This paper proposes a simple, single extension of OWL, namely maximalSubPropertyOf that allows to “localize” a given property to a fixed domain and range. As an application we show that both qualified cardinality restriction and reflexive properties can be defined using this construction in conjunction with existing OWL functionality.
When we model, we need a way to say that one or more individuals of a class can be related to one or more individuals of the same or some other class. For example, we might need to state that an individual of class Component can be part of an individual of class Machine or that an individual of class Person can be part of zero or more individuals of class Organization. We usually do this by defining a property isPartOf and then creating two subproperties, e.g. isComponentPartOfMachine (with domain range and cardinality 1) and
Component, Machine isPersonPartOfOrganization (with domain Person, range Organization and no defined cardinality). In this way, whenever we state that some of the subproperties hold for two individuals of the respective classes, we can infer that the property isPartOf also holds for the same individuals. However, anytime we state that a a we implicitly mean that Component c isPartOf Machine m c isComponentPartOfMachine m, and likewise if Person p isPartOf an Organization o we implicitly mean that p isPersonPartOfOrganization o. However, it is impossible to express such implications in the current version of OWL.
In this paper we propose a simple, single extension of OWL, namelly maximalSubPropertyOf, that allows expressing this. For example, it would be more convenient and more precise to say that isComponentPartOfMachine is the maximal subproperty of isPartOf with domain Person and range Machine, and isPersonPartOfOrganization is the maximal subproperty of isPartOf with domain Person and range Organization. This way, we can simply use isPartOf to state relations between individuals, and later on query using the more specific subproperties isComponentPartOfMachine and isPersonPartOfOrganization or vice versa. In addition, maximal subproperties can be used to address the lack of the qualified cardinality restrictions and reflexive properties in the current version of OWL.
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We propose a simple, single extension of OWL, namely maximalSubPropertyOf. The
formal semantics of maximalSubPropertyOf is defined as follows:
∃maximalSubPropertyOf(Q, P) ∧ ∃domain(Q, D) ∧ ∃range(Q, R) →
(∀x.D(x) ∧ ∀y.R(y) ∧ ∃P(x, y) → ∃Q(x, y)
where Q with domain D and range R is the maximal subproperty of the property P.
The maximal subproperty of P is the unique subproperty which is maximal among all
subproperties of P with the given domain D and range R, i.e. the domain and range of a
maximalSubPropertyOf is part of its definition. A more natural name for this
construction is the restriction of a property to the given domain D and range R, as the
maximal subproperty of P is the same as the property P between D and R, and
undefined outside. This point of view can be seen from the following example. Consider
the sine function as a property on the real numbers. Then the restriction of sine to a
function on [-π/2, π/2] with values in [
Q maximalSubPropertyOf P; domain D; range R.
(1) x Q y.
(2) x a D. y a R. x P y.
(1 => 2) trivial. (2 =>1) is precisely the maximality of the maximalSubPropertyOf.
It follows from the lemma that in the absence of further statements involving Q, decidability is as difficult as classifying individuals.
OWL allows the definition of cardinality restrictions which are used to constrain the number of values of a particular property. Suppose we want to define a building with at least 4 fire escape stairs:
Building a owl:Class .
FireEscapeStairs a owl:Class . hasFireEscapeStairs a owl:ObjectProperty ; rdfs:domain Building ; rdfs:range FireEscapeStairs .
BuildingMin4Fire a Building , [a owl:Restriction;
owl:onProperty hasFireEscapeStairs; owl:minCardinality 4
Now, suppose we want to refine our model and say that at least two of the fire escape stairs should be external fire escape stairs. To do that we need a mechanism to state that the property hasFireEscapeStairs must have owl:minCardinality 2 when its value has type FireEscapeStairs and owl:minCardinality 2 when its value has type ExternalFireEscapeStairs. In the OWL community this is known as qualified cardinality restriction and unfortunately, is unexpressable by the current version of OWL. The need for qualified cardinality restriction is motivated by Alan
One possible workaround (proposed by Guus Schreiber in [
Building a owl:Class .
FireEscapeStairs a owl:Class .
ExternalFireEscapeStairs rdfs:subClassOf FireEscapeStairs . hasFireEscapeStairs a owl:ObjectProperty ; rdfs:domain Building ; rdfs:range FireEscapeStairs . hasExternalFireEscapeStairs rdfs:subPropertyOf hasFireEscapeStairs ; rdfs:range ExternalFireEscapeStairs .
BuildingMin4FireMin2External a [a owl:Restriction; owl:onProperty hasFireEscapeStairs; owl:minCardinality 4], [a owl:Restriction; owl:onProperty hasExternalFireEscapeStairs; owl:minCardinality 2].
However, this workaround has a serious reasoning limitation. Suppose we define an instance of a Building only using the hasFireEscapeStairs property: stairs1 a FireEspaceStairs ; stairs2 a FireEscpaeStairs ; stairs3 a ExternalFireEscpaeStairs ; stairs4 a ExternalFireEscopeStairs ; aBuilding a Building ; hasFireEscapeStairs stairs1 ; hasFireEscapeStairs stairs2 ; hasFireEscapeStairs stairs3 ; hasFireEscapeStairs stairs4 .
The instance cannot be classified as BuildingMin4FireMin2External because aBuilding does not have at least two properties hasExternalFireEscapeStairs with values of type ExternalFireEscapeStairs.
Now, let us redefine the property hasExternalFireEscapeStairs maximalSubPropertyOf: using hasExternalFireEscapeStairs a owlx:maximalSubPropertyOf hasFireEscapeStairs ; rdfs:domain Building ; rdfs:range ExternalFireEscapeStairs .
BuildingMin4FireMin2External because the definition and the facts as a stairs3 a ExternalFireEscpaeStairs ; stairs4 a ExternalFireEscopeStairs ; aBuilding a Building ; hasFireEscapeStairs stairs3 ; hasFireEscapeStairs stairs4 . imply aBuilding hasExternalFireEscapeStairs stairs3 . aBuilding hasExternalFireEscapeStairs stairs4 .
The maximalSubPropertyOf allows defining reflexive properties. A reflexive property defined on a domain D is a property that holds between each individual of D and itself, i.e.:
reflexive(R) ∧ ∃domain(R, D) ∧ ∃range(R, D) → (∀x.D(x) → ∃R(x, x)) Examples of reflexive properties are “is equal to”, “is subset of”, "is less than or equal to" and “is greater than or equal to”.
If we consider R as a subset of D × D (i.e. we consider the interpretation), then R is reflexive if and only if the diagonal in D × D is contained in R. This geometric interpretation shows a natural way to define a reflexive property using maximalSubPropertyOf. The diagonal of owl:Thing × owl:Thing corresponds to owl:sameAs. To get the diagonal in D × D, we merely have to restrict the property owl:sameAs to D: sameAsOnD owlx:maximalSubPropertyOf owl:sameAs; rdfs:domain D ; rdfs:range D . aReflexiveProperty a owl:ObjectProperty; rdfs:domain D ; rdfs:range D .
sameAsOnD rdfs:subPropertyOf aReflexiveProperty .
A reflexive and transitive property P on class C is a property such that the
composition of zero or more properties P is a subproperty of P, where a composition of zero
properties P equals the restriction of owl:sameAs to the domain C (i.e., the
composition identity on C). This point of view is very convenient when dealing with
partwhole relations. This was recognized by the W3C best practices workgroup and
described in an extensive workaround[
Space a owl:Class .
Building rdfs:subClassOf Space Room rdfs:subClassOf Space .
Corner rdfs:subClassOf Space .
Printer a owl:Class . isLocatedIn a owl:ObjectProperty ;
rdfs:range Space . isPartOf a owl:TransitiveProperty ; rdfs:domain Space ; rdfs:range Space .
We create instances of rooms, corners and printers.
aBuilding a Building. aRoom a Room . aCorner a Corner . aPrinter a Printer . aRoom isPartOf aBuilding. aCorner isPartOf aRoom .
aPrinter isLocatedIn aCorner .
Now suppose we want to know whether aPrinter is in aRoom and if yes, where exactly in the room it is located. Therefore, we pose a query
isLocatedIn(aPrinter, ?x) AND isPartOf(?x, aRoom)
?x = aCorner Now, suppose that we were less accurate in describing the location of our printer and instead of
aPrinter isLocatedIn aCorner . we stated aPrinter isLocatedIn aRoom .
aRoom isPartOf aRoom .
The same query now returns no results because there is no statement (neither explicit nor inferred) i.e. the isPartOf(?x, aRoom) will never evaluate true.
We can add isPartOf property for all individuals of type Space, but clearly a more elegant solution is to make isPartOf a reflexive property.
This research is a part of the Freeband Communication projects A-Muse (http://a-muse.freeband.nl) and awareness (http://awareness.freeband.nl). Freeband Communication (http://www.freeband.nl) is sponsored by the Dutch government under contract BSIK 03025. 5 References