RDF/RDFS Reasoning Capabilities

Type inheritance through rdfs:subclassOf. For example, the facts

(rdf:type Morris Cat)
(rdfs:subClassOf Cat Mammal)

imply the fact

(rdf:type Morris Mammal)

Reflexivity of rdfs:subPropertyOf and rdfs:subclassOf. For any rdf:Property p, the fact
```
(rdfs:subPropertyOf p p)
```
is inferred. For any Class C, the fact
```
(rdfs:subClassOf C C)
```
is inferred.

Type inference through rdfs:range and rdfs:domain constraints. For example, the facts

(rdfs:domain teaches Teacher)
(rdfs:range teaches Student)
(teaches Bob Scooter)

imply the facts

(rdf:type Bob Teacher)
(rdf:type Scooter Student)

Transitivity of rdfs:subClassOf and rdfs:subPropertyOf. For example, the facts

(rdfs:subClassOf Dog Mammal)
(rdfs:subClassOf Mammal Animal)

imply the fact

(rdfs:subClassOf Dog Animal)

Similarly, the facts

(rdfs:subPropertyOf parent ancestor)
(rdfs:subPropertyOf ancestor relative)

imply the fact

(rdfs:subPropertyOf parent relative)

OWL Reasoning Capabilities

OWL reasoning capabilities include the RDF/RDFS reasoning capabilities as well as the following capabilities. In all references to reasoning with rdfs:subClassOf, the same type of reasoning is done with owl:subClassOf.

Enforcing transitivity of owl:TransitiveProperty. For example, the facts

(rdf:type ancestor owl:TransitiveProperty)
(ancestor Sue Mary)
(ancestor Mary Anne)

imply the fact

(ancestor Sue Anne)

Semantics of owl:SymmetricProperty is enforced.

Reasoning with owl:inverseOf. For example, the facts

(owl:inverseOf parentOf hasParent)
(parentOf Goldie Kate)

imply the fact

(hasParent Kate Goldie)

Inheritance of disjointness constraints. For example, the facts

(owl:disjointWith Plant Animal)
(rdfs:subClassOf Mammal Animal)

imply the fact

(owl:disjointWith Plant Mammal)

When an owl:sameAs relationship is asserted or inferred between two entities that are known to be classes, an owl:equivalentClass relationship is inferred between the classes. Similarly, when an owl:sameAs relationship is asserted or inferred between two entities that are known to be properties, an owl:equivalentProperty relationship is inferred between the classes. For example, the facts
```
(owl:sameAs Human Person)
(rdf:type Human rdfs:Class)
(rdf:type Person rdfs:Class)
```
imply the fact
```
(owl:equivalentClass Human Person)
```
All the subclasses of a given class are disjoint with the class's complement. For example, the facts
```
(owl:complementOf Animal NonAnimals)
(rdfs:subClassOf Mammal Animal)
```
imply the fact
```
(owl:disjointWith Mammal NonAnimals)
```

A complicated bit of reasoning about owl:complementOf captured by following KIF axiom.

(=> (and (owl:complementOf ?c1 ?c2)
         (rdfs:subClassOf ?c3 ?c1)
         (rdfs:subClassOf ?c4 ?c2)
         (owl:complementOf ?c4 ?c5))
    (rdfs:subClassOf ?c3 c5))

Inferring owl:sameAs relationships via owl:FunctionalProperty and owl:InverseFunctionalProperty. For example, the facts
```
(rdf:type mother owl:FunctionalProperty)
(mother Joe Margaret)
(mother Joe Maggie)
```
imply the fact
```
(owl:sameAs Margaret Maggie)
```
If a class A is owl:oneOf a list of objects, say X, Y, and Z, then each of X, Y, and Z has rdf:type A.

If an object is rdf:type an owl:hasValue owl:Restriction, then the object has the specified value for the specified property. For example, the facts

(owl:onProperty RestrictionOrangeSkin skinColor)
(owl:hasValue RestrictionOrangeSkin Orange)
(rdf:type MrOompaLoompa RestrictionOrangeSkin)

imply the fact

(skinColor MrOompaLoompa Orange)

If an owl:hasValue owl:Restriction restricts a particular property to a particular value, and an object has that value for that property, then the object has the Restriction as a type. For example, the facts
```
(owl:onProperty RestrictionOrangeSkin skinColor)
(owl:hasValue RestrictionOrangeSkin Orange)
(skinColor MrOompaLoompa Orange)
```
imply the fact
```
(rdf:type MrOompaLoompa RestrictionOrangeskin)
```
If an object is a rdf:type an owl:allValuesFrom owl:Restriction, and the object has values for the specified property, then the values are of the specified type. For example, the facts
```
(owl:onProperty RestrictionCatChildren child)
(owl:allValuesFrom RestrictionCatChildren Cat)
(rdf:type Fluffy RestrictionCatChildren)
(child Fluffy Cupcake)
```
imply the fact
```
(rdf:type Cupcake Cat)
```
If an owl:someValuesFrom owl:Restriction restricts a particular property to a particular type, and if an object has some values of the specificied type for the specified property, then that object has the Restriction as a type. For example, the facts
```
(owl:onProperty RestrictionIvyLeagueDegree degree)
(owl:someValuesFrom RestrictionIvyLeagueDegree IvyLeagueSchool)
(degree Mary Harvard)
(rdf:type Harvard IvyLeagueSchool)
```
imply the fact
```
(rdf:type Mary RestrictionIvyLeagueDegree)
```
If a property Q is owl:inverseOf of a property P, and P is an owl:TransitiveProperty, then Q is also an owl:TransitiveProperty.
All of the elements of an owl:AllDifferent are owl:differentFrom each other.

OWL Classification inferences

All classes are rdfs:subClassOf owl:Thing.
If a class A is the owl:unionOf a list of classes, say B, C, and D, then each of B, C, and D, is rdfs:subClassOf A.
If a class A is the owl:intersectionOf a list of classes, say B, C, and D, then A is rdfs:subClassOf of each of B, C, and D.
If a class A is the owl:intersectionOf a list of constituent classes and a class B is the owl:intersectionOf a list of constituent classes, then A is a rdfs:subClassOf B if every constituent of A is a rdfs:subClassOf some constituent of B.
If a class A is the owl:unionOf a list of constituent classes and a class B is the owl:unionOf a list of constituent classes, then A is a rdfs:subClassOf B if every constituent of B is a superclass of some constituent of A.
If a class A is the owl:intersectionOf a list of constituent classes and a class B is the owl:unionOf a list of constituent classes, then A is a rdfs:subClassOf B if every constituent of A is a rdfs:subClassOf every constituent of B.
An owl:hasValue owl:Restriction A subsumes another owl:hasValue restriction B if the property restricted by B is a rdfs:subPropertyOf the property restricted by A, and the restricted value is the same.
An owl:allValuesFrom owl:Restriction A subsumes another owl:allValuesFrom restriction B if the property restricted by B is a rdfs:subPropertyOf the property restricted by A, and the restricted class of B is a rdfs:subClassOf the restricted class of A. Recall that any class is a subclass of itself, and any property is a subproperty of itself.
If an object is a type of owl:maxCardinality owl:Restriction where the specified property is restricted to a maximum of 1 value, then all values of which the object has for the specified property are owl:sameAs one another.
An owl:minCardinality owl:Restriction A is disjoint from an owl:maxCardinality owl:Restriction B if the property restricted by A is a rdfs:subPropertyOf the property restricted by B, and the number of values specified by A is greater than the number of values specified by B.
An owl:minCardinality owl:Restriction A subsumes another owl:minCardinality restriction B if the property restricted by B is a rdfs:subPropertyOf the property restricted by A, and the number of values specified by A is less than or equal to the number of values specified by B.
An owl:maxCardinality owl:Restriction A subsumes another owl:maxCardinality restriction B if the property restricted by B is a rdfs:subPropertyOf the property restricted by A, and the number of values specified by A is greater than or equal to the number of values specified by B.