:B0124/S0V ('B3/S1234V', 'B4/S0234V')
46:B0124/S02V ('B3/S134V', 'B24/S0234V')
47:B0124/S01V ('B3/S234V', 'B34/S0234V')
48:B0124/S012V ('B3/S34V', 'B234/S0234V')
49:B0123/SV ('B4/S01234V', 'B/S1234V')
50:B0123/S2V ('B4/S0134V', 'B2/S1234V')
51:B0123/S1V ('B4/S0234V', 'B3/S1234V')
52:B0123/S12V ('B4/S034V', 'B23/S1234V')
53:B0123/S0V ('B4/S1234V', 'B4/S1234V')
54:B0123/S02V ('B4/S134V', 'B24/S1234V')
55:B0123/S01V ('B4/S234V', 'B34/S1234V')
56:B0123/S012V ('B4/S34V', 'B234/S1234V')
57:B01234/SV ('B/S01234V', 'B/S01234V')
58:B01234/S2V ('B/S0134V', 'B2/S01234V')
59:B01234/S1V ('B/S0234V', 'B3/S01234V')
60:B01234/S12V ('B/S034V', 'B23/S01234V')
61:B01234/S0V ('B/S1234V', 'B4/S01234V')
62:B01234/S02V ('B/S134V', 'B24/S01234V')
63:B01234/S01V ('B/S234V', 'B34/S01234V')
64:B01234/S012V ('B/S34V', 'B234/S01234V')

I am not sure in '*4V' but I am certain that finite patterns in these rules cannot escape from their bounding rectangle.

EDIT #1:

Should avoid the following 64 rules, too:

Code: Select all

$ for i in B0{,1}{,2}{,3}{,4}/S{,0}{,1}{,2}{,3}V; do echo -n "$i "; ./decodeB0rulestring.py "$i"; done | grep "'B1.*'B1" | grep -n '^'
1:B0/S3V ('B1234/S0124V', 'B1/S4V')
2:B0/S23V ('B1234/S014V', 'B12/S4V')
3:B0/S13V ('B1234/S024V', 'B13/S4V')
4:B0/S123V ('B1234/S04V', 'B123/S4V')
5:B0/S03V ('B1234/S124V', 'B14/S4V')
6:B0/S023V ('B1234/S14V', 'B124/S4V')
7:B0/S013V ('B1234/S24V', 'B134/S4V')
8:B0/S0123V ('B1234/S4V', 'B1234/S4V')
9:B04/S3V ('B123/S0124V', 'B1/S04V')
10:B04/S23V ('B123/S014V', 'B12/S04V')
11:B04/S13V ('B123/S024V', 'B13/S04V')
12:B04/S123V ('B123/S04V', 'B123/S04V')
13:B04/S03V ('B123/S124V', 'B14/S04V')
14:B04/S023V ('B123/S14V', 'B124/S04V')
15:B04/S013V ('B123/S24V', 'B134/S04V')
16:B04/S0123V ('B123/S4V', 'B1234/S04V')
17:B03/S3V ('B124/S0124V', 'B1/S14V')
18:B03/S23V ('B124/S014V', 'B12/S14V')
19:B03/S13V ('B124/S024V', 'B13/S14V')
20:B03/S123V ('B124/S04V', 'B123/S14V')
21:B03/S03V ('B124/S124V', 'B14/S14V')
22:B03/S023V ('B124/S14V', 'B124/S14V')
23:B03/S013V ('B124/S24V', 'B134/S14V')
24:B03/S0123V ('B124/S4V', 'B1234/S14V')
25:B034/S3V ('B12/S0124V', 'B1/S014V')
26:B034/S23V ('B12/S014V', 'B12/S014V')
27:B034/S13V ('B12/S024V', 'B13/S014V')
28:B034/S123V ('B12/S04V', 'B123/S014V')
29:B034/S03V ('B12/S124V', 'B14/S014V')
30:B034/S023V ('B12/S14V', 'B124/S014V')
31:B034/S013V ('B12/S24V', 'B134/S014V')
32:B034/S0123V ('B12/S4V', 'B1234/S014V')
33:B02/S3V ('B134/S0124V', 'B1/S24V')
34:B02/S23V ('B134/S014V', 'B12/S24V')
35:B02/S13V ('B134/S024V', 'B13/S24V')
36:B02/S123V ('B134/S04V', 'B123/S24V')
37:B02/S03V ('B134/S124V', 'B14/S24V')
38:B02/S023V ('B134/S14V', 'B124/S24V')
39:B02/S013V ('B134/S24V', 'B134/S24V')
40:B02/S0123V ('B134/S4V', 'B1234/S24V')
41:B024/S3V ('B13/S0124V', 'B1/S024V')
42:B024/S23V ('B13/S014V', 'B12/S024V')
43:B024/S13V ('B13/S024V', 'B13/S024V')
44:B024/S123V ('B13/S04V', 'B123/S024V')
45:B024/S03V ('B13/S124V', 'B14/S024V')
46:B024/S023V ('B13/S14V', 'B124/S024V')
47:B024/S013V ('B13/S24V', 'B134/S024V')
48:B024/S0123V ('B13/S4V', 'B1234/S024V')
49:B023/S3V ('B14/S0124V', 'B1/S124V')
50:B023/S23V ('B14/S014V', 'B12/S124V')
51:B023/S13V ('B14/S024V', 'B13/S124V')
52:B023/S123V ('B14/S04V', 'B123/S124V')
53:B023/S03V ('B14/S124V', 'B14/S124V')
54:B023/S023V ('B14/S14V', 'B124/S124V')
55:B023/S013V ('B14/S24V', 'B134/S124V')
56:B023/S0123V ('B14/S4V', 'B1234/S124V')
57:B0234/S3V ('B1/S0124V', 'B1/S0124V')
58:B0234/S23V ('B1/S014V', 'B12/S0124V')
59:B0234/S13V ('B1/S024V', 'B13/S0124V')
60:B0234/S123V ('B1/S04V', 'B123/S0124V')
61:B0234/S03V ('B1/S124V', 'B14/S0124V')
62:B0234/S023V ('B1/S14V', 'B124/S0124V')
63:B0234/S013V ('B1/S24V', 'B134/S0124V')
64:B0234/S0123V ('B1/S4V', 'B1234/S0124V')

In the above rules, all patterns will explode in all four axial directions.

Remaining 128 rules should be investigated for gliders:

Code: Select all

1:B0/SV 2:B0/S2V 3:B0/S1V 4:B0/S12V 5:B0/S0V 6:B0/S02V 7:B0/S01V 8:B0/S012V
9:B04/SV 10:B04/S2V 11:B04/S1V 12:B04/S12V 13:B04/S0V 14:B04/S02V 15:B04/S01V 16:B04/S012V
17:B03/SV 18:B03/S2V 19:B03/S1V 20:B03/S12V 21:B03/S0V 22:B03/S02V 23:B03/S01V 24:B03/S012V
25:B034/SV 26:B034/S2V 27:B034/S1V 28:B034/S12V 29:B034/S0V 30:B034/S02V 31:B034/S01V 32:B034/S012V
33:B02/SV 34:B02/S2V 35:B02/S1V 36:B02/S12V 37:B02/S0V 38:B02/S02V 39:B02/S01V 40:B02/S012V
41:B024/SV 42:B024/S2V 43:B024/S1V 44:B024/S12V 45:B024/S0V 46:B024/S02V 47:B024/S01V 48:B024/S012V
49:B023/SV 50:B023/S2V 51:B023/S1V 52:B023/S12V 53:B023/S0V 54:B023/S02V 55:B023/S01V 56:B023/S012V
57:B0234/SV 58:B0234/S2V 59:B0234/S1V 60:B0234/S12V 61:B0234/S0V 62:B0234/S02V 63:B0234/S01V 64:B0234/S012V
65:B01/S3V 66:B01/S23V 67:B01/S13V 68:B01/S123V 69:B01/S03V 70:B01/S023V 71:B01/S013V 72:B01/S0123V
73:B014/S3V 74:B014/S23V 75:B014/S13V 76:B014/S123V 77:B014/S03V 78:B014/S023V 79:B014/S013V 80:B014/S0123V
81:B013/S3V 82:B013/S23V 83:B013/S13V 84:B013/S123V 85:B013/S03V 86:B013/S023V 87:B013/S013V 88:B013/S0123V
89:B0134/S3V 90:B0134/S23V 91:B0134/S13V 92:B0134/S123V 93:B0134/S03V 94:B0134/S023V 95:B0134/S013V 96:B0134/S0123V
97:B012/S3V 98:B012/S23V 99:B012/S13V 100:B012/S123V 101:B012/S03V 102:B012/S023V 103:B012/S013V 104:B012/S0123V
105:B0124/S3V 106:B0124/S23V 107:B0124/S13V 108:B0124/S123V 109:B0124/S03V 110:B0124/S023V 111:B0124/S013V 112:B0124/S0123V
113:B0123/S3V 114:B0123/S23V 115:B0123/S13V 116:B0123/S123V 117:B0123/S03V 118:B0123/S023V 119:B0123/S013V 120:B0123/S0123V
121:B01234/S3V 122:B01234/S23V 123:B01234/S13V 124:B01234/S123V 125:B01234/S03V 126:B01234/S023V 127:B01234/S013V 128:B01234/S0123V

[Naszvadi]

Investigated all remaining rules.
The 128 candidates contain many "obsolete" ones, especially those that match their equivalent B0-less rulestring-pairs:

Code: Select all

"'B[12].*B[12]"
"'B1.*'B.*S.*1"
"'B.*S.*1.*'B1"

In the final battle, the following 32 rules appeared:

Code: Select all

1:B01234/S0123V ('B/S4V', 'B1234/S01234V')
2:B01234/S013V ('B/S24V', 'B134/S01234V')
3:B01234/S123V ('B/S04V', 'B123/S01234V')
4:B01234/S13V ('B/S024V', 'B13/S01234V')
5:B0123/S0123V ('B4/S4V', 'B1234/S1234V')
6:B0123/S013V ('B4/S24V', 'B134/S1234V')
7:B0123/S123V ('B4/S04V', 'B123/S1234V')
8:B0123/S13V ('B4/S024V', 'B13/S1234V')
9:B0124/S0123V ('B3/S4V', 'B1234/S0234V')
10:B0124/S013V ('B3/S24V', 'B134/S0234V')
11:B0124/S123V ('B3/S04V', 'B123/S0234V')
12:B0124/S13V ('B3/S024V', 'B13/S0234V')
13:B012/S0123V ('B34/S4V', 'B1234/S234V')
14:B012/S013V ('B34/S24V', 'B134/S234V')
15:B012/S123V ('B34/S04V', 'B123/S234V')
16:B012/S13V ('B34/S024V', 'B13/S234V')
17:B024/S01V ('B13/S234V', 'B34/S024V')
18:B024/S0V ('B13/S1234V', 'B4/S024V')
19:B024/S1V ('B13/S0234V', 'B3/S024V')
20:B024/SV ('B13/S01234V', 'B/S024V')
21:B02/S01V ('B134/S234V', 'B34/S24V')
22:B02/S0V ('B134/S1234V', 'B4/S24V')
23:B02/S1V ('B134/S0234V', 'B3/S24V')
24:B02/SV ('B134/S01234V', 'B/S24V')
25:B04/S01V ('B123/S234V', 'B34/S04V')
26:B04/S0V ('B123/S1234V', 'B4/S04V')
27:B04/S1V ('B123/S0234V', 'B3/S04V')
28:B04/SV ('B123/S01234V', 'B/S04V')
29:B0/S01V ('B1234/S234V', 'B34/S4V')
30:B0/S0V ('B1234/S1234V', 'B4/S4V')
31:B0/S1V ('B1234/S0234V', 'B3/S4V')
32:B0/SV ('B1234/S01234V', 'B/S4V')

Many spaceships in many neumann rules, next-to-last a linear growth (cuckoo-egg!), and finally the smallest pattern that produces possible infinite growth in B04/S1V

==> b0124s13_ship1.rle <==

Code: Select all

x = 4, y = 5, rule = B0124/S13V
bo$3o$2bo$o2bo$b2o!

==> b024s1_ship1.rle <==

Code: Select all

x = 3, y = 2, rule = B024/S1V
obo$o!

==> b024s1_ship2.rle <==

Code: Select all

x = 2, y = 5, rule = B024/S1V
bo2$2o2$bo!

==> b024s_ship1.rle <==

Code: Select all

x = 4, y = 3, rule = B024/SV
bo$obo$2b2o!

==> b02s0_ship1.rle <==

Code: Select all

x = 5, y = 3, rule = B02/S0V
bobo2$2ob2o!

==> b02sv_ship1.rle <==

Code: Select all

x = 4, y = 3, rule = B02/SV
bo$obo$2b2o!

==> b03s01.rle <==

Code: Select all

x = 41, y = 21, rule = B03/S01V
bo37bo$bo37bo$2ob2o31b2ob2o$bo2bo31bo2bo$2bo2b2obo23bob2o2bo$3bo2bo2bo
21bo2bo2bo$4bo2bo2bo19bo2bo2bo$5bo2bo2bo17bo2bo2bo$6bo2bo2bo15bo2bo2bo
$7bo2bo2bo13bo2bo2bo$8bo2bo2bo11bo2bo2bo$9bo2bo2bo9bo2bo2bo$10bo2bo2bo
7bo2bo2bo$11bo2bo2bo5bo2bo2bo$12bo2bo2bo3bo2bo2bo$13bo2bo2bobo2bo2bo$
14bo2bo2bo2bo2bo$15bo2bo3bo2bo$16bo2bobo2bo$17bo2bo2bo$18bo3bo!

==> b04s1.rle <==

Code: Select all

x = 4, y = 2, rule = B04/S1V
4o$2o!

[Naszvadi]

...

In the final battle, the following 32 rules appeared:

Code: Select all

...

Code: Select all

B012345678/S01234-e5-c6-cn7e == B01234/S0123V ('B/S4V', 'B1234/S01234V')
B012345678/S012-ei3-arkj4cjnry5qeyin6ak7e == B01234/S013V ('B/S24V', 'B134/S01234V')
B012345678/S1e2-cn3-c4-ce5-c6-cn7e == B01234/S123V ('B/S04V', 'B123/S01234V')
B012345678/S1e2ak3qeyin4jnry5qeyin6ak7e == B01234/S13V ('B/S024V', 'B13/S01234V')
B01234-e5-c6-cn7e/S01234-e5-c6-cn7e == B0123/S0123V ('B4/S4V', 'B1234/S1234V')
B01234-e5-c6-cn7e/S012-ei3-arkj4cjnry5qeyin6ak7e == B0123/S013V ('B4/S24V', 'B134/S1234V')
B01234-e5-c6-cn7e/S1e2-cn3-c4-ce5-c6-cn7e == B0123/S123V ('B4/S04V', 'B123/S1234V')
B01234-e5-c6-cn7e/S1e2ak3qeyin4jnry5qeyin6ak7e == B0123/S13V ('B4/S024V', 'B13/S1234V')
B0123-e4-jr5-qyin6-ak7c8/S01234-e5-c6-cn7e == B0124/S0123V ('B3/S4V', 'B1234/S0234V')
B0123-e4-jr5-qyin6-ak7c8/S012-ei3-arkj4cjnry5qeyin6ak7e == B0124/S013V ('B3/S24V', 'B134/S0234V')
B0123-e4-jr5-qyin6-ak7c8/S1e2-cn3-c4-ce5-c6-cn7e == B0124/S123V ('B3/S04V', 'B123/S0234V')
B0123-e4-jr5-qyin6-ak7c8/S1e2ak3qeyin4jnry5qeyin6ak7e == B0124/S13V ('B3/S024V', 'B13/S0234V')
B0123-e4-ejr5arekj6ei/S01234-e5-c6-cn7e == B012/S0123V ('B34/S4V', 'B1234/S234V')
B0123-e4-ejr5arekj6ei/S012-ei3-arkj4cjnry5qeyin6ak7e == B012/S013V ('B34/S24V', 'B134/S234V')
B0123-e4-ejr5arekj6ei/S1e2-cn3-c4-ce5-c6-cn7e == B012/S123V ('B34/S04V', 'B123/S234V')
B0123-e4-ejr5arekj6ei/S1e2ak3qeyin4jnry5qeyin6ak7e == B012/S13V ('B34/S024V', 'B13/S234V')
B01c2-ak3acrkj4-jnry5acrkj6-ak7c8/S012-ei3qcyin4cny5e == B024/S01V ('B13/S234V', 'B34/S024V')
B01c2-ak3acrkj4-jnry5acrkj6-ak7c8/S01c2cn3c4c == B024/S0V ('B13/S1234V', 'B4/S024V')
B01c2-ak3acrkj4-jnry5acrkj6-ak7c8/S1e2ak3qyin4ny5e == B024/S1V ('B13/S0234V', 'B3/S024V')
B01c2-ak3acrkj4-jnry5acrkj6-ak7c8/S == B024/SV ('B13/S01234V', 'B/S024V')
B01c2-ak3acrkj4-ejnry5arkj6ei/S012-ei3qcyin4cny5e == B02/S01V ('B134/S234V', 'B34/S24V')
B01c2-ak3acrkj4-ejnry5arkj6ei/S01c2cn3c4c == B02/S0V ('B134/S1234V', 'B4/S24V')
B01c2-ak3acrkj4-ejnry5arkj6ei/S1e2ak3qyin4ny5e == B02/S1V ('B134/S0234V', 'B3/S24V')
B01c2-ak3acrkj4-ejnry5arkj6ei/S == B02/SV ('B134/S01234V', 'B/S24V')
B01c2cn3c4ce5c6cn7c8/S012-ei3qcyin4cny5e == B04/S01V ('B123/S234V', 'B34/S04V')
B01c2cn3c4ce5c6cn7c8/S01c2cn3c4c == B04/S0V ('B123/S1234V', 'B4/S04V')
B01c2cn3c4ce5c6cn7c8/S1e2ak3qyin4ny5e == B04/S1V ('B123/S0234V', 'B3/S04V')
B01c2cn3c4ce5c6cn7c8/S == B04/SV ('B123/S01234V', 'B/S04V')
B01c2cn3c4c/S012-ei3qcyin4cny5e == B0/S01V ('B1234/S234V', 'B34/S4V')
B01c2cn3c4c/S01c2cn3c4c == B0/S0V ('B1234/S1234V', 'B4/S4V')
B01c2cn3c4c/S1e2ak3qyin4ny5e == B0/S1V ('B1234/S0234V', 'B3/S4V')
B01c2cn3c4c/S == B0/SV ('B1234/S01234V', 'B/S4V')

Let's haul 'em!

EDIT #1: faulty. Unordered rulestring parts. Fixed here:

Code: Select all

B012345678/S01234-e5-c6-cn7e == B01234/S0123V ('B/S4V', 'B1234/S01234V')
B012345678/S012-ei3-ajkr4cjnry5einqy6ak7e == B01234/S013V ('B/S24V', 'B134/S01234V')
B012345678/S1e2-cn3-c4-ce5-c6-cn7e == B01234/S123V ('B/S04V', 'B123/S01234V')
B012345678/S1e2ak3einqy4jnry5einqy6ak7e == B01234/S13V ('B/S024V', 'B13/S01234V')
B01234-e5-c6-cn7e/S01234-e5-c6-cn7e == B0123/S0123V ('B4/S4V', 'B1234/S1234V')
B01234-e5-c6-cn7e/S012-ei3-ajkr4cjnry5einqy6ak7e == B0123/S013V ('B4/S24V', 'B134/S1234V')
B01234-e5-c6-cn7e/S1e2-cn3-c4-ce5-c6-cn7e == B0123/S123V ('B4/S04V', 'B123/S1234V')
B01234-e5-c6-cn7e/S1e2ak3einqy4jnry5einqy6ak7e == B0123/S13V ('B4/S024V', 'B13/S1234V')
B0123-e4-jr5-inqy6-ak7c8/S01234-e5-c6-cn7e == B0124/S0123V ('B3/S4V', 'B1234/S0234V')
B0123-e4-jr5-inqy6-ak7c8/S012-ei3-ajkr4cjnry5einqy6ak7e == B0124/S013V ('B3/S24V', 'B134/S0234V')
B0123-e4-jr5-inqy6-ak7c8/S1e2-cn3-c4-ce5-c6-cn7e == B0124/S123V ('B3/S04V', 'B123/S0234V')
B0123-e4-jr5-inqy6-ak7c8/S1e2ak3einqy4jnry5einqy6ak7e == B0124/S13V ('B3/S024V', 'B13/S0234V')
B0123-e4-ejr5aejkr6ei/S01234-e5-c6-cn7e == B012/S0123V ('B34/S4V', 'B1234/S234V')
B0123-e4-ejr5aejkr6ei/S012-ei3-ajkr4cjnry5einqy6ak7e == B012/S013V ('B34/S24V', 'B134/S234V')
B0123-e4-ejr5aejkr6ei/S1e2-cn3-c4-ce5-c6-cn7e == B012/S123V ('B34/S04V', 'B123/S234V')
B0123-e4-ejr5aejkr6ei/S1e2ak3einqy4jnry5einqy6ak7e == B012/S13V ('B34/S024V', 'B13/S234V')
B01c2-ak3acjkr4-jnry5acjkr6-ak7c8/S012-ei3cinqy4cny5e == B024/S01V ('B13/S234V', 'B34/S024V')
B01c2-ak3acjkr4-jnry5acjkr6-ak7c8/S01c2cn3c4c == B024/S0V ('B13/S1234V', 'B4/S024V')
B01c2-ak3acjkr4-jnry5acjkr6-ak7c8/S1e2ak3inqy4ny5e == B024/S1V ('B13/S0234V', 'B3/S024V')
B01c2-ak3acjkr4-jnry5acjkr6-ak7c8/S == B024/SV ('B13/S01234V', 'B/S024V')
B01c2-ak3acjkr4-ejnry5ajkr6ei/S012-ei3cinqy4cny5e == B02/S01V ('B134/S234V', 'B34/S24V')
B01c2-ak3acjkr4-ejnry5ajkr6ei/S01c2cn3c4c == B02/S0V ('B134/S1234V', 'B4/S24V')
B01c2-ak3acjkr4-ejnry5ajkr6ei/S1e2ak3inqy4ny5e == B02/S1V ('B134/S0234V', 'B3/S24V')
B01c2-ak3acjkr4-ejnry5ajkr6ei/S == B02/SV ('B134/S01234V', 'B/S24V')
B01c2cn3c4ce5c6cn7c8/S012-ei3cinqy4cny5e == B04/S01V ('B123/S234V', 'B34/S04V')
B01c2cn3c4ce5c6cn7c8/S01c2cn3c4c == B04/S0V ('B123/S1234V', 'B4/S04V')
B01c2cn3c4ce5c6cn7c8/S1e2ak3inqy4ny5e == B04/S1V ('B123/S0234V', 'B3/S04V')
B01c2cn3c4ce5c6cn7c8/S == B04/SV ('B123/S01234V', 'B/S04V')
B01c2cn3c4c/S012-ei3cinqy4cny5e == B0/S01V ('B1234/S234V', 'B34/S4V')
B01c2cn3c4c/S01c2cn3c4c == B0/S0V ('B1234/S1234V', 'B4/S4V')
B01c2cn3c4c/S1e2ak3inqy4ny5e == B0/S1V ('B1234/S0234V', 'B3/S4V')
B01c2cn3c4c/S == B0/SV ('B1234/S01234V', 'B/S4V')

[BlinkerSpawn]

Does apgsearch support B0?

[praosylen]

Does apgsearch support B0?

No, unfortunately. I might hopefully get to that eventually, but not right now.

[Naszvadi]

Well, it seems that I invented wheel (partly at least )

According to this, some of the above spaceships have been known since 2003 or earlier: https://groups.google.com/forum/#!topic/comp.theory.cell-automata/jH5BgssD9Zw
( "Gliders in 4-neighbor semitotalistic 2D cellular automata" )

[Naszvadi]

...

EDIT #1: faulty. Unordered rulestring parts. Fixed here:

Code: Select all

B012345678/S01234-e5-c6-cn7e == B01234/S0123V ('B/S4V', 'B1234/S01234V')
B012345678/S012-ei3-ajkr4cjnry5einqy6ak7e == B01234/S013V ('B/S24V', 'B134/S01234V')
B012345678/S1e2-cn3-c4-ce5-c6-cn7e == B01234/S123V ('B/S04V', 'B123/S01234V')
B012345678/S1e2ak3einqy4jnry5einqy6ak7e == B01234/S13V ('B/S024V', 'B13/S01234V')
B01234-e5-c6-cn7e/S01234-e5-c6-cn7e == B0123/S0123V ('B4/S4V', 'B1234/S1234V')
B01234-e5-c6-cn7e/S012-ei3-ajkr4cjnry5einqy6ak7e == B0123/S013V ('B4/S24V', 'B134/S1234V')
B01234-e5-c6-cn7e/S1e2-cn3-c4-ce5-c6-cn7e == B0123/S123V ('B4/S04V', 'B123/S1234V')
B01234-e5-c6-cn7e/S1e2ak3einqy4jnry5einqy6ak7e == B0123/S13V ('B4/S024V', 'B13/S1234V')
B0123-e4-jr5-inqy6-ak7c8/S01234-e5-c6-cn7e == B0124/S0123V ('B3/S4V', 'B1234/S0234V')
B0123-e4-jr5-inqy6-ak7c8/S012-ei3-ajkr4cjnry5einqy6ak7e == B0124/S013V ('B3/S24V', 'B134/S0234V')
B0123-e4-jr5-inqy6-ak7c8/S1e2-cn3-c4-ce5-c6-cn7e == B0124/S123V ('B3/S04V', 'B123/S0234V')
B0123-e4-jr5-inqy6-ak7c8/S1e2ak3einqy4jnry5einqy6ak7e == B0124/S13V ('B3/S024V', 'B13/S0234V')
B0123-e4-ejr5aejkr6ei/S01234-e5-c6-cn7e == B012/S0123V ('B34/S4V', 'B1234/S234V')
B0123-e4-ejr5aejkr6ei/S012-ei3-ajkr4cjnry5einqy6ak7e == B012/S013V ('B34/S24V', 'B134/S234V')
B0123-e4-ejr5aejkr6ei/S1e2-cn3-c4-ce5-c6-cn7e == B012/S123V ('B34/S04V', 'B123/S234V')
B0123-e4-ejr5aejkr6ei/S1e2ak3einqy4jnry5einqy6ak7e == B012/S13V ('B34/S024V', 'B13/S234V')
B01c2-ak3acjkr4-jnry5acjkr6-ak7c8/S012-ei3cinqy4cny5e == B024/S01V ('B13/S234V', 'B34/S024V')
B01c2-ak3acjkr4-jnry5acjkr6-ak7c8/S01c2cn3c4c == B024/S0V ('B13/S1234V', 'B4/S024V')
B01c2-ak3acjkr4-jnry5acjkr6-ak7c8/S1e2ak3inqy4ny5e == B024/S1V ('B13/S0234V', 'B3/S024V')
B01c2-ak3acjkr4-jnry5acjkr6-ak7c8/S == B024/SV ('B13/S01234V', 'B/S024V')
B01c2-ak3acjkr4-ejnry5ajkr6ei/S012-ei3cinqy4cny5e == B02/S01V ('B134/S234V', 'B34/S24V')
B01c2-ak3acjkr4-ejnry5ajkr6ei/S01c2cn3c4c == B02/S0V ('B134/S1234V', 'B4/S24V')
B01c2-ak3acjkr4-ejnry5ajkr6ei/S1e2ak3inqy4ny5e == B02/S1V ('B134/S0234V', 'B3/S24V')
B01c2-ak3acjkr4-ejnry5ajkr6ei/S == B02/SV ('B134/S01234V', 'B/S24V')
B01c2cn3c4ce5c6cn7c8/S012-ei3cinqy4cny5e == B04/S01V ('B123/S234V', 'B34/S04V')
B01c2cn3c4ce5c6cn7c8/S01c2cn3c4c == B04/S0V ('B123/S1234V', 'B4/S04V')
B01c2cn3c4ce5c6cn7c8/S1e2ak3inqy4ny5e == B04/S1V ('B123/S0234V', 'B3/S04V')
B01c2cn3c4ce5c6cn7c8/S == B04/SV ('B123/S01234V', 'B/S04V')
B01c2cn3c4c/S012-ei3cinqy4cny5e == B0/S01V ('B1234/S234V', 'B34/S4V')
B01c2cn3c4c/S01c2cn3c4c == B0/S0V ('B1234/S1234V', 'B4/S4V')
B01c2cn3c4c/S1e2ak3inqy4ny5e == B0/S1V ('B1234/S0234V', 'B3/S4V')
B01c2cn3c4c/S == B0/SV ('B1234/S01234V', 'B/S4V')

Update for a while:

Observe that the B1-containing and B1-less alternate rulestring parts can be exchanged with each other, and the other order also assigned to a feasible B0* rule. And each pattern in one shifted to in another generation with different parity, will show the same behaviour and properties.

Checked by a shell oneliner:

Code: Select all

# awk '{if($4 ~ /B1/){print $4 $5 " " $3}else{print $5 $4 " " $3}}' pairs.txt | tr -d "'()," | sed "s/V/V, /" | sort | awk 'BEGIN{a="";b=""}{printf("%s ",$0);if(a==$1 && b==$2){print "complementer of the previous one"}else{print "first occurrence";a=$1;b=$2}}'
B1234/S01234V, B/S4V B01234/S0123V first occurrence
B1234/S01234V, B/S4V B0/SV complementer of the previous one
B1234/S0234V, B3/S4V B0124/S0123V first occurrence
B1234/S0234V, B3/S4V B0/S1V complementer of the previous one
B1234/S1234V, B4/S4V B0123/S0123V first occurrence
B1234/S1234V, B4/S4V B0/S0V complementer of the previous one
B1234/S234V, B34/S4V B012/S0123V first occurrence
B1234/S234V, B34/S4V B0/S01V complementer of the previous one
B123/S01234V, B/S04V B01234/S123V first occurrence
B123/S01234V, B/S04V B04/SV complementer of the previous one
B123/S0234V, B3/S04V B0124/S123V first occurrence
B123/S0234V, B3/S04V B04/S1V complementer of the previous one
B123/S1234V, B4/S04V B0123/S123V first occurrence
B123/S1234V, B4/S04V B04/S0V complementer of the previous one
B123/S234V, B34/S04V B012/S123V first occurrence
B123/S234V, B34/S04V B04/S01V complementer of the previous one
B134/S01234V, B/S24V B01234/S013V first occurrence
B134/S01234V, B/S24V B02/SV complementer of the previous one
B134/S0234V, B3/S24V B0124/S013V first occurrence
B134/S0234V, B3/S24V B02/S1V complementer of the previous one
B134/S1234V, B4/S24V B0123/S013V first occurrence
B134/S1234V, B4/S24V B02/S0V complementer of the previous one
B134/S234V, B34/S24V B012/S013V first occurrence
B134/S234V, B34/S24V B02/S01V complementer of the previous one
B13/S01234V, B/S024V B01234/S13V first occurrence
B13/S01234V, B/S024V B024/SV complementer of the previous one
B13/S0234V, B3/S024V B0124/S13V first occurrence
B13/S0234V, B3/S024V B024/S1V complementer of the previous one
B13/S1234V, B4/S024V B0123/S13V first occurrence
B13/S1234V, B4/S024V B024/S0V complementer of the previous one
B13/S234V, B34/S024V B012/S13V first occurrence
B13/S234V, B34/S024V B024/S01V complementer of the previous one

So the following 16 rules remained for further investigations:

1. B0/SV
2. B0/S1V
3. B0/S0V
4. B0/S01V
5. B04/SV
6. B04/S1V
7. B04/S0V
8. B04/S01V
9. B02/SV
10. B02/S1V
11. B02/S0V
12. B02/S01V
13. B024/SV
14. B024/S1V
15. B024/S0V
16. B024/S01V

[Naszvadi]

...

So the following 16 rules remained for further investigations:

1. B0/SV
2. B0/S1V
3. B0/S0V
4. B0/S01V
5. B04/SV
6. B04/S1V
7. B04/S0V
8. B04/S01V
9. B02/SV
10. B02/S1V
11. B02/S0V
12. B02/S01V
13. B024/SV
14. B024/S1V
15. B024/S0V
16. B024/S01V

A handy tool for classifying small patterns in the above rules: ../forums/viewtopic.php?f=9&t=45&p=45558#p45558

[Naszvadi]

...
So the following 16 rules remained for further investigations:

1. B0/SV
2. B0/S1V
3. B0/S0V
4. B0/S01V
5. B04/SV
6. B04/S1V
7. B04/S0V
8. B04/S01V
9. B02/SV
10. B02/S1V
11. B02/S0V
12. B02/S01V
13. B024/SV
14. B024/S1V
15. B024/S0V
16. B024/S01V

Automated search were performed for ALL distinct patterns that fit into a 4x4 square, all gliders and oscillators below period 20 were found (due to the bug of the investigated pattern list, there are reduntancy, duplicates) - oscillators between period 9 and 21 and all gliders listed here:

c/2 orthogonal (period 4):

Code: Select all

x = 0, y = 0, rule = B02/SV
bo$obo$2b2o16$2o$bobo$2bo16$bo$b2o$obo$2b2o16$2o$bobo$b2o$2bo!

the same results in this rule:

Code: Select all

x = 0, y = 0, rule = B024/SV
bo$obo$2b2o16$2o$bobo$2bo16$bo$b2o$obo$2b2o16$2o$bobo$b2o$2bo!

c/12d diagonal and a p12 oscillator:

Code: Select all

x = 0, y = 0, rule = B02/S1V
o$b2o$bobo$3bo16$2o2$bobo$2b2o16$2o$2bo$b2o$3bo16$2o$obo2$2b2o16$4o$o2bo$o2bo$4o!

p12 and p20 oscillators:

Code: Select all

x = 0, y = 0, rule = B02/S01V
o2bo$bo$o2bo16$2obo$3o$b3o$ob2o!

c/2 orthogonal ship with period 4 (has a symmetric variant in 5x3, published here earlier):

Code: Select all

x = 0, y = 0, rule = B024/S1V
obo$o16$2obo2$bo!

p12 (three phases) and a p10 oscillator at bottom:

Code: Select all

x = 0, y = 0, rule = B0/S1V
2bo$bo$obo$3bo16$o$bobo$2bo$bo16$obo$bobo$obo$bo16$2obo$b2o$4o$obo!

[Naszvadi]

...
So the following 16 rules remained for further investigations:

1. B0/SV
2. B0/S1V
3. B0/S0V
4. B0/S01V
5. B04/SV
6. B04/S1V
7. B04/S0V
8. B04/S01V
9. B02/SV
10. B02/S1V
11. B02/S0V
12. B02/S01V
13. B024/SV
14. B024/S1V
15. B024/S0V
16. B024/S01V

Automated search were performed for ALL distinct patterns that fit into a 4x4 square, all gliders and oscillators below period 20 were found (due to the bug of the investigated pattern list, there are reduntancy, duplicates) - oscillators between period 9 and 21 and all gliders listed here:

c/2 orthogonal (period 4):

Code: Select all

x = 0, y = 0, rule = B02/SV
bo$obo$2b2o16$2o$bobo$2bo16$bo$b2o$obo$2b2o16$2o$bobo$b2o$2bo!

the same results in this rule:

Code: Select all

x = 0, y = 0, rule = B024/SV
bo$obo$2b2o16$2o$bobo$2bo16$bo$b2o$obo$2b2o16$2o$bobo$b2o$2bo!

c/12d diagonal and a p12 oscillator:

Code: Select all

x = 0, y = 0, rule = B02/S1V
o$b2o$bobo$3bo16$2o2$bobo$2b2o16$2o$2bo$b2o$3bo16$2o$obo2$2b2o16$4o$o2bo$o2bo$4o!

p12 and p20 oscillators:hoho
ahah
hihi

Code: Select all

x = 0, y = 0, rule = B02/S01V
o2bo$bo$o2bo16$2obo$3o$b3o$ob2o!

c/2 orthogonal ship with period 4 (has a symmetric variant in 5x3, published here earlier):

Code: Select all

x = 0, y = 0, rule = B024/S1V
obo$o16$2obo2$bo!

p12 (three phases) and a p10 oscillator at bottom:

Code: Select all

x = 0, y = 0, rule = B0/S1V
2bo$bo$obo$3bo16$o$bobo$2bo$bo16$obo$bobo$obo$bo16$2obo$b2o$4o$obo!

There are at most 16 distinct vNeumann 2-state rules on E^2 that could support gliders (and another equivalent counterpart per rule, so total 32 rules).

One can recognize that blinking rules can be emulated on E^2 with 3 necessary states, where alternating generations must have the same "alive" cell value, so for example even generations has "0" and "1" cell states only as well as odd generations must NOT have cell state value "1".

I had a dream, so after reading A for awesome's post here: ../forums/viewtopic.php?f=11&t=3058 titled "3-state inner-totalistic von Neumann (3ITVN) rules", enumerated all 32 blinking 2-state rules above if they could have a trivial mapping 3ITVN rules. And only 2 (four) have such mapping listed by this code snippet:

Code: Select all

perl -e '
  while(<>){
    $previous_line=$current_line;
    $current_line=$_;m/^B(\d*)\/S(\d*)V.*one$/;
    $birth=$1;
    ($survive=$2)=~s/\d/$&<4?($&+1):""/eg;
    if($birth and ($birth==$survive)){
      print"$previous_line$current_line"
    };
  };
'   <<END
B1234/S01234V, B/S4V B01234/S0123V first occurrence
B1234/S01234V, B/S4V B0/SV complementer of the previous one
B1234/S0234V, B3/S4V B0124/S0123V first occurrence
B1234/S0234V, B3/S4V B0/S1V complementer of the previous one
B1234/S1234V, B4/S4V B0123/S0123V first occurrence
B1234/S1234V, B4/S4V B0/S0V complementer of the previous one
B1234/S234V, B34/S4V B012/S0123V first occurrence
B1234/S234V, B34/S4V B0/S01V complementer of the previous one
B123/S01234V, B/S04V B01234/S123V first occurrence
B123/S01234V, B/S04V B04/SV complementer of the previous one
B123/S0234V, B3/S04V B0124/S123V first occurrence
B123/S0234V, B3/S04V B04/S1V complementer of the previous one
B123/S1234V, B4/S04V B0123/S123V first occurrence
B123/S1234V, B4/S04V B04/S0V complementer of the previous one
B123/S234V, B34/S04V B012/S123V first occurrence
B123/S234V, B34/S04V B04/S01V complementer of the previous one
B134/S01234V, B/S24V B01234/S013V first occurrence
B134/S01234V, B/S24V B02/SV complementer of the previous one
B134/S0234V, B3/S24V B0124/S013V first occurrence
B134/S0234V, B3/S24V B02/S1V complementer of the previous one
B134/S1234V, B4/S24V B0123/S013V first occurrence
B134/S1234V, B4/S24V B02/S0V complementer of the previous one
B134/S234V, B34/S24V B012/S013V first occurrence
B134/S234V, B34/S24V B02/S01V complementer of the previous one
B13/S01234V, B/S024V B01234/S13V first occurrence
B13/S01234V, B/S024V B024/SV complementer of the previous one
B13/S0234V, B3/S024V B0124/S13V first occurrence
B13/S0234V, B3/S024V B024/S1V complementer of the previous one
B13/S1234V, B4/S024V B0123/S13V first occurrence
B13/S1234V, B4/S024V B024/S0V complementer of the previous one
B13/S234V, B34/S024V B012/S13V first occurrence
B13/S234V, B34/S024V B024/S01V complementer of the previous one
END

The rules are:

• B0/SV ( ~ B01234/S0123V)
• B02/S1V ( ~ B0124/S013V)

The last rule(pair) has a known c/12d glider, found by myself:

Code: Select all

#C c/12d
x = 4, y = 4, rule = B02/S1V
o$obo$b2o$3bo!

Its corresponding 3ITVN rulestring is with maximal number of zeroes: "020222-00000-0000-100-00-1";
Enumerated manually from this rulepair: (B134/S0234V, B3/S24V) using that such minimal 3ITVN rule has the following form "0_____-_0000-_000-_00-_0-_" meaning neighbourhoods containing both 1 and 2 value cells must extinct.

[Naszvadi]

Lemma: Filtering rules out in which a pattern cannot escape from its bounding box.

Generated all 3*2 boxes with this python2 script and then per pattern advanced 2 generations in our favourite CA software:

Code: Select all

#!/usr/bin/python
RULES='B0/SV B0/S1V B0/S0V B0/S01V B04/SV B04/S1V B04/S0V B04/S01V B02/SV'
RULES+=' B02/S1V B02/S0V B02/S01V B024/SV B024/S1V B024/S0V B024/S01V'
RULES=RULES.split()
WORDS=[bin(_)[-3:].replace('0','b').replace('1','o')for _ in range(8,16)]
S='x=0,y=0,rule=%s\n'+'bbbbob'*8+'$'+''.join(_*2for _ in WORDS)+'!'
for R in RULES:print S%R;print

Code: Select all

x=0,y=0,rule=B0/SV
bbbbobbbbbobbbbbobbbbbobbbbbobbbbbobbbbbobbbbbob$bbbbbbbbobbobobbobboobooobbobboboobooobooboooooo!

Code: Select all

x=0,y=0,rule=B0/S0V
bbbbobbbbbobbbbbobbbbbobbbbbobbbbbobbbbbobbbbbob$bbbbbbbbobbobobbobboobooobbobboboobooobooboooooo!

Code: Select all

x=0,y=0,rule=B04/SV
bbbbobbbbbobbbbbobbbbbobbbbbobbbbbobbbbbobbbbbob$bbbbbbbbobbobobbobboobooobbobboboobooobooboooooo!

Verified: all diamond with radius 2 at all boundaries: the following three rules cannot have gliders: B0/SV, B0/S0V and B04/SV.
Q.E.D.

Remaining rules are:

1. B0/S1V
2. B0/S01V
3. B04/S1V
4. B04/S0V
5. B04/S01V
6. B02/SV
7. B02/S1V
8. B02/S0V
9. B02/S01V
10. B024/SV
11. B024/S1V
12. B024/S0V
13. B024/S01V

It is confirmed that the following 4 rules have gliders: B024/S1V, B024/SV, B02/S1V, B02/SV. Remaining 9 rules are to be dis/proven to have gliders.

[APPEND#1]

Filtering out rules in which patterns cannot escape from their bounding diamonds:

Code: Select all

x=0,y=0,rule=B0/S1V
$b$2b$3b$4b$5b$5bo$7b$8b$7bob$10b$11b$10boo$13b$13bo$15b$16b$16bo$17bo$19b$19bo$19bob$22b$22bo$22boo$25b$24bob$27b$28b$27bob$29bo$31b$30bob$31bob$34b$33bob$34boo$37b$36boo$39b$40b$39boo$41bo$43b$42boo$43bob$46b$45boo$46boo$48bo$50b$51b$51bo$53b$53bo$54bo$56b$55bob$57bo$59b$58boo$60bo$61bo$63b$63bo$64bo$65bo$66bo$67bo$67bob$69bo$70bo$70boo$72bo$72bob$75b$75bo$75bob$77bo$78bo$78bob$79bob$81bo$81bob$82boo$84bo$84boo$87b$87bo$87boo$89bo$90bo$90boo$91bob$93bo$93boo$94boo!

Code: Select all

x=0,y=0,rule=B0/S01V
$b$2b$3b$4b$5b$5bo$7b$8b$7bob$10b$11b$10boo$13b$13bo$15b$16b$16bo$17bo$19b$19bo$19bob$22b$22bo$22boo$25b$24bob$27b$28b$27bob$29bo$31b$30bob$31bob$34b$33bob$34boo$37b$36boo$39b$40b$39boo$41bo$43b$42boo$43bob$46b$45boo$46boo$48bo$50b$51b$51bo$53b$53bo$54bo$56b$55bob$57bo$59b$58boo$60bo$61bo$63b$63bo$64bo$65bo$66bo$67bo$67bob$69bo$70bo$70boo$72bo$72bob$75b$75bo$75bob$77bo$78bo$78bob$79bob$81bo$81bob$82boo$84bo$84boo$87b$87bo$87boo$89bo$90bo$90boo$91bob$93bo$93boo$94boo!

Removing 2 rules: B0/S1V and B0/S01V, the remaining eleven's list:

1. B04/S1V
2. B04/S0V
3. B04/S01V
4. B02/SV
5. B02/S1V
6. B02/S0V
7. B02/S01V
8. B024/SV
9. B024/S1V
10. B024/S0V
11. B024/S01V

Used this python2 script for enumerating all diagonal boundaires:

Code: Select all

#!/usr/bin/python

RULES='B0/S1V B0/S01V B04/S1V B04/S0V B04/S01V B02/SV B02/S1V B02/S0V B02/S01V B024/SV B024/S1V B024/S0V B024/S01V'.split()

WORDS5=[bin(_)[-5:].replace('0','b').replace('1','o')for _ in range(32,64)]

S='x=0,y=0,rule=%s\n'

for n,i in enumerate(WORDS5):
    if S: S+='$'
    S+=(3*n*'b').join(['',i[0]+'$',i[1:3]+'$','b'+i[3:]])

for R in RULES:
    print (S+'!')%R
    print

[Naszvadi]

Extra respect to the author(s) of Rust Life Search browser version!

Now it is an article-quality discovery, a new glider! Found on a machine T3000 64 bit linux host, gecko-alike browser engine, the glider is c/4 orthogonal!

Code: Select all

x = 12, y = 12, rule = B04/S1V
............$
.......o....$
.....oo.o...$
.......o....$
...o........$
..o..o......$
...o........$
....ooo.....$
.......oo.o.$
............$
............$
............!

Setup:
Rust Life SearchA Game of Life pattern searcher written in Rust.

• Rule: B04/S1V
• Width: 12
• Height: 12
• Period: 8
• dx: 2
• dy: 0
• Transformation: Id
• Symmetry: C1
• Max cell count: 0
• Search order: Automatic
• Choice of state for unknown cells: Alive
• Force the first row or column to be nonempty "False"
• Reduce the max cell count when a result is found "False"

https://alephalpha.github.io/rlifesrc/

The following rules could support gliders, bolded are already confirmed having at least one:

1. B04/S1V
2. B04/S0V
3. B04/S01V
4. B02/SV
5. B02/S1V
6. B02/S0V
7. B02/S01V
8. B024/SV
9. B024/S1V
10. B024/S0V
11. B024/S01V

[Naszvadi]

Extra respect to the author(s) of Rust Life Search browser version!

Now it is an article-quality discovery, a new glider! Found on a machine T3000 64 bit linux host, gecko-alike browser engine, the glider is c/4 orthogonal!

Code: Select all

x = 12, y = 12, rule = B04/S1V
............$
.......o....$
.....oo.o...$
.......o....$
...o........$
..o..o......$
...o........$
....ooo.....$
.......oo.o.$
............$
............$
............!

Setup:
Rust Life SearchA Game of Life pattern searcher written in Rust.

• Rule: B04/S1V
• Width: 12
• Height: 12
• Period: 8
• dx: 2
• dy: 0
• Transformation: Id
• Symmetry: C1
• Max cell count: 0
• Search order: Automatic
• Choice of state for unknown cells: Alive
• Force the first row or column to be nonempty "False"
• Reduce the max cell count when a result is found "False"

https://alephalpha.github.io/rlifesrc/

The following rules could support gliders, bolded are already confirmed having at least one:

1. B04/S1V
2. B04/S0V
3. B04/S01V
4. B02/SV
5. B02/S1V
6. B02/S0V
7. B02/S01V
8. B024/SV
9. B024/S1V
10. B024/S0V
11. B024/S01V

Update - B02/S0V has at least 2 gliders: a c/2o and a 4c/8o, the latter found manually after the first one:

Code: Select all

x = 3, y = 22, rule = B02/S0V
o$obo2$obo$o12$o$2o3$2o$o!

The following rules could support gliders, bolded are already confirmed having at least one:

1. B04/S1V
2. B04/S0V
3. B04/S01V
4. B02/SV
5. B02/S1V
6. B02/S0V
7. B02/S01V
8. B024/SV
9. B024/S1V
10. B024/S0V
11. B024/S01V