Qualified cardinality restrictions (QCRs): Constraining the
number of values of a particular type for a property
NOTE: this document is NOT a W3C draft, it is intended for discussion only.
Editor's Draft 8 May 2006
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- Editors:
Alan Rector, University of Manchester
Guus Schreiber, Vrije Universiteit Amsterdam
Copyright
© 2006 W3C® (MIT, ERCIM,
Keio), All Rights Reserved. W3C liability,
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use and software
licensing rules apply.
It is often useful to express constraints such as "has
exactly four parts that are legs", "has at least two groups
that are phosphate groups", "has exactly one feature
that is temperature", etc. In each of these cases, we
want to constrain not the total number of values for a
property, but rather the number of values of a given
type. Such restrictions are called "qualified
cardinality restrictions" ("QCRs") because they are
"qualified" by the type of the value. They are
supported in most modern description logics but were omitted
from the final version of the OWL standard. This note
discusses the uses of such constraints, a partial "work
around" within the OWL standard, and a natural extension to
the OWL standard to allow it to express QCRs correctly.
Status of this Document
This section describes the status of this document at
the time of its publication. Other documents may supersede
this document. A list of current W3C publications and the
latest revision of this technical report can be found in the
W3C technical reports
index at http://www.w3.org/TR/.
This document os produced by the "Ontology Engineering and
Patterns" (OEP) Task Force of the
Semantic
Web Best Practices and Deployment Working Group.
This document is not yet a W3C Working Draft and is
expected to change. The SWBPD WG does not expect this
document to become a Recommendation. Rather, after further
development, review and refinement, it is intended to be
published and maintained as a WG Note.
We encourage public comments. Please send comments to public-swbp-wg@w3.org
Open issues, todo
items:
- Discussions over future status of QCRs with in the course of any revision of the OWL standard
- General references to QCRs in the Description Logic
literature
Publication as a draft does not imply endorsement by the
W3C Membership. This document is a draft and may be
updated, replaced or made obsolete by other documents at any
time. It is inappropriate to cite this document as other
than work in progress.
Contents
1. General issues
Cardinality restrictions are commonly used to constrain the
number of values of a particular property, irrespective of
the value type. For example, we can state that a
"Minimal Italian Dinner" contains exactly three courses:
:MinimalItalianDinner
a owl:Class ;
rdfs:subClassOf
[ a owl:Restriction ;
owl:onProperty :hasCourse ;
owl:cardinality "3"^^xsd:nonNegativeInteger ] .
NOTE: Some Italian restaurants in the US
consider you a dummy when you skip either primo or secondo,
but it is quite common in contemporary Italy
However, suppose we wanted to add the following constraints
for minimal Italian dinners:
- The dinner must include exactly one "antipasto"
(starter)
- The dinner must include exactly one "dolce"
(desert)
- The dinner must include exactly one of either "primo
piatoo" (first dish) or "secondo piatoo" (second dish)
To express these constraints we need a way to restrict the
valujes of a particular
type (e.g., starters) to a particular value
(e.g. 1). We call such
constraints "qualified cardinality restrictions", where the
term "qualified" indicates that they apply only to a
specific type of value rather than to the property
overall.
2. Use cases
Rector [Rector, 2003]
mentions a number of use cases, and others
have arisen in the course of developing ontologies for
various communities. In general the use cases occur
where there are many different kinds of parts, features,
chemical groups, legal statuses, qualities, etc. In
such cases defining a separate property or 'slot' for each
is cumbersome at best and difficult to extend.
- Anatomy:
- "The normal hand has exactly five fingers of which
one is a thumb."
- "The heart has four chambers: two atria and two
ventricles."
- Bio-ontologies and chemistry:
- "Tricarboxylic acid contains exactly three
carboxyl groups and opne acidic group."
- "Haemoglobin consists of four subunits, each of
which contains exactly one haem group, each of which
contains exactly one iron ion."
- Many legal strictures, e.g. the British
Nationality Act:
- An important concept in the act is of a "Person
who has at least one parent who is a British
citizen".
- Administrative structures:
- "A medical oversight committee must consist of at
least five members of which two must be medically
qualified, one a manager, and two members of the
public."
- Drug interactions:
- "A legal drug regimen must not contain more than
one Central Nervous System depressant, although it may
contain any number of drugs in total."
- N-ary relations
[Noy and Rector, 2006]:
- "A person may have any number of 'features' or
'qualities', but (at a given time) only one of each:
e.g. 'height', one 'weight', one 'body temperature',
etc. each of which can have a value, an undertainty on
that value, and a measure of its change or trend."
- "An employee may have many relationships with
other employees, but only one lline manager
relationship."
3. Pattern I: Use
owl:someValuesFrom
The OWL construct owl:someValuesFrom is
actually a qualified restriction in disguise. It is
equivalent to a qualified cardinality restriction with a
minimum cardinality of i.e. "this property must
have at least one value of this type.". The
Mcode>someValuesFrom constructor can be used to
constrain an Italian dinner to have at least one
antipasto:
:ItalianDinner
a owl:Class ;
rdfs:subClassOf
[ a owl:Restriction ;
owl:onProperty :hasCourse ;
owl:someValuesFrom :AntiPasto ] .
The example in use case 3 above could also be expressed
simply using owl:someValuesfrom:
:PersonWithBritishParent
a owl:Class ;
owl:equivalentClass
[ a owl:Class ;
owl:intersectionOf
( :Person
[ a owl:Restriction ;
owl:onProperty :hasParent ;
owl:someValuesFrom :BritishCitizen ] ) ] .
However, this approach is not useful for values other than
"at least one" or for imposing maximum cardinalities.
It therefore cannot deal with the other uses cases or
generalisations of 3 such as "at least two grandparents who
were British citizens".
4. Pattern II: Work around
using subPropertyOf
A common 'work around' to deal with other cardinalities
is to introduce a subproperty of the primary property and
then to introduce an unqualified cardinality restriction on
that subproperty. For example, we might represent the
"normal hand" example from use case 1 by:
:BodyPart
a owl:Class .
:Finger
a owl:Class ;
rdfs:subClassOf :BodyPart .
:Thumb
a :Class ;
rdfs:subClassOf :Finger .
:hasPart
a owl:ObjectProperty ;
rdfs:range :BodyPart .
:hasFinger
a owl:ObjectProperty ;
rdfs:range :Finger ;
rdfs:subPropertyOf :hasPart .
:hasThumb
a owl:ObjectProperty ;
rdfs:range :Thumb ;
rdfs:subPropertyOf :hasFinger .
We would then hve the property hierarchy:
hasPart
hasFinger
hasThumb
We could then represent the "normal hand" by:
:NormalHand
a owl:Class ;
rdfs:subClassOf :Hand ;
owl:equivalentClass
[ a owl:Class ;
owl:intersectionOf (
[ a owl:Restriction ;
owl:onProperty :hasFinger ;
owl:cardinality "5"^^xsd:nonNegativeInteger ]
[ a owl:Restriction ;
owl:onProperty :hasThumb ;
owl:cardinality "1"^^xsd:nonNegativeInteger ] ) ] .
This workaround can also be used to represent the
constraints on "Minimal Italian Dinner":
:AntiPasto
a owl:Class ;
rdfs:subClassOf :Course .
:PrimoOrSecundoPiatto
a owl:Class ;
rdfs:subClassOf :Course .
:Dolce
a owl:Class ;
rdfs:subClassOf :Course .
:hasAntiPasto
a owl:ObjectProperty ;
rdfs:subPropertyOf :hasCourse .
:hasPrimoOrSecundo
a owl:ObjectProperty ;
rdfs:subPropertyOf :hasCourse .
:hasDolce
a owl:ObjectProperty ;
rdfs:subPropertyOf :hasCourse .
:MinimalItalianDinner
a owl:Class ;
owl:equivalentClass
[ a owl:Class ;
owl:intersectionOf (
[ a owl:Restriction ;
owl:onProperty :hasAntiPasto ;
owl:cardinality "1"^^xsd:nonNegativeInteger ]
[ a owl:Restriction ;
owl:onProperty :hasPrimoOrSecundo ;
owl:cardinality "1"^^xsd:nonNegativeInteger ]
[ a owl:Restriction ;
owl:onProperty :hasDolce ;
owl:cardinality "1"^^xsd:nonNegativeInteger ] ) ] .
Discussion
This pattern suffices for simple cases but presents several problems:
- The constraints are incomplete because the range of
the super property must subsume the ranges of the
subproperties. Therefore, there is no way to prevent
the use of the parent property inappropriately.For example,
we could add another dessert to the "Minimal Italian
Dinner":
.....
[ a owl:Restriction ;
owl:onProperty :hasDolce ;
owl:someValuesFrom :Tiramisu ]
[ a owl:Restriction ;
owl:onProperty :hasCourse ;
owl:someValuesFrom :IceCream ]
.....
Assuming that IceCream is a kind of
Dolce is a kind of Course, this
is legal according to the OWL constraints but not their
intent, which was to restrict the "minimal" dinner to
exactly one each of starter, main, and dessert. There
is no way to constrain the range of the property
hasCourse without that constraint also
applying to its subproperties, so there is no way around
this problem except to use tools to impose additional
constraints based on annotations or meta properties which
are outside the OWL syntax.
- In cases where there are many different kinds of
things to be constrained, this pattern gives rise to vast
numbers of subproperties. For example, for the anatomy
example, there would have to be one subproperty for each
kind of body part. And again, the parent property
cannot be constrained so that there is no way of preventing
expressions analogous to the extra desert in 1 above.
- In the case of n-ary relations, a separate property is
similarly required for each feature type, which similarly
gives rise to vast numbers of properties,e.g.
hasTemperatureFeature,
hasHeightFeature, etc.
5. Pattern III: Use a non-endorsed OWL
extension
The Web Ontology Working Group has postponed the issue of
full representation of QCR [WebOnt QCR
issue], but has at the same time already suggestd an OWL
representation for them [WebOnt QCR
resolution]. It is not unlikely that this extension will
be incoporatd into a future version of the language.
OWL users who require QCRs for use cases not well served by
the work around in Pattern II may want to use this extension,
even though they are not yet endorsed. (At least one widely
used tool already supports QCRs as do many of the widely
used classifiers.)
The syntax looks like this. Qualified restrictions resemble
regular restrictions but contain one extra triple in the RDF
representation and an extra argument in the abstract syntax.
A simple cardinality constraint contains just two
triples:
owl:onProperty, pointing to the property
concerned- a cardinality constraint (
owl:cardinality,
owl:minCardinality or
owl:maxCardinality), which restricts the
number of values the property can take.
To this a qualified cardinality constraint need only add
a third triple:
owl:valuesFrom, pointing to the value
type being restricted.
In the abstract syntax, we need only add an additional
argument to the cardinality restrictions.
TODO: include ref plus discussion OWL 1.1
abstract syntax
The class "minimal italilan dinner" might be represented
as:
:MinimalItalianDinner
a owl:Class ;
owl:equivalentClass
[ a owl:Class ;
owl:intersectionOf (
[ a owl:Restriction ;
owl:onProperty :hasCourse ;
owl:cardinality "3"^^xsd:nonNegativeInteger ]
[ a owl:Restriction ;
owl:onProperty :hasCourse ;
owl:cardinality "1"^^xsd:nonNegativeInteger ;
owl:valuesFrom :AntiPasto ]
[ a owl:Restriction ;
owl:onProperty :hasCourse ;
owl:cardinality "1"^^xsd:nonNegativeInteger ;
owl:valuesFrom :PrimoOrSecundoPiatto ]]
[ a owl:Restriction ;
owl:onProperty :hasCourse ;
owl:cardinality "1"^^xsd:nonNegativeInteger
owl:valuesFrom :Dolce ] ) ] .
The normal hand example can be expressed in a similar
way:
:NormalHand
a owl:Class ;
rdfs:subClassOf :Hand ;
owl:equivalentClass
[ a owl:Class ;
owl:intersectionOf (
[ a owl:Restriction ;
owl:onProperty :hasPart ;
owl:cardinality "5"^^xsd:nonNegativeInteger ;
owl:valuesFrom :Finger ]
[ a owl:Restriction ;
owl:onProperty :hasPart ;
owl:cardinality "1"^^xsd:nonNegativeInteger ;
owl:valuesFrom :Thumb ] ) ] .
In this representation the pattern for many n-ary
relations (use case 6) is illustrated by:
TODO
where hasLevel and hasTrend are
functional properties but has_feature is not
functional. Hence any patient might have many features (of
which just two are shown here), but each Feature could have
just one level and one trend.
A still more complex case, illustrated by use case 4
would be:
:MedicalOversightCommittee
a owl:Class ;
rdfs:subClassOf :Committee ;
owl:equivalentClass
[ a owl:Class ;
owl:intersectionOf (
[ a owl:Restriction ;
owl:onProperty :hasMember ;
owl:minCardinality "5"^^xsd:nonNegativeInteger ]
[ a owl:Restriction ;
owl:onProperty :hasMember ;
owl:minCardinality "2"^^xsd:nonNegativeInteger ;
owl:valuesFrom :MedicalStaff ]
[ a owl:Restriction ;
owl:onProperty :hasMember ;
owl:cardinality "1"^^xsd:nonNegativeInteger ;
owl:valuesFrom :ManagerStaff ]]
[ a owl:Restriction ;
owl:onProperty :hasMember ;
owl:cardinality "2"^^xsd:nonNegativeInteger
owl:valuesFrom :MemberOfPublic ] ) ] .
Expressing this complex constraint using the work around in
Representation Pattern II would require defining three separate
subproperties of hasMember and would still not
fully capture the constraints required.
Discussion
- This represtation is legal in OWL Full according to
the current standard. According to the OWL Reference [OWL Reference], RDF/OWL parsers should
produce a warning message when non-endorsed OWL vocabulary
is used, but should otherwise proceed normally.
However, the semantics will only be treated correctly by
parsers and classifiers which are 'QCR aware'.
As of the time of writing, Racer is known to
support QCRs. FaCT++ is
expected to do so shortly.
Protege-OWL
and the CO-ODE
plugins provide syntactic and user
interface support for an extension to OWL including
QCRs.
- The intended semantics fully capture the intent of the
use cases given.
- Simple cardinalty restrictions can be taken as a
special case of Qualified Cardinality Restrictions where
qualifying argument is
valuesFrom(owl:Thing).
Appendix A: References
[Noy and Rector, 2006]
Defining N-ary Relations on the Semantic Web.
W3C Working Group Note 12 April 2006, Noy and Rector (eds.)
http://www.w3.org/TR/swbp-n-aryRelations/.
[OWL Reference]
OWL Web Ontology Language Reference.
Mike Dean and Guus Schreiber, Editors.
W3C Recommendation, 10 February 2004.
Latest version
available at http://www.w3.org/TR/owl-ref/.
[Rector, 2003]
Alan Rector. Message to public-webont-comments@w3.org: "Case
for Reinstatement of Qualified Cardinality Restrictions",
http://lists.w3.org/Archives/Public/public-webont-comments/2003Apr/0040.html
[WebOnt QCR issue]
Web Ontology Working Group, Issues list, November
2003. Issue 3,2 Qualified Restrictions. http://www.w3.org/2001/sw/WebOnt/webont-issues.html#I3.2-Qualified-Restrictions
[WebOnt QCR resolution]
Web Ontology Working Group, message to
www-webont-wg@w3.org: "Issue 3.2 QCR: proposal to POSTPONE"
http://lists.w3.org/Archives/Public/www-webont-wg/2003May/0072.html
Version May 8, 2006
- Examples revised; N3 syntax,
- Link to OWL 1.1
Version November 2, 2005
- Reformatted and restructured
- Examples and use cases extended
- Ordering of named arguments in abstract syntax modified for
improved readability