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ONT Re: Cactus Rules




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CR.  Note 14

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I will explain my concept of "thematization"
or "thematic extension" after I copy out the
series of Tables that is formed on its basis.
In the meantime, here is a general exposition:

| Jon Awbrey, "Differential Logic and Dynamic Systems"
| DLOG D28.  http://suo.ieee.org/ontology/msg04826.html
| DLOG D29.  http://suo.ieee.org/ontology/msg04827.html
| DLOG D30.  http://suo.ieee.org/ontology/msg04828.html
| DLOG D31.  http://suo.ieee.org/ontology/msg04829.html
| DLOG D32.  http://suo.ieee.org/ontology/msg04830.html
| DLOG D33.  http://suo.ieee.org/ontology/msg04832.html

In order to make the pattern of their construction
more evident, I have left the expressions of the
thematic extensions in their unreduced forms.

Table 10.  Thematic Extensions:  [q, r] -> [p, q, r]
o---------o------------o-----------------o---------------------o
| L_1     | L_2        | L_3             | L_4                 |
|         |            |                 |                     |
| Decimal | Binary     | Vector          | Cactus              |
o---------o------------o-----------------o---------------------o
|         |          p : 1 1 1 1 0 0 0 0 |                     |
|         |          q : 1 1 0 0 1 1 0 0 |                     |
|         |          r : 1 0 1 0 1 0 1 0 |                     |
o---------o------------o-----------------o---------------------o
|         |            |                 |                     |
| q_15    | q_00001111 | 0 0 0 0 1 1 1 1 | ((p ,    ( )    ))  |
|         |            |                 |                     |
| q_30    | q_00011110 | 0 0 0 1 1 1 1 0 | ((p ,  (q) (r)  ))  |
|         |            |                 |                     |
| q_45    | q_00101101 | 0 0 1 0 1 1 0 1 | ((p ,  (q)  r   ))  |
|         |            |                 |                     |
| q_60    | q_00111100 | 0 0 1 1 1 1 0 0 | ((p ,  (q)      ))  |
|         |            |                 |                     |
| q_75    | q_01001011 | 0 1 0 0 1 0 1 1 | ((p ,   q  (r)  ))  |
|         |            |                 |                     |
| q_90    | q_01011010 | 0 1 0 1 1 0 1 0 | ((p ,      (r)  ))  |
|         |            |                 |                     |
| q_105   | q_01101001 | 0 1 1 0 1 0 0 1 | ((p ,  (q , r)  ))  |
|         |            |                 |                     |
| q_120   | q_01111000 | 0 1 1 1 1 0 0 0 | ((p ,  (q   r)  ))  |
|         |            |                 |                     |
| q_135   | q_10000111 | 1 0 0 0 0 1 1 1 | ((p ,   q   r   ))  |
|         |            |                 |                     |
| q_150   | q_10010110 | 1 0 0 1 0 1 1 0 | ((p , ((q , r)) ))  |
|         |            |                 |                     |
| q_165   | q_10100101 | 1 0 1 0 0 1 0 1 | ((p ,       r   ))  |
|         |            |                 |                     |
| q_180   | q_10110100 | 1 0 1 1 0 1 0 0 | ((p ,  (q  (r)) ))  |
|         |            |                 |                     |
| q_195   | q_11000011 | 1 1 0 0 0 0 1 1 | ((p ,   q       ))  |
|         |            |                 |                     |
| q_210   | q_11010010 | 1 1 0 1 0 0 1 0 | ((p , ((q)  r)  ))  |
|         |            |                 |                     |
| q_225   | q_11100001 | 1 1 1 0 0 0 0 1 | ((p , ((q) (r)) ))  |
|         |            |                 |                     |
| q_240   | q_11110000 | 1 1 1 1 0 0 0 0 | ((p ,           ))  |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o

Jon Awbrey

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