Differential Analytic Turing Automata

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DATA.  Note 1

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For the purposes of the NKS Forum my aim is to chart a course from general
ideas about "transformational equivalence classes of graphs" (TECOG's) to
a notion of "differential analytic turing automata" (DATA).  It may be
a while before we get within sight of that goal, but it will help me
to name the thread after the envisioned end rather than the homely
starting place.

The basic idea here is that you have a species of graphs
and a set of transformation rules that take you from one
graph to another -- and back again, as I'm only thinking
of equational rules -- and this partitions the species of
graphs into "transformational equivalence classes" (TEC's).

There are many interesting excursions to be had here,
but I will focus mainly on logical applications, and
and so the TEC's I talk about will almost always have
the character of "logical equivalence classes" (LEC's).

An example that will figure heavily in the sequel
is given by rooted trees as the species of graphs
and a pair of equational transformation rules that
derive from the graphical calculi of C.S. Peirce,
as revived and extended by George Spencer Brown.

Here are the fundamental transformation rules,
also referred to as the "arithmetic axioms",
more precisely, the "arithmetic initials".

o-----------------------------------------------------------o
|                                                           |
|                 o   o                 o                   |
|                  \ /                  |                   |
|                   @         =         @                   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
|                ( ) ( )      =        ( )                  |
|                                                           |
o-----------------------------------------------------------o
| Axiom I_1.   Distract <---- | ----> Condense              |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
|                                                           |
|                   o                                       |
|                   |                                       |
|                   o                                       |
|                   |                                       |
|                   @         =         @                   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
|                 (( ))       =                             |
|                                                           |
o-----------------------------------------------------------o
| Axiom I_2.     Unfold <---- | ----> Refold                |
o-----------------------------------------------------------o

That should be enough to get started.

Jon Awbrey

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