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Re: a problem with comprehensive entailments

From: Jos De_Roo <jos.deroo.jd@belgium.agfa.com>
Date: Mon, 6 May 2002 00:33:16 +0200
To: pfps@research.bell-labs.com
Cc: "jjc" <jjc@hplb.hpl.hp.com>, "www-webont-wg" <www-webont-wg@w3.org>
Message-ID: <OF7A357577.10D3B117-ONC1256BB0.00796165@agfa.be>

> > 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.

OK
with the following pair of ``entailment'' rules

  { :rule9r1 .
    ?s a ?C }
    log:implies
      { ?s a [ a owl:Restriction;
               owl:onProperty rdf:type;
               owl:toClass ?C ] } .

  { :rule9r2 .
    ?s a ?C }
    log:implies
      { ?s a [ owl:complementOf [ a owl:Restriction;
                                  owl:onProperty rdf:type;
                                  owl:maxCardinality "0";
                                  owl:hasClass ?C ] ] } .

we were able to entail

  rdfs:Class a [ a owl:Restriction;
                 owl:onProperty rdf:type;
                 owl:toClass rdfs:Resource ] .

  rdfs:Class a [ owl:complementOf [ a owl:Restriction;
                                    owl:onProperty rdf:type;
                                    owl:maxCardinality "0";
                                    owl:hasClass rdfs:Class ] ] .

and we have kept some statements in appropriate N3 contexts

--
Jos
Received on Sunday, 5 May 2002 18:33:57 GMT

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