ONT Re: Differential Analytic Turing Automata

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DATA.  Note 3

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I will draw on those previously advertised resources
of notation and theory as needed, but right now I
sense the need for some concrete examples.

Let's say we have a system that is known by the name of
its state space X and we have a boolean state variable
x : X -> B, where B = {0, 1}.

We observe X for a while, relative to a discrete time frame,
and we write down the following sequence of values for x.

x

0
1
0
1
0
1
0
1
...

"Aha!" we say, and think we see the way of things,
writing down the rule x' = (x), where x' indicates
the state that comes next after x, and (x) is the
negation of x in boolean logic.

Another way to detect patterns is to write out a table
of finite differences.  For this example, we would get:

x dx d2x

0 1 0 ...
1 1 0
0 1 0
1 1 0
0 1 0
1 1 0
0 1
1 
...

And of course, all the higher order differences are 0.

This leads to thinking of X as having an extended state
<x, dx, (d^2x), ..., (d^k)x>, and this sort of language
gives us the facility of describing state transitions in
terms of the various orders of differences.  For example,
the rule x' = (x) can be expressed by the rule dx = 1.

I will leave you to think about
some of the implications of that.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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