## 4 The Relationship between the Interpretation Process and Scale
### The Entropy Reduction Process  
A way to study the change of the resulting entropy, produced by the change of the selected set of symbols, is by focusing on one or two consecutive symbols. For a single symbol the only way it can change itself while keeping invariant its character sequence, is by splitting itself into two or more symbols. For two adjacent symbols, we identify two cases in which they can vary while keeping constant the text represented by the two symbols together: symbols' boundary drifting and symbols joining. Examples of these cases are presented. _Case 1. Symbol Splitting:_ Split symbol \(Y_{j}\) into two smaller symbols. In this case, the object symbol \(Y_{j}\) is replaced by new symbols \(Y_{j}^{\prime}\) and \(Y_{j+1}^{\prime}\). The sub-indexes of all successive symbols will increase by one. The new symbolic diversity also increases from \(D\) to \(D+2\). One could think that symbol \(Y_{j}\) as the only instance of that symbol in the entire text. If that were the case, by replacing it with \(Y_{j}^{\prime}\) and \(Y_{j+1}^{\prime}\), it would generate only one additional symbol. However, this seems very unlikely because, in order to be part of a minimal entropy set of symbols, \(Y_{j}\) should appear more than once in the message. Another possibility is that the new symbols \(Y_{j}^{\prime}\) and \(Y_{j+1}^{\prime}\) already exist within the message. This is also unlikely because the minimal entropy criterion has already '_decided_' to represent the sequence \(Y_{j}^{\prime}Y_{j+1}^{\prime}\) with the '_compacted_' form \(Y_{j}\). Concluding, for Case 1 we can use \(V=D+2\) as the symbolic diversity for this message interpretation. Example 4 (E4) illustrates this situation:  
**Example 4 (E4). **_Take \(Y_{3}=\{\mathcal{O}abc\}\) from the interpretation of Example 2. Then, symbol \(Y_{3}\), could be split into two symbols called them and \(Y_{32}\). Making \(Y_{31}=\{\mathcal{O}ab\}\) and \(Y_{32}=\{c\}\) is a valid replacement for \(Y_{3}\) since they add up the original \(Y_{3}\) with no character overlapping. In this case the emergence of symbols \(Y_{31}=\{\mathcal{O}ab\}\) and \(Y_{32}=\{c\}\), which did not appear in the previous interpretation, would increase the symbolic diversity by 2 and since \(Y_{3}\) only appeared in one instance, now extinguished, the resulting diversity after the symbol splitting end up being \(D\) to \(V=D+1\). The resulting symbol set probabilities are:_  
\[P\big{(}Y_{1}={}^{\prime}\mathcal{O}abc^{\prime}\big{)}=\frac{1\cdot 2}{35},\ P \big{(}Y_{2}={}^{\prime}\mathcal{O}ab^{\prime}\big{)}=\frac{1\cdot 3}{35},\ P\big{(}Y_{3}={}^{ \prime}\mathcal{O}abc^{\prime}\big{)}=\frac{0\cdot 4}{35}=0,\ P\big{(}Y_{4}={}^{ \prime}\mathcal{O}abcd^{\prime}\big{)}=\frac{1\cdot 5}{35},\]  
\[P\big{(}Y_{5}={}^{\prime}\mathcal{O}abcde^{\prime}\big{)}=\frac{1\cdot 6}{35},\ P \big{(}Y_{6}={}^{\prime}\mathcal{O}abcdef^{\prime}\big{)}=\frac{1\cdot 7}{35},\ P \big{(}Y_{7}={}^{\prime}\mathcal{O}abcdefg^{\prime}\big{)}=\frac{1\cdot 8}{35},\]  
\[P\big{(}Y_{31}={}^{\prime}\mathcal{O}ab^{\prime}\big{)}=\frac{1\cdot 3}{35}, \ P\big{(}Y_{32}={}^{\prime}c^{\prime}\big{)}=\frac{1\cdot 1}{35}.\]  
_The resulting scale base is \(D=8\) and the computed entropy is \(h\approx 0.926\)._  
_Case 2. Symbols Boundary Drifting:_ We refer to 'boundary drifting' as the effect produced when the end and start of neighbor symbols shifts, leaving constant the total length of both symbols but adding to a symbol the same number of characters diminished on the other. The boundary between two adjacent symbols \(Y_{j}\) and \(Y_{j+1}\) moves making one symbol larger and the other smaller and resulting in new symbols \(Y_{j}^{\prime}\) and \(Y_{j+1}^{\prime}\). The sub-indexes of symbols do not shift but the new symbolic diversity increases from \(D\) to \(D+2\) due to the _birth_ of new symbols \(Y_{j}^{\prime}\) and \(Y_{j+1}^{\prime}\) which did not exist before.