## 3 Methods
### Transforming Models to Graphs  
The MGV transform is a composition of a transform from a model into a uniform GDS representation (M\(\rightarrow\)G), with a second transform from GDS into view (G\(\rightarrow\)V). Transforms into GDS require mapping each ML construct to type in the GDS category. Graphs have only two elements: _nodes_ and _edges_. GDS has only one element: _tuple_ (an abstraction and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Category** & \begin{tabular}{c} **Systems,** \\ _SYS_ \\ \end{tabular} & \multicolumn{1}{c}{**C\(\rightarrow\)M**} & \
\begin{tabular}{c} **Object–Process Modeling** \\ **Language, OPML** \\ \end{tabular} & \multicolumn{1}{c}{**M\(\rightarrow\)G**} &
\begin{tabular}{c} **Graph Data Structure,** \\ _GDS_ \\ \end{tabular} \\ \hline Types & \multicolumn{3}{c}{Model, Diagram, Spec} & & \\  & Structure & \(\rightarrow\) & Object & & \\  & Relation & \(\rightarrow\) & Process, Relation & & \\  & State & \(\rightarrow\) & State & & \\  & Operand & \(\rightarrow\) & Object & & \\ \hline  & & & & Report & \(\rightarrow\) & Tuple set \\ \hline \hline \multirow{2}{*}{Morphisms} & Behavior & \(\rightarrow\) & Process & & \\  & N/A & & modeling, specifying, rendering & & tuple operation \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mapping the category of Systems, _SYS_, to a category of OPML models, _OPML_, and to a category of Graph Data Structures, _GDS_. Figure 6: The OPM modeling language, OPML, as a category. Modeling (the morphism \(m\)) is like hopping through the instantiation of category types based on existing artifacts. For example: (a) creating a new Diagram D1 in a Model M, (b) creating an Object Ob1 in D1, (c) creating a Process P1 in D1, (d) creating a RelationRr1(Ob1,P1) from Ob1 to P1, (e) creating a new Diagram D2 from P1, (f) adding States St11 and St12 to Ob1 in D2, and so on. Every instantiation of a type also affects the Spec, and therefore all graphical elements have a mapping into Spec (the morphism \(s\)). Spec also feeds into the Rendering morphism \(r\), which renders a Report from the Spec, currently in the form of a PDF file. Diagram created using MATLAB 2019b digraph plotting capabilities [85]. extension of a node-edge-node triplet). A graph is merely a view on GDS. Such transforms are possible on graphic MLs, although graphics do not always map to simple graphs--for instance, when shapes are contained inside other shapes or overlap with other shapes (as in Venn diagrams). OPCloud's Exported Report includes the textual OPL specification and lists of the objects, processes, and relations in the model. This helps in mapping all model entities and relations and translating them to GDS tuples. The final product G is a set of tuples, which includes a relation R, a source node S, a target node T, a unique identifier U, and a valuator V, denoted as the RSTUV tuple. V is useful for various applications such as quantification, verification, validation, versioning, and configuration control. Let us revisit the lane keeping system model in Figure 2 and transform it into a GDS. The graph of the model is shown in Figure 7.  
RSTUV is sufficient for representing model data. It also captures aspects that are not specifically illustrated in the model. For example, it specifies the model's inclusion of the diagrams, as well as each diagram's inclusion of visualized objects and processes. The set of RSTUV tuples is obtained by executing the following functorial rules from \(OPML\) to \(GDS\) (also summarized in Table 5):  
1. ML types are defined as RSTUV identity tuples. This step is only done once per ML. 2. Entities are mapped to their OPM entity type (model, diagram, object, process, etc.) using a Classification relation. 3. Relations, such that E\({}_{\text{i}}\) and E\({}_{\text{j}}\) are entities connected by relation R, are mapped as is. 4. Each entity E\({}_{\text{j}}\) is mapped to any diagram D\({}_{\text{i}}\) that includes it by an Inclusion relation. 5. Relations are mapped to any diagram that includes them by an Inclusion relation. 6.