Camels14

[Yoel]

I believe that the rule B2en3aei5ac6aci7c8/S2-i3-k4c6a7c abundant with natural (2,6c)/14 camelships deserves its own thread. First, some definitions. Every pattern's evolution has its own rulespace with a given neighborhood and symmetry, which is a partially ordered finite set (unless we consider an infinitely large neighborhood) defined by the necessary birth/survival conditions enabling the pattern to evolve and all the optional conditions that still don't affect the pattern's behavior. The partial order works between rules A ≥ B, if the set of b/s conditions of B is a subset of b/s conditions of A. Thus, terminology and operations of order theory can be applied to rules and rulespaces, such as additions and subtraction by set operations between rules' conditions. Every finite rulespace has its bottom and top element, which I view as natural "signatures" of the pattern. For example, the bottom rule of the glider is B3aijn/S2ae3jnr and its top rule is B2ikn34-r5-n678/S0234-k5678. The bottom rule has the minimum of the pattern's necessary birth and survival conditions, while the top rule has the maximum of necessary and permitted conditions.

EDIT: See the discussion below. Some terms above have been clarified. This notion of bottom and top rules is applicable, for example, to 2-state rules where the states are ordered 0 and 1. Only isotropic non-totalistic rules are being discussed here (for now). Group theory rather than order theory may be better used to describe various types of multistate rules and their respective rulespaces, as well as such restricted subspaces of Like-like rules as self-complementary like Day & Night, because their "bottom rule" is B/S012345678 rather than B/S.

Camels14 is the bottom rule of the following camelship, studied (and discovered?) in some other rule by Nikro:

Code: Select all

x = 7, y = 10, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
3$3b3o$4b2o$2bob2o$2b3o!

It's abundant with natural and semi-natural guns that shoot little orthogonal c/2 ships, which can be used for various constructions:

Code: Select all

x = 400, y = 337, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
16$257b2o$253b2o2b2o$253b2o$256bo10b2o8b2o8b2o$255b2o11bo9bo9bo9b2o$
256bob3o6b2o8b2o8b2o9b2o$254bo2bob2o$254b2o$257bo2b2o$256b2o2b2o7$254b
2o2b2o$255bo3bo$252b2o3bo$252bo2b2o$254bo3bo8b2o8b2o8b2o$253b2ob3o9bo
9bo9bo4b2o$254bob3o8b2o8b2o8b2o4b2o$252bo$252b2ob2o$255b2o5$259b2o$
259b2o$253b2o$253bo2b2o$255bobobo7b2o8b2o$254b2ob3o8bo9bob2o$255bo11b
2o8b2ob2o$253bo$253b2ob2o$256b2o10$145b2o$145b2o4$146bo$145bobo111b2o
31b2o10b2o$145bobo111b2o33bo9b2o$263b2o3b2o3b2o3b2o2bo9b2o$176b2o81bob
ob2o3bo5bo3b2obobo$174bo2bo81b2o6bo7bo6b2o$174b2o83b3o5bobo3bobo5b3o$
176bo89b2ob2ob2ob2o$146bo29b3o3b2o3b2o3b2o3b2o3b2o3b2o3b2o45b3o5bobo3b
obo5b3o$145bobo28b2obo4bo4bo4bo4bo4bo4bo4bo2b2o40b2o6bo7bo6b2o$145bobo
28bobo3b2o3b2o3b2o3b2o3b2o3b2o3b2o3b2o40bobob2o3bo5bo3b2obobo$172b2o
89b2o3b2o3b2o3b2o24b2o$172b2ob2o82b2o21b2o20b2o$175b2o82b2o21b2o3$146b
o$145bobo$145bobo2$89b2ob2o$89b2ob2o$94bo$85b2obobo4b2o$85b2o2bobobobo
bo2b2o6b2o5b2o29bo$88bobo5bo5bo6bo7bo27bobo$96bo3b2o6b2o5b2o5b2o21bobo
$88b2ob2o29b2o144b2o$88b2ob2o175b2o2$205b2obo55b2o$205bob2o55b2o$146bo
66b2o4b2o5b2o34bo3bo$145bobo56b2o3b2o3bo6bo5bo6b2o26b2o3b2o5b2o9b2o10b
2o$145bobo33b2o4b2o5b2o4b2o2b2o3b2o2b2o4b2o5b2o6b2o33bo5bo11bo10bo10b
2o$174b2o4bo6bo5bo6bo3b2o3b2o51b2o3b2o5b2o9b2o10b2o10b2o$174b2o5b2o4b
2o5b2o4b2o60bo3bo$206b2obo54b2o$73b2o131bob2o54b2o$73b2o2b2o$78bo67bo
121b2o$71b2o3bo68bobo120b2o$71bo2bo2bo67bobo$73bob4o5b2o6b2o7b2o6b2o$
72b2obobo7bo8bo7bo8bo4b2o$70bo2bo2bo7b2o6b2o7b2o6b2o5b2o242b2o2b2o$70b
2o288b2o2b2o$73b2o$73b2o71bo215b2o$145bobo215bo$145bobo213bo$266bo3b2o
67b2o17b4o11b2o$267bo2b2o62b2o3bo18bo15bo17b2o$151b2o181b2o3b2o17b4o
11b2o17b2o$151b2o113b2o2bob2o87bo$142b2o3bo118b2obo3bo89bo$142b2o2b4o
112b2o6bo91b2o$146bo2bo112b2o7bo14b2o16b2o$149b2o4b2o6b2o6b2o6b2o6b2o
6b2o6b2o65bo17bo16bo12b2o40b2o2b2o$145b2ob2o7bo7bo7bo7bo7bo7bo7bo5b2o
49b2o7bo14b2o16b2o12b2o40b2o2b2o$145b2o8b2o6b2o6b2o6b2o6b2o6b2o6b2o6b
2o49b2o6bo$266b2obo3bo$149b2o115b2o2bob2o$149b2o$66b2o199bo2b2o$63b2ob
2o198bo3b2o$63b2o3bo$66bob3o7b2o7b2o7b2o7b2o7b2o7b2o4b2o$65b2ob3o8bo8b
o8bo8bo8bo8bo4b2o$66bo3bo7b2o7b2o7b2o7b2o7b2o7b2o$64bo2b2o$64b2o3bo$
71bo$70b2o282b2o$354b2o2$348b2o$348bo$344b2o$269b2o73b2o2b2o5b2o15b2o$
269b2o77bobo5bo17bo14b2o$168b2o6b2o11b2o6b2o65b2o78b2o2b2o5b2o15b2o15b
2o$168b2o6b2o11b2o6b2o65bo2bo76b2o$266b3o79bo$170b2o23b2o69bo3b2o76b2o
$170bo11b3o11bo73b2o$172bo10bo10bo66b2o7bo83b2o$135b2o22b2o11b4o15b4o
11b2o22b2o29b2o7bobo16b2o16b2o45b2o$128b2o5bo23bo15bo15bo15bo23bo5b2o
31b2obo17bo17bo12b2o$128b2o5b2o22b2o11b4o15b4o11b2o22b2o5b2o22b2o7bobo
16b2o16b2o13b2o$172bo10bo10bo66b2o7bo$39b2o129bo11b3o11bo73b2o$39b2o6b
2o121b2o23b2o69bo3b2o$47b2o217b3o$46bo121b2o6b2o11b2o6b2o65bo2bo$44b2o
122b2o6b2o11b2o6b2o65b2o$43b3o3b2o7b2o6b2o7b2o6b2o184b2o$40bo10bo7bo8b
o7bo6b2o184b2o$40bo8b2o7b2o6b2o7b2o$40b2o5$40b2o$40bo8b2o7b2o6b2o7b2o
6b2o$40bo10bo7bo8bo7bo6b2o$43b3o3b2o7b2o6b2o7b2o$44b2o109b2o27b2o$46bo
108b2o3b2ob2o3b2ob2o3b2ob2o3b2o$47b2o111b2obo5bobo5bob2o$39b2o6b2o$39b
2o75b2o3b2o19b2o19bo13bo19b2o19b2o$116b2o2bo20bo20b3o11b3o20bo20bo8b2o
$121b2o19b2o19bo13bo19b2o19b2o9b2o$265b2o4b2o$160b2obo5bobo5bob2o84b2o
4bo$155b2o3b2ob2o3b2ob2o3b2ob2o3b2o$155b2o27b2o83b2obo2b2o$269b2obob3o
$260b2o10b4o$260b2o11b2o21b2o22b2o7b2o$273b2o23bo23bo6b2o$260b2o11b2o
21b2o22b2o$260b2o10b4o$269b2obob3o$269b2obo2b2o2$265b2o4bo$265b2o4b2o
6$162b2o5b2o$163bo5bo$161b3o5b3o$162bo7bo$160bo11bo$159b2o11b2o$159b2o
11b2o$161bo9bo$162b2o5b2o$106b2o15b2o25b2o10b2o5b2o10b2o25b2o12b2o$
106b2o15bo25bo33bo25bo12b2o$123b2o25b2o8b2o9b2o8b2o25b2o81b2ob2o$160bo
11bo105b2o11b2ob2o11b2o$162b3o3b3o87b2o17bo31bo14b2o$162bobo3bobo87b2o
18b2o9b2obobob2o9b2o15b2o$165bobo121b2o2bo2b2o$162bobo3bobo121bobo$
162b3o3b3o122bo$160bo11bo$123b2o25b2o8b2o9b2o8b2o25b2o83bo$106b2o15bo
25bo33bo25bo12b2o68bobo$106b2o15b2o25b2o10b2o5b2o10b2o25b2o12b2o65b2o
2bo2b2o$162b2o5b2o87b2o18b2o9b2obobob2o9b2o15b2o$161bo9bo86b2o17bo31bo
14b2o$159b2o11b2o104b2o11b2ob2o11b2o$159b2o11b2o117b2ob2o$160bo11bo$
162bo7bo$161b3o5b3o$163bo5bo$162b2o5b2o19$308b2o7b2o3b2o7b2o$308b2o7b
2o3b2o7b2o2$313b2o3b2ob2o3b2o$315bo2b2ob2o2bo$309b2obobo11bobob2o$276b
2o30b2o2bo6bobo6bo2b2o30b2o$251b2o22bo32bo5bobobobobobobo5bo32bo25b2o$
77b2o66b2ob2o101b2o23b2o30b2o2bo6bobo6bo2b2o30b2o26b2o$77b2o8b2o27b2o
27b2ob2o27b2o27b2o9b2o90b2obobo11bobob2o$87bo28bo61bo28bo9b2o96bo2b2ob
2o2bo$87b2o27b2o25b2o5b2o25b2o27b2o105b2o3b2ob2o3b2o$143b2o5b2o$141bo
3b5o3bo154b2o7b2o3b2o7b2o$140bob2o2bobo2b2obo153b2o7b2o3b2o7b2o$139bob
o4b3o4bobo$140bob2o2bobo2b2obo$141bo3b5o3bo$143b2o5b2o$87b2o27b2o25b2o
5b2o25b2o27b2o$87bo28bo61bo28bo9b2o$77b2o8b2o27b2o27b2ob2o27b2o27b2o9b
2o$77b2o66b2ob2o18$253b2o33b2o49b2o33b2o$242b2o8bo34bo53bo34bo11b2o$
242b2o9b2o33b2o11b2o2b2ob2ob2o3b2ob2ob2o2b2o11b2o33b2o12b2o$301b2o2b2o
b2obo5bob2ob2o2b2o$313bobo$305b2o5b2ob2o5b2o$139b2o19b2o143bo6bobobo6b
o$104b2o22b2o9b2o19b2o9b2o22b2o108b2o5b2ob2o5b2o$97b2o4bo24bo43bo24bo
5b2o108bobo$97b2o5b2o22b2o8b2o5b2o7b2o5b2o8b2o22b2o6b2o96b2o2b2ob2obo
5bob2ob2o2b2o$138b2o5b2o7b2o5b2o79b2o9b2o33b2o11b2o2b2ob2ob2o3b2ob2ob
2o2b2o11b2o33b2o12b2o$242b2o8bo34bo53bo34bo11b2o$137b2o2bo2bobobo3bobo
bo2bo2b2o89b2o33b2o49b2o33b2o$136bo3b2obobobobobobobobob2o3bo$137b2o2b
o2bobobo3bobobo2bo2b2o2$138b2o5b2o7b2o5b2o$97b2o5b2o22b2o8b2o5b2o7b2o
5b2o8b2o22b2o6b2o$97b2o4bo24bo43bo24bo5b2o$104b2o22b2o9b2o19b2o9b2o22b
2o$139b2o19b2o!

Some constructed guns:

Code: Select all

x = 422, y = 316, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
43$185b2o$185b2ob2o$188b2o$294b2o$294b2o47b2o7b2o$181b2o160b2o7b2o$
181b2o$275b2o$269b2o4bo$269b2o4b2o$193b2o149b2o5b2o$193b2o149b2o5b2o2$
181b2o12b2ob2o$181b2o12b2ob2o144b2o5b2o$306b2o36b2o5b2o$181b2o2b3o9b2o
106bobo$181b2o2bobo5bo4bo106b2o37b2o5b2o$193bo3bo107b3o36b2o5b2o$193bo
2bobo149bo$193bo150b2obobob2o$344b2ob3ob2o$190b2o155b3o$166b2o22b2o$
166b2o179b3o$344b2ob3ob2o$174b2o16b2o150b2obobob2o$175bo16b2o154bo$
174b2o168b2o5b2o$191b2o151b2o5b2o$191b2o$176b2o166b2o5b2o$176b2o166b2o
5b2o2$187b2o$187b2o155b2o5b2o$327b2o15b2o5b2o$326bobo$326b2o$326b3o4$
343b2o7bobo$343b2o7b3o4$207bo$208bobo$208b3o$204b3o$86b2o6b2ob2o106bob
4o146b2o37b2o$86b2o6b2ob2o109bobo146b2o6b2o2b2ob2ob2o3b2ob2ob2o2b2o4b
2o$208b3o154b2o2b2ob2ob2o3b2ob2ob2o2b2o$209bo$81b2o4b2o40b2o247bo$81b
2o3bo42bo38bo9bo169b2o24bo2b3o2bo$87b2o40b2o36b3o7b3o167bobo28bo$82b2o
83bobobo5bobo167b2o$82b2o83b2o2b2o4b4obo164b3o15b2o2b2ob2ob2o3b2ob2ob
2o2b2o$163b2o6b2o8b3o9b2o165b2o3b2o2b2ob2ob2o3b2ob2ob2o2b2o4b2o$163b2o
12b3o13b2o111b2o51bo36b2o$177bobo126b2o4b2o2b2ob2ob2o3b2ob2ob2o2b2o16b
o4b2o$171b3o6bo13b2o116b2o3bob2ob2o3b2ob2obo3b2o15b4o$164b2o7bo20b2o
119bo19bo18bo2bo$105b2o57b2o3bo3bo140b2o6bobobobo6b2o17b2ob2o$105b2o
62b2o2bo137bo2b2o6bo2bo2bo6b2o2bo15b3o$178b2ob2o6b2o123b2o6bobobobo6b
2o16bo50b2o$98bo2b2o62b2o11b2ob2o6b2o124bo19bo14b3o20b2o2b2ob2ob2o3b2o
b2ob2o2b2o4b2o$98bo3bo3b2o57b2o145b2o3bob2ob2o3b2ob2obo3b2o11b3o20b2o
3bob2ob2o3b2ob2obo3b2o$93b2o3bo7b2o198b2o4b2o2b2ob2ob2o3b2ob2ob2o2b2o
6b2o3b3o23bo8bobo8bo$93b2o3b3o205b2o37b2o14b2o9b4o2b5o2b3o2b5o2b4o$
361b2o9bob2o2b2obo3bobo3bob2o2b2obo$107b2o263b4o2b5o2b3o2b5o2b4o$99b2o
6b2o267bo8bobo8bo$99b2o2b2o83b2o183b2o3bob2ob2o3b2ob2obo3b2o$100bobobo
83b2o175b2o6b2o2b2ob2ob2o3b2ob2ob2o2b2o4b2o$102b3o36b2o40b2o180b2o37b
2o$103bo38bo42bo3b2o$141b2o40b2o4b2o162b2o5b2o$353b2o2bo2b2o$356bobo$
173b2ob2o6b2o$173b2ob2o6b2o167b2ob3ob2o$353b2o5b2o2$218b2o133b2o2bo2b
2o$218b2o133b2obobob2o$356bobo$353b2o5b2o$353bo2b3o2bo$355b2ob2o$357bo
$355b2ob2o$353bo2b3o2bo$353b2o5b2o$356bobo$353b2obobob2o$353b2o2bo2b2o
2$353b2o5b2o$353b2ob3ob2o2$356bobo$353b2o2bo2b2o$353b2o5b2o5$352b2o7b
2o$352b2o7b2o7$297b2ob2o$297b2ob2o3$292bo2b2ob3ob2o2bo$293bobo2bobo2bo
bo$297bobobo$296b2obob2o$271b2o19b2o3bo3bo3b2o$271b2o19b2o5bo5b2o3$
289b2o$274bobo9b2o4bo20b2o3bo$274bobo9b2o2bo7bobo12b2o2bo$148b2ob2o6b
2o114bo10b3obob2o4bobo$148b2ob2o6b2o131b2o5bo17b2o$79bo212bo20bo3b2o$
79bo213bo18bobo6b2o$79bo39b2o43b2o133b2o11bobo6b2o$76b4o41bo38b3ob2o
132bobo10b2o$75b2o42b2o39bo138b7o6bobo6b2o$164bo135bob2obo6bobo6b2o$
79bobo81b2o137b2obo7bo3b2o$82b2o233b2o$82b2o$71b2o240b2o2bo$34b2o34bo
226b2o5bo5b2ob2o3bo$34b2o35b2o8b2o192b2o20b2o3bo3bo3b2o$81b2o57b2o133b
2o24b2obob2o$140b2o160bobobo$298bobo2bobo2bobo$80b2o215bo2b2ob3ob2o2bo
$80b2o57b2o$139b2o8b2o41b2o40b2o$151bo41bo42bo4b2o59b2ob2o$149b2o41b2o
40b2o5b2o59b2ob2o$138b2o$138b2o$57b2o81bobo$57bo$61bo39b2o42b2o$56b2ob
3o38bo41b4o$56b2o43b2o39bo$142bo$142bo$61b2o6b2ob2o$61b2o6b2ob2o33$
244b2o$244b2o4$222b2o3bo$224bobobo$221b3o3bobo23b2o41b2o$226bo28bo41bo
31b2o$220bo5b3o24b2o41b2o31b2o$220bo3b3o3$227b2o$223b2o2b2o$223b2o$
187b2o$187b2ob2o$190bob3o$187b2o$187b2o$184b2o5bo2bo$184b2o6b2o15b2o
25b2o$209b2ob2o22b2o$205b2o5b2o$205b2o$175b2o$175b2o40b2o41b2ob2o6b2o$
191b2o24b2o21bobo17b2ob2o6b2o$174b2o14bo49bobo$174b2o15b2o48bo$210bo
48b2o15b2o$209bobo49bo14b2o$179b2o6b2ob2o17bobo21b2o24b2o$179b2o6b2ob
2o41b2o40b2o$275b2o$245b2o$238b2o5b2o$214b2o22b2ob2o$214b2o25b2o15b2o
6b2o$257bo2bo5b2o$263b2o$263b2o$257b3obo$260b2ob2o$263b2o6$225b3o3bo$
223b3o5bo$225bo$222bobo3b3o$223bobobo$224bo3b2o4$206b2o$206b2o!

These splitters can be used to construct guns of any period greater than 84.

The following reflector can be used to construct oscillators of any period greater than 60:

Code: Select all

x = 88, y = 56, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
4$11b2o$11b2o56b2o$69b2ob2o$22b2o41b2o5b2o$22b2o41b2o$7b2o62b2o$7b2o
57b3o2b2o4b2o$17b2o28b2o17b4o7b2o$6b2o8bo30bo20b3obo$6b2o9b2o28b2o20b
2obo$69b2o$69bobo$8b2o59b3o$8b2o2bobo$12bobo$13bo2$74b2o$74b2o4$8b2o$
8b2o2$70bo$69bobo$69bobo2b2o$12b3o59b2o$12bobo$13b2o$11bob2o20b2o28b2o
9b2o$11bob3o20bo30bo8b2o$5b2o7b4o17b2o28b2o$5b2o4b2o2b3o57b2o$11b2o62b
2o$17b2o41b2o$10b2o5b2o41b2o$10b2ob2o$13b2o56b2o$71b2o!

Omniperiodicity proven by collecting non-trivial oscillators of periods 2-60:

Code: Select all

x = 599, y = 625, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
30$252b2o$189b2o22b2ob2o21b2o11bo32b2o$185b2o3bo23bob2o21b2o28b2o14b2o
$172bo12b2o11b2o27bo14b2o12bo12bo10b2o$171bo15bo10bo11bo3bob2o8bobo7b
2obobobo12b2o12b3o7b5ob2o$188b2o20b2obo3bo18b2o3bo13bo11b2o2bobo5b2obo
bob2o$184bo3b2o12bo23b3o10b2o14bo11b2o2b2o6b6o$184b2o15b2o7b2obo24bo
17b2o22b2o$210b2ob2o23b2o43b2o$173b2o77b2o14b2o13b2o$171bo2bo77b2o14b
2o$171bo15b2o$171b3o13b2o$169b2o58b2o$169b2o15b3o11b2o9b2o4b2o6b2o3bo
10b2o$186bo10bobo2bobo6bo2b2o2bo6bo11b2o2b2ob2o$186bo10b2o4b2o8bo2bo
10bo9b2o6bo10b2o15b2o9b2o$184b3o13b2o10bo2bo13bo10bo2bo13bo9b2o4bo7b2o
bo$200b2o10bobo9bo3b2o9b2obo8b2o2bo11b2o3bo2bo5b2o3bo$172b2o12b2o25bo
10b2o14bobo8b2o4bo12b3ob2o9bo3b2o$172bo2bo10b2o23bo25b2o16b3o12bobo12b
o3b2o$174b2o35b2o24b2o2b2o13bo2bo11bo13b2o$173bo67b2o15b2o$168b2o83b2o
13b2o15b2o$168bo2bo14b2o65b2o13b2o15b2o$170bo15b2o$169b2o28b2o$182b2ob
obo11b2o23b2o13b2o$182b2o2bo2b2o33bo3b2o10bo11b2o3b2o$185bobob2o11b2o
7b2o4b2o9bo9bo3b2o8bo2bo2bo$198bo3b2o7b2o4b2o8bo2bo7bobo2bo10b3o17b2o$
185b2o11b2o14b2o9bo3b2o7b2obo11bo4b2o15bo6b2o3b2o$185b2o25bo2b2o8b2o
14b2o10b3o2b2o11b3o8bo2bo2bo$212bo3bo21b2o14b3o13b2o12b3o$210bo27b2o
14bo14b3o11bo2b2o$210b2o5b2o33b2o15bo16bo$217b2o33b2o15bo12bo2bo2bo$
267bo14b2o3b2o$267b2o33$276b2o25b2o$133b2o35b2ob2o101b2o17b3o5b2o23b2o
$128b2o3b2o35b2ob2o41b2o37b2o38bobo28bo2bo$128bo2bo17b3o11b2o26b2o3b2o
14b2o2b2o10b2o25b2o35b2obobo27b3o$130bobob2o13b3o11b2o4b7o12b2ob2o3b2o
b2o11b2o14b2o16b2o24b2o3bo14b2ob4o25b3obo$130b3ob2o14bo3b2o12bob2ob2ob
o11b2o4bo4b2o45bo9b2o14b2o2b3o2bo18bo24bob3o$155bo13b7o4b2o11b3o21bo
15b3o20b2o22b2o17bo26b3o$153bo26b2o11bobo16bo2b3o16bo38bo27b4ob2o15b2o
2bo$127b2o22bo18b2ob2o19bo17b2o14b2o5bo13bobo21b2ob2o24bobob2o15b2o$
127b2o22b2o17b2ob2o53b2o20bobo23b2o24bobo$245bo2b2o45b2o5b3o$245b2o4bo
43b2o$251b2o2$151b3o$130b2o$130b2o19bobo32b2o13b2o$152bo16b2o4b2o9b2ob
2o7b2ob2o10b2o2bo54b2o$147b2o7b2o11bo5b2o13bo7bo14b2obo2bo52b2o$131b3o
13bo9bo13b4o12bo13bo17bo12bobo$131b3o36bob2o13b2ob2o5b2ob2o14b5o11b2o
40b2o20b2o$129b2o3b2o15bobo17b3o16b2o5b2o21bo11bo17b3o15b2o6bo19b2o$
129bo2bo2bo14b2ob2o14bo22b5o15bo2bo35bo16b2o3bobo53b2o$131b2obo34b2o
21b2ob2o15b2o7bo49bo3bob2o50bo2b2o$134b2o56bobobo23b2o49bob3ob2o43b2o
6bob2o$170b2o20b2ob2o54b2o19bo29b2o18b2o5bo$170b2o20b5o35bo17b2o22b2o
17b2o7b2o26bob2o$190b2o5b2o33b2o40b2o17b2o34bo2b2o$187b2ob2o5b2ob2o30b
obo63b3o28b2o$187bo13bo53b2o42b2o$150b2ob2o35bo7bo56b2o43bo$150b2ob2o
31b2ob2o7b2ob2o104b2o$186b2o13b2o47b2o55b2o$132b2o13b2o7b2o57b2o33b2o$
130bo2bo13b2o2b3o2b2o17b2o38b2o84b2o$130b2o4bo33b3o2b2o124b2o$132bo18b
obo16bobo44b2o55b2o3b2o$133b2o2bo14bo17b2o2b2o41b2o10b2o3b2o38b2o3b2o$
130b2o7bo7b2o7b2o15bobo19b2o16b3o13bo2bo2bo41bo$130b2ob2o3b2o7bo9bo14b
o20bo2bo16b2obo10bo3bobo3bo9b2o2bo24bobo$134bo34b2ob2o18bobo18b3o11b2o
7b2o9bo2b4o22b3o$132bo18bobo15b2o20bo4b2o19bo31bo6b2o18b3o$132b2o16b2o
b2o37b3ob2o18b2o13b3o14b3ob2o3bo$192b3o34bo2bo15bo18b2o17b2o$215b2o12b
2o3b2o16bo14b2o17b2o11b2o$215b2o17b2o12bo7bobo11b3o9b3o14bo3b2o$195b2o
51b2obo2bobobo10bob2o9b2obo15bo2bo23bo2b2o$195b2o55bo5b2o10b3o9b3o17bo
24b3o2bob2o$192b2o62bobo8b2o17b2o13b3ob2o19b3obo3b2o$192b2o58bo3b2o9b
2o17b2o12b3o3bo20bob3o$130b2o3b2o115b2ob2o2bo44bo3b2o18b3o$130bo4b2o
121b2o16b3o23b2o2bo2bo15b2o2bo$132b2o142b3o24bob3o17b2o$132b3o141bobo
26b2obo$132b3o142bo27bo4bo$130bo5bo137b2o3b2o17b2o9b2o$130b2o3b2o137b
2o3b2o17b2o6$275b2o25b2o3b2o$275b2o26bo3bo$301bo2bobo2bo$276b3o22b2ob
3ob2o$277bo26b3o$271b2o9b2o17b2o2bo2b2o$271b2o9b2o17b2o5b2o2$277bo17b
2ob2ob4ob4ob2ob2o$276b3o14bo2bob2obo2bobo2bob2obo2bo$293b2o6bob2ob2obo
6b2o$275b2o18b3o3b4ob4o3b3o$275b2o19b3o13b3o$295b3o3b4ob4o3b3o$293b2o
6bob2ob2obo6b2o$293bo2bob2obo2bobo2bob2obo2bo$295b2ob2ob4ob4ob2ob2o2$
301b2o5b2o$301b2o2bo2b2o$304b3o$301b2ob3ob2o$301bo2bobo2bo$303bo3bo$
302b2o3b2o24$130b2o25b2o$130b2o3b2o15b2o3b2o$135b2o15b2o29b2o53b2o82b
2o$129b3o27b2ob2o19b2o50bo2b2o14bo3b2o21b2o13b2o24b2o22b2o$131bo27bo2b
2o70b2o17bo2bob2o18bo3bo13b2o30b2o16bo$130b2o27b2o50b2o3b2o15b3o2b2o
13bo2bo21bobo48bo18b2o$130bo2b2o19b2o25b2o27b2o4b2o20b2o11bo2b2o2b2o
11b2o6bo2b2o43bo21bo$128b2o3b2o19b2o3b2o20bo2b2o24b3o38b2o5b2o11b2o9b
2o41bo2bo18bo$128b2o29b2o25bo2b2o19b2o20b2ob2o58b2o2bo26b3o2b2o14b2o$
180b2o3bo3b2o16bo2bo21b2ob2o14b2o42bo2bobo23b5o3bo14bo$180b2o2bo22b2o
42b2o44bo2bo3b2o16bo7bo18b2o$185b2o2bo88b2o16b2o6b2o16b2o5b2o18b2o$
188b2o88b2o$303b2o16b2o22bo$303b2o16b2o23bo$186b2o$186b2o2$160b2o$131b
2ob2o24b2o$129bo2bob2o105b2o10b2ob2o$129b2o23b2o2bo82b2o10b2ob2o17b2o$
134b2o18b2o2b2o49b2o5b2o12b2o19bo23b2o2b2o$134b2o23bo49b2o5b2o12bo2bo
7b2o8b2o2b2o2b2o18b2o44b2ob2o$232b2o5bob2o13b2ob2o64b2obo2bo17b2o11b2o
$128b2o24b2o7b2o39b2o3bobo3bobo3b2o6b2obo5b2o11b2o4bo22b2o14b2o11b2o
19b2o14b2ob2o11b2ob2o$128b2ob3o20b2o7b2o39b2o2bobobobobobo2b2o6b2o7bo
2bo9b2obo4b2o15b3ob2o15bo3b2ob2o3bo14b2o20b2o5b2o3b2o5b2o$131b3ob2o42b
2ob2o3b2ob2o17bobo3bobo22b2o17b2o15b2o24bobo19bo28b2o3b2o$130bo4b2o42b
2ob2o3b2ob2o37b2o23b2obo18b3ob2o11bo19bo10bo8b2o10b2o19b2o$131bobo51bo
23b2o5b2o11b2o23b2ob2o21b2o11b2o7b3o7b2o11bo7b2o10b2o19b2o$132bo51b3o
22b2o5b2o78b2o3b2ob2o3b2o12bo$279b2o12b2o7b3o7b2o9b2o4b2o23bo3bo$174b
2o19b2o78b2o2b2o12bo19bo15b2o16b2o4b2o3b2o4b2o$174b2o19b2o78b2o25bobo
42b2o3b2o5b2o3b2o$297bo3b2ob2o3bo$174b2o19b2o99b2o11b2o$174b2o19b2o$
177bo15bo153b2o3b2o5b2o3b2o$176b2o15b2o152b2o4b2o3b2o4b2o$155b2o3b2o
15bo15bo18bo21b2o3b2o10b2o3b2o96bo3bo$155b2o3b2o12b2o19b2o9b2o3bobob2o
18bo3bo11b2o3b2o$158bo15b2o19b2o9b2o2b4ob2o16bo7bo103b2o19b2o$153b2o2b
3o2b2o69b2o5b2o36b2o65b2o19b2o$153b2o7b2o10b2o19b2o37bo5bo31b2o4b2o14b
2o17b2o38b2o3b2o$132b2o40b2o19b2o9b2o24bo4bo4bo16b2o11b2o20b2o17b2o31b
2o5b2o3b2o5b2o$132b2o72b2o12b2o10b2o2bobo2b2o17bo16b2o4bo62b2ob2o11b2o
b2o$127b2o3b2o21b2o3b2o22b3o33b2o14b3o12b2o3bo21bo4b2o11b2o3b2o3b2o3b
2o15b2o19b2o11b2o$127bo4bo4bo17b2o3b2o23bo46b2o2bobo2b2o8b2o3bo7b2o10b
o7bo12bo3bo5bo3bo13b2obo4b2o$131b2o3b2o41b2ob2o3b2ob2o28b2o10bo4bo4bo
13b3o5b2o10b2o4bo20bobo19b2o6b2o$131b2o46b2ob2o3b2ob2o19b2o7bobo11bo5b
o36bo4b2o18b2ob2o$131b2o78b2o7b2o11b2o5b2o45b2o13bobobo17b2o$220bo12bo
7bo39b2o4b2o13b2ob2o17b2o$235bo3bo11b2o4b2o5b2o15b2o20bobo21b3o5bo$
234b2o3b2o10b2o4b2o5b2o31bo3bo5bo3bo15bobo4b2o$216b2o2b2o74b2o3b2o3b2o
3b2o15b2o$216b2o2b2o111b2o$294b2o17b2o9b2o4b2ob2o$294b2o17b2o9b2o4b2o
3$155b2o3b2o$155bo5bo$185b2ob2o$176b2ob2o4b2ob2o4b2ob2o$154b3o3b3o13bo
3bo13bo3bo$130b2o18b2o2bo7bo2b2o11bo6bobobo6bo$130b2o18bo3bo7bo3bo10bo
bo4bobobobo4bobo$136bo41bo6bobobo6bo$135b2o39bo3bo13bo3bo$121b2o12bo
40b2ob2o4b2ob2o4b2ob2o$121b2o17b2o8bo3bo7bo3bo18b2ob2o$127bo12b2o8b2o
2bo7bo2b2o$126b2o26b3o3b3o$126bo$131b2o$131b2o22bo5bo$155b2o3b2o40$
158b2o$94b2o25b2o35b2o103b2o$94b2ob2o22b2o29b2o76b2o28b2ob2o35b2o$97b
2o53b2o34bo2b2o28b2o7b2o28b2o33b2o3b2o34b2o15b2o$114b2ob3ob2o31bo3bo
24b2obo5bo28b2o3b2o67b2o39b2o15b2o$90b2o22b2ob3ob2o34b2o24b2obo3bo37bo
2b2o152b2o$90b2o4b3ob2o55bo28bo6b2o31b2o3b2o26bo5bo33b4o82b2o$95b2obob
2o20b2o31b2o2bo24bo4b2o2b2o64b2o3b3o35bo$91b2o2bob2o14b2o2b2o3b2o26b2o
2b2o3bo24b2o3b2o107bo88bo$91b2o2b3o15b2o3bo31b2o5b3o63b2o2b2o31b2o2b2o
b2o72b3o3b3o36bobo$117b2o67b2o35b2o2b2o31b2o5b2o31bo38bobobo3bobobo31b
2o2bo2b2o$95b2o89b2o110b2o41b3o3b3o33b2obobob2o17b2o$95b2o21b2o184b2o
21b2o11bo9bo11b2o23bo21b2o$118b2o43b2o133b2ob2ob2o21b2o13bo5bo13b2o19b
2obobob2o$163b2o133b2ob2o80b2o2bo2b2o21b3o$334bo21bo29bobo21bo$336bo
17bo30bobobo20bo5bo$333b3o19b3o25b2o5b2o18b2o2b2o$333bobobo15bobobo25b
2o5b2o21bo$333b3o19b3o56b2o$383b2o5b2o22b2o$92b2o3b2o126b2o156b2o5b2o$
92b2o3b2o19b2o105b2o158bobobo$116bo2bo37b2ob2o24b2o34b2o109b3o19b3o28b
obo$91b2o5b2o13b2ob2o3b2o34b2ob2o24bo35b2o109bobobo15bobobo25b2o2bo2b
2o$91bo7bo13b2o6b2o103bobo29b2o73b3o19b3o25b2obobob2o$117b2obo34b2o5b
2o21b2ob4o34bobo29b2o76bo17bo32bo$90b2o7b2o16bo5b2o30bo7bo18b2obo5bo
27b2o6bo106bo21bo26b2obobob2o$90b2ob2ob2ob2o19bo3bo32b5o20bo4bo31b2o
70b2o17b2o71b2o2bo2b2o$93bo3bo14b2o2bo2b2obo35b3o24bo3bo2bo70b2o26b2o
17b2o15b2o13bo5bo13b2o22bobo$93b2ob2o14b2o3bo4b2o31b2o5b2o21bo2b2o3bo
26b2o41b2ob2o59b2o11bo9bo11b2o23bo$95bo19bo39b2obobob2o21bo34bo42bobob
o19b2o25b2o25b3o3b3o$113b2o6b2o35bobo24b2o8b2o25b2o5b2o34b2o20bo6b2o3b
2ob2o3b2o6bo23bobobo3bobobo33b2o$113b2ob2o3b2o32b2o2bo2b2o24bo4bob2o
25b2o2b2ob2o32bo3b2o23bob2o3b2ob2o3b2obo30b3o3b3o35b2o$93b2o21b2o37b2o
5b2o25bo2b2o32b2o35b2ob3o23bo17bo$93b2o100bo96bo5b2obob2o5bo$191b2ob2o
94b3o2b4obobob4o2b3o$191b2o99bo5b2obob2o5bo98b2o$271b2o19bo17bo25b2o
15b2o54bo$271b2o19bob2o3b2ob2o3b2obo25b2o15b2o56b2o$123b2ob2o159bo6b2o
3b2ob2o3b2o6bo96bo$123b2ob2o159b2o25b2o94bobo$412b2o$92b2o3b2o19b2ob2o
5b2ob2o158b2o17b2o100b5o$92b2o3b2o19b2ob2o5b2ob2o158b2o17b2o98b2o4bo$
224b2o184b2o2bo3bo$91b2o5b2o20b2o3bo3b2o93b2o182bo8b2o$92bob3obo21bo3b
obo3bo277b2o$90bo2b2ob2o2bo21bo2bo2bo$90b2ob2ob2ob2o11b2o10b3o10b2o83b
2o5b2o$112b2o9bobobo9b2o83bo7bo$90b2o7b2o14b2o17b2o88b2ob2o$90b2o7b2o
11b2obo8bobo8bob2o86bobo$112b2o3bo4b2o3b2o4bo3b2o49bo2b2o26b2o4bobo4b
2o$109b2o8bo2bo2bo2bo2bo8b2o44bo5bo26b2o4b3o4b2o58b2o15b2o$109b2o5bobo
2bo2bobo2bo2bobo5b2o17b2o25bo3bo72b2o5b2o20b2o15b2o$90b2o7b2o14bob3o3b
obobo3b3obo14b2o4b2ob2o25bo6b2o68b2o5b2o$90b2o7b2o8b2o5bobo2bo2bobo2bo
2bobo5b2o8b2o4b2o26bo4b2o2b2o20b2o19b2o18b2o5bo7bo5b2o17b2o7b2o$109b2o
8bo2bo2bo2bo2bo8b2o42b2o3b2o24b2o19b2o18b2o7bo3bo7b2o17b2o7b2o$90b2ob
2ob2ob2o11b2o3bo4b2o3b2o4bo3b2o125b2o3b2o17b2o9b2ob2o9b2o$90bo2b2ob2o
2bo11b2obo8bobo8bob2o16bo30b2o23b2o27b2o46b2o10bobo10b2o$92bob3obo16b
2o17b2o19b3o6b2o20b2o23bo29bo22b2o3b2o20bo6bo5bo6bo$91b2o5b2o12b2o9bob
obo9b2o12b2o3bo3bo3b2o41b2o4bo25bo4b2o18b2o3b2o19b2o7bo3bo7b2o$112b2o
10b3o10b2o12b2o6b3o45b2o4b4o19b4o4b2o54bobo$92b2o3b2o23bo2bo2bo32bo54b
o19bo53b2o7bo3bo7b2o$92b2o3b2o21bo3bobo3bo82b4o19b4o51bo6bo5bo6bo$120b
2o3bo3b2o82bo25bo48b2o10bobo10b2o$159b2o4b2o44bo29bo46b2o9b2ob2o9b2o$
118b2ob2o5b2ob2o23b2ob2o4b2o44b2o27b2o54b2o7b2o$118b2ob2o5b2ob2o23b2o
138b2o7b2o$215b2o19b2o$123b2ob2o87b2o19b2o54b2o15b2o$123b2ob2o164b2o
15b2o2$219b2o4b3o4b2o$219b2o4bobo4b2o$225bobo$224b2ob2o$222bo7bo$222b
2o5b2o3$227b2o$227b2o27$124b2o5b2o198b2o7b2o$91b2o31b2o5b2o198b2o7b2o$
91b2o$331b2o7b2o$95b3o26b2o5b2o198b2o7b2o63b2o7b2o$92bo31b2o5b2o31b2o
169b3o67b2o7b2o$92bo5bo65b2o166b2obobob2o$92b2o2b2o24b2o9b2o34bo4b2o
156bo3bo3bo33b2o30b2o5b2o$95bo26b2ob7ob2o22b2o3b2ob2obobo3b2o10b2o4b2o
36b2o139b2ob2o30b2ob3ob2o$96b2o17b2o2b2o3bo3bo3bo3b2o2b2o15b2o3b2ob2ob
o17b2o4b2o9bob2o14b2o7b2o139b2o36bobo$96b2o17b2o2b2o2bo9bo2b2o2b2o12b
2o15bo30bo2b2o14b2o157b2o24b2obobob2o$123bo9bo20b2o7b2o4b3o14bo61b2o4b
2o124b2o19b2o3b2ob3ob2o3b2o$123bo3bobo3bo29bo4b3o16bo12bo23bob3o19b2o
4b2o9bo2b2o25b2o104b2o15b2o$123b2o7b2o20b2o7b2o4b3o18b2o7bobo26bo37bob
2o25b2o21b2ob2o27b2ob2o26b2o58b2o19b2o$123bo3bobo3bo20b2o15bo17bo9bobo
22bobobo20bo68b2ob2ob2o6b2o5b2o6b2ob2ob2o26b2o58b2o19b2o$123bo9bo23b2o
3b2ob2obo21b2o13b2o17bo23bo75bo7b2o5b2o7bo25bobo17b2o29b2o$115b2o2b2o
2bo9bo2b2o2b2o15b2o3b2ob2obobo3b2o29b2o17b3o32b2o113bobo17b2ob2ob2o17b
2ob2ob2o$115b2o2b2o3bo3bo3bo3b2o2b2o27bo4b2o53b2o28bo31b2o30b2o23b2o
25bo21b2ob2o17b2ob2o$122b2ob7ob2o29b2o54b2o7b2o28b2o7b2o21b2ob3o26bobo
21bobo20b2o71b2o3bo3bo3bo3b2o$122b2o9b2o29b2o54b2o46b2o26bo26b2o23b2o
20b2o26b4o17b4o20b2o3b2ob3ob2o3b2o$252bobo37bo105bo2bo17bo2bo28bobo$
89b2o33b2o5b2o52b2o65b3o37b2o30bo7b2o5b2o7bo22b2o25b4o17b4o25b2o2bo2b
2o$89bo34b2o5b2o52b2o13b2o99b2o15b2ob2ob2o6b2o5b2o6b2ob2ob2o16b2o75b2o
5b2o$91b2o97bobo9bo59b3o36b2o15b2ob2o27b2ob2o25b2o15b2ob2o17b2ob2o$92b
o97bobo7b2o60bobo112b2ob2o12b2ob2ob2o17b2ob2ob2o22b2o3b2o$90bobo31b2o
5b2o58bo12bo42b2o128b2o15b2o29b2o22b2o3b2o$92b2o30b2o5b2o72bo41b2o7b2o
$92b5o88b2o2bo67bo$90b2o4bo88b2obo9b2o4b2o17b2o31b2o143b2o15b2o$90b2o
2bo3bo99b2o4b2o14b2ob2o7b2o34bo63bo3bo3bo60b2o3b2ob3ob2o3b2o$88bo8b2o
121b2o10b2o33bo64b2obobob2o65b2obobob2o$88b2o132bo24b2obo84b3o71bobo$
223b2o8b2o12b2o2bo9b2o4b2o62b2o7b2o64b2ob3ob2o$222bo10b2o26b2o4b2o62b
2o7b2o64b2o5b2o$223b2o$217b2o112b2o7b2o63b2o7b2o$217b2o112b2o7b2o63b2o
7b2o2$119b2o4b2o91b2o10b2o$119b2o4b2o7bob2o80b2o7b2ob2o$133bo2b2o89b2o
$119bo$120bo$126b2o$125bo321b2o$126b2o8b2o309b2o$91b2o38bo4b2o313b3o3b
o$88b2ob2o6b2o29bobo318b3o2b2o$88b2o9b2o29bobo$123bobo323b2o$100b2o21b
obo323b2o$100b2o16b2o4bo333b2o$92bo25b2o8b2o328b2o$85b2obo2bo38bo$85b
2obob2o36b2o$88bo46bo$86bo10b2o37bo$86b2o6b2ob2o19b2o2bo$94b2o22b2obo
7b2o4b2o$129b2o4b2o27$217b2o4b2o66b2o4b2o66b2o4b2o$217b2o4b2o14bo2b2o
47b2o4b2o15bo2b2o46b2o4b2o16bo2b2o$240bob2o71bob2o71bob2o28b2o$155b2o
4b2o55bo73bo73bo52b2ob2o$155b2o4b2o13bo2b2o36bo73bo73bo53b2o$177bob2o
40b2o13b3o26b2o28b2o14b3o55b2o15b3o39b2o$156bo64bo14bobo23b2ob2o28bo
15bobo23b2ob2o4b2o21bo16bobo39b2o$155bo65b2o19b2o18b2o31b2o20b2o18b2ob
2o4b2o21b2o21b2o$81b2o4b2o70b2o12b3o66b2o27b2o44b2o73b2o35b2o$81b2o4b
2o12bo2b2o21b2o30bo13bobo95b2o65b2o5b2o82b2o$102bob2o18b2ob2o30b2o18b
2o157b2o5b2o$82bo41b2o53b2o12b2o3b2o3b2o67b2o68bo$81bo51b2o58b2o3b2o3b
2o67b2o67b3o$85b2o11b3o32b2o61bob2o$85bo12bobo96bo143b3o$85b2o17b2o28b
2o59b2o145bo81b3o$104b2o24bo3b2o202b2o5b2o71b2o4bobo$129bobo135b3o68b
2o5b2o71b2o$129bobo129b2o4bobo$216b2o43b2o74b2ob2o4b2o71b2o$216b2o19b
2o51b2o45b2ob2o4b2o71b2o$123b2o29b2o65bobo14bo23b2o26b2o20b2o50b2o62b
2o$123b2o29b2o18b2o45b3o13b2o23b2o31bobo15bo50b2o21b2o36b2ob2o$159bobo
13bo66bo28b2o22b3o14b2o55bobo16bo36b2o$124b2o33b3o12b2o65bo26b2ob2o44b
o51b3o15b2o$80b2o42b2o53bo36b2obo48b2o46bo75bo$80b2o17b2o32b2o43bo37b
2o2bo14b2o4b2o47b2obo97bo$85bobo12bo29b2ob2o19b2obo77b2o4b2o47b2o2bo
15b2o4b2o46b2obo$85b3o11b2o29b2o22b2o2bo13b2o4b2o130b2o4b2o46b2o2bo16b
2o4b2o$104bo67b2o4b2o205b2o4b2o$103bo$80b2obo$80b2o2bo12b2o4b2o$97b2o
4b2o5$167b2o$167b2o2$163b2o$163b2o2$170b2o$167b2o2bob2o$160b2o5b3o3b2o
$160b2o4$159b2o$159b2o2$160b2o$160b2o!

AND-NOT gates:

Code: Select all

x = 236, y = 117, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
16$68b2o$68b2o10$145b2o7b2o3b2o7b2o$145b2o5bo2bo3bo2bo5b2o$153bo7bo$
149bo2bo2b2ob2o2bo2bo$151bo3b2ob2o3bo$149bo15bo$19b2o7b2o3b2o7b2o107b
2o9b2o14b2o$19b2o7b2o3b2o7b2o95b2o9bo2bo7bo2bo15bo$139b2o10b2o9b2o14b
2o$24b2o3b2ob2o3b2o110bo15bo$24b2o4bobo4b2o112bo3b2ob2o3bo$26b3o5b3o
112bo2bo2b2ob2o2bo2bo33b2o$17bo4b2o2b3o5b3o2b2o4bo21b2o84bo7bo37b2o$
13b2o2b2o2bo2bobobo5bobobo2bo2b2o22bo37b2o37b2o5bo2bo3bo2bo5b2o$13b2o
2bo4b2o2b3o5b3o2b2o4bo21b2o37b2o37b2o7b2o3b2o7b2o$26b3o5b3o$24b2o4bobo
4b2o$24b2o3b2ob2o3b2o2$19b2o7b2o3b2o7b2o24b3o$19b2o7b2o3b2o7b2o24bobo
7$194bo$193bobo$193bobo9$187b2o11b2o$187b2o11b2o3$68b3o119bobo3bobo$
69bo124bo$62b2o11b2o114bobobobo$62b2o11b2o111bobo2bobo2bobo$69bo119bo
4bo4bo$68bobo116bo13bo$68bobo116b2ob2o5b2ob2o$65b2o2bo2b2o116b2o5b2o$
65b2o5b2o$67b5o118b2o5b2o$67b2ob2o115b2ob2o5b2ob2o$62b2o3b5o3b2o110bo
13bo$62b2obo7bob2o112bo4bo4bo$65b2o5b2o114bobo2bobo2bobo$191bobobobo$
65b2o5b2o120bo$62b2obo7bob2o113bobo3bobo$62b2o3b5o3b2o$67b2ob2o$67b5o
115b2o11b2o$65b2o5b2o113b2o11b2o$65b2o2bo2b2o$68bobo$68bobo$69bo$62b2o
11b2o117b2o$62b2o11b2o117b2o$69bo$68b3o3$69b2o$69b2o!

AND gates:

Code: Select all

x = 303, y = 172, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
31$121b2o7b2o3b2o7b2o48b2o7b2o3b2o7b2o$121b2o7b2o3b2o7b2o48b2o7b2o3b2o
7b2o$164b2o4b2o$131b2ob2o28b2ob2ob2o32b2ob2o$132bobo32b2o36bobo$128bob
o5bobo62bobo5bobo$127b4o5b4o10b2o36b2o10b4o5b4o$115b2o9bo2bo7bo2bo10bo
36bo10bo2bo7bo2bo9b2o$115b2o10b4o5b4o10b2o36b2o10b4o5b4o10b2o$128bobo
5bobo62bobo5bobo$132bobo70bobo$82b2o47b2ob2o68b2ob2o$82b2o$121b2o7b2o
3b2o7b2o33b2o13b2o7b2o3b2o7b2o$121b2o7b2o3b2o7b2o30b2ob2o13b2o7b2o3b2o
7b2o$176b2o10$77b2o$77b2o2$76b2o$76b2o3$32b2o7b2o3b2o7b2o$32b2o7b2o3b
2o7b2o2$37b2o3b2ob2o3b2o$37bo2bo2bobo2bo2bo$41bo5bo$30b2o2b4o4bo3bo4b
4o2b2o21b2o$26b2ob3o2bo2bo2bobo3bobo2bo2bo2b3o22bo$26b2o2b2o2b4o4bo3bo
4b4o2b2o21b2o39b2o7b2o3b2o7b2o49b2o7b2o3b2o7b2o$41bo5bo73b2o7b2o3b2o7b
2o49b2o7b2o3b2o7b2o$37bo2bo2bobo2bo2bo75bobo7bobo24b2o4b2o29bobo7bobo$
37b2o3b2ob2o3b2o73b2obo2b2ob2o2bob2o22b2ob2ob2o27b2obo2b2ob2o2bob2o$
81bo43bo3bob2ob2obo3bo25b2o30bo3bob2ob2obo3bo$32b2o7b2o3b2o7b2o23bobo
45bo9bo63bo9bo$32b2o7b2o3b2o7b2o23bobo44bo11bo13b2o31b2o13bo11bo$115b
2o11bo9bo16bo29bo16bo9bo11b2o$115b2o10bo11bo13b2o31b2o13bo11bo10b2o$
128bo9bo63bo9bo$125bo3bob2ob2obo3bo57bo3bob2ob2obo3bo$125b2obo2b2ob2o
2bob2o57b2obo2b2ob2o2bob2o$127bobo7bobo61bobo7bobo$121b2o7b2o3b2o7b2o
49b2o7b2o3b2o7b2o$121b2o7b2o3b2o7b2o49b2o7b2o3b2o7b2o10$167b2o$167b2o
2$81bo$80b3o$80b3o$74b2o11b2o$74b2o11b2o$80b3o$80bobo$80bobo$77b2ob3ob
2o$77bo7bo2$78bo2bo2bo$74b2o3bo3bo3b2o$74b2obo2b3o2bob2o$77b2o5b2o2$
77b2o5b2o$74b2obo2b3o2bob2o$74b2o3bo3bo3b2o$78bo2bo2bo2$77bo7bo$77b2ob
3ob2o$80bobo$80bobo$80b3o$74b2o11b2o$74b2o11b2o$80b3o$80b3o$81bo2$81b
2o$81b2o!

A diode:

Code: Select all

x = 46, y = 70, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
6$16b2ob2o6b2o$16b2ob2o6b2o3$7b2o23b2o$8bo23b2o$7b2o$31b2o$31b2o7$14b
2o$14b2ob2o$17b2o14$16b2ob2o6b2o$16b2ob2o6b2o3$32b2o$32b2o2$31b2o$31b
2o3$23b2o$22bo$23b2o2$14b2o$14b2ob2o$17b2o!

One of many ways to construct a memory bit:

Code: Select all

x = 35, y = 93, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
7$21bo$19b2o2$16b2o$16b3obo$20bo$20bo$17b2obo$17b4o$17b3o4$10b2o$10b2o
2$9b2o$9b2o3$6b2o$6b2o43$18b3o$18bobo!

Corderized camelship-based puffers:

Code: Select all

x = 129, y = 74, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
7$44b2ob3o$45bobob2o$44b2ob3o$43bo3bo$38b2o8bo$38b2o6bo6bo$32b2o15b2ob
ob2o$32b2o17bobobo$26b2o20b3obob2o$26b2o24bo$20b2o29bo$20b2o28b3o$14b
2o35b2o$14b2o$8b2o$8b2o2$12bo$8b2obob2o100b3o$10bobobo101b2o$7b3obob2o
93b2o4bob2o$11bo96bo5b3o$10bo91b2o6bo$9b3o90bo6b2o$10b2o84b2o6bo8b2o4b
2o$96bo6b2o8b2o4b2obo$90b2o6bo8b2o10bobobo$90bo6b2o8b2o11bobo$84b2o6bo
8b2o14bo$84bo6b2o8b2o14b2o$78b2o6bo8b2o20bo$78bo6b2o8b2o$72b2o6bo$72bo
6b2o$66b2o6bo20b2o$66b2o5b2o20b2obo$60b2o33bobobo$60b2o34bobo$54b2o14b
o2b2o18bo$54bo14bo3b2obo16b2o$48b2o6b2o15bobobo15bo$48b2o6b2o16bobo$
42b2o27bo$42bo28b2o$36b2o6b2o25bo$36b2o6b2o$30b2o$30b2o$24b2o18b2o$24b
2o18b2obo$18b2o24bobobo$18bo26bobo$20bo21bo$21bo20b2o$24b2o16bo$24bobo
$18b2o3bobo$24bo$19bo$19bo!

Period dubblers:

Code: Select all

x = 187, y = 121, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
9$133b2o$133b2o2$54b2o$54b2o18$11b2o7b2o3b2o7b2o$11b2o7b2o3b2o7b2o2$
17bo3b2ob2o3bo65b2o7b2o3b2o7b2o$16bobo2b2ob2o2bobo64b2o7b2o3b2o7b2o$
15b4o9b4o$16bo13bo11b2o61b2ob2o$5b2o8b3o11b3o12bo60b2ob2o$5b2o9bo13bo
11b2o54b2o4bo5bo4b2o23b2o$15b4o9b4o61b2o9bo2bo2bo9b2o18b2o$16bobo2b2ob
2o2bobo24b2o32b2o2bo3bobo3b2ob3ob2o3bobo3bo$17bo3b2ob2o3bo25b2ob2o29b
2o2b2o9bo2bo2bo9b2o$58b2o38b2o4bo5bo4b2o19b2o$11b2o7b2o3b2o7b2o69b2ob
2o26b2o$11b2o7b2o3b2o7b2o69b2ob2o54b2o$164b2o$95b2o7b2o3b2o7b2o$95b2o
7b2o3b2o7b2o6$54bo$53bobo$53bobo3$138b3o$138bobo$132b2o11b2o$132b2o11b
2o$47b2o11b2o77bo$47b2o11b2o75bo3bo$137bobobo2$52bobobo$51b7o$50bobobo
bobo80bo$51b2o3b2o74b2o3b5o3b2o$132b2ob2o5b2ob2o$47b2o11b2o73b2o2bo2b
2o$47b2ob2o5b2ob2o76b3o$50b2o5b2o76b2o2bo2b2o$132b2ob2o5b2ob2o$50b2o5b
2o73b2o3b5o3b2o$47b2ob2o5b2ob2o77bo$47b2o11b2o2$51b2o3b2o$50bobobobobo
78bobobo$51b7o79bo3bo$52bobobo82bo$132b2o11b2o$132b2o11b2o$47b2o11b2o
76bobo$47b2o11b2o76b3o3$139b2o$139b2o$54b2o$54b2o!

A random splitter network:

Code: Select all

x = 324, y = 291, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
34$139b2o$139b2o2$204b2ob2o6b2o$204b2ob2o6b2o3$129b2o41b2o40b2o4b2o$
131bo41bo42bo3b2o$129b2o4bobo34b2o40b2o$103b2o30bobo81b2o$101bo4b2o28b
o82b2o$103bo2b2o$100b2obob3o14b2o$100b2o16b2o2b2o$101bo16b2o$82b2o16bo
$82b2o112b2o$196b2o31b2ob2o6b2o$82b2o145b2ob2o6b2o$82b2o116b2o2bo$79b
2o114b2o3bo3bo$79b2o23b2o25b2o62b2o7bo36b2o2b2o$104b2ob2o22b2o69b3o35b
obo2b2o$100b2o5b2o132bobo$100b2o92b2o46bo2bo$70b2o34b2o32b2o52b2o6b2o
40b2o$70b2o29b2o3b2o4b2o20b2o19b2ob2o6b2o30b2o2b2o$102bo9b2o19b4obobo
14b2obo7b2o30bobobo$69b2o10bo17bo6b2obo23bo2bobo2bo18b2o36b3o$69b2o6bo
2b2o28bo21bo4bo22b2o37bo$80b2o22bo4bo21bo28b2o2bo6b2o$80b2o18bo2bobo2b
o23bob2o6bo17bo10b2o48b2o$74b2o7bob2o14bobob4o19b2o9bo81b2o32b2ob2o6b
2o$74b2o6b2ob2o19b2o20b2o4b2o3b2o29b2o83b2ob2o6bo$100b2o32b2o34b2o96bo
$140b2o78b2o41b4obo$133b2o5b2o78b2o10b2o29bo2bobo2b2o$109b2o22b2ob2o
95bo29b5o3b2o$109b2o25b2o23b2o69b2o32b3o$161b2o104b2ob2o$158b2o110b2o$
158b2o$226bo$158b2o63bo2bo$141bo16b2o64b3o34b2o$122b2o16bo83bo2bo2bo
30b2o$118b2o2b2o16b2o80b2o5b2o$50b2o6b2ob2o55b2o14b3obob2o80bob3o$50b
2o6b2ob2o71b2o2bo83b2o36b2o$105bo28b2o4bo58b2o59b2o$104bobo30b2o60b2o
22b2o$45b2o21b2o34bobo4b2o110b2o$45b2o13bo7bo41bo143bobo$61bobo4b2o41b
2o141b3o$46b2o13b3o$46b2o9b3o$58bob4o$61bobo$61b3o$62bo38b2o147b3o$
101b2o146bo$69b2o175b2o3b2o$69b2o175bo$249bo$250bo$70b2o175b2obo$33b2o
3b2o30b2o175bobo$33b2o3bo209bo$38b2o2$38b2o$33b2o3bo$33b2o3b2o2$66b2o$
66b2o5$249b2o$249b2o3$248b2o$248b2o3$242bobo$242bobo$243bo4$239bo14bob
o$237b2o15bobo$255bo$234b2o$234b3obo$238bo$238bo$235b2obo$235b4o$235b
3o17$231b2o2$230b2o3b2o$235b2o$228b3o$228b3o$234b2o$234b2o4$226b2o14bo
bo$224bo17b3o$222bobo$222bobo$223bo3bo$227bo$223bob3o26bobo$222bobobo
27b3o$223bo2bo$224bo26$228bobo$228bobo$229bo$213bo$210bobo$210b3o$214b
3o$210b4obo26bobo$210bobo29bobo$210b3o30bo$211bo3$254bobo$254bobo$255b
o24$200bo$198bobo$198bobo2bo$199bo2b3o$198bob2ob2o23bobo$197bobo28b3o$
198bobo$197b4o$195b2o$195b2o2$242bobo$242b3o3$228b2o12b2o10b2o$228b2o
12b2o10b2o!

[confocaloid]

It is relatively rare for people to begin with (attempted) definitions.

I don't view such "signatures" as "natural"; see below for details. They are already arbitrary for two-state cellular automata. They become ill-defined and misleading in more general situations.

[...] First, some definitions. Every pattern's evolution has its own rulespace with a given neighborhood and symmetry, which is a partially ordered finite set (unless we consider an infinitely large neighborhood) defined by the necessary transitions enabling the pattern to evolve and all the optional transitions that don't affect the pattern's behavior. The partial order works between rules A ≥ B, if the set of transitions of B is a subset of transitions of A. Thus, terminology and operations of order theory can be applied to rules and rulespaces, such as additions and subtraction by set operations between rules' transitions. Every finite rulespace has its bottom and top element, which I view as natural "signatures" of the pattern. For example, the bottom rule of the glider is B3aijn/S2ae3jnr and its top rule is B2ikn34-r5-n678/S0234-k5678. [...]

It is safe to use "partially ordered finite set" without defining it explicitly here, because that's standard mathematical terminology, see for example
https://en.wikipedia.org/wiki/Partially_ordered_set
https://en.wikipedia.org/wiki/Greatest_ ... st_element
(Even then, it would be nice to link to a source.)

However, words and phrases such as "rule", "rulespace", "pattern's evolution", "transition" aren't standard mathematical terminology. So, at the very least, all of them would need to be defined explicitly, if you want to define what claims such as "bottom rule of the glider is B3aijn/S2ae3jnr" mean.
Since you didn't define the local jargon, you didn't really define "bottom rule" or "top rule" either.

(I can guess what you likely meant, simply because I'm familiar with counterintuitive local jargon. However, you cannot expect the same level of familiarity with jargon from your readers.)

Additionally, many uses of those words and phrases (including those in the above post) are inconsistent in various ways with available CA-related sources. Whenever counterintuitive local jargon contradicts CA-related sources in which authors were more careful with terminology, I'm going to discard the jargon and follow the sources, as much as possible.

Additionally, I don't see how "symmetry" would be relevant here, can you explain that part? (I see relevance of neighbourhood.)

"Associated subspaces" of evolutionary sequences

As I see it, in the context of discussing CA of some specific type (for example in the context of discussing two-state isotropic cellular automata with Moore neighbourhood), for every evolutionary sequence (with cells updating their states according to the set of rules of the cellular automaton) there exists an "associated subspace" (defined here as the set of all cellular automata where that evolutionary sequence is preserved exactly, unchanged).

The "associated subspace" of an evolutionary sequence is only defined relative to the particular CA type under discussion. (For example, it will in general change if you change the neighbourhood, even if the evolutionary sequence itself is preserved.)

The "associated subspace" of an evolutionary sequence can be a single-element set (in which case the evolutionary sequence is said to be "endemic" relative to the particular CA type under discussion).

In general, the "associated subspace" of an evolutionary sequence can be fully described by writing down those CA rules that are necessary for the evolutionary sequence to proceed. Given a specific CA type and the set of "necessary rules", it is possible to derive the set of "forbidden rules" (those CA rules that contradict one of "necessary rules") and it is also possible to derive the set of "optional rules" (those CA rules that are neither "necessary rules" nor contradictory with any of "necessary rules").

In general, there is no unique natural way to define "top" or "bottom" elements in the "associated subspace" of an evolutionary sequence. With two cellstates, there are four different possible transitions (birth, nonbirth, survival, death) and four different types of CA rules. That means there are different ways to define what would count as "top" or "bottom", for example:

• Count the Life-like CA "B/S012345678" as the "bottom CA" of the empty evolutionary sequence (all optional nonbirth rules are followed; all optional survival rules are followed), and count B012345678/S as "top CA" of the empty evolutionary sequence (all optional birth rules are followed; all optional death rules are followed).
• Count the Life-like CA "B/S" as the "bottom CA" of the empty evolutionary sequence (all optional nonbirth rules are followed; all optional death rules are followed), and count B012345678/S012345678 as "top CA" of the empty evolutionary sequence (all optional birth rules are followed; all optional survival rules are followed).

With two or more possibilities like this, neither of them is automatically "more natural" than another.
(I consider it much more natural to define "bottom CA" by saying that all optional nonbirth rules are followed, and all optional survival rules are followed, and to define "top CA" by saying that all optional birth rules are followed, and all optional death rules are followed.
The underlying intuition here is that the birth rules and the death rules are those rules that prescribe state-changing transitions (births and deaths), and it makes sense to have "as inactive as possible" and "as active as possible" on two opposite ends of the "spectrum".)

With three or more cellstates, the whole idea of "top CA" and "bottom CA" becomes impossible to define in an intuitively convincing non-arbitrary way. I think it misses the point.

[...]
Existing CA-related sources (web pages, books, publications, and so on)?

The choices "rule", "transition rule", "update rule" also have the significant advantage that the same terminology can be found in multiple already existing CA-related sources. Many other people used the same words/phrases to refer to the same concepts. This terminology is already in use.

[...] To avoid misunderstanding, I'd like to clarify that this terminology is old. It is not created by someone on these forums. The wider Life/CA community uses the same words and phrases to refer to the same ideas.

So it's not quite correct to refer to this as "proposed terminology" -- the terminology is already in use.

Below I'm crossposting links to some sources (and to forum posts with more links/quotes). Those sources are CA-related, and use 'condition', 'transition rule', 'transition', 'rule' to refer to the same ideas.

[...]
https://web.archive.org/web/20230928182 ... ary_03.htm
https://conwaylife.com/w/index.php?titl ... 314#Page_2
https://web.archive.org/web/20040323172 ... ndixA.html

[...]
viewtopic.php?p=171596#p171596
viewtopic.php?p=171671#p171671
viewtopic.php?p=172091#p172091
[...]

[R2INT]

Here are some 'interesting' things that the small 'U' spaceship can do with still lives of up to 7 cells:

Code: Select all

x = 60, y = 51, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
7b2o48b2o$7b2o49bo$21b2o32b2o$2o21bo31b2o$2bo18b2o7b2o16b2o$2o26bo2bo
18bo$28b2o18b2o9$7b2o$2o5b2o12b2o33b2o$2bo20bo32bo$2o19b2o35b2o$58b2o
$30b2o$28bo2bo17b2o$28b2o21bo$49b2o2$2o5b2o$2bo4b2o$2o5$28b2o$28bo$21b
2o7bo$23bo5b2o$21b2o$9b2o$7bo2bo$7b2o$2o$2bo$2o$30b2o$31bo$28b2o$28b2o
2$21b2o$23bo$21b2o!

EDIT: For 8 cells, including 2-block constellations:

Code: Select all

x = 261, y = 76, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
69b2o31b2o$7b2o60b2o31b2o$7bo28b2o5b2o17b2o31b2o50b2o79b2o$8bo29bo4b2o
19bo32bo49b2o30b2o47b2o22b2o$2o6bo27b2o24b2o31b2o43b2o37b2o26b2o43b2o
$2bo6bo132bo29b2o33b2ob2o21b2o22b2o$2o6b2o33b2o95b2o32bo5b2o18b2o8b2o
21b2o22b2o$43b2o50b2o51b2o22b2o6b2o20bo23b2o22b2o$62b2o31b2o51b2o50b2o
26bo23bo$62b2o162b2o22b2o4$43b2o$2o34b2o5b2o$2bo35bo$2o34b2o2$11b2o30b
2o57b2o$7bob2obo30b2o57b2o$7b2o86b2o$97bo$95b2o3$100b2o148b2o$100b2o103b
2o43b2o$205b2o49b2o$70b2o152b2o30b2o$11b2o57b2o75b2o29b2o24b2o18b2o23b
2o$7bob2obo50b2o82b2o29b2o24b2o45bo$7b2o56bo74b2o29b2o9b2o13b2o32b2o16b
2o$2o61b2o77bo30bo8b2o15bo31b2o$2bo137b2o29b2o24b2o25b2o$2o145b2o77bo
$147b2o75b2o$62b2o$62b2o8$103b2o$11b2o90b2o154b2o$7bob2obo83b2o161b2o
$7b2o89bo48b2o57b2o16b2o$96b2o49b2o29b2o26b2o16b2o$2o138b2o36b2o77b2o
$2bo67b2o70bo28b2o8b2o23b2o24b2o23b2o$2o68b2o30b2o36b2o6b2o23bo7b2o23b
2o24b2o16b2o$63b2o37b2o44b2o21b2o26b2o24b2o25bo$65bo135bo25bo22b2o$63b
2o134b2o24b2o3$61b2o$61b2o6$229b2o$229b2o$148b2o27b2o53b2o$104b2o42b2o
27b2o2b2o19b2o28b2o$104b2o35b2o27b2o9b2o19b2o21b2o$97b2o44bo6b2o20bo54b
o$99bo41b2o7b2o18b2o34b2o17b2o$97b2o107b2o$199b2o$104b2o95bo$104b2o93b
2o!

[Yoel]

Additionally, I don't see how "symmetry" would be relevant here, can you explain that part? (I see relevance of neighbourhood.)

That's because the sets of transitions and thus rulespaces are different in different symmetries. There are only 2^18 outer-totalistic Life-like CAs, 2^102 isotropic non-totalistic Life-like CAs, 2^512 anisotropic Life-like CAs, etc.

With two cellstates, there are four different possible transitions (birth, nonbirth, survival, death) and four different types of CA rules. That means there are different ways to define what would count as "top" or "bottom", for example:

• Count the Life-like CA "B/S012345678" as the "bottom CA" of the empty evolutionary sequence (all optional nonbirth rules are followed; all optional survival rules are followed), and count B012345678/S as "top CA" of the empty evolutionary sequence (all optional birth rules are followed; all optional death rules are followed).
• Count the Life-like CA "B/S" as the "bottom CA" of the empty evolutionary sequence (all optional nonbirth rules are followed; all optional death rules are followed), and count B012345678/S012345678 as "top CA" of the empty evolutionary sequence (all optional birth rules are followed; all optional survival rules are followed).

One can certainly define different orders in rulespaces, However, if the states are ordered as 0 and 1 (dead cells are "inferior" to live cells), as it is the case in traditionally viewed Like-like rules, the only straightforward way to define the zero or bottom rule of the entire Life-like rulespace is B/S.

With three or more cellstates, the whole idea of "top CA" and "bottom CA" becomes impossible to define in an intuitively convincing non-arbitrary way. I think it misses the point.

If N states are ordered (enumerated), then all the evolutional transitions could in principle also be ordered, e.g. simply by encoding them as N-base numbers. I agree it's counter-intuitive though. For example, rulespaces of my cyclical multistate rules could be better described as groups of transitions where the identity element is the rule itself, thus applying group theory rather than order theory.

However, you're right that my definitions are only applicable to Life-like rules. To me they make sense. I've discovered many rules by applying logical and order operations to transitions. B3aijn/S2ae3jnr clearly describes what transitions exactly make a glider evolve, while B2ikn34-r5-n678/S0234-k5678 describes the absolute maximum of transitions that still allow a glider to evolve. Such ranges are mentioned on every page of Lifewiki dedicated to various Life or Life-like objects.

Rulespaces of some restrictes subtypes of Life-llke rules like self-complementary may also be better described as groups because the states are intertwined and there is no B/S, only B/S012345678. I'll add some clarifications to the post.

[confocaloid]

[...] Every pattern's evolution has its own rulespace with a given neighborhood and symmetry, [...]

Additionally, I don't see how "symmetry" would be relevant here, can you explain that part? (I see relevance of neighbourhood.)

That's because the sets of conditions and thus rulespaces are different in different symmetries. There are only 2^18 outer-totalistic Life-like CAs, [...]

The phrase "Every pattern's evolution has its own rulespace with a given neighborhood and symmetry, ..." doesn't tell that.

Interpreting "pattern's evolution" in the sense "evolutionary sequence", by itself it doesn't determine any symmetry in the sense you mentioned here.
For example,

• the R-pentomino evolutionary sequence can be seen as happening in a Life-like cellular automaton,
• or alternatively it can be seen as happening in a two-state isotropic cellular automaton with Moore neighbourhood,
• or alternatively the same evolutionary sequence can be seen as happening in a two-state (not necessarily isotropic) cellular automaton with Moore neighbourhood,
• etc. etc.

"Symmetry" in this sense isn't a property of the evolutionary sequence.

With two cellstates, there are four different possible transitions (birth, nonbirth, survival, death) and four different types of CA rules. That means there are different ways to define what would count as "top" or "bottom", for example:

• Count the Life-like CA "B/S012345678" as the "bottom CA" of the empty evolutionary sequence (all optional nonbirth rules are followed; all optional survival rules are followed), and count B012345678/S as "top CA" of the empty evolutionary sequence (all optional birth rules are followed; all optional death rules are followed).
• Count the Life-like CA "B/S" as the "bottom CA" of the empty evolutionary sequence (all optional nonbirth rules are followed; all optional death rules are followed), and count B012345678/S012345678 as "top CA" of the empty evolutionary sequence (all optional birth rules are followed; all optional survival rules are followed).

Well, one can certainly define different orders in rulespaces, However, if the states themselves are ordered as 0 vs.1, as it is the case in Like-like rules, the only straightforward way to define the zero or bottom rule of the entire Life-like rulespace is B/S.

It isn't unique, as long as the difference between the character string (notation in use) and the encoded set of rules of the cellular automaton is understood. Instead of considering birth rules and survival rules, it suffices to consider birth rules and death rules.
Write 0 if a state-preserving rule is followed, and write 1 if a state-changing rule is followed:

• For birth rules and nonbirth rules, write 0 if a nonbirth rule is followed and 1 if a birth rule is followed.
• For survival rules and death rules, write 0 if a survival rule is followed and 1 if a death rule is followed.

Then B/S012345678 becomes "all zeros" because every followed rule preserves the state of the middle cell, and B012345678/S becomes "all ones" because every followed rule changes the state of the middle cell.

I see this as a simpler and more natural approach.

[...] However, you're right that my definitions are only applicable in Life-like rules. [...]

Not only; the same idea also can be applied to two-state cellular automata that aren't Life-like CA (for example two-state isotropic CA with Moore neighbourhood).

One of main issues is that so far there were almost no clear definitions. Instead, people are attempting to use local jargon, apparently without pausing and double-checking what it could mean (and whether it even means anything in particular at all). The local jargon is counterintuitive, and it is inconsistent with terminology used in relevant CA-related sources (I linked to examples).

Another issue is that, despite what is claimed in the above posts, "B3aijn/S2ae3jnr" fails to tell what exactly makes a glider evolve. It only tells part of the information that would be needed to know what is necessary for the glider to evolve. This may mean that the approach is misleading; it hides (obscures) important information that needs to be seen and understood.

[Yoel]

"Symmetry" in this sense isn't a property of the evolutionary sequence.

It is a property of rulespaces, not of evolutionary sequences. For example, there are many non-isotropic rules that allow a glider to evolve, if it's positioned in a particular way. But this is not what we usually view as rulespaces, because if we reflect or turn the same glider, the sequence may not work. We may also consider rules where the evolutionary sequence of a glider is followed, if it's turned 90 degrees, but not if it is reflected, etc. If we don't consider symmetry, the normative concept of the glider falls apart.

I see this as a simpler and more natural approach.

I see your point. However, there seems to be already an established tradition to describe rulespaces in LikeWiki, Calagolue and this forum the way I've described. It sound natural to me, but I can see how your approach can be viewed more natural. For one thing, it solves the apparent difference between Like-like rules in general vs. self-complementary Life-like rules, where B/S012345678 is the natural "zero".

One of main issues is that so far there were almost no clear definitions. Instead, people are attempting to use local jargon, apparently without pausing and double-checking what it could mean (and whether it even means anything in particular at all). The local jargon is counterintuitive, and it is inconsistent with terminology used in relevant CA-related sources (I linked to examples).

I believe that algebraic approach to rules, whether applying order theory or group theory (for some restricted symmetries), can be useful and productive. I totally agree that clear definitions are very important. I'll think about your approach; as you know, I dedicate much research to self-complementary rules where your view is clearly natural.

Another issue is that, despite what is claimed here, "B3aijn/S2ae3jnr" fails to tell what exactly makes a glider evolve. It only tells part of the information that would be needed to know what is necessary for the glider to evolve. This may mean that the approach is misleading; it hides (obscures) important information that needs to be seen and understood.

Yes, two lines are necessary for the full description of the glider: B3aijn/S2ae3jnr and B2ikn34-r5-n678/S0234-k5678. I would expect the bottom rule to be the least chaotic, merely allowing the glider to evolve, and the top rule to be the most explosive.

If we rewrite these lines as B3aijn/S0234-k5678 and B2ikn34-r5-n678/S2ae3jnr according to your approach, such a set also constitutes a partial order. However, both B3aijn/S0234-k5678 and B2ikn34-r5-n678/S2ae3jnr exhibit infinite growth, while non-explosive rules lie somewhere as optimal points in the middle of this range. I personally find such an ordering highly counter-intuitive (unless applied to self-complementary rules).

[Yoel]

What software (if any) did you use to produce this collection? I've found stable reflectors and splitters by playing manually with still-lifes. I suspect much more efficient ones could be found automatically.

Here are some 'interesting' things that the small 'U' spaceship can do with still lives of up to 7 cells:

Code: Select all

x = 60, y = 51, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
7b2o48b2o$7b2o49bo$21b2o32b2o$2o21bo31b2o$2bo18b2o7b2o16b2o$2o26bo2bo
18bo$28b2o18b2o9$7b2o$2o5b2o12b2o33b2o$2bo20bo32bo$2o19b2o35b2o$58b2o
$30b2o$28bo2bo17b2o$28b2o21bo$49b2o2$2o5b2o$2bo4b2o$2o5$28b2o$28bo$21b
2o7bo$23bo5b2o$21b2o$9b2o$7bo2bo$7b2o$2o$2bo$2o$30b2o$31bo$28b2o$28b2o
2$21b2o$23bo$21b2o!

EDIT: For 8 cells, including 2-block constellations:

Code: Select all

x = 261, y = 76, rule = B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
69b2o31b2o$7b2o60b2o31b2o$7bo28b2o5b2o17b2o31b2o50b2o79b2o$8bo29bo4b2o
19bo32bo49b2o30b2o47b2o22b2o$2o6bo27b2o24b2o31b2o43b2o37b2o26b2o43b2o
$2bo6bo132bo29b2o33b2ob2o21b2o22b2o$2o6b2o33b2o95b2o32bo5b2o18b2o8b2o
21b2o22b2o$43b2o50b2o51b2o22b2o6b2o20bo23b2o22b2o$62b2o31b2o51b2o50b2o
26bo23bo$62b2o162b2o22b2o4$43b2o$2o34b2o5b2o$2bo35bo$2o34b2o2$11b2o30b
2o57b2o$7bob2obo30b2o57b2o$7b2o86b2o$97bo$95b2o3$100b2o148b2o$100b2o103b
2o43b2o$205b2o49b2o$70b2o152b2o30b2o$11b2o57b2o75b2o29b2o24b2o18b2o23b
2o$7bob2obo50b2o82b2o29b2o24b2o45bo$7b2o56bo74b2o29b2o9b2o13b2o32b2o16b
2o$2o61b2o77bo30bo8b2o15bo31b2o$2bo137b2o29b2o24b2o25b2o$2o145b2o77bo
$147b2o75b2o$62b2o$62b2o8$103b2o$11b2o90b2o154b2o$7bob2obo83b2o161b2o
$7b2o89bo48b2o57b2o16b2o$96b2o49b2o29b2o26b2o16b2o$2o138b2o36b2o77b2o
$2bo67b2o70bo28b2o8b2o23b2o24b2o23b2o$2o68b2o30b2o36b2o6b2o23bo7b2o23b
2o24b2o16b2o$63b2o37b2o44b2o21b2o26b2o24b2o25bo$65bo135bo25bo22b2o$63b
2o134b2o24b2o3$61b2o$61b2o6$229b2o$229b2o$148b2o27b2o53b2o$104b2o42b2o
27b2o2b2o19b2o28b2o$104b2o35b2o27b2o9b2o19b2o21b2o$97b2o44bo6b2o20bo54b
o$99bo41b2o7b2o18b2o34b2o17b2o$97b2o107b2o$199b2o$104b2o95bo$104b2o93b
2o!

[confocaloid]

Note: your post contains some damaged quotes with incorrectly specified author (making it appear as if you were talking to yourself).

"Symmetry" in this sense isn't a property of the evolutionary sequence.

It is a property of rulespaces, not of evolutionary sequences. [...]

It is one of properties of a CA type. What are the symmetries in this sense (i.e. which equivalence relation is defined on the set of all neighbourhood configurations), depends on the CA type.

For Life-like cellular automata, there are 18 equivalence classes of neighbourhood configurations (that is, 18 fully-specified cell conditions).

For two-state isotropic cellular automata with Moore neighbourhood, there are 102 equivalence classes of neighbourhood configurations (that is, 102 fully-specified cell conditions).

For (not necessarily isotropic) two-state CA with Moore neighbourhood, there are 512 fully-specified cell conditions (in this particular case, fully-specified cell condition ~ neighbourhood configuration).

(Obviously one can continue for other types of cellular automata.)

[...] For example, there are many non-isotropic rules that allow a glider to evolve, if it's positioned in a particular way. But this is not what we usually view as rulespaces, [...]

While recent exploration on these forums is focused on two-dimensional isotropic cellular automata (and on a small subset of those), I don't see a good reason why one would arbitrarily limit the notion of a "rulespace" (however it would be defined) to exclude non-isotropic CA.

(Spaceships can meaningfully exist in non-isotropic CA, too.)

[...] I believe that algebraic approach to rules, whether applying order theory or group theory (for some restricted symmetries), can be useful and productive. I totally agree that clear definitions are very important. [...]
[...] I would expect the bottom rule to be the least chaotic, merely allowing the glider to evolve, and the top rule to be the most explosive. [...]

I'm not sure how much of that "algebraic approach" remains useful, by the time when one begins caring about "explosivity".

I think defining "bottom" and "top" for subspaces of cellular automata makes more sense for the purpose of doing further calculations, automated searches, etc., as well as for gaining better understanding. In that context, having "as few state-changing rules as possible, as many state-preserving rules as possible" and "as many state-changing rules as possible, as few state-preserving rules as possible" at two opposite ends seems to make more sense.

For the purposes of explaining, defining or measuring explosivity, neither of those two choices leads to a particularly "nice" situation, seemingly minor changes in the update rules of the cellular automaton can have totally unexpected consequences, regardless of how you define coordinates. (And neither of possible "top/bottom" approaches tells what to do in the general case when there are more than two cellstates.)

[Yoel]

Note: your post contains some damaged quotes with incorrectly specified author (making it appear as if you were talking to yourself).

Sorry for that! Fixed.

While recent exploration on these forums is focused on two-dimensional isotropic cellular automata (and on a small subset of those), I don't see a good reason why one would arbitrarily limit the notion of a "rulespace" (however it would be defined) to exclude non-isotropic CA.

(Spaceships can meaningfully exist in non-isotropic CA, too.)

Because we intuitively expect an object that looks, say, like a rocket, to fly like a rocket in every direction. Because it's much harder to construct something complex in a rule where guns, reflectors and other common elements may need to have 8 different versions shooting different spaceships in different directions. Because the non-isotropic space is too large and we are still only scratching the surface of isotropic CAs. In theory, of course, one should not limit such notions to non-isotropic CAs.

I'm not sure how much of that "algebraic approach" remains useful, by the time when one begins caring about "explosivity".

What I mean by "algebraic approach" is that addition, subtraction, in some cases inversion and other operations over rules and rulespaces may provide us with useful tools for designing rules that exhibit some desired behavior.

Explosivity should also be defined and measured rigorously, although I have not seen research on this subject. One somewhat related example is Chaitin's constant, which measures the probability that a random program will halt or run forever - an extremely chaotic thing, yet rigorously definable and measurable.

For the purposes of explaining, defining or measuring explosivity, neither of those two choices leads to a particularly "nice" situation, seemingly minor changes in the update rules of the cellular automaton can have totally unexpected consequences, regardless of how you define coordinates. (And neither of possible "top/bottom" approaches tells what to do in the general case when there are more than two cellstates.)

That's true, of course. Undecidability and non-trivial behavior is one of the things that make CAs exciting and Turing-complete, for that matter. Often enough removing some birth transitions make a rule explosive, because it causes something unexpected like a replicator or a breeder to emerge.

[confocaloid]

[...] I don't see a good reason why one would arbitrarily limit the notion of a "rulespace" (however it would be defined) to exclude non-isotropic CA. [...]

Because we intuitively expect an object that looks, say, like a rocket, to fly like a rocket in every direction. [...] In theory, of course, one should not limit such notions to non-isotropic CAs. [...]

Why "in theory" and not "in practice"?

Again: there is no debate that things are easier to understand, when the rules of the cellular automaton are isotropic. There is no debate that isotropic cellular automata are easier to explore.

However, that doesn't really provide support for your claim "But this is not what we usually view as rulespaces", and that doesn't give any good reasons why one would want to exclude non-isotropic cellular automata when defining what "rulespace" is supposed to mean. (And arbitrary restrictions can make ideas less intuitive and harder to understand.)

[...] Often enough removing some birth transitions make a rule explosive, [...]

Those aren't "transitions", those are either rules (birth rules, survival rules, death rules, non-birth rules), or birth/survival conditions (if you want to stay closer to the "rulestring notation" representation).

For terminology, I suggest to check the sources I linked before: viewtopic.php?p=209512#p209512

Following the terminology from relevant sources, there is only one birth transition in a two-state cellular automaton (and there are at most four transitions in total). If you "remove" or "forbid" the birth transition, then no cell will be born under any circumstances; an initially nonempty pattern would either gradually die out or settle into a still-life.

(Following the local jargon from these forums would lead to nowhere, simply because the local jargon is counterintuitive and sometimes self-contradictory, and it was never well-defined or even very well understood.)

[Yoel]

[...] I don't see a good reason why one would arbitrarily limit the notion of a "rulespace" (however it would be defined) to exclude non-isotropic CA. [...]

Because we intuitively expect an object that looks, say, like a rocket, to fly like a rocket in every direction. [...] In theory, of course, one should not limit such notions to non-isotropic CAs. [...]

Why "in theory" and not "in practice"?

Here is a Common Lisp program I just have written that translates Hensel rulelines into potentially anisotropic ruletables:

Code: Select all

;; Ruletable generator that turns isotropic Life-like cellular automata into potentially anisotropic ones

;; By Yoel Matveyev, 2025

(defun reflect (l)
  (list
   (car l)
   (nth 7 l)
   (nth 6 l)
   (nth 5 l)
   (nth 4 l)
   (nth 3 l)
   (nth 2 l)
   (nth 1 l)))

(defun rotate (l)
  (list
   (nth 6 l)
   (nth 7 l)
   (car l)
   (nth 1 l)
   (nth 2 l)
   (nth 3 l)
   (nth 4 l)
   (nth 5 l)))

(defun rotate4reflect (l)
  (remove-duplicates
   (append
    (loop for x from 0 to 3 collect l do (setf l (rotate l)))
    (progn
      (setf l (reflect l))
      (loop for x from 0 to 3 collect l do (setf l (rotate l)))))
   :test #'equal))

(defun split-bs (ln)
  (let (b s)
  (setf ln (remove #\/ (string-downcase ln))
	b (position #\b ln)
	s (position #\s ln))
  (if (< b s)
      (list (subseq ln (1+ b) s) (subseq ln (1+ s)))
      (list (subseq ln (1+ b)) (subseq ln (1+ s) b)))))

(defun find-digits (ln)
  (sort (remove nil
		(loop for x from 0 to 9 collect
					(position (digit-char x) ln)))
	#'<))

(defun find-subconds (n ln)
  (let ((p (position (digit-char n) ln))
	(d (reverse (find-digits ln))))
    (if p
	(subseq ln (1+ p)
		(car (last
		      (loop for x in d while (< p x) collect x))))
	nil)))

  
(defparameter conds
  '(((nil (0 0 0 0 0 0 0 0))) 
    ((#\c (0 1 0 0 0 0 0 0))
     (#\e (1 0 0 0 0 0 0 0)))
    ((#\a (1 1 0 0 0 0 0 0))
     (#\c (0 1 0 1 0 0 0 0))
     (#\e (1 0 1 0 0 0 0 0))
     (#\i (1 0 0 0 1 0 0 0))
     (#\k (1 0 0 1 0 0 0 0))
     (#\n (0 1 0 0 0 1 0 0)))
    ((#\a (1 1 1 0 0 0 0 0))
     (#\c (0 1 0 1 0 1 0 0))
     (#\e (1 0 1 0 1 0 0 0))
     (#\i (0 1 1 1 0 0 0 0))
     (#\j (1 0 1 1 0 0 0 0))
     (#\k (1 0 1 0 0 1 0 0))
     (#\n (1 1 0 1 0 0 0 0))
     (#\q (1 1 0 0 0 1 0 0))
     (#\r (1 1 0 0 1 0 0 0))
     (#\y (1 0 0 1 0 1 0 0)))
    ((#\a (1 1 1 1 0 0 0 0))
     (#\c (0 1 0 1 0 1 0 1))
     (#\e (1 0 1 0 1 0 1 0))
     (#\i (0 1 1 0 0 0 1 1))
     (#\j (0 1 1 0 1 0 1 0))
     (#\k (1 1 0 1 0 0 1 0))
     (#\n (0 1 1 1 0 0 0 1))
     (#\q (1 1 1 0 0 1 0 0))
     (#\r (1 1 1 0 1 0 0 0))
     (#\t (1 0 0 1 1 1 0 0))
     (#\w (1 1 0 0 0 1 1 0))
     (#\y (1 1 0 1 0 1 0 0))
     (#\z (1 1 0 0 1 1 0 0)))
    ((#\a (0 1 1 1 1 1 0 0))
     (#\c (1 1 1 0 1 0 1 0))
     (#\e (1 1 0 1 0 1 0 1))
     (#\i (1 1 1 0 0 0 1 1))
     (#\j (1 1 1 1 0 1 0 0))
     (#\k (1 1 0 1 0 1 1 0))
     (#\n (1 1 1 0 1 0 0 1))
     (#\q (1 1 1 0 1 1 0 0))
     (#\r (1 1 0 1 1 1 0 0))
     (#\y (1 0 1 1 0 1 1 0)))
    ((#\a (1 1 1 1 1 1 0 0))
     (#\c (1 0 1 1 1 1 1 0))
     (#\e (0 1 1 1 1 1 0 1))
     (#\i (0 1 1 1 0 1 1 1))
     (#\k (1 1 1 1 0 1 1 0))
     (#\n (1 1 1 0 1 1 1 0)))
    ((#\c (1 1 1 1 1 1 1 0))
     (#\e (0 1 1 1 1 1 1 1)))
   ((nil (1 1 1 1 1 1 1 1)))))  


(defun minus-cond (n ln)
  (let ((cn (loop for x in (elt conds n) collect (car x)))
	(cn- (loop for x across ln collect x)))
    (loop for x in cn- do (setf cn (remove x cn)))
    cn))

(defun find-subcond-list (n s)
  (let (l l2)
    (if s
	(if (equal s "")
	    (loop for x in (elt conds n) do (push (cadr x) l))
	    (progn
	      (if (find #\- s)
		  (setf l2 (minus-cond n (subseq s 1)))
		  (setf l2 (loop for x across s collect x)))
	      (loop for x in l2 do
		(loop for y in (elt conds n) do
		  (if (equal (car y) x) (push (cadr y) l)))))))
    (reverse l)))

;; Use Hensel notation here and name your rule

(defun print-ruletable (ln name)
  (let 
      ((bs (split-bs ln))
       w bl sl p)
    (loop for x from 0 to 8 do
      (setf w (find-subconds x (car bs)))
      (push w bl))
    (setf bl (reverse bl))
    (loop for x from 0 to 8 do
      (setf w (find-subconds x (cadr bs)))
      (push w sl))
    (setf sl (reverse sl))
    (list bl sl)
    (format t "@RULE ~a~%" name)
    (format t "@COLORS~%1 255 255 255~%")
    (format t "@TABLE~%n_states:2~%neighborhood:Moore~%symmetries:none~%")
    (format t "var A={0,1}~%var B=A~%var C=A~%var D=A~%var E=A~%var F=A~%var G=A~%var H=A~%")
    (format t "# Birth conditions~%")
    (loop for x from 0 to 8 do
      (setf p
	    (find-subcond-list x (elt bl x)))
      (if p
	  (loop for y in p do
	    (loop for z in (rotate4reflect y) do
	      (format t "0,~{~a~^,~},1~%" z)))))
    (format t "# Survival conditions~%")
     (loop for x from 0 to 8 do
      (setf p
	    (find-subcond-list x (elt sl x)))
      (if p
	  (loop for y in p do
	    (loop for z in (rotate4reflect y) do
	      (format t "1,~{~a~^,~},1~%" z)))))
    (format t "1,A,B,C,D,E,F,G,H,0")))

Let's reproduce CGOL and insert the survival condition 1,0,1,0,1,0,1,1,1,1 into it. The Snark still works, the Gosper glider gun works too. There is still horizontal reflection symmetry. However, complex things act wacky:

Code: Select all

x = 901, y = 735, rule = AlmostLife-S1010101111
90$541bo$541bo2$541bo$540bobo2$540b3o3$540b3o2$540bobo$541bo2$541bo$
541bo$540b2o3b2o3bo2bo3b2o$540b2o3b5o4b5o$539b2o4b2o3bo2bo3b2o$541bo$
414bo$414bo2$414bo$413bobo2$404bo8b3o$403bobo2$413b3o$403b3o$403b3o7bo
bo$404bo9bo2$414bo$404bo9bo$403b3o7b2o3b2o3bo2bo3b2o$403b3o7b2o3b5o4b
5o$412b2o4b2o3bo2bo3b2o$414bo$403bobo$404bo$388bo2bob2obo2bo$387b2o2bo
4bo2b2o$388bo2bob2obo2bo6$385b8o$385bob4obo$385b8o3$415bo$398b3o15b2o
84bobo$397bo3bo13b2o85b2o$397bo3bo101bo77b2o$398b3o180bobo$581bo9b2o6b
2o$589bo4bo2bo4bo$589bo4bo2bo4bo$589bo4bo2bo4bo$398b3o190b2o6b2o$397bo
3bo$397bo3bo$398b3o184b3o$584bo3bo$584bo3bo$585b3o5$585b3o$584bo3bo$
584bo3bo$454b2o129b3o$454bobo$454bo7$483b2o3bo2bo3b2o$483b5o4b5o$483b
2o3bo2bo3b2o2$224b3o2$223bo3bo$223bo3bo239bo$466b2o$224b3o230bo8bobo
80b3o$458bo2bo17b2o6b2o62bo$439b2o6b2o7b3o2bobo13bo4bo2bo4bo59bo$224b
3o211bo2bo4bo2bo11b2o14bo4bo2bo4bo$437b6o2b6o26bo4bo2bo4bo$223bo3bo
210bo2bo4bo2bo29b2o6b2o$223bo3bo211b2o6b2o2$224b3o$231b2o6b2o$230bo2bo
4bo2bo$230bo2bo4bo2bo$230bo2bo4bo2bo196b2o6b2o12bo13bo$231b2o6b2o196bo
2bo4bo2bo9b2o14bo$222bo213b6o2b6o9b2o$221bo215bo2bo4bo2bo25bo$221b3o
214b2o6b2o25bobo2$473b3o$434b3o2$433bo3bo35b3o$433bo3bo$473bobo$434b3o
37bo2$474bo$434b3o37bo$502bo2bob2obo2bo$433bo3bo64b4ob2ob4o$433bo3bo
64bo2bob2obo2bo2$434b3o$470bobo24b3o$471b2o$420bo2bob2obo2bo39bo24bo3b
o$420b4ob2ob4o56b2o6bo3bo$420bo2bob2obo2bo57b2o$488bo8b3o$179b3o$179b
3o$180bo316b3o8b2o$180bo328b2o$180bo315bo3bo7bo$179bobo314bo3bo2$259b
2o236b3o$179bobo77bobo$180bo78bo$180bo$180bo269bobo$179b3o269b2o$179b
3o269bo3$167bobo2bobo$163b2obo2bo2bo2bob4obo296b2o6b2o$167bobo2bobo4b
3o295bo4bo2bo4bo$181bo295bo4bo2bo4bo$477bo4bo2bo4bo$479b2o6b2o2$448b2o
6b2o35b3o$447bo2bo4bo2bo35bo$446b6o2b6o34bo$447bo2bo4bo2bo34b3o$448b2o
6b2o5bo$463bo29b3o$462b3o28b3o2$493b3o$462b3o29bo$463bo30bo$463bo29b3o
$463bo$463bo$462b3o3$462b3o$463bo$463bo3$287b3o$288bo$288bo$287b3o11bo
$300b2o$287b3o10bobo$287b3o2$287b3o$219bo68bo$220b2o66bo$219b2o66b3o2$
294b2o6b2o$292bo4bo2bo4bo$292bo4bo2bo4bo$292bo4bo2bo4bo$250b3o41b2o6b
2o$250b3o68bo2bob2obo2bo$251bo69b4ob2ob4o$251bo69bo2bob2obo2bo$251bo$
250bobo$261b3o32bo19b3o$295b2o$250bobo7bo3bo30bobo17bo3bo$251bo8bo3bo
50bo3bo$251bo$251bo9b3o52b3o$250b3o$250b3o$261b3o52b3o2$260bo3bo50bo3b
o$260bo3bo50bo3bo2$261b3o52b3o2$275bobo2bobo$271b2obo2bo2bo2bob2o$275b
obo2bobo$292b2o$293b2o346b3o$292bo348bobo$641b3o6bo$259bo381b3o5b3o$
257b2o382b3o$258b2o381b3o$641bobo5b3o$327bo2bo4bo2bo302b3o$261bo63b3o
2b6o2b3o308bobo$262bo64bo2bo4bo2bo310bobo$260b3o$271bobo375b3o$272b2o
49bo303b2o6b2o$260b2o10bo49b3o301bo2bo4bo2bo$259bobo59bobobo300bo2bo4b
o2bo11b3o$242bo18bo59bobobo300bo2bo4bo2bo12bo$241bo80b3o302b2o6b2o$
241b3o79bo$653bo2b2o4b2o2bo$309b2o341bo3b3o2b3o3bo$310b2o11bo329bo2b2o
4b2o2bo$263bo45bo12b3o$262b3o8bo47bobobo$261bobobo6b3o46bobobo$261bobo
bo5bobobo46b3o$262b3o6bobobo47bo$263bo8b3o$273bo381bo$655b3o4bobo2bobo
$263bo390bob4obo2bo2bo2bob2o$262b3o8bo362bobo23bobo2bobo$261bobobo6b3o
361b2o$261bobobo5bobobo361bo$262b3o6bobobo379b3o$263bo8b3o5b2o6b2o365b
3o$273bo4bo4bo2bo4bo268b3o93bo$278bo4bo2bo4bo269bo94bo$278bo4bo2bo4bo
269bo94bo$280b2o6b2o270b3o92bobo2$199b3o92b3o263b3o$199b3o93bo264b3o
92bobo$200bo94bo360bo$200bo93b3o263b3o93bo$200bo360bo94bo$199bobo92b3o
264bo93b3o$294b3o263b3o92b3o2$199bobo92b3o270b2o6b2o$200bo94bo269bo4bo
2bo4bo$200bo94bo269bo4bo2bo4bo$200bo93b3o268bo4bo2bo4bo4bo$199b3o365b
2o6b2o5b3o8bo$199b3o379bobobo6b3o$219bo361bobobo5bobobo$219b2o361b3o6b
obobo$187bobo2bobo23bobo362bo8b3o$183b2obo2bo2bo2bob4obo390bo$187bobo
2bobo4b3o$201bo381bo$582b3o8bo$533bo47bobobo6b3o$532b3o46bobobo5bobobo
$531bobobo46b3o6bobobo$531bobobo47bo8b3o$532b3o12bo45bo$190bo2b2o4b2o
2bo329bo11b2o$189bo3b3o2b3o3bo341b2o$190bo2b2o4b2o2bo$533bo79b3o$220b
2o6b2o302b3o80bo$206bo12bo2bo4bo2bo300bobobo59bo18bo$205b3o11bo2bo4bo
2bo300bobobo59bobo$219bo2bo4bo2bo301b3o49bo10b2o$220b2o6b2o303bo49b2o$
205b3o375bobo$594b3o$205bobo310bo2bo4bo2bo64bo$205bobo308b3o2b6o2b3o
63bo$213b3o302bo2bo4bo2bo$205b3o5bobo$213b3o381b2o$213b3o382b2o$205b3o
5b3o381bo$206bo6b3o$213bobo348bo$213b3o346b2o$563b2o$574bobo2bobo$570b
2obo2bo2bo2bob2o$574bobo2bobo2$538b3o52b3o2$537bo3bo50bo3bo$537bo3bo
50bo3bo2$538b3o52b3o$604b3o$604b3o$538b3o52b3o9bo$605bo$537bo3bo50bo3b
o8bo$537bo3bo17bobo30bo3bo7bobo$560b2o$538b3o19bo32b3o$604bobo$605bo$
524bo2bob2obo2bo69bo$524b4ob2ob4o69bo$524bo2bob2obo2bo68b3o$553b2o6b2o
41b3o$551bo4bo2bo4bo$551bo4bo2bo4bo$551bo4bo2bo4bo$553b2o6b2o2$567b3o
66b2o$568bo66b2o$568bo68bo$567b3o2$567b3o$554bobo10b3o$555b2o$555bo11b
3o$568bo$568bo$567b3o3$393bo$393bo$392b3o3$392b3o$393bo$393bo$361b3o
29bo$362bo30bo$362bo29b3o$361b3o2$361b3o28b3o$361b3o29bo$393bo5b2o6b2o
$361b3o34bo2bo4bo2bo$362bo34b6o2b6o$362bo35bo2bo4bo2bo$361b3o35b2o6b2o
2$368b2o6b2o$366bo4bo2bo4bo$366bo4bo2bo4bo295bo$366bo4bo2bo4bo295b3o4b
obo2bobo$368b2o6b2o296bob4obo2bo2bo2bob2o$682bobo2bobo3$405bo269b3o$
404b2o269b3o$404bobo269bo$676bo$597bo78bo$595bobo77bobo$357b3o236b2o2$
356bo3bo314bobo$348bo7bo3bo315bo$346b2o328bo$347b2o8b3o316bo$675b3o$
675b3o$357b3o8bo$366b2o57bo2bob2obo2bo$356bo3bo6b2o56b4ob2ob4o$356bo3b
o24bo39bo2bob2obo2bo$384b2o$357b3o24bobo$420b3o2$343bo2bob2obo2bo64bo
3bo$343b4ob2ob4o64bo3bo$343bo2bob2obo2bo$382bo37b3o$382bo2$382bo37b3o$
381bobo$419bo3bo$381b3o35bo3bo2$420b3o$381b3o2$381bobo25b2o6b2o214b3o$
382bo25bo2bo4bo2bo215bo$396b2o9b6o2b6o213bo$382bo14b2o9bo2bo4bo2bo196b
2o6b2o$382bo13bo12b2o6b2o196bo2bo4bo2bo$615bo2bo4bo2bo$615bo2bo4bo2bo$
616b2o6b2o$630b3o2$408b2o6b2o211bo3bo$368b2o6b2o29bo2bo4bo2bo210bo3bo$
366bo4bo2bo4bo26b6o2b6o$366bo4bo2bo4bo14b2o11bo2bo4bo2bo211b3o$306bo
59bo4bo2bo4bo13bobo2b3o7b2o6b2o$305bo62b2o6b2o17bo2bo$305b3o80bobo8bo
230b3o$389b2o$389bo239bo3bo$629bo3bo2$630b3o2$360b2o3bo2bo3b2o$360b5o
4b5o$360b2o3bo2bo3b2o7$402bo$400bobo$269b3o129b2o$268bo3bo$268bo3bo$
269b3o5$269b3o$268bo3bo$268bo3bo$269b3o184b3o$455bo3bo$455bo3bo$256b2o
6b2o190b3o$254bo4bo2bo4bo$254bo4bo2bo4bo$254bo4bo2bo4bo$256b2o6b2o9bo$
273bobo180b3o$274b2o77bo101bo3bo$353b2o85b2o13bo3bo$352bobo84b2o15b3o$
441bo3$464b8o$464bob4obo$464b8o6$457bo2bob2obo2bo$456b2o2bo4bo2b2o$
457bo2bob2obo2bo$452bo$451bobo$442bo$425b2o3bo2bo3b2o4b2o$425b5o4b5o3b
2o7b3o$425b2o3bo2bo3b2o3b2o7b3o$442bo9bo$442bo2$442bo9bo$441bobo7b3o$
451b3o$441b3o2$451bobo$441b3o8bo2$441bobo$442bo2$442bo$442bo$315bo$
298b2o3bo2bo3b2o4b2o$298b5o4b5o3b2o$298b2o3bo2bo3b2o3b2o$315bo$315bo2$
315bo$314bobo2$314b3o3$314b3o2$314bobo$315bo2$315bo$315bo!
@RULE AlmostLife-S1010101111
@COLORS
1 255 255 255
@TABLE
n_states:2
neighborhood:Moore
symmetries:none
var A={0,1}
var B=A
var C=A
var D=A
var E=A
var F=A
var G=A
var H=A
# Birth conditions
0,1,0,0,1,0,1,0,0,1
0,0,0,1,0,0,1,0,1,1
0,0,1,0,0,1,0,0,1,1
0,0,1,0,1,0,0,1,0,1
0,1,1,0,0,1,0,0,0,1
0,0,0,1,1,0,0,1,0,1
0,1,0,0,0,1,1,0,0,1
0,0,0,1,0,0,0,1,1,1
0,1,0,0,0,1,0,0,1,1
0,0,1,1,0,0,0,1,0,1
0,1,0,0,1,1,0,0,0,1
0,0,0,1,0,0,1,1,0,1
0,1,1,0,0,0,1,0,0,1
0,0,0,1,1,0,0,0,1,1
0,0,1,0,0,1,1,0,0,1
0,0,0,0,1,0,0,1,1,1
0,1,0,0,1,0,0,0,1,1
0,0,1,1,0,0,1,0,0,1
0,0,0,0,1,1,0,0,1,1
0,0,1,0,0,0,1,1,0,1
0,1,1,0,1,0,0,0,0,1
0,0,0,1,1,0,1,0,0,1
0,0,0,0,0,1,1,0,1,1
0,0,1,0,0,0,0,1,1,1
0,1,0,0,0,0,1,0,1,1
0,0,1,1,0,0,0,0,1,1
0,0,1,0,1,1,0,0,0,1
0,0,0,0,1,0,1,1,0,1
0,1,0,0,1,0,0,1,0,1
0,1,0,1,0,0,1,0,0,1
0,0,0,1,0,1,0,0,1,1
0,0,1,0,0,1,0,1,0,1
0,1,0,1,1,0,0,0,0,1
0,0,0,1,0,1,1,0,0,1
0,0,0,0,0,1,0,1,1,1
0,1,1,0,0,0,0,1,0,1
0,1,0,0,0,0,1,1,0,1
0,1,0,1,0,0,0,0,1,1
0,0,1,1,0,1,0,0,0,1
0,0,0,0,1,1,0,1,0,1
0,0,0,0,0,0,1,1,1,1
0,1,1,0,0,0,0,0,1,1
0,0,1,1,1,0,0,0,0,1
0,0,0,0,1,1,1,0,0,1
0,1,0,0,0,1,0,1,0,1
0,1,0,1,0,0,0,1,0,1
0,1,0,1,0,1,0,0,0,1
0,0,0,1,0,1,0,1,0,1
0,0,0,0,1,0,1,0,1,1
0,0,1,0,0,0,1,0,1,1
0,0,1,0,1,0,0,0,1,1
0,0,1,0,1,0,1,0,0,1
0,1,0,0,0,0,0,1,1,1
0,1,1,1,0,0,0,0,0,1
0,0,0,1,1,1,0,0,0,1
0,0,0,0,0,1,1,1,0,1

# Survival conditions
1,0,0,0,1,0,0,0,1,1
1,0,1,0,0,0,1,0,0,1
1,1,0,0,1,0,0,0,0,1
1,0,0,1,0,0,1,0,0,1
1,0,0,0,0,1,0,0,1,1
1,0,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,0,0,1
1,0,0,1,0,0,0,0,1,1
1,0,1,0,0,1,0,0,0,1
1,0,0,0,1,0,0,1,0,1
1,1,0,0,0,1,0,0,0,1
1,0,0,1,0,0,0,1,0,1
1,1,0,0,0,0,0,1,0,1
1,1,0,1,0,0,0,0,0,1
1,0,0,1,0,1,0,0,0,1
1,0,0,0,0,1,0,1,0,1
1,0,0,0,0,0,1,0,1,1
1,0,1,0,0,0,0,0,1,1
1,0,1,0,1,0,0,0,0,1
1,0,0,0,1,0,1,0,0,1
1,1,1,0,0,0,0,0,0,1
1,0,0,1,1,0,0,0,0,1
1,0,0,0,0,1,1,0,0,1
1,0,0,0,0,0,0,1,1,1
1,1,0,0,0,0,0,0,1,1
1,0,1,1,0,0,0,0,0,1
1,0,0,0,1,1,0,0,0,1
1,0,0,0,0,0,1,1,0,1
1,1,0,0,1,0,1,0,0,1
1,0,0,1,0,0,1,0,1,1
1,0,1,0,0,1,0,0,1,1
1,0,1,0,1,0,0,1,0,1
1,1,1,0,0,1,0,0,0,1
1,0,0,1,1,0,0,1,0,1
1,1,0,0,0,1,1,0,0,1
1,0,0,1,0,0,0,1,1,1
1,1,0,0,0,1,0,0,1,1
1,0,1,1,0,0,0,1,0,1
1,1,0,0,1,1,0,0,0,1
1,0,0,1,0,0,1,1,0,1
1,1,1,0,0,0,1,0,0,1
1,0,0,1,1,0,0,0,1,1
1,0,1,0,0,1,1,0,0,1
1,0,0,0,1,0,0,1,1,1
1,1,0,0,1,0,0,0,1,1
1,0,1,1,0,0,1,0,0,1
1,0,0,0,1,1,0,0,1,1
1,0,1,0,0,0,1,1,0,1
1,1,1,0,1,0,0,0,0,1
1,0,0,1,1,0,1,0,0,1
1,0,0,0,0,1,1,0,1,1
1,0,1,0,0,0,0,1,1,1
1,1,0,0,0,0,1,0,1,1
1,0,1,1,0,0,0,0,1,1
1,0,1,0,1,1,0,0,0,1
1,0,0,0,1,0,1,1,0,1
1,1,0,0,1,0,0,1,0,1
1,1,0,1,0,0,1,0,0,1
1,0,0,1,0,1,0,0,1,1
1,0,1,0,0,1,0,1,0,1
1,1,0,1,1,0,0,0,0,1
1,0,0,1,0,1,1,0,0,1
1,0,0,0,0,1,0,1,1,1
1,1,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,1,0,1
1,1,0,1,0,0,0,0,1,1
1,0,1,1,0,1,0,0,0,1
1,0,0,0,1,1,0,1,0,1
1,0,0,0,0,0,1,1,1,1
1,1,1,0,0,0,0,0,1,1
1,0,1,1,1,0,0,0,0,1
1,0,0,0,1,1,1,0,0,1
1,1,0,0,0,1,0,1,0,1
1,1,0,1,0,0,0,1,0,1
1,1,0,1,0,1,0,0,0,1
1,0,0,1,0,1,0,1,0,1
1,0,0,0,1,0,1,0,1,1
1,0,1,0,0,0,1,0,1,1
1,0,1,0,1,0,0,0,1,1
1,0,1,0,1,0,1,0,0,1
1,1,0,0,0,0,0,1,1,1
1,1,1,1,0,0,0,0,0,1
1,0,0,1,1,1,0,0,0,1
1,0,0,0,0,1,1,1,0,1

1,0,1,0,1,0,1,1,1,1

1,A,B,C,D,E,F,G,H,0

In practice such rules look weird and highly counter-intuitive. One notable example is the famous Rule110. Since it's one-dimensional, its asymmetry is relatively easy to comprehend.

EDIT1:

This rule has the following common extremely messy upward-moving puffer, somewhat similar to explosive patterns of B3/S237e:

Code: Select all

x = 19, y = 13, rule = AlmostLife-S01111111
5$6b3o$5bo2bo$5bo3bo$4b2obobo$4b2ob2o$5b3o!
@RULE AlmostLife-S1010101111
@COLORS
1 255 255 255
@TABLE
n_states:2
neighborhood:Moore
symmetries:none
var A={0,1}
var B=A
var C=A
var D=A
var E=A
var F=A
var G=A
var H=A
# Birth conditions
0,1,0,0,1,0,1,0,0,1
0,0,0,1,0,0,1,0,1,1
0,0,1,0,0,1,0,0,1,1
0,0,1,0,1,0,0,1,0,1
0,1,1,0,0,1,0,0,0,1
0,0,0,1,1,0,0,1,0,1
0,1,0,0,0,1,1,0,0,1
0,0,0,1,0,0,0,1,1,1
0,1,0,0,0,1,0,0,1,1
0,0,1,1,0,0,0,1,0,1
0,1,0,0,1,1,0,0,0,1
0,0,0,1,0,0,1,1,0,1
0,1,1,0,0,0,1,0,0,1
0,0,0,1,1,0,0,0,1,1
0,0,1,0,0,1,1,0,0,1
0,0,0,0,1,0,0,1,1,1
0,1,0,0,1,0,0,0,1,1
0,0,1,1,0,0,1,0,0,1
0,0,0,0,1,1,0,0,1,1
0,0,1,0,0,0,1,1,0,1
0,1,1,0,1,0,0,0,0,1
0,0,0,1,1,0,1,0,0,1
0,0,0,0,0,1,1,0,1,1
0,0,1,0,0,0,0,1,1,1
0,1,0,0,0,0,1,0,1,1
0,0,1,1,0,0,0,0,1,1
0,0,1,0,1,1,0,0,0,1
0,0,0,0,1,0,1,1,0,1
0,1,0,0,1,0,0,1,0,1
0,1,0,1,0,0,1,0,0,1
0,0,0,1,0,1,0,0,1,1
0,0,1,0,0,1,0,1,0,1
0,1,0,1,1,0,0,0,0,1
0,0,0,1,0,1,1,0,0,1
0,0,0,0,0,1,0,1,1,1
0,1,1,0,0,0,0,1,0,1
0,1,0,0,0,0,1,1,0,1
0,1,0,1,0,0,0,0,1,1
0,0,1,1,0,1,0,0,0,1
0,0,0,0,1,1,0,1,0,1
0,0,0,0,0,0,1,1,1,1
0,1,1,0,0,0,0,0,1,1
0,0,1,1,1,0,0,0,0,1
0,0,0,0,1,1,1,0,0,1
0,1,0,0,0,1,0,1,0,1
0,1,0,1,0,0,0,1,0,1
0,1,0,1,0,1,0,0,0,1
0,0,0,1,0,1,0,1,0,1
0,0,0,0,1,0,1,0,1,1
0,0,1,0,0,0,1,0,1,1
0,0,1,0,1,0,0,0,1,1
0,0,1,0,1,0,1,0,0,1
0,1,0,0,0,0,0,1,1,1
0,1,1,1,0,0,0,0,0,1
0,0,0,1,1,1,0,0,0,1
0,0,0,0,0,1,1,1,0,1

# Survival conditions
1,0,0,0,1,0,0,0,1,1
1,0,1,0,0,0,1,0,0,1
1,1,0,0,1,0,0,0,0,1
1,0,0,1,0,0,1,0,0,1
1,0,0,0,0,1,0,0,1,1
1,0,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,0,0,1
1,0,0,1,0,0,0,0,1,1
1,0,1,0,0,1,0,0,0,1
1,0,0,0,1,0,0,1,0,1
1,1,0,0,0,1,0,0,0,1
1,0,0,1,0,0,0,1,0,1
1,1,0,0,0,0,0,1,0,1
1,1,0,1,0,0,0,0,0,1
1,0,0,1,0,1,0,0,0,1
1,0,0,0,0,1,0,1,0,1
1,0,0,0,0,0,1,0,1,1
1,0,1,0,0,0,0,0,1,1
1,0,1,0,1,0,0,0,0,1
1,0,0,0,1,0,1,0,0,1
1,1,1,0,0,0,0,0,0,1
1,0,0,1,1,0,0,0,0,1
1,0,0,0,0,1,1,0,0,1
1,0,0,0,0,0,0,1,1,1
1,1,0,0,0,0,0,0,1,1
1,0,1,1,0,0,0,0,0,1
1,0,0,0,1,1,0,0,0,1
1,0,0,0,0,0,1,1,0,1
1,1,0,0,1,0,1,0,0,1
1,0,0,1,0,0,1,0,1,1
1,0,1,0,0,1,0,0,1,1
1,0,1,0,1,0,0,1,0,1
1,1,1,0,0,1,0,0,0,1
1,0,0,1,1,0,0,1,0,1
1,1,0,0,0,1,1,0,0,1
1,0,0,1,0,0,0,1,1,1
1,1,0,0,0,1,0,0,1,1
1,0,1,1,0,0,0,1,0,1
1,1,0,0,1,1,0,0,0,1
1,0,0,1,0,0,1,1,0,1
1,1,1,0,0,0,1,0,0,1
1,0,0,1,1,0,0,0,1,1
1,0,1,0,0,1,1,0,0,1
1,0,0,0,1,0,0,1,1,1
1,1,0,0,1,0,0,0,1,1
1,0,1,1,0,0,1,0,0,1
1,0,0,0,1,1,0,0,1,1
1,0,1,0,0,0,1,1,0,1
1,1,1,0,1,0,0,0,0,1
1,0,0,1,1,0,1,0,0,1
1,0,0,0,0,1,1,0,1,1
1,0,1,0,0,0,0,1,1,1
1,1,0,0,0,0,1,0,1,1
1,0,1,1,0,0,0,0,1,1
1,0,1,0,1,1,0,0,0,1
1,0,0,0,1,0,1,1,0,1
1,1,0,0,1,0,0,1,0,1
1,1,0,1,0,0,1,0,0,1
1,0,0,1,0,1,0,0,1,1
1,0,1,0,0,1,0,1,0,1
1,1,0,1,1,0,0,0,0,1
1,0,0,1,0,1,1,0,0,1
1,0,0,0,0,1,0,1,1,1
1,1,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,1,0,1
1,1,0,1,0,0,0,0,1,1
1,0,1,1,0,1,0,0,0,1
1,0,0,0,1,1,0,1,0,1
1,0,0,0,0,0,1,1,1,1
1,1,1,0,0,0,0,0,1,1
1,0,1,1,1,0,0,0,0,1
1,0,0,0,1,1,1,0,0,1
1,1,0,0,0,1,0,1,0,1
1,1,0,1,0,0,0,1,0,1
1,1,0,1,0,1,0,0,0,1
1,0,0,1,0,1,0,1,0,1
1,0,0,0,1,0,1,0,1,1
1,0,1,0,0,0,1,0,1,1
1,0,1,0,1,0,0,0,1,1
1,0,1,0,1,0,1,0,0,1
1,1,0,0,0,0,0,1,1,1
1,1,1,1,0,0,0,0,0,1
1,0,0,1,1,1,0,0,0,1
1,0,0,0,0,1,1,1,0,1

1,0,1,0,1,0,1,1,1,1

1,A,B,C,D,E,F,G,H,0

Code: Select all

x = 16, y = 16, rule = B3/S237e
boobbbooobbooobb$
oobooooobbbooobb$
obobobbbbobbbbbo$
oobbbbbboobobbbo$
bbbobbobbbbobobb$
bboooobbobobboob$
bobbboobbbboobbb$
bbobbobboboboooo$
obobobboobobbbbb$
ooboobobobbbobbo$
boooboobbooobobb$
oobbobbobbobobbo$
oobobobobbbboooo$
oobbobbbbbobbbbb$
bbobboooobbooobb$
obobbobobobboobb!

[confocaloid]

[...] In practice such rules look weird and highly counter-intuitive. One notable example is the famous Rule110. Since it's one-dimensional, its asymmetry is relatively easy to comprehend.

Nothing requires breaking every symmetry. For example, it's possible to leave rotational symmetries:

At least some of these cellular automata can be defined using a notation other than MAP rulestrings.

I think a weighted definition is more illuminating in this case. I used the following neighbourhood weights:

Code: Select all

5 5 1
1 0 1
1 5 5

Code: Select all

x = 106, y = 104, rule = R1,C0,S2-3,6-7,10-11,15,B3,7,11,15,18,NW551101155
52b3o$51bo2bo$54bo$50bo3bo$50bo3bo$54bo$51bobo4$52b3o$51bo2bo$54bo$50b
o3bo$54bo$51bobo4$52b3o$51bo2bo$38b2o14bo$38bobo13bo$39b3o9bobo$39b2o$
34b2o7$79bo$79bo3$82b2o$38b3o24b3o12b2obo$38bo28bo12b3o$39bo26bo14bo4$
53bo$52bobo$52bobo$53bo3$92bo8b2o$6o4b5o4b4o23b2o5bo5b2o21bo2bo4bo3bo
4bo4bo$o5bo3bo4bo3bo3bo21bo2bo3b3o3bo2bo24bo8bo9bo$o9bo8bo26b2o5bo5b2o
21bo3bo3bo4bo3bo5bo$bo4bo4bo3bo4bo2bo59b4o4b5o4b6o$3b2o8bo2$53bo$52bob
o$52bobo$53bo$23bo$22b3o$21bob2o$21b2o$39bo26bo$38bo28bo$25bo12b3o24b
3o$25bo11$66b2o$61b2o$52bobo5b3o$51bo9bobo$51bo10b2o$51bo2bo$51b3o4$
52bobo$51bo$51bo3bo$51bo$51bo2bo$51b3o4$52bobo$51bo$51bo3bo$51bo3bo$
51bo$51bo2bo$51b3o!

Without the living condition B18, it would be plain Conway's Life:

Code: Select all

x = 3, y = 3, rule = R1,C2,S2-3,6-7,10-11,15,B3,7,11,15,NW551101155
b2o$2o$bo!
#C [[ ZOOM 4 STOP 1103 ]]

To preserve both 90-degree rotation symmetry of the rules and the four standard spaceships, each of the following sets of living conditions can either be completely included or completely excluded:

Code: Select all

B3,B15
B4,B20
B9,B21
B14,B22
B18
B24
S4,S20
S9,S21

[Yoel]

Very cool! Translating into a ruletable:

Code: Select all

x = 120, y = 111, rule = B36ces23-x6ak
3$61b3o$60bo2bo$63bo$59bo3bo$59bo3bo$63bo$60bobo4$61b3o$60bo2bo$63bo$
59bo3bo$63bo$60bobo4$61b3o$60bo2bo$47b2o14bo$47bobo13bo$48b3o9bobo$48b
2o$43b2o7$88bo$88bo3$91b2o$47b3o24b3o12b2obo$47bo28bo12b3o$48bo26bo14b
o4$62bo$61bobo$61bobo$62bo3$101bo8b2o$9b6o4b5o4b4o23b2o5bo5b2o21bo2bo
4bo3bo4bo4bo$9bo5bo3bo4bo3bo3bo21bo2bo3b3o3bo2bo24bo8bo9bo$9bo9bo8bo
26b2o5bo5b2o21bo3bo3bo4bo3bo5bo$10bo4bo4bo3bo4bo2bo59b4o4b5o4b6o$12b2o
8bo2$62bo$61bobo$61bobo$62bo$32bo$31b3o$30bob2o$30b2o$48bo26bo$47bo28b
o$34bo12b3o24b3o$34bo11$75b2o$70b2o$61bobo5b3o$60bo9bobo$60bo10b2o$60b
o2bo$60b3o4$61bobo$60bo$60bo3bo$60bo$60bo2bo$60b3o4$61bobo$60bo$60bo3b
o$60bo3bo$60bo$60bo2bo$60b3o!
@RULE B36ces23-x6ak
@COLORS
1 255 255 255
@TABLE
n_states:2
neighborhood:Moore
symmetries:none
var A={0,1}
var B=A
var C=A
var D=A
var E=A
var F=A
var G=A
var H=A
# Birth conditions
0,1,0,0,1,0,1,0,0,1
0,0,0,1,0,0,1,0,1,1
0,0,1,0,0,1,0,0,1,1
0,0,1,0,1,0,0,1,0,1
0,1,1,0,0,1,0,0,0,1
0,0,0,1,1,0,0,1,0,1
0,1,0,0,0,1,1,0,0,1
0,0,0,1,0,0,0,1,1,1
0,1,0,0,0,1,0,0,1,1
0,0,1,1,0,0,0,1,0,1
0,1,0,0,1,1,0,0,0,1
0,0,0,1,0,0,1,1,0,1
0,1,1,0,0,0,1,0,0,1
0,0,0,1,1,0,0,0,1,1
0,0,1,0,0,1,1,0,0,1
0,0,0,0,1,0,0,1,1,1
0,1,0,0,1,0,0,0,1,1
0,0,1,1,0,0,1,0,0,1
0,0,0,0,1,1,0,0,1,1
0,0,1,0,0,0,1,1,0,1
0,1,1,0,1,0,0,0,0,1
0,0,0,1,1,0,1,0,0,1
0,0,0,0,0,1,1,0,1,1
0,0,1,0,0,0,0,1,1,1
0,1,0,0,0,0,1,0,1,1
0,0,1,1,0,0,0,0,1,1
0,0,1,0,1,1,0,0,0,1
0,0,0,0,1,0,1,1,0,1
0,1,0,0,1,0,0,1,0,1
0,1,0,1,0,0,1,0,0,1
0,0,0,1,0,1,0,0,1,1
0,0,1,0,0,1,0,1,0,1
0,1,0,1,1,0,0,0,0,1
0,0,0,1,0,1,1,0,0,1
0,0,0,0,0,1,0,1,1,1
0,1,1,0,0,0,0,1,0,1
0,1,0,0,0,0,1,1,0,1
0,1,0,1,0,0,0,0,1,1
0,0,1,1,0,1,0,0,0,1
0,0,0,0,1,1,0,1,0,1
0,0,0,0,0,0,1,1,1,1
0,1,1,0,0,0,0,0,1,1
0,0,1,1,1,0,0,0,0,1
0,0,0,0,1,1,1,0,0,1
0,1,0,0,0,1,0,1,0,1
0,1,0,1,0,0,0,1,0,1
0,1,0,1,0,1,0,0,0,1
0,0,0,1,0,1,0,1,0,1
0,0,0,0,1,0,1,0,1,1
0,0,1,0,0,0,1,0,1,1
0,0,1,0,1,0,0,0,1,1
0,0,1,0,1,0,1,0,0,1
0,1,0,0,0,0,0,1,1,1
0,1,1,1,0,0,0,0,0,1
0,0,0,1,1,1,0,0,0,1
0,0,0,0,0,1,1,1,0,1
0,0,1,0,1,1,1,1,1,1
0,1,1,0,1,0,1,1,1,1
0,1,1,1,1,0,1,0,1,1
0,0,1,1,1,1,1,0,1,1
0,1,0,1,1,1,1,1,0,1
0,1,0,1,0,1,1,1,1,1
0,1,1,1,0,1,0,1,1,1
0,1,1,1,1,1,0,1,0,1

# Adding rotation-symmetrical 6ak

0,0,1,1,1,1,0,1,1,1
0,1,1,0,1,1,1,1,0,1
0,1,0,1,1,0,1,1,1,1
0,1,1,1,0,1,1,0,1,1
0,0,0,1,1,1,1,1,1,1
0,1,1,0,0,1,1,1,1,1
0,1,1,1,1,0,0,1,1,1
0,1,1,1,1,1,1,0,0,1

# Survival conditions
1,0,0,0,1,0,0,0,1,1
1,0,1,0,0,0,1,0,0,1
1,1,0,0,1,0,0,0,0,1
1,0,0,1,0,0,1,0,0,1
1,0,0,0,0,1,0,0,1,1
1,0,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,0,0,1
1,0,0,1,0,0,0,0,1,1
1,0,1,0,0,1,0,0,0,1
1,0,0,0,1,0,0,1,0,1
1,1,0,0,0,1,0,0,0,1
1,0,0,1,0,0,0,1,0,1
1,1,0,0,0,0,0,1,0,1
1,1,0,1,0,0,0,0,0,1
1,0,0,1,0,1,0,0,0,1
1,0,0,0,0,1,0,1,0,1
1,0,0,0,0,0,1,0,1,1
1,0,1,0,0,0,0,0,1,1
1,0,1,0,1,0,0,0,0,1
1,0,0,0,1,0,1,0,0,1
1,1,1,0,0,0,0,0,0,1
1,0,0,1,1,0,0,0,0,1
1,0,0,0,0,1,1,0,0,1
1,0,0,0,0,0,0,1,1,1
1,1,0,0,0,0,0,0,1,1
1,0,1,1,0,0,0,0,0,1
1,0,0,0,1,1,0,0,0,1
1,0,0,0,0,0,1,1,0,1
1,1,0,0,1,0,1,0,0,1
1,0,0,1,0,0,1,0,1,1
1,0,1,0,0,1,0,0,1,1
1,0,1,0,1,0,0,1,0,1
1,1,1,0,0,1,0,0,0,1
1,0,0,1,1,0,0,1,0,1
1,1,0,0,0,1,1,0,0,1
1,0,0,1,0,0,0,1,1,1
1,1,0,0,0,1,0,0,1,1
1,0,1,1,0,0,0,1,0,1
1,1,0,0,1,1,0,0,0,1
1,0,0,1,0,0,1,1,0,1
1,1,1,0,0,0,1,0,0,1
1,0,0,1,1,0,0,0,1,1
1,0,1,0,0,1,1,0,0,1
1,0,0,0,1,0,0,1,1,1
1,1,0,0,1,0,0,0,1,1
1,0,1,1,0,0,1,0,0,1
1,0,0,0,1,1,0,0,1,1
1,0,1,0,0,0,1,1,0,1
1,1,1,0,1,0,0,0,0,1
1,0,0,1,1,0,1,0,0,1
1,0,0,0,0,1,1,0,1,1
1,0,1,0,0,0,0,1,1,1
1,1,0,0,0,0,1,0,1,1
1,0,1,1,0,0,0,0,1,1
1,0,1,0,1,1,0,0,0,1
1,0,0,0,1,0,1,1,0,1
1,1,0,0,1,0,0,1,0,1
1,1,0,1,0,0,1,0,0,1
1,0,0,1,0,1,0,0,1,1
1,0,1,0,0,1,0,1,0,1
1,1,0,1,1,0,0,0,0,1
1,0,0,1,0,1,1,0,0,1
1,0,0,0,0,1,0,1,1,1
1,1,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,1,0,1
1,1,0,1,0,0,0,0,1,1
1,0,1,1,0,1,0,0,0,1
1,0,0,0,1,1,0,1,0,1
1,0,0,0,0,0,1,1,1,1
1,1,1,0,0,0,0,0,1,1
1,0,1,1,1,0,0,0,0,1
1,0,0,0,1,1,1,0,0,1
1,1,0,0,0,1,0,1,0,1
1,1,0,1,0,0,0,1,0,1
1,1,0,1,0,1,0,0,0,1
1,0,0,1,0,1,0,1,0,1
1,0,0,0,1,0,1,0,1,1
1,0,1,0,0,0,1,0,1,1
1,0,1,0,1,0,0,0,1,1
1,0,1,0,1,0,1,0,0,1
1,1,0,0,0,0,0,1,1,1
1,1,1,1,0,0,0,0,0,1
1,0,0,1,1,1,0,0,0,1
1,0,0,0,0,1,1,1,0,1
1,A,B,C,D,E,F,G,H,0

EDIT1: A non-reflective p4 oscillator:

Code: Select all

x = 12, y = 21, rule = B36ces23-x6ak
$3b3o$3b3o$7b3o$7b3o12$7b3o$7b3o$3b3o$3b3o!
@RULE B36ces23-x6ak
@COLORS
1 255 255 255
@TABLE
n_states:2
neighborhood:Moore
symmetries:none
var A={0,1}
var B=A
var C=A
var D=A
var E=A
var F=A
var G=A
var H=A
# Birth conditions
0,1,0,0,1,0,1,0,0,1
0,0,0,1,0,0,1,0,1,1
0,0,1,0,0,1,0,0,1,1
0,0,1,0,1,0,0,1,0,1
0,1,1,0,0,1,0,0,0,1
0,0,0,1,1,0,0,1,0,1
0,1,0,0,0,1,1,0,0,1
0,0,0,1,0,0,0,1,1,1
0,1,0,0,0,1,0,0,1,1
0,0,1,1,0,0,0,1,0,1
0,1,0,0,1,1,0,0,0,1
0,0,0,1,0,0,1,1,0,1
0,1,1,0,0,0,1,0,0,1
0,0,0,1,1,0,0,0,1,1
0,0,1,0,0,1,1,0,0,1
0,0,0,0,1,0,0,1,1,1
0,1,0,0,1,0,0,0,1,1
0,0,1,1,0,0,1,0,0,1
0,0,0,0,1,1,0,0,1,1
0,0,1,0,0,0,1,1,0,1
0,1,1,0,1,0,0,0,0,1
0,0,0,1,1,0,1,0,0,1
0,0,0,0,0,1,1,0,1,1
0,0,1,0,0,0,0,1,1,1
0,1,0,0,0,0,1,0,1,1
0,0,1,1,0,0,0,0,1,1
0,0,1,0,1,1,0,0,0,1
0,0,0,0,1,0,1,1,0,1
0,1,0,0,1,0,0,1,0,1
0,1,0,1,0,0,1,0,0,1
0,0,0,1,0,1,0,0,1,1
0,0,1,0,0,1,0,1,0,1
0,1,0,1,1,0,0,0,0,1
0,0,0,1,0,1,1,0,0,1
0,0,0,0,0,1,0,1,1,1
0,1,1,0,0,0,0,1,0,1
0,1,0,0,0,0,1,1,0,1
0,1,0,1,0,0,0,0,1,1
0,0,1,1,0,1,0,0,0,1
0,0,0,0,1,1,0,1,0,1
0,0,0,0,0,0,1,1,1,1
0,1,1,0,0,0,0,0,1,1
0,0,1,1,1,0,0,0,0,1
0,0,0,0,1,1,1,0,0,1
0,1,0,0,0,1,0,1,0,1
0,1,0,1,0,0,0,1,0,1
0,1,0,1,0,1,0,0,0,1
0,0,0,1,0,1,0,1,0,1
0,0,0,0,1,0,1,0,1,1
0,0,1,0,0,0,1,0,1,1
0,0,1,0,1,0,0,0,1,1
0,0,1,0,1,0,1,0,0,1
0,1,0,0,0,0,0,1,1,1
0,1,1,1,0,0,0,0,0,1
0,0,0,1,1,1,0,0,0,1
0,0,0,0,0,1,1,1,0,1
0,0,1,0,1,1,1,1,1,1
0,1,1,0,1,0,1,1,1,1
0,1,1,1,1,0,1,0,1,1
0,0,1,1,1,1,1,0,1,1
0,1,0,1,1,1,1,1,0,1
0,1,0,1,0,1,1,1,1,1
0,1,1,1,0,1,0,1,1,1
0,1,1,1,1,1,0,1,0,1

# Adding rotation-symmetrical 6ak

0,0,1,1,1,1,0,1,1,1
0,1,1,0,1,1,1,1,0,1
0,1,0,1,1,0,1,1,1,1
0,1,1,1,0,1,1,0,1,1
0,0,0,1,1,1,1,1,1,1
0,1,1,0,0,1,1,1,1,1
0,1,1,1,1,0,0,1,1,1
0,1,1,1,1,1,1,0,0,1

# Survival conditions
1,0,0,0,1,0,0,0,1,1
1,0,1,0,0,0,1,0,0,1
1,1,0,0,1,0,0,0,0,1
1,0,0,1,0,0,1,0,0,1
1,0,0,0,0,1,0,0,1,1
1,0,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,0,0,1
1,0,0,1,0,0,0,0,1,1
1,0,1,0,0,1,0,0,0,1
1,0,0,0,1,0,0,1,0,1
1,1,0,0,0,1,0,0,0,1
1,0,0,1,0,0,0,1,0,1
1,1,0,0,0,0,0,1,0,1
1,1,0,1,0,0,0,0,0,1
1,0,0,1,0,1,0,0,0,1
1,0,0,0,0,1,0,1,0,1
1,0,0,0,0,0,1,0,1,1
1,0,1,0,0,0,0,0,1,1
1,0,1,0,1,0,0,0,0,1
1,0,0,0,1,0,1,0,0,1
1,1,1,0,0,0,0,0,0,1
1,0,0,1,1,0,0,0,0,1
1,0,0,0,0,1,1,0,0,1
1,0,0,0,0,0,0,1,1,1
1,1,0,0,0,0,0,0,1,1
1,0,1,1,0,0,0,0,0,1
1,0,0,0,1,1,0,0,0,1
1,0,0,0,0,0,1,1,0,1
1,1,0,0,1,0,1,0,0,1
1,0,0,1,0,0,1,0,1,1
1,0,1,0,0,1,0,0,1,1
1,0,1,0,1,0,0,1,0,1
1,1,1,0,0,1,0,0,0,1
1,0,0,1,1,0,0,1,0,1
1,1,0,0,0,1,1,0,0,1
1,0,0,1,0,0,0,1,1,1
1,1,0,0,0,1,0,0,1,1
1,0,1,1,0,0,0,1,0,1
1,1,0,0,1,1,0,0,0,1
1,0,0,1,0,0,1,1,0,1
1,1,1,0,0,0,1,0,0,1
1,0,0,1,1,0,0,0,1,1
1,0,1,0,0,1,1,0,0,1
1,0,0,0,1,0,0,1,1,1
1,1,0,0,1,0,0,0,1,1
1,0,1,1,0,0,1,0,0,1
1,0,0,0,1,1,0,0,1,1
1,0,1,0,0,0,1,1,0,1
1,1,1,0,1,0,0,0,0,1
1,0,0,1,1,0,1,0,0,1
1,0,0,0,0,1,1,0,1,1
1,0,1,0,0,0,0,1,1,1
1,1,0,0,0,0,1,0,1,1
1,0,1,1,0,0,0,0,1,1
1,0,1,0,1,1,0,0,0,1
1,0,0,0,1,0,1,1,0,1
1,1,0,0,1,0,0,1,0,1
1,1,0,1,0,0,1,0,0,1
1,0,0,1,0,1,0,0,1,1
1,0,1,0,0,1,0,1,0,1
1,1,0,1,1,0,0,0,0,1
1,0,0,1,0,1,1,0,0,1
1,0,0,0,0,1,0,1,1,1
1,1,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,1,0,1
1,1,0,1,0,0,0,0,1,1
1,0,1,1,0,1,0,0,0,1
1,0,0,0,1,1,0,1,0,1
1,0,0,0,0,0,1,1,1,1
1,1,1,0,0,0,0,0,1,1
1,0,1,1,1,0,0,0,0,1
1,0,0,0,1,1,1,0,0,1
1,1,0,0,0,1,0,1,0,1
1,1,0,1,0,0,0,1,0,1
1,1,0,1,0,1,0,0,0,1
1,0,0,1,0,1,0,1,0,1
1,0,0,0,1,0,1,0,1,1
1,0,1,0,0,0,1,0,1,1
1,0,1,0,1,0,0,0,1,1
1,0,1,0,1,0,1,0,0,1
1,1,0,0,0,0,0,1,1,1
1,1,1,1,0,0,0,0,0,1
1,0,0,1,1,1,0,0,0,1
1,0,0,0,0,1,1,1,0,1
1,A,B,C,D,E,F,G,H,0

[...] In practice such rules look weird and highly counter-intuitive. One notable example is the famous Rule110. Since it's one-dimensional, its asymmetry is relatively easy to comprehend.

Nothing requires breaking every symmetry. For example, it's possible to leave rotational symmetries:

At least some of these cellular automata can be defined using a notation other than MAP rulestrings.

I think a weighted definition is more illuminating in this case. I used the following neighbourhood weights:

Code: Select all

5 5 1
1 0 1
1 5 5

Code: Select all

x = 106, y = 104, rule = R1,C0,S2-3,6-7,10-11,15,B3,7,11,15,18,NW551101155
52b3o$51bo2bo$54bo$50bo3bo$50bo3bo$54bo$51bobo4$52b3o$51bo2bo$54bo$50b
o3bo$54bo$51bobo4$52b3o$51bo2bo$38b2o14bo$38bobo13bo$39b3o9bobo$39b2o$
34b2o7$79bo$79bo3$82b2o$38b3o24b3o12b2obo$38bo28bo12b3o$39bo26bo14bo4$
53bo$52bobo$52bobo$53bo3$92bo8b2o$6o4b5o4b4o23b2o5bo5b2o21bo2bo4bo3bo
4bo4bo$o5bo3bo4bo3bo3bo21bo2bo3b3o3bo2bo24bo8bo9bo$o9bo8bo26b2o5bo5b2o
21bo3bo3bo4bo3bo5bo$bo4bo4bo3bo4bo2bo59b4o4b5o4b6o$3b2o8bo2$53bo$52bob
o$52bobo$53bo$23bo$22b3o$21bob2o$21b2o$39bo26bo$38bo28bo$25bo12b3o24b
3o$25bo11$66b2o$61b2o$52bobo5b3o$51bo9bobo$51bo10b2o$51bo2bo$51b3o4$
52bobo$51bo$51bo3bo$51bo$51bo2bo$51b3o4$52bobo$51bo$51bo3bo$51bo3bo$
51bo$51bo2bo$51b3o!

Without the living condition B18, it would be plain Conway's Life:

Code: Select all

x = 3, y = 3, rule = R1,C2,S2-3,6-7,10-11,15,B3,7,11,15,NW551101155
b2o$2o$bo!
#C [[ ZOOM 4 STOP 1103 ]]

To preserve both 90-degree rotation symmetry of the rules and the four standard spaceships, each of the following sets of living conditions can either be completely included or completely excluded:

Code: Select all

B3,B15
B4,B20
B9,B21
B14,B22
B18
B24
S4,S20
S9,S21

[confocaloid]

[...] EDIT: See the discussion below. Some terms above have been clarified. [...]

Note that most of cellular automata mentioned here aren't Life-like.
A Life-like cellular automaton is defined as a two-state outer-totalistic cellular automaton with the (range-1) Moore neighbourhood.
Conway's Life, Day & Night, Seeds, LFoD are examples of Life-like cellular automata.
However, neither of the following is a Life-like CA:

Code: Select all

B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
B3aijn/S2ae3jnr
B2ikn34-r5-n678/S0234-k5678

[...] Let's reproduce CGOL and insert the survival condition 1,0,1,0,1,0,1,1,1,1 into it. The Snark still works, the Gosper glider gun works too. There is still horizontal reflection symmetry. However, complex things act wacky: [...]

Note: the full line (containing 10 numbers)

Code: Select all

1,0,1,0,1,0,1,1,1,1

tells more than just the "condition" part. In the RuleLoader ruletable format, any such line tells a rule. See also:
https://web.archive.org/web/20040323172 ... ndixA.html

The rule "1,0,1,0,1,0,1,1,1,1" can be explained in a more verbose way, as follows:

If the cell is currently alive and its Moore neighbours currently have states matching the pattern shown below, then the cell will remain alive in the next generation:

Code: Select all

1 0 1       ? ? ?
1 1 0  ==>  ? 1 ?
1 0 1       ? ? ?

This rule prescribes the transition 1->1 (survival). In other words, the rule "1,0,1,0,1,0,1,1,1,1" is a survival rule.
The "condition" part of the rule is only the first 9 numbers in the line, telling the cell's current state and current states of its neighbours:

Code: Select all

1,0,1,0,1,0,1,1,1

[Yoel]

[...] EDIT: See the discussion below. Some terms above have been clarified. [...]

Note that most of cellular automata mentioned here aren't Life-like.
A Life-like cellular automaton is defined as a two-state outer-totalistic cellular automaton with the (range-1) Moore neighbourhood.
Conway's Life, Day & Night, Seeds, LFoD are examples of Life-like cellular automata.
However, neither of the following is a Life-like CA:

Code: Select all

B2en3aei5ac6aci7c8/S2-i3-k4c6a7c
B3aijn/S2ae3jnr
B2ikn34-r5-n678/S0234-k5678

Ok, isotropic non-totalistic. Added the link to LifeWiki,

[...] Let's reproduce CGOL and insert the survival condition 1,0,1,0,1,0,1,1,1,1 into it. The Snark still works, the Gosper glider gun works too. There is still horizontal reflection symmetry. However, complex things act wacky: [...]

Note: the full line (containing 10 numbers)

Code: Select all

1,0,1,0,1,0,1,1,1,1

tells more than just the "condition" part. In the RuleLoader ruletable format, any such line tells a rule. See also:
https://web.archive.org/web/20040323172 ... ndixA.html

I did write down the full line. Frankly, the RuleLoader ruletable format terminology is confusing and should be improved. The entire ruletable represents a rule; its lines should be then called subrules or something.

The rule "1,0,1,0,1,0,1,1,1,1" can be explained in a more verbose way, as follows:

If the cell is currently alive and its Moore neighbours currently have states matching the pattern shown below, then the cell will remain alive in the next generation:

Code: Select all

1 0 1       ? ? ?
1 1 0  ==>  ? 1 ?
1 0 1       ? ? ?

This rule prescribes the transition 1->1 (survival). In other words, the rule "1,0,1,0,1,0,1,1,1,1" is a survival rule.
The "condition" part of the rule is only the first 9 numbers in the line, telling the cell's current state and current states of its neighbours:

Code: Select all

1,0,1,0,1,0,1,1,1

It sure can be explained in a more verbose way, but the line alone should be clear enough for people who read and write ruletables by hand. We are playing around here, not spending time on explaining things, don't we? Factual errors and incorrect terminology is another thing that must be agreed upon and fixed.

[confocaloid]

To quote from an (otherwise mostly unrelated) old discussion,

[...] Fun can be had when multiple people play a game, all using a commonly agreed upon set of rules. [...]

In relation to the current discussion, when the jargon becomes confusing enough (making it hard to impossible to understand what people are talking about and what would count as nonsense), I think it's worth pausing and thinking about the underlying concepts, and checking earlier relevant sources (articles, publications) where people were careful enough to clearly distinguish between those concepts.

[...] I did write down the full line. Frankly, the RuleLoader ruletable format terminology is confusing and should be improves. The entire ruletable represent a rule; its like should be called subrules or something. [...]

[...] It sure can be explained in a more verbose way, but the line alone should be clear enough for people who read and write ruletables by hand. We are playing around here, not spending time on explaining things, don't we? Factual errors and incorrect terminology is another thing that must be agrees upon and fixed.

Yes I agree that terminology needs to be agreed upon.

Re: "spending time on explaining things" -- I don't consider explaining things as spent time, as long as it leads to better understanding. Forum rules (e.g. the forum rule 5b) also mention the need for descriptions/explanations.

Sometimes, it's worth explaining something in a more verbose way, if only just to make sure that everyone involved is on the same page, and to make sure that people know what exactly they're talking about.

The entire RuleLoader ruletable represents a cellular automaton.
Individual lines (such as "1, 0,1,0,1,0,1,1,1, 1") represent specific update rules of the cellular automaton.

[Yoel]

Since you already have answered, I move my edit to the previous post here:

Here is another ruletable, Day & Night with a slight non-reflective change. Explaining it in a more verbose way, as you wish:

Code: Select all

0 0 1       ? ? ?
1 0 1  ==>  ? 1 ?
1 0 0       ? ? ?

1 1 0       ? ? ?
0 0 0  ==>  ? 1 ?
0 1 1       ? ? ?

1 0 0       ? ? ?
1 0 1  ==>  ? 0 ?
0 0 1       ? ? ?

0 1 1       ? ? ?
0 0 0  ==>  ? 0 ?
1 1 0       ? ? ?

0 0 1       ? ? ?
1 1 1  ==>  ? 0 ?
1 1 0       ? ? ?

1 1 0       ? ? ?
0 1 0  ==>  ? 0 ?
0 1 1       ? ? ?

1 0 0       ? ? ?
1 1 1  ==>  ? 1 ?
0 0 1       ? ? ?

0 1 1       ? ? ?
0 1 0  ==>  ? 1 ?
1 1 0       ? ? ?

Code: Select all

x = 53, y = 21, rule = DN-x4z
3$33b2o6b2o$3bo10bo$3bobo8b3o17b2o4b2o$5bo10bo5$5bo10bo$3bobo8b3o$3bo
10bo$33b2o6b2o$34bo6bo$34b2o4b2o!
@RULE DN-x4z
@COLORS
1 255 255 255
@TABLE
n_states:2
neighborhood:Moore
symmetries:none
var A={0,1}
var B=A
var C=A
var D=A
var E=A
var F=A
var G=A
var H=A
# Birth conditions
0,1,0,0,0,0,0,1,1,1
0,1,1,1,0,0,0,0,0,1
0,0,0,1,1,1,0,0,0,1
0,0,0,0,0,1,1,1,0,1
0,0,0,0,1,0,1,0,1,1
0,0,1,0,0,0,1,0,1,1
0,0,1,0,1,0,0,0,1,1
0,0,1,0,1,0,1,0,0,1
0,1,0,0,0,1,0,1,0,1
0,1,0,1,0,0,0,1,0,1
0,1,0,1,0,1,0,0,0,1
0,0,0,1,0,1,0,1,0,1
0,0,0,0,0,0,1,1,1,1
0,1,1,0,0,0,0,0,1,1
0,0,1,1,1,0,0,0,0,1
0,0,0,0,1,1,1,0,0,1
0,1,0,1,1,0,0,0,0,1
0,0,0,1,0,1,1,0,0,1
0,0,0,0,0,1,0,1,1,1
0,1,1,0,0,0,0,1,0,1
0,1,0,0,0,0,1,1,0,1
0,1,0,1,0,0,0,0,1,1
0,0,1,1,0,1,0,0,0,1
0,0,0,0,1,1,0,1,0,1
0,1,0,0,1,0,0,1,0,1
0,1,0,1,0,0,1,0,0,1
0,0,0,1,0,1,0,0,1,1
0,0,1,0,0,1,0,1,0,1
0,1,1,0,1,0,0,0,0,1
0,0,0,1,1,0,1,0,0,1
0,0,0,0,0,1,1,0,1,1
0,0,1,0,0,0,0,1,1,1
0,1,0,0,0,0,1,0,1,1
0,0,1,1,0,0,0,0,1,1
0,0,1,0,1,1,0,0,0,1
0,0,0,0,1,0,1,1,0,1
0,1,1,0,0,0,1,0,0,1
0,0,0,1,1,0,0,0,1,1
0,0,1,0,0,1,1,0,0,1
0,0,0,0,1,0,0,1,1,1
0,1,0,0,1,0,0,0,1,1
0,0,1,1,0,0,1,0,0,1
0,0,0,0,1,1,0,0,1,1
0,0,1,0,0,0,1,1,0,1
0,1,1,0,0,1,0,0,0,1
0,0,0,1,1,0,0,1,0,1
0,1,0,0,0,1,1,0,0,1
0,0,0,1,0,0,0,1,1,1
0,1,0,0,0,1,0,0,1,1
0,0,1,1,0,0,0,1,0,1
0,1,0,0,1,1,0,0,0,1
0,0,0,1,0,0,1,1,0,1
0,1,0,0,1,0,1,0,0,1
0,0,0,1,0,0,1,0,1,1
0,0,1,0,0,1,0,0,1,1
0,0,1,0,1,0,0,1,0,1

# Adding non-reflective B4z

0,1,0,0,1,1,0,0,1,1
0,0,1,1,0,0,1,1,0,1

0,1,1,1,1,1,1,0,0,1
0,0,0,1,1,1,1,1,1,1
0,1,1,0,0,1,1,1,1,1
0,1,1,1,1,0,0,1,1,1
0,1,0,0,1,1,1,1,1,1
0,1,1,1,0,0,1,1,1,1
0,1,1,1,1,1,0,0,1,1
0,0,1,1,1,1,1,1,0,1
0,1,0,1,1,1,1,1,0,1
0,1,0,1,0,1,1,1,1,1
0,1,1,1,0,1,0,1,1,1
0,1,1,1,1,1,0,1,0,1
0,0,1,0,1,1,1,1,1,1
0,1,1,0,1,0,1,1,1,1
0,1,1,1,1,0,1,0,1,1
0,0,1,1,1,1,1,0,1,1
0,0,1,1,1,0,1,1,1,1
0,1,1,0,1,1,1,0,1,1
0,1,1,1,1,0,1,1,0,1
0,1,0,1,1,1,1,0,1,1
0,0,1,1,0,1,1,1,1,1
0,1,1,0,1,1,0,1,1,1
0,1,0,1,1,0,1,1,1,1
0,1,1,1,0,1,1,0,1,1
0,0,1,1,1,1,0,1,1,1
0,1,1,0,1,1,1,1,0,1
0,1,0,1,1,1,0,1,1,1
0,1,1,1,0,1,1,1,0,1
0,1,0,1,1,1,1,1,1,1
0,1,1,1,0,1,1,1,1,1
0,1,1,1,1,1,0,1,1,1
0,1,1,1,1,1,1,1,0,1
0,0,1,1,1,1,1,1,1,1
0,1,1,0,1,1,1,1,1,1
0,1,1,1,1,0,1,1,1,1
0,1,1,1,1,1,1,0,1,1
0,1,1,1,1,1,1,1,1,1
# Survival conditions
1,1,0,0,0,0,0,1,1,1
1,1,1,1,0,0,0,0,0,1
1,0,0,1,1,1,0,0,0,1
1,0,0,0,0,1,1,1,0,1
1,0,0,0,1,0,1,0,1,1
1,0,1,0,0,0,1,0,1,1
1,0,1,0,1,0,0,0,1,1
1,0,1,0,1,0,1,0,0,1
1,1,0,0,0,1,0,1,0,1
1,1,0,1,0,0,0,1,0,1
1,1,0,1,0,1,0,0,0,1
1,0,0,1,0,1,0,1,0,1
1,0,0,0,0,0,1,1,1,1
1,1,1,0,0,0,0,0,1,1
1,0,1,1,1,0,0,0,0,1
1,0,0,0,1,1,1,0,0,1
1,1,0,1,1,0,0,0,0,1
1,0,0,1,0,1,1,0,0,1
1,0,0,0,0,1,0,1,1,1
1,1,1,0,0,0,0,1,0,1
1,1,0,0,0,0,1,1,0,1
1,1,0,1,0,0,0,0,1,1
1,0,1,1,0,1,0,0,0,1
1,0,0,0,1,1,0,1,0,1
1,1,0,0,1,0,0,1,0,1
1,1,0,1,0,0,1,0,0,1
1,0,0,1,0,1,0,0,1,1
1,0,1,0,0,1,0,1,0,1
1,1,1,0,1,0,0,0,0,1
1,0,0,1,1,0,1,0,0,1
1,0,0,0,0,1,1,0,1,1
1,0,1,0,0,0,0,1,1,1
1,1,0,0,0,0,1,0,1,1
1,0,1,1,0,0,0,0,1,1
1,0,1,0,1,1,0,0,0,1
1,0,0,0,1,0,1,1,0,1
1,1,1,0,0,0,1,0,0,1
1,0,0,1,1,0,0,0,1,1
1,0,1,0,0,1,1,0,0,1
1,0,0,0,1,0,0,1,1,1
1,1,0,0,1,0,0,0,1,1
1,0,1,1,0,0,1,0,0,1
1,0,0,0,1,1,0,0,1,1
1,0,1,0,0,0,1,1,0,1
1,1,1,0,0,1,0,0,0,1
1,0,0,1,1,0,0,1,0,1
1,1,0,0,0,1,1,0,0,1
1,0,0,1,0,0,0,1,1,1
1,1,0,0,0,1,0,0,1,1
1,0,1,1,0,0,0,1,0,1
1,1,0,0,1,1,0,0,0,1
1,0,0,1,0,0,1,1,0,1
1,1,0,0,1,0,1,0,0,1
1,0,0,1,0,0,1,0,1,1
1,0,1,0,0,1,0,0,1,1
1,0,1,0,1,0,0,1,0,1
1,1,1,1,1,0,0,0,0,1
1,0,0,1,1,1,1,0,0,1
1,0,0,0,0,1,1,1,1,1
1,1,1,0,0,0,0,1,1,1
1,1,0,0,0,0,1,1,1,1
1,1,1,1,0,0,0,0,1,1
1,0,1,1,1,1,0,0,0,1
1,0,0,0,1,1,1,1,0,1
1,0,1,0,1,0,1,0,1,1
1,1,0,1,0,1,0,1,0,1
1,0,1,1,0,0,0,1,1,1
1,1,1,0,1,1,0,0,0,1
1,0,0,1,1,0,1,1,0,1
1,1,0,0,0,1,1,0,1,1
1,0,1,1,0,1,0,1,0,1
1,1,0,0,1,1,0,1,0,1
1,1,0,1,0,0,1,1,0,1
1,1,0,1,0,1,0,0,1,1
1,0,0,1,0,1,0,1,1,1
1,1,1,0,0,1,0,1,0,1
1,1,0,1,1,0,0,1,0,1
1,1,0,1,0,1,1,0,0,1
1,1,1,0,1,0,0,1,0,1
1,1,0,1,1,0,1,0,0,1
1,0,0,1,0,1,1,0,1,1
1,0,1,0,0,1,0,1,1,1
1,1,0,1,0,0,1,0,1,1
1,0,1,1,0,1,0,0,1,1
1,0,1,0,1,1,0,1,0,1
1,1,0,0,1,0,1,1,0,1
1,0,1,1,1,0,0,0,1,1
1,0,1,0,1,1,1,0,0,1
1,0,0,0,1,0,1,1,1,1
1,1,1,0,0,0,1,0,1,1
1,0,1,0,0,0,1,1,1,1
1,1,1,0,1,0,0,0,1,1
1,0,1,1,1,0,1,0,0,1
1,0,0,0,1,1,1,0,1,1
1,1,0,0,1,0,0,1,1,1
1,1,1,1,0,0,1,0,0,1
1,0,0,1,1,1,0,0,1,1
1,0,1,0,0,1,1,1,0,1
1,1,1,1,0,1,0,0,0,1
1,0,0,1,1,1,0,1,0,1
1,1,0,0,0,1,1,1,0,1
1,1,0,1,0,0,0,1,1,1
1,1,0,0,0,1,0,1,1,1
1,1,1,1,0,0,0,1,0,1
1,1,0,1,1,1,0,0,0,1
1,0,0,1,0,1,1,1,0,1
1,1,0,0,1,1,1,0,0,1
1,0,0,1,0,0,1,1,1,1
1,1,1,0,0,1,0,0,1,1
1,0,1,1,1,0,0,1,0,1
1,1,0,1,1,0,0,0,1,1
1,0,1,1,0,1,1,0,0,1
1,0,0,0,1,1,0,1,1,1
1,1,1,0,0,0,1,1,0,1
1,1,1,0,1,0,1,0,0,1
1,0,0,1,1,0,1,0,1,1
1,0,1,0,0,1,1,0,1,1
1,0,1,0,1,0,0,1,1,1
1,1,0,0,1,0,1,0,1,1
1,0,1,1,0,0,1,0,1,1
1,0,1,0,1,1,0,0,1,1
1,0,1,0,1,0,1,1,0,1

# Making S4z non-reflective 

1,1,1,0,0,1,1,0,0,1
1,0,0,1,1,0,0,1,1,1

1,1,1,1,1,1,1,0,0,1
1,0,0,1,1,1,1,1,1,1
1,1,1,0,0,1,1,1,1,1
1,1,1,1,1,0,0,1,1,1
1,1,0,0,1,1,1,1,1,1
1,1,1,1,0,0,1,1,1,1
1,1,1,1,1,1,0,0,1,1
1,0,1,1,1,1,1,1,0,1
1,1,0,1,1,1,1,1,0,1
1,1,0,1,0,1,1,1,1,1
1,1,1,1,0,1,0,1,1,1
1,1,1,1,1,1,0,1,0,1
1,0,1,0,1,1,1,1,1,1
1,1,1,0,1,0,1,1,1,1
1,1,1,1,1,0,1,0,1,1
1,0,1,1,1,1,1,0,1,1
1,0,1,1,1,0,1,1,1,1
1,1,1,0,1,1,1,0,1,1
1,1,1,1,1,0,1,1,0,1
1,1,0,1,1,1,1,0,1,1
1,0,1,1,0,1,1,1,1,1
1,1,1,0,1,1,0,1,1,1
1,1,0,1,1,0,1,1,1,1
1,1,1,1,0,1,1,0,1,1
1,0,1,1,1,1,0,1,1,1
1,1,1,0,1,1,1,1,0,1
1,1,0,1,1,1,0,1,1,1
1,1,1,1,0,1,1,1,0,1
1,1,0,1,1,1,1,1,1,1
1,1,1,1,0,1,1,1,1,1
1,1,1,1,1,1,0,1,1,1
1,1,1,1,1,1,1,1,0,1
1,0,1,1,1,1,1,1,1,1
1,1,1,0,1,1,1,1,1,1
1,1,1,1,1,0,1,1,1,1
1,1,1,1,1,1,1,0,1,1
1,1,1,1,1,1,1,1,1,1
1,A,B,C,D,E,F,G,H,0

Yes I agree that terminology needs to be agreed upon.

Re: "spending time on explaining thing" -- I don't consider explaining things as spent time. Forum rules (e.g. the forum rule 5b) also mention the need for descriptions/explanations.

I am aware of the forum rules. However, this forum is made for people who have practical experience with CAs, ruletables and presumably some math and computer science. In my opinion, these lines of 10 constants or variables can be easily read imagining the neighborhood. The forum rule 5b recommends to briefly explain what the posted pattern does, not serve as a detailed coding manual.

The entire RuleLoader ruletable represents a cellular automaton.
Individual lines (such as "1, 0,1,0,1,0,1,1,1, 1") represent specific update rules of the cellular automaton.

Then again we have a contradiction of terms. Either people on this forum should call their rules CAs and not rules, or we should call the ruletable lines subrules or something like that. Since everyone here does call their CAs rules and in ruletables they are also called @RULE, this confusing terminology, IMO, should better be revised.

[confocaloid]

[...]

The entire RuleLoader ruletable represents a cellular automaton.
Individual lines (such as "1, 0,1,0,1,0,1,1,1, 1") represent specific update rules of the cellular automaton.

Then again we have a contradiction of terms. Either people on this forum should call their rules CAs and not rules, or we should call the ruletable lines subrules or something like that. [...]

I don't see a direct contradiction in this case.

It remains a consistent approach to "merge" or "combine" all rules of the CA into a single rule (for example expressed as a mathematical function).

All of the rules of Conway's Life can be packed into just a single rule, for example as a single expression in some syntax, like this:

Code: Select all

1 if n == 3 else (c if n == 2 else 0)

where c stands for the current state of a cell, and n stands for the number of currently-alive neighbours of the cell.

Interactive Python session to illustrate the idea:

Code: Select all

>>>> def cgol(c, n):
....     return 1 if n == 3 else (c if n == 2 else 0)
.... 
>>>> [cgol(0, n) for n in range(9)]
[0, 0, 0, 1, 0, 0, 0, 0, 0]
>>>> [cgol(1, n) for n in range(9)]
[0, 0, 1, 1, 0, 0, 0, 0, 0]
>>>>

Even if you view all rules of a CA as a single rule, that doesn't tell everything about the CA. It remains to specify what is the neighbourhood, what is the number of cellstates.

It is not necessary to write down an explicit expression. One can consider the set of rules (the ruleset, collectively covering all possible conditions that can occur), or alternatively one can consider them as a single rule that covers all possible conditions by itself.

[Yoel]

[...]

The entire RuleLoader ruletable represents a cellular automaton.
Individual lines (such as "1, 0,1,0,1,0,1,1,1, 1") represent specific update rules of the cellular automaton.

Then again we have a contradiction of terms. Either people on this forum should call their rules CAs and not rules, or we should call the ruletable lines subrules or something like that. [...]

I don't see a direct contradiction in this case.

It remains a consistent approach to "merge" or "combine" all rules of the CA into a single rule (for example expressed as a mathematical function).

Ok, agreed, like a composition of functions is a function.

All of the rules of Conway's Life can be packed into just a single rule, for example as a single expression in some syntax, like this:

Code: Select all

1 if n == 3 else (c if n == 2 else 0)

where c stands for the current state of a cell, and n stands for the number of currently-alive neighbours of the cell.

Interactive Python session to illustrate the idea:

Code: Select all

>>>> def cgol(c, n):
....     return 1 if n == 3 else (c if n == 2 else 0)
.... 
>>>> [cgol(0, n) for n in range(9)]
[0, 0, 0, 1, 0, 0, 0, 0, 0]
>>>> [cgol(1, n) for n in range(9)]
[0, 0, 1, 1, 0, 0, 0, 0, 0]
>>>>

My CGOL metacell in FireWorld is built from parts of my Izhora computer (registers, counters and integer comparison units) that do actually what you describe, computing this very function.

BTW, here is a 90c/180 spaceship in the modified Day & Night that turns into a puffer when flipped:

Code: Select all

x = 74, y = 129, rule = DN-x4z
20$16bo43bo$16bo2b2o35b2obobo$15bo5bo27bo6b6o$14bo2b7obobob7o5bo4b6o3b
ob5obo$12b2ob12ob16o3b7o2b9o$12b54o$12b3o2b49o$12b3o4b12o2b3obob10o2b
14o$22b2o3bob3o3bo3b8o3bob9o2bo$21bo3bo2bo11b2ob2o6b8obobo$40bo10b2ob
2o4bo$55bo73$55bo$40bo10b2ob2o4bo$21bo3bo2bo11b2ob2o6b8obobo$22b2o3bob
3o3bo3b8o3bob9o2bo$12b3o4b12o2b3obob10o2b14o$12b3o2b49o$12b54o$12b2ob
12ob16o3b7o2b9o$14bo2b7obobob7o5bo4b6o3bob5obo$15bo5bo27bo6b6o$16bo2b
2o35b2obobo$16bo43bo!

After more research I'll write about it in the Miscellaneous Discoveries thread. I agree with you that non-isotropic rules deserve more attention.