ONT Re: Differential Analytic Turing Automata
DATA. Note 15
| 'Tis a derivative from me to mine,
| And only that I stand for.
| Winter's Tale, 3.2.43-44
We've talked about differentials long enough
that I think it's past time we met with some.
When the term is being used with its more exact sense,
a "differential" is a locally linear approximation to
a function, in the context of this logical discussion,
then, a locally linear approximation to a proposition.
I think that it would be best to just go ahead and
exhibit the simplest form of a differential dF for
the current example of a logical transformation F,
after which the majority of the easiest questions
will've been answered in visually intuitive terms.
For F = <f, g> we have dF = <df, dg>, and so we can proceed
componentwise, patching the pieces back together at the end.
We have prepared the ground already by computing these terms:
Ef = ((u + du)(v + dv))
Eg = ((u + du, v + dv))
Df = ((u)(v)) + ((u + du)(v + dv))
Dg = ((u, v)) + ((u + du, v + dv))
As a matter of fact, computing the symmetric differences
Df = f + Ef and Dg = g + Eg has already taken care of the
"localizing" part of the task by subtracting out the forms
of f and g from the forms of Ef and Eg, respectively. Thus
all we have left to do is to decide what linear propositions
best approximate the difference maps Df and Dg, respectively.
This raises the question: What is a linear proposition?
The answer that makes the most sense in this context is this:
A proposition is just a boolean-valued function, so a linear
proposition is a linear function into the boolean space B.
In particular, the linear functions that we want will be
linear functions in the differential variables du and dv.
As it turns out, there are just four linear propositions
in the associated "differential universe" dU% = [du, dv],
and these are the propositions that are commonly denoted:
0, du, dv, du + dv, in other words, (), du, dv, (du, dv).
To be continued ...
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