[Inquiry] Re: Cactus Rules

[Jon Awbrey]

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CR.  Note 6

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Within each space of boolean functions {f : B^k -> B},
altogether ranking a cardinality of 2^(2^k) functions,
there are several standard subsets of cardinality 2^k
that rate special mention and study.  One such subset
is the space of linear functions, known algebraically
as the set of "homomorphisms" {hom : B^k -> B} or the
"dual space" X*, because it is dual to the coordinate
space X of "points" or "vectors" in B^k.

In the present setting, where k = 3, we may expect to find
2^3 = 8 linear functions of the abstract type h : B^3 -> B.

Table 2 shows the q_j that are linear functions, together
with their boolean complements or their logical negations.

Table 2.  Linear Propositions and Their Complements
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_0     | q_00000000 | 0 0 0 0 0 0 0 0 |        ( )        |
|         |            |                 |                   |
| q_240   | q_11110000 | 1 1 1 1 0 0 0 0 |    p              |
|         |            |                 |                   |
| q_204   | q_11001100 | 1 1 0 0 1 1 0 0 |         q         |
|         |            |                 |                   |
| q_170   | q_10101010 | 1 0 1 0 1 0 1 0 |              r    |
|         |            |                 |                   |
| q_60    | q_00111100 | 0 0 1 1 1 1 0 0 |   (p ,  q)        |
|         |            |                 |                   |
| q_90    | q_01011010 | 0 1 0 1 1 0 1 0 |   (p ,       r)   |
|         |            |                 |                   |
| q_102   | q_01100110 | 0 1 1 0 0 1 1 0 |        (q ,  r)   |
|         |            |                 |                   |
| q_150   | q_10010110 | 1 0 0 1 0 1 1 0 |   (p , (q ,  r))  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_255   | q_11111111 | 1 1 1 1 1 1 1 1 |       (( ))       |
|         |            |                 |                   |
| q_15    | q_00001111 | 0 0 0 0 1 1 1 1 |   (p)             |
|         |            |                 |                   |
| q_51    | q_00110011 | 0 0 1 1 0 0 1 1 |        (q)        |
|         |            |                 |                   |
| q_85    | q_01010101 | 0 1 0 1 0 1 0 1 |             (r)   |
|         |            |                 |                   |
| q_195   | q_11000011 | 1 1 0 0 0 0 1 1 |  ((p ,  q))       |
|         |            |                 |                   |
| q_165   | q_10100101 | 1 0 1 0 0 1 0 1 |  ((p ,       r))  |
|         |            |                 |                   |
| q_153   | q_10011001 | 1 0 0 1 1 0 0 1 |       ((q ,  r))  |
|         |            |                 |                   |
| q_105   | q_01101001 | 0 1 1 0 1 0 0 1 |  ((p , (q ,  r))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

The Figures that follow give a representative selection
of the corresponding cacti in all their greenest glory.

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `( )` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_0` ` ` ` | ` ` ` ` | ` ` ` q_255 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` `(p)` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_240 ` ` ` | ` ` ` ` | ` ` ` q_15` ` ` ` |
o-------------------o ` ` ` ` o-------------------o

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p , q)` ` ` | ` ` ` ` | ` ` ((p , q)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_60` ` ` ` | ` ` ` ` | ` ` ` q_195 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` r ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o---o ` ` |
| ` ` ` ` q ` r ` ` | ` ` ` ` | ` ` ` p `\ /` ` ` |
| ` ` ` ` o---o ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p `\ /` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p , (q , r)) ` | ` ` ` ` | `((p , (q , r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_150 ` ` ` | ` ` ` ` | ` ` ` q_105 ` ` ` |
o-------------------o ` ` ` ` o-------------------o

Beannachtaí na Féile Pádraig oraibh go leir!

Jon Awbrey

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