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Existential restrictions


Qualified existential restrictions



ObjectProperty: r
Class: D
    EquivalentTo: r some C
Class: C

the semantics of r some C is the set of individuals such that for each individual x there is at least 1 individual y of type C that is linked to x via the object property r.

Based on this semantics, a possible world adhering to our initial equivalence axiom may be:

In this Venn diagram we assume individuals are black dots. Thus, our world consists of 7 individuals, with only 2 classes, namely C and D, as well 2 object properties, namely r and q. In this world, D and thus the class r some C, consist of only 2 individuals. D and r some C consist of only 2 individuals because these are the only individuals linked via object property r to at least 1 individual respectively in C.


In the following we define a pet owner as someone that owns at least 1 pet.

ObjectProperty: owns
Class: PetOwner
    EquivalentTo: owns some Pet
Class: Pet

If we want to introduce the class DogOwner, assuming we can only use the class Pet and the object property owns (assuming we have not defined PetOwner), we could say that a dog owner is a subset of pet owners:

ObjectProperty: owns
Class: DogOwner
    SubClassOf: owns some Pet
Class: Pet

In this case we use SubClassOf instead of EquivalentTo because not every pet owner necessarily owns a dog. This is equivalent to stating:

ObjectProperty: owns
Class: PetOwner
    EquivalentTo: owns some Pet
Class: Pet
Class: DogOwner 
    SubClassOf: PetOwner

Variations on existential restrictions

Unqualified existential restrictions

In the previous section we modeled a PetOwner as owns some Pet. In the expression owns some Pet Pet is referred to as the filler of owns and more specifically we say Pet is the owns-filler.

The PetOwner EquivalentTo: owns some Pet state that pet owners are those individuals that own a pet and ignore all other owns-fillers that are not pets. How can we define arbitrary ownership?

ObjectProperty: owns
Class: Owner
    EquivalentTo: owns some owl:Thing

Value restrictions

We can base restrictions on having a relation to a specific named individual, i.e.:

Individual: UK
ObjectProperty: citizenOf
Class: UKCitizen
    EquivalentTo: citizenOf hasValue UK

Existential restrictions on data properties

This far we have only considered existential restrictions based on object properties, but it is possible to define existential restrictions based on data properties. As an example, we all expect that persons have at least 1 name. This could be expressed as follows:

DataProperty: name
Class: Person
    SubClassOf: name some xsd:string

When to use SubClassOf vs EquivalentTo with existential restrictions

In our example of Person SubClassOf: name some xsd:string, why did we use SubClassOf rather than EquivalentTo? That is, why did we not use Person EquivalentTo: name some xsd:string? With using the EquivalentTo axiom, any individual that has a name, will be inferred to be an instance of Person. However, there are many things in the world that have names that are not persons. Some examples are pets, places, regions, etc:

Compare this with, for example, DogOwner:

ObjectProperty: owns
Class: Dog
Class: DogOwner
    EquivalentTo: owns some Dog