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.
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.
4 chamber Systems
Consider a 4 chamber system with all possibilities of picking the converting agents.
If we apply rule 1, we get the following three possibilities as depicted in each row:
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.
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.
|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.
java Rosen [start] [end] [pop] [print]
[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
7: e f g h b c d
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.