[rule30: wrap? Or no wrap?] - A New Kind of Science: The NKS Forum

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rule30: wrap? Or no wrap?

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Posted by: Philip Ronald Dutton

Rule 30, when run, produces a shifting effect when you do not 'wrap' the edges after which it settles into a vertical line (either on or off) pattern. What are the implications of wrapping or not wrapping in general? (I was using MJCell applet)


My intuition, first, tells me that since we are limited to a finite amount of cells (within our computers) that we should not wrap our CA edges.


Posted by: Jason Cawley

In general, systems of limited size go to class 2 behavior. There are a finite number of possible states, and the systems are deterministic, so if they revisit a state hit once before, they go through the same sequence (that follows from that state) again. If the system is of any appreciable size, though, this period can be extremely long, so that it is not encountered in practice.

Sometimes you will see it with certain widths of initial conditions but not with nearby ones, for essentially number-theoretic reasons. E.g. a width that has lots of divisors has more potential sub-cycles than a system whose width is prime. In general, the more regularity there is in the underlying behavior, the easier it is to encounter subcycles that "go periodic" or simple, early enough to see them (as opposed to the astronomical maximum possible repetition periods).

In the case of rule 30, the left edge of the growing pattern from a single black cell is quite regular. There is a texture transition line that moves to the left at less than "light speed", to the right of which and below, the behavior is class 3, while to the left and above, it is a set of regular stripes. When wrapped, this pattern hits the random area, but typically gets swallowed into that randomness. The system will eventually repeat, but the repetition period can be quite long. And it can vary significantly with the exact width used. There is a plot of the repetition period of rule 30 from a single black cell with wrapped boundary, as a function of the width, on page 260 of the NKS book -

http://www.wolframscience.com/nksonline/page-260

There is an early portion of that graph that is flat, level at 1 - that is probably what you noticed. As you can see, it doesn't stay that way, and the repetition period can get quite large. It might also give a sense of how many cases were examined for the production of the NKS book, and the importance of exhaustive search to NKS methods generally.

If you want to avoid this issue, it is easy enough. Just make the pattern width twice the number of steps you evaluate, or more - well, 2t+1 if t is the number of steps. Since the pattern can't grow more than one cell in either direction at each step, it won't have time to wrap. If you want to investigate what happens when left moving structures hit right moving ones in a given rule, sometimes you'll want to deliberately allow wrapping by keeping the width small relative to the number of steps evaluated, to watch what it does.

It is easy enough to experiment with these relations using NKSX. Just fiddle with the width and step parameters, for a variety of rules. A relevant section of the NKS book to understand what can happen is the section on class 2 behavior and systems of limited size, starting on page 255 -

http://www.wolframscience.com/nksonline/page-255


Posted by: Philip Ronald Dutton

Lately when looking at pictures of the rule 30 CA I can not focus on much other than the interesting mix of regularity (left side) and randomness (right side) in the unbounded evolution (before we begin to wrap the left side to the right side). It seems the norm is for us to wrap the CA at our specified bound (length, or row length, etc.).

Has anyone simply wrapped the CA within the context of unbounded growth? In other words, pretend there is an infinite grid but go ahead and wrap the sides so that information can flow from the sides.

What would be the result? Would the CA even be able to progress from an initial single black cell? How would the pattern be affected?

I haven't thought too much about this but surely it would be interesting to see the random portion interacting with that regular portion (in an additional way)???


Posted by: RichMacDonald

For anyone still interested in rule 30 wrap-around and/or the effect of cell width on repeatability, you might be interested in my notes at http://www.clevercaboose.com/rich/c...ith_rule30.pdf.

Basically, I duplicated Wolfram's rule 30 plot on p260 ( http://www.wolframscience.com/nksonline/page-260), but also ran some more random studies. I found that repeat periods can actually be very low




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