Re: Cactus Rules

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

CR.  Note 18

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

Table 14 shows the propositions q_i : B^3 -> B whose "fibers of truth",
that is, whose pre-images of 1, have the form of a single point in B^3
together with the three points that make up its immediate neighborhood.
Here I use the alternative syntax "x + y" for the exclusive-or (x , y).

Table 14.  Proximal Propositions
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_23    | q_00010111 | 0 0 0 1 0 1 1 1 | (p)(q)(r) + ((p),(q),(r)) |
|         |            |                 |                           |
| q_43    | q_00101011 | 0 0 1 0 1 0 1 1 | (p)(q) r  + ((p),(q), r ) |
|         |            |                 |                           |
| q_77    | q_01001101 | 0 1 0 0 1 1 0 1 | (p) q (r) + ((p), q ,(r)) |
|         |            |                 |                           |
| q_142   | q_10001110 | 1 0 0 0 1 1 1 0 | (p) q  r  + ((p), q , r ) |
|         |            |                 |                           |
| q_113   | q_01110001 | 0 1 1 1 0 0 0 1 |  p (q)(r) + ( p ,(q),(r)) |
|         |            |                 |                           |
| q_178   | q_10110010 | 1 0 1 1 0 0 1 0 |  p (q) r  + ( p ,(q), r ) |
|         |            |                 |                           |
| q_212   | q_11010100 | 1 1 0 1 0 1 0 0 |  p  q (r) + ( p , q ,(r)) |
|         |            |                 |                           |
| q_232   | q_11101000 | 1 1 1 0 1 0 0 0 |  p  q  r  + ( p , q , r ) |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o