## Oscillator Universality

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83bismuth38
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### Oscillator Universality

Dump the most universal oscillators here! a universal oscillator (man that is hard to type) is an oscillator that can be found in all rules. of course, this doesn't exist, since there are (8+the number of modifers)!^2 rules, if my math is correct. anyways, dump em' here, guys!

Code: Select all

x = 8, y = 10, rule = B3/S23
3b2o$3b2o$2b3o$4bobo$2obobobo$3bo2bo$2bobo2bo$2bo4bo$2bo4bo$2bo! No football of any dui mauris said that. blah Posts: 248 Joined: April 9th, 2016, 7:22 pm ### Re: Oscillator Universality I don't think your maths is correct. I don't know where you got the factorial from, and I think you put the exponent on the wrong end; there are 2^18 = 262,144 lifelike rules, and 2^58 (58 being 8+the number of modifiers if I counted them right) isotropic rules. Anyway, this oscillator is pretty universal: Code: Select all x = 0, y = 0, rule = B3/S23 ! I call it "Empty Space". It's a p1 oscillator in 50% of rules (anything without B0), a p2 oscillator in 25% of rules (with B0 but without S8), and the immediate predecessor of a still life in the other 25% of rules (with B0 and S8). You never said it couldn't be trivial. :^) succ Gamedziner Posts: 796 Joined: May 30th, 2016, 8:47 pm Location: Milky Way Galaxy: Planet Earth ### Re: Oscillator Universality blah wrote:I don't think your maths is correct. I don't know where you got the factorial from, and I think you put the exponent on the wrong end; there are 2^18 = 262,144 lifelike rules, and 2^58 (58 being 8+the number of modifiers if I counted them right) isotropic rules. Anyway, this oscillator is pretty universal: Code: Select all x = 0, y = 0, rule = B3/S23 ! I call it "Empty Space". It's a p1 oscillator in 50% of rules (anything without B0), a p2 oscillator in 25% of rules (with B0 but without S8), and the immediate predecessor of a still life in the other 25% of rules (with B0 and S8). You never said it couldn't be trivial. :^) Except a so called "p1 oscillator" is actually a still life and nothing more. Code: Select all x = 81, y = 96, rule = LifeHistory 58.2A$58.2A3$59.2A17.2A$59.2A17.2A3$79.2A$79.2A2$57.A$56.A$56.3A4$27.
A$27.A.A$27.2A21$3.2A$3.2A2.2A$7.2A18$7.2A$7.2A2.2A$11.2A11$2A$2A2.2A
$4.2A18$4.2A$4.2A2.2A$8.2A!

blah
Posts: 248
Joined: April 9th, 2016, 7:22 pm

### Re: Oscillator Universality

Gamedziner wrote:Except a so called "p1 oscillator" is actually a still life and nothing more.
It's still a p2 oscillator in 25% of rules. But in all seriousness, I think 'Checkers' is pretty universal:

Code: Select all

x = 2, y = 2, rule = B2/S
bo$o! In totalistic terms, it requires B2 is present, B1 is not present, and S1 is not present. That's three conditions, which means this exists in one eighth of lifelike rules ((1/2^3)2^18 = 32768). In isotropic terms, it requires lack of B1e, B1c, S1c, and presence of B2e, which would count as four terms, putting it in one 16th of isotropic rules. Unless you count the inverted version: Code: Select all x = 16, y = 16, rule = B7/S01234578 16o$16o$16o$16o$16o$16o$16o$7ob8o$8ob7o$16o$16o$16o$16o$16o$16o$16o!
succ

83bismuth38
Posts: 461
Joined: March 2nd, 2017, 4:23 pm
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### Re: Oscillator Universality

A for awesome wrote:
BlinkerSpawn wrote:The most universal oscillator would be the p2 doublet, which only requires B2e and prohibits B01/S1c.
A single dot is a p1 oscillator requiring S0 and prohibiting B01. A block is a p1 oscillator requiring S3a and prohibiting B01c2a. Both work in 1/16 of all rules, whereas the duoplet works in only 1/32 of all rules.

I recognize that p1 oscillators might not count, though.

EDIT: For p3 oscillators the best I can find is requiring B2i3a4e/S2c and prohibiting B012ce4c/S03i4e:

Code: Select all

x = 3, y = 3, rule = B2i3a4e/S2c
bo$obo$bo!
(1/8192)

For p4s it requires B2k3a/S0 and prohibits B012a/S2a3j:

Code: Select all

x = 3, y = 2, rule = B2k3a/S0
o$2bo! (1/512) all credit to him for these. they were on an unrelated subject. Code: Select all x = 8, y = 10, rule = B3/S23 3b2o$3b2o$2b3o$4bobo$2obobobo$3bo2bo$2bobo2bo$2bo4bo$2bo4bo$2bo!
No football of any dui mauris said that.

83bismuth38
Posts: 461
Joined: March 2nd, 2017, 4:23 pm
Location: Still sitting around in Sagittarius A...
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### Re: Oscillator Universality

Ooh a new twist: what is the most universal RULE? (it has the most oscillators, ships, guns, still lifes, replicators, agars, etc.)

Code: Select all

x = 8, y = 10, rule = B3/S23
3b2o$3b2o$2b3o$4bobo$2obobobo$3bo2bo$2bobo2bo$2bo4bo$2bo4bo$2bo! No football of any dui mauris said that. Gamedziner Posts: 796 Joined: May 30th, 2016, 8:47 pm Location: Milky Way Galaxy: Planet Earth ### Re: Oscillator Universality 83bismuth38 wrote:Ooh a new twist: what is the most universal RULE? (it has the most oscillators, ships, guns, still lifes, replicators, agars, etc.) In terms of replicators, the Replicator rule definitely takes the cake. Code: Select all x = 1, y = 1, rule = B1357/S1357 1000o! Code: Select all x = 81, y = 96, rule = LifeHistory 58.2A$58.2A3$59.2A17.2A$59.2A17.2A3$79.2A$79.2A2$57.A$56.A$56.3A4$27.
A$27.A.A$27.2A21$3.2A$3.2A2.2A$7.2A18$7.2A$7.2A2.2A$11.2A11$2A$2A2.2A
$4.2A18$4.2A$4.2A2.2A$8.2A!

83bismuth38
Posts: 461
Joined: March 2nd, 2017, 4:23 pm
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### Re: Oscillator Universality

yeah, it has inf replicators, but 0 still lifes, agars, guns, fuses, oscillators, ships, methuselahs, or really anything else, so... it's not very universal.

Code: Select all

x = 8, y = 10, rule = B3/S23
3b2o$3b2o$2b3o$4bobo$2obobobo$3bo2bo$2bobo2bo$2bo4bo$2bo4bo$2bo! No football of any dui mauris said that. Saka Posts: 3186 Joined: June 19th, 2015, 8:50 pm Location: In the kingdom of Sultan Hamengkubuwono X Contact: ### Re: Oscillator Universality The most unuversal rule would be b3/s23 because it has been studied the most Code: Select all x = 17, y = 10, rule = B3/S23 b2ob2obo5b2o$11b4obo$2bob3o2bo2b3o$bo3b2o4b2o$o2bo2bob2o3b4o$bob2obo5b
o2b2o$2b2o4bobo2b3o$bo3b5ob2obobo$2bo5bob2o$4bob2o2bobobo!
(Check gen 2)

83bismuth38
Posts: 461
Joined: March 2nd, 2017, 4:23 pm
Location: Still sitting around in Sagittarius A...
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### Re: Oscillator Universality

Saka wrote:The most unuversal rule would be b3/s23 because it has been studied the most
but really, it is incredibly universal, but most likely not the most.

Code: Select all

x = 8, y = 10, rule = B3/S23
3b2o$3b2o$2b3o$4bobo$2obobobo$3bo2bo$2bobo2bo$2bo4bo$2bo4bo$2bo! No football of any dui mauris said that. Sphenocorona Posts: 484 Joined: April 9th, 2013, 11:03 pm ### Re: Oscillator Universality One issue with rule universality is that of defining when a rule has 'more' of some type of object than another when both have infinitely many objects of that type. Importantly, they can't be distinguished by cardinality anymore as the list of all possible [insert object here] has a cardinality of ℵ₀; That is, any infinite set of objects that exist in a rule has exactly the same "amount" of objects as any other, because they can always be mapped one-to-one with the natural numbers (0), 1, 2, 3, 4, ... The reason this is important is because B3/S23 already has multiple infinite series of spaceship velocities alone, though not all of them are necessarily minimal period or such... yet clearly there's a huge number of those we haven't managed to produce at even low multiples of minimal period. There are certainly other ways out of this to define this, but it means it's not exactly straight forward how one should measure this sort of thing in a satisfactory way. A for awesome Posts: 1993 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 Contact: ### Re: Oscillator Universality Sphenocorona wrote:the list of all possible [insert object here] has a cardinality of ℵ₀ Are you sure? It seems like in some rules, some categories of objects may have cardinalities of ℵ₀, but wouldn't cardinalities more commonly be finite or ℵ₁? x₁=ηx V ⃰_η=c²√(Λη) K=(Λu²)/2 Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt) $$x_1=\eta x$$ $$V^*_\eta=c^2\sqrt{\Lambda\eta}$$ $$K=\frac{\Lambda u^2}2$$ $$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$ http://conwaylife.com/wiki/A_for_all Aidan F. Pierce dvgrn Moderator Posts: 6459 Joined: May 17th, 2009, 11:00 pm Location: Madison, WI Contact: ### Re: Oscillator Universality A for awesome wrote: Sphenocorona wrote:the list of all possible [insert object here] has a cardinality of ℵ₀ Are you sure? It seems like in some rules, some categories of objects may have cardinalities of ℵ₀, but wouldn't cardinalities more commonly be finite or ℵ₁? I think you're pretty much stuck with ℵ₀ when you're enumerating objects in any CA rule. Any infinite set of objects that can be ordered by apgcode is going to be ℵ₀, for example. Seems like you can't put any set of CA objects in one-to-one correspondence with real numbers, unless 100% of the objects have infinite population (ugh). Some categories are certainly finite, but by and large they don't seem like the interesting ones. Often you need arbitrary limitations -- "stable reflectors that fit inside a 12x12 bounding box", etc. And the larger context of Sphenocorona's post specifically excludes that case: Sphenocorona wrote:One issue with rule universality is that of defining when a rule has 'more' of some type of object than another when both have infinitely many objects of that type. So, yeah, it seems like you need some kind of measurement of the density of occurrences of interesting objects -- appearance in random soups, maybe, or findability by some kind of standardized search mechanism -- to be able to compare one rule to another. And rules that come out on top via one measurement method may not rank very high if you switch another metric. A for awesome Posts: 1993 Joined: September 13th, 2014, 5:36 pm Location: 0x-1 Contact: ### Re: Oscillator Universality dvgrn wrote:Seems like you can't put any set of CA objects in one-to-one correspondence with real numbers, unless 100% of the objects have infinite population (ugh). Okay, now I guess I understand — this is a pedantic distinction between infinite sets of objects each with finite information content and infinite sets of objects each with infinite information content. I would personally consider infinite objects to fall into the same categories as finite objects, but I can see why someone else wouldn't. x₁=ηx V ⃰_η=c²√(Λη) K=(Λu²)/2 Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt) $$x_1=\eta x$$ $$V^*_\eta=c^2\sqrt{\Lambda\eta}$$ $$K=\frac{\Lambda u^2}2$$ $$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$ http://conwaylife.com/wiki/A_for_all Aidan F. Pierce 83bismuth38 Posts: 461 Joined: March 2nd, 2017, 4:23 pm Location: Still sitting around in Sagittarius A... Contact: ### Re: Oscillator Universality what i meant is objects show up more commonly. not that there are physically 'more' of that object. Code: Select all x = 8, y = 10, rule = B3/S23 3b2o$3b2o$2b3o$4bobo$2obobobo$3bo2bo$2bobo2bo$2bo4bo$2bo4bo$2bo!
No football of any dui mauris said that.

blah
Posts: 248
Joined: April 9th, 2016, 7:22 pm

### Re: Oscillator Universality

I missed a requirement for the rules in which the 'Checkers' oscillator exists. It requires that there is no B0. That halves its commonness, putting it in (1/2^4)2^18 = 16384 lifelike rules (unless you account for B0 rules that have it anyway, by coincidence)
succ