# Language and Reasoning by Entropy Fractals
## 4 Tests and Results
### Rule R2  
The information is carried in sentences with the analysis centered on nouns (**S**) and verbs (V), change the amount of HS (entropy of the sentence S), and needs to be efficient to get a target word (the goal of the 20Q game). Rule 2 explains that sentences work not in a linear transmission of H:  
There is a complement between V and N, though not perfect, they balance each other, and \(H\) has a certain rhythm and cycle. In this context, rhythm is considered as its original sense: A regular movement or pattern of movements, and a regular pattern of change [29]. A cycle is a group of events that happen in a particular order, one following the other, and are often repeated [29]. For the first (rhythm) \(H\) is considered as the pattern information, and the second (cycle) is a scaling of the original pattern, from a fractal perspective (see [30]). The scaling is measured with fractal dimension (\(D\)). Entropy is calculated as usual [31]:  
\[ET(X)=H(X)=-\sum_{i=1}^{k}p_{i}\log_{2}(p_{i}) \tag{5}\]  
With:  
\(X\) as the property to be measured. In the present context, it could be **cantV** (or #V, the number of verbs in a sentence), **cantN** (or #_N,_ the number of nouns), or **cantA** (the number of attributes that describe the concept). \(P_{i}\) represents the probability of that property (taken as the number of words of the type under study over the total). \(k\) is the number of sentences in the game (usually a positive integer less than 30). The standard entropy \(H\) is named in this paper with convenient labels like \(ET\) for total entropy, \(ET(V)\) for entropy due to verbs, \(ET(N)\) entropy due to nouns, and \(ET(A)\) entropy due to attributes. In certain tests, ET is evaluated sentence by sentence with a subscript notation as in \(ET_{q}\). Analyzing the entropy by Verbs and nouns there is a behavior in all tests: \(ET(V)\), the variation of ET due to Verbs, is higher than \(ET(N)\) and \(ET(A)\), variations for Nouns and descriptors respectively. Also, the curve \(ET(\textbf{N})\) is typically the lower one. Figure 3 shows the case of game 9. Figure 3: Entropy for game 9. _ET_ shows a relationship with the result of the game. Table 4 shows that **ET** is in a specific interval out of scope of the cases when the ANN loses (Succeed = N) and similar intervals when it wins (Succeed = Y) or gets the concept but not the word (Succeed = C). The entropy conveyed through questions is centered here by using just nouns, verbs, and descriptors (adjectives, adverbs, etc.). Take Equation (5) and replace \(p_{1}\) by \(p(\textbf{s})\):  
\[p(s)=\frac{s}{T} \tag{6}\]  
where:  
\(s\): number of verbs (\(V\)), nouns (\(N\)), or descriptive word (\(A\)). \(T\): total number of words. Regarding the fractal dimension **D** is defined as [31]:  
\[\text{D}=\frac{\log_{x}N}{\log_{x}(1/r)} \tag{7}\]  
With:  
\(N\): number of partitions. change rate. When D is defined with:  
\(x=2\)  
\(N=\Delta\text{ET}=\text{ET}_{\text{q}}-\text{ET}_{\text{q}}-1\)  
\(r=\text{ET}(\text{A})\text{q}\)  
Where \(ET_{q}\) is the \(ET\) in question \(q\), \(ET_{q-1}\) is \(ET\) in question (\(\text{q}-1\)), and ET(A) is ET due to Attributes. The plots of D along every game show a unique pulse like in Figure 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Succeed** & **MIN** & **MAX** & **AVG** & **DEV** & **INTERVAL** \\ \hline Y & 6.06 & 10.64 & 8.24 & 1.44 & [6.79] & [9.68] \\ C & 6.77 & 13.23 & 9.40 & 2.61 & [6.80] & [9.40] \\ N & 9.83 & 10.77 & 10.30 & 0.40 & [9.91] & [10.30] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: **ET** values for games that ANN wins (Y), get close (C), or loses (N). Figure 4: D for test #2. After analyzing the sentences, it is clear that the minimum represents a status where the concept represented by the target word is obtained by discarding other concepts. The peak, a fast increase of the entropy, fits after the class of object determination, decreasing as the specificity increases.