ONT Re: Differential Analytic Turing Automata

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DATA.  Note 13

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I think that it ought to be clear at this point that we
need a more systematic symbolic method for computing the
differentials of logical transformations, using the term
"differential" in a loose way at present for all sorts of
finite differences and derivatives, leaving it to another
discussion to sharpen up its more exact technical senses.

For convenience of reference, let's recast our current
example in the form F = <f, g> = <((u)(v)), ((u, v))>.

In their application to this logical transformation the operators
E and D respectively produce the "enlarged map" EF = <Ef, Eg> and
the "difference map" DF = <Df, Dg>, whose components can be given
as follows, if the reader, in the absence of a special format for
logical parentheses, can forgive syntactically 2-lingual phrases:

Ef  =  ((u + du)(v + dv))

Eg  =  ((u + du, v + dv))

Df  =  ((u)(v))  +  ((u + du)(v + dv))

Dg  =  ((u, v))  +  ((u + du, v + dv))

But these initial formulas are purely definitional,
and help us little to understand either the purpose
of the operators or the significance of the results.
Working symbolically, let's apply a more systematic
method to the separate components of the mapping F.

A sketch of this work is presented in the following series
of Figures, where each logical proposition is expanded over
the basic cells uv, u(v), (u)v, (u)(v) of the 2-dimensional
universe of discourse U% = [u, v].

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

Figure 1 shows how f = ((u)(v)) expands over [u, v]
to give the exclusive disjunction uv + u(v) + (u)v.

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Figure 1.  f = ((u)(v))

Figure 2 shows how Ef = ((u + du)(v + dv)) expands over [u, v] as:

uv.(du dv) + u(v).(du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv))

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Figure 2.  Ef = ((u + du)(v + dv))

Figure 3 shows how Df = f + Ef expands over [u, v] to give:

uv.du dv + u(v).du(dv) + (u)v.(du)dv + (u)(v).((du)(dv))

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Figure 3.  Df = f + Ef

I'll break this here in case anyone wants
to try and do the work for g on their own.

Jon Awbrey

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