# Cactus Rules

## Note 1

```With an eye toward the aims of the NKS Forum, I've begun to work out
a translation of the "elementary cellular automaton rules" (ECAR's),
in effect, just the boolean functions of abstract type q : B^3 -> B,
into cactus language, and I'll post a selection of my working notes
here.  By way of the briefest possible reminder, this cactus syntax,
in its existential interpretation and its traverse-string redaction,
uses just two series of k-adic connectives, first, the concatenation
a bracket of the form (e_1, ..., e_k) is read to say that exactly one
of the k expressions e_1, ..., e_k is false.  I may sometimes refer to
this bracket as a k-adic "boundary operator" or a k-place "cactus lobe".

Reference Material:

http://atlas.wolfram.com/
http://atlas.wolfram.com/01/01/
http://atlas.wolfram.com/01/01/views/3/TableView.html
http://atlas.wolfram.com/01/01/views/87/TableView.html
http://atlas.wolfram.com/01/01/views/172/TableView.html

Incidental Musement:

http://www.pinball.com/games/cactus/
```

## Note 2

```One of the first things I note is that several whole families
of otherwise enigmatic and obscurely 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 {And, Or, Xor, Not}, for
example, as shown in the "Wolfram Atlas of Simple Programs":

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

## Note 3

```Here are the parse-graph portraits of the family of cacti
that we examined last time, listed in complementary pairs.

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p q r       |
|                   |         |       o-o-o       |
|       p q r       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ( p , q , r )   |         |  (( p , q , r ))  |
o-------------------o         o-------------------o
|       q_104       |         |       q_151       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |       p           |
|                   |         |       o           |
|       p           |         |       | q r       |
|       o           |         |       o-o-o       |
|       | q r       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ((p), q , r )   |         |  (((p), q , r ))  |
o-------------------o         o-------------------o
|       q_134       |         |       q_121       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |         q         |
|                   |         |         o         |
|         q         |         |       p | r       |
|         o         |         |       o-o-o       |
|       p | r       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ( p ,(q), r )   |         |  (( p ,(q), r ))  |
o-------------------o         o-------------------o
|       q_146       |         |       q_109       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |           r       |
|                   |         |           o       |
|           r       |         |       p q |       |
|           o       |         |       o-o-o       |
|       p q |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ( p , q ,(r))   |         |  (( p , q ,(r)))  |
o-------------------o         o-------------------o
|       q_148       |         |       q_107       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |       p q         |
|                   |         |       o o         |
|       p q         |         |       | | r       |
|       o o         |         |       o-o-o       |
|       | | r       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ((p),(q), r )   |         |  (((p),(q), r ))  |
o-------------------o         o-------------------o
|       q_41        |         |       q_214       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |       p   r       |
|                   |         |       o   o       |
|       p   r       |         |       | q |       |
|       o   o       |         |       o-o-o       |
|       | q |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ((p), q ,(r))   |         |  (((p), q ,(r)))  |
o-------------------o         o-------------------o
|       q_73        |         |       q_182       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |         q r       |
|                   |         |         o o       |
|         q r       |         |       p | |       |
|         o o       |         |       o-o-o       |
|       p | |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ( p ,(q),(r))   |         |  (( p ,(q),(r)))  |
o-------------------o         o-------------------o
|       q_97        |         |       q_158       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |       p q r       |
|                   |         |       o o o       |
|       p q r       |         |       | | |       |
|       o o o       |         |       o-o-o       |
|       | | |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ((p),(q),(r))   |         |  (((p),(q),(r)))  |
o-------------------o         o-------------------o
|       q_22        |         |       q_233       |
o-------------------o         o-------------------o

As I work through the 256 ECAR's or functions q_j : B^3 -> B,
I will keep an updated copy of my worksheet as an attachment
to the first posting on this thread at the NKS Forum website:

The interested reader is invited to help check this work,
as errors are almost inevitable in this type of exercise.
Plus, I can't always get expressions that are as elegant
as I might like, and it may be that other eyes would see
forms more economical than the ones that strike me first.
```

## Note 4

```Given the novelty of the cactus calculus, it is probably
wise to run through a representative sample of the forms
just set down, to note some principles of interpretation,
and to pick up a few clues as to their ordinary language
renderings.  Throughout the rest of this reading it will
be good to recall that "truth", or a boolean valaue of 1,
is represented by a blank string or a blank-labeled node,
while "falsity", or a boolean value of 0, is rendered as
the string "()" or an unlabeled terminal edge, a "spike".

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p q r       |
|                   |         |       o-o-o       |
|       p q r       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ( p , q , r )   |         |  (( p , q , r ))  |
o-------------------o         o-------------------o
|       q_104       |         |       q_151       |
o-------------------o         o-------------------o

The function q_104 : B^3 -> B is a basic 3-lobe,
interpreted as the "just one false" operator on
three boolean variables, and the function q_151
is its boolean complement or its exact negation.

o-------------------o         o-------------------o
|                   |         |       p           |
|                   |         |       o           |
|       p           |         |       | q r       |
|       o           |         |       o-o-o       |
|       | q r       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ((p), q , r )   |         |  (((p), q , r ))  |
o-------------------o         o-------------------o
|       q_134       |         |       q_121       |
o-------------------o         o-------------------o

The operation of q_134 can be understood by asking
what happens if p is true, in effect, if the label
"p" disappears, leaving only its supporting spike.
That spike, the unique false argument on the lobe,
punctures the lobe beneath, if you will, and what
abides is the statement "q r", that is, "q and r".
On the other hand, if p is (), then the branch (p)
appears to be (()), which reduces to true, and so
it disappears instead, leaving just (q, r), which
is tantamount to stating that q is not equal to r.
In sum the cases are:  p q r, (p) q (r), (p)(q) r.
Once again, q_121 is just the complement of q_134.

o-------------------o         o-------------------o
|                   |         |       p q r       |
|                   |         |       o o o       |
|       p q r       |         |       | | |       |
|       o o o       |         |       o-o-o       |
|       | | |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ((p),(q),(r))   |         |  (((p),(q),(r)))  |
o-------------------o         o-------------------o
|       q_22        |         |       q_233       |
o-------------------o         o-------------------o

The rest of this gang can be dispatched by the same method.
But I want to single out for special mention the form q_22,
the "just one true" operator that is especially handy when
the time comes to specify a partition of the universe into
a number of mutually exclusive and exhaustive territories,
here envisioned to salute the flags p, q, r, respectively.
```

## Note 5

```So long as we're seeing the sights at Cactus Junction,
we might as well take a gander at a computational way
to assay the import of any ole cactus expression that
comes down the pike.  Way out here, and elsewhere, too,
the computational clarification of a formal expression
is claimed to yield its canonical or its "normal" form.
Finer distinctions can be weighed, of course, and there
is always the problem of just how, exactly, and, indeed,
even whether such forms will be forthcoming from a given
cut of syntax for a given objective domain, or any other
wide open space.  But the notion of a "normal form" is
cast in the right direction, and so it'll do for now.

By way of example, let's examine the subtype of cactoid expression
that is typified by q_97 and its complement q_158, and that hardly
got its just deserts in the way of attention the last time around.

o-------------------o         o-------------------o
|                   |         |         q r       |
|                   |         |         o o       |
|         q r       |         |       p | |       |
|         o o       |         |       o-o-o       |
|       p | |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
o-------------------o         o-------------------o
|   ( p ,(q),(r))   |         |  (( p ,(q),(r)))  |
o-------------------o         o-------------------o
|       q_97        |         |       q_158       |
o-------------------o         o-------------------o

Cactus forms of the generic shape (g, (s_1), ..., (s_k))
are those that arise when we have a "genus and species"
or a "pie chart" arrangement of logical features, where
g is the genus and the k species are s_1 through s_k,
or g is the whole pie and the slices are the s_j.

o-------------------------------------------------o
|                                                 |
|                       s_1   s_k                 |
|                        o     o                  |
|                  g     |     |                  |
|                  o-----o-...-o                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o

We can reason out the meaning of all such expressions
by using the case analysis tactic that we used before.
If g is true, then it's just like "g" wasn't there at
all, and the expression comes down to the case below:

o-------------------------------------------------o
|                                                 |
|                   s_1     s_k                   |
|                    o       o                    |
|                    |       |                    |
|                    o--...--o                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o

But this expresses the "just one true" condition that partitions
the remaining space, that is to say, the space where g is true,
into k sectors where each of the s_j in its own turn is true.

On the other hand, in the case that g is false, we are left
with a (k+1)-lobe that is known to bear this one bare spike:

o-------------------------------------------------o
|                                                 |
|                       s_1   s_k                 |
|                  o     o     o                  |
|                  |     |     |                  |
|                  o-----o-...-o                  |
|                   \         /                   |
|                    \       /                    |
|                     \     /                     |
|                      \   /                      |
|                       \ /                       |
|                        @                        |
|                                                 |
o-------------------------------------------------o

If that expression as a whole is going to turn out to be true,
then there can be only one expression that evaluates to false
on its argument list, and since we already have it in custody,
we know that the remaining arguments, (s_1), ..., (s_k), will
all have to be true.  In effect, the spike collapses the lobe
to a node, leaving a conjunction of the negations of the s_j.

o-------------------------------------------------o
|                                                 |
|                  s_1     s_k                    |
|                   o  ...  o                     |
|                    \  |  /                      |
|                     \ | /                       |
|                      \|/                        |
|                       @                         |
|                                                 |
o-------------------------------------------------o

In summation, we have the following interpretation:
If g is true, then exactly one of the s_j is true;
if g is false, then all of the s_j are false, too.

That is not yet a method that would be amenable to
computational routine, but it does get us part way.
```

## Note 6

```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!
```

## Note 7

```Had I been thinking ahead, I might have mentioned this first,
but now that aspects of algebra and geometry have intruded on
our logical paradise, in the guise of the dual space X*, let's
give belated notice to one family of propositions that have been
basic to our enterprise all along, whether we noticed them or not.

In a k-dimensional universe of discourse X% = [x_1, ..., x_k] the
position space X = <|x_1, ..., x_k|> is isomorphic to B^k and the
proposition space X^ = (X -> B) = {f : X -> B} bears the abstract
type B^k -> B.  In algebra and geometry, as a rule, one tends to
take position spaces and function spaces together in pairs, and
so we assign the universe X% a "stereotype" of <B^k, B^k -> B>,
or B^k +-> B, for short.  I like to think of these spaces as
the "paint layer" X and "draw layer" X^ of the universe X%.

What I need to make a point of at this point is that the k-set
of logical features !X! = {x_1, ..., x_k} that we invoke as the
basis of the universe of discourse also constitutes an important
family of propositions x_j : B^k -> B, for j = 1 to k.  These are
called by any one of several different names:  "basic propositions",
"coordinate projections", or "simple propositions".

Table 0 accords this family of simple propositions their
formal recognition, for the present case of 3 dimensions.

Table 0.  Simple 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_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    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Of course, we've already seen this 3-set of basic propositions
numbered among the (2^3)-set of linear propositions in Table 2.

Additional discussion of these underpinnings can be found here:

| Jon Awbrey, "Differential Logic and Dynamic Systems"

Especially:

DLOG D2.  http://stderr.org/pipermail/inquiry/2003-May/000480.html
DLOG D5.  http://stderr.org/pipermail/inquiry/2003-May/000483.html

With that out of the way, I'll try to
get back to the main event next time.
```

## Note 8

```In any k-dimensional universe of discourse X% = [x_1, ..., x_k]
there are two other (2^k)-clans of propositions that ordinarily
merit special attention.  These are the "positive" propositions
and the "singular" propositions, tabulated for the present case
k = 3 in Tables 3 and 4, respectively, as usual throwing in the
logical complements just for good measure.

Table 3.  Positive 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_255   | q_11111111 | 1 1 1 1 1 1 1 1 |       (( ))       |
|         |            |                 |                   |
| 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_192   | q_11000000 | 1 1 0 0 0 0 0 0 |    p    q         |
|         |            |                 |                   |
| q_160   | q_10100000 | 1 0 1 0 0 0 0 0 |    p         r    |
|         |            |                 |                   |
| q_136   | q_10001000 | 1 0 0 0 1 0 0 0 |         q    r    |
|         |            |                 |                   |
| q_128   | q_10000000 | 1 0 0 0 0 0 0 0 |    p    q    r    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_0     | q_00000000 | 0 0 0 0 0 0 0 0 |        ( )        |
|         |            |                 |                   |
| 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_63    | q_00111111 | 0 0 1 1 1 1 1 1 |   (p    q)        |
|         |            |                 |                   |
| q_95    | q_01011111 | 0 1 0 1 1 1 1 1 |   (p         r)   |
|         |            |                 |                   |
| q_119   | q_01110111 | 0 1 1 1 0 1 1 1 |        (q    r)   |
|         |            |                 |                   |
| q_127   | q_01111111 | 0 1 1 1 1 1 1 1 |   (p    q    r)   |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Table 4.  Singular 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_1     | q_00000001 | 0 0 0 0 0 0 0 1 |   (p)  (q)  (r)   |
|         |            |                 |                   |
| q_2     | q_00000010 | 0 0 0 0 0 0 1 0 |   (p)  (q)   r    |
|         |            |                 |                   |
| q_4     | q_00000100 | 0 0 0 0 0 1 0 0 |   (p)   q   (r)   |
|         |            |                 |                   |
| q_8     | q_00001000 | 0 0 0 0 1 0 0 0 |   (p)   q    r    |
|         |            |                 |                   |
| q_16    | q_00010000 | 0 0 0 1 0 0 0 0 |    p   (q)  (r)   |
|         |            |                 |                   |
| q_32    | q_00100000 | 0 0 1 0 0 0 0 0 |    p   (q)   r    |
|         |            |                 |                   |
| q_64    | q_01000000 | 0 1 0 0 0 0 0 0 |    p    q   (r)   |
|         |            |                 |                   |
| q_128   | q_10000000 | 1 0 0 0 0 0 0 0 |    p    q    r    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_254   | q_11111110 | 1 1 1 1 1 1 1 0 |  ((p)  (q)   r))  |
|         |            |                 |                   |
| q_253   | q_11111101 | 1 1 1 1 1 1 0 1 |  ((p)  (q)   r )  |
|         |            |                 |                   |
| q_251   | q_11111011 | 1 1 1 1 1 0 1 1 |  ((p)   q   (r))  |
|         |            |                 |                   |
| q_247   | q_11110111 | 1 1 1 1 0 1 1 1 |  ((p)   q    r )  |
|         |            |                 |                   |
| q_239   | q_11101111 | 1 1 1 0 1 1 1 1 |  ( p   (q)  (r))  |
|         |            |                 |                   |
| q_223   | q_11011111 | 1 1 0 1 1 1 1 1 |  ( p   (q)   r )  |
|         |            |                 |                   |
| q_191   | q_10111111 | 1 0 1 1 1 1 1 1 |  ( p    q   (r))  |
|         |            |                 |                   |
| q_127   | q_01111111 | 0 1 1 1 1 1 1 1 |  ( p    q    r )  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
```

## Note 9

```In the language of cacti, as in Peirce's existential graphs,
the implication p => q takes the form (p (q)), which can be
parsed in a revealing manner as "not p without q".  Thus it
forms the counterpoint to its counter-exemplary form, p (q),
which may be parsed as "p without q", or just "p and not q".

The parse-graph of (p (q)) is a particular type of tree,
that my school of thought in graph theory nomenclates as
a "painted and rooted tree" (PART).  The symbols from the
alphabet !X! of logical marks, in our case, "p", "q", "r",
are called "paints" as a way of signifying that one can put
as many of them as one likes on a node, or none at all, and
that there is no requirement to use all of the paints of the
given palette !X! on any particular graph.  In my etchings,
the root node is singled out with the amphora sign "@".

The graph of a simple implication can be drawn in any way that
a free rooted tree can be, but it is frequently convenient to
portray it as we see below, partly because of how often we
find ourselves linking implications in stepwise series.

o-------------------------------------------------o
|                                                 |
|                  p           q                  |
|                  o-----------o                  |
|                   \                             |
|                    \                            |
|                     \                           |
|                      \                          |
|                       \                         |
|                        @                        |
|                                                 |
o-------------------------------------------------o
|                    ( p ( q ))                   |
o-------------------------------------------------o

Table 5 shows a number of ECAR's that have the form
of simple implications or their logical complements.

Table 5.  Variations on a Theme of Implication
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_207   | q_11001111 | 1 1 0 0 1 1 1 1 |   (p   (q))       |
|         |            |                 |                   |
| q_175   | q_10101111 | 1 0 1 0 1 1 1 1 |   (p        (r))  |
|         |            |                 |                   |
| q_187   | q_10111011 | 1 0 1 1 1 0 1 1 |        (q   (r))  |
|         |            |                 |                   |
| q_243   | q_11110011 | 1 1 1 1 0 0 1 1 |  ((p)   q)        |
|         |            |                 |                   |
| q_245   | q_11110101 | 1 1 1 1 0 1 0 1 |  ((p)        r)   |
|         |            |                 |                   |
| q_221   | q_11011101 | 1 1 0 1 1 1 0 1 |       ((q)   r)   |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_48    | q_00110000 | 0 0 1 1 0 0 0 0 |    p   (q)        |
|         |            |                 |                   |
| q_80    | q_01010000 | 0 1 0 1 0 0 0 0 |    p        (r)   |
|         |            |                 |                   |
| q_68    | q_01000100 | 0 1 0 0 0 1 0 0 |         q   (r)   |
|         |            |                 |                   |
| q_12    | q_00001100 | 0 0 0 0 1 1 0 0 |   (p)   q         |
|         |            |                 |                   |
| q_10    | q_00001010 | 0 0 0 0 1 0 1 0 |   (p)        r    |
|         |            |                 |                   |
| q_34    | q_00100010 | 0 0 1 0 0 0 1 0 |        (q)   r    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
```

## Note 10

```Table 6.  More Variations on a Theme of Implication
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_176   | q_10110000 | 1 0 1 1 0 0 0 0 |    p   (q   (r))  |
|         |            |                 |                   |
| q_208   | q_11010000 | 1 1 0 1 0 0 0 0 |    p   (r   (q))  |
|         |            |                 |                   |
| q_11    | q_00001011 | 0 0 0 0 1 0 1 1 |   (p)  (q   (r))  |
|         |            |                 |                   |
| q_13    | q_00001101 | 0 0 0 0 1 1 0 1 |   (p)  (r   (q))  |
|         |            |                 |                   |
| q_140   | q_10001100 | 1 0 0 0 1 1 0 0 |    q   (p   (r))  |
|         |            |                 |                   |
| q_196   | q_11000100 | 1 1 0 0 0 1 0 0 |    q   (r   (p))  |
|         |            |                 |                   |
| q_35    | q_00100011 | 0 0 1 0 0 0 1 1 |   (q)  (p   (r))  |
|         |            |                 |                   |
| q_49    | q_00110001 | 0 0 1 1 0 0 0 1 |   (q)  (r   (p))  |
|         |            |                 |                   |
| q_138   | q_10001010 | 1 0 0 0 1 0 1 0 |    r   (p   (q))  |
|         |            |                 |                   |
| q_162   | q_10100010 | 1 0 1 0 0 0 1 0 |    r   (q   (p))  |
|         |            |                 |                   |
| q_69    | q_01000101 | 0 1 0 0 0 1 0 1 |   (r)  (p   (q))  |
|         |            |                 |                   |
| q_81    | q_01010001 | 0 1 0 1 0 0 0 1 |   (r)  (q   (p))  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_79    | q_01001111 | 0 1 0 0 1 1 1 1 |  ( p   (q   (r))) |
|         |            |                 |                   |
| q_47    | q_00101111 | 0 0 1 0 1 1 1 1 |  ( p   (r   (q))) |
|         |            |                 |                   |
| q_244   | q_11110100 | 1 1 1 1 0 1 0 0 |  ((p)  (q   (r))) |
|         |            |                 |                   |
| q_242   | q_11110010 | 1 1 1 1 0 0 1 0 |  ((p)  (r   (q))) |
|         |            |                 |                   |
| q_115   | q_01110011 | 0 1 1 1 0 0 1 1 |  ( q   (p   (r))) |
|         |            |                 |                   |
| q_59    | q_00111011 | 0 0 1 1 1 0 1 1 |  ( q   (r   (p))) |
|         |            |                 |                   |
| q_220   | q_11011100 | 1 1 0 1 1 1 0 0 |  ((q)  (p   (r))) |
|         |            |                 |                   |
| q_206   | q_11001110 | 1 1 0 0 1 1 1 0 |  ((q)  (r   (p))) |
|         |            |                 |                   |
| q_117   | q_01110101 | 0 1 1 1 0 1 0 1 |  ( r   (p   (q))) |
|         |            |                 |                   |
| q_93    | q_01011101 | 0 1 0 1 1 1 0 1 |  ( r   (q   (p))) |
|         |            |                 |                   |
| q_186   | q_10111010 | 1 0 1 1 1 0 1 0 |  ((r)  (p   (q))) |
|         |            |                 |                   |
| q_174   | q_10101110 | 1 0 1 0 1 1 1 0 |  ((r)  (q   (p))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
```

## Note 11

```Table 7.  Conjunctive Implications 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_139   | q_10001011 | 1 0 0 0 1 0 1 1 |   (p (q))(q (r))  |
|         |            |                 |                   |
| q_141   | q_10001101 | 1 0 0 0 1 1 0 1 |   (p (r))(r (q))  |
|         |            |                 |                   |
| q_177   | q_10110001 | 1 0 1 1 0 0 0 1 |   (q (r))(r (p))  |
|         |            |                 |                   |
| q_163   | q_10100011 | 1 0 1 0 0 0 1 1 |   (q (p))(p (r))  |
|         |            |                 |                   |
| q_197   | q_11000101 | 1 1 0 0 0 1 0 1 |   (r (p))(p (q))  |
|         |            |                 |                   |
| q_209   | q_11010001 | 1 1 0 1 0 0 0 1 |   (r (q))(q (p))  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_116   | q_01110100 | 0 1 1 1 0 1 0 0 |  ((p (q))(q (r))) |
|         |            |                 |                   |
| q_114   | q_01110010 | 0 1 1 1 0 0 1 0 |  ((p (r))(r (q))) |
|         |            |                 |                   |
| q_78    | q_01001110 | 0 1 0 0 1 1 1 0 |  ((q (r))(r (p))) |
|         |            |                 |                   |
| q_92    | q_01011100 | 0 1 0 1 1 1 0 0 |  ((q (p))(p (r))) |
|         |            |                 |                   |
| q_58    | q_00111010 | 0 0 1 1 1 0 1 0 |  ((r (p))(p (q))) |
|         |            |                 |                   |
| q_46    | q_00101110 | 0 0 1 0 1 1 1 0 |  ((r (q))(q (p))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
```

## Note 12

```In the language of cacti, unlike Peirce's alpha graphs,
it is possible to represent the logical functions that
correspond to the difference in truth value and the
equality in truth value of two logical variables
in forms that mention each variable only once.

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

We have already noted the initial variations on the themes
of difference and equality among the forms in Table 2 that
gave the linear propositions and their logical complements.
Table 8 enumerates a few more variations along these lines.

Table 8.  More Variations on Difference and Equality
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_96    | q_01100000 | 0 1 1 0 0 0 0 0 |    p   (q ,  r)   |
|         |            |                 |                   |
| q_72    | q_01001000 | 0 1 0 0 1 0 0 0 |    q   (p ,  r)   |
|         |            |                 |                   |
| q_40    | q_00101000 | 0 0 1 0 1 0 0 0 |    r   (p ,  q)   |
|         |            |                 |                   |
| q_144   | q_10010000 | 1 0 0 1 0 0 0 0 |    p  ((q ,  r))  |
|         |            |                 |                   |
| q_132   | q_10000100 | 1 0 0 0 0 1 0 0 |    q  ((p ,  r))  |
|         |            |                 |                   |
| q_130   | q_10000010 | 1 0 0 0 0 0 1 0 |    r  ((p ,  q))  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_6     | q_00000110 | 0 0 0 0 0 1 1 0 |   (p)  (q ,  r)   |
|         |            |                 |                   |
| q_18    | q_00010010 | 0 0 0 1 0 0 1 0 |   (q)  (p ,  r)   |
|         |            |                 |                   |
| q_20    | q_00010100 | 0 0 0 1 0 1 0 0 |   (r)  (p ,  q)   |
|         |            |                 |                   |
| q_9     | q_00001001 | 0 0 0 0 1 0 0 1 |   (p) ((q ,  r))  |
|         |            |                 |                   |
| q_33    | q_00100001 | 0 0 1 0 0 0 0 1 |   (q) ((p ,  r))  |
|         |            |                 |                   |
| q_65    | q_01000001 | 0 1 0 0 0 0 0 1 |   (r) ((p ,  q))  |
|         |            |                 |                   |
o=========o============o=================o===================o
|         |            |                 |                   |
| q_159   | q_10011111 | 1 0 0 1 1 1 1 1 |   (p   (q ,  r))  |
|         |            |                 |                   |
| q_183   | q_10110111 | 1 0 1 1 0 1 1 1 |   (q   (p ,  r))  |
|         |            |                 |                   |
| q_215   | q_11010111 | 1 1 0 1 0 1 1 1 |   (r   (p ,  q))  |
|         |            |                 |                   |
| q_111   | q_01101111 | 0 1 1 0 1 1 1 1 |   (p  ((q ,  r))) |
|         |            |                 |                   |
| q_123   | q_01111011 | 0 1 1 1 1 0 1 1 |   (q  ((p ,  r))) |
|         |            |                 |                   |
| q_125   | q_01111101 | 0 1 1 1 1 1 0 1 |   (r  ((p ,  q))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_249   | q_11111001 | 1 1 1 1 1 0 0 1 |  ((p)  (q ,  r))  |
|         |            |                 |                   |
| q_237   | q_11101101 | 1 1 1 0 1 1 0 1 |  ((q)  (p ,  r))  |
|         |            |                 |                   |
| q_235   | q_11101011 | 1 1 1 0 1 0 1 1 |  ((r)  (p ,  q))  |
|         |            |                 |                   |
| q_246   | q_11110110 | 1 1 1 1 0 1 1 0 |  ((p) ((q ,  r))) |
|         |            |                 |                   |
| q_222   | q_11011110 | 1 1 0 1 1 1 1 0 |  ((q) ((p ,  r))) |
|         |            |                 |                   |
| q_190   | q_10111110 | 1 0 1 1 1 1 1 0 |  ((r) ((p ,  q))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
```

## Note 13

```Table 9.  Conjunctive Differences and Equalities
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_24    | q_00011000 | 0 0 0 1 1 0 0 0 |   (p, q)  (p, r)   |
|         |            |                 |                    |
| q_36    | q_00100100 | 0 0 1 0 0 1 0 0 |   (p, q)  (q, r)   |
|         |            |                 |                    |
| q_66    | q_01000010 | 0 1 0 0 0 0 1 0 |   (p, r)  (q, r)   |
|         |            |                 |                    |
| q_129   | q_10000001 | 1 0 0 0 0 0 0 1 |  ((p, q))((q, r))  |
|         |            |                 |                    |
o---------o------------o-----------------o--------------------o
|         |            |                 |                    |
| q_231   | q_11100111 | 1 1 1 0 0 1 1 1 | ( (p, q)  (p, r) ) |
|         |            |                 |                    |
| q_219   | q_11011011 | 1 1 0 1 1 0 1 1 | ( (p, q)  (q, r) ) |
|         |            |                 |                    |
| q_189   | q_10111101 | 1 0 1 1 1 1 0 1 | ( (p, r)  (q, r) ) |
|         |            |                 |                    |
| q_126   | q_01111110 | 0 1 1 1 1 1 1 0 | (((p, q))((q, r))) |
|         |            |                 |                    |
o---------o------------o-----------------o--------------------o
```

## Note 14

```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
```

## Note 15

```Table 11.  Thematic Extensions:  [p, 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_51    | q_00110011 | 0 0 1 1 0 0 1 1 | ((q ,    ( )    ))  |
|         |            |                 |                     |
| q_54    | q_00110110 | 0 0 1 1 0 1 1 0 | ((q ,  (p) (r)  ))  |
|         |            |                 |                     |
| q_57    | q_00111001 | 0 0 1 1 1 0 0 1 | ((q ,  (p)  r   ))  |
|         |            |                 |                     |
| q_60    | q_00111100 | 0 0 1 1 1 1 0 0 | ((q ,  (p)      ))  |
|         |            |                 |                     |
| q_99    | q_01100011 | 0 1 1 0 0 0 1 1 | ((q ,   p  (r)  ))  |
|         |            |                 |                     |
| q_102   | q_01100110 | 0 1 1 0 0 1 1 0 | ((q ,      (r)  ))  |
|         |            |                 |                     |
| q_105   | q_01101001 | 0 1 1 0 1 0 0 1 | ((q ,  (p , r)  ))  |
|         |            |                 |                     |
| q_108   | q_01101100 | 0 1 1 0 1 1 0 0 | ((q ,  (p   r)  ))  |
|         |            |                 |                     |
| q_147   | q_10010011 | 1 0 0 1 0 0 1 1 | ((q ,   p   r   ))  |
|         |            |                 |                     |
| q_150   | q_10010110 | 1 0 0 1 0 1 1 0 | ((q , ((p , r)) ))  |
|         |            |                 |                     |
| q_153   | q_10011001 | 1 0 0 1 1 0 0 1 | ((q ,       r   ))  |
|         |            |                 |                     |
| q_156   | q_10011100 | 1 0 0 1 1 1 0 0 | ((q ,  (p  (r)) ))  |
|         |            |                 |                     |
| q_195   | q_11000011 | 1 1 0 0 0 0 1 1 | ((q ,   p       ))  |
|         |            |                 |                     |
| q_198   | q_11000110 | 1 1 0 0 0 1 1 0 | ((q , ((p)  r)  ))  |
|         |            |                 |                     |
| q_201   | q_11001001 | 1 1 0 0 1 0 0 1 | ((q , ((p) (r)) ))  |
|         |            |                 |                     |
| q_204   | q_11001100 | 1 1 0 0 1 1 0 0 | ((q ,           ))  |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o
```

## Note 16

```Table 12.  Thematic Extensions:  [p, q] -> [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_85    | q_01010101 | 0 1 0 1 0 1 0 1 | ((r ,    ( )    ))  |
|         |            |                 |                     |
| q_86    | q_01010110 | 0 1 0 1 0 1 1 0 | ((r ,  (p) (q)  ))  |
|         |            |                 |                     |
| q_89    | q_01011001 | 0 1 0 1 1 0 0 1 | ((r ,  (p)  q   ))  |
|         |            |                 |                     |
| q_90    | q_01011010 | 0 1 0 1 1 0 1 0 | ((r ,  (p)      ))  |
|         |            |                 |                     |
| q_101   | q_01100101 | 0 1 1 0 0 1 0 1 | ((r ,   p  (q)  ))  |
|         |            |                 |                     |
| q_102   | q_01100110 | 0 1 1 0 0 1 1 0 | ((r ,      (q)  ))  |
|         |            |                 |                     |
| q_105   | q_01101001 | 0 1 1 0 1 0 0 1 | ((r ,  (p , q)  ))  |
|         |            |                 |                     |
| q_106   | q_01101010 | 0 1 1 0 1 0 1 0 | ((r ,  (p   q)  ))  |
|         |            |                 |                     |
| q_149   | q_10010101 | 1 0 0 1 0 1 0 1 | ((r ,   p   q   ))  |
|         |            |                 |                     |
| q_150   | q_10010110 | 1 0 0 1 0 1 1 0 | ((r , ((p , q)) ))  |
|         |            |                 |                     |
| q_153   | q_10011001 | 1 0 0 1 1 0 0 1 | ((r ,       q   ))  |
|         |            |                 |                     |
| q_154   | q_10011010 | 1 0 0 1 1 0 1 0 | ((r ,  (p  (q)) ))  |
|         |            |                 |                     |
| q_165   | q_10100101 | 1 0 1 0 0 1 0 1 | ((r ,   p       ))  |
|         |            |                 |                     |
| q_166   | q_10100110 | 1 0 1 0 0 1 1 0 | ((r , ((p)  q)  ))  |
|         |            |                 |                     |
| q_169   | q_10101001 | 1 0 1 0 1 0 0 1 | ((r , ((p) (q)) ))  |
|         |            |                 |                     |
| q_170   | q_10101010 | 1 0 1 0 1 0 1 0 | ((r ,           ))  |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o
```

## Note 17

```Table 13.  Differences & Equalities Conjoined with Implications
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_44    | q_00101100 | 0 0 1 0 1 1 0 0 |   (p, q)   (p (r))  |
|         |            |                 |                     |
| q_52    | q_00110100 | 0 0 1 1 0 1 0 0 |   (p, q)   ((p) r)  |
|         |            |                 |                     |
| q_56    | q_00111000 | 0 0 1 1 1 0 0 0 |   (p, q)   (q (r))  |
|         |            |                 |                     |
| q_28    | q_00011100 | 0 0 0 1 1 1 0 0 |   (p, q)   ((q) r)  |
|         |            |                 |                     |
| q_131   | q_10000011 | 1 0 0 0 0 0 1 1 |  ((p, q))  (p (r))  |
|         |            |                 |                     |
| q_193   | q_11000001 | 1 1 0 0 0 0 0 1 |  ((p, q))  ((p) r)  |
|         |            |                 |                     |
|         |            |                 |                     |
| q_74    | q_01001010 | 0 1 0 0 1 0 1 0 |   (p, r)   (p (q))  |
|         |            |                 |                     |
| q_82    | q_01010010 | 0 1 0 1 0 0 1 0 |   (p, r)   ((p) q)  |
|         |            |                 |                     |
| q_26    | q_00011010 | 0 0 0 1 1 0 1 0 |   (p, r)   (q (r))  |
|         |            |                 |                     |
| q_88    | q_01011000 | 0 1 0 1 1 0 0 0 |   (p, r)   ((q) r)  |
|         |            |                 |                     |
| q_133   | q_10000101 | 1 0 0 0 0 1 0 1 |  ((p, r))  (p (q))  |
|         |            |                 |                     |
| q_161   | q_10100001 | 1 0 1 0 0 0 0 1 |  ((p, r))  ((p) q)  |
|         |            |                 |                     |
|         |            |                 |                     |
| q_70    | q_01000110 | 0 1 0 0 0 1 1 0 |   (q, r)   (p (q))  |
|         |            |                 |                     |
| q_98    | q_01100010 | 0 1 1 0 0 0 1 0 |   (q, r)   ((p) q)  |
|         |            |                 |                     |
| q_38    | q_00100110 | 0 0 1 0 0 1 1 0 |   (q, r)   (p (r))  |
|         |            |                 |                     |
| q_100   | q_01100100 | 0 1 1 0 0 1 0 0 |   (q, r)   ((p) r)  |
|         |            |                 |                     |
| q_137   | q_10001001 | 1 0 0 0 1 0 0 1 |  ((q, r))  (p (q))  |
|         |            |                 |                     |
| q_145   | q_10010001 | 1 0 0 1 0 0 0 1 |  ((q, r))  ((p) q)  |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o
|         |            |                 |                     |
| q_211   | q_11010011 | 1 1 0 1 0 0 1 1 |  ((p, q)   (p (r))) |
|         |            |                 |                     |
| q_203   | q_11001011 | 1 1 0 0 1 0 1 1 |  ((p, q)   ((p) r)) |
|         |            |                 |                     |
| q_199   | q_11000111 | 1 1 0 0 0 1 1 1 |  ((p, q)   (q (r))) |
|         |            |                 |                     |
| q_227   | q_11100011 | 1 1 1 0 0 0 1 1 |  ((p, q)   ((q) r)) |
|         |            |                 |                     |
| q_124   | q_01111100 | 0 1 1 1 1 1 0 0 | (((p, q))  (p (r))) |
|         |            |                 |                     |
| q_62    | q_00111110 | 0 0 1 1 1 1 1 0 | (((p, q))  ((p) r)) |
|         |            |                 |                     |
|         |            |                 |                     |
| q_181   | q_10110101 | 1 0 1 1 0 1 0 1 |  ((p, r)   (p (q))) |
|         |            |                 |                     |
| q_173   | q_10101101 | 1 0 1 0 1 1 0 1 |  ((p, r)   ((p) q)) |
|         |            |                 |                     |
| q_229   | q_11100101 | 1 1 1 0 0 1 0 1 |  ((p, r)   (q (r))) |
|         |            |                 |                     |
| q_167   | q_10100111 | 1 0 1 0 0 1 1 1 |  ((p, r)   ((q) r)) |
|         |            |                 |                     |
| q_122   | q_01111010 | 0 1 1 1 1 0 1 0 | (((p, r))  (p (q))) |
|         |            |                 |                     |
| q_94    | q_01011110 | 0 1 0 1 1 1 1 0 | (((p, r))  ((p) q)) |
|         |            |                 |                     |
|         |            |                 |                     |
| q_185   | q_10111001 | 1 0 1 1 1 0 0 1 |  ((q, r)   (p (q))) |
|         |            |                 |                     |
| q_157   | q_10011101 | 1 0 0 1 1 1 0 1 |  ((q, r)   ((p) q)) |
|         |            |                 |                     |
| q_217   | q_11011001 | 1 1 0 1 1 0 0 1 |  ((q, r)   (p (r))) |
|         |            |                 |                     |
| q_155   | q_10011011 | 1 0 0 1 1 0 1 1 |  ((q, r)   ((p) r)) |
|         |            |                 |                     |
| q_118   | q_01110110 | 0 1 1 1 0 1 1 0 | (((q, r))  (p (q))) |
|         |            |                 |                     |
| q_110   | q_01101110 | 0 1 1 0 1 1 1 0 | (((q, r))  ((p) q)) |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o
```

## Note 18

```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
```

## Note 19

```Table 15.  Differences and Equalities between Simples and Boundaries
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_152   | q_10011000 | 1 0 0 1 1 0 0 0 |  p + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_164   | q_10100100 | 1 0 1 0 0 1 0 0 |  q + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_194   | q_11000010 | 1 1 0 0 0 0 1 0 |  r + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_230   | q_11100110 | 1 1 1 0 0 1 1 0 |  p + ((p), (q), (r))      |
|         |            |                 |                           |
| q_218   | q_11011010 | 1 1 0 1 1 0 1 0 |  q + ((p), (q), (r))      |
|         |            |                 |                           |
| q_188   | q_10111100 | 1 0 1 1 1 1 0 0 |  r + ((p), (q), (r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_103   | q_01100111 | 0 1 1 0 0 1 1 1 |  p = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_91    | q_01011011 | 0 1 0 1 1 0 1 1 |  q = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_61    | q_00111101 | 0 0 1 1 1 1 0 1 |  r = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_25    | q_00011001 | 0 0 0 1 1 0 0 1 |  p = ((p), (q), (r))      |
|         |            |                 |                           |
| q_37    | q_00100101 | 0 0 1 0 0 1 0 1 |  q = ((p), (q), (r))      |
|         |            |                 |                           |
| q_67    | q_01000011 | 0 1 0 0 0 0 1 1 |  r = ((p), (q), (r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
```

## Note 20

```Table 16.  Paisley 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_216   | q_11011000 | 1 1 0 1 1 0 0 0 |   (p, q)(p, r)  +  p q    |
|         |            |                 |                           |
| q_184   | q_10111000 | 1 0 1 1 1 0 0 0 |   (p, q)(p, r)  +  p r    |
|         |            |                 |                           |
| q_228   | q_11100100 | 1 1 1 0 0 1 0 0 |   (p, q)(q, r)  +  p q    |
|         |            |                 |                           |
| q_172   | q_10101100 | 1 0 1 0 1 1 0 0 |   (p, q)(q, r)  +  q r    |
|         |            |                 |                           |
| q_226   | q_11100010 | 1 1 1 0 0 0 1 0 |   (p, r)(q, r)  +  p r    |
|         |            |                 |                           |
| q_202   | q_11001010 | 1 1 0 0 1 0 1 0 |   (p, r)(q, r)  +  q r    |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_39    | q_00100111 | 0 0 1 0 0 1 1 1 |   (p, q)(p, r)  =  p q    |
|         |            |                 |                           |
| q_71    | q_01000111 | 0 1 0 0 0 1 1 1 |   (p, q)(p, r)  =  p r    |
|         |            |                 |                           |
| q_27    | q_00011011 | 0 0 0 1 1 0 1 1 |   (p, q)(q, r)  =  p q    |
|         |            |                 |                           |
| q_83    | q_01010011 | 0 1 0 1 0 0 1 1 |   (p, q)(q, r)  =  q r    |
|         |            |                 |                           |
| q_29    | q_00011101 | 0 0 0 1 1 1 0 1 |   (p, r)(q, r)  =  p r    |
|         |            |                 |                           |
| q_53    | q_00110101 | 0 0 1 1 0 1 0 1 |   (p, r)(q, r)  =  q r    |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
```

## Note 21

```Table 17 gives another way of writing the "paisley propositions"
that makes their symmetry class more manifest.  The venn diagram
that follows the Table may provide an idea of why I chose to dub
them that, at least, until I can think of a Greek or Latin label.

Table 17.  Paisley 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_216   | q_11011000 | 1 1 0 1 1 0 0 0 |   p + pq + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_184   | q_10111000 | 1 0 1 1 1 0 0 0 |   p + pr + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_228   | q_11100100 | 1 1 1 0 0 1 0 0 |   q + pq + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_172   | q_10101100 | 1 0 1 0 1 1 0 0 |   q + qr + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_226   | q_11100010 | 1 1 1 0 0 0 1 0 |   r + pr + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_202   | q_11001010 | 1 1 0 0 1 0 1 0 |   r + qr + pqr + (p, q, r)   |
|         |            |                 |                              |
o---------o------------o-----------------o------------------------------o
|         |            |                 |                              |
| q_39    | q_00100111 | 0 0 1 0 0 1 1 1 | 1 + p + pq + pqr + (p, q, r) |
|         |            |                 |                              |
| q_71    | q_01000111 | 0 1 0 0 0 1 1 1 | 1 + p + pr + pqr + (p, q, r) |
|         |            |                 |                              |
| q_27    | q_00011011 | 0 0 0 1 1 0 1 1 | 1 + q + pq + pqr + (p, q, r) |
|         |            |                 |                              |
| q_83    | q_01010011 | 0 1 0 1 0 0 1 1 | 1 + q + qr + pqr + (p, q, r) |
|         |            |                 |                              |
| q_29    | q_00011101 | 0 0 0 1 1 1 0 1 | 1 + r + pr + pqr + (p, q, r) |
|         |            |                 |                              |
| q_53    | q_00110101 | 0 0 1 1 0 1 0 1 | 1 + r + qr + pqr + (p, q, r) |
|         |            |                 |                              |
o---------o------------o-----------------o------------------------------o

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /%%%%%%%%%%%%%%%\                |
|               /%%%%%%%%%%%%%%%%%\               |
|              /%%%%%%%%%%%%%%%%%%%\              |
|             /%%%%%%%%%%%%%%%%%%%%%\             |
|            o%%%%%%%%%%%%%%%%%%%%%%%o            |
|            |%%%%%%%%%% P %%%%%%%%%%|            |
|            |%%%%%%%%%%%%%%%%%%%%%%%|            |
|            |%%%%%%%%%%%%%%%%%%%%%%%|            |
|        o---o---------o%%%o---------o---o        |
|       /     \%%%%%%%%%\%/         /     \       |
|      /       \%%%%%%%%%o         /       \      |
|     /         \%%%%%%%/%\       /         \     |
|    /           \%%%%%/%%%\     /           \    |
|   o             o---o-----o---o             o   |
|   |                 |%%%%%|                 |   |
|   |                 |%%%%%|                 |   |
|   |        Q        |%%%%%|        R        |   |
|   o                 o%%%%%o                 o   |
|    \                 \%%%/                 /    |
|     \                 \%/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_216.  p + p q + p q r + (p, q, r)
```

## Note 22

```I'm puzzled by the blind-spot that prevented me
from seeing this very simple and natural family
counted a third of their number.  At any rate,
here they be, and modulo the usual number of
corrections I think that these complete the
set of 256 propositions on three variables.

Table 18.  Desultory Junctions 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_224   | q_11100000 | 1 1 1 0 0 0 0 0 |        p   ((q)(r))       |
|         |            |                 |                           |
| q_200   | q_11001000 | 1 1 0 0 1 0 0 0 |        q   ((p)(r))       |
|         |            |                 |                           |
| q_168   | q_10101000 | 1 0 1 0 1 0 0 0 |        r   ((p)(q))       |
|         |            |                 |                           |
| q_14    | q_00001110 | 0 0 0 0 1 1 1 0 |       (p)  ((q)(r))       |
|         |            |                 |                           |
| q_50    | q_00110010 | 0 0 1 1 0 0 1 0 |       (q)  ((p)(r))       |
|         |            |                 |                           |
| q_84    | q_01010100 | 0 1 0 1 0 1 0 0 |       (r)  ((p)(q))       |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_31    | q_00011111 | 0 0 0 1 1 1 1 1 |       (p   ((q)(r)))      |
|         |            |                 |                           |
| q_55    | q_00110111 | 0 0 1 1 0 1 1 1 |       (q   ((p)(r)))      |
|         |            |                 |                           |
| q_87    | q_01010111 | 0 1 0 1 0 1 1 1 |       (r   ((p)(q)))      |
|         |            |                 |                           |
| q_241   | q_11110001 | 1 1 1 1 0 0 0 1 |      ((p)  ((q)(r)))      |
|         |            |                 |                           |
| q_205   | q_11001101 | 1 1 0 0 1 1 0 1 |      ((q)  ((p)(r)))      |
|         |            |                 |                           |
| q_171   | q_10101011 | 1 0 1 0 1 0 1 1 |      ((r)  ((p)(q)))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
```

## Note 23

```For ease of viewing, I am placing
copies of the Cactus Rules Table
at a couple of other sites:

Table 256.  http://stderr.org/pipermail/inquiry/2004-April/001314.html
Table 256.  http://suo.ieee.org/ontology/msg05512.html
```

## Note 24a

```Here is a set of representative cactus graphs
for the 256 propositions on three variables.

To make some cactus graphs easier to draw in Ascii,
I will occasionally be forced to "stretch a point",
drawing the root node "@" as @=@, @=@=@, and so on,
and the regular nodes "o" as o=o, o=o=o, and so on.

(I will keep adding to this after Easter,
but right now I've got spikes in my eyes.)

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 q r       |
|                   |         |       o o o       |
|       p q r       |         |        \|/        |
|       o o o       |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p)(q)(r)     |         |    ((p)(q)(r))    |
o-------------------o         o-------------------o
|        q_1        |         |       q_254       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p   q       |
|                   |         |       o   o       |
|       p   q       |         |        \ /        |
|       o   o       |         |         o r       |
|        \ /        |         |         |         |
|         @ r       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p)(q) r      |         |    ((p)(q) r)     |
o-------------------o         o-------------------o
|        q_2        |         |       q_253       |
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_3        |         |       q_252       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p   r       |
|                   |         |       o   o       |
|       p   r       |         |        \ /        |
|       o   o       |         |         o q       |
|        \ /        |         |         |         |
|         @ q       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p) q (r)     |         |    ((p) q (r))    |
o-------------------o         o-------------------o
|        q_4        |         |       q_251       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p   r       |
|                   |         |       o   o       |
|       p   r       |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (p) (r)      |         |     ((p) (r))     |
o-------------------o         o-------------------o
|        q_5        |         |       q_250       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p q r       |
|                   |         |       o o-o       |
|       p q r       |         |        \|/        |
|       o o-o       |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p)(q, r)     |         |    ((p)(q, r))    |
o-------------------o         o-------------------o
|        q_6        |         |       q_249       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p  q r      |
|                   |         |       o   o       |
|       p  q r      |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p) (q r)     |         |    ((p) (q r))    |
o-------------------o         o-------------------o
|        q_7        |         |       q_248       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         p         |
|                   |         |         o         |
|         p         |         |         |         |
|         o         |         |         o q r     |
|         |         |         |         |         |
|         @ q r     |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (p) q r      |         |     ((p) q r)     |
o-------------------o         o-------------------o
|        q_8        |         |       q_247       |
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_9        |         |       q_246       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         p         |
|                   |         |         o         |
|         p         |         |         |         |
|         o         |         |         o r       |
|         |         |         |         |         |
|         @ r       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       (p) r       |         |      ((p) r)      |
o-------------------o         o-------------------o
|        q_10       |         |       q_245       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           r       |
|                   |         |           o       |
|           r       |         |       p   |       |
|           o       |         |       o   o q     |
|       p   |       |         |        \ /        |
|       o   o q     |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    (p) (q (r))    |         |   ((p) (q (r)))   |
o-------------------o         o-------------------o
|        q_11       |         |       q_244       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         p         |
|                   |         |         o         |
|         p         |         |         |         |
|         o         |         |         o q       |
|         |         |         |         |         |
|         @ q       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       (p) q       |         |      ((p) q)      |
o-------------------o         o-------------------o
|        q_12       |         |       q_243       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           q       |
|                   |         |           o       |
|           q       |         |       p   |       |
|           o       |         |       o   o r     |
|       p   |       |         |        \ /        |
|       o   o r     |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    (p) ((q) r)    |         |   ((p) ((q) r))   |
o-------------------o         o-------------------o
|        q_13       |         |       q_242       |
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_14       |         |       q_241       |
o-------------------o         o-------------------o

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

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       q   r       |
|                   |         |       o   o       |
|       q   r       |         |        \ /        |
|       o   o       |         |       p o         |
|        \ /        |         |         |         |
|       p @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      p (q)(r)     |         |     (p (q)(r))    |
o-------------------o         o-------------------o
|        q_16       |         |       q_239       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       q   r       |
|                   |         |       o   o       |
|       q   r       |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (q) (r)      |         |     ((q) (r))     |
o-------------------o         o-------------------o
|        q_17       |         |       q_238       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p r q       |
|                   |         |       o-o o       |
|       p r q       |         |        \|/        |
|       o-o o       |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p, r)(q)     |         |    ((p, r)(q))    |
o-------------------o         o-------------------o
|        q_18       |         |       q_237       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |      p r  q       |
|                   |         |       o   o       |
|      p r  q       |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p r) (q)     |         |    ((p r) (q))    |
o-------------------o         o-------------------o
|        q_19       |         |       q_236       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p q r       |
|                   |         |       o-o o       |
|       p q r       |         |        \|/        |
|       o-o o       |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p, q)(r)     |         |    ((p, q)(r))    |
o-------------------o         o-------------------o
|        q_20       |         |       q_235       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |      p q  r       |
|                   |         |       o   o       |
|      p q  r       |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p q) (r)     |         |    ((p q) (r))    |
o-------------------o         o-------------------o
|        q_21       |         |       q_234       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p q r       |
|                   |         |       o o o       |
|       p q r       |         |       | | |       |
|       o o o       |         |       o-o-o       |
|       | | |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((p),(q),(r))   |         |  (((p),(q),(r)))  |
o-------------------o         o-------------------o
|        q_22       |         |       q_233       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |          p q r    |
|                   |         |          o o o    |
|          p q r    |         |    p q r | | |    |
|          o o o    |         |    o o o o-o-o    |
|    p q r | | |    |         |     \|/   \ /     |
|    o o o o-o-o    |         |      o-----o      |
|     \|/   \ /     |         |       \   /       |
|      o-----o      |         |        \ /        |
|       \   /       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|  ( (p) (q) (r)    |         | (( (p) (q) (r)    |
|  ,((p),(q),(r)))  |         |  ,((p),(q),(r)))) |
o-------------------o         o-------------------o
|        q_23       |         |       q_232       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |      p q p r      |
|                   |         |      o-o o-o      |
|      p q p r      |         |       \| |/       |
|      o-o o-o      |         |        o=o        |
|       \| |/       |         |         |         |
|        @=@        |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, q) (p, r)   |         |  ((p, q) (p, r))  |
o-------------------o         o-------------------o
|        q_24       |         |       q_231       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p q r     |         |                   |
|         o o o     |         |                   |
|         | | |     |         |         p q r     |
|         o-o-o     |         |         o o o     |
|       p  \ /      |         |         | | |     |
|       o---o       |         |         o-o-o     |
|        \ /        |         |       p  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((      p         |         |  (      p         |
|  ,((p),(q),(r)))) |         |  ,((p),(q),(r)))  |
o-------------------o         o-------------------o
|        q_25       |         |       q_230       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           r       |
|                   |         |           o       |
|           r       |         |       p r |       |
|           o       |         |       o-o o q     |
|       p r |       |         |        \|/        |
|       o-o o q     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, r)(q (r))   |         |  ((p, r)(q (r)))  |
o-------------------o         o-------------------o
|        q_26       |         |       q_229       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        p q q r    |         |                   |
|        o-o o-o    |         |                   |
|    p q  \| |/     |         |        p q q r    |
|      o---o=o      |         |        o-o o-o    |
|       \   /       |         |    p q  \| |/     |
|        \ /        |         |      o---o=o      |
|         o         |         |       \   /       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (( p      q       |         |  ( p      q       |
|  ,(p, q) (q, r))) |         |  ,(p, q) (q, r))  |
o-------------------o         o-------------------o
|        q_27       |         |       q_228       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           q       |
|                   |         |           o       |
|           q       |         |       p q |       |
|           o       |         |       o-o o r     |
|       p q |       |         |        \|/        |
|       o-o o r     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, q)((q) r)   |         |  ((p, q)((q) r))  |
o-------------------o         o-------------------o
|        q_28       |         |       q_227       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        p r q r    |         |                   |
|        o-o o-o    |         |                   |
|    p r  \| |/     |         |        p r q r    |
|      o---o=o      |         |        o-o o-o    |
|       \   /       |         |    p r  \| |/     |
|        \ /        |         |      o---o=o      |
|         o         |         |       \   /       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (( p      r       |         |  ( p      r       |
|  ,(p, r) (q, r))) |         |  ,(p, r) (q, r))  |
o-------------------o         o-------------------o
|        q_29       |         |       q_226       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         q   r     |         |                   |
|         o   o     |         |                   |
|       p  \ /      |         |         q   r     |
|       o---o       |         |         o   o     |
|        \ /        |         |       p  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((p, (q) (r)))  |         |    (p, (q) (r))   |
o-------------------o         o-------------------o
|        q_30       |         |       q_225       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       q   r       |         |                   |
|       o   o       |         |                   |
|        \ /        |         |       q   r       |
|         o         |         |       o   o       |
|         |         |         |        \ /        |
|       p o         |         |         o         |
|         |         |         |         |         |
|         @         |         |       p @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    (p ((q)(r)))   |         |     p ((q)(r))    |
o-------------------o         o-------------------o
|        q_31       |         |       q_224       |
o-------------------o         o-------------------o
```

## Note 24b

```o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         q         |
|                   |         |         o         |
|         q         |         |         |         |
|         o         |         |       p o r       |
|         |         |         |         |         |
|       p @ r       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      p (q) r      |         |     (p (q) r)     |
o-------------------o         o-------------------o
|        q_32       |         |       q_223       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |     p   r         |
|                   |         |     o---o         |
|     p   r         |         |      \ /  q       |
|     o---o         |         |       o   o       |
|      \ /  q       |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p, r))(q)    |         |   (((p, r))(q))   |
o-------------------o         o-------------------o
|        q_33       |         |       q_222       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         q         |
|                   |         |         o         |
|         q         |         |         |         |
|         o         |         |         o r       |
|         |         |         |         |         |
|         @ r       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       (q) r       |         |      ((q) r)      |
o-------------------o         o-------------------o
|        q_34       |         |       q_221       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       r           |
|                   |         |       o           |
|       r           |         |       |   q       |
|       o           |         |     p o   o       |
|       |   q       |         |        \ /        |
|     p o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p (r))(q)    |         |    ((p (r))(q))   |
o-------------------o         o-------------------o
|        q_35       |         |       q_220       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |      p q q r      |
|                   |         |      o-o o-o      |
|      p q q r      |         |       \| |/       |
|      o-o o-o      |         |        o=o        |
|       \| |/       |         |         |         |
|        @=@        |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, q) (q, r)   |         |  ((p, q) (q, r))  |
o-------------------o         o-------------------o
|        q_36       |         |       q_219       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p q r     |         |                   |
|         o o o     |         |                   |
|         | | |     |         |         p q r     |
|         o-o-o     |         |         o o o     |
|       q  \ /      |         |         | | |     |
|       o---o       |         |         o-o-o     |
|        \ /        |         |       q  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((      q         |         |  (      q         |
|  ,((p),(q),(r)))) |         |  ,((p),(q),(r)))  |
o-------------------o         o-------------------o
|        q_37       |         |       q_218       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           r       |
|                   |         |           o       |
|           r       |         |       q r |       |
|           o       |         |       o-o o p     |
|       q r |       |         |        \|/        |
|       o-o o p     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (q, r)(p (r))   |         |  ((q, r)(p (r)))  |
o-------------------o         o-------------------o
|        q_38       |         |       q_217       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        p q p r    |         |                   |
|        o-o o-o    |         |                   |
|    p q  \| |/     |         |        p q p r    |
|      o---o=o      |         |        o-o o-o    |
|       \   /       |         |    p q  \| |/     |
|        \ /        |         |      o---o=o      |
|         o         |         |       \   /       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (( p      q       |         |  ( p      q       |
|  ,(p, q) (p, r))) |         |  ,(p, q) (p, r))  |
o-------------------o         o-------------------o
|        q_39       |         |       q_216       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p   q       |
|                   |         |       o---o       |
|       p   q       |         |        \ /        |
|       o---o       |         |         o r       |
|        \ /        |         |         |         |
|         @ r       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p, q) r      |         |    ((p, q) r)     |
o-------------------o         o-------------------o
|        q_40       |         |       q_215       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p q         |
|                   |         |       o o         |
|       p q         |         |       | | r       |
|       o o         |         |       o-o-o       |
|       | | r       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((p),(q), r )   |         |  (((p),(q), r ))  |
o-------------------o         o-------------------o
|       q_41        |         |       q_214       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |        p q        |
|                   |         |         o         |
|        p q        |         |         |         |
|         o         |         |         o r       |
|         |         |         |         |         |
|         @ r       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (p q) r      |         |     ((p q) r)     |
o-------------------o         o-------------------o
|        q_42       |         |       q_213       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |          p q      |
|                   |         |          o o      |
|          p q      |         |    p q   | | r    |
|          o o      |         |    o o   o-o-o    |
|    p q   | | r    |         |     \|    \ /     |
|    o o   o-o-o    |         |    r o-----o      |
|     \|    \ /     |         |       \   /       |
|    r o-----o      |         |        \ /        |
|       \   /       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|  ( (p) (q)  r     |         | (( (p) (q)  r     |
|  ,((p),(q), r ))  |         |  ,((p),(q), r ))) |
o-------------------o         o-------------------o
|        q_43       |         |       q_212       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           r       |
|                   |         |           o       |
|           r       |         |       p q |       |
|           o       |         |       o-o o p     |
|       p q |       |         |        \|/        |
|       o-o o p     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, q)(p (r))   |         |  ((p, q)(p (r)))  |
o-------------------o         o-------------------o
|        q_44       |         |       q_211       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|           q       |         |                   |
|           o       |         |                   |
|       p   |       |         |           q       |
|       o---o r     |         |           o       |
|        \ /        |         |       p   |       |
|         o         |         |       o---o r     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((p, (q) r))    |         |    (p, (q) r)     |
o-------------------o         o-------------------o
|        q_45       |         |       q_210       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       p   q       |         |                   |
|       o   o       |         |                   |
|       |   |       |         |       p   q       |
|     q o   o r     |         |       o   o       |
|        \ /        |         |       |   |       |
|         o         |         |     q o   o r     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (((p) q) ((q) r)) |         |  ((p) q) ((q) r)  |
o-------------------o         o-------------------o
|        q_46       |         |       q_209       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         q         |         |                   |
|         o         |         |                   |
|         |         |         |         q         |
|         o r       |         |         o         |
|         |         |         |         |         |
|       p o         |         |         o r       |
|         |         |         |         |         |
|         @         |         |       p @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    (p ((q) r))    |         |     p ((q) r)     |
o-------------------o         o-------------------o
|        q_47       |         |       q_208       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         q         |
|                   |         |         o         |
|         q         |         |         |         |
|         o         |         |       p o         |
|         |         |         |         |         |
|       p @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       p (q)       |         |      (p (q))      |
o-------------------o         o-------------------o
|        q_48       |         |       q_207       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p           |
|                   |         |       o           |
|       p           |         |       |   q       |
|       o           |         |     r o   o       |
|       |   q       |         |        \ /        |
|     r o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p) r) (q)    |         |   (((p) r) (q))   |
o-------------------o         o-------------------o
|        q_49       |         |       q_206       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |     p   r         |
|                   |         |     o   o         |
|     p   r         |         |      \ /  q       |
|     o   o         |         |       o   o       |
|      \ /  q       |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((p) (r)) (q)   |         |  (((p) (r)) (q))  |
o-------------------o         o-------------------o
|        q_50       |         |       q_205       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         q         |         |                   |
|         o         |         |                   |
|         |         |         |         q         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|        (q)        |         |         q         |
o-------------------o         o-------------------o
|        q_51       |         |       q_204       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           p       |
|                   |         |           o       |
|           p       |         |       p q |       |
|           o       |         |       o-o o r     |
|       p q |       |         |        \|/        |
|       o-o o r     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, q)((p) r)   |         |  ((p, q)((p) r))  |
o-------------------o         o-------------------o
|        q_52       |         |       q_203       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        p r q r    |         |                   |
|        o-o o-o    |         |                   |
|    q r  \| |/     |         |        p r q r    |
|      o---o=o      |         |        o-o o-o    |
|       \   /       |         |    q r  \| |/     |
|        \ /        |         |      o---o=o      |
|         o         |         |       \   /       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (( q      r       |         |  ( q      r       |
|  ,(p, r) (q, r))) |         |  ,(p, r) (q, r))  |
o-------------------o         o-------------------o
|        q_53       |         |       q_202       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p   r     |         |                   |
|         o   o     |         |                   |
|       q  \ /      |         |         p   r     |
|       o---o       |         |         o   o     |
|        \ /        |         |       q  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((q, (p)(r)))   |         |    (q, (p)(r))    |
o-------------------o         o-------------------o
|        q_54       |         |       q_201       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       p   r       |         |                   |
|       o   o       |         |                   |
|        \ /        |         |       p   r       |
|         o         |         |       o   o       |
|         |         |         |        \ /        |
|         o q       |         |         o         |
|         |         |         |         |         |
|         @         |         |         @ q       |
|                   |         |                   |
o-------------------o         o-------------------o
|   (((p)(r)) q)    |         |    ((p)(r)) q     |
o-------------------o         o-------------------o
|        q_55       |         |       q_200       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           r       |
|                   |         |           o       |
|           r       |         |       p q |       |
|           o       |         |       o-o o q     |
|       p q |       |         |        \|/        |
|       o-o o q     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, q)(q (r))   |         |  ((p, q)(q (r)))  |
o-------------------o         o-------------------o
|        q_56       |         |       q_199       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|           p       |         |                   |
|           o       |         |                   |
|       q   |       |         |           p       |
|       o---o r     |         |           o       |
|        \ /        |         |       q   |       |
|         o         |         |       o---o r     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((q, (p) r))    |         |    (q, (p) r)     |
o-------------------o         o-------------------o
|        q_57       |         |       q_198       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       q   p       |         |                   |
|       o   o       |         |                   |
|       |   |       |         |       q   p       |
|     p o   o r     |         |       o   o       |
|        \ /        |         |       |   |       |
|         o         |         |     p o   o r     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((p (q)) ((p) r)) |         |  (p (q)) ((p) r)  |
o-------------------o         o-------------------o
|        q_58       |         |       q_197       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p         |         |                   |
|         o         |         |                   |
|         |         |         |         p         |
|         o r       |         |         o         |
|         |         |         |         |         |
|         o q       |         |         o r       |
|         |         |         |         |         |
|         @         |         |         @ q       |
|                   |         |                   |
o-------------------o         o-------------------o
|    (((p) r) q)    |         |     ((p) r) q     |
o-------------------o         o-------------------o
|        q_59       |         |       q_196       |
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
|                   |         |                   |
|         p q r     |         |                   |
|         o-o-o     |         |                   |
|       r  \ /      |         |         p q r     |
|       o---o       |         |         o-o-o     |
|        \ /        |         |       r  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((r, (p, q, r ))) |         |  (r, (p, q, r ))  |
o-------------------o         o-------------------o
|        q_61       |         |       q_194       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|     p   q   p     |         |                   |
|     o---o   o     |         |                   |
|      \ /   /      |         |     p   q   p     |
|       o   o r     |         |     o---o   o     |
|        \ /        |         |      \ /   /      |
|         o         |         |       o   o r     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (((p, q))((p) r)) |         |  ((p, q))((p) r)  |
o-------------------o         o-------------------o
|        q_62       |         |       q_193       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        p q        |         |                   |
|         o         |         |                   |
|         |         |         |        p q        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       (p q)       |         |        p q        |
o-------------------o         o-------------------o
|        q_63       |         |       q_192       |
o-------------------o         o-------------------o
```

## Note 24c

```o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         r         |
|                   |         |         o         |
|         r         |         |         |         |
|         o         |         |     p q o         |
|         |         |         |         |         |
|     p q @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     p q (r)       |         |    (p q (r))      |
o-------------------o         o-------------------o
|        q_64       |         |       q_191       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |     p   q         |
|                   |         |     o---o         |
|     p   q         |         |      \ /  r       |
|     o---o         |         |       o   o       |
|      \ /  r       |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p, q))(r)    |         |   (((p, q))(r))   |
o-------------------o         o-------------------o
|        q_65       |         |       q_190       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |      p r q r      |
|                   |         |      o-o o-o      |
|      p r q r      |         |       \| |/       |
|      o-o o-o      |         |        o=o        |
|       \| |/       |         |         |         |
|        @=@        |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, r) (q, r)   |         |  ((p, r) (q, r))  |
o-------------------o         o-------------------o
|        q_66       |         |       q_189       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p q r     |         |                   |
|         o o o     |         |                   |
|         | | |     |         |         p q r     |
|         o-o-o     |         |         o o o     |
|       r  \ /      |         |         | | |     |
|       o---o       |         |         o-o-o     |
|        \ /        |         |       r  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((      r         |         |  (      r         |
|  ,((p),(q),(r)))) |         |  ,((p),(q),(r)))  |
o-------------------o         o-------------------o
|        q_67       |         |       q_188       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         r         |
|                   |         |         o         |
|         r         |         |         |         |
|         o         |         |       q o         |
|         |         |         |         |         |
|       q @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       q (r)       |         |      (q (r))      |
o-------------------o         o-------------------o
|        q_68       |         |       q_187       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       q           |
|                   |         |       o           |
|       q           |         |       |   r       |
|       o           |         |     p o   o       |
|       |   r       |         |        \ /        |
|     p o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p (q))(r)    |         |    ((p (q))(r))   |
o-------------------o         o-------------------o
|        q_69       |         |       q_186       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           q       |
|                   |         |           o       |
|           q       |         |       q r |       |
|           o       |         |       o-o o p     |
|       q r |       |         |        \|/        |
|       o-o o p     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (q, r)(p (q))   |         |  ((q, r)(p (q)))  |
o-------------------o         o-------------------o
|        q_70       |         |       q_185       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        p q p r    |         |                   |
|        o-o o-o    |         |                   |
|    p r  \| |/     |         |        p q p r    |
|      o---o=o      |         |        o-o o-o    |
|       \   /       |         |    p r  \| |/     |
|        \ /        |         |      o---o=o      |
|         o         |         |       \   /       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (( p      r       |         |  ( p      r       |
|  ,(p, q) (p, r))) |         |  ,(p, q) (p, r))  |
o-------------------o         o-------------------o
|        q_71       |         |       q_184       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p   r       |
|                   |         |       o---o       |
|       p   r       |         |        \ /        |
|       o---o       |         |         o q       |
|        \ /        |         |         |         |
|         @ q       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p, r) q      |         |    ((p, r) q)     |
o-------------------o         o-------------------o
|        q_72       |         |       q_183       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p   r       |
|                   |         |       o   o       |
|       p   r       |         |       | q |       |
|       o   o       |         |       o-o-o       |
|       | q |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((p), q ,(r))   |         |  (((p), q ,(r)))  |
o-------------------o         o-------------------o
|       q_73        |         |       q_182       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           q       |
|                   |         |           o       |
|           q       |         |       p r |       |
|           o       |         |       o-o o p     |
|       p r |       |         |        \|/        |
|       o-o o p     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, r)(p (q))   |         |  ((p, r)(p (q)))  |
o-------------------o         o-------------------o
|        q_74       |         |       q_181       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|           r       |         |                   |
|           o       |         |                   |
|       p   |       |         |           r       |
|       o---o q     |         |           o       |
|        \ /        |         |       p   |       |
|         o         |         |       o---o q     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p, q (r)))   |         |     (p, q (r))    |
o-------------------o         o-------------------o
|        q_75       |         |       q_180       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |        p r        |
|                   |         |         o         |
|        p r        |         |         |         |
|         o         |         |         o q       |
|         |         |         |         |         |
|         @ q       |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (p r) q      |         |     ((p r) q)     |
o-------------------o         o-------------------o
|        q_76       |         |       q_179       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |          p   r    |
|                   |         |          o   o    |
|          p   r    |         |    p r   | q |    |
|          o   o    |         |    o o   o-o-o    |
|    p r   | q |    |         |     \|    \ /     |
|    o o   o-o-o    |         |    q o-----o      |
|     \|    \ /     |         |       \   /       |
|    q o-----o      |         |        \ /        |
|       \   /       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|  ( (p)  q  (r)    |         | (( (p)  q  (r)    |
|  ,((p), q ,(r)))  |         |  ,((p), q ,(r)))) |
o-------------------o         o-------------------o
|        q_77       |         |       q_178       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       p   r       |         |                   |
|       o   o       |         |                   |
|       |   |       |         |       p   r       |
|     r o   o q     |         |       o   o       |
|        \ /        |         |       |   |       |
|         o         |         |     r o   o q     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (((p) r) (q (r))) |         |  ((p) r) (q (r))  |
o-------------------o         o-------------------o
|        q_78       |         |       q_177       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         r         |         |                   |
|         o         |         |                   |
|         |         |         |         r         |
|       q o         |         |         o         |
|         |         |         |         |         |
|       p o         |         |       q o         |
|         |         |         |         |         |
|         @         |         |       p @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    (p (q (r)))    |         |     p (q (r))     |
o-------------------o         o-------------------o
|        q_79       |         |       q_176       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         r         |
|                   |         |         o         |
|         r         |         |         |         |
|         o         |         |       p o         |
|         |         |         |         |         |
|       p @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       p (r)       |         |      (p (r))      |
o-------------------o         o-------------------o
|        q_80       |         |       q_175       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p           |
|                   |         |       o           |
|       p           |         |       |   r       |
|       o           |         |     q o   o       |
|       |   r       |         |        \ /        |
|     q o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p) q)(r)     |         |   (((p) q)(r))    |
o-------------------o         o-------------------o
|        q_81       |         |       q_174       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           p       |
|                   |         |           o       |
|           p       |         |       p r |       |
|           o       |         |       o-o o q     |
|       p r |       |         |        \|/        |
|       o-o o q     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, r)((p) q)   |         |  ((p, r)((p) q))  |
o-------------------o         o-------------------o
|        q_82       |         |       q_173       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        p q q r    |         |                   |
|        o-o o-o    |         |                   |
|    q r  \| |/     |         |        p q q r    |
|      o---o=o      |         |        o-o o-o    |
|       \   /       |         |    q r  \| |/     |
|        \ /        |         |      o---o=o      |
|         o         |         |       \   /       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (( q      r       |         |  ( q      r       |
|  ,(p, q) (q, r))) |         |  ,(p, q) (q, r))  |
o-------------------o         o-------------------o
|        q_83       |         |       q_172       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |     p   q         |
|                   |         |     o   o         |
|     p   q         |         |      \ /  r       |
|     o   o         |         |       o   o       |
|      \ /  r       |         |        \ /        |
|       o   o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p)(q))(r)    |         |   (((p)(q))(r))   |
o-------------------o         o-------------------o
|        q_84       |         |       q_171       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         r         |         |                   |
|         o         |         |                   |
|         |         |         |         r         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|        (r)        |         |         r         |
o-------------------o         o-------------------o
|        q_85       |         |       q_170       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p   q     |         |                   |
|         o   o     |         |                   |
|       r  \ /      |         |         p   q     |
|       o---o       |         |         o   o     |
|        \ /        |         |       r  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((r, (p)(q)))   |         |    (r, (p)(q))    |
o-------------------o         o-------------------o
|        q_86       |         |       q_169       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       p   q       |         |                   |
|       o   o       |         |                   |
|        \ /        |         |       p   q       |
|         o         |         |       o   o       |
|         |         |         |        \ /        |
|         o r       |         |         o         |
|         |         |         |         |         |
|         @         |         |         @ r       |
|                   |         |                   |
o-------------------o         o-------------------o
|   (((p)(q)) r)    |         |    ((p)(q)) r     |
o-------------------o         o-------------------o
|        q_87       |         |       q_168       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           q       |
|                   |         |           o       |
|           q       |         |       p r |       |
|           o       |         |       o-o o r     |
|       p r |       |         |        \|/        |
|       o-o o r     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (p, r)((q) r)   |         |  ((p, r)((q) r))  |
o-------------------o         o-------------------o
|        q_88       |         |       q_167       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|           p       |         |                   |
|           o       |         |                   |
|       r   |       |         |           p       |
|       o---o q     |         |           o       |
|        \ /        |         |       r   |       |
|         o         |         |       o---o q     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((r, (p) q))    |         |    (r, (p) q)     |
o-------------------o         o-------------------o
|        q_89       |         |       q_166       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p   r       |
|                   |         |       o---o       |
|       p   r       |         |        \ /        |
|       o---o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (p , r)      |         |     ((p , r))     |
o-------------------o         o-------------------o
|        q_90       |         |       q_165       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p q r     |         |                   |
|         o-o-o     |         |                   |
|       q  \ /      |         |         p q r     |
|       o---o       |         |         o-o-o     |
|        \ /        |         |       q  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((q, (p, q, r)))  |         |  (q, (p, q, r))   |
o-------------------o         o-------------------o
|        q_91       |         |       q_164       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       r   p       |         |                   |
|       o   o       |         |                   |
|       |   |       |         |       r   p       |
|     p o   o q     |         |       o   o       |
|        \ /        |         |       |   |       |
|         o         |         |     p o   o q     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((p (r)) ((p) q)) |         |  (p (r)) ((p) q)  |
o-------------------o         o-------------------o
|        q_92       |         |       q_163       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p         |         |                   |
|         o         |         |                   |
|         |         |         |         p         |
|         o q       |         |         o         |
|         |         |         |         |         |
|         o r       |         |         o q       |
|         |         |         |         |         |
|         @         |         |         @ r       |
|                   |         |                   |
o-------------------o         o-------------------o
|    (((p) q) r)    |         |     ((p) q) r     |
o-------------------o         o-------------------o
|        q_93       |         |       q_162       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|     p   r   p     |         |                   |
|     o---o   o     |         |                   |
|      \ /   /      |         |     p   r   p     |
|       o   o q     |         |     o---o   o     |
|        \ /        |         |      \ /   /      |
|         o         |         |       o   o q     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (((p, r))((p) q)) |         |  ((p, r))((p) q)  |
o-------------------o         o-------------------o
|        q_94       |         |       q_161       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        p r        |         |                   |
|         o         |         |                   |
|         |         |         |        p r        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       (p r)       |         |        p r        |
o-------------------o         o-------------------o
|        q_95       |         |       q_160       |
o-------------------o         o-------------------o
```

## Note 24d

```o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       q   r       |
|                   |         |       o---o       |
|       q   r       |         |        \ /        |
|       o---o       |         |       p o         |
|        \ /        |         |         |         |
|       p @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      p (q, r)     |         |     (p (q, r))    |
o-------------------o         o-------------------o
|        q_96       |         |       q_159       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |         q r       |
|                   |         |         o o       |
|         q r       |         |       p | |       |
|         o o       |         |       o-o-o       |
|       p | |       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    (p, (q),(r))   |         |   ((p, (q),(r)))  |
o-------------------o         o-------------------o
|        q_97       |         |       q_158       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           p       |
|                   |         |           o       |
|           p       |         |       q r |       |
|           o       |         |       o-o o q     |
|       q r |       |         |        \|/        |
|       o-o o q     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (q, r)((p) q)   |         |  ((q, r)((p) q))  |
o-------------------o         o-------------------o
|        q_98       |         |       q_157       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|           r       |         |                   |
|           o       |         |                   |
|       q   |       |         |           r       |
|       o---o p     |         |           o       |
|        \ /        |         |       q   |       |
|         o         |         |       o---o p     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((q, p (r)))   |         |     (q, p (r))    |
o-------------------o         o-------------------o
|        q_99       |         |       q_156       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |           p       |
|                   |         |           o       |
|           p       |         |       q r |       |
|           o       |         |       o-o o r     |
|       q r |       |         |        \|/        |
|       o-o o r     |         |         o         |
|        \|/        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   (q, r)((p) r)   |         |  ((q, r)((p) r))  |
o-------------------o         o-------------------o
|       q_100       |         |       q_155       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|           q       |         |                   |
|           o       |         |                   |
|       r   |       |         |           q       |
|       o---o p     |         |           o       |
|        \ /        |         |       r   |       |
|         o         |         |       o---o p     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((r, p (q)))   |         |     (r, p (q))    |
o-------------------o         o-------------------o
|       q_101       |         |       q_154       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       q   r       |
|                   |         |       o---o       |
|       q   r       |         |        \ /        |
|       o---o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (q , r)      |         |     ((q , r))     |
o-------------------o         o-------------------o
|       q_102       |         |       q_153       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         p q r     |         |                   |
|         o-o-o     |         |                   |
|       p  \ /      |         |         p q r     |
|       o---o       |         |         o-o-o     |
|        \ /        |         |       p  \ /      |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((p, (p, q, r)))  |         |  (p, (p, q, r))   |
o-------------------o         o-------------------o
|       q_103       |         |       q_152       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |       p q r       |
|                   |         |       o-o-o       |
|       p q r       |         |        \ /        |
|       o-o-o       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|     (p, q, r)     |         |    ((p, q, r))    |
o-------------------o         o-------------------o
|       q_104       |         |       q_151       |
o-------------------o         o-------------------o

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

o-------------------o         o-------------------o
|                   |         |                   |
|          p q      |         |                   |
|           o       |         |                   |
|       r   |       |         |          p q      |
|       o---o       |         |           o       |
|        \ /        |         |       r   |       |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((r, (p q)))    |         |    (r, (p q))     |
o-------------------o         o-------------------o
|       q_106       |         |       q_149       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|           r       |         |                   |
|           o       |         |                   |
|       p q |       |         |           r       |
|       o-o-o       |         |           o       |
|        \ /        |         |       p q |       |
|         o         |         |       o-o-o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p, q, (r)))  |         |     (p, q, (r))   |
o-------------------o         o-------------------o
|       q_107       |         |       q_148       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|          p r      |         |                   |
|           o       |         |                   |
|       q   |       |         |          p r      |
|       o---o       |         |           o       |
|        \ /        |         |       q   |       |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((q, (p r)))    |         |    (q, (p r))     |
o-------------------o         o-------------------o
|       q_108       |         |       q_147       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         q         |         |                   |
|         o         |         |                   |
|       p | r       |         |         q         |
|       o-o-o       |         |         o         |
|        \ /        |         |       p | r       |
|         o         |         |       o-o-o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((p, (q), r))   |         |    (p, (q), r)    |
o-------------------o         o-------------------o
|       q_109       |         |       q_146       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|     p   q   r     |         |                   |
|     o   o---o     |         |                   |
|      \   \ /      |         |     p   q   r     |
|     q o   o       |         |     o   o---o     |
|        \ /        |         |      \   \ /      |
|         o         |         |     q o   o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (((p) q)((q, r))) |         |  ((p) q)((q, r))  |
o-------------------o         o-------------------o
|       q_110       |         |       q_145       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       q   r       |         |                   |
|       o---o       |         |                   |
|        \ /        |         |       q   r       |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|       p o         |         |         o         |
|         |         |         |         |         |
|         @         |         |       p @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    (p ((q, r)))   |         |     p ((q, r))    |
o-------------------o         o-------------------o
|       q_111       |         |       q_144       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |        q r        |
|                   |         |         o         |
|        q r        |         |         |         |
|         o         |         |       p o         |
|         |         |         |         |         |
|       p @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      p (q r)      |         |     (p (q r))     |
o-------------------o         o-------------------o
|       q_112       |         |       q_143       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|                   |         |            q r    |
|                   |         |            o o    |
|            q r    |         |    q r   p | |    |
|            o o    |         |    o o   o-o-o    |
|    q r   p | |    |         |     \|    \ /     |
|    o o   o-o-o    |         |    p o-----o      |
|     \|    \ /     |         |       \   /       |
|    p o-----o      |         |        \ /        |
|       \   /       |         |         o         |
|        \ /        |         |         |         |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|  (  p  (q) (r)    |         | ((  p  (q) (r)    |
|  ,( p ,(q),(r)))  |         |  ,( p ,(q),(r)))) |
o-------------------o         o-------------------o
|       q_113       |         |       q_142       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       r   q       |         |                   |
|       o   o       |         |                   |
|       |   |       |         |       r   q       |
|     p o   o r     |         |       o   o       |
|        \ /        |         |       |   |       |
|         o         |         |     p o   o r     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((p (r)) (r (q))) |         |  (p (r)) (r (q))  |
o-------------------o         o-------------------o
|       q_114       |         |       q_141       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         r         |         |                   |
|         o         |         |                   |
|         |         |         |         r         |
|       p o         |         |         o         |
|         |         |         |         |         |
|       q o         |         |       p o         |
|         |         |         |         |         |
|         @         |         |       q @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p (r)) q)    |         |     (p (r)) q     |
o-------------------o         o-------------------o
|       q_115       |         |       q_140       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       q   r       |         |                   |
|       o   o       |         |                   |
|       |   |       |         |       q   r       |
|     p o   o q     |         |       o   o       |
|        \ /        |         |       |   |       |
|         o         |         |     p o   o q     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((p (q)) (q (r))) |         |  (p (q)) (q (r))  |
o-------------------o         o-------------------o
|       q_116       |         |       q_139       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|         q         |         |                   |
|         o         |         |                   |
|         |         |         |         q         |
|       p o         |         |         o         |
|         |         |         |         |         |
|       r o         |         |       p o         |
|         |         |         |         |         |
|         @         |         |       r @         |
|                   |         |                   |
o-------------------o         o-------------------o
|    ((p (q)) r)    |         |     (p (q)) r     |
o-------------------o         o-------------------o
|       q_117       |         |       q_138       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|     q   q   r     |         |                   |
|     o   o---o     |         |                   |
|      \   \ /      |         |     q   q   r     |
|     p o   o       |         |     o   o---o     |
|        \ /        |         |      \   \ /      |
|         o         |         |     p o   o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((p (q))((q, r))) |         |  (p (q))((q, r))  |
o-------------------o         o-------------------o
|       q_118       |         |       q_137       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|        q r        |         |                   |
|         o         |         |                   |
|         |         |         |        q r        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|       (q r)       |         |        q r        |
o-------------------o         o-------------------o
|       q_119       |         |       q_136       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|          q r      |         |                   |
|           o       |         |                   |
|       p   |       |         |          q r      |
|       o---o       |         |           o       |
|        \ /        |         |       p   |       |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|   ((p, (q r)))    |         |    (p, (q r))     |
o-------------------o         o-------------------o
|       q_120       |         |       q_135       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       p           |         |                   |
|       o           |         |                   |
|       | q r       |         |       p           |
|       o-o-o       |         |       o           |
|        \ /        |         |       | q r       |
|         o         |         |       o-o-o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|  (((p), q, r))    |         |   ((p), q, r)     |
o-------------------o         o-------------------o
|       q_121       |         |       q_134       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|     q   p   r     |         |                   |
|     o   o---o     |         |                   |
|      \   \ /      |         |     q   p   r     |
|     p o   o       |         |     o   o---o     |
|        \ /        |         |      \   \ /      |
|         o         |         |     p o   o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| ((p (q))((p, r))) |         |  (p (q))((p, r))  |
o-------------------o         o-------------------o
|       q_122       |         |       q_133       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       p   r       |         |                   |
|       o---o       |         |                   |
|        \ /        |         |       p   r       |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         o q       |         |         o         |
|         |         |         |         |         |
|         @         |         |         @ q       |
|                   |         |                   |
o-------------------o         o-------------------o
|   (((p, r)) q)    |         |    ((p, r)) q     |
o-------------------o         o-------------------o
|       q_123       |         |       q_132       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|     p   q   r     |         |                   |
|     o---o   o     |         |                   |
|      \ /   /      |         |     p   q   r     |
|       o   o p     |         |     o---o   o     |
|        \ /        |         |      \ /   /      |
|         o         |         |       o   o p     |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (((p, q))(p (r))) |         |  ((p, q))(p (r))  |
o-------------------o         o-------------------o
|       q_124       |         |       q_131       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       p   q       |         |                   |
|       o---o       |         |                   |
|        \ /        |         |       p   q       |
|         o         |         |       o---o       |
|         |         |         |        \ /        |
|         o r       |         |         o         |
|         |         |         |         |         |
|         @         |         |         @ r       |
|                   |         |                   |
o-------------------o         o-------------------o
|   (((p, q)) r)    |         |    ((p, q)) r     |
o-------------------o         o-------------------o
|       q_125       |         |       q_130       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|     p q   q r     |         |                   |
|     o-o   o-o     |         |                   |
|      \|   |/      |         |     p q   q r     |
|       o   o       |         |     o-o   o-o     |
|        \ /        |         |      \|   |/      |
|         o         |         |       o   o       |
|         |         |         |        \ /        |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
| (((p,q)) ((q,r))) |         |  ((p,q)) ((q,r))  |
o-------------------o         o-------------------o
|       q_126       |         |       q_129       |
o-------------------o         o-------------------o

o-------------------o         o-------------------o
|                   |         |                   |
|       p q r       |         |                   |
|         o         |         |                   |
|         |         |         |       p q r       |
|         @         |         |         @         |
|                   |         |                   |
o-------------------o         o-------------------o
|      (p q r)      |         |       p q r       |
o-------------------o         o-------------------o
|       q_127       |         |       q_128       |
o-------------------o         o-------------------o
```

## Note 24e

```I'm attaching here a text file copy of the current set
of cactus graphs for propositions on three variables,
and I have placed additional copies at the following
two sites:

CR 24.  http://stderr.org/pipermail/inquiry/2004-April/001322.html
CR 24.  http://suo.ieee.org/ontology/msg05518.html

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

CR.  Cactus Rules -- Jon Awbrey

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

Table 256.  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_0     | q_00000000 | 0 0 0 0 0 0 0 0 |            ( )            |
|         |            |                 |                           |
| q_1     | q_00000001 | 0 0 0 0 0 0 0 1 |       (p)  (q)  (r)       |
|         |            |                 |                           |
| q_2     | q_00000010 | 0 0 0 0 0 0 1 0 |       (p)  (q)   r        |
|         |            |                 |                           |
| q_3     | q_00000011 | 0 0 0 0 0 0 1 1 |       (p)  (q)            |
|         |            |                 |                           |
| q_4     | q_00000100 | 0 0 0 0 0 1 0 0 |       (p)   q   (r)       |
|         |            |                 |                           |
| q_5     | q_00000101 | 0 0 0 0 0 1 0 1 |       (p)       (r)       |
|         |            |                 |                           |
| q_6     | q_00000110 | 0 0 0 0 0 1 1 0 |       (p)  (q ,  r)       |
|         |            |                 |                           |
| q_7     | q_00000111 | 0 0 0 0 0 1 1 1 |       (p)  (q    r)       |
|         |            |                 |                           |
| q_8     | q_00001000 | 0 0 0 0 1 0 0 0 |       (p)   q    r        |
|         |            |                 |                           |
| q_9     | q_00001001 | 0 0 0 0 1 0 0 1 |       (p) ((q ,  r))      |
|         |            |                 |                           |
| q_10    | q_00001010 | 0 0 0 0 1 0 1 0 |       (p)        r        |
|         |            |                 |                           |
| q_11    | q_00001011 | 0 0 0 0 1 0 1 1 |       (p)  (q   (r))      |
|         |            |                 |                           |
| q_12    | q_00001100 | 0 0 0 0 1 1 0 0 |       (p)   q             |
|         |            |                 |                           |
| q_13    | q_00001101 | 0 0 0 0 1 1 0 1 |       (p) ((q)   r)       |
|         |            |                 |                           |
| q_14    | q_00001110 | 0 0 0 0 1 1 1 0 |       (p) ((q)  (r))      |
|         |            |                 |                           |
| q_15    | q_00001111 | 0 0 0 0 1 1 1 1 |       (p)                 |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_16    | q_00010000 | 0 0 0 1 0 0 0 0 |        p   (q)  (r)       |
|         |            |                 |                           |
| q_17    | q_00010001 | 0 0 0 1 0 0 0 1 |            (q)  (r)       |
|         |            |                 |                           |
| q_18    | q_00010010 | 0 0 0 1 0 0 1 0 |       (p ,  r)  (q)       |
|         |            |                 |                           |
| q_19    | q_00010011 | 0 0 0 1 0 0 1 1 |       (p    r)  (q)       |
|         |            |                 |                           |
| q_20    | q_00010100 | 0 0 0 1 0 1 0 0 |       (p ,  q)  (r)       |
|         |            |                 |                           |
| q_21    | q_00010101 | 0 0 0 1 0 1 0 1 |       (p    q)  (r)       |
|         |            |                 |                           |
| q_22    | q_00010110 | 0 0 0 1 0 1 1 0 |      ((p), (q), (r))      |
|         |            |                 |                           |
| q_23    | q_00010111 | 0 0 0 1 0 1 1 1 | (p)(q)(r) + ((p),(q),(r)) |
|         |            |                 |                           |
| q_24    | q_00011000 | 0 0 0 1 1 0 0 0 |       (p, q) (p, r)       |
|         |            |                 |                           |
| q_25    | q_00011001 | 0 0 0 1 1 0 0 1 |  p = ((p), (q), (r))      |
|         |            |                 |                           |
| q_26    | q_00011010 | 0 0 0 1 1 0 1 0 |       (p, r) (q (r))      |
|         |            |                 |                           |
| q_27    | q_00011011 | 0 0 0 1 1 0 1 1 |   (p, q)(q, r)  =  p q    |
|         |            |                 |                           |
| q_28    | q_00011100 | 0 0 0 1 1 1 0 0 |       (p, q)((q) r)       |
|         |            |                 |                           |
| q_29    | q_00011101 | 0 0 0 1 1 1 0 1 |   (p, r)(q, r)  =  p r    |
|         |            |                 |                           |
| q_30    | q_00011110 | 0 0 0 1 1 1 1 0 |      ((p , (q)  (r)))     |
|         |            |                 |                           |
| q_31    | q_00011111 | 0 0 0 1 1 1 1 1 |       (p  ((q)  (r)))     |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_32    | q_00100000 | 0 0 1 0 0 0 0 0 |        p   (q)   r        |
|         |            |                 |                           |
| q_33    | q_00100001 | 0 0 1 0 0 0 0 1 |      ((p ,  r)) (q)       |
|         |            |                 |                           |
| q_34    | q_00100010 | 0 0 1 0 0 0 1 0 |            (q)   r        |
|         |            |                 |                           |
| q_35    | q_00100011 | 0 0 1 0 0 0 1 1 |       (p   (r)) (q)       |
|         |            |                 |                           |
| q_36    | q_00100100 | 0 0 1 0 0 1 0 0 |       (p, q) (q, r)       |
|         |            |                 |                           |
| q_37    | q_00100101 | 0 0 1 0 0 1 0 1 |  q = ((p), (q), (r))      |
|         |            |                 |                           |
| q_38    | q_00100110 | 0 0 1 0 0 1 1 0 |       (q, r) (p (r))      |
|         |            |                 |                           |
| q_39    | q_00100111 | 0 0 1 0 0 1 1 1 |   (p, q)(p, r)  =  p q    |
|         |            |                 |                           |
| q_40    | q_00101000 | 0 0 1 0 1 0 0 0 |       (p ,  q)   r        |
|         |            |                 |                           |
| q_41    | q_00101001 | 0 0 1 0 1 0 0 1 |      ((p), (q),  r)       |
|         |            |                 |                           |
| q_42    | q_00101010 | 0 0 1 0 1 0 1 0 |       (p    q)   r        |
|         |            |                 |                           |
| q_43    | q_00101011 | 0 0 1 0 1 0 1 1 | (p)(q) r  + ((p),(q), r ) |
|         |            |                 |                           |
| q_44    | q_00101100 | 0 0 1 0 1 1 0 0 |       (p, q) (p (r))      |
|         |            |                 |                           |
| q_45    | q_00101101 | 0 0 1 0 1 1 0 1 |      ((p , (q)   r))      |
|         |            |                 |                           |
| q_46    | q_00101110 | 0 0 1 0 1 1 1 0 |      ((r (q))(q (p)))     |
|         |            |                 |                           |
| q_47    | q_00101111 | 0 0 1 0 1 1 1 1 |       (p  ((q)   r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_48    | q_00110000 | 0 0 1 1 0 0 0 0 |        p   (q)            |
|         |            |                 |                           |
| q_49    | q_00110001 | 0 0 1 1 0 0 0 1 |      ((p)   r)  (q)       |
|         |            |                 |                           |
| q_50    | q_00110010 | 0 0 1 1 0 0 1 0 |      ((p)  (r)) (q)       |
|         |            |                 |                           |
| q_51    | q_00110011 | 0 0 1 1 0 0 1 1 |            (q)            |
|         |            |                 |                           |
| q_52    | q_00110100 | 0 0 1 1 0 1 0 0 |       (p, q)((p) r)       |
|         |            |                 |                           |
| q_53    | q_00110101 | 0 0 1 1 0 1 0 1 |   (p, r)(q, r)  =  q r    |
|         |            |                 |                           |
| q_54    | q_00110110 | 0 0 1 1 0 1 1 0 |      ((q , (p)  (r)))     |
|         |            |                 |                           |
| q_55    | q_00110111 | 0 0 1 1 0 1 1 1 |     (((p)  (r))  q)       |
|         |            |                 |                           |
| q_56    | q_00111000 | 0 0 1 1 1 0 0 0 |       (p, q) (q (r))      |
|         |            |                 |                           |
| q_57    | q_00111001 | 0 0 1 1 1 0 0 1 |      ((q , (p)   r))      |
|         |            |                 |                           |
| q_58    | q_00111010 | 0 0 1 1 1 0 1 0 |      ((r (p))(p (q)))     |
|         |            |                 |                           |
| q_59    | q_00111011 | 0 0 1 1 1 0 1 1 |     (((p)   r)   q)       |
|         |            |                 |                           |
| q_60    | q_00111100 | 0 0 1 1 1 1 0 0 |       (p ,  q)            |
|         |            |                 |                           |
| q_61    | q_00111101 | 0 0 1 1 1 1 0 1 |  r = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_62    | q_00111110 | 0 0 1 1 1 1 1 0 |    (((p, q)) ((p) r))     |
|         |            |                 |                           |
| q_63    | q_00111111 | 0 0 1 1 1 1 1 1 |       (p    q)            |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_64    | q_01000000 | 0 1 0 0 0 0 0 0 |        p    q   (r)       |
|         |            |                 |                           |
| q_65    | q_01000001 | 0 1 0 0 0 0 0 1 |      ((p ,  q)) (r)       |
|         |            |                 |                           |
| q_66    | q_01000010 | 0 1 0 0 0 0 1 0 |       (p, r) (q, r)       |
|         |            |                 |                           |
| q_67    | q_01000011 | 0 1 0 0 0 0 1 1 |  r = ((p), (q), (r))      |
|         |            |                 |                           |
| q_68    | q_01000100 | 0 1 0 0 0 1 0 0 |             q   (r)       |
|         |            |                 |                           |
| q_69    | q_01000101 | 0 1 0 0 0 1 0 1 |       (p   (q)) (r)       |
|         |            |                 |                           |
| q_70    | q_01000110 | 0 1 0 0 0 1 1 0 |       (q, r) (p (q))      |
|         |            |                 |                           |
| q_71    | q_01000111 | 0 1 0 0 0 1 1 1 |   (p, q)(p, r)  =  p r    |
|         |            |                 |                           |
| q_72    | q_01001000 | 0 1 0 0 1 0 0 0 |       (p ,  r)   q        |
|         |            |                 |                           |
| q_73    | q_01001001 | 0 1 0 0 1 0 0 1 |      ((p),  q , (r))      |
|         |            |                 |                           |
| q_74    | q_01001010 | 0 1 0 0 1 0 1 0 |       (p, r) (p (q))      |
|         |            |                 |                           |
| q_75    | q_01001011 | 0 1 0 0 1 0 1 1 |      ((p ,  q   (r)))     |
|         |            |                 |                           |
| q_76    | q_01001100 | 0 1 0 0 1 1 0 0 |       (p    r)   q        |
|         |            |                 |                           |
| q_77    | q_01001101 | 0 1 0 0 1 1 0 1 | (p) q (r) + ((p), q ,(r)) |
|         |            |                 |                           |
| q_78    | q_01001110 | 0 1 0 0 1 1 1 0 |      ((q (r))(r (p)))     |
|         |            |                 |                           |
| q_79    | q_01001111 | 0 1 0 0 1 1 1 1 |       (p   (q   (r)))     |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_80    | q_01010000 | 0 1 0 1 0 0 0 0 |        p        (r)       |
|         |            |                 |                           |
| q_81    | q_01010001 | 0 1 0 1 0 0 0 1 |      ((p)   q)  (r)       |
|         |            |                 |                           |
| q_82    | q_01010010 | 0 1 0 1 0 0 1 0 |       (p, r)((p) q)       |
|         |            |                 |                           |
| q_83    | q_01010011 | 0 1 0 1 0 0 1 1 |   (p, q)(q, r)  =  q r    |
|         |            |                 |                           |
| q_84    | q_01010100 | 0 1 0 1 0 1 0 0 |      ((p)  (q)) (r)       |
|         |            |                 |                           |
| q_85    | q_01010101 | 0 1 0 1 0 1 0 1 |                 (r)       |
|         |            |                 |                           |
| q_86    | q_01010110 | 0 1 0 1 0 1 1 0 |      ((r , (p)  (q)))     |
|         |            |                 |                           |
| q_87    | q_01010111 | 0 1 0 1 0 1 1 1 |     (((p)  (q))  r)       |
|         |            |                 |                           |
| q_88    | q_01011000 | 0 1 0 1 1 0 0 0 |       (p, r)((q) r)       |
|         |            |                 |                           |
| q_89    | q_01011001 | 0 1 0 1 1 0 0 1 |      ((r , (p)   q))      |
|         |            |                 |                           |
| q_90    | q_01011010 | 0 1 0 1 1 0 1 0 |       (p ,       r)       |
|         |            |                 |                           |
| q_91    | q_01011011 | 0 1 0 1 1 0 1 1 |  q = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_92    | q_01011100 | 0 1 0 1 1 1 0 0 |      ((q (p))(p (r)))     |
|         |            |                 |                           |
| q_93    | q_01011101 | 0 1 0 1 1 1 0 1 |     (((p)   q)   r)       |
|         |            |                 |                           |
| q_94    | q_01011110 | 0 1 0 1 1 1 1 0 |    (((p, r)) ((p) q))     |
|         |            |                 |                           |
| q_95    | q_01011111 | 0 1 0 1 1 1 1 1 |       (p         r)       |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_96    | q_01100000 | 0 1 1 0 0 0 0 0 |        p   (q ,  r)       |
|         |            |                 |                           |
| q_97    | q_01100001 | 0 1 1 0 0 0 0 1 |       (p , (q), (r))      |
|         |            |                 |                           |
| q_98    | q_01100010 | 0 1 1 0 0 0 1 0 |       (q, r)((p) q)       |
|         |            |                 |                           |
| q_99    | q_01100011 | 0 1 1 0 0 0 1 1 |      ((q ,  p   (r)))     |
|         |            |                 |                           |
| q_100   | q_01100100 | 0 1 1 0 0 1 0 0 |       (q, r)((p) r)       |
|         |            |                 |                           |
| q_101   | q_01100101 | 0 1 1 0 0 1 0 1 |      ((r ,  p   (q)))     |
|         |            |                 |                           |
| q_102   | q_01100110 | 0 1 1 0 0 1 1 0 |            (q ,  r)       |
|         |            |                 |                           |
| q_103   | q_01100111 | 0 1 1 0 0 1 1 1 |  p = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_104   | q_01101000 | 0 1 1 0 1 0 0 0 |       (p ,  q ,  r)       |
|         |            |                 |                           |
| q_105   | q_01101001 | 0 1 1 0 1 0 0 1 |      ((p , (q ,  r)))     |
|         |            |                 |                           |
| q_106   | q_01101010 | 0 1 1 0 1 0 1 0 |      ((r , (p    q)))     |
|         |            |                 |                           |
| q_107   | q_01101011 | 0 1 1 0 1 0 1 1 |      ((p ,  q , (r)))     |
|         |            |                 |                           |
| q_108   | q_01101100 | 0 1 1 0 1 1 0 0 |      ((q , (p    r)))     |
|         |            |                 |                           |
| q_109   | q_01101101 | 0 1 1 0 1 1 0 1 |      ((p , (q),  r))      |
|         |            |                 |                           |
| q_110   | q_01101110 | 0 1 1 0 1 1 1 0 |     (((p) q)((q, r)))     |
|         |            |                 |                           |
| q_111   | q_01101111 | 0 1 1 0 1 1 1 1 |       (p  ((q ,  r)))     |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_112   | q_01110000 | 0 1 1 1 0 0 0 0 |        p   (q    r)       |
|         |            |                 |                           |
| q_113   | q_01110001 | 0 1 1 1 0 0 0 1 |  p (q)(r) + ( p ,(q),(r)) |
|         |            |                 |                           |
| q_114   | q_01110010 | 0 1 1 1 0 0 1 0 |      ((p (r))(r (q)))     |
|         |            |                 |                           |
| q_115   | q_01110011 | 0 1 1 1 0 0 1 1 |      ((p   (r))  q)       |
|         |            |                 |                           |
| q_116   | q_01110100 | 0 1 1 1 0 1 0 0 |      ((p (q))(q (r)))     |
|         |            |                 |                           |
| q_117   | q_01110101 | 0 1 1 1 0 1 0 1 |      ((p   (q))  r)       |
|         |            |                 |                           |
| q_118   | q_01110110 | 0 1 1 1 0 1 1 0 |     (((q, r))(p (q)))     |
|         |            |                 |                           |
| q_119   | q_01110111 | 0 1 1 1 0 1 1 1 |            (q    r)       |
|         |            |                 |                           |
| q_120   | q_01111000 | 0 1 1 1 1 0 0 0 |      ((p , (q    r)))     |
|         |            |                 |                           |
| q_121   | q_01111001 | 0 1 1 1 1 0 0 1 |     (((p),  q ,  r))      |
|         |            |                 |                           |
| q_122   | q_01111010 | 0 1 1 1 1 0 1 0 |     (((p, r))(p (q)))     |
|         |            |                 |                           |
| q_123   | q_01111011 | 0 1 1 1 1 0 1 1 |     (((p ,  r))  q)       |
|         |            |                 |                           |
| q_124   | q_01111100 | 0 1 1 1 1 1 0 0 |     (((p, q))(p (r)))     |
|         |            |                 |                           |
| q_125   | q_01111101 | 0 1 1 1 1 1 0 1 |     (((p ,  q))  r)       |
|         |            |                 |                           |
| q_126   | q_01111110 | 0 1 1 1 1 1 1 0 |    (((p, q)) ((q, r)))    |
|         |            |                 |                           |
| q_127   | q_01111111 | 0 1 1 1 1 1 1 1 |       (p    q    r)       |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_128   | q_10000000 | 1 0 0 0 0 0 0 0 |        p    q    r        |
|         |            |                 |                           |
| q_129   | q_10000001 | 1 0 0 0 0 0 0 1 |     ((p, q)) ((q, r))     |
|         |            |                 |                           |
| q_130   | q_10000010 | 1 0 0 0 0 0 1 0 |      ((p ,  q))  r        |
|         |            |                 |                           |
| q_131   | q_10000011 | 1 0 0 0 0 0 1 1 |     ((p, q))  (p (r))     |
|         |            |                 |                           |
| q_132   | q_10000100 | 1 0 0 0 0 1 0 0 |      ((p ,  r))  q        |
|         |            |                 |                           |
| q_133   | q_10000101 | 1 0 0 0 0 1 0 1 |     ((p, r))  (p (q))     |
|         |            |                 |                           |
| q_134   | q_10000110 | 1 0 0 0 0 1 1 0 |      ((p),  q ,  r)       |
|         |            |                 |                           |
| q_135   | q_10000111 | 1 0 0 0 0 1 1 1 |      ((p ,  q    r))      |
|         |            |                 |                           |
| q_136   | q_10001000 | 1 0 0 0 1 0 0 0 |             q    r        |
|         |            |                 |                           |
| q_137   | q_10001001 | 1 0 0 0 1 0 0 1 |     ((q, r))  (p (q))     |
|         |            |                 |                           |
| q_138   | q_10001010 | 1 0 0 0 1 0 1 0 |       (p   (q))  r        |
|         |            |                 |                           |
| q_139   | q_10001011 | 1 0 0 0 1 0 1 1 |       (p (q))(q (r))      |
|         |            |                 |                           |
| q_140   | q_10001100 | 1 0 0 0 1 1 0 0 |       (p   (r))  q        |
|         |            |                 |                           |
| q_141   | q_10001101 | 1 0 0 0 1 1 0 1 |       (p (r))(r (q))      |
|         |            |                 |                           |
| q_142   | q_10001110 | 1 0 0 0 1 1 1 0 | (p) q  r  + ((p), q , r ) |
|         |            |                 |                           |
| q_143   | q_10001111 | 1 0 0 0 1 1 1 1 |       (p   (q    r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_144   | q_10010000 | 1 0 0 1 0 0 0 0 |        p  ((q ,  r))      |
|         |            |                 |                           |
| q_145   | q_10010001 | 1 0 0 1 0 0 0 1 |      ((p) q)((q, r))      |
|         |            |                 |                           |
| q_146   | q_10010010 | 1 0 0 1 0 0 1 0 |       (p , (q),  r)       |
|         |            |                 |                           |
| q_147   | q_10010011 | 1 0 0 1 0 0 1 1 |      ((q ,  p    r))      |
|         |            |                 |                           |
| q_148   | q_10010100 | 1 0 0 1 0 1 0 0 |       (p ,  q , (r))      |
|         |            |                 |                           |
| q_149   | q_10010101 | 1 0 0 1 0 1 0 1 |      ((r ,  p    q))      |
|         |            |                 |                           |
| q_150   | q_10010110 | 1 0 0 1 0 1 1 0 |       (p , (q ,  r))      |
|         |            |                 |                           |
| q_151   | q_10010111 | 1 0 0 1 0 1 1 1 |      ((p ,  q ,  r))      |
|         |            |                 |                           |
| q_152   | q_10011000 | 1 0 0 1 1 0 0 0 |  p + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_153   | q_10011001 | 1 0 0 1 1 0 0 1 |           ((q ,  r))      |
|         |            |                 |                           |
| q_154   | q_10011010 | 1 0 0 1 1 0 1 0 |      ((r , (p   (q))))    |
|         |            |                 |                           |
| q_155   | q_10011011 | 1 0 0 1 1 0 1 1 |      ((q, r)((p) r))      |
|         |            |                 |                           |
| q_156   | q_10011100 | 1 0 0 1 1 1 0 0 |      ((q , (p   (r))))    |
|         |            |                 |                           |
| q_157   | q_10011101 | 1 0 0 1 1 1 0 1 |      ((q, r)((p) q))      |
|         |            |                 |                           |
| q_158   | q_10011110 | 1 0 0 1 1 1 1 0 |      ((p , (q), (r)))     |
|         |            |                 |                           |
| q_159   | q_10011111 | 1 0 0 1 1 1 1 1 |       (p   (q ,  r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_160   | q_10100000 | 1 0 1 0 0 0 0 0 |        p         r        |
|         |            |                 |                           |
| q_161   | q_10100001 | 1 0 1 0 0 0 0 1 |     ((p, r)) ((p) q)      |
|         |            |                 |                           |
| q_162   | q_10100010 | 1 0 1 0 0 0 1 0 |      ((p)   q)   r        |
|         |            |                 |                           |
| q_163   | q_10100011 | 1 0 1 0 0 0 1 1 |       (q (p))(p (r))      |
|         |            |                 |                           |
| q_164   | q_10100100 | 1 0 1 0 0 1 0 0 |  q + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_165   | q_10100101 | 1 0 1 0 0 1 0 1 |      ((p ,       r))      |
|         |            |                 |                           |
| q_166   | q_10100110 | 1 0 1 0 0 1 1 0 |      ((r ,((p)   q)))     |
|         |            |                 |                           |
| q_167   | q_10100111 | 1 0 1 0 0 1 1 1 |      ((p, r)((q) r))      |
|         |            |                 |                           |
| q_168   | q_10101000 | 1 0 1 0 1 0 0 0 |      ((p)  (q))  r        |
|         |            |                 |                           |
| q_169   | q_10101001 | 1 0 1 0 1 0 0 1 |      ((r ,((p)  (q))))    |
|         |            |                 |                           |
| q_170   | q_10101010 | 1 0 1 0 1 0 1 0 |                  r        |
|         |            |                 |                           |
| q_171   | q_10101011 | 1 0 1 0 1 0 1 1 |     (((p)  (q)) (r))      |
|         |            |                 |                           |
| q_172   | q_10101100 | 1 0 1 0 1 1 0 0 |   (p, q)(q, r)  +  q r    |
|         |            |                 |                           |
| q_173   | q_10101101 | 1 0 1 0 1 1 0 1 |      ((p, r)((p) q))      |
|         |            |                 |                           |
| q_174   | q_10101110 | 1 0 1 0 1 1 1 0 |     (((p)   q)  (r))      |
|         |            |                 |                           |
| q_175   | q_10101111 | 1 0 1 0 1 1 1 1 |       (p        (r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_176   | q_10110000 | 1 0 1 1 0 0 0 0 |        p   (q   (r))      |
|         |            |                 |                           |
| q_177   | q_10110001 | 1 0 1 1 0 0 0 1 |       (q (r))(r (p))      |
|         |            |                 |                           |
| q_178   | q_10110010 | 1 0 1 1 0 0 1 0 |  p (q) r  + ( p ,(q), r ) |
|         |            |                 |                           |
| q_179   | q_10110011 | 1 0 1 1 0 0 1 1 |      ((p    r)   q)       |
|         |            |                 |                           |
| q_180   | q_10110100 | 1 0 1 1 0 1 0 0 |      ((p , (q   (r))))    |
|         |            |                 |                           |
| q_181   | q_10110101 | 1 0 1 1 0 1 0 1 |      ((p, r) (p (q)))     |
|         |            |                 |                           |
| q_182   | q_10110110 | 1 0 1 1 0 1 1 0 |     (((p),  q , (r)))     |
|         |            |                 |                           |
| q_183   | q_10110111 | 1 0 1 1 0 1 1 1 |      ((p ,  r)   q        |
|         |            |                 |                           |
| q_184   | q_10111000 | 1 0 1 1 1 0 0 0 |   (p, q)(p, r)  +  p r    |
|         |            |                 |                           |
| q_185   | q_10111001 | 1 0 1 1 1 0 0 1 |      ((q, r) (p (q)))     |
|         |            |                 |                           |
| q_186   | q_10111010 | 1 0 1 1 1 0 1 0 |      ((p   (q)) (r))      |
|         |            |                 |                           |
| q_187   | q_10111011 | 1 0 1 1 1 0 1 1 |            (q   (r))      |
|         |            |                 |                           |
| q_188   | q_10111100 | 1 0 1 1 1 1 0 0 |  r + ((p), (q), (r))      |
|         |            |                 |                           |
| q_189   | q_10111101 | 1 0 1 1 1 1 0 1 |      ((p, r) (q, r))      |
|         |            |                 |                           |
| q_190   | q_10111110 | 1 0 1 1 1 1 1 0 |     (((p ,  q)) (r))      |
|         |            |                 |                           |
| q_191   | q_10111111 | 1 0 1 1 1 1 1 1 |       (p    q   (r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_192   | q_11000000 | 1 1 0 0 0 0 0 0 |        p    q             |
|         |            |                 |                           |
| q_193   | q_11000001 | 1 1 0 0 0 0 0 1 |     ((p, q)) ((p) r)      |
|         |            |                 |                           |
| q_194   | q_11000010 | 1 1 0 0 0 0 1 0 |  r + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_195   | q_11000011 | 1 1 0 0 0 0 1 1 |      ((p ,  q))           |
|         |            |                 |                           |
| q_196   | q_11000100 | 1 1 0 0 0 1 0 0 |      ((p)   r)   q        |
|         |            |                 |                           |
| q_197   | q_11000101 | 1 1 0 0 0 1 0 1 |       (r (p))(p (q))      |
|         |            |                 |                           |
| q_198   | q_11000110 | 1 1 0 0 0 1 1 0 |      ((q ,((p)   r)))     |
|         |            |                 |                           |
| q_199   | q_11000111 | 1 1 0 0 0 1 1 1 |      ((p, q) (q (r)))     |
|         |            |                 |                           |
| q_200   | q_11001000 | 1 1 0 0 1 0 0 0 |      ((p)  (r))  q        |
|         |            |                 |                           |
| q_201   | q_11001001 | 1 1 0 0 1 0 0 1 |      ((q ,((p)  (r))))    |
|         |            |                 |                           |
| q_202   | q_11001010 | 1 1 0 0 1 0 1 0 |   (p, r)(q, r)  +  q r    |
|         |            |                 |                           |
| q_203   | q_11001011 | 1 1 0 0 1 0 1 1 |     ((p, q) ((p) r))      |
|         |            |                 |                           |
| q_204   | q_11001100 | 1 1 0 0 1 1 0 0 |             q             |
|         |            |                 |                           |
| q_205   | q_11001101 | 1 1 0 0 1 1 0 1 |     (((p)  (r)) (q))      |
|         |            |                 |                           |
| q_206   | q_11001110 | 1 1 0 0 1 1 1 0 |     (((p)   r)  (q))      |
|         |            |                 |                           |
| q_207   | q_11001111 | 1 1 0 0 1 1 1 1 |       (p   (q))           |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_208   | q_11010000 | 1 1 0 1 0 0 0 0 |        p  ((q)   r)       |
|         |            |                 |                           |
| q_209   | q_11010001 | 1 1 0 1 0 0 0 1 |       (r (q))(q (p))      |
|         |            |                 |                           |
| q_210   | q_11010010 | 1 1 0 1 0 0 1 0 |      ((p ,((q)   r)))     |
|         |            |                 |                           |
| q_211   | q_11010011 | 1 1 0 1 0 0 1 1 |      ((p, q) (p (r)))     |
|         |            |                 |                           |
| q_212   | q_11010100 | 1 1 0 1 0 1 0 0 |  p  q (r) + ( p , q ,(r)) |
|         |            |                 |                           |
| q_213   | q_11010101 | 1 1 0 1 0 1 0 1 |      ((p    q)   r)       |
|         |            |                 |                           |
| q_214   | q_11010110 | 1 1 0 1 0 1 1 0 |     (((p), (q),  r))      |
|         |            |                 |                           |
| q_215   | q_11010111 | 1 1 0 1 0 1 1 1 |      ((p ,  q)   r)       |
|         |            |                 |                           |
| q_216   | q_11011000 | 1 1 0 1 1 0 0 0 |   (p, q)(p, r)  +  p q    |
|         |            |                 |                           |
| q_217   | q_11011001 | 1 1 0 1 1 0 0 1 |      ((q, r) (p (r)))     |
|         |            |                 |                           |
| q_218   | q_11011010 | 1 1 0 1 1 0 1 0 |  q + ((p), (q), (r))      |
|         |            |                 |                           |
| q_219   | q_11011011 | 1 1 0 1 1 0 1 1 |      ((p, q) (q, r))      |
|         |            |                 |                           |
| q_220   | q_11011100 | 1 1 0 1 1 1 0 0 |      ((p   (r)) (q))      |
|         |            |                 |                           |
| q_221   | q_11011101 | 1 1 0 1 1 1 0 1 |           ((q)   r)       |
|         |            |                 |                           |
| q_222   | q_11011110 | 1 1 0 1 1 1 1 0 |     (((p ,  r)) (q))      |
|         |            |                 |                           |
| q_223   | q_11011111 | 1 1 0 1 1 1 1 1 |       (p   (q)   r)       |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_224   | q_11100000 | 1 1 1 0 0 0 0 0 |        p  ((q)  (r))      |
|         |            |                 |                           |
| q_225   | q_11100001 | 1 1 1 0 0 0 0 1 |       (p,  (q)  (r))      |
|         |            |                 |                           |
| q_226   | q_11100010 | 1 1 1 0 0 0 1 0 |   (p, r)(q, r)  +  p r    |
|         |            |                 |                           |
| q_227   | q_11100011 | 1 1 1 0 0 0 1 1 |      ((p, q)((q) r))      |
|         |            |                 |                           |
| q_228   | q_11100100 | 1 1 1 0 0 1 0 0 |   (p, q)(q, r)  +  p q    |
|         |            |                 |                           |
| q_229   | q_11100101 | 1 1 1 0 0 1 0 1 |      ((p, r) (q (r)))     |
|         |            |                 |                           |
| q_230   | q_11100110 | 1 1 1 0 0 1 1 0 |  p + ((p), (q), (r))      |
|         |            |                 |                           |
| q_231   | q_11100111 | 1 1 1 0 0 1 1 1 |      ((p, q) (p, r))      |
|         |            |                 |                           |
| q_232   | q_11101000 | 1 1 1 0 1 0 0 0 |  p  q  r  + ( p , q , r ) |
|         |            |                 |                           |
| q_233   | q_11101001 | 1 1 1 0 1 0 0 1 |     (((p), (q), (r)))     |
|         |            |                 |                           |
| q_234   | q_11101010 | 1 1 1 0 1 0 1 0 |      ((p    q)  (r))      |
|         |            |                 |                           |
| q_235   | q_11101011 | 1 1 1 0 1 0 1 1 |      ((p,   q)  (r))      |
|         |            |                 |                           |
| q_236   | q_11101100 | 1 1 1 0 1 1 0 0 |      ((p    r)  (q))      |
|         |            |                 |                           |
| q_237   | q_11101101 | 1 1 1 0 1 1 0 1 |      ((p,   r)  (q))      |
|         |            |                 |                           |
| q_238   | q_11101110 | 1 1 1 0 1 1 1 0 |           ((q)  (r))      |
|         |            |                 |                           |
| q_239   | q_11101111 | 1 1 1 0 1 1 1 1 |       (p   (q)  (r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_240   | q_11110000 | 1 1 1 1 0 0 0 0 |        p                  |
|         |            |                 |                           |
| q_241   | q_11110001 | 1 1 1 1 0 0 0 1 |      ((p) ((q)  (r)))     |
|         |            |                 |                           |
| q_242   | q_11110010 | 1 1 1 1 0 0 1 0 |      ((p) ((q)   r))      |
|         |            |                 |                           |
| q_243   | q_11110011 | 1 1 1 1 0 0 1 1 |      ((p)   q)            |
|         |            |                 |                           |
| q_244   | q_11110100 | 1 1 1 1 0 1 0 0 |      ((p)  (q   (r)))     |
|         |            |                 |                           |
| q_245   | q_11110101 | 1 1 1 1 0 1 0 1 |      ((p)        r)       |
|         |            |                 |                           |
| q_246   | q_11110110 | 1 1 1 1 0 1 1 0 |      ((p) ((q,   r)))     |
|         |            |                 |                           |
| q_247   | q_11110111 | 1 1 1 1 0 1 1 1 |      ((p)   q    r)       |
|         |            |                 |                           |
| q_248   | q_11111000 | 1 1 1 1 1 0 0 0 |      ((p)  (q    r))      |
|         |            |                 |                           |
| q_249   | q_11111001 | 1 1 1 1 1 0 0 1 |      ((p)  (q,   r))      |
|         |            |                 |                           |
| q_250   | q_11111010 | 1 1 1 1 1 0 1 0 |      ((p)       (r))      |
|         |            |                 |                           |
| q_251   | q_11111011 | 1 1 1 1 1 0 1 1 |      ((p)   q   (r))      |
|         |            |                 |                           |
| q_252   | q_11111100 | 1 1 1 1 1 1 0 0 |      ((p)  (q))           |
|         |            |                 |                           |
| q_253   | q_11111101 | 1 1 1 1 1 1 0 1 |      ((p)  (q)   r)       |
|         |            |                 |                           |
| q_254   | q_11111110 | 1 1 1 1 1 1 1 0 |      ((p)  (q)  (r))      |
|         |            |                 |                           |
| q_255   | q_11111111 | 1 1 1 1 1 1 1 1 |           (( ))           |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
```

## Work Area 1

```o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \         \ /         /     \       |
|      /       \         o         /       \      |
|     /         \       / \       /         \     |
|    /           \     /   \     /           \    |
|   o             o---o-----o---o             o   |
|   |                 |     |                 |   |
|   |                 |     |                 |   |
|   |        Q        |     |        R        |   |
|   o                 o     o                 o   |
|    \                 \   /                 /    |
|     \                 \ /                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
Figure 0.  Null Universe

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|`````````` P ``````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/`````\`````````\`/`````````/`````\```````|
|``````/```````\`````````o`````````/```````\``````|
|`````/`````````\```````/`\```````/`````````\`````|
|````/```````````\`````/```\`````/```````````\````|
|```o`````````````o---o-----o---o`````````````o```|
|```|`````````````````|`````|`````````````````|```|
|```|`````````````````|`````|`````````````````|```|
|```|``````` Q ```````|`````|``````` R ```````|```|
|```o`````````````````o`````o`````````````````o```|
|````\`````````````````\```/`````````````````/````|
|`````\`````````````````\`/`````````````````/`````|
|``````\`````````````````o`````````````````/``````|
|```````\```````````````/`\```````````````/```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
Figure 1.  Full Universe
```

## Work Area 2

```Table 1.  Boundaries 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_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

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |```````````P```````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /`````\         \`/         /`````\       |
|      /```````\         o         /```````\      |
|     /`````````\       / \       /`````````\     |
|    /```````````\     /   \     /```````````\    |
|   o```````````` o---o-----o---o`````````````o   |
|   |`````````````````|     |`````````````````|   |
|   |`````````````````|     |`````````````````|   |
|   |``````` Q ```````|     |``````` R ```````|   |
|   o`````````````````o     o`````````````````o   |
|    \`````````````````\   /`````````````````/    |
|     \`````````````````\ /`````````````````/     |
|      \`````````````````o`````````````````/      |
|       \```````````````/ \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_22.  ((p),(q),(r))

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /`````\`````````\ /`````````/`````\       |
|      /```````\`````````o`````````/```````\      |
|     /`````````\```````/`\```````/`````````\     |
|    /```````````\`````/```\`````/```````````\    |
|   o```````````` o---o-----o---o`````````````o   |
|   |`````````````````|     |`````````````````|   |
|   |`````````````````|     |`````````````````|   |
|   |``````` Q ```````|     |``````` R ```````|   |
|   o`````````````````o     o`````````````````o   |
|    \`````````````````\   /`````````````````/    |
|     \`````````````````\ /`````````````````/     |
|      \`````````````````o`````````````````/      |
|       \```````````````/ \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_25.  p + ((p),(q),(r))

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \         \ /`````````/`````\       |
|      /       \         o`````````/```````\      |
|     /         \       / \```````/`````````\     |
|    /           \     /   \`````/```````````\    |
|   o             o---o-----o---o`````````````o   |
|   |                 |`````|`````````````````|   |
|   |                 |`````|`````````````````|   |
|   |        Q        |`````|``````` R ```````|   |
|   o                 o`````o`````````````````o   |
|    \                 \```/`````````````````/    |
|     \                 \`/`````````````````/     |
|      \                 o`````````````````/      |
|       \               / \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_42.  p + q + ((p),(q),(r))

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \`````````\ /`````````/     \       |
|      /       \`````````o`````````/       \      |
|     /         \```````/ \```````/         \     |
|    /           \`````/   \`````/           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_104.  (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |`````````` P ``````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /     \         \`/         /     \       |
|      /       \         o         /       \      |
|     /         \       /`\       /         \     |
|    /           \     /```\     /           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_152.  p + (p, q, r)

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/               \````````````````|
|```````````````/                 \```````````````|
|``````````````/                   \``````````````|
|`````````````/                     \`````````````|
|````````````o                       o````````````|
|````````````|           P           |````````````|
|````````````|                       |````````````|
|````````````|                       |````````````|
|````````o---o---------o   o---------o---o````````|
|```````/     \         \ /`````````/     \```````|
|``````/       \         o`````````/       \``````|
|`````/         \       / \```````/         \`````|
|````/           \     /   \`````/           \````|
|```o             o---o-----o---o             o```|
|```|                 |`````|                 |```|
|```|                 |`````|                 |```|
|```|        Q        |`````|        R        |```|
|```o                 o`````o                 o```|
|````\                 \```/                 /````|
|`````\                 \`/                 /`````|
|``````\                 o                 /``````|
|```````\               /`\               /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_41.  ((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_216   |            | 1 1 0 1 1 0 0 0 |                           |
|         |            |                 |                           |
| q_217   |            | 1 1 0 1 1 0 0 1 |    p + ((p),(q), r)       |
|         |            |                 |                           |
| q_131   |            | 1 0 0 0 0 0 1 1 |    r + ((p),(q), r)       |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |```````````P```````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /`````\`````````\`/         /`````\       |
|      /```````\`````````o         /```````\      |
|     /`````````\```````/`\       /`````````\     |
|    /```````````\`````/```\     /```````````\    |
|   o```````````` o---o-----o---o`````````````o   |
|   |`````````````````|     |`````````````````|   |
|   |`````````````````|     |`````````````````|   |
|   |``````` Q ```````|     |``````` R ```````|   |
|   o`````````````````o     o`````````````````o   |
|    \`````````````````\   /`````````````````/    |
|     \`````````````````\ /`````````````````/     |
|      \`````````````````o`````````````````/      |
|       \```````````````/ \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_214.  pq + ((p),(q),(r))

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|`````````` P ``````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/     \`````````\`/         /     \```````|
|``````/       \`````````o         /       \``````|
|`````/         \```````/`\       /         \`````|
|````/           \`````/```\     /           \````|
|```o             o---o-----o---o             o```|
|```|                 |`````|                 |```|
|```|                 |`````|                 |```|
|```|        Q        |`````|        R        |```|
|```o                 o`````o                 o```|
|````\                 \```/                 /````|
|`````\                 \`/                 /`````|
|``````\                 o                 /``````|
|```````\               /`\               /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_217.  p + ((p),(q), r)

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/               \````````````````|
|```````````````/                 \```````````````|
|``````````````/                   \``````````````|
|`````````````/                     \`````````````|
|````````````o                       o````````````|
|````````````|           P           |````````````|
|````````````|                       |````````````|
|````````````|                       |````````````|
|````````o---o---------o   o---------o---o````````|
|```````/     \         \ /         /`````\```````|
|``````/       \         o         /```````\``````|
|`````/         \       /`\       /`````````\`````|
|````/           \     /```\     /```````````\````|
|```o             o---o-----o---o`````````````o```|
|```|                 |     |`````````````````|```|
|```|                 |     |`````````````````|```|
|```|        Q        |     |``````` R ```````|```|
|```o                 o     o`````````````````o```|
|````\                 \   /`````````````````/````|
|`````\                 \ /`````````````````/`````|
|``````\                 o`````````````````/``````|
|```````\               /`\```````````````/```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_131.  r + ((p),(q), r)
```

## Work Area 3

```o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |`````````` P ``````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /     \         \`/         /     \       |
|      /       \         o         /       \      |
|     /         \       / \       /         \     |
|    /           \     /   \     /           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_24.  (p, q) (p, r)

q_24.  p + p q r + (p, q, r)

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|```````````P```````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/     \         \`/         /     \```````|
|``````/       \         o         /       \``````|
|`````/         \       / \       /         \`````|
|````/           \     /   \     /           \````|
|```o             o---o-----o---o             o```|
|```|                 |`````|                 |```|
|```|                 |`````|                 |```|
|```|        Q        |`````|        R        |```|
|```o                 o`````o                 o```|
|````\                 \```/                 /````|
|`````\                 \`/                 /`````|
|``````\                 o                 /``````|
|```````\               /`\               /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_25.

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|`````````` P ``````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/     \         \`/         /`````\```````|
|``````/       \         o         /```````\``````|
|`````/         \       / \       /`````````\`````|
|````/           \     /   \     /```````````\````|
|```o             o---o-----o---o`````````````o```|
|```|                 |`````|`````````````````|```|
|```|                 |`````|`````````````````|```|
|```|        Q        |`````|``````` R ```````|```|
|```o                 o`````o`````````````````o```|
|````\                 \```/`````````````````/````|
|`````\                 \`/`````````````````/`````|
|``````\                 o`````````````````/``````|
|```````\               /`\```````````````/```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_27.

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|`````````` P ``````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/`````\         \`/         /     \```````|
|``````/```````\         o         /       \``````|
|`````/`````````\       / \       /         \`````|
|````/```````````\     /   \     /           \````|
|```o`````````````o---o-----o---o             o```|
|```|`````````````````|`````|                 |```|
|```|`````````````````|`````|                 |```|
|```|``````` Q ```````|`````|        R        |```|
|```o`````````````````o`````o                 o```|
|````\`````````````````\```/                 /````|
|`````\`````````````````\`/                 /`````|
|``````\`````````````````o                 /``````|
|```````\```````````````/`\               /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_29.

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/               \````````````````|
|```````````````/                 \```````````````|
|``````````````/                   \``````````````|
|`````````````/                     \`````````````|
|````````````o                       o````````````|
|````````````|           Q           |````````````|
|````````````|                       |````````````|
|````````````|                       |````````````|
|````````o---o---------o   o---------o---o````````|
|```````/`````\`````````\ /         /     \```````|
|``````/```````\`````````o         /       \``````|
|`````/`````````\```````/ \       /         \`````|
|````/```````````\`````/   \     /           \````|
|```o`````````````o---o-----o---o             o```|
|```|`````````````````|`````|                 |```|
|```|`````````````````|`````|                 |```|
|```|````````P````````|`````|        R        |```|
|```o`````````````````o`````o                 o```|
|````\`````````````````\```/                 /````|
|`````\`````````````````\`/                 /`````|
|``````\`````````````````o                 /``````|
|```````\```````````````/`\               /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_113.

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_97    | q_01100001 | 0 1 1 0 0 0 0 1 | ( p ,  (q), (r))          |
|         |            |                 |                           |
| q_225   | q_11100001 | 1 1 1 0 0 0 0 1 | ((p , ((q)  (r)) ))       |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/               \````````````````|
|```````````````/                 \```````````````|
|``````````````/                   \``````````````|
|`````````````/                     \`````````````|
|````````````o                       o````````````|
|````````````|           P           |````````````|
|````````````|                       |````````````|
|````````````|                       |````````````|
|````````o---o---------o   o---------o---o````````|
|```````/     \`````````\ /`````````/     \```````|
|``````/       \`````````o`````````/       \``````|
|`````/         \```````/ \```````/         \`````|
|````/           \`````/   \`````/           \````|
|```o             o---o-----o---o             o```|
|```|                 |     |                 |```|
|```|                 |     |                 |```|
|```|        Q        |     |        R        |```|
|```o                 o     o                 o```|
|````\                 \   /                 /````|
|`````\                 \ /                 /`````|
|``````\                 o                 /``````|
|```````\               /`\               /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
Genus and Species q_97.  (p, (q),(r))

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/               \````````````````|
|```````````````/                 \```````````````|
|``````````````/                   \``````````````|
|`````````````/                     \`````````````|
|````````````o                       o````````````|
|````````````|           P           |````````````|
|````````````|                       |````````````|
|````````````|                       |````````````|
|````````o---o---------o   o---------o---o````````|
|```````/     \`````````\ /`````````/     \```````|
|``````/       \`````````o`````````/       \``````|
|`````/         \```````/`\```````/         \`````|
|````/           \`````/```\`````/           \````|
|```o             o---o-----o---o             o```|
|```|                 |     |                 |```|
|```|                 |     |                 |```|
|```|        Q        |     |        R        |```|
|```o                 o     o                 o```|
|````\                 \   /                 /````|
|`````\                 \ /                 /`````|
|``````\                 o                 /``````|
|```````\               /`\               /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
Thematic Extension q_225.  ((p, ((q)(r)) ))
```

## Work Area 4

```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_112   | q_01110000 | 0 1 1 1 0 0 0 0 |     p   (q    r)    |
|         |            |                 |                     |
| q_76    | q_01001100 | 0 1 0 0 1 1 0 0 |     q   (p    r)    |
|         |            |                 |                     |
| q_42    | q_00101010 | 0 0 1 0 1 0 1 0 |     r   (p    q)    |
|         |            |                 |                     |
| q_7     | q_00000111 | 0 0 0 0 0 1 1 1 |    (p)  (q    r)    |
|         |            |                 |                     |
| q_19    | q_00010011 | 0 0 0 1 0 0 1 1 |    (p    r)  (q)    |
|         |            |                 |                     |
| q_21    | q_00010101 | 0 0 0 1 0 1 0 1 |    (p    q)  (r)    |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o
|         |            |                 |                     |
| q_143   | q_10001111 | 1 0 0 0 1 1 1 1 |    (p   (q    r))   |
|         |            |                 |                     |
| q_179   | q_10110011 | 1 0 1 1 0 0 1 1 |    (q   (p    r))   |
|         |            |                 |                     |
| q_213   | q_11010101 | 1 1 0 1 0 1 0 1 |    (r   (p    q))   |
|         |            |                 |                     |
| q_248   | q_11111000 | 1 1 1 1 1 0 0 0 |   ((p)  (q    r))   |
|         |            |                 |                     |
| q_236   | q_11101100 | 1 1 1 0 1 1 0 0 |   ((q)  (p    r))   |
|         |            |                 |                     |
| q_234   | q_11101010 | 1 1 1 0 1 0 1 0 |   ((r)  (p    q))   |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o
```

## Appendices

```Table 0.  Simple 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_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    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

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

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

Table 3.  Positive 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_255   | q_11111111 | 1 1 1 1 1 1 1 1 |       (( ))       |
|         |            |                 |                   |
| 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_192   | q_11000000 | 1 1 0 0 0 0 0 0 |    p    q         |
|         |            |                 |                   |
| q_160   | q_10100000 | 1 0 1 0 0 0 0 0 |    p         r    |
|         |            |                 |                   |
| q_136   | q_10001000 | 1 0 0 0 1 0 0 0 |         q    r    |
|         |            |                 |                   |
| q_128   | q_10000000 | 1 0 0 0 0 0 0 0 |    p    q    r    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_0     | q_00000000 | 0 0 0 0 0 0 0 0 |        ( )        |
|         |            |                 |                   |
| 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_63    | q_00111111 | 0 0 1 1 1 1 1 1 |   (p    q)        |
|         |            |                 |                   |
| q_95    | q_01011111 | 0 1 0 1 1 1 1 1 |   (p         r)   |
|         |            |                 |                   |
| q_119   | q_01110111 | 0 1 1 1 0 1 1 1 |        (q    r)   |
|         |            |                 |                   |
| q_127   | q_01111111 | 0 1 1 1 1 1 1 1 |   (p    q    r)   |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Table 4.  Singular 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_1     | q_00000001 | 0 0 0 0 0 0 0 1 |   (p)  (q)  (r)   |
|         |            |                 |                   |
| q_2     | q_00000010 | 0 0 0 0 0 0 1 0 |   (p)  (q)   r    |
|         |            |                 |                   |
| q_4     | q_00000100 | 0 0 0 0 0 1 0 0 |   (p)   q   (r)   |
|         |            |                 |                   |
| q_8     | q_00001000 | 0 0 0 0 1 0 0 0 |   (p)   q    r    |
|         |            |                 |                   |
| q_16    | q_00010000 | 0 0 0 1 0 0 0 0 |    p   (q)  (r)   |
|         |            |                 |                   |
| q_32    | q_00100000 | 0 0 1 0 0 0 0 0 |    p   (q)   r    |
|         |            |                 |                   |
| q_64    | q_01000000 | 0 1 0 0 0 0 0 0 |    p    q   (r)   |
|         |            |                 |                   |
| q_128   | q_10000000 | 1 0 0 0 0 0 0 0 |    p    q    r    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_254   | q_11111110 | 1 1 1 1 1 1 1 0 |  ((p)  (q)   r))  |
|         |            |                 |                   |
| q_253   | q_11111101 | 1 1 1 1 1 1 0 1 |  ((p)  (q)   r )  |
|         |            |                 |                   |
| q_251   | q_11111011 | 1 1 1 1 1 0 1 1 |  ((p)   q   (r))  |
|         |            |                 |                   |
| q_247   | q_11110111 | 1 1 1 1 0 1 1 1 |  ((p)   q    r )  |
|         |            |                 |                   |
| q_239   | q_11101111 | 1 1 1 0 1 1 1 1 |  ( p   (q)  (r))  |
|         |            |                 |                   |
| q_223   | q_11011111 | 1 1 0 1 1 1 1 1 |  ( p   (q)   r )  |
|         |            |                 |                   |
| q_191   | q_10111111 | 1 0 1 1 1 1 1 1 |  ( p    q   (r))  |
|         |            |                 |                   |
| q_127   | q_01111111 | 0 1 1 1 1 1 1 1 |  ( p    q    r )  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Table 5.  Variations on a Theme of Implication
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_207   | q_11001111 | 1 1 0 0 1 1 1 1 |   (p   (q))       |
|         |            |                 |                   |
| q_175   | q_10101111 | 1 0 1 0 1 1 1 1 |   (p        (r))  |
|         |            |                 |                   |
| q_187   | q_10111011 | 1 0 1 1 1 0 1 1 |        (q   (r))  |
|         |            |                 |                   |
| q_243   | q_11110011 | 1 1 1 1 0 0 1 1 |  ((p)   q)        |
|         |            |                 |                   |
| q_245   | q_11110101 | 1 1 1 1 0 1 0 1 |  ((p)        r)   |
|         |            |                 |                   |
| q_221   | q_11011101 | 1 1 0 1 1 1 0 1 |       ((q)   r)   |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_48    | q_00110000 | 0 0 1 1 0 0 0 0 |    p   (q)        |
|         |            |                 |                   |
| q_80    | q_01010000 | 0 1 0 1 0 0 0 0 |    p        (r)   |
|         |            |                 |                   |
| q_68    | q_01000100 | 0 1 0 0 0 1 0 0 |         q   (r)   |
|         |            |                 |                   |
| q_12    | q_00001100 | 0 0 0 0 1 1 0 0 |   (p)   q         |
|         |            |                 |                   |
| q_10    | q_00001010 | 0 0 0 0 1 0 1 0 |   (p)        r    |
|         |            |                 |                   |
| q_34    | q_00100010 | 0 0 1 0 0 0 1 0 |        (q)   r    |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Table 6.  More Variations on a Theme of Implication
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_176   | q_10110000 | 1 0 1 1 0 0 0 0 |    p   (q   (r))  |
|         |            |                 |                   |
| q_208   | q_11010000 | 1 1 0 1 0 0 0 0 |    p   (r   (q))  |
|         |            |                 |                   |
| q_11    | q_00001011 | 0 0 0 0 1 0 1 1 |   (p)  (q   (r))  |
|         |            |                 |                   |
| q_13    | q_00001101 | 0 0 0 0 1 1 0 1 |   (p)  (r   (q))  |
|         |            |                 |                   |
| q_140   | q_10001100 | 1 0 0 0 1 1 0 0 |    q   (p   (r))  |
|         |            |                 |                   |
| q_196   | q_11000100 | 1 1 0 0 0 1 0 0 |    q   (r   (p))  |
|         |            |                 |                   |
| q_35    | q_00100011 | 0 0 1 0 0 0 1 1 |   (q)  (p   (r))  |
|         |            |                 |                   |
| q_49    | q_00110001 | 0 0 1 1 0 0 0 1 |   (q)  (r   (p))  |
|         |            |                 |                   |
| q_138   | q_10001010 | 1 0 0 0 1 0 1 0 |    r   (p   (q))  |
|         |            |                 |                   |
| q_162   | q_10100010 | 1 0 1 0 0 0 1 0 |    r   (q   (p))  |
|         |            |                 |                   |
| q_69    | q_01000101 | 0 1 0 0 0 1 0 1 |   (r)  (p   (q))  |
|         |            |                 |                   |
| q_81    | q_01010001 | 0 1 0 1 0 0 0 1 |   (r)  (q   (p))  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_79    | q_01001111 | 0 1 0 0 1 1 1 1 |  ( p   (q   (r))) |
|         |            |                 |                   |
| q_47    | q_00101111 | 0 0 1 0 1 1 1 1 |  ( p   (r   (q))) |
|         |            |                 |                   |
| q_244   | q_11110100 | 1 1 1 1 0 1 0 0 |  ((p)  (q   (r))) |
|         |            |                 |                   |
| q_242   | q_11110010 | 1 1 1 1 0 0 1 0 |  ((p)  (r   (q))) |
|         |            |                 |                   |
| q_115   | q_01110011 | 0 1 1 1 0 0 1 1 |  ( q   (p   (r))) |
|         |            |                 |                   |
| q_59    | q_00111011 | 0 0 1 1 1 0 1 1 |  ( q   (r   (p))) |
|         |            |                 |                   |
| q_220   | q_11011100 | 1 1 0 1 1 1 0 0 |  ((q)  (p   (r))) |
|         |            |                 |                   |
| q_206   | q_11001110 | 1 1 0 0 1 1 1 0 |  ((q)  (r   (p))) |
|         |            |                 |                   |
| q_117   | q_01110101 | 0 1 1 1 0 1 0 1 |  ( r   (p   (q))) |
|         |            |                 |                   |
| q_93    | q_01011101 | 0 1 0 1 1 1 0 1 |  ( r   (q   (p))) |
|         |            |                 |                   |
| q_186   | q_10111010 | 1 0 1 1 1 0 1 0 |  ((r)  (p   (q))) |
|         |            |                 |                   |
| q_174   | q_10101110 | 1 0 1 0 1 1 1 0 |  ((r)  (q   (p))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Table 7.  Conjunctive Implications 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_139   | q_10001011 | 1 0 0 0 1 0 1 1 |   (p (q))(q (r))  |
|         |            |                 |                   |
| q_141   | q_10001101 | 1 0 0 0 1 1 0 1 |   (p (r))(r (q))  |
|         |            |                 |                   |
| q_177   | q_10110001 | 1 0 1 1 0 0 0 1 |   (q (r))(r (p))  |
|         |            |                 |                   |
| q_163   | q_10100011 | 1 0 1 0 0 0 1 1 |   (q (p))(p (r))  |
|         |            |                 |                   |
| q_197   | q_11000101 | 1 1 0 0 0 1 0 1 |   (r (p))(p (q))  |
|         |            |                 |                   |
| q_209   | q_11010001 | 1 1 0 1 0 0 0 1 |   (r (q))(q (p))  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_116   | q_01110100 | 0 1 1 1 0 1 0 0 |  ((p (q))(q (r))) |
|         |            |                 |                   |
| q_114   | q_01110010 | 0 1 1 1 0 0 1 0 |  ((p (r))(r (q))) |
|         |            |                 |                   |
| q_78    | q_01001110 | 0 1 0 0 1 1 1 0 |  ((q (r))(r (p))) |
|         |            |                 |                   |
| q_92    | q_01011100 | 0 1 0 1 1 1 0 0 |  ((q (p))(p (r))) |
|         |            |                 |                   |
| q_58    | q_00111010 | 0 0 1 1 1 0 1 0 |  ((r (p))(p (q))) |
|         |            |                 |                   |
| q_46    | q_00101110 | 0 0 1 0 1 1 1 0 |  ((r (q))(q (p))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Table 8.  More Variations on Difference and Equality
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_96    | q_01100000 | 0 1 1 0 0 0 0 0 |    p   (q ,  r)   |
|         |            |                 |                   |
| q_72    | q_01001000 | 0 1 0 0 1 0 0 0 |    q   (p ,  r)   |
|         |            |                 |                   |
| q_40    | q_00101000 | 0 0 1 0 1 0 0 0 |    r   (p ,  q)   |
|         |            |                 |                   |
| q_144   | q_10010000 | 1 0 0 1 0 0 0 0 |    p  ((q ,  r))  |
|         |            |                 |                   |
| q_132   | q_10000100 | 1 0 0 0 0 1 0 0 |    q  ((p ,  r))  |
|         |            |                 |                   |
| q_130   | q_10000010 | 1 0 0 0 0 0 1 0 |    r  ((p ,  q))  |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_6     | q_00000110 | 0 0 0 0 0 1 1 0 |   (p)  (q ,  r)   |
|         |            |                 |                   |
| q_18    | q_00010010 | 0 0 0 1 0 0 1 0 |   (q)  (p ,  r)   |
|         |            |                 |                   |
| q_20    | q_00010100 | 0 0 0 1 0 1 0 0 |   (r)  (p ,  q)   |
|         |            |                 |                   |
| q_9     | q_00001001 | 0 0 0 0 1 0 0 1 |   (p) ((q ,  r))  |
|         |            |                 |                   |
| q_33    | q_00100001 | 0 0 1 0 0 0 0 1 |   (q) ((p ,  r))  |
|         |            |                 |                   |
| q_65    | q_01000001 | 0 1 0 0 0 0 0 1 |   (r) ((p ,  q))  |
|         |            |                 |                   |
o=========o============o=================o===================o
|         |            |                 |                   |
| q_159   | q_10011111 | 1 0 0 1 1 1 1 1 |   (p   (q ,  r))  |
|         |            |                 |                   |
| q_183   | q_10110111 | 1 0 1 1 0 1 1 1 |   (q   (p ,  r))  |
|         |            |                 |                   |
| q_215   | q_11010111 | 1 1 0 1 0 1 1 1 |   (r   (p ,  q))  |
|         |            |                 |                   |
| q_111   | q_01101111 | 0 1 1 0 1 1 1 1 |   (p  ((q ,  r))) |
|         |            |                 |                   |
| q_123   | q_01111011 | 0 1 1 1 1 0 1 1 |   (q  ((p ,  r))) |
|         |            |                 |                   |
| q_125   | q_01111101 | 0 1 1 1 1 1 0 1 |   (r  ((p ,  q))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o
|         |            |                 |                   |
| q_249   | q_11111001 | 1 1 1 1 1 0 0 1 |  ((p)  (q ,  r))  |
|         |            |                 |                   |
| q_237   | q_11101101 | 1 1 1 0 1 1 0 1 |  ((q)  (p ,  r))  |
|         |            |                 |                   |
| q_235   | q_11101011 | 1 1 1 0 1 0 1 1 |  ((r)  (p ,  q))  |
|         |            |                 |                   |
| q_246   | q_11110110 | 1 1 1 1 0 1 1 0 |  ((p) ((q ,  r))) |
|         |            |                 |                   |
| q_222   | q_11011110 | 1 1 0 1 1 1 1 0 |  ((q) ((p ,  r))) |
|         |            |                 |                   |
| q_190   | q_10111110 | 1 0 1 1 1 1 1 0 |  ((r) ((p ,  q))) |
|         |            |                 |                   |
o---------o------------o-----------------o-------------------o

Table 9.  Conjunctive Differences and Equalities
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_24    | q_00011000 | 0 0 0 1 1 0 0 0 |   (p, q)  (p, r)   |
|         |            |                 |                    |
| q_36    | q_00100100 | 0 0 1 0 0 1 0 0 |   (p, q)  (q, r)   |
|         |            |                 |                    |
| q_66    | q_01000010 | 0 1 0 0 0 0 1 0 |   (p, r)  (q, r)   |
|         |            |                 |                    |
| q_129   | q_10000001 | 1 0 0 0 0 0 0 1 |  ((p, q))((q, r))  |
|         |            |                 |                    |
o---------o------------o-----------------o--------------------o
|         |            |                 |                    |
| q_231   | q_11100111 | 1 1 1 0 0 1 1 1 | ( (p, q)  (p, r) ) |
|         |            |                 |                    |
| q_219   | q_11011011 | 1 1 0 1 1 0 1 1 | ( (p, q)  (q, r) ) |
|         |            |                 |                    |
| q_189   | q_10111101 | 1 0 1 1 1 1 0 1 | ( (p, r)  (q, r) ) |
|         |            |                 |                    |
| q_126   | q_01111110 | 0 1 1 1 1 1 1 0 | (((p, q))((q, r))) |
|         |            |                 |                    |
o---------o------------o-----------------o--------------------o

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

Table 11.  Thematic Extensions:  [p, 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_51    | q_00110011 | 0 0 1 1 0 0 1 1 | ((q ,    ( )    ))  |
|         |            |                 |                     |
| q_54    | q_00110110 | 0 0 1 1 0 1 1 0 | ((q ,  (p) (r)  ))  |
|         |            |                 |                     |
| q_57    | q_00111001 | 0 0 1 1 1 0 0 1 | ((q ,  (p)  r   ))  |
|         |            |                 |                     |
| q_60    | q_00111100 | 0 0 1 1 1 1 0 0 | ((q ,  (p)      ))  |
|         |            |                 |                     |
| q_99    | q_01100011 | 0 1 1 0 0 0 1 1 | ((q ,   p  (r)  ))  |
|         |            |                 |                     |
| q_102   | q_01100110 | 0 1 1 0 0 1 1 0 | ((q ,      (r)  ))  |
|         |            |                 |                     |
| q_105   | q_01101001 | 0 1 1 0 1 0 0 1 | ((q ,  (p , r)  ))  |
|         |            |                 |                     |
| q_108   | q_01101100 | 0 1 1 0 1 1 0 0 | ((q ,  (p   r)  ))  |
|         |            |                 |                     |
| q_147   | q_10010011 | 1 0 0 1 0 0 1 1 | ((q ,   p   r   ))  |
|         |            |                 |                     |
| q_150   | q_10010110 | 1 0 0 1 0 1 1 0 | ((q , ((p , r)) ))  |
|         |            |                 |                     |
| q_153   | q_10011001 | 1 0 0 1 1 0 0 1 | ((q ,       r   ))  |
|         |            |                 |                     |
| q_156   | q_10011100 | 1 0 0 1 1 1 0 0 | ((q ,  (p  (r)) ))  |
|         |            |                 |                     |
| q_195   | q_11000011 | 1 1 0 0 0 0 1 1 | ((q ,   p       ))  |
|         |            |                 |                     |
| q_198   | q_11000110 | 1 1 0 0 0 1 1 0 | ((q , ((p)  r)  ))  |
|         |            |                 |                     |
| q_201   | q_00000000 | 1 1 0 0 1 0 0 1 | ((q , ((p) (r)) ))  |
|         |            |                 |                     |
| q_204   | q_00000000 | 1 1 0 0 1 1 0 0 | ((q ,           ))  |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o

Table 12.  Thematic Extensions:  [p, q] -> [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_85    | q_01010101 | 0 1 0 1 0 1 0 1 | ((r ,    ( )    ))  |
|         |            |                 |                     |
| q_86    | q_01010110 | 0 1 0 1 0 1 1 0 | ((r ,  (p) (q)  ))  |
|         |            |                 |                     |
| q_89    | q_01011001 | 0 1 0 1 1 0 0 1 | ((r ,  (p)  q   ))  |
|         |            |                 |                     |
| q_90    | q_01011010 | 0 1 0 1 1 0 1 0 | ((r ,  (p)      ))  |
|         |            |                 |                     |
| q_101   | q_01100101 | 0 1 1 0 0 1 0 1 | ((r ,   p  (q)  ))  |
|         |            |                 |                     |
| q_102   | q_01100110 | 0 1 1 0 0 1 1 0 | ((r ,      (q)  ))  |
|         |            |                 |                     |
| q_105   | q_01101001 | 0 1 1 0 1 0 0 1 | ((r ,  (p , q)  ))  |
|         |            |                 |                     |
| q_106   | q_01101010 | 0 1 1 0 1 0 1 0 | ((r ,  (p   q)  ))  |
|         |            |                 |                     |
| q_149   | q_10010101 | 1 0 0 1 0 1 0 1 | ((r ,   p   q   ))  |
|         |            |                 |                     |
| q_150   | q_10010110 | 1 0 0 1 0 1 1 0 | ((r , ((p , q)) ))  |
|         |            |                 |                     |
| q_153   | q_10011001 | 1 0 0 1 1 0 0 1 | ((r ,       q   ))  |
|         |            |                 |                     |
| q_154   | q_10011010 | 1 0 0 1 1 0 1 0 | ((r ,  (p  (q)) ))  |
|         |            |                 |                     |
| q_165   | q_10100101 | 1 0 1 0 0 1 0 1 | ((r ,   p       ))  |
|         |            |                 |                     |
| q_166   | q_10100110 | 1 0 1 0 0 1 1 0 | ((r , ((p)  q)  ))  |
|         |            |                 |                     |
| q_169   | q_10101001 | 1 0 1 0 1 0 0 1 | ((r , ((p) (q)) ))  |
|         |            |                 |                     |
| q_170   | q_10101010 | 1 0 1 0 1 0 1 0 | ((r ,           ))  |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o

Table 13.  Differences & Equalities Conjoined with Implications
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_44    | q_00101100 | 0 0 1 0 1 1 0 0 |   (p, q)   (p (r))  |
|         |            |                 |                     |
| q_52    | q_00110100 | 0 0 1 1 0 1 0 0 |   (p, q)   ((p) r)  |
|         |            |                 |                     |
| q_56    | q_00111000 | 0 0 1 1 1 0 0 0 |   (p, q)   (q (r))  |
|         |            |                 |                     |
| q_28    | q_00011100 | 0 0 0 1 1 1 0 0 |   (p, q)   ((q) r)  |
|         |            |                 |                     |
| q_131   | q_10000011 | 1 0 0 0 0 0 1 1 |  ((p, q))  (p (r))  |
|         |            |                 |                     |
| q_193   | q_11000001 | 1 1 0 0 0 0 0 1 |  ((p, q))  ((p) r)  |
|         |            |                 |                     |
|         |            |                 |                     |
| q_74    | q_01001010 | 0 1 0 0 1 0 1 0 |   (p, r)   (p (q))  |
|         |            |                 |                     |
| q_82    | q_01010010 | 0 1 0 1 0 0 1 0 |   (p, r)   ((p) q)  |
|         |            |                 |                     |
| q_26    | q_00011010 | 0 0 0 1 1 0 1 0 |   (p, r)   (q (r))  |
|         |            |                 |                     |
| q_88    | q_01011000 | 0 1 0 1 1 0 0 0 |   (p, r)   ((q) r)  |
|         |            |                 |                     |
| q_133   | q_10000101 | 1 0 0 0 0 1 0 1 |  ((p, r))  (p (q))  |
|         |            |                 |                     |
| q_161   | q_10100001 | 1 0 1 0 0 0 0 1 |  ((p, r))  ((p) q)  |
|         |            |                 |                     |
|         |            |                 |                     |
| q_70    | q_01000110 | 0 1 0 0 0 1 1 0 |   (q, r)   (p (q))  |
|         |            |                 |                     |
| q_98    | q_01100010 | 0 1 1 0 0 0 1 0 |   (q, r)   ((p) q)  |
|         |            |                 |                     |
| q_38    | q_00100110 | 0 0 1 0 0 1 1 0 |   (q, r)   (p (r))  |
|         |            |                 |                     |
| q_100   | q_01100100 | 0 1 1 0 0 1 0 0 |   (q, r)   ((p) r)  |
|         |            |                 |                     |
| q_137   | q_10001001 | 1 0 0 0 1 0 0 1 |  ((q, r))  (p (q))  |
|         |            |                 |                     |
| q_145   | q_10010001 | 1 0 0 1 0 0 0 1 |  ((q, r))  ((p) q)  |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o
|         |            |                 |                     |
| q_211   | q_11010011 | 1 1 0 1 0 0 1 1 |  ((p, q)   (p (r))) |
|         |            |                 |                     |
| q_203   | q_11001011 | 1 1 0 0 1 0 1 1 |  ((p, q)   ((p) r)) |
|         |            |                 |                     |
| q_199   | q_11000111 | 1 1 0 0 0 1 1 1 |  ((p, q)   (q (r))) |
|         |            |                 |                     |
| q_227   | q_11100011 | 1 1 1 0 0 0 1 1 |  ((p, q)   ((q) r)) |
|         |            |                 |                     |
| q_124   | q_01111100 | 0 1 1 1 1 1 0 0 | (((p, q))  (p (r))) |
|         |            |                 |                     |
| q_62    | q_00111110 | 0 0 1 1 1 1 1 0 | (((p, q))  ((p) r)) |
|         |            |                 |                     |
|         |            |                 |                     |
| q_181   | q_10110101 | 1 0 1 1 0 1 0 1 |  ((p, r)   (p (q))) |
|         |            |                 |                     |
| q_173   | q_10101101 | 1 0 1 0 1 1 0 1 |  ((p, r)   ((p) q)) |
|         |            |                 |                     |
| q_229   | q_11100101 | 1 1 1 0 0 1 0 1 |  ((p, r)   (q (r))) |
|         |            |                 |                     |
| q_167   | q_10100111 | 1 0 1 0 0 1 1 1 |  ((p, r)   ((q) r)) |
|         |            |                 |                     |
| q_122   | q_01111010 | 0 1 1 1 1 0 1 0 | (((p, r))  (p (q))) |
|         |            |                 |                     |
| q_94    | q_01011110 | 0 1 0 1 1 1 1 0 | (((p, r))  ((p) q)) |
|         |            |                 |                     |
|         |            |                 |                     |
| q_185   | q_10111001 | 1 0 1 1 1 0 0 1 |  ((q, r)   (p (q))) |
|         |            |                 |                     |
| q_157   | q_10011101 | 1 0 0 1 1 1 0 1 |  ((q, r)   ((p) q)) |
|         |            |                 |                     |
| q_217   | q_11011001 | 1 1 0 1 1 0 0 1 |  ((q, r)   (p (r))) |
|         |            |                 |                     |
| q_155   | q_10011011 | 1 0 0 1 1 0 1 1 |  ((q, r)   ((p) r)) |
|         |            |                 |                     |
| q_118   | q_01110110 | 0 1 1 1 0 1 1 0 | (((q, r))  (p (q))) |
|         |            |                 |                     |
| q_110   | q_01101110 | 0 1 1 0 1 1 1 0 | (((q, r))  ((p) q)) |
|         |            |                 |                     |
o---------o------------o-----------------o---------------------o

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

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|```````````P```````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/`````\         \`/         /`````\```````|
|``````/```````\         o         /```````\``````|
|`````/`````````\       / \       /`````````\`````|
|````/```````````\     /   \     /```````````\````|
|```o```````````` o---o-----o---o`````````````o```|
|```|`````````````````|     |`````````````````|```|
|```|`````````````````|     |`````````````````|```|
|```|``````` Q ```````|     |``````` R ```````|```|
|```o`````````````````o     o`````````````````o```|
|````\`````````````````\   /`````````````````/````|
|`````\`````````````````\ /`````````````````/`````|
|``````\`````````````````o`````````````````/``````|
|```````\```````````````/`\```````````````/```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_23.  (p)(q)(r) + ((p),(q),(r))

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \`````````\ /`````````/     \       |
|      /       \`````````o`````````/       \      |
|     /         \```````/`\```````/         \     |
|    /           \`````/```\`````/           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_232.  p q r + (p, q, r)

Table 15.  Differences and Equalities between Simples and Boundaries
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_152   | q_10011000 | 1 0 0 1 1 0 0 0 |  p + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_164   | q_10100100 | 1 0 1 0 0 1 0 0 |  q + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_194   | q_11000010 | 1 1 0 0 0 0 1 0 |  r + ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_230   | q_11100110 | 1 1 1 0 0 1 1 0 |  p + ((p), (q), (r))      |
|         |            |                 |                           |
| q_218   | q_11011010 | 1 1 0 1 1 0 1 0 |  q + ((p), (q), (r))      |
|         |            |                 |                           |
| q_188   | q_10111100 | 1 0 1 1 1 1 0 0 |  r + ((p), (q), (r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_103   | q_01100111 | 0 1 1 0 0 1 1 1 |  p = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_91    | q_01011011 | 0 1 0 1 1 0 1 1 |  q = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_61    | q_00111101 | 0 0 1 1 1 1 0 1 |  r = ( p ,  q ,  r )      |
|         |            |                 |                           |
| q_25    | q_00011001 | 0 0 0 1 1 0 0 1 |  p = ((p), (q), (r))      |
|         |            |                 |                           |
| q_37    | q_00100101 | 0 0 1 0 0 1 0 1 |  q = ((p), (q), (r))      |
|         |            |                 |                           |
| q_67    | q_01000011 | 0 1 0 0 0 0 1 1 |  r = ((p), (q), (r))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |`````````` P ``````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /     \         \`/         /     \       |
|      /       \         o         /       \      |
|     /         \       /`\       /         \     |
|    /           \     /```\     /           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_152.  p + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /`````\         \ /`````````/     \       |
|      /```````\         o`````````/       \      |
|     /`````````\       /`\```````/         \     |
|    /```````````\     /```\`````/           \    |
|   o`````````````o---o-----o---o             o   |
|   |`````````````````|     |                 |   |
|   |`````````````````|     |                 |   |
|   |``````` Q ```````|     |        R        |   |
|   o`````````````````o     o                 o   |
|    \`````````````````\   /                 /    |
|     \`````````````````\ /                 /     |
|      \`````````````````o                 /      |
|       \```````````````/ \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_164.  q + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \`````````\ /         /`````\       |
|      /       \`````````o         /```````\      |
|     /         \```````/`\       /`````````\     |
|    /           \`````/```\     /```````````\    |
|   o             o---o-----o---o`````````````o   |
|   |                 |     |`````````````````|   |
|   |                 |     |`````````````````|   |
|   |        Q        |     |``````` R ```````|   |
|   o                 o     o`````````````````o   |
|    \                 \   /`````````````````/    |
|     \                 \ /`````````````````/     |
|      \                 o`````````````````/      |
|       \               / \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_194.  r + (p, q, r)

o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/               \````````````````|
|```````````````/                 \```````````````|
|``````````````/                   \``````````````|
|`````````````/                     \`````````````|
|````````````o                       o````````````|
|````````````|           P           |````````````|
|````````````|                       |````````````|
|````````````|                       |````````````|
|````````o---o---------o   o---------o---o````````|
|```````/     \         \ /         /     \```````|
|``````/       \         o         /       \``````|
|`````/         \       /`\       /         \`````|
|````/           \     /```\     /           \````|
|```o             o---o-----o---o             o```|
|```|                 |     |                 |```|
|```|                 |     |                 |```|
|```|        Q        |     |        R        |```|
|```o                 o     o                 o```|
|````\                 \   /                 /````|
|`````\                 \ /                 /`````|
|``````\                 o                 /``````|
|```````\               /`\               /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_129.  ((p, q))((q, r))

Table 16.  Paisley 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_216   | q_11011000 | 1 1 0 1 1 0 0 0 |   (p, q)(p, r)  +  p q    |
|         |            |                 |                           |
| q_184   | q_10111000 | 1 0 1 1 1 0 0 0 |   (p, q)(p, r)  +  p r    |
|         |            |                 |                           |
| q_228   | q_11100100 | 1 1 1 0 0 1 0 0 |   (p, q)(q, r)  +  p q    |
|         |            |                 |                           |
| q_172   | q_10101100 | 1 0 1 0 1 1 0 0 |   (p, q)(q, r)  +  q r    |
|         |            |                 |                           |
| q_226   | q_11100010 | 1 1 1 0 0 0 1 0 |   (p, r)(q, r)  +  p r    |
|         |            |                 |                           |
| q_202   | q_11001010 | 1 1 0 0 1 0 1 0 |   (p, r)(q, r)  +  q r    |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_39    | q_00100111 | 0 0 1 0 0 1 1 1 |   (p, q)(p, r)  =  p q    |
|         |            |                 |                           |
| q_71    | q_01000111 | 0 1 0 0 0 1 1 1 |   (p, q)(p, r)  =  p r    |
|         |            |                 |                           |
| q_27    | q_00011011 | 0 0 0 1 1 0 1 1 |   (p, q)(q, r)  =  p q    |
|         |            |                 |                           |
| q_83    | q_01010011 | 0 1 0 1 0 0 1 1 |   (p, q)(q, r)  =  q r    |
|         |            |                 |                           |
| q_29    | q_00011101 | 0 0 0 1 1 1 0 1 |   (p, r)(q, r)  =  p r    |
|         |            |                 |                           |
| q_53    | q_00110101 | 0 0 1 1 0 1 0 1 |   (p, r)(q, r)  =  q r    |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o

Table 17.  Paisley 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_216   | q_11011000 | 1 1 0 1 1 0 0 0 |   p + pq + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_184   | q_10111000 | 1 0 1 1 1 0 0 0 |   p + pr + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_228   | q_11100100 | 1 1 1 0 0 1 0 0 |   q + pq + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_172   | q_10101100 | 1 0 1 0 1 1 0 0 |   q + qr + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_226   | q_11100010 | 1 1 1 0 0 0 1 0 |   r + pr + pqr + (p, q, r)   |
|         |            |                 |                              |
| q_202   | q_11001010 | 1 1 0 0 1 0 1 0 |   r + qr + pqr + (p, q, r)   |
|         |            |                 |                              |
o---------o------------o-----------------o------------------------------o
|         |            |                 |                              |
| q_39    | q_00100111 | 0 0 1 0 0 1 1 1 | 1 + p + pq + pqr + (p, q, r) |
|         |            |                 |                              |
| q_71    | q_01000111 | 0 1 0 0 0 1 1 1 | 1 + p + pr + pqr + (p, q, r) |
|         |            |                 |                              |
| q_27    | q_00011011 | 0 0 0 1 1 0 1 1 | 1 + q + pq + pqr + (p, q, r) |
|         |            |                 |                              |
| q_83    | q_01010011 | 0 1 0 1 0 0 1 1 | 1 + q + qr + pqr + (p, q, r) |
|         |            |                 |                              |
| q_29    | q_00011101 | 0 0 0 1 1 1 0 1 | 1 + r + pr + pqr + (p, q, r) |
|         |            |                 |                              |
| q_53    | q_00110101 | 0 0 1 1 0 1 0 1 | 1 + r + qr + pqr + (p, q, r) |
|         |            |                 |                              |
o---------o------------o-----------------o------------------------------o

o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o-------------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `/%%%%%%%%%%%%%%%\` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `/%%%%%%%%%%%%%%%%%%%\` ` ` ` ` ` ` |
| ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` |
| ` ` ` ` ` `o%%%%%%%%%%%%%%%%%%%%%%%o` ` ` ` ` ` |
| ` ` ` ` ` `|%%%%%%%%%% P %%%%%%%%%%|` ` ` ` ` ` |
| ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` |
| ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` |
| ` ` ` `o---o---------o%%%o---------o---o` ` ` ` |
| ` ` ` / ` ` \%%%%%%%%%\%/ ` ` ` ` / ` ` \ ` ` ` |
| ` ` `/` ` ` `\%%%%%%%%%o` ` ` ` `/` ` ` `\` ` ` |
| ` ` / ` ` ` ` \%%%%%%%/%\ ` ` ` / ` ` ` ` \ ` ` |
| ` `/` ` ` ` ` `\%%%%%/%%%\` ` `/` ` ` ` ` `\` ` |
| ` o ` ` ` ` ` ` o---o-----o---o ` ` ` ` ` ` o ` |
| ` | ` ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` ` | ` |
| ` | ` ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` ` | ` |
| ` | ` ` ` `Q` ` ` ` |%%%%%| ` ` ` `R` ` ` ` | ` |
| ` o ` ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` ` o ` |
| ` `\` ` ` ` ` ` ` ` `\%%%/` ` ` ` ` ` ` ` `/` ` |
| ` ` \ ` ` ` ` ` ` ` ` \%/ ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` /`\ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `o-------------o` `o-------------o` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
q_216.  p + p q + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |`````````` P ``````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /     \         \`/         /     \       |
|      /       \         o         /       \      |
|     /         \       / \       /         \     |
|    /           \     /   \     /           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_24.  (p, q)(p, r)

q_24.   p + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |`````````` P ``````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /     \`````````\`/         /     \       |
|      /       \`````````o         /       \      |
|     /         \```````/`\       /         \     |
|    /           \`````/```\     /           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_216.  (p, q)(p, r) + p q

q_216.   p + p q + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /```````````````\                |
|               /`````````````````\               |
|              /```````````````````\              |
|             /`````````````````````\             |
|            o```````````````````````o            |
|            |`````````` P ``````````|            |
|            |```````````````````````|            |
|            |```````````````````````|            |
|        o---o---------o```o---------o---o        |
|       /     \         \`/`````````/     \       |
|      /       \         o`````````/       \      |
|     /         \       /`\```````/         \     |
|    /           \     /```\`````/           \    |
|   o             o---o-----o---o             o   |
|   |                 |`````|                 |   |
|   |                 |`````|                 |   |
|   |        Q        |`````|        R        |   |
|   o                 o`````o                 o   |
|    \                 \```/                 /    |
|     \                 \`/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_184.  (p, q)(p, r) + p r

q_184.   p + p r + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /`````\         \ /`````````/     \       |
|      /```````\         o`````````/       \      |
|     /`````````\       / \```````/         \     |
|    /```````````\     /   \`````/           \    |
|   o`````````````o---o-----o---o             o   |
|   |`````````````````|     |                 |   |
|   |`````````````````|     |                 |   |
|   |``````` Q ```````|     |        R        |   |
|   o`````````````````o     o                 o   |
|    \`````````````````\   /                 /    |
|     \`````````````````\ /                 /     |
|      \`````````````````o                 /      |
|       \```````````````/ \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_36.  (p, q)(q, r)

q_36.  q + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /`````\`````````\ /`````````/     \       |
|      /```````\`````````o`````````/       \      |
|     /`````````\```````/`\```````/         \     |
|    /```````````\`````/```\`````/           \    |
|   o`````````````o---o-----o---o             o   |
|   |`````````````````|     |                 |   |
|   |`````````````````|     |                 |   |
|   |``````` Q ```````|     |        R        |   |
|   o`````````````````o     o                 o   |
|    \`````````````````\   /                 /    |
|     \`````````````````\ /                 /     |
|      \`````````````````o                 /      |
|       \```````````````/ \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_228.  (p, q)(q, r) + p q

q_228.  q + p q + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /`````\         \ /`````````/     \       |
|      /```````\         o`````````/       \      |
|     /`````````\       /`\```````/         \     |
|    /```````````\     /```\`````/           \    |
|   o`````````````o---o-----o---o             o   |
|   |`````````````````|`````|                 |   |
|   |`````````````````|`````|                 |   |
|   |``````` Q ```````|`````|        R        |   |
|   o`````````````````o`````o                 o   |
|    \`````````````````\```/                 /    |
|     \`````````````````\`/                 /     |
|      \`````````````````o                 /      |
|       \```````````````/ \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_172.  (p, q)(q, r) + q r

q_172.  q + q r + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \`````````\ /         /`````\       |
|      /       \`````````o         /```````\      |
|     /         \```````/ \       /`````````\     |
|    /           \`````/   \     /```````````\    |
|   o             o---o-----o---o`````````````o   |
|   |                 |     |`````````````````|   |
|   |                 |     |`````````````````|   |
|   |        Q        |     |``````` R ```````|   |
|   o                 o     o`````````````````o   |
|    \                 \   /`````````````````/    |
|     \                 \ /`````````````````/     |
|      \                 o`````````````````/      |
|       \               / \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_66.  (p, r)(q, r)

q_66.  r + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \`````````\ /`````````/`````\       |
|      /       \`````````o`````````/```````\      |
|     /         \```````/`\```````/`````````\     |
|    /           \`````/```\`````/```````````\    |
|   o             o---o-----o---o`````````````o   |
|   |                 |     |`````````````````|   |
|   |                 |     |`````````````````|   |
|   |        Q        |     |``````` R ```````|   |
|   o                 o     o`````````````````o   |
|    \                 \   /`````````````````/    |
|     \                 \ /`````````````````/     |
|      \                 o`````````````````/      |
|       \               / \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_226.  (p, r)(q, r) + p r

q_266.  r + p r + p q r + (p, q, r)

o-------------------------------------------------o
|                                                 |
|                                                 |
|                 o-------------o                 |
|                /               \                |
|               /                 \               |
|              /                   \              |
|             /                     \             |
|            o                       o            |
|            |           P           |            |
|            |                       |            |
|            |                       |            |
|        o---o---------o   o---------o---o        |
|       /     \`````````\ /         /`````\       |
|      /       \`````````o         /```````\      |
|     /         \```````/`\       /`````````\     |
|    /           \`````/```\     /```````````\    |
|   o             o---o-----o---o`````````````o   |
|   |                 |`````|`````````````````|   |
|   |                 |`````|`````````````````|   |
|   |        Q        |`````|``````` R ```````|   |
|   o                 o`````o`````````````````o   |
|    \                 \```/`````````````````/    |
|     \                 \`/`````````````````/     |
|      \                 o`````````````````/      |
|       \               / \```````````````/       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
q_202.  (p, r)(q, r) + q r

q_202.  r + q r + p q r + (p, q, r)

Table 18.  Desultory Junctions 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_224   | q_11100000 | 1 1 1 0 0 0 0 0 |        p   ((q)(r))       |
|         |            |                 |                           |
| q_200   | q_11001000 | 1 1 0 0 1 0 0 0 |        q   ((p)(r))       |
|         |            |                 |                           |
| q_168   | q_10101000 | 1 0 1 0 1 0 0 0 |        r   ((p)(q))       |
|         |            |                 |                           |
| q_14    | q_00001110 | 0 0 0 0 1 1 1 0 |       (p)  ((q)(r))       |
|         |            |                 |                           |
| q_50    | q_00110010 | 0 0 1 1 0 0 1 0 |       (q)  ((p)(r))       |
|         |            |                 |                           |
| q_84    | q_01010100 | 0 1 0 1 0 1 0 0 |       (r)  ((p)(q))       |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
|         |            |                 |                           |
| q_31    | q_00011111 | 0 0 0 1 1 1 1 1 |       (p   ((q)(r)))      |
|         |            |                 |                           |
| q_55    | q_00110111 | 0 0 1 1 0 1 1 1 |       (q   ((p)(r)))      |
|         |            |                 |                           |
| q_87    | q_01010111 | 0 1 0 1 0 1 1 1 |       (r   ((p)(q)))      |
|         |            |                 |                           |
| q_241   | q_11110001 | 1 1 1 1 0 0 0 1 |      ((p)  ((q)(r)))      |
|         |            |                 |                           |
| q_205   | q_11001101 | 1 1 0 0 1 1 0 1 |      ((q)  ((p)(r)))      |
|         |            |                 |                           |
| q_171   | q_10101011 | 1 0 1 0 1 0 1 1 |      ((r)  ((p)(q)))      |
|         |            |                 |                           |
o---------o------------o-----------------o---------------------------o
```

## Discussion Note

```Just by way of incidental kibitzing,
I notice that Rule 73 has the form of
a "genus and species" or "pie-chart"
proposition, where q is the genus
and p and r are the species.

The cactus expression and
cactus graph are as follows:

o-------------------o
|                   |
|                   |
|       p   r       |
|       o   o       |
|       | q |       |
|       o-o-o       |
|        \ /        |
|         @         |
o-------------------o
|   ((p), q ,(r))   |
o-------------------o
|       q_73        |
o-------------------o

See the discussion in and
around Cactus Rules Note 5.

```