a problem with comprehensive entailments

From: "Jeremy Carroll" <jjc@hplb.hpl.hp.com>
Subject: DTTF: another summary
Date: Fri, 3 May 2002 10:45:34 +0100

> I believe we have three live proposals:
> 
> - comprehensive entailments: Jeremy

Unfortunately, this proposal is fatally flawed.

In the empty KB, we can apply the kitchen sink comprehension rule

(forall (?v)
(=> (and (Type ?p Property)
         (Type ?card NonNegativeInteger)
         (Type ?min NonNegativeInteger)
         (Type ?max NonNegativeInteger)
         (Type ?c1 rdfs:Class)
         (Type ?c2 rdfs:Class))
     (exists (?r) (and (Type ?r Restriction)
                      (PropertyValue onProperty ?r ?p)
                      (PropertyValue minCardinality ?r ?min)
                      (PropertyValue maxCardinality ?r ?max)
                      (PropertyValue cardinality ?r ?card)
                      (PropertyValue hasClass ?r ?c1)
                      (PropertyValue toClass ?r ?c2)  ) ) ) )

with ?p as rdf:type
     ?card as 0
     ?min as 0
     ?max as 0
     ?c1 as rdfs:Class
     ?c2 as rdfs:Resource
     ?v as rdf:type (?v is not used in the rule)

to obtain the following restriction (written in n-triples notation)

   _:r rdf:type owl:Restriction .
   _:r owl:onProperty rdf:type .
   _:r owl:minCardinality 0 .
   _:r owl:maxCardinality 0 .
   _:r owl:cardinality 0 .
   _:r owl:hasClass rdfs:Class .
   _:r owl:toClass rdfs:Resource .

Now consider whether rdfs:Class is a member of this restriction.

Because _:r has both owl:onProperty and owl:toClass, an object belongs to it
exactly when all its rdf:type fillers belong to rdfs:Resource.  But this is
true of all objects in all interpretations.  Therefore rdfs:Class belongs
to _:r.

Because _:r has both owl:onProperty and owl:maxCardinality, an object
belongs to it exactly when it has at most 0 rdf:type fillers.  But this is
not true of rdfs:Class, because rdfs:Class has rdfs:Class as an rdf:type.
Therefore rdfs:Class does not belong to _:r.

Therefore an empty knowledge base implies that some object both belongs and
does not belong to some class.


peter

Received on Friday, 3 May 2002 09:53:35 UTC