GFO Part I Basic Principles
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3.3 Abstract Core Level

This section presents the meta-level in the architecture that is formed by abstract core ontologies. The abstract core level of GFO exhibits the upper part of GFO, in the same way as a domain core ontology is the upper part of a domain ontology.

Apart from pragmatic aspects, ACOs must first be determined by their main entity types and the relations among them, for which a certain vocabulary must be introduced. Secondly, logical interdependences of those entities and their relations need to be specified. The latter exemplify the formalization of several types of interdependence using axioms of first-order logic.

We start from the idea that the entities of the (real) world - being represented on the ATO-level by the items - are divided into categories and individuals, i.e., everything in an ontology is either a category or an individual, and individuals instantiate ($\Ginst$) categories. Moreover, among individuals we distinguish objects, attributes, roles and relators. Objects are entities that have attributes, and play certain roles with respect to other entities. Objects are to be understood in the same way as the notion of ``object'' in object-oriented analysis. In particular, objects comprise animate and inanimate things like humans, trees or cars, as well as processes, like this morning's sunrise.

Examples of attributes are particular weights, forms and colors. A sentence like ``This rose is red.'' refers to a particular object, a rose, and to a particular attribute, red. Another basic relation is needed in order to connect objects and attributes. The phrases ``having attributes'' and ``playing a role'' used above are included in the basic relation of inherence, meaning that an attribute or a role inheres in some object. This relation illustrates the dependence of attributes and roles on entities in which they can inhere.

The difference between attributes and roles is that roles are interdependent (36). Examples of roles are available through terms like parent, child or neighbor. Here, parent and child would be considered as a pair of interdependent roles. Apparently, these examples easily remind one of relations like ``is-child-of''. Indeed, a composition of interdependent roles is a relator, i.e., an entity that connects several other entities. The formation of relators from roles further involves the basic relation, role-of.

By introducing a vocabulary for the considered entities we obtain the following signature:

$\Sigma = (\GCat, \GOCat, \GP, \GRCat, \GR;
\GInd, \GObj, \GAtt, \GRol, \GRela;
{\it =, \Ginst , \Ginh, \Groleof} )$

$\GCat$ denotes the meta-category of all categories, $\GOCat$ represents the category of all object categories, $\GP$ indicates the category of all properties, and $\GR$ identifies the category of all relations. $\GInd$ is the category of all individuals, $\GObj$ designates the category of all objects, $\GAtt$ represents the category of all individual attributes, $\GRol$ identifies the category of all roles, and $\GRela$ denotes the category of all relators. These categories are all presented as predicates, i.e., they occur on the ATO-level as sets of items. We present, as an example, a simple axiomatic fragment using the vocabulary that is related to a taxonomy of the unary predicates. 10

$\forall x y (\Ginh(x,y) \Limp (\GAtt(x) \Land \GObj(y)))$

$\forall x (\GObj(x) \Limp \exists y ( \GAtt(y) \Land

$\forall x (\GInd(x) \Liff \Lnot \GCat(x)$

$\forall x (\GObj (x) \Liff
\exists y ( \GOCat(y) \Land x \Ginst y))$

$\forall x (\GAtt(x) \Liff
\exists y (\GP(y) \Land x \Ginst y))$

$\forall x (\GPrimCat(x) \Liff \GCat(x) \Land
\exists y ( y \Ginst x ) \Land
\forall z ( z \Ginst x \Limp {\it\GInd(z)}))$

The core vocabulary $\Sigma$ can be extended by categories that classify types, and by categories of individuals capturing its formal structure. The type is the most simple structural feature a category may possess. We start with the primitive type (the initial type), which is denoted by the symbol $i$ (for individuals). Every primitive type is a type. If $t_{1}, \ldots, t_{n}$ are types, then t_1, ..., t_n is a type. Nothing is a type unless it follows the conditions mentioned. A category is said to be well-founded if it has a type. Two categories $C_1$ and $C_2$, are said to be extensional equivalent if they have the same instances. We may introduce a cross-level relation connecting categories with sets by postulating that for every category $C$, there is a set $X$ such that $\forall u ( u \in X \leftrightarrow u \Ginst C)$. Such an axiom influences the structure of the ATO-level; if there are categories which are not well-founded, then the cross-level axiom implies the existence of hyper-sets.

The basic signature $\Sigma$ of the ACO level may be extended by adding a number of meta-categories. One extension is created by adding for any finite type $\tau$ a meta-category $C \{ \tau \}$ whose instances are just all categories of type $\tau$. A special case are primitive categories, whose instances are individuals. Non-primitive categories can be found in every sufficiently complex field, for example, in the biological domain. Means of expressing categories of higher type have also found their way into UML, in the form of the UML elements metaclass and powertype (48).

Robert Hoehndorf 2006-10-18


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