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Name	NegativePropertyAssertions
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Author(s)
SubmittedBy	OlafNoppens

Description

Motivation	The motivation of this pattern is to model negative property assertions (NPAs) in ontology languages such as OWL 1 [1] that do not provide a special constructor for expressing it. It is worth mentioning that not all knowledge base systems can be migrated to OWL 2 [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 constraint 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. [1] Patel-Schneider, P.F., Hayes, P., Horrocks, I.: OWL Web Ontology Language Semantics and Abstract Syntax, W3C Recommendation 10 February 2004. [2] Motik, B., Patel-Schneider, P.F., Parsia, B.: OWL 2 Structural Speciﬁcation and Functional-Style Syntax. W3C Candidate Recommendation 11 June 2009, 2009.
Aim	Expressing NPAs in ontologies prior to OWL 2 as well as given an transformation rule when using OWL 2.
Solution description	NegativeObjectPropertyAssertion(prop i1 i2) is equivalent to (using OWL 2 Abstract Syntax): SubClassOf(ObjectOneOf(i1), ObjectComplementOf(ObjectSomeValuesFrom(prop, ObjectOneOf(i2)))) 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., '( SubClassOf( C ObjectComplementOf(D) ). The equivalence is correct because of the duality of disjointness, equivalence, and unsatisfiability: C is subsumed by D if, and only if, ObjectIntersectionOf( C ObjectComplementOf(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 ObjectSomeValuesFrom(prop C) is the set of individuals i which are connected to an individual j that is in the extension of the concept C, by the property prop. Let NegativePropertyAssertion(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 ObjectSomeValuesFrom( p ObjectOneOf(b) ) which contain all individuals that are connected to b by p must not contain a. This is true, if, and only if ObjectOneOf(a) is disjoint to ObjectSomeValuesFrom( p ObjectOneOf(b) )
Elements	Individiual i1 Individual i2 ObjectProperty prop
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	Submission to event WOP2009:Main