Thread for basic non-CGOL questions

[WhiteHawk]

Is there a faster/more compact stable 180-degree glider reflector in Pedestrian Life EDIT: That satisfies the requirement that returning reflected gliders must not collide with later input gliders?

Code: Select all

x = 37, y = 50, rule = B38/S23
11bo$10bobo$10bobo$8b3ob2o$7bo$8b3ob2o$10bob2o7$9b2o$9b2o5$18bo$17bo$
17b3o2$16bo$15bobo$2o13bobo$bo14bo$bobo$2b2o2$33b2o$33bobo$35bo$35b2o
8$25b2o$25bobo$27bo$11b2o14b2o$10bobo$10bo$9b2o5bo$15bobo$16bo!

EDIT: Looking into SKOPs in Pedestrian Life; as the Rectifier loop fails in this rule, different stable 180 glider reflectors (or Snark Loops) must be used for high-period prime SKOPs.

[confocaloid]

edit: I'm not aware of anything with either lower population or lower repeat time than the reflector you posted. So answering the question in an "interesting" way probably needs new discoveries. Presumably the forum thread was already scanned?

I think you'll need to add the requirement that returning reflected gliders must not collide with later input gliders, otherwise it's impossible to construct a glider loop oscillator.

The Jubjub reflector doesn't work for even-period loops, either. It's a same-lane reflector, which means a returning reflected glider will collide with the next input glider. Hence only a family of glider shuttles is possible with Jubjubs.

Is there a faster/more compact stable 180-degree glider reflector in Pedestrian Life (Excluding the Jubjub, which doesn't work for odd period loops)?

Code: Select all

x = 37, y = 50, rule = B38/S23
11bo$10bobo$10bobo$8b3ob2o$7bo$8b3ob2o$10bob2o7$9b2o$9b2o5$18bo$17bo$
17b3o2$16bo$15bobo$2o13bobo$bo14bo$bobo$2b2o2$33b2o$33bobo$35bo$35b2o
8$25b2o$25bobo$27bo$11b2o14b2o$10bobo$10bo$9b2o5bo$15bobo$16bo!

[b-engine]

What are known examples of "misleading symmetry" appearing in common evolutionary sequences?

One could engineer a pattern that does this, like destroying a beehive from a honeyfarm that's almost formed, and reconstruct the beehive via some way:

Code: Select all

x = 45, y = 44, rule = B3/S23
23b2o$23b2o4$20b2obo$21b3o$22bo$bo$ob2o7b2o$b2o8bo31$42b3o$42bo$43bo!

I don't know any examples, but it may be possible by engineering a rule which behaves like B3/S23, but symmetric predecessors always result in a common pattern - this means that any such constellation formed in a different way would be considered a different patterns by APGsearch.

[confocaloid]

I think you misunderstood the question. I was asking about commonly occurring non-symmetric predecessors (asymmetric predecessors), which "just so happen" to evolve into highly symmetric constellations, without any "good reason" for why the final configuration must be symmetric.

As an example, I described an imaginary scenario, where an asymmetric seed evolves to leave two differently-oriented beehives, then continues to evolve to leave a third beehive, and then the remainder of the active reaction would evolve into the fourth beehive, all four beehives forming the honey farm constellation, without any obvious reason for the resulting symmetry.

What are known examples of "misleading symmetry" appearing in common evolutionary sequences?

What I mean by "misleading symmetry" here: one can imagine a common asymmetric predecessor that eventually evolves into a highly symmetric constellation (like traffic light or honey farm in Conway's Life), except the symmetric constellation forms in an asymmetric way (for example, the honey farm constellation might form by leaving first two beehives oriented differently, then leaving a third beehive, and only later leaving the fourth beehive, resulting in an entirely asymmetric formation of a highly symmetric pattern).

One can expect a "simple" way of engineering a solution with more than two states, so this question is probably limited to two-state cellular automata.
Also the CA must be isotropic (rotations/reflections that preserve the grid lines must also preserve the reactions).
Other than that, any tilings (square/hexagonal/triangular) and any neighbourhoods/ranges are accepted.

Code: Select all

x = 13, y = 13, rule = B3/S23
6bo$5bobo$5bobo$6bo2$b2o7b2o$o2bo5bo2bo$b2o7b2o2$6bo$5bobo$5bobo$6bo!

What are known examples of "misleading symmetry" appearing in common evolutionary sequences?

One could engineer a pattern that does this, like destroying a beehive from a honeyfarm that's almost formed, and reconstruct the beehive via some way:

Code: Select all

x = 45, y = 44, rule = B3/S23
23b2o$23b2o4$20b2obo$21b3o$22bo$bo$ob2o7b2o$b2o8bo31$42b3o$42bo$43bo!

I don't know any examples, but it may be possible by engineering a rule which behaves like B3/S23, but symmetric predecessors always result in a common pattern - this means that any such constellation formed in a different way would be considered a different patterns by APGsearch.

[Haycat2009]

Can someone please help me fix the parity mismatch for this XOR gate? The top-center OR gate has to be compatible with both bottom and right inputs.

Code: Select all

x = 69, y = 76, rule = B2a3i/S12History
37.A6.A15.A6.A$38.A4.A17.A4.A2$26.A$25.A2$38.A22.A$37.A16.A5.A$53.A2$
10.A15.A$9.A15.A5$6.A6.A8.A6.A30.A6.A$7.A4.A10.A4.A32.A4.A5$12.A15.A37.
A$13.A15.A37.A5$25.A30.A8.A$26.A30.A6.A$51.A$52.A3$6.A6.A8.A6.A21.A6.
A2.A6.A$7.A4.A10.A4.A23.A4.A4.A4.A2$.A44.A$A44.A2$7.A15.A28.A9.A$6.A15.
A28.A9.A3$22.A6.A$23.A4.A$55.A$48.A5.A$49.A2$28.A$29.A$51.A6.A2.A6.A$
52.A4.A4.A4.A$25.2A$24.A2.A3$57.A9.A$58.A9.A5$37.A6.A6.A6.A$38.A4.A8.
A4.A2$64.A$63.A2$43.A13.A$44.A13.A3$40.2D$39.D2.D!

[hotcrystal0]

Are there any rules with a very common (3,2) ship?

[LaundryPizza03]

Are there any rules with a very common (3,2) ship?

This rule with small (3,2)c/8 and 4c/26o spaceships probably answers your question:

Code: Select all

x = 6, y = 16, rule = B2k3-qr4q5en6ck/S2-kn3-cky4wy5aijq6k
2bo$b2obo$2bob2o$3bo9$obo$2b2o$obo$bo!

EDIT: For this one, B36cn8/S2ace3ijnr4aijnrwz5ainy6ak is also interesting.

Code: Select all

x = 3, y = 5, rule = B36acn8/S2-k3ijknr4aijnrwz5ainy6ak
2o$bo$b2o$2o$o!

[Layz Boi]

Can someone please help me fix the parity mismatch for this XOR gate? The top-center OR gate has to be compatible with both bottom and right inputs.

I hope this is what you needed. Changes surrounded in red.
Note it does add some delay.

Code: Select all

x = 73, y = 78, rule = B2a3i/S12History
64.9D$64.D7.D$37.A6.A19.D2.A4.D$38.A4.A20.D.A5.D$64.D7.D$26.A37.D5.A.
D$25.A38.D4.A2.D$64.D7.D$38.A25.D.A5.D$37.A16.A9.D2.A4.D$53.A10.D7.D$
64.9D$10.A15.A$9.A15.A5$6.A6.A8.A6.A30.A6.A$7.A4.A10.A4.A32.A4.A5$12.
A15.A37.A$13.A15.A37.A5$25.A30.A8.A$26.A30.A6.A$51.A$52.A3$6.A6.A8.A6.
A21.A6.A2.A6.A$7.A4.A10.A4.A23.A4.A4.A4.A2$.A44.A$A44.A2$7.A15.A28.A9.
A$6.A15.A28.A9.A3$22.A6.A$23.A4.A$55.A$48.A5.A$49.A2$28.A$29.A$51.A6.
A2.A6.A$52.A4.A4.A4.A$25.2A$24.A2.A3$57.A9.A$58.A9.A5$37.A6.A6.A6.A$38.
A4.A8.A4.A2$64.A$63.A2$43.A13.A$44.A13.A3$40.2D$39.D2.D!

Some extra stuff that changes parity/color: (not optimized at all)

Code: Select all

x = 75, y = 87, rule = B2a3i/S12
24bo$25bo$25bo$24bo10$36bo$34bo2bo4bo$35bo5bo$24bo$25bo$25bo$24bo18bo
$44bo$41bo$40bo5$38bo$39bo10$36bo$37bo2$41bo2bo$34bo5bo2bo$35bo$24bo$
25bo$25bo$24bo18bo$42bo$37bo$38bo5$o$bo$bo$o7$39bo15bo$38bo15bo2$61bo
$60bo5bo$67bo$40bo8bo4bo$41bo6bo6bo2bo12bo2bo$44bo10bobo6bo5bo2bo$24b
o18bo10bo10bo$25bo$25bo$24bo$35bo5bo31bo$34bo2bo4bo6bo4bo17bo$36bo11b
o6bo11bo$68bo4$52bo$51bo!

[H. H. P. M. P. Cole]

There is a known 56-cell periodic pattern in Life that is endemic assuming the INT rulespace.

However, other rulespaces have smaller endemic patterns. Take this p2 oscillator, for example:

Code: Select all

x = 3, y = 1, rule = R1,C2,S0-1,B2,NW000101000
obo!

In the R1 1D OT rulespace, it queries:
- B0, B1, B2, S0 at even generations
- B0, B1, S1, S2 at odd generations

making it endemic to the 2^6-size range-1 OT 1D rulespace.

Can we make collections of the smallest endemic objects within certain small rulespaces? (There should be some in my 1D ship collections, but unfortunately I am on mobile and I cannot access them for now.)

[WhiteHawk]

Is there a way to quickly turn an R into a glider in order to "Fix" the p21 gun in Pedestrian Life?

Code: Select all

x = 38, y = 21, rule = B38/S23
14b2o$14bobo9b2o5b2o$15bo10b2o5b2o$19bo$18b3o$17bo2b2o4b3o3b3o$17bo3b
o2bo4bobo4bo$19bobo7bobo$23bo3bobobobo3bo$16b2ob2o3bo2bobobobo2bo$4b2o
3b2o7bo6b3obobob3o$2b3obobob3o6bo7b2o3b2o$bo2bobobobo2bo3b2ob2o$o3bob
obobo3bo$6bobo7bobo$bo4bobo4bo2bo3bo$3b3o3b3o4b2o2bo$17b3o$18bo$3b2o5b
2o$3b2o5b2o!

EDIT: There is, so is there a way to insert some spark as shown?

Code: Select all

x = 36, y = 25, rule = B38/S23
13b2o$13bo4b2o5b2o5b2o$14b2o4bo4b2o5b2o$18bo$21bo$15bo9b2o5b2o$16b2ob
o4bo2bo3bo2bo$18bo5bo3bobo3bo$24bo3bobo3bo$19bo3bo2bobobobo2bo$4bo3bo
8b3o4bobobobobobo$bobobobobobo4b3o8bo3bo$o2bobobobo2bo3bo$bo3bobo3bo$
bo3bobo3bo5bo$bo2bo3bo2bo4bob2o$2b2o5b2o9bo$14bo$17bo$2b2o5b2o4bo2bo$
2b2o5b2o5b3o2$13b2o$13b3o$13b3o!

EDIT 2: 34p7 works, so is there anything smaller?

Code: Select all

x = 36, y = 38, rule = B38/S23
13b2o$13bo4b2o5b2o5b2o$14b2o4bo4b2o5b2o$18bo$21bo$15bo9b2o5b2o$16b2ob
o4bo2bo3bo2bo$18bo5bo3bobo3bo$24bo3bobo3bo$19bo3bo2bobobobo2bo$4bo3bo
8b3o4bobobobobobo$bobobobobobo4b3o8bo3bo$o2bobobobo2bo3bo$bo3bobo3bo$
bo3bobo3bo5bo$bo2bo3bo2bo4bob2o$2b2o5b2o9bo$14bo$17bo$2b2o5b2o4bo2bo$
2b2o5b2o5b3o2$13b2o$12b4o$11b2obobo$10b2o3b2o$11bo$9bo$10b3o$8b2o2b2o
$11b2o2b2o$12b3o$15bo$13bo$8b2o3b2o$8bobob2o$9b4o$10b2o!

[H. H. P. M. P. Cole]

Is it possible to make a complete list of spaceships in the NW00000001000000000000050000000000001900000001051900190501000000190000000000000500000000000001000000 rulespace where there are more than 2^233 rules that the spaceship "works" in?

The bound of 2^234 is so that I can include these spaceships:

Code: Select all

x = 73, y = 7, rule = R3,C2,S26,B30,55,NW00000001000000000000050000000000001900000001051900190501000000190000000000000500000000000001000000
o13bo13bo13bo13bo13bo$o13bobo9bobobo11bo13bobo9bobobo2$o13bo13bo13bo13b
o13bo3$42bo13bo13bo!

The two rightmost spaceships work in two different 2^234-large subrulespaces.

[confocaloid]

Is it possible to make a complete list of spaceships in the NW[......] rulespace where there are more than 2^234 rules that the spaceship "works" in?

The bound of 2^234 is so that I can include these spaceships: [...] The two rightmost spaceships work in two different 2^234-large subrulespaces.

I'm not sure if I understand the question. If you specify "more than 2^234 cases", then everything that works in exactly 2^234 cases is going to be excluded, right?

It's worth noting that the entire space includes 2^250 cellular automata (0 through 1 the current state of the cell, 0 through 4 alive range-1 neighbours, 0 through 4 alive range-2 neighbours, 0 through 4 alive range-3 neighbours -> 2 x 5 x 5 x 5 = 250 distinguishable fully-specified cell conditions). Here are the neighbour weights in decimal:

Code: Select all

-- -- -- 01 -- -- --
-- -- -- 05 -- -- --
-- -- -- 25 -- -- --
01 05 25 -- 25 05 01
-- -- -- 25 -- -- --
-- -- -- 05 -- -- --
-- -- -- 01 -- -- --

The bound of 2^234 is so that I can include these spaceships:

Code: Select all

x = 73, y = 7, rule = R3,C2,S26,B30,55,NW00000001000000000000050000000000001900000001051900190501000000190000000000000500000000000001000000
o13bo13bo13bo13bo13bo$o13bobo9bobobo11bo13bobo9bobobo2$o13bo13bo13bo13b
o13bo3$42bo13bo13bo!

The two rightmost spaceships work in two different 2^234-large subrulespaces.

I think that entire collection works in 2^230 cellular automata of the same kind and with the same neighbour weights:

Code: Select all

optional birth conditions (113): 2-4, 7-9, 11-24, 27-29, 32-49, 52-54, 56-124
optional survival conditions (117): 0, 2-4, 8-25, 27-29, 31-34, 36-39, 41-124

And here are what I believe to be the optional living conditions for every single object in the collection, going from left to right:

• (2^239 CA) optional birth conditions (117): 2-4, 7-24, 27-29, 31-54, 56-124, optional survival conditions (122): 0-5, 7-25, 27-29, 31-124
• (2^236 CA) optional birth conditions (115): 2-4, 7-9, 11-24, 27-29, 31-49, 51-54, 56-124, optional survival conditions (121): 0-4, 7-25, 27-34, 36-124
• (2^236 CA) optional birth conditions (115): 2-4, 7-9, 11-24, 27-29, 31-50, 52-54, 56-124, optional survival conditions (121): 0-4, 7-25, 27-39, 41-124
• (2^238 CA) optional birth conditions (117): 2-4, 7-24, 26-29, 32-54, 56-124, optional survival conditions (121): 0, 2-6, 8-25, 27-29, 31-124
• (2^234 CA) optional birth conditions (114): 2-4, 7-9, 11-24, 27-29, 32-49, 51-54, 56-124, optional survival conditions (120): 0, 2-4, 6, 8-25, 27-34, 36-124
• (2^234 CA) optional birth conditions (114): 2-4, 7-9, 11-24, 27-29, 32-50, 52-54, 56-124, optional survival conditions (120): 0, 2-4, 6, 8-25, 27-39, 41-124

[H. H. P. M. P. Cole]

Is it possible to make a complete list of spaceships in the NW[......] rulespace where there are more than 2^233 rules that the spaceship "works" in?

The bound of 2^234 is so that I can include these spaceships: [...] The two rightmost spaceships work in two different 2^234-large subrulespaces.

I'm not sure if I understand the question. If you specify "more than 2^234 cases", then everything that works in exactly 2^234 cases is going to be excluded, right?

My bad, I modified my question so that anything that works in exactly 2^234 CA is included, but anything that works in 2^233 CA or lower is excluded.

[confocaloid]

Is it possible to make a complete list of spaceships [...]

I think the following infinite family of spaceships proves that it's impossible to make a complete list of spaceships existing in at least 2^234 CA of the specified kind:

Code: Select all

x = 15, y = 71, rule = R3,C0,S31,56-57,61-62,B31,56,NW00000001000000000000050000000000001900000001051900190501000000190000000000000500000000000001000000
ob6o10$ob7o10$ob8o10$ob9o10$ob10o10$ob11o10$ob12o10$ob13o!

This family exists in 2^235 cellular automata with the same weighted neighbourhood:

Code: Select all

optional birth conditions (117): 2-4, 7-24, 27-30, 32-55, 57-124
optional survival conditions (118): 0-5, 7-30, 32-35, 37-55, 58-60, 63-124

[H. H. P. M. P. Cole]

Is it possible to make a complete list of spaceships [...]

I think the following infinite family of spaceships proves that it's impossible to make a complete list of spaceships existing in at least 2^234 CA of the specified kind:

Code: Select all

x = 15, y = 71, rule = R3,C0,S31,56-57,61-62,B31,56,NW00000001000000000000050000000000001900000001051900190501000000190000000000000500000000000001000000
ob6o10$ob7o10$ob8o10$ob9o10$ob10o10$ob11o10$ob12o10$ob13o!

This family exists in 2^235 cellular automata with the same weighted neighbourhood:

Code: Select all

optional birth conditions (117): 2-4, 7-24, 27-30, 32-55, 57-124
optional survival conditions (118): 0-5, 7-30, 32-35, 37-55, 58-60, 63-124

I think this family exists in a 2^238-large rulespace, assuming my calculation of the optional birth conditions and optional survival conditions is correct:

Code: Select all

x = 21, y = 49, rule = R3,C2,S,B26,51-52,NW00000001000000000000050000000000001900000001051900190501000000190000000000000500000000000001000000
o3bobobo8$o3bobobobo8$o3bobobobobo8$o3bobobobobobo8$o3bobobobobobobo8$
o3bobobobobobobobo8$o3bobobobobobobobobo!
12

Code: Select all

optional birth conditions (116): 2-4, 6-9, 11-24, 28-50, 53-124
optional survival conditions (122): 1-4, 6-9, 11-124

[confocaloid]

[...] I think this family exists in a 2^238-large rulespace, assuming my calculation of the optional birth conditions and optional survival conditions is correct:

Code: Select all

x = 21, y = 49, rule = R3,C2,S,B26,51-52,NW00000001000000000000050000000000001900000001051900190501000000190000000000000500000000000001000000
o3bobobo8$o3bobobobo8$o3bobobobobo8$o3bobobobobobo8$o3bobobobobobobo8$
o3bobobobobobobobo8$o3bobobobobobobobobo!
12

Code: Select all

optional birth conditions (116): 2-4, 6-9, 11-24, 28-50, 53-124
optional survival conditions (122): 1-4, 6-9, 11-124

Yes, I get the same answer for the space where those spaceships can be found. And of course one can modify the neighbourhood to add more objects:

Code: Select all

x = 76, y = 161, rule = R3,C0,S170-298,B86,171-172,256-298,440-451,NW00000501050000000500050005000500555555000501055500550501050055555500050005000500050000000501050000
o3bobobo8$o3bobobobo8$o3bobobobobo8$o3bobobobobobo8$o3bobobobobobobo8$
o3bobobobobobobobo8$o3bobobobobobobobobo18$18b3o$16bo4bo25b2o$17b2o3bo
24b3o$16bo4bo25b2o$18b3o15$5b2o19bo$3bo4bo15bo3bo16bo2bo25bo$9bo19bo
19bo25bo$3bo5bo14bo4bo15bo3bo23b3o$4b6o15b5o16b4o16$4bobobobobobobobob
o17$2bobo3bobo3bobo3bo$6bo5bo5bo17$3bo5bo5bo5bo$bo3bobo3bobo3bobo$3bo
5bo5bo5bo17$6bo5bo5bo$2bobo3bobo3bobo3bo!

Code: Select all

x = 48, y = 89, rule = R3,C0,S11,20-28,100-108,180-191,262-272,342-352,423-434,505-518,596-681,B11,20,109,190-211,270-273,289-290,351-354,433,514,596-695,NW00510001005100515151015151510051090909510001010900090101005109090951005151510151515100510001005100
ob6o10$ob7o10$ob8o10$ob9o10$ob10o10$ob11o10$ob12o10$ob13o14$16b3o$14bo
4bo26bo$15b2o3bo26bo$14bo4bo25b3o$16b3o!

[muzik]

Are there any high-period statorless hexagonal grid oscillators with C6 symmetry and D12 time symmetry that don't use B0?

Here's one with B0:

Code: Select all

x = 27, y = 27, rule = R2,C0,S0-5,10,B0,2-3,6-7,NH
10bo3$o7$23bo6$3bo7$26bo3$16bo!

Here's a square grid example of something similar to what I'm looking for:

Code: Select all

x = 17, y = 17, rule = B3-r4cekz5aei6-ae78/S2ae3-aei4-cekz5r678
7b5o$9bo3bo$7b5o$bob2ob8o$3b11o$ob11o$ob12o$15obo$ob13obo$ob15o$3b12obo$4b11o
bo$3b11o$3b8ob2obo$5b5o$3bo3bo$5b5o!

[confocaloid]

Previous discussion of what appears to be a closely-related question:
viewtopic.php?p=191040#p191040 Re: Hexagonal self-complementary rules

Are there any high-period statorless hexagonal grid oscillators with C6 symmetry and D12 time symmetry that don't use B0? [...]

[confocaloid]

I think there were some known 'interesting' examples of the following, but I can't remember where they were posted.

Imagine an oscillator that repeatedly creates and destroys some compact stationary "junk" (a small object or a compact constellation), for example a block, as in the following illustration. The state-2 region on the left must contain the same configuration of dead/alive cells as the state-2 region on the right. The left side and the right side are intended to be two phases of a period-(T1+T2) oscillator that takes T1 ticks to get from the left phase to the right phase (creating the "junk"), takes T2 ticks to get from the right phase to the left phase (removing the "junk"), and additionally T1 /= T2 and gcd(T1, T2) = 1.

Code: Select all

x = 47, y = 15, rule = LifeHistory
23.F$23.F$13B10.F6.13B$13B10.F6.13B$13B2.2D6.F6.13B2.2C$13B2.2D6.F6.
13B2.2C$13B10.F6.13B$13B10.F6.13B$13B10.F6.13B$13B10.F6.13B$13B10.F6.
13B$13B10.F6.13B$13B10.F6.13B$23.F$23.F!

What are known non-engineered examples in two-state isotropic CA with Moore neighbourhood?
Are there known examples on the hexagonal tiling (or any interesting examples of this for that matter)?

[islptng]

I still wonder that why some rule do not explode on asymmetry but is explosive on D2_+1.
For example, this rule:

Someone (on discord) told me that there exists some rule that explodes on D2_x and D2_+2, but the two rule is just a simple linear growth, unlike some natural D2_+1-explosive rules(which is quadratic).

[confocaloid]

When the rules of a CA are isotropic (the same regardless of how the pattern is rotated/reflected), a pattern with some single-generation symmetries cannot lose those symmetries during evolution. For example an axis of symmetry cannot disappear during evolution. Locally, this limits possible 3-by-3 configurations that are crossed by the axis, and can change the probabilities of specific cell conditions occurring near the axis.

Additionally, events that would be otherwise very unlikely (such as two lightweight spaceships colliding with each other at 90 degrees) can become much more likely (e.g. in D2_x for this example) because once one of two lightweight spaceships appears and moves to the axis without any other reactions interfering with it, the other LWSS is guaranteed to appear and the collision is guaranteed to happen.

The details may vary depending on what are the rules, what are common evolutionary sequences, how symmetries of a reaction affect the situation.

There is no reason why more symmetric reactions should necessarily be "more explosive" or "more chaotic". Sometimes it is the other way round, and apgsearch works for symmetric soups, but stucks/consumes unreasonable amount of memory in the C1 case.

I still wonder that why some rule do not explode on asymmetry but is explosive on D2_+1.

Someone (on discord) told me that there exists some rule that explodes on D2_x and D2_+2, but the two rule is just a simple linear growth, unlike some natural D2_+1-explosive rules(which is quadratic).

[amling]

It's easy to make rules with no oscillators and there are known rules which are omniperiodic, but what lies between?

Are there any rules for which the complete classification of periods for which there exist finite oscillators is known, but for which the answer isn't eventually-all-yes (i.e. omniperiodic but for a finite number of somehow unattainable periods) or eventually-all-no (i.e. only a finite number of attainable periods)?

I think a Wang tile style argument can produce a CA (with a large neighborhood distance, a large state set, and requiring orientation) whose attainable periods are any recursively enumerable set and of course a simple brute force search shows any set of any CA is itself recursively enumerable. Some embedding arguments I think can get the CA requirements down to something like (2-state INT with large neighborhood) or (many state INT with distance 1 neighborhood).

I'm especially interested if there are any rules with interesting period sets which are plain 2-state 1-range INT or even 2-state 1-range OT.

[confocaloid]

I think "many states, distance 1 neighborhood" should work as follows, are there any flaws here?
Define a four-state isotropic CA with Moore neighbourhood, and with these restrictions on the update rules:

• A state-1 cell never remains state-1 in the next generation.
• A state-2 cell never remains state-2 in the next generation.
• A state-3 cell always remains state-3 in the next generation.
• Every possible interaction between state 1 and state 2 creates a state-3 cell.
• When any cell has at least one state-3 neighbour, it becomes state-3 in the next generation.
• Otherwise, patterns entirely made out of state-1 cells either instantly vanish or evolve into patterns entirely made out of state-2 cells; and vice versa, patterns entirely made out of state-2 cells either instantly vanish or evolve into patterns entirely made out of state-1 cells.

With such restrictions, I think odd-period finite oscillators are impossible. Any finite oscillator must be either a single object that alternates between being entirely state-1 and being entirely state-2, or is a constellation made out of such objects not interacting with each other. Any interaction between states 1 and 2 will cause unstoppable growth.

At the same time, it should be possible to achieve all even periods (for example, make the state-1 patterns follow B3/S23 when evolving into state-2 patterns, and make state-2 patterns follow B3/S23 when evolving into state-1 patterns).

Code: Select all

x = 44, y = 21, rule = Test_post205931
.A$2.A$3A16$15.10A2$34.10B!

@RULE Test_post205931

https://conwaylife.com/forums/viewtopic.php?p=205931#p205931

@TABLE
n_states:4
neighborhood:Moore
symmetries:permute

var p={0,1,2,3}
var q=p
var r=p
var s=p
var t=p
var u=p
var v=p
var w=p
var x=p

3, p,q,r,s,t,u,v,w, 3
x, p,q,r,s,t,u,v,3, 3
1, p,q,r,s,t,u,v,2, 3
2, p,q,r,s,t,u,v,1, 3
0, p,q,r,s,t,u,1,2, 3

0, 1,1,1,0,0,0,0,0, 2
1, 1,1,1,0,0,0,0,0, 2
1, 1,1,0,0,0,0,0,0, 2
0, 2,2,2,0,0,0,0,0, 1
2, 2,2,2,0,0,0,0,0, 1
2, 2,2,0,0,0,0,0,0, 1

x, p,q,r,s,t,u,v,w, 0

EDIT: there is at least one flaw, see the post by FWKnightship. To eliminate loopable oscillators like this, one would probably need to disallow "oscillators surrounded by empty space", and only allow oscillators on patches alternating between two different nonzero "background states".

[...] But state 1 and 2 in this rule simulates B2k3aeiry4ct5ckq6c8_S2-in3ijkqr4cinq5acikr6cin8, which has a p232(29 * 8 ) RRO.
So, we can still make a p29 (odd period) in this rule: [...]

[FWKnightship]

Are there any flaws here?

This 4 state rule:

• A state-1 cell never remains state-1 in the next generation.
• A state-2 cell never remains state-2 in the next generation.
• A state-3 cell always remains state-3 in the next generation.
• Every possible interaction between state 1 and state 2 creates a state-3 cell.
• When any cell has at least one state-3 neighbour, it becomes state-3 in the next generation.
• Otherwise, patterns entirely made out of state-1 cells either instantly vanish or evolve into patterns entirely made out of state-2 cells; and vice versa, patterns entirely made out of state-2 cells either instantly vanish or evolve into patterns entirely made out of state-1 cells.

But state 1 and 2 in this rule simulates B2k3aeiry4ct5ckq6c8_S2-in3ijkqr4cinq5acikr6cin8, which has a p232(29 * 8 ) RRO.
So, we can still make a p29 (odd period) in this rule:

Code: Select all

x = 33, y = 33, rule = p29
9.A.A11.2B$8.A2.A10.B4.B$7.2A13.B2.B$8.4A10.2B$9.A.A$.B2$2.B27.A$B28.
3A$B2.B24.2A2.A$.3B25.A$28.2A.2A10$2A.2A$3.A25.3B$A2.2A24.B2.B$.3A28.
B$2.A27.B2$31.B$21.A.A$9.2B10.4A$7.B2.B13.2A$5.B4.B10.A2.A$8.2B11.A.A
!
@RULE p29

********************************
**** COMPILED FROM NUTSHELL ****
****         v0.6.3         ****
********************************

@TABLE
neighborhood: Moore
symmetries: rotate4reflect
n_states: 4

var any.0 = {0,1,2,3}
var any.1 = any.0
var any.2 = any.0
var any.3 = any.0
var any.4 = any.0
var any.5 = any.0
var any.6 = any.0
var any.7 = any.0
var any.8 = any.0

0, 0, 0, 0, 0, 1, 0, 0, 1, 2
0, 0, 0, 0, 0, 1, 1, 1, 0, 2
0, 0, 0, 1, 0, 1, 0, 1, 0, 2
0, 0, 0, 0, 0, 0, 1, 1, 1, 2
0, 0, 0, 1, 0, 0, 1, 1, 0, 2
0, 0, 0, 1, 0, 0, 1, 0, 1, 2
0, 0, 1, 0, 1, 0, 1, 0, 1, 2
0, 0, 0, 1, 0, 0, 1, 1, 1, 2
0, 1, 0, 1, 0, 1, 0, 1, 1, 2
0, 0, 1, 0, 1, 1, 0, 1, 1, 2
0, 0, 0, 1, 1, 1, 0, 1, 1, 2
0, 1, 0, 1, 0, 1, 1, 1, 1, 2
0, 1, 1, 1, 1, 1, 1, 1, 1, 2
1, 0, 0, 0, 0, 0, 1, 0, 1, 2
1, 0, 0, 0, 0, 1, 0, 1, 0, 2
1, 0, 0, 0, 0, 1, 0, 0, 1, 2
1, 0, 0, 0, 0, 0, 0, 1, 1, 2
1, 0, 0, 0, 0, 0, 1, 1, 1, 2
1, 0, 0, 0, 1, 1, 0, 1, 0, 2
1, 0, 0, 1, 0, 1, 0, 0, 1, 2
1, 0, 0, 0, 1, 1, 0, 0, 1, 2
1, 0, 0, 1, 0, 0, 1, 1, 0, 2
1, 0, 1, 0, 1, 0, 1, 0, 1, 2
1, 0, 0, 1, 1, 0, 1, 1, 0, 2
1, 0, 0, 0, 1, 0, 1, 1, 1, 2
1, 0, 0, 1, 1, 1, 0, 0, 1, 2
1, 0, 0, 0, 1, 1, 1, 1, 1, 2
1, 1, 0, 1, 0, 1, 0, 1, 1, 2
1, 0, 0, 1, 1, 1, 1, 1, 0, 2
1, 0, 1, 0, 1, 1, 0, 1, 1, 2
1, 0, 1, 1, 0, 0, 1, 1, 1, 2
1, 1, 0, 1, 0, 1, 1, 1, 1, 2
1, 0, 1, 1, 1, 0, 1, 1, 1, 2
1, 1, 0, 1, 1, 1, 0, 1, 1, 2
1, 1, 1, 1, 1, 1, 1, 1, 1, 2
0, 0, 0, 0, 0, 2, 0, 0, 2, 1
0, 0, 0, 0, 0, 2, 2, 2, 0, 1
0, 0, 0, 2, 0, 2, 0, 2, 0, 1
0, 0, 0, 0, 0, 0, 2, 2, 2, 1
0, 0, 0, 2, 0, 0, 2, 2, 0, 1
0, 0, 0, 2, 0, 0, 2, 0, 2, 1
0, 0, 2, 0, 2, 0, 2, 0, 2, 1
0, 0, 0, 2, 0, 0, 2, 2, 2, 1
0, 2, 0, 2, 0, 2, 0, 2, 2, 1
0, 0, 2, 0, 2, 2, 0, 2, 2, 1
0, 0, 0, 2, 2, 2, 0, 2, 2, 1
0, 2, 0, 2, 0, 2, 2, 2, 2, 1
0, 2, 2, 2, 2, 2, 2, 2, 2, 1
2, 0, 0, 0, 0, 0, 2, 0, 2, 1
2, 0, 0, 0, 0, 2, 0, 2, 0, 1
2, 0, 0, 0, 0, 2, 0, 0, 2, 1
2, 0, 0, 0, 0, 0, 0, 2, 2, 1
2, 0, 0, 0, 0, 0, 2, 2, 2, 1
2, 0, 0, 0, 2, 2, 0, 2, 0, 1
2, 0, 0, 2, 0, 2, 0, 0, 2, 1
2, 0, 0, 0, 2, 2, 0, 0, 2, 1
2, 0, 0, 2, 0, 0, 2, 2, 0, 1
2, 0, 2, 0, 2, 0, 2, 0, 2, 1
2, 0, 0, 2, 2, 0, 2, 2, 0, 1
2, 0, 0, 0, 2, 0, 2, 2, 2, 1
2, 0, 0, 2, 2, 2, 0, 0, 2, 1
2, 0, 0, 0, 2, 2, 2, 2, 2, 1
2, 2, 0, 2, 0, 2, 0, 2, 2, 1
2, 0, 0, 2, 2, 2, 2, 2, 0, 1
2, 0, 2, 0, 2, 2, 0, 2, 2, 1
2, 0, 2, 2, 0, 0, 2, 2, 2, 1
2, 2, 0, 2, 0, 2, 2, 2, 2, 1
2, 0, 2, 2, 2, 0, 2, 2, 2, 1
2, 2, 0, 2, 2, 2, 0, 2, 2, 1
2, 2, 2, 2, 2, 2, 2, 2, 2, 1
0, 1, 2, any.0, any.1, any.2, any.3, any.4, any.5, 3
0, 1, any.0, 2, any.1, any.2, any.3, any.4, any.5, 3
0, 1, any.0, any.1, 2, any.2, any.3, any.4, any.5, 3
0, 1, any.0, any.1, any.2, 2, any.3, any.4, any.5, 3
0, 2, 1, any.0, any.1, any.2, any.3, any.4, any.5, 3
0, 2, any.0, any.1, 1, any.2, any.3, any.4, any.5, 3
0, any.0, 1, any.1, 2, any.2, any.3, any.4, any.5, 3
0, any.0, 1, any.1, any.2, any.3, 2, any.4, any.5, 3
1, 2, any.0, any.1, any.2, any.3, any.4, any.5, any.6, 3
1, any.0, 2, any.1, any.2, any.3, any.4, any.5, any.6, 3
2, 1, any.0, any.1, any.2, any.3, any.4, any.5, any.6, 3
2, any.0, 1, any.1, any.2, any.3, any.4, any.5, any.6, 3
3, any.0, any.1, any.2, any.3, any.4, any.5, any.6, any.7, 3
any.0, 3, any.1, any.2, any.3, any.4, any.5, any.6, any.7, 3
any.0, any.1, 3, any.2, any.3, any.4, any.5, any.6, any.7, 3
any.0, any.1, any.2, any.3, any.4, any.5, any.6, any.7, any.8, 0

[confocaloid]

[...] Some embedding arguments I think can get the CA requirements down to something like (2-state INT with large neighborhood) or (many state INT with distance 1 neighborhood). [...]

I think "many states, distance 1 neighborhood" should work as follows, are there any flaws here? [...]

Are there any flaws here?

This 4 state rule: [...]
But state 1 and 2 in this rule simulates B2k3aeiry4ct5ckq6c8_S2-in3ijkqr4cinq5acikr6cin8, which has a p232(29 * 8 ) RRO.
So, we can still make a p29 (odd period) in this rule: [...]

Well, I think the following should work, but of course it would be interesting to know why if it doesn't:

Code: Select all

x = 270, y = 137, rule = Test_post206089
156.C4.C$154.2C.4C.2C$156.C4.C$8.A$7.A.A$8.A2$6.5A$5.A4.A20.2A36.C3.C
14.A3.A$4.A2.A23.A37.5C14.5A$.A2.A.2A21.A.A$A.A.A5.A18.2A$.A2.A4.A.A$
4.2A2.A2.A$9.2A$40.2A$40.A89.2C$38.A.A90.2C$38.2A90.C$116.C4.C$20.2A
92.2C.4C.2C$20.2A94.C4.C5$72.2C14.2A$72.2C14.2A$74.2C10.2A$74.2C10.2A
$43.A8.A$44.2A5.A.A$43.2A6.A.A$30.A21.A$29.A.A6.2A$29.A.A5.2A$30.A8.A
101.A4.A30.3C$130.2A7.2A.4A.2A28.C3D$131.2A8.A4.A31.C2D$130.A48.2D$
116.A4.A58.D$114.2A.4A.2A$116.A4.A3$61.2A$61.2A2$43.2A$42.A.A$42.A$
41.2A$72.2A$71.A2.A2.2A$71.A.A4.A2.A$52.2A18.A5.A.A.A$51.A.A21.2A.A2.
A$51.A23.A2.A$50.2A20.A4.A$72.5A2$74.A$73.A.A$74.A71$267.3A$267.A$
268.A!

@RULE Test_post206089

https://conwaylife.com/forums/viewtopic.php?p=206089#p206089

Cellstates:
0 = quiescent background, two non-forbidden transitions: 0->0, 0->5.
1 = live A, three non-forbidden transitions: 1->3 (survival A->B), 1->4 (death A->B), 1->5.
2 = dead A, three non-forbidden transitions: 2->3 (birth A->B), 2->4 (abstain A->B), 2->5.
3 = live B, three non-forbidden transitions: 3->1 (survival B->A), 3->2 (death B->A), 3->5.
4 = dead B, three non-forbidden transitions: 4->1 (birth B->A), 4->2 (abstain B->A), 4->5.
5 = kaboom, one non-forbidden transition: only 5->5.

Connected configurations involving only cellstates 0, 1, 2 follow CGoL (B3/S23)
and one tick later evolve into configurations involving only cellstates 0, 3, 4.

Connected configurations involving only cellstates 0, 3, 4 follow CGoL (B3/S23)
and one tick later evolve into configurations involving only cellstates 0, 1, 2.

Every possible interaction involving at least one of cellstates 1, 2 and
at least one of cellstates 3, 4 creates a state-5 cell. State-5 cells never change,
and all neighbours of a state-5 cell always become state-5 one tick later.

@COLORS

0 48 48 48
1 48 240 48
2 48 120 48
3 48 48 240
4 48 48 120
5 240 240 240

@TABLE
n_states:6
neighborhood:Moore
symmetries:permute

var nonzero={1,2,3,4,5}

var a={0,2}
var a1=a
var a2=a
var a3=a
var a4=a
var a5=a
var a6=a
var a7=a
var a8=a

var b={0,4}
var b1=b
var b2=b
var b3=b
var b4=b
var b5=b
var b6=b
var b7=b
var b8=b

var x={0,1,2,3,4,5}
var n=x
var r=x
var e=x
var h=x
var s=x
var v=x
var w=x
var l=x

# quiescent background:

0, 1,1,1,1,1,1,1,1, 0
0, 1,1,1,1,1,1,1,a8, 0
0, 1,1,1,1,1,1,a7,a8, 0
0, 1,1,1,1,1,a6,a7,a8, 0
0, 1,1,1,1,a5,a6,a7,a8, 0
0, 1,1,1,a4,a5,a6,a7,a8, 3  # birth A->B
0, 1,1,a3,a4,a5,a6,a7,a8, 0
0, 1,a2,a3,a4,a5,a6,a7,a8, 0
0, a1,a2,a3,a4,a5,a6,a7,a8, 0

0, 3,3,3,3,3,3,3,3, 0
0, 3,3,3,3,3,3,3,b8, 0
0, 3,3,3,3,3,3,b7,b8, 0
0, 3,3,3,3,3,b6,b7,b8, 0
0, 3,3,3,3,b5,b6,b7,b8, 0
0, 3,3,3,b4,b5,b6,b7,b8, 1  # birth B->A
0, 3,3,b3,b4,b5,b6,b7,b8, 0
0, 3,b2,b3,b4,b5,b6,b7,b8, 0
0, b1,b2,b3,b4,b5,b6,b7,b8, 0

# rules prescribing either "survival A->B" or "death A->B":
1, 1,1,1,1,1,1,1,1, 4
1, 1,1,1,1,1,1,1,a8, 4
1, 1,1,1,1,1,1,a7,a8, 4
1, 1,1,1,1,1,a6,a7,a8, 4
1, 1,1,1,1,a5,a6,a7,a8, 4
1, 1,1,1,a4,a5,a6,a7,a8, 3  # survival A->B
1, 1,1,a3,a4,a5,a6,a7,a8, 3  # survival A->B
1, 1,a2,a3,a4,a5,a6,a7,a8, 4
1, a1,a2,a3,a4,a5,a6,a7,a8, 4

# rules prescribing either "birth A->B" or "abstain A->B":
2, 1,1,1,1,1,1,1,1, 4
2, 1,1,1,1,1,1,1,a8, 4
2, 1,1,1,1,1,1,a7,a8, 4
2, 1,1,1,1,1,a6,a7,a8, 4
2, 1,1,1,1,a5,a6,a7,a8, 4
2, 1,1,1,a4,a5,a6,a7,a8, 3  # birth A->B
2, 1,1,a3,a4,a5,a6,a7,a8, 4
2, 1,a2,a3,a4,a5,a6,a7,a8, 4
2, a1,a2,a3,a4,a5,a6,a7,a8, 4

# rules prescribing either "survival B->A" or "death B->A":
3, 3,3,3,3,3,3,3,3, 2
3, 3,3,3,3,3,3,3,b8, 2
3, 3,3,3,3,3,3,b7,b8, 2
3, 3,3,3,3,3,b6,b7,b8, 2
3, 3,3,3,3,b5,b6,b7,b8, 2
3, 3,3,3,b4,b5,b6,b7,b8, 1  # survival B->A
3, 3,3,b3,b4,b5,b6,b7,b8, 1  # survival B->A
3, 3,b2,b3,b4,b5,b6,b7,b8, 2
3, b1,b2,b3,b4,b5,b6,b7,b8, 2

# rules prescribing either "birth B->A" or "abstain B->A":
4, 3,3,3,3,3,3,3,3, 2
4, 3,3,3,3,3,3,3,b8, 2
4, 3,3,3,3,3,3,b7,b8, 2
4, 3,3,3,3,3,b6,b7,b8, 2
4, 3,3,3,3,b5,b6,b7,b8, 2
4, 3,3,3,b4,b5,b6,b7,b8, 1  # birth B->A
4, 3,3,b3,b4,b5,b6,b7,b8, 2
4, 3,b2,b3,b4,b5,b6,b7,b8, 2
4, b1,b2,b3,b4,b5,b6,b7,b8, 2

# otherwise, kaboom:
nonzero, n,r,e,h,s,v,w,l, 5
0, n,r,e,h,s,v,w,nonzero, 5