An OWL model theory layered on RDF $Id: owlsem55.txt,v 1.2 2002/06/28 17:41:12 connolly Exp $ 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 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 | is in IEXT(I(rdf:type)) }. Each of the following terms is in the vocabulary (V) of every owl interpretation: * 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, the URI references http://www.w3.org/2001/10/daml+oil#complementOf , http://www.w3.org/2001/10/daml+oil#differentFrom , and so on are in V; 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: is in IEXT(I(ont:complementOf)), iff ICEXT(?s) is the complement of ICEXT(?o) relative to IR. differentFrom: is in IEXT(I(ont:differentFrom)), iff ?s is not equal to ?o. disjointWith: is in IEXT(I(ont:disjointWith)), iff ICEXT(?s) is disjoint from ICEXT(?o). equivalentTo: is in IEXT(I(ont:equivalentTo)), iff IEXT(?s) = IEXT(?o). onProperty/hasClass: if is in IEXT(I(ont:onProperty)) and is in IEXT(I(ont:hasClass)), then ICEXT(?r) = { ?s : in IEXT(?p) for some ?x in ICEXT(?a) } onProperty/hasValue: if is in IEXT(I(ont:onProperty)) and is in IEXT(I(ont:hasValue)), then the set ICEXT(?r) = { ?s: in IEXT(?p) } imports: none. imports doesn't constrain interpretations (other than having the subproperty relationship with rdfs:seeAlso). intersectionOf1/intersectionOf2: if is in IEXT(I(ont:intersectionOf1)) and is in IEXT(I(ont:intersectionOf2)) and ?x is in ICEXT(?b) and in ICEXT(?c) then ?x is in ICEXT(?a). conversely, if is in IEXT(I(ont:intersectionOf1)) and is in IEXT(I(ont:intersectionOf2)) and ?x is in ICEXT(?a). then ?x is in ICEXT(?b) and in ICEXT(?c) inverseOf: is in IEXT(I(ont:intersectionOf)) iff IEXT(?p) is the relational inverse of IEXT(?q); IEXT(?p) = { | in IEXT(?q) }. oneOf1/oneOf2: if is in IEXT(I(ont:oneOf1)) and is in IEXT(I(ont:oneOf2)) then ?b is in ICEXT(?a) and ICEXT(?c) is a subset of ICEXT(?a). conversely, if is in IEXT(I(ont:oneOf1)) and 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: is in IEXT(I(ont:sameClassAs)), iff ICEXT(?s) = ICEXT(?o). samePropertyAs: is in IEXT(I(samePropertyAs)), iff IEXT(?s) = IEXT(?o). toClass: if in IEXT(I(ont:toClass)) and in IEXT(I(ont:onProperty)) and 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 in IEXT(?p) and in IEXT(?p) then in IEXT(?p). manyToOneOver: in IEXT(I(ont:manyToOneOver)) iff for every ?x, ?y, and ?z in IR, if ?x in ICEXT(?c) and in IEXT(?p) and in IEXT(?p) then ?y = ?z. ManyToOneProperty: ?p in IEXT(I(ont:ManyToOneProperty)) iff for every ?x, ?y, and ?z in IR, if in IEXT(?p) and in IEXT(?p) then ?y = ?z. manyToOneOver: in IEXT(I(ont:manyToOneOver)) iff for every ?x, ?y, and ?z in IR, if ?x in ICEXT(?c) and in IEXT(?p) and in IEXT(?p) then ?y = ?z. OneToManyProperty: ?p in IEXT(I(ont:OneToManyProperty)) iff for every ?x, ?y, and ?z in IR, if in IEXT(?p) and in IEXT(?p) then ?y = ?z. oneToManyOver: in IEXT(I(ont:oneToManyOver)) iff for every ?x, ?y, and ?z in IR, if ?x in ICEXT(?c) and in IEXT(?p) and in IEXT(?p) then ?y = ?z. unionOf1/unionOf2: if is in IEXT(I(ont:unionOf1)) and is in IEXT(I(ont:unionOf2)) and ?x is in ICEXT(?b) or in ICEXT(?c) then ?x is in ICEXT(?a). conversely, if is in IEXT(I(ont:unionOf1)) and 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. This model theory takes a fairly conservative position on "5.10 DAML+OIL semantics is too weak" http://www.w3.org/2001/sw/WebOnt/webont-issues.html#I5.10-DAML-OIL-semantics-is-too-weak cf # DDTF/layering: weak class theory seems good enough (5.3, 5.10) Dan Connolly (Tue, May 28 2002) http://lists.w3.org/Archives/Public/www-webont-wg/2002May/0235.html This model theory doesn't address issue 5.6 daml:imports as magic syntax http://www.w3.org/2001/sw/WebOnt/webont-issues.html#I5.6-daml:imports-as-magic-syntax Jos DeRoo reported some initial successful implementation experience: From: Jos De_Roo (jos.deroo.jd@belgium.agfa.com) Date: Thu, Jun 20 2002 http://lists.w3.org/Archives/Public/www-webont-wg/2002Jun/0171.html [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