Restrictive Rulespaces

[Docsy]

This thread is for rulespaces defined in a way that limits the ability to form classes of interesting patterns (such as oscillators, spaceships, guns, puffers, rakes, linear/quadratic growth), and thus prompt the question on what classes of patterns exist at all in the entire rule space, rather than any specific rule.

Ones explored so far include:
B2c3a/S* (Rules where B2c3a are the only birth conditions. When looking at using only the minimal birth conditions needed for spaceships, this is seemingly the hardest one to find spaceships in.)
B*-(012a)/S (Non-B0 sublight speed rules without survival conditions. All spaceships have to be phoenixes, that can't keep their fronts alive by moving at light speed.)
B2ce/S* (Similar to B2c3a in concept, but considerably more active, and some basic rules have simple spaceships. However, adding just a few extra restrictions can make this one tough, with a lot of interest in removing INT rules contained within S012)

The original intro to this thread:

This thread is for the rule space of INT rules that have only B2c3a as their birth rules (and nothing else) and have arbitrary survival rules, which has gotten some attention on the Discord lately. Inspired by a desire to prove that in non-B0 INT rules the set of B1c, B1e, B2a, B2ce, B2c3a, B2e3i, and B3ai are not just required for expanding patterns, but sufficient as sole birth rules if all survival rules are game. It turned out that B2c3a/S* is somewhat difficult to find mobile patterns in, making the ones that exist anywhere in the space of interest. Oscillators to a lesser degree can also be constrained by this and can be intriguing as well.

Early discoveries in B2c3a/S*:
Here is a document of many of the discoveries of mobile patterns made so far. Feel free to download a local copy if you wish.
https://docs.google.com/document/d/1HUQ ... sp=sharing

Some highlights:
Small c/5 spaceship found by AforAmpere

Code: Select all

x = 8, y = 9, rule = B2c3a/S1c2a3-cj4aejnryz5-aiq6-c78
4bo2$4bo$ob4obo2$ob4obo$4bo2$4bo!

orthogonal wickstretcher found by me

Code: Select all

x = 8, y = 7, rule = B2c3a/S1c2ai3-cj4aejnryz5-inq6-ci78
2o4b2o$b2o2b2o$8o2$8o$b2o2b2o$2o4b2o!

c/8d spaceship found by Rabbit

Code: Select all

x = 7, y = 7, rule = B2c3a/S3-j4aejnw5-i678
2bo2b2o$3b4o$ob4o$3b3o$4bobo2$4bo!

thick diagonal wickstretcher found by Rabbit

Code: Select all

x = 9, y = 9, rule = B2c3a/S3-j4aejnw5-i678
2ob2o$5o$b3o$5o$2ob4o$4b4o$4b5o$5b2o$6bo!

spacefiller found by me

Code: Select all

x = 13, y = 13, rule = B2c3a/S2cn3acin4acq5acejn6-i78
obobo3bobobo$b3o5b3o$4o5b4o$b4o3b4o$o2b2o3b2o2bo$5bobo$6bo$5bobo$o2b2o
3b2o2bo$b4o3b4o$4o5b4o$b3o5b3o$obobo3bobobo!

dynamic quadratic growth pattern found by me

Code: Select all

x = 13, y = 7, rule = B2c3a/S2cn3acin4aq5acejn6-i78
obobo3bobobo$b3o5b3o$4o5b4o$b4o3b4o$o2b2o3b2o2bo$5bobo$6bo!

wobbling c/12 spaceship found by AforAmpere

Code: Select all

x = 6, y = 7, rule = B2c3a/S2ck3ijn4knty5-ckry6ace78
bo$b2o$4o$b4o$ob4o$3bo$bobo!

Some questions for B2c3a/S*:
Are there puffers?
Are there guns or rakes?
Are there waves that can be stabilized?
Can you go faster than c/5? What speeds exist generally?
Is there an oblique spaceship, or growth pattern more broadly?
How many distinct speed/direction combos (as in, either can be different, not necessarily both) can be in a single rule?
Is there in general a quadratic growth pattern that grows cells of both parities as it goes? (to be precise with technicalities, a pattern where you can group cells into a rectangular tessellation such that the number of rectangles containing cells of both parities grows quadratically)

[wildmyron]

Edit Please ignore this post - irrelevant results from a different rulespace /Edit

That rulespace is so vast and clearly includes a variety of interesting behaviour, so I'm confident that the answer to most of your questions is Yes! with the following exceptions:

What speeds exist generally?

Pretty much all of them that also exist elsewhere in the B2 rulespace except maybe those that could only be found in a small number of rules - e.g. something like 9c/10 (which there is no known example of but almost certainly exists somewhere)

How many distinct speed/direction combos (as in, either can be different, not necessarily both) can be in a single rule?

Infinitely many - there's definitely a rule somewhere in this space with an adjustable spaceship design

To more constructively answer one of those questions, here are some faster spaceships (found with LLS (Logic_Life_Search)):

c/4 orthogonal, p4

Code: Select all

x = 9, y = 11, rule = B2a3c/S1e2cei3-cqry4air5in6k
5bo$o4b4o$2o$2o2bo$4b2ob2o2$4b2ob2o$2o2bo$2o$o4b4o$5bo!

c/3 orthogonal, p3

Code: Select all

x = 11, y = 10, rule = B2a3c/S1e2e3acinq4jkqr
7bo2$4b2obob2o$3b3o4bo$2bo3bobobo$3o3bobobo$bob3o4bo$4b2obob2o2$7bo!

c/2 orthogonal, p2

Code: Select all

x = 8, y = 8, rule = B2a3c/S2aei3acery4y
6bo$8o2$3bo$o2bo2b2o$6bo$5bo$3b3o!

p1 photon (already known)

Code: Select all

x = 2, y = 5, rule = B2a3c/S2e
bo$2o2$2o$bo!

Code: Select all

x = 3, y = 5, rule = B2a3c/S2e
bo$2o2$b2o$2bo!

Edit
c/1 puffer, p2

Code: Select all

x = 5, y = 3, rule = B2a3c/S01e3a4ar
o$2b3o$2b3o!

[islptng]

Code: Select all

rule = B2a3c/S...

Oops! It's B2c3a, not B2a3c.

And to keep it ontopic, I'm running a TRS search:

Code: Select all

r: B2c3a/S B2c3a/S012345678
p: 2 30
x: 1 -1
y: 0 -1
==start
bo$3o!
==end

but sadly, no results found in 25m rules.

EDIT: Er, I'm ... sorry...

Code: Select all

$ python3 lls -c -b 9 9 -s p2 x1 y0 -r "pB2c-aeikn3a-ceiknyqjr/S012345678"
Getting search pattern...
Done

Preprocessing...
Done

Width: 11
Height: 11
Duration: 3

Number of undetermined cells: 153
Number of variables: 286
Number of clauses: 75073

Unsatisfiable

Total solver time: 0.16274666786193848

EDIT 2: That c/5 is the fastest & smallest ship, proven with LLS...

[wildmyron]

Code: Select all

rule = B2a3c/S...

Oops! It's B2c3a, not B2a3c.

Oops indeed! I really should have checked when those easy results turned up.
That certainly tempers my confidence about the affirmative answers.

EDIT 2: That c/5 is the fastest & smallest ship, proven with LLS...

Do you mean that's the fastest ship of that size? I'm not sure you can completely rule out faster than c/5 with LLS.

[Docsy]

Thank you for looking anyway wildmyron! Fun ships in that rule even if it wasn't the topic.

Also no worries islptng, this space is not very forgiving, allowing any survival condition you want is really required to make it interesting haha.

AforAmpere did discover some other speeds, like:

Code: Select all

x = 4, y = 9, rule = B2c3a/S012ae3-ekr4ejnrwy5-inr6en
bo2$bobo$b2o$2obo$b2o$bobo2$bo!

Code: Select all

x = 7, y = 9, rule = B2c3a/S01c2aek3-jkr4acenqry5acej6ain8
3bo2$3bo$3b2obo$o$3b2obo$3bo2$3bo!

and also posted some other spaceships found in bulk, of various speeds:
13, B2c3a/S012ae3-ekr4ejnrwy5-inr6en, -1, 0, 10, 2bo2$obo$b2o$ob2o$b2o$obo2$2bo!
11, B2c3a/S01c2aek3-jkr4-ijktwz5acej6ain8, -1, 0, 14, 3bo2$3bo$ob2o$6bo$ob2o$3bo2$3bo!
17, B2c3a/S2a3ai4aenrwy5acjn6cek, 1, 0, 5, 2b2o$b2o$4o$bo$4o$b2o$2b2o!
17, B2c3a/S2ae3ai4aenrwy5acjn6cek, 1, 0, 5, 2b2o$b2o$4o$bo$4o$b2o$2b2o!
17, B2c3a/S2a3ain4aenrwy5acjn6cek, 1, 0, 5, 2b2o$b2o$4o$bo$4o$b2o$2b2o!
13, B2c3a/S01c2ae3-ekr4-aciqtz5-inr6cen, -1, 0, 10, 2bo2$obo$b2o$ob2o$b2o$obo2$2bo!
13, B2c3a/S012ae3-ekr4-aciqtz5-inr6cen, -1, 0, 10, 2bo2$obo$b2o$ob2o$b2o$obo2$2bo!
11, B2c3a/S01e2ae3aknr4acnry5acjqr6ae7e, 3, 0, 24, 4bo2$4bo$4b2obo$o$4b2obo$4bo2$4bo!
12, B2c3a/S02ae3aeikn4acnry5-ikry6en78, 2, 0, 12, 4bo2$4bo$o3b2obo2$o3b2obo$4bo2$4bo!
11, B2c3a/S02ae3aeikq4acnry5cejq6an7e8, 2, 0, 12, 2bo2$2bo$2b2obo$o$2b2obo$2bo2$2bo!
13, B2c3a/S02a3aikr4acjnry5cejqy7e8, 1, 0, 6, 2bo2$2bo$ob2obo$bo$ob2obo$2bo2$2bo!
13, B2c3a/S01e2ae3aiknr4acjnry5-aik67e8, 3, 0, 18, 5bo2$5bo$bo3b2obo$o$bo3b2obo$5bo2$5bo!
17, B2c3a/S01e2a3aiknr4-eiqtwz5-ikr67e8, 2, 0, 12, 5bo2$4b2o$3b4obo$o$3b4obo$4b2o2$5bo!
12, B2c3a/S02ae3-ceqr4-eiqtz5-ik6-ci78, 1, 0, 20, bo2$bo$b2obo$2o$b2obo$bo2$bo!
12, B2c3a/S02ae3aijkn4-eiqtz5-ik6-i78, 1, 0, 22, bo2$bo$b2obo$2o$b2obo$bo2$bo!
11, B2c3a/S02ae3aeikq4acnqry5cej6an8, 2, 0, 12, 2bo2$2bo$2b2obo$o$2b2obo$2bo2$2bo!
13, B2c3a/S012aek3aiknq4-ceqtwz5-ikny6-ci78, 1, 0, 15, 4bo2$4bobo$bob2o$o$bob2o$4bobo2$4bo!
11, B2c3a/S01c2aek3acen4anqry5cej6an8, 1, 0, 14, 3bo2$3bo$3b2obo$o$3b2obo$3bo2$3bo!

[Docsy]

When I checked early on for rules that support multiple speeds and/or directions of travel, I couldn't find any compatible spaceships that didn't share the same speed and direction, but with all these new spaceships it seems plausible that there is some rule that contains at least two speeds.

[Docsy]

A lot of interesting discoveries have been made in the B2c3a/S* rulespace since the last update, here's a bit of a showcase:
Oscillators by Rabbit:

Code: Select all

x = 6, y = 6, rule = B2c3a/S3ij4nqry5-c678
3bo$2b3o$b5o$5o$b3o$2bo!

Code: Select all

x = 10, y = 10, rule = B2c3a/S1c3aijn4anqy5-i6-n78
5bo$6bo$2b2ob2o$2b7o$3b5obo$ob5o$b7o$3b2ob2o$3bo$4bo!

Code: Select all

x = 7, y = 4, rule = B2c3a/S34aejnr5-ir678
4b2o$2b5o$7o$2obob2o!

Code: Select all

x = 7, y = 7, rule = B2c3a/S3ir4-aqr5678
bo$3o$b3o$2b3o$3b3o$4b2o$6bo!

Code: Select all

x = 13, y = 13, rule = B2c3a/S3ir4-aqr5678
11bo$10b3o$9bobo$8b3o$7b3o$3b2ob3o$4b4o$3b4o$3b5o$2b4obo$b3o$3o$bo!

Code: Select all

x = 13, y = 23, rule = B2c3a/S3ir4-aqr5678
3o$4o$5o$b5o$2b3o$3bo5$11bo$10b3o$9bobo$8b3o$7b3o$3b2ob3o$4b4o$3b4o$3b
5o$2b4obo$b3o$3o$bo!

Code: Select all

x = 12, y = 12, rule = B2c3a/S3in4-aqrt5678
o2$o2$5bo$4b3o$3b5o$2b5o$2b4o$2b3o2$9bobo!

Code: Select all

x = 29, y = 29, rule = B2c3a/S2n3cin4iknw5-ky6-i78
3o23b3o$4o21b4o$5o19b5o$b5o17b5o$2b3o19b3o$3bo10bo10bo2$14bo7$5bobo13b
obo7$14bo2$3bo10bo10bo$2b3o19b3o$b5o17b5o$5o19b5o$4o21b4o$3o23b3o!

Code: Select all

x = 29, y = 29, rule = B2c3a/S2n3cin4iknw5-ky6-i78
3o23b3o$4o21b4o$5o19b5o$b5o17b5o$2b3o19b3o$3bo21bo5$14bo$12b5o$11bob3o
bo$11b2o3b2o$10b3o3b3o$11b2o3b2o$11bob3obo$12b5o$14bo5$3bo21bo$2b3o19b
3o$b5o17b5o$5o19b5o$4o21b4o$3o23b3o!

Slow orthogonal spaceships by Ampere:

Code: Select all

x = 4, y = 9, rule = B2c3a/S012ae3-ekr4ejnrwy5-inr6en
bo2$bobo$b2o$2obo$b2o$bobo2$bo!

Code: Select all

x = 50, y = 50, rule = B2c3a/S012aek3aiknq4-ceqtwz5-ikny6-ci78
4bo2$4bobo$bob2o$o$bob2o$4bobo2$4bo!

Code: Select all

x = 50, y = 50, rule = B2c3a/S02ae3aijkn4-eiqtz5-ik6-i78
bo2$bo$b2obo$2o$b2obo$bo2$bo!

Small diagonal spaceship by Ampere:

Code: Select all

x = 6, y = 6, rule = B2c3a/S12a3ainq4ijqrw5aeikq6-ik7e
bo2$bo$3o$2b2obo$2bo!

c/7d spaceship by Rabbit:

Code: Select all

x = 6, y = 6, rule = B2c3a/S1c3i4-a5678
3o$4o$5o$b5o$2b3o$3bo!

c/7d is thought to be the diagonal speed limit (and c/5o the orthogonal speed limit)
Other c/7d's by Ampere:

Code: Select all

x = 6, y = 6, rule = B2c3a/S12ikn3iknqy4-a5-cq6-e7e
o$2o$b2o$2b2o$3b2o$4b2o!

Code: Select all

x = 9, y = 9, rule = B2c3a/S2a3-cekq4nry5aejr6-kn78
3bo$3b2o$3b3o$7o$8o$2b7o$3b3o$4b2o$4b2o!

Very slow diagonal spaceships by Ampere:

Code: Select all

x = 5, y = 5, rule = B2c3a/S2ekn3inr4jkwy5aeijn6-n78
2o$b2o$4o$b4o$2bobo!

Code: Select all

x = 7, y = 7, rule = B2c3a/S2n3cin4iknw5-ky6-i78
6bo$4b2o$3b3o$2b3o$b3o$3o$bo!

Code: Select all

x = 16, y = 16, rule = B2c3a/S1e2an3cikny4cijknwy5aejr678
10bobo$11b3o$10b5o$9b7o$10b5o$9bob3obo$10bobo2$obo5$7bo2$7bo!

Code: Select all

x = 8, y = 8, rule = B2c3a/S1e2akn3cin4-aeqrz5aej6-n78
2bobo2$2bobo$b3obobo$5o$b5obo$ob3o$2obo!

c/1288:

Code: Select all

x = 8, y = 8, rule = B2c3a/S1e2a3ciknq4ceijkny5aej6-n78
2bobo2$2bobo$b3obobo$5o$b5obo$ob3o$2obo!

Large diagonal wickstretcher by Rabbit:

Code: Select all

x = 7, y = 7, rule = B2c3a/S2a3-aej4-aqrt5-cnq6-n78
bobo$2b3o$b5o$7o$b5o$ob3obo$bobo!

Extensible spaceship based on above by Ampere:

Code: Select all

x = 28, y = 28, rule = B2c3a/S2an3cin4knqwy5aeijr6-n78
22bo2$22bobo$21b3obo$20b5o$19b7obo$18b7o$17b7o$16b7o$15b7o$7bo6b7o$11b
ob7o$7bobo2b7o$6b12o$3bob12o$6b10o$5b10obo$6b9o$6b10o$5b10o$2bob12obo$
3b12o$9o2bobo$b7o$b6o6bo$6obo$b5o$2bo2bo!

Two coexisting spaceships of different speeds, c/31d by Sylvie:

Code: Select all

x = 14, y = 11, rule = B2c3a/S2c3aeij4acinq5-ikqy6-n78
b2o$4o$b4o$2b3o$3bo7b2o$10b4o$9b5o$8b5o$7b5o$9b2o$9bo!

Coexisting orthogonal spaceships:

Code: Select all

x = 9, y = 21, rule = B2c3a/S02a3aiknr4acjknry5-ikr67e8
5bo2$5bo$3bob2obo$4bo$3bob2obo$5bo2$5bo4$5bo2$4b2o$3b4obo$o$3b4obo$4b
2o2$5bo!

4 coexisting diagonal spaceships:

Code: Select all

x = 26, y = 26, rule = B2c3a/S2c3aci4anq5acejn6-in78
o$bobo$3o$b3o$obobo$10bo$9bobo$7bo2bobo$8b4obo$7b4obo$8b3o$7bobobo$17b
o2$14bo2bo$15b4o$14b4obo$15b3o$14bobobo3$21bo$22b2o$21b4o$22b3o$21bob
obo!

Spaceship reflector by Rabbit (Allows for non-margolus oscillators of arbitrarily large period):

Code: Select all

x = 13, y = 13, rule = B2c3a/S3ai4aen5acejn6-in78
11b2o$11b2o3$5bo$4b3o$4b4o$5b4o$6b2o3$2o$2o!

Quirky diagonal spaceships by Sylvie:

Code: Select all

x = 8, y = 8, rule = B2c3a/S2ae3ik4jknwy5-n67
3bo2$3bo$ob2o$3b3obo$4bo2$4bo!

Code: Select all

x = 7, y = 7, rule = B2c3a/S1e2en3-ckry4jnrtw5-ky6-ik7c8
2bo$bob2o$6o$o2b3o$bo2bobo$2bob2o$3b2o!

[ThePlayzr]

This one is nearly explosive, and has a c/5d:

Code: Select all

x = 4, y = 4, rule = B2cek3ajqr4-k5-ciny/S
obo$bobo$2bo$3bo!

Code: Select all

x = 0, y = 0, rule = B2cek3ajqr4-k5-ciny/S
!
[[ RANDOMIZE STEP 64 AUTOSTART ]]

B2c margolus oscillators work.
Are odd period oscillators possible?
While testing if b2e margolus oscillators work, I found this p8 which evolves from a length-5 diagonal line:

Code: Select all

x = 9, y = 9, rule = B2cek3ajqr4-k5-ciny/S
obobobobo$b2obob2o$3obob3o$3b3o$4ob4o$3b3o$3obob3o$b2obob2o$obobobobo
!

Pattern collection:

Code: Select all

x = 50, y = 78, rule = B2cek3ajqr4-k5-ciny/S
48bo$37bo10b2o$4b3o25bo5b2o6bo$6bo4bo9bo4bo3bo5b2o8bo2bo$4b3o3bo3bobo
2bo4bo8bo6b2o2bo2bo$4bo15bo5bo4bo6bo9bo$4b3o32bo5bo2bo$47bo$47bo2$4bo
bo$4bobo4bobobobo$4b3o$6bo4bobobobo$6bo6$4b3o14bo$4bo15bo$4b3o3bobobo
bo6bo$4bobo13bo$4b3o14bo7$13bobobobobo$14b2obob2o$4b3o6b3obob3o9bo$4b
obo9b3o14bo$4b3o6b4ob4o8b2o$4bobo9b3o13bobo$4b3o6b3obob3o8bobo$14b2ob
ob2o$13bobobobobo7$15bo$15bo$2bob3o6bobobo$2bobobo6bob2o$2bobobo4bobo
b3o$2bobobo3bo$2bob3o5bobobo$12bobo6$2bobobo$2bobobo$2bob3o3bobobobob
obo2bobobobobobobobo$2bo3bo$2bo3bo10$3ob3o$o5bo$3ob3o3bobobobobobobob
obobo$obobo$3ob3o!

Margolus oscillators above length 10 are not shown.
Can someone apgsearch it?

[rabbit]

Relatively active B2ce/S* rule with many c/2 spaceships and a c/4 diagonal:

Code: Select all

x = 86, y = 5, rule = B2ce/S1c2an3c4-ir5aj
2bo47bo21bo9bo$bobo47bo19bobo7bobo$o3bo47bo17bo3bo9bo$bobo49bo21bo9bo$
2bo!

Adding S5k turns one of the provided patterns into a p24 eight barreled gun.

[rabbit]

A variant of the above rule in the B2ce/S* space has this c/13 orthogonal engine that nearly replicates:

Code: Select all

x = 8, y = 11, rule = B2ce/S1c2ain3cinq4-inq5acjHistory
2C4.2C$.C4.C$C.C2.C.C6$D.D2.D.D$.D4.D$2D4.2D!

This version makes hamburgers:

Code: Select all

x = 8, y = 3, rule = B2ce/S1c2ain3cnq4aeqrwy5ajq
2o4b2o$bo4bo$obo2bobo!

EDIT: Here is a really good rule with a failed replicator which can be disturbed:

Code: Select all

x = 10, y = 4, rule = B2ce/S1c2an3ceij4aenqw5ajk6ek
o8bo$bo$2bo6bo$3bo!

Natural gun in the above rule:

Code: Select all

x = 16, y = 16, rule = B2ce/S1c2an3ceij4aenqw5ajk6ek
obo2b4o4b2o$o2b4ob2o$4bo2bo4b4o$2obo3bo3b5o$2o4b2ob3obobo$b3ob2o2bo2b
2obo$2o2bo4b2o2bo$bo2b2obo2bob3o$4b2o4bob2o$b2o2b8o2bo$5obobo2b4o$o2b
2o5b3o2bo$obob2ob3o2bobo$3ob6o3bo$o3bobo3bobobo$b2ob6o2bob2o!

[yyh_baboon]

Puffrake engine in the rule:

Code: Select all

 x = 24, y = 4, rule = B2ce/S1c2an3ceij4aenqw5ajk6ek
o22bo$bo20bo$2bo18bo$3bo16bo!