Beyond Wolfram's Classification

[Yoel]

As it is well known, Stephen Wolfram defined four main categories of CA. His classification is not rigorous and relies on human intuition. However, most rules are easy to put into one of the 4 classes.

If we define a rule that allows coexistence of several "sub-behaviors", it may exhibit, depending in the pattern, the behavior of 2 or or more classes (the basic classification remains true though for a particular subset of patterns).

One example I already have demonstrated here.

Two more examples. B3/S0123 is an exploding rule with average density roughly twice higher than Seeds. If we transform into a 3-color 4-state cyclical rule, it still explodes ("islands" of each color independently). However, if we augment the 3-color version with the following additional rule, patterns tend to stabilize after a color collision and random soups usually stabilize rather quickly:

The additional rule: a new cell of a certain color is born, of surrounded by 2 cells of a different color and 1 cell of the third color, T3a12c21c-0n1n2n3n is m rule table generator's notation.

Code: Select all

x = 344, y = 101, rule = Triple_Swamps
7$11.3C$13.C$11.C.C$12.2C80$328.B!

Note that the stabilization process, if started after a sufficient 1 color chaotic growth, produces a structure that resembles very realistic maps of lakes and marshes (compare the output of this pattern to the map of Northern Canada or Finland). Better to see it in Golly; A nice map emerges after 15000 generations.

If we transform the standard Conway's Game of Life into a 3-color cyclical rule, we get a chaotic behavior of approximately the same density as Seeds, if we add the following rule and make two colors collide:

A cell of color A is born, if surrounded by 1 cell of the color A and 2 cells of another color:

Code: Select all

x = 18, y = 7, rule = Triple_Behavior_Life
2$2.A12.B$3.2A9.B$2.2A10.3B!

Now, this rule table actually contains a couple of other rules enacted when all 3 color collide, which produces a slowly growing homogeneous stable circular pattern (it's stable in itself, but grows due to the exploding 2-color collision on the edges):

1. A cell of color R is born, if surrounded by 1 R, 2 G and 2 B cells. Since the rule is defined cyclically, it implies that G is born, if surrounded by 1 G, 2 B and 2 R; B is born, if surrounded by 1 B, 2 R and 2 G.

2. A cell survives, if surrounded by 2 or 3 neighbors of any color (3-color extension of the original Life rule)

3. A cell survives, if surrounded by 4,5,6,7 or 8 neighbors, if all 3 colors are present among these neighbors.

Code: Select all

x = 21, y = 22, rule = Triple_Behavior_Life
2$6.A$7.A9.B.B$5.3A9.2B$18.B11$2.3C$4.C$3.C!

There are many other types of behavior that I've encountered while playing with such rules. For example, some rules produce methuselahs of enormous longevity (10s of millions of generations) from random soups, because color collisions produce glider emitting "kernels" (not guns. kind of chaotic things) that constantly revitalize the pattern.

[Hunting]

To me this is still under Wolfram's Classification, TripleSwamps* as a chaotic rule, TBL as a chaotic rule. Just that an infinite number of patterns happens to be outside of this behaviour. They are classified with unbiased random soups AFAIK. It's easy to create infinitely many insta-dies in B34/S34, but it's still class 3.

*In case you wonder, TBL is not causing any trouble while TS can also mean TwilightSparkle.

[Yoel]

To me this is still under Wolfram's Classification, TripleSwamps* as a chaotic rule, TBL as a chaotic rule. Just that an infinite number of patterns happens to be outside of this behaviour. They are classified with unbiased random soups AFAIK. It's easy to create infinitely many insta-dies in B34/S34, but it's still class 3.

*In case you wonder, TBL is not causing any trouble while TS can also mean TwilightSparkle.

Random soups in Triple_Swamps tend to stabilize very quickly. I ran a test of 100 soups, 100*100 each, filled to 33%. 72 soups stabilized very quickly (50-500 generations), 7 stabilized slowly (1000-5000 generations), 6 or 7 definitely exploded due to an emerging breeder or a very messy puffer, and the rest - unknown. They may eventually stabilize or not, I didn't have patience to watch more than 20000 generations. Anyhow, over 70% random soups, so-called overwhelming majority, clearly exhibit Class 2 pattern (the patterns are stable, but there is some p2 blinking going on), while 1 color soups are almost 100% chaotic Class 3. If we disregard 1-color soups as biased, then it's Class 2 bordering in a substational minority of cases on Class 3.

There is a small, but significant probability of at least 2 natural puffers in B3/S23 too, as one can see in apgsearch results. But we judge by majority, don't we?

I may agree that TBL is chaotic, although I find its behavior of having a stable kernel surrounded by slowly growing "puddles" of chaos not exactly fitting into Wolfram's scheme. Anyhow, it's fun to play with such "multi-layer" rules.

I'll check out TwilightSparkle, Never heard of this rule, thank you for informing me about it!

EDIT:

Eeeh... is TwilightSparkle a rule or one of your usernames?

[bubblegum]

Eeeh... is TwilightSparkle a rule or one of your usernames?

Twilight Sparkle comes from MLP (My Little Pony), something Hunting is weirdly obsessed with. And no, no one has ever referred to Twilight Sparkle with TS.

(REIMAGINE :the game: contained a joke where Edward Cullen was also considered to be "Twilight Sparkle" but it's not that)

[Hunting]

To me this is still under Wolfram's Classification, TripleSwamps* as a chaotic rule, TBL as a chaotic rule. Just that an infinite number of patterns happens to be outside of this behaviour. They are classified with unbiased random soups AFAIK. It's easy to create infinitely many insta-dies in B34/S34, but it's still class 3.

*In case you wonder, TBL is not causing any trouble while TS can also mean TwilightSparkle.

Random soups in Triple_Swamps tend to stabilize very quickly.

Eeeh... is TwilightSparkle a rule or one of your usernames?

1. Sorry, I meant oscillating (class 2).

2. Both. I suspect you might be an active member in MAWP...

Twilight Sparkle comes from MLP (My Little Pony), something Hunting is weirdly obsessed with. And no, no one has ever referred to Twilight Sparkle with TS.

Weirdly? I don't think so.

It is very common in Chinese MLP community AFAIK.