Wednesday, March 16, 2011

Is there something in-between?

Lately we have been goofing around a LOT with Cellular Automata and Agent-Based simulations.  Of course we're working on the Genetic Algorithms cases study, but some students have asked about other applications of these really interesting parts of computational science.

So today I was reading Cities and Complexity and came to the part where they shift from strict rule-based, deterministic cellular automata simulations, to simulations that include random elements.  I immediately remembered something that I had thought of a while back while working on some of Wolfram's one-dimensional CA studies (and then, for some reason, I completely forgot about it again until this morning).  Here's the thought:

When we think of a typical CA simulation, we use a single set of rules, like, say, Rule 110 for the one-dimensional CA that Wolfram had studied.  That rule is a strict one-to-one mapping of situation to response, so there is NO randomness at all. 

Naturally, one gets curious about what adding random elements would do to the patterns.  But then I thought of something else, and it really got me to thinking about determinism and randomness.  Let me just show you what I was thinking:

Rule 30 from Wolfram is the base 10 name for this binary mapping:

0 0 0 1 1 1 1 0

Now of course that tells each cell what to do (whether to be "on" or "off" in the next phase) for each of the eight possible situations it could be in. 

So, if we wanted to, we could randomize it by replacing one of those digits with a coin flip (f):

0 0 f 1 1 1 1 0

Here, in the third digit from the left, we have switched from "always turn off" to "turn off with a 50% chance. "  By that we mean something like, "flip a coin, dude, and heads you are on, tails you are off."

But then I remembered something I had toyed around with when thinking up some AI patterns and state machine stuff for a simple game I was working on.  What if, instead of something like "choose 1 80% of the time, chose 0 the other 20%" was handled differently?  What if we instead put an array of numbers into that particular position, with a string of results in it:

0 0 [ 1 1 1 1 0 ] 1 1 1 1 0

Now, instead of an 80% CHANCE of choosing 1, we simply tell the agent that she should respond with the number in the array--and she should increment by one each successive turn.  Meaning, using the example above, when the 3rd position in the array first comes up, she chooses 1.  She does that the 2nd, 3rd, and 4th time, also.  However, the fifth time she chooses 0.  After that she simply loops around and starts over at the beginning.  So you can see that she will choose 1 exactly 80% of the time, but there's no chance involved.  It's completely deterministic, and yet it takes the whole question of one-dimensional CA to a whole new level of, well, complexity.  Or does it?  I really haven't had much time to think about this in detail, but it seems strangely different than randomness. 

I guess what it really makes me think about is how attractive the finite set of 256 rules that Wolfram studied seemed to me at first.  How ... cozy it seemed that the structure was so, well, structured.  Now I'm thinking that there's an infinity of rules--all strictly deterministic--that could be looked at for good ol' one-dimensional CA. 

Time to hit the books....

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