The concept that is not defined but referred is called atomic concept, e.g., Person and Female in the above demonstration, while a complex concept is defined using other concepts (complex or atomic), e.g., Woman and Man. The atomic concept in SWCLOS exists as object but does not have slot values except rdfs:subClassOf and owl:equivalentClass.

12.3  OWL Property Restrictions

A complex concept is often composed of concepts and restrictions. A restriction, which is an instance of owl:Restriction, works as constraint for a specific property and a specific concept. A property restriction is anonymous and has only two slots of owl:onPropertyOf and restriction types, i.e., owl:allValuesFrom, owl:someValuesFrom, owl:hasValue, owl:maxCardinality, owl:minCardinality, and owl:cardinality.

In the following example, the concept Mother is defined as the intersection of Woman and the restriction such that the property hasChild must exist in Woman individuals and one value of the property hasChild must be an instance of Person at least.

The property restriction in the intersection is useful to make a sub concept that is specialized by the property restriction. In the above case, the concept Mother is defined as Woman who has the slot hasChild, in which one slot value at least must be an instance of a Person. Exactly speaking on the above demonstration, it is possible to have something but Person in the hasChild slot in the above definition. Note that it is needed to add the value restriction (owl:allValuesFrom) on hasChild for Person, if you want to restrict that all of children must be a Person, exactly in OWL semantics.

Note that owl:Restriction in OWL is specialized to subclass restrictions in SWCLOS, i.e., owl:hasValueRestriction, owl:someValuesFromRestriction, owl:allValuesFromRestriction, or owl:cardinalityRestriction. The reason is mainly for human readability of an anonymous object of owl:Restriction. Using this subclassing of owl:Restriction, SWCLOS provides the customization of printing restrictions. See the followings.

gx-user(15): (-> Father owl:intersectionOf)
(#<owl:Class Man> #<��hasChild.Person>)


In CLOS, we have the type option in the slot definition. Therefore, we have implemented the allValuesFrom functionality and the someValuesFrom functionality through this type option in the slot definition as well as the range value constraint in RDFS. Furthermore, in order to implement the cardinality constraint, we set two new slot option, maxcardinality and mincardinality in the slot definition as OwlProperty-slot-definition.

12.4  Substantial and non-Substantial Relationship

Whereas the subsumption relationship is decided with only rdfs:subClassOf in RDFS, the subsumption decidability is very complicated in OWL. There are many properties that rule subsumption, such as rdfs:subClassOf, owl:intersectionOf, owl:unionOf, owl:equivalentClass, owl:equivalentProperty on rdf:type, rdfs:subPropertyOf on rdf:type, etc. From the viewpoint of DL, they have same strength for subsumption decidability. However, from the viewpoint of Ontology Engineering, generally we have to discriminate substantial ones and non-substantial ones for subsumption. For instance, the wife and husband relation is weaker than man-person or woman-person relation. The former is not substantial but the latter is substantial.

On the other hand, from the viewpoint of CLOS, rdfs:subClassOf relation is mapped onto class-subclass relation, and a CLOS object as rdfs:subClassOf property value is also placed in the direct-superclasses list slot in the class metaobject. Then, in case that a property prop1 is a subproperty of rdfs:subClassOf, or a equivalent property of rdfs:subClassOf, whether or not should we place the property value into the direct-superclasses list in the class to which the property prop1 is attached? In other words, what property should cause the structural variation in CLOS superclass-subclass relation, and what property should not cause the structural variation? For instance, food:Wine is equivalent to vin:Wine, and food:Wine in Food Ontology does not have any slots, and vin:Wine in Wine Ontology has many slots. If we add vin:Wine into the direct-superclasses list of food:Wine, then food:Wine inherits all slot definitions in the superclasses of vin:Wine. So, should we put the same slots as vin:Wine to food:Wine or not? In SWCLOS, we specified that rdfs:subClassOf, owl:intersectionOf, and owl:unionOf should cause the structural variation of CLOS, but other properties such as owl:equivalentClass, owl:equivalentProperty, including subproperties and equivalent properties of rdfs:subClass, owl:intersectionOf, and owl:unionOf, should not cause the structural variation. The predicate gx:subtypep supports the subsumption by class-subclass relation in RDFS semantics, and subsumed-p reasons the subsumption using other subsumption-related properties in OWL.

From the standpoint of ontology engineering, the effects of substantial properties may propagate to other entities, but the result of non-substantial properties should not cause any side effects in ontology. Thus, such discrimination rule of substantial and non-substantial subsumption allows ones to add and delete the relation of rdfs:subPropertyOf, owl:equivalentClass, etc., and keeps easy to maintain ontologies.

12.5  Axiomatic Complete Relations

Among many properties in OWL, only four properties, i.e., owl:intersectionOf, owl:unionOf, owl:complementOf, and owl:oneOf make axiomatic complete relations as if and only if condition. In other words, these properties define complete equivalency upon the binary relation of concepts. For example, the following states the definition of WhiteBordeaux, and if something is a Bordeaux and WhiteWine, it should be concluded to be a WhiteBordeaux.

Similarly, the following assertion defines WineColor, which has the enumerative membership of White, Rose, and Red of complete relation, so that it states that the instance of WineColor is exactly one of the three, and not to be the others.

Therefore, it is not necessary to mind Open World Assumption upon such axiomatic complete relation properties. If we find the contents of the definition body matches the database, then we may conclude the definition is applicable to the database without a worry about other statements.

See the following example. SWCLOS concludes that QueenElizabethII should be a woman, because there is a terminology that a person who has gender female is a woman, and it is asserted that QueenElizabethII is an instance of Person and hasGender Female. Here, note that SWCLOS proactively made the entailment without demand or query from users.