It has long been recognized that patterns can be a useful and important tool when building models. This is reflected by their adoption in the practice of ontology and software engineering. Similarly, visual representations of information are often seen as beneficial with, for example, software engineering making use of the suite of diagrammatic notations forming the UML. Likewise, ontology engineering has seen the development of a variety of different visualizations for classes and properties. This paper combines these two strands of work, making visual (diagrammatic) patterns available to ontology engineers.
This section provides an introduction to concept diagrams, designed as part of
the Ontology Engineering with Diagrams project (www.ontologyengineering.org).
Readers interested in the full notation and its formalization should see [
Fig. 1 asserts that the class B is subsumed by A and both A and B are disjoint from C. The spatial properties of inclusion and exclusion correspond to the semantic properties of set inclusion and disjointness. The solid arrow P asserts that individuals in A are only related to elements in C: the target of the arrow is the set of things to which the elements in A are related and this set is subsumed by C. The solid arrow R , where R is the inverse of R, asserts that all things are, between them, related to exactly the individuals in C under R ; the source of this arrow is the boundary rectangle which represents the set of all things, often denoted ⊤. Lastly, the dashed arrow provides partial information about property Q: under Q, the individuals in B are, between them, related to at least the elements in C. Under some circumstances, we may not wish to assert disjointness and subsumption relationships. Concept diagrams make this readily achievable, whilst avoiding clutter, by using multiple rectangles. Fig. 2 visualizes the same information as Fig. 1 except for the disjointness of C with A and B. Spatial relationships only convey information within a single rectangle. 3
We now demonstrate how to express nine commonly occurring axioms using patterns, contrasting with Description Logic (DL). Common constraints that are imposed on classes are subsumption (subset), disjointness and equivalence. To express that one class is subsumed by another class, concept diagrams use curve containment, reflected in our first pattern; to express class subsumption in DL, one asserts C2 ⊑ C1. Similarly, concept diagrams use curve disjointness (i.e. non-overlapping curves) to express class disjointness, captured in DL by C1 ⊓ C2 ⊑ ⊥.
Pattern 1: Class Subsumption Class C1 subsumes class C2, Fig. 3. Pattern 2: Class Disjointness Classes C1 and C2 are disjoint, Fig. 4.
Pattern 2 has an obvious generalization to (concisely) assert that n classes are pairwise disjoint (that is, any pair of the n classes are disjoint). Using DL, one axiom is required for each pair of classes to capture this disjointness information: C1 ⊓ C2 ⊑ ⊥,...,C1 ⊓ Cn ⊑ ⊥, ..., Cn 1 ⊓ Cn ⊑ ⊥; OWL has a more succinct representation: DisjointClasses(C1; :::; Cn). In the diagram below, the ellipsis indicates the presence of a further n − 3 circles labelled in the obvious fashion.
Pattern 3: General Class Disjointness C1; ::; Cn are pairwise disjoint, Fig. 5.
Ontology engineers often want to express that two classes are equivalent. As with the other patterns, there are many semantically equivalent, but syntactically different, concept diagrams that express class equivalence. The following pattern employs two overlaying (completely concurrent) curves. Pattern 4: Class Equivalence Classes C1 and C2 are equivalent, Fig. 6.
In DL, Class Equivalence can be expressed by C1 ≡ C2. Again, the Class Equivalence pattern has an obvious generalization to the n-class case: to express that n classes are equivalent draw n overlaying curves. The number of DL axioms to express many classes are all equivalent to each other increases rapidly, whereas only a single diagram is needed, omitted for space reasons. A common property restriction is to enforce ‘All Values From’ and ‘Some Values From’ constraints. Pattern 5: All Values From All individuals in class C1 have all values, under property P, from class C2, Fig. 7
The arrow in the above diagram formally asserts that the image of the property P (considering P as a binary relation), when its domain is restricted to C1, is a subset of C2. In other words, the only things that individuals in C1 are related to, under P, must be in C2. The use of multiple bounding boxes ensures that no unintended disjointness information between classes is asserted. In DL, the All Values From pattern is captured by C1 ⊑ ∀P:C2. We also present a pattern for ‘Some Values From’, expressed in DL by C1 ⊑ ∃P:C2.
Pattern 6: Some Values From Individuals in class C1 have at least one value, under property P, from class C2, Fig. 8.
The above diagram makes use of the inverse of property P. To justify the correctness of the Some Values From pattern, consider an individual, c1, in the class C1. The pattern must ensure that c1 has a value, c2, from C2 under property P. Well, c1 has such a value, c2, if and only if c2 has the value c1 under P . Equivalently, the image of P when its domain is restricted to C2 includes at least all of the individuals in C1, captured by the dashed arrow.
Our last three patterns concern domains and ranges of properties. Firstly, consider the domain, D, of property P. The domain of P is D if and only if the range of the inverse, P , of P is D.
Pattern 7: Domain of a Property The domain of property P is D, Fig. 9.
The corresponding DL formalization of this pattern is ∀P:⊤ ⊑ D; the construction ∀P:⊤ builds the pre-image of the property P. The Domain of a Property pattern employs the same style of construction: the arrow builds the pre-image of P. In DL, the range is typically defined by ⊤ ⊑ ∀P:R. The range can also be defined in DL by constructing the pre-image of the inverse, P , of P and asserting that this pre-image is subsumed by R: ∀P :⊤ ⊑ R. Our Range of a Property pattern constructs the image of P, using an arrow, and asserts that this image is subsumed by the range, R.
Pattern 8: Range of a Property The range of property P is R, Fig. 10. Pattern 9: Domain and Range of a Property The domain and range of property P are D and R respectively, Fig. 11. 4
We demonstrate the application of the patterns to the Semantic Sensor Networks
(SSN) Ontology [
Fig. 16. Some Val. From Situation and Event. There is a property, includesEvent, with domain Situation and range Event shown in Fig. 17. 5
We have presented nine diagrammatic patterns for defining constraints that
occur frequently in ontology engineering. These patterns are all formal, since
concept diagrams have a fully defined syntax and semantics [