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Directory talk:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 01:32, 14 July 2008
, 01:32, 14 July 2008→Table 9: cleanup
===Reality at the Threshold of Logic===
===Reality at the Threshold of Logic===
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 5. A Bridge Over Troubled Waters'''
|+ '''Table 5. A Bridge Over Troubled Waters'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| align="center" | <math>\mbox{Linear Space}\!</math>
| align="center" | <math>\mbox{Liminal Space}\!</math>
| align="center" | <math>\mbox{Logical Space}\!</math>
|-
|-
|
|
<font face="lucida calligraphy">X</font><br>
<math>\begin{matrix}
{''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>}<br>
\mathcal{X}
& = & \{x_1, \ldots, x_n\} \\
\end{matrix}</math>
|
|
<font face="lucida calligraphy"><u>X</u></font><br>
<math>\begin{matrix}
{<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>}<br>
\underline\mathcal{X}
& = & \{\underline{x}_1, \ldots, \underline{x}_n\} \\
\end{matrix}</math>
|
|
<font face="lucida calligraphy">A</font><br>
<math>\begin{matrix}
{''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}<br>
\mathcal{A}
& = & \{a_1, \ldots, a_n\} \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
X_i
& = & \langle x_i \rangle \\
& \cong & \mathbb{K} \\
\end{matrix}</math>
|
|
<u>''X''</u><sub>''i''</sub><br>
<math>\begin{matrix}
{(<u>''x''</u><sub>''i''</sub>), <u>''x''</u><sub>''i''</sub>}<br>
\underline{X}_i
& = & \{(\underline{x}_i), \underline{x}_i \} \\
& \cong & \mathbb{B} \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
{(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}<br>
A_i
& = & \{(a_i), a_i \} \\
& \cong & \mathbb{B} \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
X \\
= & \langle \mathcal{X} \rangle \\
= & \langle x_1, \ldots, x_n \rangle \\
= & X_1 \times \ldots \times X_n \\
∏<sub>''i''</sub> ''X''<sub>''i''</sub><br>
= & \prod_{i=1}^n X_i \\
\cong & \mathbb{K}^n \\
\end{matrix}</math>
|
|
<u>''X''</u><br>
<math>\begin{matrix}
\underline{X} \\
= & \langle \underline\mathcal{X} \rangle \\
{‹<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>›}<br>
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\
= & \underline{X}_1 \times \ldots \times \underline{X}_n \\
∏<sub>''i''</sub> <u>''X''</u><sub>''i''</sub><br>
= & \prod_{i=1}^n \underline{X}_i \\
\cong & \mathbb{B}^n \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
A \\
= & \langle \mathcal{A} \rangle \\
= & \langle a_1, \ldots, a_n \rangle \\
= & A_1 \times \ldots \times A_n \\
∏<sub>''i''</sub> ''A''<sub>''i''</sub><br>
= & \prod_{i=1}^n A_i \\
\cong & \mathbb{B}^n \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
(hom : ''X'' → '''K''')<br>
X^*
& = & (\ell : X \to \mathbb{K}) \\
& \cong & \mathbb{K}^n \\
\end{matrix}</math>
|
|
<u>''X''</u>*<br>
<math>\begin{matrix}
(hom : <u>''X''</u> → '''B''')<br>
\underline{X}^*
& = & (\ell : \underline{X} \to \mathbb{B}) \\
& \cong & \mathbb{B}^n \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
(hom : ''A'' → '''B''')<br>
A^*
& = & (\ell : A \to \mathbb{B}) \\
& \cong & \mathbb{B}^n \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
(''X'' → '''K''')<br>
X^\uparrow
& = & (X \to \mathbb{K}) \\
('''K'''<sup>''n''</sup> → '''K''')
& \cong & (\mathbb{K}^n \to \mathbb{K}) \\
\end{matrix}</math>
|
|
<u>''X''</u>^<br>
<math>\begin{matrix}
(<u>''X''</u> → '''B''')<br>
\underline{X}^\uparrow
& = & (\underline{X} \to \mathbb{B}) \\
('''B'''<sup>''n''</sup> → '''B''')
& \cong & (\mathbb{B}^n \to \mathbb{B}) \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
(''A'' → '''B''')<br>
A^\uparrow
& = & (A \to \mathbb{B}) \\
('''B'''<sup>''n''</sup> → '''B''')
& \cong & (\mathbb{B}^n \to \mathbb{B}) \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
[<font face="lucida calligraphy">X</font>]<br>
X^\circ \\
[''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>]<br>
= & [\mathcal{X}] \\
(''X'', ''X''^)<br>
= & [x_1, \ldots, x_n] \\
(''X'' +→ '''K''')<br>
= & (X, X^\uparrow) \\
(''X'', (''X'' → '''K'''))<br>
= & (X\ +\!\to \mathbb{K}) \\
= & (X, (X \to \mathbb{K})) \\
('''K'''<sup>''n''</sup>, ('''K'''<sup>''n''</sup> → '''K'''))<br>
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K})) \\
('''K'''<sup>''n''</sup> +→ '''K''')<br>
= & (\mathbb{K}^n\ +\!\to \mathbb{K}) \\
['''K'''<sup>''n''</sup>]
= & [\mathbb{K}^n] \\
\end{matrix}</math>
|
|
<u>''X''</u><sup>•</sup><br>
<math>\begin{matrix}
[<font face="lucida calligraphy"><u>X</u></font>]<br>
\underline{X}^\circ \\
[<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>]<br>
= & [\underline\mathcal{X}] \\
(<u>''X''</u>, <u>''X''</u>^)<br>
= & [\underline{x}_1, \ldots, \underline{x}_n] \\
(<u>''X''</u> +→ '''B''')<br>
= & (\underline{X}, \underline{X}^\uparrow) \\
(<u>''X''</u>, (<u>''X''</u> → '''B'''))<br>
= & (\underline{X}\ +\!\to \mathbb{B}) \\
= & (\underline{X}, (\underline{X} \to \mathbb{B})) \\
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\
('''B'''<sup>''n''</sup> +→ '''B''')<br>
= & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\
['''B'''<sup>''n''</sup>]
= & [\mathbb{B}^n] \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
[<font face="lucida calligraphy">A</font>]<br>
A^\circ \\
[''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>]<br>
= & [\mathcal{A}] \\
(''A'', ''A''^)<br>
= & [a_1, \ldots, a_n] \\
(''A'' +→ '''B''')<br>
= & (A, A^\uparrow) \\
(''A'', (''A'' → '''B'''))<br>
= & (A\ +\!\to \mathbb{B}) \\
= & (A, (A \to \mathbb{B})) \\
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\
('''B'''<sup>''n''</sup> +→ '''B''')<br>
= & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\
['''B'''<sup>''n''</sup>]
= & [\mathbb{B}^n] \\
\end{matrix}</math>
|}<br>
|}<br>
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
===The Extended Universe of Discourse===
====Table 8====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 8. Differential Extension : Basic Notation'''
|- style="background:ghostwhite"
! Symbol
! Notation
! Description
! Type
|-
| <math>\operatorname{d}\mathfrak{A}</math>
| <math>\lbrace\!</math> “<math>\operatorname{d}a_1</math>” <math>, \ldots,\!</math> “<math>\operatorname{d}a_n</math>” <math>\rbrace\!</math>
| Alphabet of<br>
differential<br>
symbols
| <math>[n] = \mathbf{n}</math>
|-
| <math>\operatorname{d}\mathcal{A}</math>
| <math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
| Basis of<br>
differential<br>
features
| <math>[n] = \mathbf{n}</math>
|-
| <math>\operatorname{d}A_i</math>
| <math>\{ (\operatorname{d}a_i), \operatorname{d}a_i \}</math>
| Differential<br>
dimension <math>i\!</math>
| <math>\mathbb{D}</math>
|-
| <math>\operatorname{d}A</math>
| <math>\langle \operatorname{d}\mathcal{A} \rangle</math><br>
<math>\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle</math><br>
<math>\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}</math><br>
<math>\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n</math><br>
<math>\textstyle \prod_i \operatorname{d}A_i</math>
| Tangent space<br>
at a point:<br>
Set of changes,<br>
motions, steps,<br>
tangent vectors<br>
at a point
| <math>\mathbb{D}^n</math>
|-
| <math>\operatorname{d}A^*</math>
| <math>(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})</math>
| Linear functions<br>
on <math>\operatorname{d}A</math>
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>
|-
| <math>\operatorname{d}A^\uparrow</math>
| <math>(\operatorname{d}A \to \mathbb{B})</math>
| Boolean functions<br>
on <math>\operatorname{d}A</math>
| <math>\mathbb{D}^n \to \mathbb{B}</math>
|-
| <math>\operatorname{d}A^\circ</math>
| <math>[\operatorname{d}\mathcal{A}]</math><br>
<math>(\operatorname{d}A, \operatorname{d}A^\uparrow)</math><br>
<math>(\operatorname{d}A\ +\!\to \mathbb{B})</math><br>
<math>(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))</math><br>
<math>[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]</math>
| Tangent universe<br>
at a point of <math>A^\circ,</math><br>
based on the<br>
tangent features<br>
<math>\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}</math>
| <math>(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))</math><br>
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{D}^n]</math>
|}<br>
====Table 9====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 9. Higher Order Differential Features'''
|
<p><math>\begin{array}{lllll}
\operatorname{d}^0 \mathcal{A}
& = & \{a_1, \ldots, a_n\}
& = & \mathcal{A} \\
\operatorname{d}^1 \mathcal{A}
& = & \{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\}
& = & \operatorname{d}\mathcal{A} \\
\end{array}</math></p>
<p><math>\begin{array}{lll}
\operatorname{d}^k \mathcal{A}
& = & \{\operatorname{d}^k a_1, \ldots, \operatorname{d}^k a_n\} \\
\operatorname{d}^* \mathcal{A}
& = & \{\operatorname{d}^0 \mathcal{A}, \ldots, \operatorname{d}^k \mathcal{A}, \ldots \} \\
\end{array}</math></p>
|
<p><math>\begin{array}{lll}
\operatorname{E}^0 \mathcal{A}
& = & \operatorname{d}^0 \mathcal{A} \\
\operatorname{E}^1 \mathcal{A}
& = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \mathcal{A} \\
\operatorname{E}^k \mathcal{A}
& = & \operatorname{d}^0 \mathcal{A}\ \cup\ \ldots\ \cup\ \operatorname{d}^k \mathcal{A} \\
\operatorname{E}^\infty \mathcal{A}
& = & \bigcup\ \operatorname{d}^* \mathcal{A} \\
\end{array}</math></p>
|}<br>