Oscillator Universality

[83bismuth38]

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!

[blah]

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. :^)

[Gamedziner]

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.

[blah]

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!

[83bismuth38]

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.

[83bismuth38]

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

[Gamedziner]

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!

[83bismuth38]

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.

[Saka]

The most unuversal rule would be b3/s23 because it has been studied the most

[83bismuth38]

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.

[Sphenocorona]

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]

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?

[dvgrn]

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:

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]

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.

[83bismuth38]

what i meant is objects show up more commonly. not that there are physically 'more' of that object.

[blah]

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)