Qualified cardinality restrictions

Qualified cardinality restrictions (QCRs): constraing the number of property values of a particular type

Guus Schreiber, first draft , 25 May 2004

General issue

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:

Class(MinimailItalianDinner,
  subClassOf(Restriction(hasCourse, cardinality(3))))

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" (dessert)
The dinner must include exactly one of either "primo piatto"
(first dish) or "secondo piatto" (second dish).

Thus, we need a way of saying that the number of values of a particular type (e.g., a starter) is restricted (e.g., to 1). We call these "qualified cardinality restrictions", where the term "qualified" means that we do not express restrictions on the overall number of valuesof a property, but only on the number of values of a certain type (i.e., class, datatype).

Use cases

Rector [1] mentions, amongst others, the following use cases.

Human 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 - andt also an acidic group".
"Haemoglobin consists of four subunits each of which contains exactly one haem group each of which contains exactly one iron ion"

British Nationality Act:

Important concept in this act: "Person who has at least one parent who is a British Citizen"

Administrative structures

"Medical oversight committees consists of at least 5 members of which two must be medically qualified, one a manager, and two members of the public"

Drug interactions:

"Regime containing more than one Central Nervous System depressant". The regime of course may consist of any number of drugs, the issue is that at least two of them are Central Nervous System Depressants.

Typically, qualified cardinality restrictions are used to specify the componen types in some part-of structure such as the human body, an administrative body, an artifact or a cemical compound.

Modelling approaches

Approach 1: use owl:someValuesFrom

The OWL construct someValuesFrom is actually a qualified restriction in disguise. It is equivalent to a qualified cardinality restriction with a minimal cardinality of 1: "this property should have at least one value of this type".

The construct can be used to describe the constraint that an Italian dinner should contain at least one antipasto:

  Class(ItalianDinner,
    subClassOf(Restriction(hasCourse, someCaluesFrom(AntiPasto)))

The "British citizen" use case can also be expressed with this OWL construct:

  Class(PersonWithBritishParent
    intersectionOf(Person,
                   Restriction(hasParent someValuesFrom(BritishCitizen))))

Approach 2: workaround with subPropertyOf

The first approach only works for QCRs with a cardinality of "1 or more". It does not handle, for example, the constraints on the "MinimalItalianDinner", which are all of the "precisely 1" type. For these situations one will need to use a workaround.

A common woraround is to introduece a subproperty for which the restriction cane be formuletaed as an isolated cardinality and valuerestrictions.

The "normal hand" use case can be expressed with this workaround. e.g.:

Property(hasFinger)
Property(hasThumb
  subPropertyOf(hasFinger))
Class(NormalHand
  subClassOf(Restriction(hasFinger cardinality(5)))
  subClassOf(Restriction(hasThumbe cardinality(1))))

The constraints formulated for minimal Italian dinners can also be expressed with this workaround:

Property(hasStarter
  range(Antipasto)
  subPropertyOf(hasCourse))
Property(hasMainCourse
  range(unionOf(PrimoPiatto SecondoPiatto))
  subPropertyOf(hasCourse))
Property(hasDessert
  range(Dolce
  subPropertyOf(hasCourse))

Class(MinimailItalianDinner
  subClassOf(Restriction(hasStarter cardinality(1)))
  subClassOf(Restriction(hasMainCourse cardinality(1)))  
  subClassOf(Restriction(hasDessert cardinality(1))))

The woraraound is probably OK for simple cases. The workaraound becomes problematic in cases where complicated part-of relations dominate the ontology. Examples could be ellaboarte anatomical structures, the structure of chemical compounds or the structure of large artifacts such ships or airplanes.Introducing subporprty relations for all the variants is cumbersome.

Approach 3: using a nonendorsed OWL extension

The Web Ontology Working Group has postponed the issue of full representation of QCRs [2], but hasat the same time already suggested an OWL representation for them [3]. It is not unlikely that a future reincarnation of the OWL group will inculde this extension into the language. OWL users who require QCRSs for describing complex part-whole relations may want to use this extensions, even though these are not yet endorsed.

The syntax for QCRS looks like this [3]. Qualified restrictions resemble regular restrictions, but, whereas the latter only contain two triples (an owl:onProperty one and some constraint such as owl:allValuesFrom or owl:cardinality), qualified restrictions contain three triples, namely:

owl:onProperty, pointing to the property concerned
owl:valuesFrom, pointing to the value type being restricted
a cardinality constraint (owl:cardinality, owl:minCardinality or owl:maxCardinality), which restricts the number of values the property can take for this type

The exampole below shows an this approach for the MinimalItalianDinner, listing one o the "regular' constraints and one of the qualified ons:

Class(MinimalItalianDinner
  subclassOf(Restriction(
    hasCourse cardinality(3)))
  subclassOf(QualifiedRestriction(
    hasCourse valuesFrom(Antipaso) cardinality(1))))

The "NormalHand" example can be represented in a similar way:

  Class(NormalHand
    subClassOf(Restriction(
      hasFinger) cardinality(5)))
    subClassOf(QualifiedRestriction(
      hasFinger valuesFrom(Thumb) cardinality(1))))

NOTE: The resulting ontology will be an OWL Full ontology. According to the OWL Reference, RDF/OWL parsers should produce a warning message when n onendorsed OWL vocabulary is used, but should otherwise proceed normally.

Acknowledgments

This noet is based on discussion on the QCR requirement in the Web Ontology Working Group and on the OWL public comments list after comments from Rector [1].

References

http://lists.w3.org/Archives/Public/public-webont-comments/2003Apr/0040.html
http://www.w3.org/2001/sw/WebOnt/webont-issues.html#I3.2-Qualified-Restrictions
http://lists.w3.org/Archives/Public/www-webont-wg/2003May/0072.html