Theory - Foundations

Towards a General Ontology Theory (ONMA)

Foundations

We define a Knowledge Graph (KG) as a three-dimensional data structure which is spanned by the dimensions Generalization, Aggregation and Particularization shown in Figure layers. Each KG represents an ontology O. In the example, the set of classes is C = {C1, ..., C6}, the set of attributes is A = {A1, ..., A6}, the set of binary relationship types is R = {(C4, ◊OP, C2)} and the set of particulars is E = {P1, ..., P7). The class hierarchy (C , ≤_∧), the relationship type hierarchy (R , ≤_◊), and the attribute type hierarchy (A , ≤_.) are organized along the Generalization dimension. Each class aggregates data property and object property definitions along the Aggregation dimension. In the particulars layer the attribute-value assignments to entities and the connections of entities to others are also organized along the Aggregation dimension. The generalization-aggregation plane constitutes the Schema Layer (SL). The particulars p ∈ E which are lying in the Particulars Layer (M₀ = PL) are connected to their classes in the Schema Layer (M₁ = SL) with, e.g., (P7, »pof, C6). Thus, particulars are instantiated along the Particularization dimension into the particulars layer plane.

Ontology Layers

Data Properties (DP) represent intrinsic properties of concepts and have a an Atomic Data Type adt ∈ ADT = {:String, :Integer, :Decimal, :Float, :Boolean, :Date, :Time, :anyURI, ...}. A class C aggregates declarations for data property A in the form of the Data Property Definition (DPD) dpd = (C, A, adt) along the Aggregation dimension of the SL layer. In a DPD the class C plays the role of the domain and the adt plays the role of the range.

Object Properties (OP) represent extrinsic properties of concepts. Object Property Definitions (OPD) between the domain class C and the range class D are also defined in the SL layer in the form opd = (C, ◊op, D). Object Property Instantiations (OPI) can take place between particulars P and Q in the PL Layer, or between a particular P and a class C between the PL and the SL layer. The former takes the form (P, »op, Q) and the latter takes the form (P, »op, C) or (C, »op^-1, P). The blue arrow from C4 to C2 in figure layers indicates an opd = (C4, ◊op, C2) while the purple arrow indicates an opi = (P4, »op, P2). The light blue arrow between the opi and the opd indicates, that the opi is an instantiation of the opd which we note with (opi, »opi, opd). This state of affairs can be expressed with ((P4, »op, P2), »opi, (C4, ◊op, C2)). As with the particularization of particulars of classes, the instantiation of object properties takes place between the SL and the PL layers.

Extension: deriver.app

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Source: taoke.de — Theory - Foundations.