ONT Re: Differential Logic
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note D17
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Life on Easy Street
| Failing to fetch me at first keep encouraged,
| Missing me one place search another,
| I stop some where waiting for you
|
| Walt Whitman, 'Leaves of Grass', [Whi, 88]
The finite character of the extended universe [E!A!] makes the problem
of solving differential propositions relatively straightforward, at least,
in principle. The solution set of the differential proposition q : EA -> B
is the set of models (q^(-1))(1) in EA. Finding all of the models of q, the
extended interpretations in EA that satisfy q, can be carried out by a finite
search. Being in possession of complete algorithms for propositional calculus
theorem proving makes the analytic task fairly simple in principle, though the
question of efficiency in the face of arbitrary complexity may always remain
another matter entirely. While the fact that propositional satisfiability
is NP-complete may be discouraging for the prospects of a single efficient
algorithm that covers the whole space of [E!A!] with equal facility, there
appears to be much room for improvement in classifying special forms and
in developing algorithms that are tailored to their practical processing.
In view of these constraints and contingencies, my focus shifts to the tasks of
approximation and interpretation that support intuition, especially in dealing
with the natural kinds of differential propositions that arise in applications,
and in the effort to understand, in succinct and adaptive forms, their dynamic
implications. In the absence of direct insights, these tasks are partially
carried out by forging analogies with the familiar situations and customary
routines of ordinary calculus. But the indirect approach, going by way of
specious analogy and intuitive habit, forces us to remain on guard against
the circumstance that occurs when the word "forging" takes on its shadier
nuance, indicting the constant risk of a counterfeit in the proportion.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o