From: Dan Connolly <connolly@w3.org>

Date: 19 Jun 2002 16:26:58 -0500

To: www-webont-wg@w3.org

Message-Id: <1024522018.26122.422.camel@dirk>

Date: 19 Jun 2002 16:26:58 -0500

To: www-webont-wg@w3.org

Message-Id: <1024522018.26122.422.camel@dirk>

This is an attempt to address feedback on: layering (5.3,5.10): a same-syntax model theory From: Dan Connolly (connolly@w3.org) Date: Thu, May 30 2002 http://lists.w3.org/Archives/Public/www-webont-wg/2002May/0264.html in particular, issues around lists and recursive definitions. This leaves lists out altogether. I think the result is a little bit klunky, so I'd like to figure out how to formalize lists properly, but until I understand more about how semantics for recursion/induction/whatever works, I'd like to know if this is a coherent model theory for OWL: As in [RDFMT], an interpretation I of a vocabulary V is 1. A non-empty set IR of resources, called the domain or universe of I. 2. A mapping IEXT from IR into the powerset of IR x (IR union LV) i.e. the set of sets of pairs <x,y> with x in IR and y in IR or LV 3. A mapping IS from V into IR Every owl interpretations is an RDF interpretations (i.e. IR contains IS(rdf:type)) and an RDFS interpretation (i.e. the RDFS closure rules apply). Recall that ICEXT(x) abbreviates {y | <y,x> is in IEXT(I(rdf:type)) }. Additionally, owl reserves the following vocabulary: * complementOf * differentFrom * disjointWith * equivalentTo * hasClass * hasValue * imports * intersectionOf1 * intersectionOf2 * inverseOf * oneOf1 * oneOf2 * onProperty * Ontology * Restriction * sameClassAs * samePropertyAs * toClass * TransitiveProperty * ManyToOneProperty * manyToOneOver * unionOf1 * unionOf2 * OneToManyProperty * oneToManyOver * versionInfo That is, it reserves the URI references http://www.w3.org/2001/10/daml+oil#complementOf , http://www.w3.org/2001/10/daml+oil#differentFrom , and so on; in this document, we'll abbreviate these as ont:cardinality, ont:complementOf and so on. The following closure rules apply: ont:imports rdfs:subPropertyOf rdfs:seeAlso. ont:versionInfo rdfs:subPropertyOf rdfs:comment. The following constraints on interpretations apply: complementOf: <?s, ?o> is in IEXT(I(ont:complementOf)), iff ICEXT(?s) is the complement of ICEXT(?o) relative to IR. differentFrom: <?s, ?o> is in IEXT(I(ont:differentFrom)), iff ?s is not equal to ?o. disjointWith: <?s, ?o> is in IEXT(I(ont:disjointWith)), iff ICEXT(?s) is disjoint from ICEXT(?o). equivalentTo: <?s, ?o> is in IEXT(I(ont:equivalentTo)), iff IEXT(?s) = IEXT(?o). onProperty/hasClass: if <?r, ?p> is in IEXT(I(ont:onProperty)) and <?r, ?a> is in IEXT(I(ont:hasClass)), then the set { ?o: for some ?s in ICEXT(?r), <?s, ?o> is in IEXT(?p) and ?o is in ICEXT(?a) } has at least one element. onProperty/hasValue: if <?r, ?p> is in IEXT(I(ont:onProperty)) and <?r, ?a> is in IEXT(I(ont:hasValue)), then the set { ?o: for some ?s in ICEXT(?r), <?s, ?a> is in IEXT(?p) } isn't empty. imports: none. imports doesn't constrain interpretations (other than having the subproperty relationship with rdfs:seeAlso). intersectionOf1/intersectionOf2: if <?a, ?b> is in IEXT(I(ont:intersectionOf1)) and <?a, ?c> is in IEXT(I(ont:intersectionOf2)) and ?x is in ICEXT(?b) and in ICEXT(?c) then ?x is in ICEXT(?a). conversely, if <?a, ?b> is in IEXT(I(ont:intersectionOf1)) and <?a, ?c> is in IEXT(I(ont:intersectionOf2)) and ?x is in ICEXT(?a). then ?x is in ICEXT(?b) and in ICEXT(?c) inverseOf: <?p, ?q> is in IEXT(I(ont:intersectionOf)) iff IEXT(?p) is the relational inverse of IEXT(?q); IEXT(?p) = { <y, x> | <x, y> in IEXT(?q) }. oneOf1/oneOf2: if <?a, ?b> is in IEXT(I(ont:oneOf1)) and <?a, ?c> is in IEXT(I(ont:oneOf2)) then ?b is in ICEXT(?a) and ICEXT(?c) is a subset of ICEXT(?a). conversely, if <?a, ?b> is in IEXT(I(ont:oneOf1)) and <?a, ?c> is in IEXT(I(ont:oneOf2)) and ?x is in ICEXT(?a) then either ?x = ?b or ?x is in ICEXT(?c). onProperty: [see hasClass, hasValue, toClass] Ontology: none. There are no constraints on interpretation specific to Ontology. Restriction: none. sameClassAs: <?s, ?o> is in IEXT(I(ont:sameClassAs)), iff ICEXT(?s) = ICEXT(?o). samePropertyAs: <?s, ?o> is in IEXT(I(samePropertyAs)), iff IEXT(?s) = IEXT(?o). toClass: if <?a, ?c> in IEXT(I(ont:toClass)) and <?a, ?p> in IEXT(I(ont:onProperty)) and <?x, ?y> in IEXT(?p) and ?x in ICEXT(?a) then ?y in ICEXT(?c). TransitiveProperty: ?p in ICEXT(I(ont:TransitiveProperty)) iff for every ?x, ?y, and ?z in IR, if <?x, ?y> in IEXT(?p) and <?y, ?z> in IEXT(?p) then <?x, ?z> in IEXT(?p). manyToOneOver: <?p, ?c> in IEXT(I(ont:manyToOneOver)) iff for every ?x, ?y, and ?z in IR, if ?x in ICEXT(?c) and <?x, ?y> in IEXT(?p) and <?x, ?z> in IEXT(?p) then ?y = ?z. ManyToOneProperty: ?p in IEXT(I(ont:ManyToOneProperty)) iff for every ?x, ?y, and ?z in IR, if <?x, ?y> in IEXT(?p) and <?x, ?z> in IEXT(?p) then ?y = ?z. manyToOneOver: <?p, ?c> in IEXT(I(ont:manyToOneOver)) iff for every ?x, ?y, and ?z in IR, if ?x in ICEXT(?c) and <?x, ?y> in IEXT(?p) and <?x, ?z> in IEXT(?p) then ?y = ?z. OneToManyProperty: ?p in IEXT(I(ont:OneToManyProperty)) iff for every ?x, ?y, and ?z in IR, if <?y, ?x> in IEXT(?p) and <?z, ?x> in IEXT(?p) then ?y = ?z. oneToManyOver: <?p, ?c> in IEXT(I(ont:oneToManyOver)) iff for every ?x, ?y, and ?z in IR, if ?x in ICEXT(?c) and <?y, ?x> in IEXT(?p) and <?z, ?x> in IEXT(?p) then ?y = ?z. unionOf1/unionOf2: if <?a, ?b> is in IEXT(I(ont:unionOf1)) and <?a, ?c> is in IEXT(I(ont:unionOf2)) and ?x is in ICEXT(?b) or in ICEXT(?c) then ?x is in ICEXT(?a). conversely, if <?a, ?b> is in IEXT(I(ont:unionOf1)) and <?a, ?c> is in IEXT(I(ont:unionOf2)) and ?x is in ICEXT(?a). then ?x is in ICEXT(?b) or in ICEXT(?c) versionInfo: none. Notes: dropped per WG decision: * cardinalityQ * hasClassQ * maxCardinalityQ * minCardinalityQ dropped re 'uniform treatment of literals' * Datatype * DatatypeProperty * DatatypeRestriction * ObjectClass * ObjectProperty * ObjectRestriction * sameIndividualAs * differentIndividualFrom (renamed differentFrom) already specified by RDF MT: * domain * Property * range * Class * subClassOf * subPropertyOf I have adopted the suggestion to drop... * disjointUnionOf cf suggestion from Mike Dean 30 May http://lists.w3.org/Archives/Public/www-webont-wg/2002May/0262.html and I got a little lazy around cardinality, cutting out * cardinality * maxCardinality * minCardinality leaving just OneToMany, ManyToOne, oneToManyOver, and ManyToOneOver. i.e. leaving just the maxcardinality 1 idioms. [RDFMT] RDF Model Theory W3C Working Draft 29 April 2002 http://www.w3.org/TR/2002/WD-rdf-mt-20020429/ esp section http://www.w3.org/TR/2002/WD-rdf-mt-20020429/#sinterp -- Dan Connolly, W3C http://www.w3.org/People/Connolly/Received on Wednesday, 19 June 2002 17:26:46 GMT

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