Let us consider the following 3 rules:
rule 1: those systems where each chamber definition must contain three different letters. All of these systems satisfy the closure principle of manufacturing the agents that convert or manufacture other entities.
rule 2: those systems where the letters present in the first chamber can be used to generate letters in other chambers forming a
chain reaction until all the reactions have taken place.
rule 3: those systems where no two chambers contain the same two letters.
Rule 2 and
rule 3 are extra constraints on
rule 1. Systems that satisfy either
rule 2 or
rule 3 also satisfy
rule 1.
It should be noted that there are
systems with self-entailing boxes that are not constrained by rule 1.
3 Chamber Systems
Let us review how the 3-chamber system was chosen. Consider all they ways to pick the agents that convert the following inputs to outputs.
| :a->b |
:b->c |
:c->d |
from |
{b,c,d} |
Applying
rule 1, there is only one choice.
This system also happens to satisfy
rule 2 because
c:a->b has letters
b and
c present that could be used in reaction
b:c->d. Then letters
d and
b are present that could be used in reaction
d:b->c.
 |
|
Satisfying rule 2 (chain reaction) |
It does not satisfy
rule 3 because as can be seen below (with the yellow highlights on
b and
c) there exists some pairs of letters that show up in more than one chamber. No systems satisfying
rule 2 also satisfy
rule 3.
4 Chamber Systems
Consider a 4 chamber system with all possibilities of picking the converting agents.
| :a->b |
:b->c |
:c->d |
:d->e |
from |
{b,c,d,e} |
If we apply
rule 1, we get the following three possibilities as depicted in each row:
| c:a->b |
d:b->c |
e:c->d |
b:d->e |
| d:a->b |
e:b->c |
b:c->d |
c:d->e |
| e:a->b |
d:b->c |
b:c->d |
c:d->e |
None of these systems satisfy
rule 2. For example, in looking at the first chamber of the top row, consider letters
b and
c. In no other chamber in the top row does
b operate on
c to allow the chain reaction to continue. A similar thing can be seen for the letters applicable for the other rows as well.
Additionally, none of these systems satisfy
rule 3. For instance, consider chambers with both letters
b and
c. All three systems contain 2 chambers with these highlights. There are duplicates of other letter pairs as well.
| c:a->b |
d:b->c |
e:c->d |
b:d->e |
| d:a->b |
e:b->c |
b:c->d |
c:d->e |
| e:a->b |
d:b->c |
b:c->d |
c:d->e |
Computer program
With the aid of a computer program, systems of higher numbers of chambers can be tested for adherence to
rule 1,
rule 2, and
rule 3.
Population 1 those systems satisfying
rule 1
Population 2 those systems satisfying
rule 2
Population 3 those systems satisfying
rule 3
The chart below shows the results of running a computer program to test for and count the members in each population for systems with up to 12 chambers. Empty cells represent the value zero.
| N |
Population |
| 1 |
2 |
3 |
| 1 |
|
|
|
| 2 |
|
|
|
| 3 |
1 |
1 |
|
| 4 |
3 |
|
|
| 5 |
16 |
1 |
|
| 6 |
96 |
|
|
| 7 |
675 |
1 |
2 |
| 8 |
5413 |
|
12 |
| 9 |
48800 |
1 |
208 |
| 10 |
488592 |
|
2942 |
| 11 |
5379333 |
1 |
37446 |
| 12 |
64595975 |
|
506112 |
| 13 |
** |
1 |
7221416 |
|
|
Click here for the source code. Save the file as Rosen.java and compile with the Java Development Kit downloadable from http://java.sun.com.
Usage:
java Rosen [start] [end] [pop] [print]
where
[start] = starting component number
[end] = ending component number
[pop] = which population to determine
[print] = whether or not to print out the various systems
The printout will also indicate the number in each population for each chamber count. For example,
java Rosen 7 7 2 y
will produce
7: e f g h b c d
7 1
which are the agents selected from the possible choices for population 2 of a 7-chamber system.
|
** The calculations required for (N=13, population 1) were so extensive that the program was aborted before completion.
The next page will show the two seven chamber diagrams from
population 3.
Created on 03/28/2009 08:33 AM by admin
Updated on 05/17/2009 07:38 AM by jprideaux
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