ONT Re: Differential Analytic Turing Automata

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DATA.  Note 8

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I am going to tip-toe in silence/consilience past many
questions of a philosophical nature/nurture that might
be asked at this juncture, no doubt to revisit them at
some future opportunity/importunity, however the cases
happen to align in the course of their inevitable fall.

Instead, let's "keep it concrete and simple", taking up the
consideration of an incrementally more complex example, but
having a slightly more general character than the orders of
sequential transformations that we've been discussing up to
this point.

The types of logical transformations that I have in mind can
be thought of as "transformations of discourse" because they
map a universe of discourse into a universe of discourse by
way of logical equations between the qualitative features
or logical variables in the source and target universes.

The sequential transformations or state transitions that we have
been considering so far are actually special cases of these more
general logical transformations, specifically, they are the ones
that have a single universe of discourse, as it happens to exist
at different moments in time, in the role of both the source and
the target universes of the transformation in question.

Onward and upward to Flatland, the differential analysis of
transformations between 2-dimensional universes of discourse.

Consider the transformation from the universe U% = [u, v] to the
universe X% = [x, y] that is defined by this system of equations:

x   =   f<u, v>   =   ((u)(v))

y   =   g<u, v>   =   ((u, v))

The parenthetical expressions on the right are the cactus forms for
the boolean functions that correspond to inclusive disjunction and
logical equivalence, respectively.  By way of a reminder, consult
Table 1 on the page at this location:

DLOG D1.  http://stderr.org/pipermail/inquiry/2003-May/000478.html

The component notation F = <F_1, F_2> = <f, g> : U% -> X% allows
us to give a name and a type to this transformation, and permits
us to define it by means of the compact description that follows:

<x, y>   =   F<u, v>   =   <((u)(v)), ((u, v))>

The information that defines the logical transformation F
can be represented in the form of a truth table, as below.

u v | f g
----+----
0 0 | 0 1
0 1 | 1 0
1 0 | 1 0
1 1 | 1 1

A more complete framework of discussion and a fuller development of
this example can be found in the neighborhood of the following site:

DLOG D73.  http://stderr.org/pipermail/inquiry/2003-June/000557.html

Jon Awbrey

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