A New Kind of Science: The NKS Forum > Pure NKS > Getting Class 3 and 4 Behavior from a Class 2 Three Color Rule
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Registered: Jan 2004
Posts: 350

Getting Class 3 and 4 Behavior from a Class 2 Three Color Rule

On page 262 of the NKS book there are depicted patterns produced by elementary rule 22 starting from random initial conditions and from an initial condition containing a single black cell.

The single black cell pattern has a simple nested class 2 structure.
The other has the random behavior of a class 3 cellular automaton.

Here is an example of a modulo 3 algebraic rule, Mod[2(q+r-pr+qs), 3], that has a simple nested class 2 structure when a single value 2 cell is used as initial conditions. See figure 1 in the attached gif.

But what if the initial conditions are neither simple nor random?
In this example they are base 3 palindrome digit expansions of the following 81 integers:

{0,81,162,738,819,900,1476,1557,1638,2460,2541,2622,3198,3279,3360,3936,4017,4098,4920,5001,5082,5658,5739,5820,6396,6477,6558,6562,6643,6724,7300,7381,7462,8038,8119,8200,9022,9103,9184,9760,9841,9922,10498,10579,10660,11482,11563,11644,12220,12301,12382,12958,13039,13120,13124,13205,13286,13862,13943,14024,14600,14681,14762,15584,15665,15746,16322,16403,16484,17060,17141,17222,18044,18125,18206,18782,18863,18944,19520,19601,19682}.

The result is class 4 sporadic bifurcating behavior on a background of class 3 random triangular structures. See figure 2 in the attached gif.

After 50,000 steps the two classes of behavior were still present, although for brief periods the class 4 behavior appeared to have died out only to subsequently come alive once again.

The colors in the graphs are: {0->Cornsilk, 1->CornflowerBlue, 2->Coral}.

Lawrence J. Thaden has attached this image:

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05-07-2004 01:06 AM
Tony Smith
Meme Media
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Registered: Oct 2003
Posts: 167

I'd say it's a Class 4 Rule

Lawrence, Class 4 rules produce Class 1/2 or Class 3 outcomes, depending sensitively on initial conditions. In fact the closest thing we have to a Class 4 only outcome is not stopping, and that is more of a non outcome.

The cute little animation I have attached is an undisputably Class 4 rule, Conway's Life constrained to a circumference 14 tube, with a particular (quite simple) set of initial conditions.

So I would instictively classify your "modulo 3 algebraic rule, Mod[2(q+r-pr+qs), 3]" as Class 4, with its behaviour from simple starting conditions exactly what we might expect.

Tony Smith has attached this image:

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05-07-2004 03:37 AM

Registered: Jan 2004
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Tony,

Let me see if I got this straight.

So classification is from observation of behavior.

And rules are assigned classification numbers according to how complex a behavior is possible given the "right" initial conditions.

Is that it?

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05-07-2004 01:02 PM
Tony Smith
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I hate quoting myself, but

Lawrence, back in 1983 Wolfram defined Classes 1-4 based on his exhaustive study of the 256 nearest neighbour 1 dimensional cellular automata and his realisation that this classification appeared to extend sensibly to other complex systems.

1D CAs are naturally shown with time down the page, so you can see their evolution at a glance. For 2D CAs and many other systems, it is usually easiest to print a snapshot at a given time which does not capture the dynamics of the process. Of course computers make it possible to view animations which can't be done justice in print.

I wrote a web paper some months back which is very much concerned with better appreciating the four Wolfram classes, and will just quote a couple of paragraphs here to encourage you to try to see past my often convoluted sentence structures and the site's regretable colour scheme:

We are naturally inclined to judge a process by its outcomes because they are accessible whereas the process is at best something we can descibe as an algorithm. The problem here is to understand the relationship between Class 1 through 4 processes and the corresponding outcomes which can really only be described as Class 1/2 (stable, cyclic, nested) or Class 3 (random). There is no corresponding Class 4 outcome because a distinguishing feature of Class 4 processes is that they can each produce many different outcomes given various input. Yet we most easily judge such processes by their outcomes, so the boundary zone can appear elusive.

(...)

It only makes sense to interpret the definition of Class 4 inclusively. When at least one rule within a particular cellular automata form is known to be Class 4, it makes sense to say that that cellular automata form is also Class 4 without in any way implying that other rules of that form will be Class 4. By extension, we know that Life is Class 4, not because all or many data input values lead to complex behaviour but because some do. Exploration of Class 4 involves finding data input values which produce complex behaviour and using what is found to hypothesise and test for even more complex behaviour.

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Last edited by Tony Smith on 02-17-2005 at 12:19 AM

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05-07-2004 02:15 PM

Registered: Jan 2004
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Why can't things just be simple?

Tony,

Someone once said that inadequacies tend to their own reversal.

I am reminded of that when I think about the 4 classes. There must be a more satisfying way to describe and explain the processes and behavior.

I read your paper, Beating the Retreat. The ideas come through even though hampered by the framework and references to the 4 classes.

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05-07-2004 05:34 PM
Tony Smith
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I know it was a rhetorical question

Why can't things just be simple?
Lawrence, if things were "simple" we would not be having this conversation, we would not be having any conversation, we would not even be.

Our own complexity is a symptom and an instance of the universal complexity that is required for anything interesting to even exist. What was that old Chinese curse?

It is almost paradoxical that even complex adaptive systems such as ourselves can only really adapt to patterns which are non-complex, "Class 1/2" even if you don't like that terminology.

Even if we were to accept a watered down version of Wolfram's PCE, Class 4 => universal, that still begs the question as to whether universal => adaptive, and I suspect that question is the real holy grail of those who wander these parts.

Let me add a postscript, seeing I only posted the above a minute ago, but I was just reminded that the NKS book does indeed provide detailed discussion of the four classes.

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Last edited by Tony Smith on 05-08-2004 at 01:46 AM

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05-08-2004 01:39 AM

Registered: Jan 2004
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More on Mod[2(q+r-pr+qs), 3]

Given the right initial conditions the modulo 3 algebraic rule, Mod[2(q+r-pr+qs), 3], evolves a cellular automaton that has a horizontal symmetric structure.

The color assigned to 0 values appears symmetric as well. But the colors assigned to values 1 and 2 take up positions on opposite sides of the symmetric structure.

Initial conditions are a replication of the pair (1, 2) together with one or more adjacent zeros inserted anywhere on the line of cells.

In the attached notebook there are graphs from evolutions that started with a single zero followed by 67 pairs of (1, 2).

The variables in the rule were assigned values from the initial conditions as follows:

p assigned from left next nearest neighbor,
q assigned from left nearest neighbor,
r assigned from right nearest neighbor,
s assigned from right next nearest neighbor.

The colors in the graphs are: {0->Cornsilk, 1->CornflowerBlue, 2->Coral}.

After each evolution of 256 steps the initial conditions were rotated right 5 positions and then the evolution was repeated until there were a total of 27 graphs.

If you download the attached notebook and animate the graphs you will see a horizontally symmetric image shifting to the right and wrapping around on itself endlessly.

Attachment: rule 720 rotateright initial conditions.zip

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05-10-2004 05:28 AM

Registered: Jan 2004
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Attached is the Mathematica notebook with code to make the graphs referred in the previous reply. Try changing the initial conditions to get other symmetric structures.

Attachment: marquee cas.nb