ONT Re: Differential Analytic Turing Automata

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

DATA.  Note 18

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Let's push on with the analysis of the transformation:

F : <u, v> ~> <f<u, v>, g<u, v>> = <((u)(v)), ((u, v))>.

For ease of comparison and computation, I will collect
the Figures that we need for the remainder of the work
together on one page.

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

Figure 1.1 expands f = ((u)(v)) over [u, v] to produce
the equivalent exclusive disjunction uv + u(v) + (u)v.

Figure 1.2 expands Ef = ((u + du)(v + dv)) over [u, v] to arrive at
Ef = uv (du dv) + u(v) (du (dv)) + (u)v ((du) dv) + (u)(v)((du)(dv)).

Ef 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 f is true.
In this case, where the prevailing proposition f is ((u)(v)), the
indication uv (du dv) of Ef tells you this:  If u and v are both
true where you are, then just don't change both u and v, and you
will end up in a place where ((u)(v)) is true.

Figure 1.3 expands Df over [u, v] to end up with the formula:
Df = uv du dv + u(v) du(dv) + (u)v (du)dv + (u)(v)((du)(dv)).

Df 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 f, that is, if you want to get to a place where the value of f
is different from what it is where you are.  In the present case,
where the reigning proposition f is ((u)(v)), the term uv du dv
of Df tells you this:  If u and v are both true where you are,
then you would have to change both u and v in order to reach
a place where the value of f is different from what it is
where you are.

Figure 1.4 approximates Df by the linear form
df = uv 0 + u(v) du + (u)v dv + (u)(v)(du, dv).

Figure 1.5 shows what remains of the difference map Df
when the first order linear contribution df is removed:
rf = uv du dv + u(v) du dv + (u)v du dv + (u)(v) du dv.
This form can be written more succinctly as rf = du dv.

o---------------------------------------o
|```````````````````````````````````````|
|```````````````````o```````````````````|
|``````````````````/%\``````````````````|
|`````````````````/%%%\`````````````````|
|````````````````/%%%%%\````````````````|
|```````````````o%%%%%%%o```````````````|
|``````````````/%\%%%%%/%\``````````````|
|`````````````/%%%\%%%/%%%\`````````````|
|````````````/%%%%%\%/%%%%%\````````````|
|```````````o%%%%%%%o%%%%%%%o```````````|
|``````````/%\%%%%%/%\%%%%%/%\``````````|
|`````````/%%%\%%%/%%%\%%%/%%%\`````````|
|````````/%%%%%\%/%%%%%\%/%%%%%\````````|
|```````o%%%%%%%o%%%%%%%o%%%%%%%o```````|
|``````/%\%%%%%/%\%%%%%/%\%%%%%/%\``````|
|`````/%%%\%%%/%%%\%%%/%%%\%%%/%%%\`````|
|````/%%%%%\%/%%%%%\%/%%%%%\%/%%%%%\````|
|```o%%%%%%%o%%%%%%%o%%%%%%%o%%%%%%%o```|
|```|\%%%%%/%\%%%%%/`\%%%%%/%\%%%%%/|```|
|```|`\%%%/%%%\%%%/```\%%%/%%%\%%%/`|```|
|```|``\%/%%%%%\%/`````\%/%%%%%\%/``|```|
|```|```o%%%%%%%o```````o%%%%%%%o```|```|
|```|```|\%%%%%/`\`````/`\%%%%%/|```|```|
|```|```|`\%%%/```\```/```\%%%/`|```|```|
|```|`u`|``\%/`````\`/`````\%/``|`v`|```|
|```o---+---o```````o```````o---+---o```|
|```````|````\`````/`\`````/````|```````|
|```````|`````\```/```\```/`````|```````|
|```````|`du```\`/`````\`/```dv`|```````|
|```````o-------o```````o-------o```````|
|````````````````\`````/````````````````|
|`````````````````\```/`````````````````|
|``````````````````\`/``````````````````|
|```````````````````o```````````````````|
|```````````````````````````````````````|
o---------------------------------------o
Figure 1.1.  f = ((u)(v))

o---------------------------------------o
|```````````````````````````````````````|
|```````````````````o```````````````````|
|``````````````````/%\``````````````````|
|`````````````````/%%%\`````````````````|
|````````````````/%%%%%\````````````````|
|```````````````o%%%%%%%o```````````````|
|``````````````/%\%%%%%/%\``````````````|
|`````````````/%%%\%%%/%%%\`````````````|
|````````````/%%%%%\%/%%%%%\````````````|
|```````````o%%%%%%%o%%%%%%%o```````````|
|``````````/%\%%%%%/`\%%%%%/%\``````````|
|`````````/%%%\%%%/```\%%%/%%%\`````````|
|````````/%%%%%\%/`````\%/%%%%%\````````|
|```````o%%%%%%%o```````o%%%%%%%o```````|
|``````/%\%%%%%/%\`````/%\%%%%%/%\``````|
|`````/%%%\%%%/%%%\```/%%%\%%%/%%%\`````|
|````/%%%%%\%/%%%%%\`/%%%%%\%/%%%%%\````|
|```o%%%%%%%o%%%%%%%o%%%%%%%o%%%%%%%o```|
|```|\%%%%%/`\%%%%%/%\%%%%%/`\%%%%%/|```|
|```|`\%%%/```\%%%/%%%\%%%/```\%%%/`|```|
|```|``\%/`````\%/%%%%%\%/`````\%/``|```|
|```|```o```````o%%%%%%%o```````o```|```|
|```|```|\`````/%\%%%%%/%\`````/|```|```|
|```|```|`\```/%%%\%%%/%%%\```/`|```|```|
|```|`u`|``\`/%%%%%\%/%%%%%\`/``|`v`|```|
|```o---+---o%%%%%%%o%%%%%%%o---+---o```|
|```````|````\%%%%%/`\%%%%%/````|```````|
|```````|`````\%%%/```\%%%/`````|```````|
|```````|`du```\%/`````\%/```dv`|```````|
|```````o-------o```````o-------o```````|
|````````````````\`````/````````````````|
|`````````````````\```/`````````````````|
|``````````````````\`/``````````````````|
|```````````````````o```````````````````|
|```````````````````````````````````````|
o---------------------------------------o
Figure 1.2.  Ef = ((u + du)(v + dv))

o---------------------------------------o
|```````````````````````````````````````|
|```````````````````o```````````````````|
|``````````````````/`\``````````````````|
|`````````````````/```\`````````````````|
|````````````````/`````\````````````````|
|```````````````o```````o```````````````|
|``````````````/`\`````/`\``````````````|
|`````````````/```\```/```\`````````````|
|````````````/`````\`/`````\````````````|
|```````````o```````o```````o```````````|
|``````````/`\`````/%\`````/`\``````````|
|`````````/```\```/%%%\```/```\`````````|
|````````/`````\`/%%%%%\`/`````\````````|
|```````o```````o%%%%%%%o```````o```````|
|``````/`\`````/`\%%%%%/`\`````/`\``````|
|`````/```\```/```\%%%/```\```/```\`````|
|````/`````\`/`````\%/`````\`/`````\````|
|```o```````o```````o```````o```````o```|
|```|\`````/%\`````/%\`````/%\`````/|```|
|```|`\```/%%%\```/%%%\```/%%%\```/`|```|
|```|``\`/%%%%%\`/%%%%%\`/%%%%%\`/``|```|
|```|```o%%%%%%%o%%%%%%%o%%%%%%%o```|```|
|```|```|\%%%%%/%\%%%%%/%\%%%%%/|```|```|
|```|```|`\%%%/%%%\%%%/%%%\%%%/`|```|```|
|```|`u`|``\%/%%%%%\%/%%%%%\%/``|`v`|```|
|```o---+---o%%%%%%%o%%%%%%%o---+---o```|
|```````|````\%%%%%/`\%%%%%/````|```````|
|```````|`````\%%%/```\%%%/`````|```````|
|```````|`du```\%/`````\%/```dv`|```````|
|```````o-------o```````o-------o```````|
|````````````````\`````/````````````````|
|`````````````````\```/`````````````````|
|``````````````````\`/``````````````````|
|```````````````````o```````````````````|
|```````````````````````````````````````|
o---------------------------------------o
Figure 1.3.  Difference Map Df = f + Ef

o---------------------------------------o
|```````````````````````````````````````|
|```````````````````o```````````````````|
|``````````````````/`\``````````````````|
|`````````````````/```\`````````````````|
|````````````````/`````\````````````````|
|```````````````o```````o```````````````|
|``````````````/`\`````/`\``````````````|
|`````````````/```\```/```\`````````````|
|````````````/`````\`/`````\````````````|
|```````````o```````o```````o```````````|
|``````````/`\`````/`\`````/`\``````````|
|`````````/```\```/```\```/```\`````````|
|````````/`````\`/`````\`/`````\````````|
|```````o```````o```````o```````o```````|
|``````/`\`````/%\`````/%\`````/`\``````|
|`````/```\```/%%%\```/%%%\```/```\`````|
|````/`````\`/%%%%%\`/%%%%%\`/`````\````|
|```o```````o%%%%%%%o%%%%%%%o```````o```|
|```|\`````/%\%%%%%/`\%%%%%/%\`````/|```|
|```|`\```/%%%\%%%/```\%%%/%%%\```/`|```|
|```|``\`/%%%%%\%/`````\%/%%%%%\`/``|```|
|```|```o%%%%%%%o```````o%%%%%%%o```|```|
|```|```|\%%%%%/%\`````/%\%%%%%/|```|```|
|```|```|`\%%%/%%%\```/%%%\%%%/`|```|```|
|```|`u`|``\%/%%%%%\`/%%%%%\%/``|`v`|```|
|```o---+---o%%%%%%%o%%%%%%%o---+---o```|
|```````|````\%%%%%/`\%%%%%/````|```````|
|```````|`````\%%%/```\%%%/`````|```````|
|```````|`du```\%/`````\%/```dv`|```````|
|```````o-------o```````o-------o```````|
|````````````````\`````/````````````````|
|`````````````````\```/`````````````````|
|``````````````````\`/``````````````````|
|```````````````````o```````````````````|
|```````````````````````````````````````|
o---------------------------------------o
Figure 1.4.  Linear Proxy df for Df

o---------------------------------------o
|```````````````````````````````````````|
|```````````````````o```````````````````|
|``````````````````/`\``````````````````|
|`````````````````/```\`````````````````|
|````````````````/`````\````````````````|
|```````````````o```````o```````````````|
|``````````````/`\`````/`\``````````````|
|`````````````/```\```/```\`````````````|
|````````````/`````\`/`````\````````````|
|```````````o```````o```````o```````````|
|``````````/`\`````/%\`````/`\``````````|
|`````````/```\```/%%%\```/```\`````````|
|````````/`````\`/%%%%%\`/`````\````````|
|```````o```````o%%%%%%%o```````o```````|
|``````/`\`````/%\%%%%%/%\`````/`\``````|
|`````/```\```/%%%\%%%/%%%\```/```\`````|
|````/`````\`/%%%%%\%/%%%%%\`/`````\````|
|```o```````o%%%%%%%o%%%%%%%o```````o```|
|```|\`````/`\%%%%%/%\%%%%%/`\`````/|```|
|```|`\```/```\%%%/%%%\%%%/```\```/`|```|
|```|``\`/`````\%/%%%%%\%/`````\`/``|```|
|```|```o```````o%%%%%%%o```````o```|```|
|```|```|\`````/`\%%%%%/`\`````/|```|```|
|```|```|`\```/```\%%%/```\```/`|```|```|
|```|`u`|``\`/`````\%/`````\`/``|`v`|```|
|```o---+---o```````o```````o---+---o```|
|```````|````\`````/`\`````/````|```````|
|```````|`````\```/```\```/`````|```````|
|```````|`du```\`/`````\`/```dv`|```````|
|```````o-------o```````o-------o```````|
|````````````````\`````/````````````````|
|`````````````````\```/`````````````````|
|``````````````````\`/``````````````````|
|```````````````````o```````````````````|
|```````````````````````````````````````|
o---------------------------------------o
Figure 1.5.  Remainder rf = Df + df

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

Exercise for the Reader.

Jon Awbrey

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