Larger Systems

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.

 c :a->b  d :b->c  b :c->d

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.

c:a->b d:b->c b:c->d

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.

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