Concept Partition Pattern

1 Introduction

The Partition Pattern is a logical pattern that introduces axioms which model a partition of concepts. A partition is a general structure which is divided into several disjoint parts. With respect to ontologies the structure is a concept which is divided into several pair-wise disjoint concepts. This pattern reflects the simplest case where a named concept is defined as a partition of concepts. 2.1

Pattern

Problem

The Partition Pattern describes how to model a partition, i. e., a named concept which is divided into several disjoint concepts. Applying this pattern to an ontology will introduce the necessary axioms. 2.2

Solution

Let P be a named concept that is the partition which is divided into several concepts Ci. Then the partition is defined by introducing the following axioms (expressed in the standard DL syntax) as also be shown in Figure 1: P ≡ C0 C1 · · · Cn and Ci Cj = ⊥ for 0 ≤ i, j ≤ n, i = j.

Note that some ontology languages such as OWL 2 [ 1 ] provides disjointness axioms as syntactical sugar to make the definition of pair-wise disjointness easier. In OWL 2 the pattern can be translated into the following axioms (expressed in OWL 2 abstract syntax): Consider a world where only humans and animals live. Then the inhabitants of this world are partitioned into humans and animals. The following two axioms EquivalentClasses(Inhabitant ObjectUnionOf(Human, Animal)) DisjointClasses(Human, Animal) describe the partitions of inhabitants into human and animals.

3 Pattern Usage

In an ontology about family relationship we defined concepts such as Person, Aunt and ParentOfSon which are characterized by a relationships such as hasChild (resp. the inverse relationship hasParent), hasSibling, married-with as well as by the gender of people (Male respectively Female). There are a lot of similar ontologies about family relationships. Our version can be downloaded at http://www.informatik.uni-ulm.de/ki/Noppens/generation.owl. The concept Gender is partitioned in Male and Female. Applying this pattern results in the following axioms: EquivalentClasses(Gender, ObjectUnionOf(Male Female)) DisjointClasses(Male Female)

4 Summary and Future Work

The Partition Pattern describes how to model a partition. The pattern reflects the simplest case where a named concept is the partition of (arbitrary) concepts. Future work will be concerned with a more general variant of this pattern: in some cases, the partition concept is not explicitly named (i. e., is not a named concept) but implicitly used, for instance, as value range of quantifications. In other words, no equivalent class axiom will be used but the value range is the union of the pair-wise disjoint parts of the partition.

References

1. Motik , B. , Patel-Schneider , P.F. , Parsia , B. : OWL 2 Structural Specification and Functional-Style Syntax . W3C Candidate Recommendation 11 June 2009 ( June 2009 )