Difference between revisions of "Directory:Jon Awbrey/Papers/Cactus Rules"
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{{DISPLAYTITLE:Cactus Rules}} | {{DISPLAYTITLE:Cactus Rules}} | ||
| + | |||
| + | ==Note 1== | ||
<pre> | <pre> | ||
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With an eye toward the aims of the NKS Forum, I've begun to work out | 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), | a translation of the "elementary cellular automaton rules" (ECAR's), | ||
| Line 39: | Line 27: | ||
http://www.pinball.com/games/cactus/ | http://www.pinball.com/games/cactus/ | ||
| + | </pre> | ||
| − | + | ==Note 2== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
One of the first things I note is that several whole families | One of the first things I note is that several whole families | ||
of otherwise enigmatic and obscurely expressed rules take on | of otherwise enigmatic and obscurely expressed rules take on | ||
| Line 109: | Line 95: | ||
http://atlas.wolfram.com/01/01/views/172/TableView.html | http://atlas.wolfram.com/01/01/views/172/TableView.html | ||
| + | </pre> | ||
| − | + | ==Note 3== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Here are the parse-graph portraits of the family of cacti | Here are the parse-graph portraits of the family of cacti | ||
that we examined last time, listed in complementary pairs. | that we examined last time, listed in complementary pairs. | ||
| Line 250: | Line 234: | ||
as I might like, and it may be that other eyes would see | as I might like, and it may be that other eyes would see | ||
forms more economical than the ones that strike me first. | forms more economical than the ones that strike me first. | ||
| + | </pre> | ||
| − | + | ==Note 4== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Given the novelty of the cactus calculus, it is probably | Given the novelty of the cactus calculus, it is probably | ||
wise to run through a representative sample of the forms | wise to run through a representative sample of the forms | ||
| Line 335: | Line 317: | ||
a number of mutually exclusive and exhaustive territories, | a number of mutually exclusive and exhaustive territories, | ||
here envisioned to salute the flags p, q, r, respectively. | here envisioned to salute the flags p, q, r, respectively. | ||
| + | </pre> | ||
| − | + | ==Note 5== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
So long as we're seeing the sights at Cactus Junction, | So long as we're seeing the sights at Cactus Junction, | ||
we might as well take a gander at a computational way | we might as well take a gander at a computational way | ||
| Line 459: | Line 439: | ||
That is not yet a method that would be amenable to | That is not yet a method that would be amenable to | ||
computational routine, but it does get us part way. | computational routine, but it does get us part way. | ||
| + | </pre> | ||
| − | + | ==Note 6== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Within each space of boolean functions {f : B^k -> B}, | Within each space of boolean functions {f : B^k -> B}, | ||
altogether ranking a cardinality of 2^(2^k) functions, | altogether ranking a cardinality of 2^(2^k) functions, | ||
| Line 586: | Line 564: | ||
Beannachtaí na Féile Pádraig oraibh go leir! | Beannachtaí na Féile Pádraig oraibh go leir! | ||
| + | </pre> | ||
| − | + | ==Note 7== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Had I been thinking ahead, I might have mentioned this first, | Had I been thinking ahead, I might have mentioned this first, | ||
but now that aspects of algebra and geometry have intruded on | but now that aspects of algebra and geometry have intruded on | ||
| Line 653: | Line 629: | ||
With that out of the way, I'll try to | With that out of the way, I'll try to | ||
get back to the main event next time. | get back to the main event next time. | ||
| + | </pre> | ||
| − | + | ==Note 8== | |
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| − | |||
| − | |||
| − | |||
| + | <pre> | ||
In any k-dimensional universe of discourse X% = [x_1, ..., x_k] | In any k-dimensional universe of discourse X% = [x_1, ..., x_k] | ||
there are two other (2^k)-clans of propositions that ordinarily | there are two other (2^k)-clans of propositions that ordinarily | ||
| Line 760: | Line 734: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o-------------------o | o---------o------------o-----------------o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 9== | |
| − | |||
| − | |||
| − | + | <pre> | |
| − | + | In the language of cacti, as in Peirce's existential graphs, | |
| − | In the language of cacti, as in Peirce's existential graphs, | ||
the implication p => q takes the form (p (q)), which can be | the implication p => q takes the form (p (q)), which can be | ||
parsed in a revealing manner as "not p without q". Thus it | parsed in a revealing manner as "not p without q". Thus it | ||
| Line 844: | Line 816: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o-------------------o | o---------o------------o-----------------o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 10== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Table 6. More Variations on a Theme of Implication | Table 6. More Variations on a Theme of Implication | ||
o---------o------------o-----------------o-------------------o | o---------o------------o-----------------o-------------------o | ||
| Line 913: | Line 883: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o-------------------o | o---------o------------o-----------------o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 11== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Table 7. Conjunctive Implications and Their Complements | Table 7. Conjunctive Implications and Their Complements | ||
o---------o------------o-----------------o-------------------o | o---------o------------o-----------------o-------------------o | ||
| Line 958: | Line 926: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o-------------------o | o---------o------------o-----------------o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 12== | |
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| − | |||
| − | |||
| − | |||
| + | <pre> | ||
In the language of cacti, unlike Peirce's alpha graphs, | In the language of cacti, unlike Peirce's alpha graphs, | ||
it is possible to represent the logical functions that | it is possible to represent the logical functions that | ||
| Line 1,056: | Line 1,022: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o-------------------o | o---------o------------o-----------------o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 13== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Table 9. Conjunctive Differences and Equalities | Table 9. Conjunctive Differences and Equalities | ||
o---------o------------o-----------------o--------------------o | o---------o------------o-----------------o--------------------o | ||
| Line 1,093: | Line 1,057: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o--------------------o | o---------o------------o-----------------o--------------------o | ||
| + | </pre> | ||
| − | + | ==Note 14== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
I will explain my concept of "thematization" | I will explain my concept of "thematization" | ||
or "thematic extension" after I copy out the | or "thematic extension" after I copy out the | ||
| Line 1,161: | Line 1,123: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| + | </pre> | ||
| − | + | ==Note 15== | |
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| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Table 11. Thematic Extensions: [p, r] -> [p, q, r] | Table 11. Thematic Extensions: [p, r] -> [p, q, r] | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| Line 1,212: | Line 1,172: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| + | </pre> | ||
| − | + | ==Note 16== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Table 12. Thematic Extensions: [p, q] -> [p, q, r] | Table 12. Thematic Extensions: [p, q] -> [p, q, r] | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| Line 1,263: | Line 1,221: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| + | </pre> | ||
| − | + | ==Note 17== | |
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| + | <pre> | ||
Table 13. Differences & Equalities Conjoined with Implications | Table 13. Differences & Equalities Conjoined with Implications | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| Line 1,360: | Line 1,316: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| + | </pre> | ||
| − | + | ==Note 18== | |
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| − | |||
| − | |||
| + | <pre> | ||
Table 14 shows the propositions q_i : B^3 -> B whose "fibers of truth", | 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 | that is, whose pre-images of 1, have the form of a single point in B^3 | ||
| Line 1,400: | Line 1,354: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| + | </pre> | ||
| − | + | ==Note 19== | |
| − | |||
| − | |||
| − | + | <pre> | |
| − | + | Table 15. Differences and Equalities between Simples and Boundaries | |
| − | Table 15. Differences and Equalities between Simples and Boundaries | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| L_1 | L_2 | L_3 | L_4 | | | L_1 | L_2 | L_3 | L_4 | | ||
| Line 1,445: | Line 1,397: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| + | </pre> | ||
| − | + | ==Note 20== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Table 16. Paisley Propositions | Table 16. Paisley Propositions | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| Line 1,490: | Line 1,440: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| + | </pre> | ||
| − | + | ==Note 21== | |
| − | |||
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| − | |||
| − | |||
| + | <pre> | ||
Table 17 gives another way of writing the "paisley propositions" | Table 17 gives another way of writing the "paisley propositions" | ||
that makes their symmetry class more manifest. The venn diagram | that makes their symmetry class more manifest. The venn diagram | ||
| Line 1,572: | Line 1,520: | ||
o-------------------------------------------------o | o-------------------------------------------------o | ||
q_216. p + p q + p q r + (p, q, r) | q_216. p + p q + p q r + (p, q, r) | ||
| + | </pre> | ||
| − | + | ==Note 22== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
I'm puzzled by the blind-spot that prevented me | I'm puzzled by the blind-spot that prevented me | ||
from seeing this very simple and natural family | from seeing this very simple and natural family | ||
| Line 1,625: | Line 1,571: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| + | </pre> | ||
| − | + | ==Note 23== | |
| − | |||
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| − | |||
| − | |||
| + | <pre> | ||
For ease of viewing, I am placing | For ease of viewing, I am placing | ||
copies of the Cactus Rules Table | copies of the Cactus Rules Table | ||
| Line 1,638: | Line 1,582: | ||
Table 256. http://stderr.org/pipermail/inquiry/2004-April/001314.html | Table 256. http://stderr.org/pipermail/inquiry/2004-April/001314.html | ||
Table 256. http://suo.ieee.org/ontology/msg05512.html | Table 256. http://suo.ieee.org/ontology/msg05512.html | ||
| + | </pre> | ||
| − | + | ==Note 24a== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Here is a set of representative cactus graphs | Here is a set of representative cactus graphs | ||
for the 256 propositions on three variables. | for the 256 propositions on three variables. | ||
| Line 2,167: | Line 2,109: | ||
| q_31 | | q_224 | | | q_31 | | q_224 | | ||
o-------------------o o-------------------o | o-------------------o o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 24b== | |
| − | |||
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| − | |||
| + | <pre> | ||
o-------------------o o-------------------o | o-------------------o o-------------------o | ||
| | | | | | | | | | ||
| Line 2,706: | Line 2,646: | ||
| q_63 | | q_192 | | | q_63 | | q_192 | | ||
o-------------------o o-------------------o | o-------------------o o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 24c== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
o-------------------o o-------------------o | o-------------------o o-------------------o | ||
| | | | | | | | | | ||
| Line 3,245: | Line 3,183: | ||
| q_95 | | q_160 | | | q_95 | | q_160 | | ||
o-------------------o o-------------------o | o-------------------o o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 24d== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
o-------------------o o-------------------o | o-------------------o o-------------------o | ||
| | | | | | | | | | ||
| Line 3,783: | Line 3,719: | ||
| q_127 | | q_128 | | | q_127 | | q_128 | | ||
o-------------------o o-------------------o | o-------------------o o-------------------o | ||
| + | </pre> | ||
| − | + | ==Note 24e== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
I'm attaching here a text file copy of the current set | I'm attaching here a text file copy of the current set | ||
of cactus graphs for propositions on three variables, | of cactus graphs for propositions on three variables, | ||
| Line 4,358: | Line 4,292: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| + | </pre> | ||
| − | + | ==Work Area 1== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
o-------------------------------------------------o | o-------------------------------------------------o | ||
| | | | | | ||
| Line 4,428: | Line 4,360: | ||
o-------------------------------------------------o | o-------------------------------------------------o | ||
Figure 1. Full Universe | Figure 1. Full Universe | ||
| + | </pre> | ||
| − | + | ==Work Area 2== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Table 1. Boundaries and Their Complements | Table 1. Boundaries and Their Complements | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| Line 4,787: | Line 4,717: | ||
o-------------------------------------------------o | o-------------------------------------------------o | ||
q_131. r + ((p),(q), r) | q_131. r + ((p),(q), r) | ||
| + | </pre> | ||
| − | + | ==Work Area 3== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
o-------------------------------------------------o | o-------------------------------------------------o | ||
| | | | | | ||
| Line 5,031: | Line 4,959: | ||
o-------------------------------------------------o | o-------------------------------------------------o | ||
Thematic Extension q_225. ((p, ((q)(r)) )) | Thematic Extension q_225. ((p, ((q)(r)) )) | ||
| + | </pre> | ||
| − | + | ==Work Area 4== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| L_1 | L_2 | L_3 | L_4 | | | L_1 | L_2 | L_3 | L_4 | | ||
| Line 5,075: | Line 5,001: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------o | o---------o------------o-----------------o---------------------o | ||
| + | </pre> | ||
| − | + | ==Appendices== | |
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Table 0. Simple Propositions | Table 0. Simple Propositions | ||
o---------o------------o-----------------o-------------------o | o---------o------------o-----------------o-------------------o | ||
| Line 6,468: | Line 6,392: | ||
| | | | | | | | | | | | ||
o---------o------------o-----------------o---------------------------o | o---------o------------o-----------------o---------------------------o | ||
| + | </pre> | ||
| − | + | ==Discussion Note== | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| + | <pre> | ||
Just by way of incidental kibitzing, | Just by way of incidental kibitzing, | ||
I notice that Rule 73 has the form of | I notice that Rule 73 has the form of | ||
| Line 6,507: | Line 6,425: | ||
http://forum.wolframscience.com/showthread.php?postid=830#post830 | http://forum.wolframscience.com/showthread.php?postid=830#post830 | ||
| + | </pre> | ||
| − | + | ==Document History== | |
| + | <pre> | ||
CR. Cactus Rules | CR. Cactus Rules | ||
| Line 6,604: | Line 6,524: | ||
01. http://forum.wolframscience.com/showthread.php?postid=901#post901 | 01. http://forum.wolframscience.com/showthread.php?postid=901#post901 | ||
| − | |||
| − | |||
</pre> | </pre> | ||
Revision as of 13:05, 22 May 2009
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 of k expressions is read as their k-adic logical conjunction, second, 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: Re: http://forum.wolframscience.com/showthread.php?postid=810#post810 In: http://forum.wolframscience.com/showthread.php?threadid=256 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"
| http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
| http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
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 of propositions, especially since I had already 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. http://forum.wolframscience.com/showthread.php?postid=830#post830
Document History
CR. Cactus Rules Ontology List 01. http://suo.ieee.org/ontology/msg05486.html 02. http://suo.ieee.org/ontology/msg05487.html 03. http://suo.ieee.org/ontology/msg05488.html 04. http://suo.ieee.org/ontology/msg05489.html 05. http://suo.ieee.org/ontology/msg05490.html 06. http://suo.ieee.org/ontology/msg05491.html 07. http://suo.ieee.org/ontology/msg05492.html 08. http://suo.ieee.org/ontology/msg05493.html 09. http://suo.ieee.org/ontology/msg05494.html 10. http://suo.ieee.org/ontology/msg05495.html 11. http://suo.ieee.org/ontology/msg05496.html 12. http://suo.ieee.org/ontology/msg05498.html 13. http://suo.ieee.org/ontology/msg05499.html 14. http://suo.ieee.org/ontology/msg05500.html 15. http://suo.ieee.org/ontology/msg05501.html 16. http://suo.ieee.org/ontology/msg05502.html 17. http://suo.ieee.org/ontology/msg05503.html 18. http://suo.ieee.org/ontology/msg05507.html 19. http://suo.ieee.org/ontology/msg05508.html 20. http://suo.ieee.org/ontology/msg05509.html 21. http://suo.ieee.org/ontology/msg05510.html 22. http://suo.ieee.org/ontology/msg05511.html 23. http://suo.ieee.org/ontology/msg05512.html 24. http://suo.ieee.org/ontology/msg05518.html Inquiry List 00. http://stderr.org/pipermail/inquiry/2004-March/thread.html#1265 00. http://stderr.org/pipermail/inquiry/2004-April/thread.html#1305 01. http://stderr.org/pipermail/inquiry/2004-March/001265.html 02. http://stderr.org/pipermail/inquiry/2004-March/001266.html 03. http://stderr.org/pipermail/inquiry/2004-March/001267.html 04. http://stderr.org/pipermail/inquiry/2004-March/001268.html 05. http://stderr.org/pipermail/inquiry/2004-March/001269.html 06. http://stderr.org/pipermail/inquiry/2004-March/001270.html 07. http://stderr.org/pipermail/inquiry/2004-March/001271.html 08. http://stderr.org/pipermail/inquiry/2004-March/001272.html 09. http://stderr.org/pipermail/inquiry/2004-March/001273.html 10. http://stderr.org/pipermail/inquiry/2004-March/001274.html 11. http://stderr.org/pipermail/inquiry/2004-March/001275.html 12. http://stderr.org/pipermail/inquiry/2004-March/001277.html 13. http://stderr.org/pipermail/inquiry/2004-March/001278.html 14. http://stderr.org/pipermail/inquiry/2004-March/001279.html 15. http://stderr.org/pipermail/inquiry/2004-March/001280.html 16. http://stderr.org/pipermail/inquiry/2004-March/001281.html 17. http://stderr.org/pipermail/inquiry/2004-March/001290.html 18. http://stderr.org/pipermail/inquiry/2004-April/001305.html 19. http://stderr.org/pipermail/inquiry/2004-April/001306.html 20. http://stderr.org/pipermail/inquiry/2004-April/001307.html 21. http://stderr.org/pipermail/inquiry/2004-April/001308.html 22. http://stderr.org/pipermail/inquiry/2004-April/001312.html 23. http://stderr.org/pipermail/inquiry/2004-April/001314.html 24. http://stderr.org/pipermail/inquiry/2004-April/001322.html NKS Forum 00. http://forum.wolframscience.com/showthread.php?threadid=256 01. http://forum.wolframscience.com/showthread.php?postid=810#post810 02. http://forum.wolframscience.com/showthread.php?postid=818#post818 03. http://forum.wolframscience.com/showthread.php?postid=826#post826 04. http://forum.wolframscience.com/showthread.php?postid=829#post829 05. http://forum.wolframscience.com/showthread.php?postid=830#post830 06. http://forum.wolframscience.com/showthread.php?postid=831#post831 07. http://forum.wolframscience.com/showthread.php?postid=832#post832 08. http://forum.wolframscience.com/showthread.php?postid=834#post834 09. http://forum.wolframscience.com/showthread.php?postid=835#post835 10. http://forum.wolframscience.com/showthread.php?postid=838#post838 11. http://forum.wolframscience.com/showthread.php?postid=840#post840 12. http://forum.wolframscience.com/showthread.php?postid=841#post841 13. http://forum.wolframscience.com/showthread.php?postid=842#post842 14. http://forum.wolframscience.com/showthread.php?postid=843#post843 15. http://forum.wolframscience.com/showthread.php?postid=844#post844 16. http://forum.wolframscience.com/showthread.php?postid=845#post845 17. http://forum.wolframscience.com/showthread.php?postid=854#post854 18. http://forum.wolframscience.com/showthread.php?postid=891#post891 19. http://forum.wolframscience.com/showthread.php?postid=894#post894 20. http://forum.wolframscience.com/showthread.php?postid=897#post897 21. http://forum.wolframscience.com/showthread.php?postid=898#post898 22. http://forum.wolframscience.com/showthread.php?postid=902#post902 23. http://forum.wolframscience.com/showthread.php?postid=909#post909 24a. http://forum.wolframscience.com/showthread.php?postid=927#post927 24b. http://forum.wolframscience.com/showthread.php?postid=928#post928 24c. http://forum.wolframscience.com/showthread.php?postid=929#post929 24d. http://forum.wolframscience.com/showthread.php?postid=933#post933 24e. http://forum.wolframscience.com/showthread.php?postid=934#post934 CR. Cactus Rules -- Discussion 01. http://forum.wolframscience.com/showthread.php?postid=901#post901
