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{{DISPLAYTITLE:Differential Analytic Turing Automata}}

{{DISPLAYTITLE:Differential Analytic Turing Automata}}

==DATA.  Note 1==

==Note 1==

For the purposes of the NKS Forum my aim is to chart a course from general ideas about ''transformational equivalence classes of graphs'' 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 provide some measure of motivation to name the thread after the envisioned end rather than the more homely starting place.

For the purposes of the NKS Forum my aim is to chart a course from

general ideas about "transformational equivalence classes of graphs"

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 provide

some measure of motivation to name the thread after the envisioned end

rather than the more homely starting place.

The basic idea here is that you have a species of graphs

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'' (TECs).

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,

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

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

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.

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,

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

also referred to as the "arithmetic axioms",

more precisely, the "arithmetic initials".

o-----------------------------------------------------------o

o-----------------------------------------------------------o

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| Axiom I_1.` `Distract <---- | ----> Condense` ` ` ` ` ` ` |

| Axiom I_1.` `Distract <---- | ----> Condense` ` ` ` ` ` ` |

o-----------------------------------------------------------o

o-----------------------------------------------------------o

o-----------------------------------------------------------o

o-----------------------------------------------------------o

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| Axiom I_2.` ` `Unfold <---- | ----> Refold` ` ` ` ` ` ` ` |

| Axiom I_2.` ` `Unfold <---- | ----> Refold` ` ` ` ` ` ` ` |

o-----------------------------------------------------------o

o-----------------------------------------------------------o

That should be enough to get started.

That should be enough to get started.

==DATA.  Note 2==

==DATA.  Note 2==