Changes

Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)

Revision as of 02:20, 12 March 2009

248 bytes added , 02:20, 12 March 2009

→‎Computation Summary : <math>g(u, v) = \texttt{((u,~v))}</math>: markup

Line 1,452: Line 1,452:

==Note 19==

−

===Computation Summary : <math>g(u, v) = \texttt{((u,~v))}</math>===

+

===Computation Summary : <math>g(u, v) = \texttt{((u, v))}</math>===

−

Figure 2.1 shows the expansion of <math>g = \texttt{((u,~v))}</math> over <math>[u, v]\!</math> to produce the expression:

+

Figure 2.1 shows the expansion of <math>g = \texttt{((u, v))}</math> over <math>[u, v]\!</math> to produce the expression:

{| align="center" cellpadding="8" width="90%"

Line 1,460: Line 1,460:

|}

−

Figure 2.2 shows the expansion of <math>\operatorname{E}g = \texttt{((u + du, ~v + dv))}</math> over <math>[u, v]\!</math> to produce the expression:

+

Figure 2.2 shows the expansion of <math>\operatorname{E}g = \texttt{((u + du, v + dv))}</math> over <math>[u, v]\!</math> to produce the expression:

{| align="center" cellpadding="8" width="90%"

| <math>\texttt{uv} \cdot \texttt{((du, dv))} + \texttt{u(v)} \cdot \texttt{(du, dv)} + \texttt{(u)v} \cdot \texttt{(du, dv)} + \texttt{(u)(v)} \cdot \texttt{((du, dv))}</math>

|}

+

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

<pre>

−

~~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).

Jon Awbrey

12,122

edits