GFO Part I Basic Principles
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2.3 Representation of Ontologies

An ontology $\mathcal{O}$ - understood as a formal knowledge base - is given by an ``explicit specification of a conceptualization'' (24). This specification - understood as a formal one - has to be expressed and presented in a formal language, and there are a variety of formal specification systems. A point to be clarified is what representation means. In the analysis of this notion we use the notion of denotation and symbolic structure. This indicates that in a representation one kind of category cannot be avoided, the category of symbolic structure. A main distinction may be drawn between logical languages with model-theoretical semantics and formalisms using graph-theoretical notations. We sketch some ideas about both types of formalisms.

2.3.1 Model-theoretical Languages

A model-theoretic language consists of a structured vocabulary $V(\mathcal{O})$ called ontological signature, and a set of axioms $Ax(\mathcal{O})$ about $V(\mathcal{O})$ which are formulated in a formal language $L(\mathcal{O})$. Hence, an ontology (understood as a formal object) is then a system $\mathcal{O}=(L,V,Ax)$; the symbols of $V$ denote categories and relations between categories or between their instances. $L$ can be understood as an operator which associates to a vocabulary $V$ a set $L(V)$ of expressions which are usually declarative formulas. We assume the following conditions: $V \subseteq V'$ implies $L(V) \subseteq L(V')$, and $L(L(V)) = L(V)$. An ontology may be augmented by a derivability relation, denoted by $\derive$, and by a semantic consequence relation, denoted by $\models$. Then, such an ontology takes the form of a knowledge system $(L,V,Ax,Mod,
\derive, \models)$ which includes a class $Mod(V)$ of interpretations which serves as a semantics for the language $L(V)$.

2.3.2 Graph-theoretical Systems

Graph-based formalisms for ontologies, as they are common for biological ontologies or at least related to medical terminologies, can be understood in the following way. Such an ontology $\mathcal{O}$ is a structure $\mathcal{O} = (Tm, C, Rel, Def, G)$. Terms $Tm$ usually cover natural language aspects and are assigned to concepts $C$ and relations $Rel$. Moreover, the relations connect concepts, which yields a labelled graph structure $G$ over concepts, such that edges are labelled by relations. The definitions $Def$ which are held in such systems, if any, are usually natural language definitions, sometimes in a semi-structured format. Particular systems of this kind can vary in several respects, e.g. focusing on the distinction between terms and concepts, the extent to which definitions are provided, the number of relations available, etc.; a corresponding overview and classification in the field of medical terminologies can be found in (19).

Robert Hoehndorf 2006-10-18


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