[Inquiry] Re: Cactus Rules

[Jon Awbrey]

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CR.  Note 2

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One of the first things I note is that several whole families of
otherwise mystifying and obfuscaciously expressed rules take on
remarkably simple and transparently related expressions in the
cactus syntax.

For example, Table 1 exhibits the cactus syntax for
an especially interesting family of ECAR's, that is,
boolean maps of the concrete shape [p, q, r] -> [q],
or the abstract type q_j : B^3 -> B.

Table 1.  A Family of Propositional Forms On Three Variables
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_22 ` `| q_00010110 | 0 0 0 1 0 1 1 0 | `((p), (q), (r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_41 ` `| q_00101001 | 0 0 1 0 1 0 0 1 | `((p), (q), `r ) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_73 ` `| q_01001001 | 0 1 0 0 1 0 0 1 | `((p), `q , (r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_134 ` | q_10000110 | 1 0 0 0 0 1 1 0 | `((p), `q , `r ) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_97 ` `| q_01100001 | 0 1 1 0 0 0 0 1 | `( p , (q), (r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_146 ` | q_10010010 | 1 0 0 1 0 0 1 0 | `( p , (q), `r ) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_148 ` | q_10010100 | 1 0 0 1 0 1 0 0 | `( p , `q , (r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_104 ` | q_01101000 | 0 1 1 0 1 0 0 0 | `( p , `q , `r ) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_233 ` | q_11101001 | 1 1 1 0 1 0 0 1 | (((p), (q), (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_214 ` | q_11010110 | 1 1 0 1 0 1 1 0 | (((p), (q), `r )) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_182 ` | q_10110110 | 1 0 1 1 0 1 1 0 | (((p), `q , (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_121 ` | q_01111001 | 0 1 1 1 1 0 0 1 | (((p), `q , `r )) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_158 ` | q_10011110 | 1 0 0 1 1 1 1 0 | (( p , (q), (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_109 ` | q_01101101 | 0 1 1 0 1 1 0 1 | (( p , (q), `r )) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_107 ` | q_01101011 | 0 1 1 0 1 0 1 1 | (( p , `q , (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_151 ` | q_10010111 | 1 0 0 1 0 1 1 1 | (( p , `q , `r )) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o

I invite the Reader to compare these expressions with their
corresponding numbers, the same boolean functions expressed
in terms of operators from the set {Not, And, Or, Xor}, for
example, as shown in the "Wolfram Atlas of Simple Programs":

http://atlas.wolfram.com/01/01/views/172/TableView.html

Jon Awbrey

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