[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D78

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Transformations of Type B^2 -> B^2 (cont.)

Given the alphabets !U! = {u, v} and !X! = {x, y},
along with the corresponding universes of discourse
U% and X% ~=~ [B^2], how many logical transformations
of the general form G = <G_1, G_2> : U% -> X% are there?

Since G_1 and G_2 can be any propositions of the type B^2 -> B,
there are 2^4 = 16 choices for each of the maps G_1 and G_2, and
thus there are 2^4 * 2^4 = 2^8 = 256 different mappings altogether
of the form G : U% -> X%.  The set of all functions of a given type
is customarily denoted by placing its type indicator in parentheses,
in the present instance writing (U% -> X%) = {G : U% -> X%}, and so
the cardinality of this "function space" can be most conveniently
summed up by writing |(U% -> X%)| = |(B^2 -> B^2)| = 4^4 = 256.

Given any transformation G = <G_1, G_2> : U% -> X% of this type,
one can define a couple of further transformations, related to G,
that operate between the extended universes, EU% and EX%, of its
source and target domains.

First, the enlargement map (or the secant transformation)
EG = <EG_1, EG_2> : EU% -> EX% is defined by the following
set of component equations:

o-------------------------------------------------o
|                                                 |
|   EG_i  =  G_i <u + du, v + dv>                 |
|                                                 |
o-------------------------------------------------o

Second, the difference map (or the chordal transformation)
DG = <DG_1, DG_2> : EU% -> EX% is defined in component-wise
fashion as the boolean sum (or mod 2 sum) of the initial and
the enlarged propositions:

o-------------------------------------------------o
|                                                 |
|   DG_i  =  G_i <u, v>  +  EG_i <u, v, du, dv>   |
|                                                 |
|         =  G_i <u, v>  +  G_i <u + du, v + dv>  |
|                                                 |
o-------------------------------------------------o

Maintaining a strict analogy with ordinary difference calculus
would perhaps have us write DG_i = EG_i - G_i, but the sum and
difference operations are the same thing in boolean arithmetic.
It is more often natural in the logical context to consider an
initial proposition q, then to compute the enlargement Eq, and
finally to determine the difference Dq = q + Eq, so we let the
variant order of terms reflect this sequence of considerations.

Viewed in this light the difference operator D is imagined to be a function
of very wide scope and polymorphic application, one that is able to realize
the association between each transformation G and its difference map DG, for
instance, taking the function space (U% -> X%) into (EU% -> EX%).  Given the
interpretive flexibility of contexts in which we are allowing a proposition
to appear, it should be clear that an operator of this scope is not at all
a trivial matter to define properly, and may take some trouble to work out.
For the moment, let's content ourselves with returning to particular cases.

Jon Awbrey

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