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ONT Re: Differential Analytic Turing Automata




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DATA.  Note 19

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I'd never rob readers of exercise ...
but for my ain sense of an ending ---

Computation Summary for g<u, v> = ((u, v))

Figure 2.1 expands g = ((u, v)) over [u, v] to get
the equivalent exclusive disjunction u v + (u)(v).

Figure 2.2 expands Eg = ((u + du, v + dv)) over [u, v] to arrive at
Eg = uv((du, dv)) + u(v)(du, dv) + (u)v (du, dv) + (u)(v)((du, dv)).

Eg tells you what you would have to do, from where you are in the
universe [u, v], if you want to end up in a place where g is true.
In this case, where the prevailing proposition g is ((u, v)), the
component uv((du, dv)) of Eg tells you this:  If u and v are both
true where you are, then change either both or neither u and v at
the same time, and you will attain a place where ((u, v)) is true.

Figure 2.3 expands Dg over [u, v] to obtain the following formula:
Dg = uv (du, dv) + u(v)(du, dv) + (u)v (du, dv) + (u)(v) (du, dv).

Dg tells you what you would have to do, from where you are in the
universe [u, v], if you want to bring about a change in the value
of g, that is, if you want to get to a place where the value of g
is different from what it is where you are.  In the present case,
where the ruling proposition g is ((u, v)), the term uv (du, dv)
of Dg tells you this:  If u and v are both true where you are,
then you would have to change one or the other but not both
u and v in order to reach a place where the value of g is
different from what it is where you are.

Figure 2.4 approximates Dg in the proxy of the linear proposition
dg = uv (du, dv) + u(v)(du, dv) + (u)v (du, dv) + (u)(v) (du, dv).
Noting the caste of the constant factor (du, dv) distributed over
the expansion of a tautology, this may be digested as dg = (u, v).

Figure 2.5 shows what remains of the difference map Dg
when the first order linear contribution dg is removed,
and this is nothing but nothing at all, leaving rg = 0.

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Figure 2.1.  g = ((u, v))

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Figure 2.2.  Eg = ((u + du, v + dv))

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Figure 2.3.  Difference Map Dg = g + Eg

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Figure 2.4.  Linear Proxy dg for Dg

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Figure 2.5.  Remainder rg = Dg + dg

| Have I carved enough, my lord --
| Child, you are a bone.
|
| Leonard Cohen, "Teachers" (1967)

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