Oscillator Universality
 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!
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:
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. :^)
Anyway, this oscillator is pretty universal:
Code: Select all
x = 0, y = 0, rule = B3/S23
!
You never said it couldn't be trivial. :^)
succ

 Posts: 796
 Joined: May 30th, 2016, 8:47 pm
 Location: Milky Way Galaxy: Planet Earth
Re: Oscillator Universality
Except a so called "p1 oscillator" is actually a still life and nothing more.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: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).Code: Select all
x = 0, y = 0, rule = B3/S23 !
You never said it couldn't be trivial. :^)
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$8.2A!
Re: Oscillator Universality
It's still a p2 oscillator in 25% of rules. But in all seriousness, I think 'Checkers' is pretty universal:Gamedziner wrote:Except a so called "p1 oscillator" is actually a still life and nothing more.
Code: Select all
x = 2, y = 2, rule = B2/S
bo$o!
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$16o!
succ
 83bismuth38
 Posts: 466
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Re: Oscillator Universality
all credit to him for these. they were on an unrelated subject.A for awesome wrote:A single dot is a p1 oscillator requiring S0 and prohibiting B01. A block is a p1 oscillator requiring S3a and prohibiting B01c2a. Both work in 1/16 of all rules, whereas the duoplet works in only 1/32 of all rules.BlinkerSpawn wrote:The most universal oscillator would be the p2 doublet, which only requires B2e and prohibits B01/S1c.
I recognize that p1 oscillators might not count, though.
EDIT: For p3 oscillators the best I can find is requiring B2i3a4e/S2c and prohibiting B012ce4c/S03i4e:(1/8192)Code: Select all
x = 3, y = 3, rule = B2i3a4e/S2c bo$obo$bo!
For p4s it requires B2k3a/S0 and prohibits B012a/S2a3j:(1/512)Code: Select all
x = 3, y = 2, rule = B2k3a/S0 o$2bo!
Code: Select all
x = 8, y = 10, rule = B3/S23
3b2o$3b2o$2b3o$4bobo$2obobobo$3bo2bo$2bobo2bo$2bo4bo$2bo4bo$2bo!
 83bismuth38
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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!

 Posts: 796
 Joined: May 30th, 2016, 8:47 pm
 Location: Milky Way Galaxy: Planet Earth
Re: Oscillator Universality
In terms of replicators, the Replicator rule definitely takes the cake.83bismuth38 wrote: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 = 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: 466
 Joined: March 2nd, 2017, 4:23 pm
 Location: Still sitting around in Sagittarius A...
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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$2bo!
 Saka
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Re: Oscillator Universality
The most unuversal rule would be b3/s23 because it has been studied the most
Code: Select all
o3b2ob2obo3b2o2b2o$bo3b2obob3o3bo2bo$2bo2b3o5b3ob4o$3o3bo2bo2b3o3b3o$
4bo4bobo4bo$2o2b2o2b4obo2bo3bo$2ob4o3bo2bo2bo2bo$b2o3bobob2o$3bobobo5b
obobobo$3bobobob2o3bo2bobo!
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 83bismuth38
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Re: Oscillator Universality
but really, it is incredibly universal, but most likely not the most.Saka wrote:The most unuversal rule would be b3/s23 because it has been studied the most
Code: Select all
x = 8, y = 10, rule = B3/S23
3b2o$3b2o$2b3o$4bobo$2obobobo$3bo2bo$2bobo2bo$2bo4bo$2bo4bo$2bo!

 Posts: 485
 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 onetoone 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.
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
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Re: Oscillator Universality
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?Sphenocorona wrote:the list of all possible [insert object here] has a cardinality of ℵ₀
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
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
Re: Oscillator Universality
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 onetoone correspondence with real numbers, unless 100% of the objects have infinite population (ugh).A for awesome wrote: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?Sphenocorona wrote:the list of all possible [insert object here] has a cardinality of ℵ₀
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:
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.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.
 A for awesome
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Re: Oscillator Universality
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.dvgrn wrote:Seems like you can't put any set of CA objects in onetoone correspondence with real numbers, unless 100% of the objects have infinite population (ugh).
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
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: 466
 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!
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