[Alternating Rules] - A New Kind of Science: The NKS Forum

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Alternating Rules

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Posted by: Alastair Hewitt

Has anyone generated CAs with alternating rules? I was interested to see what would happen if you did one line with rule 110 and then the next with rule 30.

I hacked up a small Java program that takes a list of rules and applies them in sequence. The above example ended up looking the same as the original rule 30; in fact mixing rule 30 into anything makes it look like rule 30!

I've attached a 3000 step CA generated using 24 rule 110s followed by a single rule 30 (then repeated 120 times). It looks very much like rule 30, but with the 110 shape. I'll try some more and see if I find any interesting hybrids.


Posted by: Alastair Hewitt

Try attachment again



Posted by: Jason Cawley

Yes certainly. Cyclic rules, the much more general case of "GCAs" (CAs with global control, which switch the rule applied depending on some global variable of the evolution up to that point), and "ICAs" (interactive CAs, which are "told" which rule to use at each step, out of their rule list, by a global "bit" fed to them at each step) - have all been investigated. Look for threads on ICAs in particular. Programming these is straightforward in Mathematica, a matter of a couple of lines. You make a step function and then use a FoldList to run them together.


Posted by: Alastair Hewitt

Thanks Jason, I'll take a closer look. I was interested in using a class 3 CA like rule 30 to select between two rules to generate a highly random pattern.

I also attached a 110 followed by 124 (the right hand version of 110). You can go the other way (124 then 110), but it is just the mirror image. It looks like a pure class 3, basically a rule 30 but the steeper edges.


Posted by: Jesse Nochella

Neat!

I like these pictures. I didn't guess the case for either the 110-30 or the 110-124 combination. Thanks Alastair.

This helps shed a little more light on why there so many class 3 rules in higher dimensions. These ICA's here can be viewed as a color-flattened slice of a two-dimensional rule that in effect applies two separate one-dimensional rules which, say in the case of the 110-124 combination, happen to be adjacent to eachother and feed off of eachother's input on alternating steps. It would look something like this, with 110's color being "1" and 124's color being "2":

0000 0000 0000 0020 0100
0000 0020 0100 0000 0000
0100 0020 0000 0000 0000 ...
0000 0000 0100 0020 0100
0000 0000 0000 0000 0100

So with all their forms they might actually be common enough to find in nature if we know what we're looking for.

That class 3 world is getting pretty big. I wonder how far we can go in dividing those rules that lay in the domain.

As a generalization, any CA setup specifies slices of CA's with greater range, color, and/or dimension. So there is no setup that doesn't have to do with some part of some ordinary rule.

Hmm... I wonder how easy, in nature this is to do. If in some case behavior is all that's needed to achieve something, is it then easier for, say some genetic program, to just change the rule instead of the conditions it's working on?


Posted by: Jason Cawley

Mathematica code for the general case can be found on the following thread.

http://forum.wolframscience.com/sho...s=&threadid=500

I'll see about posting a notebook with all the necessary functions tomorrow. I've got hundreds of megabytes on these things.


Posted by: Jason Cawley

Jesse, one fun question to consider is point mutation paths through a rule space being selected according to some characteristic, that is itself only a perhaps weak reflection of the underlying varying thing. This is a sort of minimal idealization of evolutionary processes, noticing that typically selection can't grab on to the actual microstate driving the system, only to some crude symptom of it.

So e.g. you can look at the ECA and imagine trying to select for density. A rule that goes to all black after one step is clearly "fittest" on this measure. If I start with some smaller population of rules and just test their density from random initials after n steps, and keep whichever ones did better than average or above some threshold, and occasionally flip one bit in their rule table, will I "walk" through the space of rules to the "all black" rule, and have it take over the population? Or, if that is an easy task, what if I try a harder one, like - get the most level distribution of block frequences, or, get an average density as close as possible to 0.6?

You could do the same thing with ICAs by treating their interaction condition as a varying environment. Or, more naturally for the ICA set up, you can look at intelligent adaptation by keeping the ICA rule fixed (essentially that means, having a limited, fixed set of rules to select among), and making the point mutations to the interaction condition (like choosing a different "move vector" in game theory - on step 8 I'll "use" rule 45...)


Posted by: Jason Cawley

Here is a notebook to expore the basic ICAs (made out of 2 ECAs). Of course the set up is general, but these are a reasonable place to start.

To get an alternating rule, just feed them a periodic interaction condition. There is a function in the notebook that builds those automatically, for any period (others that use random ones with any probability etc).




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