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Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)
Revision as of 17:54, 10 March 2009
, 17:54, 10 March 2009→Note 15: recall example
When the term is being used with its more exact sense, a ''differential'' is a locally linear approximation to a function, in the context of this logical discussion, then, a locally linear approximation to a proposition.
When the term is being used with its more exact sense, a ''differential'' is a locally linear approximation to a function, in the context of this logical discussion, then, a locally linear approximation to a proposition.
I think that it would be best to just go ahead and exhibit the simplest form of a differential <math>\operatorname{d}F\!</math> for the current example of a logical transformation <math>F,\!</math> after which the majority of the easiest questions will have been answered in visually intuitive terms.
Recall the form of the current example:
{| align="center" cellpadding="8" width="90%"
|
<math>\begin{array}{lllll}
F & = & (f, g) & = & ( ~\texttt{((u)(v))}~ , ~\texttt{((u,~v))}~ ).
\end{array}</math>
|}
To speed things along, I will skip a mass of motivating discussion and just exhibit the simplest form of a differential <math>\operatorname{d}F\!</math> for the current example of a logical transformation <math>F,\!</math> after which the majority of the easiest questions will have been answered in visually intuitive terms.
For <math>F = (f, g)\!</math> we have <math>\operatorname{d}F = (\operatorname{d}f, \operatorname{d}g),</math> and so we can proceed componentwise, patching the pieces back together at the end.
For <math>F = (f, g)\!</math> we have <math>\operatorname{d}F = (\operatorname{d}f, \operatorname{d}g),</math> and so we can proceed componentwise, patching the pieces back together at the end.