From: Peter F. Patel-Schneider (pfps@research.bell-labs.com)
Date: 08/29/01
Here are some thoughts on rdfs:subClassOf. peter Notes on the new definition of rdfs:subClassOf The RDF Core working group has modified the definition of rdfs:subClassOf to indicate that rdfs:subClassOf means proper subclass. In his model theory for RDF(S) Pat Hayes gives this the meaning that if A is a subclass of B then in each model of the entire knowledge base the extension of A must be a proper subclass of the extension of B. This definition of rdfs:subClassOf has several unfortunate consequences. These unfortunate consequences do not appear to affect RDF(S), they only arise if one wants to create a more-expressive formalism that uses rdfs:subClass as its generalization primitive, such as DAML+OIL. 1/ If A is an rdfs:subClassOf of B and if B is necessarily empty then the entire knowledge base is inconsistent. For example, any knowledge base containing the two following classes <daml:Class rdf:ID="Confused"> <rdfs:subClassOf> <daml:Restriction daml:minCardinalty="2"> <daml:onProperty rdf:resource="children"> </daml:Restriction> </rdfs:subClassOf> <rdfs:subClassOf> <daml:Restriction daml:maxCardinalty="1"> <daml:onProperty rdf:resource="children"> </daml:Restriction> </rdfs:subClassOf> </daml:Class> <daml:Class rdf:ID="SubConfused"> <rdfs:subClassOf rdf:resource="#Confused"> </daml:Class> is inconsistent. It is strange that the definitions of two classes, which may not be used anywhere else in the knowledge base can cause the entire knowledge base to become inconsistent. This is a consequence of any reasonable definition that makes rdfs:subClassOf a proper subclass, not just Pat Hayes's model theory. 2/ If A is an rdfs:subClassOf of B then B has at least one member in every model. This means that B cannot even be contingently empty. Thus from <daml:Class rdf:ID="Confused"> </daml:Class> <daml:Class rdf:ID="SubConfused"> <rdfs:subClassOf rdf:resource="#Confused"> </daml:Class> it follows that Confused has cardinality at least one. Again, it is strange that simply having a subclass provides a minimum cardinality for a class. This is a consequence of Pat Hayes's model theory, and need not be true in other definitions that make rdfs:subClassOf be a proper subclass. 3/ If Ai is an rdfs:subClassOf of Ai+1 for i running from 0 to n, then An has at least n members in every model. As a consequence, the minimum number of resources in a model for a knowledge base with an rdfs:subClass chain of length n is n. This is a consequence of Pat Hayes's model theory, and need not be true in other definitions that make rdfs:subClassOf be a proper subclass. Peter F. Patel-Schneider
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