fractal fairy tales

[Jon Awbrey]

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

seqfan folks,

it doesn't look like i'll get time to think about this anytime soon,
so i thought i'd ask and see if anybody here has a quick explanation.
i have noticed the self-similar patterns in the following tables that
i made in the process of studying "higher order propositions", really
just the boolean functions of type m : (B^k -> B) -> B.  here are the
tables in question, which are really just the incidence matrices for
the implication relation (and its converse) between boolean functions
f_i, f_j : B^2 -> B.  for example, in table 11, f_0 (constant false)
implies every other function.

context at:

http://suo.ieee.org/ontology/msg04247.html

Table 10.  Qualifiers of Implication Ordering:  !a!_i f  =  !Y!(f_i => f)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|  | x | 1100 |    f     |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |
|  | y | 1010 |          |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |
| f \  |      |          |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|      |      |          |                                               |
| f_0  | 0000 |    ()    |                                             1 |
|      |      |          |                                               |
| f_1  | 0001 |  (x)(y)  |                                          1  1 |
|      |      |          |                                               |
| f_2  | 0010 |  (x) y   |                                       1     1 |
|      |      |          |                                               |
| f_3  | 0011 |  (x)     |                                    1  1  1  1 |
|      |      |          |                                               |
| f_4  | 0100 |   x (y)  |                                 1           1 |
|      |      |          |                                               |
| f_5  | 0101 |     (y)  |                              1  1        1  1 |
|      |      |          |                                               |
| f_6  | 0110 |  (x, y)  |                           1     1     1     1 |
|      |      |          |                                               |
| f_7  | 0111 |  (x  y)  |                        1  1  1  1  1  1  1  1 |
|      |      |          |                                               |
| f_8  | 1000 |   x  y   |                     1                       1 |
|      |      |          |                                               |
| f_9  | 1001 | ((x, y)) |                  1  1                    1  1 |
|      |      |          |                                               |
| f_10 | 1010 |      y   |               1     1                 1     1 |
|      |      |          |                                               |
| f_11 | 1011 |  (x (y)) |            1  1  1  1              1  1  1  1 |
|      |      |          |                                               |
| f_12 | 1100 |   x      |         1           1           1           1 |
|      |      |          |                                               |
| f_13 | 1101 | ((x) y)  |      1  1        1  1        1  1        1  1 |
|      |      |          |                                               |
| f_14 | 1110 | ((x)(y)) |   1     1     1     1     1     1     1     1 |
|      |      |          |                                               |
| f_15 | 1111 |   (())   |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
|      |      |          |                                               |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

Table 11.  Qualifiers of Implication Ordering:  !b!_i f  =  !Y!(f => f_i)
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|  | x | 1100 |    f     |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |
|  | y | 1010 |          |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 |
| f \  |      |          |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
|      |      |          |                                               |
| f_0  | 0000 |    ()    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 |
|      |      |          |                                               |
| f_1  | 0001 |  (x)(y)  |   1     1     1     1     1     1     1     1 |
|      |      |          |                                               |
| f_2  | 0010 |  (x) y   |      1  1        1  1        1  1        1  1 |
|      |      |          |                                               |
| f_3  | 0011 |  (x)     |         1           1           1           1 |
|      |      |          |                                               |
| f_4  | 0100 |   x (y)  |            1  1  1  1              1  1  1  1 |
|      |      |          |                                               |
| f_5  | 0101 |     (y)  |               1     1                 1     1 |
|      |      |          |                                               |
| f_6  | 0110 |  (x, y)  |                  1  1                    1  1 |
|      |      |          |                                               |
| f_7  | 0111 |  (x  y)  |                     1                       1 |
|      |      |          |                                               |
| f_8  | 1000 |   x  y   |                        1  1  1  1  1  1  1  1 |
|      |      |          |                                               |
| f_9  | 1001 | ((x, y)) |                           1     1     1     1 |
|      |      |          |                                               |
| f_10 | 1010 |      y   |                              1  1        1  1 |
|      |      |          |                                               |
| f_11 | 1011 |  (x (y)) |                                 1           1 |
|      |      |          |                                               |
| f_12 | 1100 |   x      |                                    1  1  1  1 |
|      |      |          |                                               |
| f_13 | 1101 | ((x) y)  |                                       1     1 |
|      |      |          |                                               |
| f_14 | 1110 | ((x)(y)) |                                          1  1 |
|      |      |          |                                               |
| f_15 | 1111 |   (())   |                                             1 |
|      |      |          |                                               |
o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o

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