[Inquiry] Re: Differential Analytic Turing Automata

[Jon Awbrey]

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DATA.  Note 10

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It is time to formulate the differential analysis of
a logical transformation, or a "mapping of discourse".
It is wise to begin with the first order differentials.

We are considering an abstract logical transformation
F = <f, g> : [u, v] -> [x, y] that can be interpreted
in a number of different ways.  Let's fix on a couple
of major variants that might be indicated as follows:

Alias Map.  <x , y >  =  F<u, v>  =  <((u)(v)), ((u, v))>

Alibi Map.  <u', v'>  =  F<u, v>  =  <((u)(v)), ((u, v))>

F is just one example among -- well, now that I think of it --
how many other logical transformations from the same source
to the same target universe?  In the light of that question,
maybe it would be advisable to contemplate the character of
F within the fold of its most closely akin transformations.

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 of this type, G : U% -> X%, the (first order)
differential analysis of G is based on the definition of a couple of
further transformations, derived by way of operators on G, that ply
between the (first order) extended universes, EU% = [u, v, du, dv]
and EX% = [x, y, dx, dy], of G's own source and target universes.

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

EG_1  =  G_1 <u + du, v + dv>

EG_2  =  G_2 <u + du, v + dv>

Second, the "difference map" (or the "chordal transformation")
DG = <DG_1, DG_2> : EU% -> EX% is defined in a component-wise
fashion as the boolean sum of the initial proposition G_j and
the enlarged or the "shifted" proposition EG_j, for j = 1, 2,
in accord with following pair of equations:

DG_1  =  G_1 <u, v>  +  EG_1 <u, v, du, dv>

      =  G_1 <u, v>  +  G_1 <u + du, v + dv>

DG_2  =  G_2 <u, v>  +  EG_2 <u, v, du, dv>

      =  G_2 <u, v>  +  G_2 <u + du, v + dv>

Maintaining a strict analogy with ordinary difference calculus
would perhaps have us write DG_j = EG_j - G_j, 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.

Given these general considerations about the operators E and D,
let's return to particular cases, and carry out the first order
analysis of the transformation F<u, v>  =  <((u)(v)), ((u, v))>.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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