=Paper=
{{Paper
|id=Vol-516/paper-13
|storemode=property
|title=Negative Property Assertion Pattern (NPAs)
|pdfUrl=https://ceur-ws.org/Vol-516/pat06.pdf
|volume=Vol-516
|dblpUrl=https://dblp.org/rec/conf/semweb/Noppens09
}}
==Negative Property Assertion Pattern (NPAs)==
https://ceur-ws.org/Vol-516/pat06.pdf
Negative Property Assertion Pattern (NPAs)
http://
ontologydesignpatterns.org/wiki/Submissions:NegativePropertyAssertions
Olaf Noppens
Inst. of Artificial Intelligence
Ulm University
Germany
olaf.noppens@uni-ulm.de
1 Introduction
The basic building blocks of Description Logics (DLs) in general are concept and
role constructors (e. g. , intersection, union, nominals, negation etc.), termino-
logical as well as individual axioms. Combining axioms, meaningful statements
can be expressed. Additional axiom constructors (so-called syntactical sugar)
has been introduced in most ontology language (such as OWL and OWL 2 [1])
in order to simplify ontology modeling, understanding and maintenance. These
syntactical sugar axioms can be reduced to basic axioms. A simple example is
disjointness of concepts that can be reduced to GCI axioms. However, even ex-
perts have often difficulties to understand the reduced forms in comparison to
syntactical sugar forms. From the modeling perspective, patterns can support
the user in modeling logical meaning that is not directly supported in an on-
tology language. For instance, OWL 2 provides a syntactical form for modeling
negative property assertions whereas using the predecessor OWL 1 one have to
model it by a more or less complicated inclusion axiom (or, alternatively, disjoint
axiom).
The motivation of this pattern is to model negative property assertions (NPAs)
in ontology languages such as OWL 1 that do not provide a special construct for
it. It is worth mentioning that not all knowledge base systems can be migrated
to OWL 2 for several reasons. On the other hand, NPAs modeled according to
this pattern can be migrated to OWL 2 using the newly introduced constructor.
A negative property assertion as defined in the upcoming OWL 2 states that a
given individual i is never connected to a given individual j by a given property
expression P . In other words, asserting that i is connected to j by P results in
an inconsistent ontology. In this sense this assertion can be considered as a con-
straint that should not be violated. In contrast, considering an ontology where
it cannot be inferred that i is connected to j by p does not necessarily mean
that there cannot be such a connection – in fact, it is merely not modeled.
2 Pattern
In this section we describe the Negative Property Assertion Pattern in detail.
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2.1 Problem
Prior to OWL 2 it is difficult to model negative property assertions and, if they
are contained in an ontology, difficult to understand because OWL 1 does not
provide a specialized constructor for it. In addition, with help of this pattern
one can also migrate OWL 1 ontologies containing axioms describing NPAs into
the NPA constructor of OWL 2.
Intent This patterns allows to express NPAs in ontologies written in an ontology
language such as OWL 1 that does not allow NPA directly. The pattern can also
be understood as a transformation rule for transforming axioms modeling NPAs
into the direct NPA axiom constructor of OWL 2.
2.2 Solution
A negative property assertion that states that an individual Individual1 is not
connected to an individual Individual2 by a property property can be modeled
as the following inclusion axiom: {Individual1 } ¬ (∃property {Individual2 })
1
. Here we are using the German DL syntax where {x} denotes an enumeration
class with a single individual x (aka. ‘one-of’). A graphical representation of this
pattern is given in Figure 1.
Fig. 1. Graphical representation of the pattern using the UML-based notation proposed
in [2].
In the following we proof the equivalence between the axiom produced by
the pattern and NPA as defined in OWL 2.
1
Note that this can also be modeled with help of a disjointness axiom. However, not
all languages provide a special disjointness constructor.
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Proof Let C and D be concepts. Then C and D are disjoint if, and only if,
C is subsumed by the complement of D, i. e. , (C ¬D). The equivalence is
correct because of the duality of disjointness, equivalence, and unsatisfiability:
C is subsumed by D if, and only if, C ¬D is unsatisfiable, and the intersection
of C and D is unsatisfiable if, and only if, C and D are disjoint.
One also reminds that the extension of the concept ∃ prop.C is the set of indi-
viduals i which are connected to an individual j that is in the extension of the
concept C, by the property prop.
Let N P A (p a b) be a negative property assertion axiom, i. e. , the individual a is
not related to b by the property p. Then the extension of ∃ p. {b} which contains
all individuals that are connected to b by p must not contain a. This is true, if,
and only if, {a} is disjoint to ∃ p. {b}.
Consequences Applying this pattern to an ontology will add the logical mean-
ing of a NPA to the ontology. There are not any restrictions or limitations of the
solution besides that nominals and unrestricted existential quantification must
be supported in the target ontology language.
2.3 Example
Consider a social network containing facts about people and their relationships.
Let Adam and Eve be two persons and like a property (’A likes B’). Furthermore
we know that Adam does not like Eve but we have no dislike relationship. More-
over, our language (such as OWL 1) does not have any NPA axiom constructor.
The sample ontology is interpreted with respect to the open-world semantics,
i. e. , one can not infer the dislike merely from the lack of a property assertion
axiom Adam like Eve. Then this fact can be expressed with the following axiom
in OWL 1: {Adam} ¬ (∃likes {Eve}).
3 Pattern Usage
The NPA pattern is useful in all situations where one has to model a negative
property assertion but the language does not allow it directly. One real-world
example here is the one mentioned in Section 2.3 where we had modeled a social
network structure and need a possibility to express that one person does not like
another and to ensure that one person does not know another person. With help
of NPAs we have the possibility to distinguish between ‘Person A does not know
Person” and “We do not know whether Person A knows Person B or not”.
4 Summary
The Negative Property Assertion Pattern allows to express negative property
assertions in languages such as OWL 1 that do not a build-in constructor for
this kind of assertions. In addition, when migrating OWL 1 ontologies to OWL
2 ontologies this pattern can be used to find negative property assertions and to
transform them into the special NPA constructor of OWL 2.
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References
1. Motik, B., Patel-Schneider, P.F., Parsia, B.: OWL 2 Structural Specification and
Functional-Style Syntax. W3C Candidate Recommendation 11 June 2009 (June
2009)
2. Borckmans, S., Volz, R., Eberhart, A., Löffler, P.: Visual Modeling of OWL DL
Ontologies Using UML. 198–213
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