Equal relationship between ontology properties

Question

I have 2 properties, e.g. 'hasColor' and 'hasFinish'. I want to express with ontology that in case of ontology class A are properties 'hasColor' and 'hasFinish' equal (owl:equivalentProperty). But in case of ontology class B the properties 'hasColor' and 'hasFinish' are NOT equal.

How can I achieve this?

One way probably is to create 'hasColor' and 'hasFinish' properties with class A as a range and set them to be equal. Then create another ones 'hasColor' and 'hasFinish' properties with class B as a range and without equal relationship. But is it right approach?

Answer 1

It's not entirely clear yet what you mean by making two properties un-equal. By default, two properties can be distinct, so you don't really have to do anything special to let them be unequal. If the question is clarified, though, perhaps more information can be added about that.

It's not entirely trivial to say that, e.g.,

∀a,x . A(a) → (hasFinish(a,x) ⇔ hasColor(a,x))

in OWL, but you can do it. If you want to say this for all classes, you can, as you pointed out, use owl:equivalentProperty. Now, when you say that p is an equivalent property to r, you could also say that

p ⊑ r
r ⊑ p

that is, that each of p and r are subproperties of the other. In OWL 2 (but, unfortunately, not OWL 2 DL, as Antoine Zimmermann pointed out in the comments), you can assert that a given property is a superproperty of a chain of properties, e.g.,

hasFather • hasBrother ⊑ hasUncle

which says that if someone has a father who has a brother, then that father's brother is that person's uncle.

There's also a concept called rolification, which has been described more in OWL 2 rolification, which is the process of creating a property corresponding to a class that relates each individual of that class to itself. For you class A, there would be a relation R_A that relates each A to itself, and only relates those instances. If you then look at a property chain like

R_A • hasFinish

you'll notice that it's really the subproperty of hasFinish where the first argument is an A. This means that you can say that hasFinish and hasColor are the same for the class A by making two subproperty assertions:

R_A • hasColor ⊑ hasFinish
R_A • hasFinish ⊑ hasColor

These assume that individuals of type A are the subjects of these statements. If A is actually the range here, then you'd just use

∀a,x . A(a) → (hasFinish(x,a) ⇔ hasColor(x,a))
hasColor • R_A ⊑ hasFinish
hasFinish • R_A ⊑ hasColor

To get the property R_A, you need to add the definition

A ≡ ∃R_A.Self

to your ontology. In Protégé, this would look like:

and the resulting ontology looks like (in RDF/XML):

<rdf:RDF
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:owl="http://www.w3.org/2002/07/owl#"
    xmlns:xsd="http://www.w3.org/2001/XMLSchema#"
    xmlns="http://example.org/ep#"
    xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#">
  <owl:Ontology rdf:about="http://example.org/ep"/>
  <owl:Class rdf:about="http://example.org/ep#A">
    <owl:equivalentClass>
      <owl:Restriction>
        <owl:onProperty>
          <owl:ObjectProperty rdf:about="http://example.org/ep#R_A"/>
        </owl:onProperty>
        <owl:hasSelf rdf:datatype="http://www.w3.org/2001/XMLSchema#boolean"
        >true</owl:hasSelf>
      </owl:Restriction>
    </owl:equivalentClass>
  </owl:Class>
  <owl:ObjectProperty rdf:about="http://example.org/ep#hasFinish">
    <owl:propertyChainAxiom rdf:parseType="Collection">
      <owl:ObjectProperty rdf:about="http://example.org/ep#R_A"/>
      <owl:ObjectProperty rdf:about="http://example.org/ep#hasColor"/>
    </owl:propertyChainAxiom>
  </owl:ObjectProperty>
  <owl:ObjectProperty rdf:about="http://example.org/ep#hasColor">
    <owl:propertyChainAxiom rdf:parseType="Collection">
      <owl:ObjectProperty rdf:about="http://example.org/ep#R_A"/>
      <owl:ObjectProperty rdf:about="http://example.org/ep#hasFinish"/>
    </owl:propertyChainAxiom>
  </owl:ObjectProperty>
</rdf:RDF>

and in Turtle:

@prefix :      <http://example.org/ep#> .
@prefix rdfs:  <http://www.w3.org/2000/01/rdf-schema#> .
@prefix owl:   <http://www.w3.org/2002/07/owl#> .
@prefix xsd:   <http://www.w3.org/2001/XMLSchema#> .
@prefix rdf:   <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .

:hasFinish  a                   owl:ObjectProperty ;
        owl:propertyChainAxiom  ( :R_A :hasColor ) .

:A      a                    owl:Class ;
        owl:equivalentClass  [ a               owl:Restriction ;
                               owl:hasSelf     true ;
                               owl:onProperty  :R_A
                             ] .

:hasColor  a                    owl:ObjectProperty ;
        owl:propertyChainAxiom  ( :R_A :hasFinish ) .

<http://example.org/ep>
        a       owl:Ontology .

:R_A    a       owl:ObjectProperty .