ONT Re: Differential Analytic Turing Automata

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DATA.  Note 6

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One more example may serve to suggest just how much dynamic
complexity can be built on a universe of discourse that has
but a single logical feature at its base.

But first, let me introduce a few more elements of general
notation that I'll be using to describe finite dimensional
universes of discourse and the qualitative dynamics that
we envision occurring in them.

Let !X! = {x_1, ..., x_n} be the "alphabet" of logical "features"
or "variables" that we use to describe the n-dimensional universe
of discourse X% = [!X!] = [x_1, ..., x_n].  Picturesquely viewed,
one may think of a venn diagram with n overlapping "circles" that
are labeled with the feature names in the set !X!.  Staying with
this picture, one visualizes the universe of discourse X% = [!X!]
as having two layers:  (1) the set X = <|!X!|> = <|x_1, ..., x_n|>
of "points" or "cells" -- in another sense of the word than when
we speak of "cellular automata" -- (2) the set X^ = (X -> B) of
"propositions", boolean-valued functions, or maps from X to B.
Thus, we may speak of the universe of discourse X% as being
an ordered pair <X, X^>, with 2^n points in the underlying
space X and 2^(2^n) propositions in the function space X^.
That's just life in Ascii-land.  It ain't Chicago.

A more complete table setting out these notations can be found here:

DLOG D2.  http://stderr.org/pipermail/inquiry/2003-May/000480.html

Now, to the Example.

Once again, let us begin with a 1-feature alphabet !X! = {x_1} = {x}.
In the discussion that follows I will consider a class of trajectories
that are ruled by the constraint that d^k.x = 0 for all k greater than
some fixed m, and I will indulge in the use of some picturesque speech
to describes salient classes of such curves.  Given this finite order
condition, there is a highest order non-zero difference d^m.x that is
exhibited at each point in the course of any determinate trajectory.
Relative to any point of the corresponding orbit or curve, let us
call this highest order differential feature d^m.x the "drive"
at that point.  Curves of constant drive d^m.x are then
referred to as "m^th gear curves".

One additional piece of notation will be needed here.
Starting from the base alphabet !X! = {x}, we define
and notate E^j.!X! = {x, d^1.x, d^2.x, ..., d^j.x}
as the "j^th order extended alphabet over !X!".

Let us now consider the family of 4^th gear curves through
the extended space E^4.X = <|x, dx, d^2.x, d^3.x, d^4.x|>.
These are the trajectories that are generated subject to
the law d^4.x = 1, where it is understood in making such
a statement that all higher order differences equal 0.

Since d^4.x and all higher order d^j.x are fixed, the entire dynamics
can be plotted in the extended space E^3.X = <|x, dx, d^2.x, d^3.x|>.
Thus, there is just enough room in a planar venn diagram to plot all
of these orbits and to show how they partition the points of E^3.X.
As it turns out, there are exactly two possible orbits, of eight
points each, as illustrated in Figures 16-a and 16-b.  See here:

DLOG D23.  http://stderr.org/pipermail/inquiry/2003-May/000502.html

Zoom^4 ...

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