A map of (half of) all OTCA rules!

[Heavpoot]

overlay.png (14.38 KiB) Viewed 3720 times

Each pixel (3x3 square, this has been upscaled by a factor of 3) represents 1 rule!
Purple - Spaceships are proven to not exist, also not explosive (heuristic)
Red - Spaceships are proven to not exist, also explosive (heuristic)
Blue - Spaceships are not disproved from existence, also explosive (heuristic)
Green - Spaceships are not disproved from existence, also not explosive (heuristic)

B0 rules are excluded (due to them having spaceships being hard to theorize about).

How the arrangements of rules works:
Take the X position of the pixel. (divided by 3, due to the upscaling)
Convert it to a 9 bit binary number (e.g 3 becomes 000000011)
Each bit now represents a birth transition (the first bit represents B0, 3 would be B78)
Do the same for the Y position to obtain your rule. For example, B3/S23 can be found at 32, 96 (or due to the upscaling, 96, 228)

The heuristic for whether a rule is explosive or not is not infallible. For example, you may notice a solitary purple dot in the large red area. This is the "fredkin replicator rule", or B1357/S1357. In this rule, every pattern is a replicator, however population experiences regular sharp drops, and the heuristic interprets this as the rule stabilizing (and thus it is not explosive).

Please let me know if you have any suggestions or questions!

[bubblegum]

Code: Select all

x = 1, y = 1, rule = B1357/S1357
o!

This is the replicator rule you mentioned, aptly named "Replicator". Replicator is a rule where everything is a replicator, but you already know that.
So if Replicator (170, 170) is purple, then why not B1357/S02468 (170, 341)?

Code: Select all

x = 1, y = 1, rule = B1357/S02468
o!

[Macbi]

You could use the Gray code so that the rules for adjacent pixels only ever differ by one transition.

EDIT: If you wanted you could also order the bits of the Gray code so that the least significant bits (that change most often) represent the transitions that are least likely to affect the colour of a pixel. That way the rules of the same colour would tend to group together.

[Heavpoot]

I don't know the rules for extending the gray code, nor can I find them, though this would be very interesting!

[Macbi]

Have a look at the 'Converting to and from Gray code' section on that Wikipedia page. You actually only need the very simple BinaryToGray function they give there, which is just n^(n>>1) where ^ is XOR and >>1 is rightshift by 1 (I don't know what you're coding in, but Python would be the same as C). Then for each pixel (x,y), give it the colour that you would have given to (BinaryToGray(x),BinaryToGray(y)).

[Heavpoot]

I wrote a Processing program to rearrange the pixels of the original, and it looks like this:

output.png (19.63 KiB) Viewed 3653 times

Interestingly, everything is more "square". CGOL can be found at 144, 240 (yes, after the upscaling). Here it is shown in the outlined box.

Screenshot_20200319_132620.png (2.92 KiB) Viewed 3653 times

[LaundryPizza03]

I wrote a Processing program to rearrange the pixels of the original, and it looks like this:output.png
Interestingly, everything is more "square". CGOL can be found at 144, 240 (yes, after the upscaling). Here it is shown in the outlined box.Screenshot_20200319_132620.png

How do I navigate this version of the map?

[LaundryPizza03]

Just a few comments.

Spaceships are also known not to exist in rules with B23/S0, S123456, B34/S12345, B345/S1234, and S234567 without B2.

For rules with B0, see this post.

[wzkchem5]

Nice work!
A small suggestion: I've read from somewhere about a method that can differentiate Class 4 (potentially Turing complete) automata from other, more "trivial" ones, though I forgot where the original source was. Generate many pairs of random soups on a torus consisting of M cells, so that each pair of soups differ by only a single cell. Simulate the soups for a sufficiently long period of time, and observe how many cells differ in each pair of soups; we denote the latter number by N. Then for Class 1 automata (every pattern eventually die out), N=0; for Class 2 (only still lives and oscillators), N=O(1); for Class 3 (chaotic), N=O(M); and for Class 4, N obeys a continuous distribution from O(1) to O(M). Is it possible to classify the CAs using this method? This will yield information about the potential universality of the CAs, which I believe is more important than the presence or absence of spaceships or explosion.

[Hunting]

Nice work!
A small suggestion: I've read from somewhere about a method that can differentiate Class 4 (potentially Turing complete) automata from other, more "trivial" ones, though I forgot where the original source was. Generate many pairs of random soups on a torus consisting of M cells, so that each pair of soups differ by only a single cell. Simulate the soups for a sufficiently long period of time, and observe how many cells differ in each pair of soups; we denote the latter number by N. Then for Class 1 automata (every pattern eventually die out), N=0; for Class 2 (only still lives and oscillators), N=O(1); for Class 3 (chaotic), N=O(M); and for Class 4, N obeys a continuous distribution from O(1) to O(M). Is it possible to classify the CAs using this method? This will yield information about the potential universality of the CAs, which I believe is more important than the presence or absence of spaceships or explosion.

Nice method! It is easy to make a script based on that.

I have to disagree on the last statement, though. Most of the interesting rules won't show anything interesting after a random Ctrl-5 soup.

[wzkchem5]

Nice method! It is easy to make a script based on that.

I have to disagree on the last statement, though. Most of the interesting rules won't show anything interesting after a random Ctrl-5 soup.

Yes, but the point is that the method can detect if the CA is Class 4 without needing to construct a universal computer, or even simply a reflector or something. The method just probes whether a small perturbation in the initial condition can generate either small scale, medium scale, or large scale changes in the final pattern. Class 1 and 2 automata do not allow for long-range communication, and you'll see N=O(1); Class 3 automata are so chaotic and experience the butterfly effect, that is even a small perturbation will generate a wholly different final outcome, and you'll see N=O(M). Only in Class 4 automata do you see information flows that can both easily transmit over long distances and easily be stopped, so that you have a broad distribution of N.
Of course, Class 4 phenomena does not imply universality, but as Stephan Wolfram famously pointed out, there is empirically a very strong correlation between the presence of Class 4 phenomena and universality. And that's what I mean by the "potential" universality of the CAs.

[Hunting]

Nice method! It is easy to make a script based on that.

I have to disagree on the last statement, though. Most of the interesting rules won't show anything interesting after a random Ctrl-5 soup.

Yes, but the point is that the method can detect if the CA is Class 4 without needing to construct a universal computer, or even simply a reflector or something. The method just probes whether a small perturbation in the initial condition can generate either small scale, medium scale, or large scale changes in the final pattern. Class 1 and 2 automata do not allow for long-range communication, and you'll see N=O(1); Class 3 automata are so chaotic and experience the butterfly effect, that is even a small perturbation will generate a wholly different final outcome, and you'll see N=O(M). Only in Class 4 automata do you see information flows that can both easily transmit over long distances and easily be stopped, so that you have a broad distribution of N.
Of course, Class 4 phenomena does not imply universality, but as Stephan Wolfram famously pointed out, there is empirically a very strong correlation between the presence of Class 4 phenomena and universality. And that's what I mean by the "potential" universality of the CAs.

Your method will classify Snowflakes, LeapLife, ATPP and Dominoplex as class 2.

[wzkchem5]

Your method will classify Snowflakes, LeapLife, ATPP and Dominoplex as class 2.

Sorry but I'm not familiar with the names of these rules. I just learned about Snowflakes and found that it supports guns; therefore chances are that perturbing a single cell of a soup (that will not produce a gun otherwise) may produce a gun, and the gun can induce changes of N>O(1) cells. Did I miss anything?

[toroidalet]

The problem here is choosing the right scale. For example, Life is class 4 with practical soup sizes, but on extremely impractical scales (2^millions), it becomes class 3 due to the formation of junk-clearing replicators and class-3 metacells. The same thing happens with Snowflakes, which is normally class 2, eventually becomes class 4 (due to guns and stuff), and probably transitions into class 3 as well. There is also the possibility of a rule that appears to be class 4 but turns out to just be a really active class 2 rule.

Also, in my experience class-4-ish rules tend to be pretty boring and difficult to search (I used to look for them based on the idea that they would be interesting). What is more indicative of Turing-completeness is the presence of small spaceships and multiple small high-period oscillators. (Also, we don't actually know any ways for something to be universal without spaceships or signals)

[Hunting]

P. S. Non-explosive rules supports soup-searching, which is usually the most fruitful search method.

[googoIpIex]

This is probably unreasonable, but is there a way to do this with non-totalistic rules?

also, bump

[Heavpoot]

Hello, sorry, I keep forgetting to check the forums!

This is probably unreasonable, but is there a way to do this with non-totalistic rules?

also, bump

Well, technically, but you would need a 2251799813685248x2251799813685248 image (still excluding B0), and that seems very unreasonable (if memory serves, there are 51 survival and 51 birth conditions in INT rules!).

Just a few comments.

Spaceships are also known not to exist in rules with B23/S0, S123456, B34/S12345, B345/S1234, and S234567 without B2.

For rules with B0, see this post.

Alright, thank you! I may post an updated version of this soon

How do I navigate this version of the map?

Very late, sorry!
First of all, you need to convert the X and Y position to binary, from greycode (EDIT: downscale them first!). Then, just follow the instructions on the original post:

Take the X position of the pixel. (divided by 3, due to the upscaling)
Convert it to a 9 bit binary number (e.g 3 becomes 000000011)
Each bit now represents a birth transition (the first bit represents B0, 3 would be B78)
Do the same for the Y position to obtain your rule. For example, B3/S23 can be found at 32, 96 (or due to the upscaling, 96, 228)

[Heavpoot]

UPDATE:
Included more axioms, and all of the B0 axioms (thanks laundrypizza03!) which were very tedious to copy, so hopefully I didn't make any mistakes!

WITHOUT GRAYCODE:

(without graycode)

img.png (27.1 KiB) Viewed 2554 times

WITH GRAYCODE:

(with graycode)

output.png (34.75 KiB) Viewed 2554 times