ONT Re: Differential Analytic Turing Automata
DATA. Note 20
In my work on "Differential Logic and Dynamic Systems",
I found it useful to develop several different ways of
visualizing logical transformations, indeed, I devised
four distinct styles of picture for the job. Thus far
in our work on the mapping F : [u, v] -> [u, v], we've
been making use of what I call the "areal view" of the
extended universe of discourse, [u, v, du, dv], but as
the number of dimensions climbs beyond four, it's time
to bid this genre adieu, and look for a style that can
scale a little better. At any rate, before we proceed
any further, let's first assemble the information that
we have gathered about F from several different angles,
and see if it can be fitted into a coherent picture of
the transformation F : <u, v> ~> <((u)(v)), ((u, v))>.
In our first crack at the transformation F, we simply
plotted the state transitions and applied the utterly
stock technique of calculating the finite differences.
Orbit 1. u v
| ` ` | d d |
| u v | u v |
| 1 1 | 0 0 |
| " " | " " |
A quick inspection of the first Table suggests a rule
to cover the case when u = v = 1, namely, du = dv = 0.
To put it another way, the Table characterizes Orbit 1
by means of the data: <u, v, du, dv> = <1, 1, 0, 0>.
Last but not least, yet another way to convey the same
information is by means of the (first order) extended
proposition: u v (du)(dv).
Orbit 2. (u v)
| ` ` | ` ` | d d |
| ` ` | d d | 2 2 |
| u v | u v | u v |
| 0 0 | 0 1 | 1 0 |
| 0 1 | 1 1 | 1 1 |
| 1 0 | 0 0 | 0 0 |
| " " | " " | " " |
A more fine combing of the second Table brings to mind
a rule that partly covers the remaining cases, that is,
du = v, dv = (u). To vary the formulation, this Table
characterizes Orbit 2 by means of the following vector
equation: <du, dv> = <v, (u)>. This much information
about Orbit 2 is also encapsulated by the (first order)
extended proposition, (uv)((du, v))(dv, u), which says
that u and v are not both true at the same time, while
du is equal in value to v, and dv is the opposite of u.
To be continued ...
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