[Inquiry] Re: Differential Logic
Jon Awbrey
jawbrey at oakland.edu
Thu May 15 14:10:53 CDT 2003
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DLOG. Note D23
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Example 2. Drives and Their Vicissitudes
| I open my scuttle at night and see the far-sprinkled systems,
| And all I see, multiplied as high as I can cipher, edge but
| the rim of the farther systems.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 81]
Before we leave the one-feature case let's look at a more substantial example,
one that illustrates a general class of curves that can be charted through
the extended feature spaces and that provides an opportunity to discuss
a number of important themes concerning their structure and dynamics.
Again, let !X! = {x_1} = {A}. In the discussion that follows I will consider
a class of trajectories having the property that d^k.A = 0 for all k greater
than some fixed m, and I indulge in the use of some picturesque terms that
describe salient classes of such curves. Given the finite order condition,
there is a highest order non-zero difference d^m.A exhibited at each point
in the course of any determinate trajectory that one may wish to consider.
With respect to any point of the corresponding orbit or curve let us call
this highest order differential feature d^m.A the "drive" at that point.
Curves of constant drive d^m.A are then referred to as "m^th gear curves".
Scholium. The fact that a difference calculus can be developed
for boolean functions is well known [Fuji], [Koh, sec. 8-4] and
was probably familiar to Boole, who was an expert in difference
equations before he turned to logic. And of course there is the
strange but true story of how the Turin machines of the 1840's
prefigured the Turing machines of the 1940's [Men, 225-297].
At the very outset of general purpose, mechanized computing
we find that the motive power driving the Analytical Engine
of Babbage, the kernel of an idea behind all of his wheels,
was exactly his notion that difference operations, suitably
trained, can serve as universal joints for any conceivable
computation [M&M], [Mel, ch. 4].
Given this language, the particular Example that I take up here can be described
as the family of 4^th gear curves through E^4.X = <|A, dA, d^2.A, d^3.A, d^4.A|>.
These are the trajectories generated subject to the dynamic law d^4.A = 1, where
it is understood in such a statement that all higher order differences are equal
to 0. Since d^4.A and all higher d^k.A are fixed, the temporal or transitional
conditions (initial, mediate, terminal -- transient or stable states) vary only
with respect to their projections as points of E^3.X = <|A, dA, d^2.A, d^3.A|>.
Thus, there is just enough space in a planar venn diagram to plot all of these
orbits and to show how they partition the points of E^3.X. It turns out that
there are exactly two possible orbits, of eight points each, as illustrated
in Figures 16-a and 16-b. (NB. I leave it as an exercise for the reader
to connect the dots in the second figure.)
o-------------------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| o o o |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o 5 o 7 o o |
| / \ ^| / \ ^| / \ / \ |
| / \/ | / \/ | / \ / \ |
| / /\ | / /\ | / \ / \ |
| / / \|/ / \|/ \ / \ |
| o 4<---|----/----|----3 o o |
| |\ /|\ / /|\ ^ / \ /| |
| | \ / | \/ / | \/ / \ / | |
| | \ / | /\ / | /\ / \ / | |
| | \ / v/ \ / |/ \ / \ / | |
| | o 6 o | o o | |
| | |\ / \ /| / \ /| | |
| | | \ / \/ | / \ / | | |
| | | \ / /\ | / \ / | | |
| | d^0.A \ / / \|/ \ / d^1.A | |
| o----+----o 2<---|----1 o----+----o |
| | \ /|\ ^ / | |
| | \ / | \/ / | |
| | \ / | /\ / | |
| | d^2.A \ / v/ \ / d^3.A | |
| o---------o 0 o---------o |
| \ / |
| \ / |
| \ / |
| \ / |
| o |
| |
o-------------------------------------------------o
Figure 16-a. A Couple of Fourth Gear Orbits: 1
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| |
| o |
| / \ |
| / \ |
| / \ |
| / \ |
| o 0 o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| o 5 o 2 o |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o o 6 o |
| / \ / \ / \ / \ |
| / \ / \ / \ / \ |
| / \ / \ / \ / \ |
| / \ / \ / \ / \ |
| o o 7 o o 4 o |
| |\ / \ / \ / \ /| |
| | \ / \ / \ / \ / | |
| | \ / \ / \ / \ / | |
| | \ / \ / \ / \ / | |
| | o o 3 o 1 o | |
| | |\ / \ / \ /| | |
| | | \ / \ / \ / | | |
| | | \ / \ / \ / | | |
| | d^0.A \ / \ / \ / d^1.A | |
| o----+----o o o----+----o |
| | \ / \ / | |
| | \ / \ / | |
| | \ / \ / | |
| | d^2.A \ / \ / d^3.A | |
| o---------o o---------o |
| \ / |
| \ / |
| \ / |
| \ / |
| o |
| |
o-------------------------------------------------o
Figure 16-b. A Couple of Fourth Gear Orbits: 2
Jon Awbrey
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