# Ontology Modeling Using Fractal and Fuzzy Concepts to Optimize Metadata Management
## 2 Literature Review
### Fuzzy Logic in Ontology and Data Management  
Existing ontologies and data systems operate on the basis of clear Boolean logic, which assumes binary truths and precise data values. However, real-world data, especially in industrial settings, are often inaccurate, uncertain, or incomplete. Fuzzy logic provides a framework for reasoning under these conditions by allowing for class-based membership within sets rather than absolute inclusion [21; 34]. In regard to ontology design, fuzzy logic introduces key features, such as linguistic variables and membership functions, which allow natural language expressions, such as 'hot' or 'low efficiency,' to be quantified [35]. Fuzzy logic also supports fuzzy IF-THEN rules and fuzzy inference systems, which enable the encoding of expert knowledge with soft boundaries and gradual transitions [36]. Fuzzy ontology is a research field that combines classical ontology and fuzzy logic. Fuzzy ontology often involves extending OWL to accommodate fuzzy relationships, data types, or integrating it with fuzzy reasoners [35; 37]. This integration helps overcome the limitations of binary representations by enabling sophisticated modeling, the integration of data from uncertain sources, and reasoning when faced with ambiguity. These capabilities are key to FFOM's goal of representing dynamic and uncertain industrial environments.  
To conceptualize visually how fuzziness is generally incorporated into ontology models, Figure 1 presents an example of a fuzzy ontology structure used in the context of a recommender system [38]. The figure shows how the concept 'BOOK' is related to various types and specific options. For example, 'BOOK' is related to several types, such as type1 or type2, and each relationship is represented by a numerical value, such as 0.2 or 0.3. These values represent the strength of the association or the degree of uncertainty. In addition, individual books, Book1 and Book2, are represented with numerical values that reflect the uncertainty of the user ratings. This fuzzy ontology structure shows that relationships between concepts and entities in an ontology can be represented more flexibly through continuous values, such as the strength of membership or preference in a class, rather than simply true or false. This approach contributes to effectively introducing a certain level of fuzziness. However, the FFOM framework takes this approach a step further by structurally integrating fuzzy semantics, using triangular fuzzy numbers (TFNs). While existing fuzzy ontology models usually express the uncertainty of a relationship through a single scalar value, FFOM stores the entire TFN, consisting of a lower bound, a center, and an upper bound, in the ontology. Looking at the current research trends, it seems rare to find a case where the TFN itself is directly integrated into the ontology's relationship definition in this way, which is one of the distinctive features of FFOM. Through this, FFOM can express not only the degree of uncertainty, but also its distribution range and central tendency, enabling more sophisticated inferences, and also supporting complex fuzzy operations, such as fuzzy multiplication. In addition, by utilizing a timestamp, it can effectively track changes in relationships among fractals over time. These functions are powerful means of dealing with uncertainty more richly and quantitatively in environments where data ambiguity and complexity coexist, such as in the manufacturing industry.