## 3 Results: AGA Logic Variables for Scene Description and Dialog Communications
### Representation of the Scene of Action by Natural Language, Numerical Codes, and Truth Levels
The chain-mapping scheme is disclosed by Figure 6b and responds to sequential clarification of content by the finite chain of conjugated vocabularies, \(V_{1,m}^{L}\)\(\rightarrow\)\(V_{2,m}^{L}\)\(\rightarrow\)\(V_{3,m}^{L}\)\(\ldots\)\(\rightarrow\)\(V_{K,m}^{L}\), where \(V_{1,m}^{L}\) is a root vocabulary. Such a method is used to provide the unified protocol for all robots, exploiting simple and advanced hardware platforms, where the first part of the message can contain coded words defined closer to root vocabularies. Then, the overall actuality of information can be quickly estimated by all agents, even those capable of interpreting only part of the message. In such a scheme, agents can undertake actual action immediately, before the transfer of all details. The formation of content root vocabulary, \(V^{L}\), is given in Table 3, which describes words of natural language. Every term (or word combination) of natural language is included in some vocabulary. Every vocabulary, \(V_{v,m}^{L}\), contains up to \(K\) different words, \(w_{v,m}^{L}\). Numbers \(N=1\div M\) refer to root vocabularies. Values \(N\) from \(K+1\) up to \(2K\) refer to the second one in the chain mapping of vocabularies. For the sake of brevity, only two chain vocabularies, \(V_{v,2}^{K}\), are shown in Table 3.  
Figure 6: Description of the scene of action by root and sequential chain vocabularies. (**a**) Mapping of arbitrarily selected root vocabularies, \(V_{1,1}^{L}\)\(\rightarrow\)\(V_{1,1}^{N}\)\(\rightarrow\)\(V_{1,1}^{K}\), transferring natural language terminology onto scales of natural numbers and logic truth levels. (**b**) Chain mapping of vocabularies like \(V_{1,1}^{L}\)\(\rightarrow\)\(V_{2,1}^{L}\)\(\rightarrow\)\(V_{3,1}^{L}\)... is convenient for the content clarification and formation of structured word phrases. Every vocabulary, \(V_{v,m}^{L}\), further has double index for compact matrix notation. \begin{table}
\begin{tabular}{p{42.7 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**N** & \ \begin{tabular}{} \end{tabular}  & **Content** & **Class of Word in Natural Language** & \begin{tabular}{} \end{tabular}  \\ \hline  
The first field in Table 3 is the initiation (or welcome) word of dialog, also indicating the type of communication format for messages and the number of parts in the phrase, which may differ for various sectors of robotics. Every agent, tool, and identified object in the scene of action can have only one name. The name of addressers and senders may include several bytes or chain elements, defined by the matrix, \(V^{K}_{v,m}\), taken with two parameters, \(v,m\ \leq\ K\). For the sake of simplicity, we avoid using the third index for the description of the number of a word in the vocabulary. AGA modeling of the scene of action is based on three equivalent representations of vocabularies, \(V^{L}_{v,m}\to V^{N}_{v,m}\to V^{K}_{v,m}\), as shown in Table 4. Such parallel use of natural language description, natural-number code, and truth-level code is convenient for human experts, as it corresponds to the situation, when one first explains orally the work of the designed robotic system and, after this, turns to number codes and AGA models. Correlated variables can be written (see Table 4) as a sequence of \(w^{K}_{i,j,k}\), sequentially raising the precision of space coordinates, but following the restriction given in Section 2.4. The scheme of logic transformations between natural-numbers code and truth levels ones is given in Figure 7, which shows the initial mapping of \(w^{N}_{i,j}\) data onto the logic representation scale \(w^{K}_{i,j}\). Also, the reverse mapping to natural numbers is given, as arithmetic calculations of check coefficients are to use Boolean logic. For convenient process, the maximal logic value, \(K-1\), and its natural numbers equivalent are reserved for possible minimization procedures in AGA [75]. By default, zero value indicates the unused word in a vocabulary, also providing zero in corresponding product terms.