## 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
\begin{table}
\begin{tabular}{c p{284.5 pt}} \hline \hline
**Notation** & **Definition** \\ \hline \multirow{3}{*}{\(K\)} & Finite set of truth levels, \(K\), is defined in AGA [70], where \(k=\{0,\ 1,\ 2,\ldots,\ K-1\}\), \(K\subset N\), and N is the set of natural numbers. Only finite sets are intended for basic logic AGA modeling and the mapping of words vocabularies. \\ \hline \multirow{3}{*}{\(V^{N}\)} & Finite subset, \(v^{N}\), of the set of natural numbers, \(v^{N}=\{1,2,\ldots,K\}\), \(V^{N}\subset N\). It is intended for auxilliary representation and mapping of words in vocabularies. \\ \hline \multirow{3}{*}{\(V^{L}\)} & The finite set (or the vocabulary) of all selected different robotic terms, given by words and word collocations of natural language. It describes the scene of action, robotic tasks, and mathematical and verification procedures. \\ \hline \multirow{3}{*}{\(V^{K}_{v}\)} & Modeling of the scene of action uses \(v\) subsets of truth levels, \(v=\{1,\ldots,K\}\), obtained by arbitrarily chosen bijective mapping, \(V^{N}{\rightarrow}K\), i.e., onto the scale of truth levels, \(k=\{0,1,\ldots,K\}\), defined in AGA. \\ \hline \multirow{3}{*}{\(V^{N}_{v}\)} & Subsets \(V^{N}_{v}\) of vocabulary \(V^{N}\) are obtained by arbitrarily chosen mapping of \(V^{L}{\rightarrow}V^{N}\), i.e., onto the scale of natural numbers \(n\in N=\{1,2,\ldots,K\}\), where the set \(V^{N}\) consists of \(v\) subsets \(V^{N}_{v}\in V^{N},v=\{1,\ldots,K\}\); \\ \hline \multirow{3}{*}{\(V^{L}_{v}\)} & Subset \(V^{L}_{v}\) of the vocabulary \(V^{L}\) determines one of \(v\) given classes of robotic terms, containing only different elements, \(V^{L}_{v}\subset V^{L}\), \(v=1,\ldots,V,\ V\leq K\), where \(K\) is the maximal number of truth levels. In the majority of tasks, convenient classes of robotic terms correspond to classes of words (noun, verb, adverbial modifiers of place, time, etc.) \\ \hline \multirow{3}{*}{\(w^{L}_{v}\)} & Element of a subset (or vocabulary) \(V^{L}_{v}\) can be given by a word or a collocation, representing in natural language some robotic term. \\ \hline \multirow{3}{*}{\(w^{N}_{v}\)} & Mapping of \(w^{L}_{v}\) onto the subset of natural numbers \(V^{N}\), providing representation of a robotic term in natural-number code. \\ \hline \multirow{3}{*}{\(w^{K}_{v}\)} & Mapping of \(w^{L}_{v}\) onto the set \(K\), providing equivalent logic representation of a robotic term. \\ \cline{2-2}  & Word phrase is a vector, \(\mathbf{W}_{z}=(w_{1},\,w_{2},\,\ldots,\,w_{Z})\), with declared fixed position of elements according to their numbers, v = \(\{1,\ldots,Z\}\), \(Z\leq V\leq K\). Word phrase is a sampling of only one word from the next vocabulary. For various types of processions, it can be written by elements \(w^{L}_{v},w^{N}_{v}\), \(w^{K}_{v}\) of subsets \(V^{L}_{v}\), \(V^{N}_{v}\), \(V^{N}_{v}\). Note that the list of selected vocabularies and message format can differ for various classes of tasks. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic formal parameters for the description of the scene of action. The specific situation of mapping on \(V_{v}^{K}\) in Figure 6a is that we can randomly assign truth levels to words or conveniently choose them for the classification of objects [75]. As the simplification (and shortening) of AGA expressions is based on the consensus method of minimization [70], it can regroup logic product terms according to assigned logic constants. A substantial feature of the used subsets, \(V_{v}^{L}\), \(V_{v}^{N}\), \(V_{v}^{K}\), is that the number of elements in them is less or equal to \(K\), and unused vacancies in subsets are to receive zero values. Mappings \(V^{L}\)\(\rightarrow\)\(V^{N}\) and \(V^{N}\)\(\rightarrow\)\(V^{K}\) are bijective: each element of a subset maps to only one of elements in another subset, providing simple forward and backward translation by transcoding tables.