## 4 The Relationship between the Interpretation Process and Scale
### A Model to Quantify the Information of Different Interpretations
Even more, assume the symbols \(Y_{j}\) are such that the resulting entropy is minimal, we can write:  
\[h_{\mathcal{A}D}^{*}=\ -\sum_{j=1}^{D}P\big{(}Y_{j}\big{)}\ \log_{D}P\big{(}Y_{j}\big{)}, \tag{3}\]  
where \(h_{\mathcal{A}D}^{*}\) represents the minimum possible entropy associated to the process of interpreting the message encoded by means of alphabet \(\mathcal{A}\). In order to comply with the character neighborhood condition when forming symbols \(Y\), any symbol \(Y_{j}\) must comply with  
\[Y_{j}=A_{i}\ A_{i+1}\ A_{i+2}\dots\ A_{i+S-1}, \tag{4}\]  
where the sub-index \(S\) represents the length of symbol \(Y_{j}\). The set of symbols \(Y\) is useful to read and interpret message \(\mathcal{M}\), then we consider it a language. Since using language \(Y\) is a better way of reading \(\mathcal{M}\), we replace the messages formerly expressed in (1) model with:  
\[\mathcal{M}:\ P(Y)=\big{\{}P(Y=Y_{1}),P(Y=Y_{2}),\ \dots,\ P\big{(}Y=Y_{j} ),\dots,\ P(Y=Y_{D})\big{\}}. \tag{5}\]  
Notice that now symbols \(Y_{j}\) are of different sizes. Therefore, the probability \(P()\) of encountering a symbol \(Y_{j}\) within the message, should account for the frequency of appearances \(f_{Y_{j}}\), the size of the symbol \(S_{Y_{j}}\) and the length \(L_{\mathcal{A}}\) of the message measured in characters. Thus, this probability can be expressed in terms of the frequency \(f_{Y_{j}}\) and the number of times the symbol of size \(S_{Y_{j}}\) fits into a \(L_{\mathcal{A}}\) characters long message, as  
\[P\big{(}Y_{j}\big{)}=\frac{f_{Y_{j}}\cdot S_{Y_{j}}}{L_{\mathcal{A}}}. \tag{6}\]  
Equation (3) for minimal entropy can now be rewritten in terms of the symbols of language \(Y\) built over the characters of alphabet \(\mathcal{A}\). We obtain:  
\[h_{\mathcal{A}D}^{*}=-\sum_{j=1}^{D}\ \frac{f_{Y_{j}}\cdot S_{Y_{j}}}{L_{ \mathcal{A}}}\ \log_{D}\frac{f_{Y_{j}}\cdot S_{Y_{j}}}{L_{\mathcal{A}}}. \tag{7}\]  
To illustrate these figures, we present examples consisting of different observations of the same description and their entropy evaluation. Consider the following character-sequence as the description, where in order to ease the reading, the character '\(\mathcal{O}\)' appears in place of the'space-character':  
\[\mathcal{O}a\ \mathcal{O}ab\mathcal{O}abc\ \mathcal{O}abcd\ \mathcal{O}abcde\ \mathcal{O}abcdef\ \mathcal{O}abcdefg\]  
Using \(Y\) as the random variable representing the probability of encountering a specific symbol within the description, three examples are shown below:  
**Example 1 (E1). **_Focusing on single characters as the symbols of this description, leads to the following probabilities based on their frequency:_  
\[P\big{(}Y_{1}={}^{\prime}\mathcal{O}^{\prime}\big{)}=\frac{7}{35},\ \ \ \ P\big{(}Y_{2}={}^{\prime}a^{\prime}\big{)}=\frac{7}{35},\ \ \ \ P\big{(}Y_{3}={}^{\prime}b^{\prime}\big{)}=\frac{6}{35},\ \ \ \ P\big{(}Y_{4}={}^{\prime}c^{\prime}\big{)}=\frac{5}{35},\]  
\[P\big{(}Y_{5}={}^{\prime}d^{\prime}\big{)}=\frac{4}{35},\ \ \ \ P\big{(}Y_{6}={}^{\prime}e^{\prime}\big{)}=\frac{3}{35},\ \ \ \ P\big{(}Y_{7}={}^{\prime}f^{\prime}\big{)}=\frac{2}{35},\ \ \ \ P\big{(}Y_{8}={}^{\prime}g^{\prime}\big{)}= \frac{1}{35}.\]_Since there are eight different symbols, the scale base is \(D=8\) and the computed entropy, applying Equation (3), is \(h\approx 0.937\)._  
**Example 2 (E2). **_A second interpretation of the same description assumes the message was expressed with a natural language and therefore regards as symbols those character strings which look as words. Those are the symbol strings beginning or ending with a'space' (' \(O\) ').