# Ontology Modeling Using Fractal and Fuzzy Concepts to Optimize Metadata Management
## 3 The FFOM Methodology
### Foundational Concepts and Formal Definition
\begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline \hline \multicolumn{1}{c}{**Primitive**} & \multicolumn{1}{c}{**Application**} \\ \hline \multirow{4}{*}{Fractal Representation} & \multirow{4}{*}{\(\bullet\)} & Depicted as a rounded rectangle. The top area contains the fractal name, while the bottom area lists its data properties (e.g., name, weight, etc.) \\  & & Labels and cardinalities (e.g., “1...N”) are shown inside the rectangle or as separate notations next to it \\  & & Denote quantitative or descriptive attributes within fractals or instances (e.g., temperature, capacity, etc.) \\  & & Shown in the lower portion of the fractal or instance box and labeled with the data type (e.g., ssd-string) \\ \hline \multirow{4}{*}{Instance Representation} & \multirow{4}{*}{\(\bullet\)} & Each named individual is linked to its fractal with a dashed arrow labeled “Instance of”, typically shown below the relevant fractal \\  & & The instance box may display both the instance Name (top) and its data properties (bottom) \\ \hline \multirow{4}{*}{Hierarch and Inheritance} & \multirow{4}{*}{\(\bullet\)} & Fractal \(\rightarrow\) subfractal relationships are indicated with a clear arrow labeled “SubFractalOf” or “Inheritance” \\  & & Fractals appear above (or to the left) with subfractals below (or to the right) for clarity \\ \hline \multirow{4}{*}{Object Properties} & \multirow{4}{*}{\(\bullet\)} & Represented by solid arrows between fractals or instances (e.g., “hasPart,” “connectedTo”) \\  & & Additional annotations (fuzzy\_value, timestamp) can be included to handle dynamic or uncertain relationships \\  & & Indicates whether two fractals (or entities) can share overlapping instances (Joint) or must remain mutually exclusive (Disjoint) \\  & & Represented by a solid line connecting the fractals, labeled either “Joint” or “Disjoint” \\  & & Labels such as ‘FractalID’, ‘Object Properties’, and ‘Data Properties’ are used within the fractal, and relationships are defined with cardinalities to represent structural \\  & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Definition of graphical primitives used in the FFOM ontology. FFOM's approach is supported by a graphical notation, designed to intuitively represent its core structural and semantic features. Table 3 summarizes these visual elements. A fundamental terminological clarification throughout FFOM's conceptual model, and reflected in Table 3, is the use of the term \"Fractal\" to denote the primary structural components. Consequently, identifiers are referred to as \"FractalID,\" and hierarchical relationships between these components are termed \"SubFractalOf.\" This conceptual \"Fractal\" is then mapped to an owl:Class during the OWL transformation process. The primitives presented in Table 3, such as 'Fractal Representation' and 'Instance Representation', while building on standard ontological symbols, are employed by FFOM to explicitly visualize its novel fuzzy and fractal characteristics. A key aspect of FFOM's notation, detailed in Table 3 under 'Object Properties', is the representation of fuzzy relational characteristics. Relationships between instances of fractals can be augmented with annotations to handle dynamic or uncertain aspects. Specifically, the strength of a relationship among fractals is represented as a TFN. For practical OWL implementation or certain reasoning tasks, these TFNs are typically defuzzified to scalar values. Furthermore, timestamps are associated with these fuzzy relationships to capture their temporal validity. As defined in Table 3, timestamps are represented in a date and time format YYYY-MM-DDThh:mm:ssZ (e.g., \"2024-01-01T12:00:00Z\"), where YYYY is the four-digit year, MM is the month, DD is the day, T' is the delimiter, hh is the hour (00-23), mm is the minute, and ss is the second, and Z indicates Universal Time Coordinated (UTC).