ONT Re: Differential Analytic Turing Automata

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DATA.  Note 12

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Way back in DATA Note 3, we guessed, or "abduced",
as a line of logicians from Aristotle to Peirce
and beyond would say, the form of a rule that
adequately accounts for the finite protocol
of states that we observed the system X
pass through, as spied in the light of
its boolean state variable x : X -> B,
and that rule is well-formulated in
any of these styles of notation:

1.1.  f : B -> B such that f : x ~> (x)

1.2.  x' = (x)

1.3.  x := (x)

1.4.  dx =  1

In the current example, having read the manual first,
I guess, we already know in advance the program that
generates the state transitions, and it is a rule of
the following equivalent and easily derivable forms:

2.1.  F : B^2 -> B^2 such that F : <u, v> ~> <((u)(v)), ((u, v))>

2.2.  u' = ((u)(v)),  v' = ((u, v))

2.3.  u := ((u)(v)),  v := ((u, v))

2.4.  ???

Well, the last one is not such a fall off the log,
but that is exactly the purpose for which we have
been developing all of the foregoing machinations.

Here is what I got when I just went ahead and
calculated the finite differences willy-nilly:

Incipit 1.  <u, v> = <0, 0> 

d d | d d | d d | d d | d d | ...
0 0 | 1 1 | 2 2 | 3 3 | 4 4 | ...
u v | u v | u v | u v | u v | ...
----+-----+-----+-----+-----+-----
0 0 | 0 1 | 1 0 | 0 1 | 1 0 | ...
0 1 | 1 1 | 1 1 | 1 1 | 1 1 | ...
1 0 | 0 0 | 0 0 | 0 0 | 0 0 | ...
1 0 | 0 0 | 0 0 | 0 0 | 0 0 | ...
" " | " " | " " | " " | " " |

Incipit 2.  <u, v> = <1, 1> 

d d | d d | d d | d d |
0 0 | 1 1 | 2 2 | 3 3 |
u v | u v | u v | u v |
----+-----+-----+-----+
1 1 | 0 0 | 0 0 | = = |
1 1 | 0 0 | 0 0 | = = |
" " | " " | " " | " " |

To be honest, I have never thought of trying
to hack the problem in such a brute-force way
until just now, and so I know enough to expect
a not unappreciable probability of error about
all that I have taken the risk of writing here,
but let me forge ahead and see what I can see.

What we are looking for is, well, one rule to rule them all,
that is, a rule that works at every state and at every time.

What we see at first sight in the tables above are patterns
of differential features that attach to the states in each
orbit of the dynamics.  Looked at locally to these orbits,
the isolated fixed point at <1, 1> is no problem, as the
rule du = dv = 0 describes it pithily enough.  When it
comes to the other orbit, the first thing that comes
to mind is to write out the law du = v, dv = (u).

I am going to take a nap to clear my head.

Jon Awbrey

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