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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)

Revision as of 18:22, 12 July 2007

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→‎Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>: wiki table
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Table&nbsp;65 extends this scheme from single cells to arbitrary regions of the source and target universes, and illustrates a form of computation that can be used to determine how a logical transformation acts on all of the propositions in the universe.  The way that a transformation of positions affects the propositions, or any other structure that can be built on top of the positions, is normally called the ''induced action'' of the given transformation on the system of structures in question.
 
Table&nbsp;65 extends this scheme from single cells to arbitrary regions of the source and target universes, and illustrates a form of computation that can be used to determine how a logical transformation acts on all of the propositions in the universe.  The way that a transformation of positions affects the propositions, or any other structure that can be built on top of the positions, is normally called the ''induced action'' of the given transformation on the system of structures in question.
   −
<pre>
+
<br><font face="courier new">
−
Table 65.  Induced Transformation on Propositions
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
−
o------------o---------------------------------o------------o
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|+ Table 65.  Induced Transformation on Propositions
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|     X%    |   <---  F = <f , g>   <---  |     U%    |
+
|- style="background:paleturquoise"
−
o------------o----------o-----------o----------o------------o
+
| ''X''<sup>&nbsp;&bull;</sup>
−
|           |     u = | 1 1 0 0 | = u      |           |
+
| colspan="3" |
−
|           |     v = | 1 0 1 0  | = v     |           |
+
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"
−
| f_i <x, y> o----------o-----------o----------o f_j <u, v> |
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| &larr;
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|           |     x = | 1 1 1 0 | = f<u,v> |           |
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| ''F''&nbsp;=&nbsp;‹''f''&nbsp;,&nbsp;''g''›
−
|           |     y = | 1 0 0 1  | = g<u,v> |           |
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| &larr;
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o------------o----------o-----------o----------o------------o
+
|}
−
|           |         |           |         |           |
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| ''U''<sup>&nbsp;&bull;</sup>
−
|   f_0    |   ()    | 0 0 0 0  |   ()    |   f_0    |
+
|- style="background:paleturquoise"
−
|           |         |           |          |            |
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| rowspan="2" | ''f''<sub>''i''</sub>‹''x'',&nbsp;''y''›
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|   f_1    |  (x)(y) | 0 0 0 1  |   ()   |   f_0    |
+
|
−
|           |         |           |         |            |
+
{| align="right" style="background:paleturquoise; text-align:right"
−
|   f_2    |  (x) y  | 0 0 1 0  | (u)(v)  |   f_1    |
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| ''u'' =
−
|           |         |           |         |           |
+
|-
−
|   f_3    | (x)    |  0 0 1 1 | (u)(v)  |   f_1    |
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| ''v'' =
−
|           |         |           |         |           |
+
|}
−
|   f_4    |   x (y) | 0 1 0 0  | (u, v) |    f_6    |
+
|
−
|           |          |          |          |            |
+
{| align="center" style="background:paleturquoise; text-align:center"
−
|   f_5    |    (y) | 0 1 0 1  | (u, v) |    f_6    |
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| 1 1 0 0
−
|           |          |          |          |            |
+
|-
−
|   f_6    |  (x, y) | 0 1 1 0  | (u v) |    f_7    |
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| 1 0 1 0
−
|           |          |          |          |            |
+
|}
−
|   f_7    |  (x  y) | 0 1 1 1  | (u v) |   f_7    |
+
|
−
|           |         |           |         |           |
+
{| align="left" style="background:paleturquoise; text-align:left"
−
o------------o----------o-----------o----------o------------o
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| = ''u''
−
|           |         |           |         |           |
+
|-
−
|   f_8    |   x  y  | 1 0 0 0  |   u  v  |   f_8    |
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| = ''v''
−
|           |         |           |         |           |
+
|}
−
|   f_9    | ((x, y)) | 1 0 0 1  |   u  v  |   f_8    |
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| rowspan="2" | ''f''<sub>''j''</sub>‹''u'',&nbsp;''v''›
−
|           |         |           |         |           |
+
|- style="background:paleturquoise"
−
|   f_10    |     y   | 1 0 1 0  | ((u, v)) |   f_9    |
+
|
−
|           |         |           |         |           |
+
{| align="right" style="background:paleturquoise; text-align:right"
−
|   f_11    | (x (y)) | 1 0 1 1  | ((u, v)) |   f_9    |
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| ''x'' =
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|           |         |           |         |           |
+
|-
−
|   f_12    |   x      | 1 1 0 0 | ((u)(v)) |   f_14    |
+
| ''y'' =
−
|           |         |           |         |           |
+
|}
−
|   f_13    | ((x) y) | 1 1 0 1  | ((u)(v)) |    f_14    |
+
|
−
|           |          |          |          |            |
+
{| align="center" style="background:paleturquoise; text-align:center"
−
|    f_14    | ((x)(y)) | 1 1 1 0  |   (())   |    f_15    |
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| 1 1 1 0
−
|           |          |          |          |            |
+
|-
−
|   f_15    |  (())   | 1 1 1 1  |   (())   |   f_15    |
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| 1 0 0 1
−
|           |         |           |         |           |
+
|}
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o------------o----------o-----------o----------o------------o
+
|
−
</pre>
+
{| align="left" style="background:paleturquoise; text-align:left"
 +
| = ''f''‹''u'',&nbsp;''v''›
 +
|-
 +
| = ''g''‹''u'',&nbsp;''v''›
 +
|}
 +
|-
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ''f''<sub>0</sub>
 +
|-
 +
| ''f''<sub>1</sub>
 +
|-
 +
| ''f''<sub>2</sub>
 +
|-
 +
| ''f''<sub>3</sub>
 +
|-
 +
| ''f''<sub>4</sub>
 +
|-
 +
| ''f''<sub>5</sub>
 +
|-
 +
| ''f''<sub>6</sub>
 +
|-
 +
| ''f''<sub>7</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ()
 +
|-
 +
| &nbsp;(''x'')(''y'')&nbsp;
 +
|-
 +
| &nbsp;(''x'')&nbsp;''y''&nbsp;&nbsp;
 +
|-
 +
| &nbsp;(''x'')&nbsp;&nbsp;&nbsp;&nbsp;
 +
|-
 +
| &nbsp;&nbsp;''x''&nbsp;(''y'')&nbsp;
 +
|-
 +
| &nbsp;&nbsp;&nbsp;&nbsp;(''y'')&nbsp;
 +
|-
 +
| &nbsp;(''x'',&nbsp;''y'')&nbsp;
 +
|-
 +
| &nbsp;(''x''&nbsp;&nbsp;''y'')&nbsp;
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| 0 0 0 0
 +
|-
 +
| 0 0 0 1
 +
|-
 +
| 0 0 1 0
 +
|-
 +
| 0 0 1 1
 +
|-
 +
| 0 1 0 0
 +
|-
 +
| 0 1 0 1
 +
|-
 +
| 0 1 1 0
 +
|-
 +
| 0 1 1 1
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ()
 +
|-
 +
| ()
 +
|-
 +
| &nbsp;(''u'')(''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u'')(''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u'',&nbsp;''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u'',&nbsp;''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u''&nbsp;&nbsp;''v'')&nbsp;
 +
|-
 +
| &nbsp;(''u''&nbsp;&nbsp;''v'')&nbsp;
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ''f''<sub>0</sub>
 +
|-
 +
| ''f''<sub>0</sub>
 +
|-
 +
| ''f''<sub>1</sub>
 +
|-
 +
| ''f''<sub>1</sub>
 +
|-
 +
| ''f''<sub>6</sub>
 +
|-
 +
| ''f''<sub>6</sub>
 +
|-
 +
| ''f''<sub>7</sub>
 +
|-
 +
| ''f''<sub>7</sub>
 +
|}
 +
|-
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ''f''<sub>8</sub>
 +
|-
 +
| ''f''<sub>9</sub>
 +
|-
 +
| ''f''<sub>10</sub>
 +
|-
 +
| ''f''<sub>11</sub>
 +
|-
 +
| ''f''<sub>12</sub>
 +
|-
 +
| ''f''<sub>13</sub>
 +
|-
 +
| ''f''<sub>14</sub>
 +
|-
 +
| ''f''<sub>15</sub>
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| &nbsp;&nbsp;''x''&nbsp;&nbsp;''y''&nbsp;&nbsp;
 +
|-
 +
| ((''x'',&nbsp;''y''))
 +
|-
 +
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''y''&nbsp;&nbsp;
 +
|-
 +
| &nbsp;(''x''&nbsp;(''y''))
 +
|-
 +
| &nbsp;&nbsp;''x''&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 +
|-
 +
| ((''x'')&nbsp;''y'')&nbsp;
 +
|-
 +
| ((''x'')(''y''))
 +
|-
 +
| (())
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| 1 0 0 0
 +
|-
 +
| 1 0 0 1
 +
|-
 +
| 1 0 1 0
 +
|-
 +
| 1 0 1 1
 +
|-
 +
| 1 1 0 0
 +
|-
 +
| 1 1 0 1
 +
|-
 +
| 1 1 1 0
 +
|-
 +
| 1 1 1 1
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;
 +
|-
 +
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;
 +
|-
 +
| ((''u'',&nbsp;''v''))
 +
|-
 +
| ((''u'',&nbsp;''v''))
 +
|-
 +
| ((''u'')(''v''))
 +
|-
 +
| ((''u'')(''v''))
 +
|-
 +
| (())
 +
|-
 +
| (())
 +
|}
 +
|
 +
{| cellpadding="2" style="background:lightcyan"
 +
| ''f''<sub>8</sub>
 +
|-
 +
| ''f''<sub>8</sub>
 +
|-
 +
| ''f''<sub>9</sub>
 +
|-
 +
| ''f''<sub>9</sub>
 +
|-
 +
| ''f''<sub>14</sub>
 +
|-
 +
| ''f''<sub>14</sub>
 +
|-
 +
| ''f''<sub>15</sub>
 +
|-
 +
| ''f''<sub>15</sub>
 +
|}
 +
|}
 +
</font><br>
    
Given the alphabets <font face="lucida calligraphy">U</font>&nbsp;=&nbsp;{''u'',&nbsp;''v''} and <font face="lucida calligraphy">X</font>&nbsp;=&nbsp;{''x'',&nbsp;''y''}, along with the corresponding universes of discourse ''U''<sup>&nbsp;&bull;</sup> and ''X''<sup>&nbsp;&bull;</sup>&nbsp;<math>\cong</math>&nbsp;['''B'''<sup>2</sup>], how many logical transformations of the general form ''G''&nbsp;=&nbsp;‹''G''<sub>1</sub>,&nbsp;''G''<sub>2</sub>›&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup> are there?
 
Given the alphabets <font face="lucida calligraphy">U</font>&nbsp;=&nbsp;{''u'',&nbsp;''v''} and <font face="lucida calligraphy">X</font>&nbsp;=&nbsp;{''x'',&nbsp;''y''}, along with the corresponding universes of discourse ''U''<sup>&nbsp;&bull;</sup> and ''X''<sup>&nbsp;&bull;</sup>&nbsp;<math>\cong</math>&nbsp;['''B'''<sup>2</sup>], how many logical transformations of the general form ''G''&nbsp;=&nbsp;‹''G''<sub>1</sub>,&nbsp;''G''<sub>2</sub>›&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup> are there?
Jon Awbrey
12,080

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