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# Differential Analytic Turing Automata • Overview

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Author: Jon Awbrey

The task ahead is to chart a course from general ideas about transformational equivalence classes of graphs to a comprehensive concept of differential analytic turing automata.  Getting within sight of that goal will take some time but I thought it made for a better measure of motivation to name the thread after its envisioned end instead of its more prosaic starting place.

The basic idea is as follows.  One has a set ${\displaystyle {\mathcal {G}}}$ of graphs and a set ${\displaystyle {\mathcal {T}}}$ of transformation rules, and each rule ${\displaystyle \mathrm {t} \in {\mathcal {T}}}$ has the effect of transforming graphs into graphs, ${\displaystyle \mathrm {t} :{\mathcal {G}}\to {\mathcal {G}}.}$  In the cases we shall be studying this set of transformation rules partitions the set of graphs into transformational equivalence classes (TECs).

There are many interesting excursions to be had from this point but I will be focusing on logical applications, so the transformational equivalence classes of interest here will almost always have the character of logical equivalence classes (LECs).

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

Here are the fundamental transformation rules, commonly known as the arithmetic axioms or more precisely as the arithmetic initials.

 (1) (2)

That should be enough to get started.