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{{DISPLAYTITLE:Cactus Rules}}
{{DISPLAYTITLE:Cactus Rules}}
==Note 1==
<pre>
<pre>
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),
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
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.
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
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
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,
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
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==
<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
| | | | |
| | | | |
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
| | | | |
| | | | |
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
| | | | |
| | | | |
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
| | | | |
| | | | |
o---------o------------o-----------------o-------------------o
o---------o------------o-----------------o-------------------o
</pre>
==Note 12==
<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
| | | | |
| | | | |
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
| | | | |
| | | | |
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
| | | | |
| | | | |
o---------o------------o-----------------o---------------------o
o---------o------------o-----------------o---------------------o
</pre>
==Note 15==
<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
| | | | |
| | | | |
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
| | | | |
| | | | |
o---------o------------o-----------------o---------------------o
o---------o------------o-----------------o---------------------o
</pre>
==Note 17==
<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
| | | | |
| | | | |
o---------o------------o-----------------o---------------------o
o---------o------------o-----------------o---------------------o
</pre>
==Note 18==
<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
| | | | |
| | | | |
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 |
| | | | |
| | | | |
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
| | | | |
| | | | |
o---------o------------o-----------------o---------------------------o
o---------o------------o-----------------o---------------------------o
</pre>
==Note 21==
<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
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
| | | | |
| | | | |
o---------o------------o-----------------o---------------------------o
o---------o------------o-----------------o---------------------------o
</pre>
==Note 23==
<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
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.
| q_31 | | q_224 |
| q_31 | | q_224 |
o-------------------o o-------------------o
o-------------------o o-------------------o
</pre>
==Note 24b==
<pre>
o-------------------o o-------------------o
o-------------------o o-------------------o
| | | |
| | | |
| 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
| | | |
| | | |
| 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
| | | |
| | | |
| 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,
| | | | |
| | | | |
o---------o------------o-----------------o---------------------------o
o---------o------------o-----------------o---------------------------o
</pre>
==Work Area 1==
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
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
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
| |
| |
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 |
| | | | |
| | | | |
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
| | | | |
| | | | |
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
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
01. http://forum.wolframscience.com/showthread.php?postid=901#post901
01. http://forum.wolframscience.com/showthread.php?postid=901#post901
</pre>
</pre>
