For many starting out with OWL ontologies understanding the exact meaning of property restrictions can be challenging. In this post I will use visual representations to explain the meaning of existential property restrictions. For the purpose of this post, we start out with a simple ontology for modelling the relation between a person and their pets. You can model this in the free Protégé ontology editor.

ObjectProperty: owns
        
Class: Cat
  SubClassOf: Pet
    
Class: Dog
  SubClassOf: Pet
    
Class: Person
  DisjointWith: Pet
        
Class: Pet
  DisjointUnionOf: Cat, Dog
  DisjointWith: Person

I have specified this ontology using the OWL 2 Manchester syntax. Other syntax can be used as well, notably the functional-style syntax, which is used to define the direct semantics of OWL 2. In this post I use the Manchester syntax because it tends to be more intuitive to non-logicians, but it has some weaknesses which I will point out when we get to it. Also for clarity and compactness I have omitted the prefixes.

This ontology states that in our domain we have persons (Class: Person) and pets (Class: Pet). Persons cannot be pets, and pets cannot be persons (Class: Pet DisjointWith: Person). Cats (Class: Cat SubClassOf: Pet) and dogs (Class: Dog SubClassOf: Pet) are pets. We further assume that cats and dogs are the only pets in our domain, and a cat is not a dog and vice versa (Class: Pet DisjointUnionOf: Cat, Dog). I have also defined an owns object property, which
will be the basis of our discussions on existential property restrictions. You can run the reasoner to confirm for yourself that this ontology is indeed consistent, that is, it contains no logical contradictions.

In this post I will explain through visual representations what is the meaning of

1. owns some owl:Thing,
2. C\nTolass: AnimalLover SubClassOf: owns some owl:Thing,
3. owns some owl:Thing SubClassOf: AnimalLover, and
4. Class: AnimalLover EquivalentTo: owns some Pet.

owns some owl:Thing

One of the first things one has to realize working with OWL ontologies is that OWL describes sets (in OWL called classes) and relations (in OWL called properties) between sets for some domain of interest. A domain of interest consists of elements (in OWL called individuals) that can belong to classes and/or form part of properties. The domain of interest represents the largest set of individuals that we are interested in, which in OWL is represented by owl:Thing. The smallest set we are interested in is the empty set, which in OWL is represented by owl:Nothing.

In Figure 1 I have illustrated an example domain consisting of some individuals (the small circles) and some owns properties (the arrows) that exist between individuals. At this stage I have not indicated the Person or Pet classes as yet. The meaning of owns some owl:Thing is that it represents the set of those individuals that we know owns something. In our example it consists of the individuals represented by the green circles (i.e. 5 individuals). Note that due to the open world assumption, we cannot assume that individuals that do not form part of the owns property, necessarily do not own anything. Rather, OWL reasoners assume that it is not know whether these individuals own something or do not own something. Because owns some owl:Thing represents a set OWL reasoners and ontology editors some times refer to an expression like owns some owl:Thing as an anonymous class. I.e. it is a class just like Person, but unlike Person it does not have name.

To refer specifically to owners of pets we have to define our existential property restriction as owns some Pet. This is illustrated in Figure 2. As you can see owns some Pet is a subset (subclass) of owns some owl:Thing.

Class: AnimalLover SubClassOf: owns some Pet

Assume now we want to model a person that loves animals as someone who has at least 1 pet. We can model this as follows:

Class: AnimalLover
  SubClassOf: 
    Person,
    owns some Pet

To test our ontology we create an individual without a pet.

Individual: anAnimalLoverWithoutAPet
  Types: AnimalLover

Since our anAnimalLoverWithoutAPet individual is defined as belonging to the AnimalLover class and we have not stated that it owns a pet, we may expect that the reasoner will find our ontology inconsistent. However, this is not the case. The reason for this is again due to the open world assumption: there is nothing in our ontology that states that the anAnimalLoverWithoutAPet individual has no pets. To make explicit that anAnimalLoverWithoutAPet owns no pets, we change our definition of anAnimalLoverWithoutAPet as follows

Individual: anAnimalLoverWithoutAPet
  Types: 
    AnimalLover, 
    owns max 0 Pet

which states that anAnimalLoverWithoutAPet has a maximum of zero pets. If we now run the reasoner it will give an inconsistency. When your ontology gives an inconsistency (even when you expect it), it is good idea to check whether it gives an inconsistency for the correct reasons. This will help you to confirm whether you designed your ontology correctly for your desired outcomes. In this case the explanation for the inconsistency is given in Figure 3: It states that the inconsistency is due to the following reasons:

1. anAnimalLoverWithoutAPet is an animal lover (anAnimalLoverWithoutAPet Types AnimalLover),
2. who does not own a pet anAnimalLoverWithoutAPet Types owns max 0 owl:Thing,
3. but the expectation is that an animal lover should own a pet (AnimalLover SubClassOf owns some Pet).

Explanations are minimal. That means that if you remove any 1 of the reasons given in an explanation, it is no longer an explanation for the inconsistency (or an entailment). Hence,

anAnimalLoverWithoutAPet Type AnimalLover
AnimalLover owns some Pet

is not an explanation for the inconsistency. This gives us a hint on how to remove an inconsistency from an ontology. If we can change our ontology such that any of the reasons given in an explanation for an inconsistency no longer holds, our ontology will be consistent. In this case we can (again) remove the owns max 0 Pet type from anAnimalLoverWithoutAPet.

An incorrect assumption we may make based on the design of our ontology is that when ever an individual has a pet, the reasoner will infer that the individual is an AnimalLover. However, as I said, this assumption is incorrect. Thus, changing our ontology as follows

Individual: aCat
  Types: Cat

Individual: aPetOwner
  Facts: 
    owns aCat

will not result in inferring that aPetOwner is an AnimalLover.

To understand this see Figure 4. Because we have defined AnimalLover as a subclass of owns some Pet it means the following possibilities exist:

1. There may be persons who have pets who do not love animals.
2. There may be individuals that have pets who are not persons.

Hence, from our current ontology, the reasoner can infer nothing more.

owns some Pet SubClassOf: AnimalLover

Now in this section, instead of defining AnimalLover as a subclass of owns some Pet, let us use a general class axiom and define owns some Pet SubClassOf: AnimalLover with AnimalLover defined as follows:

Class: AnimalLover
  SubClassOf: Person

Note that if you now save your ontology in Manchester syntax, you will loose the owns some Pet SubClassOf: AnimalLover general class axiom. Rather save it in say RDF/XML or OWL/XML syntax. We also add the following individuals:

Individual: aCat

Individual: aPetOwner
  Facts: 
    owns aCat

Ensure that you have specified no type information for aCat and aPetOwner. If you run the reasoner it will not infer that aCat is of type Cat. This is because aCat is not of type Cat. If you state that aCat is of type Cat, it will infer that aPetOwner is an AnimalLover.

One reason why this design may not be ideal is that if we again consider the case where an individual anAnimalLoverWithoutAPet of type AnimalLover does not have a pet (as defined in the previous section), it will not give an inconsistency. To understand the reason why the reasoner cannot make this inference, see Figure 5. Because we defined owns some Pet SubClassOf: AnimalLover the possibility exists that there are AnimalLovers who do not own any pets.

Class: AnimalLover EquivalentTo: owns some Pet

From the previous 2 sections we have noted that defining AnimalLover as AnimalLover SubClassOf: owns some Pet and own some Pet SubClassOf: AnimalLover each have benefits and downsides. If only we could define AnimalLover in terms of both. This can be achieved by defining AnimalLover as follows:

Class: AnimalLover
  EquivalentTo: owns some Pet

With this definition in place, an individual who owns a pet will be inferred to be of type AnimalLover. If an individual is of type AnimalLover, but it is known to have no pets, it will cause an inconsistency. The reason way this definition is not suffering of the problems we encountered with our previous approaches is because the AnimalLover class is now equivalent to the owns some Pet class.

Does this mean it is better to define classes in terms of equivalence rather than subclasses? No. In this instance it turned out to be the case only because we made the assumption that AnimalLovers are equivalent to someone owning a pet. Clearly in reality there may be people owning pets but who do not like pets. Hence, given other assumptions a different ontology may have been a better choice.

Conclusion

The purpose of this post is to explain the meaning of existential property restrictions. For this reason I did not consider using domain and range restrictions to enable inferring that someone who owns a pet is necessarily an AnimalLover. This can easily be achieved by stating

ObjectProperty: owns
  Domain: AnimalLover
  Range: Pet

However, domain and range restrictions are really just syntactical sugar that can be expressed in terms of existential property restrictions (or universal property restrictions).

The example ontologies of this post can be found in github.