An Axiomatic Semantics for RDF, RDF-Schema, and DAML+OIL Richard Fikes Deborah McGuinness

1. An Axiomatic Semantics for RDF, RDF-Schema, and DAML+OIL Richard Fikes Deborah McGuinness Knowledge Systems Laboratory Stanford University www.ksl.stanford.edu 2/12/01

2. What’s Inferable From Semantic Markup • Formal specification of intended meaning of semantic markup • RDF, RDF Schema, and DAML+OIL • Translation of semantic markup into first order logic (FOL) • Produces a logically equivalent ontology in FOL • FOL language is KIF (Knowledge Interchange Format) • Provides axioms that specify legal inferences and constraints [137] • Facilitates query answering and constraint checking • By traditional theorem provers and problem solvers • Provides specs for special purpose reasoners • Provides basis for discussing language changes and extensions

3. Translation Method • Translate markup into RDF statements I.e., into “Property P of resource R has value V” E.g., “Property parent of resource Joe has value John” • Translate each RDF statement into a FOL sentence I.e., into “(PropertyValue P R V)” E.g., “(PropertyValue parent Joe John)” • Simplify property typing sentences using relation type I.e., simplify “(PropertyValue type R V)” to “(type R V)” E.g., simplify “(PropertyValue type Joe Person)” to “(type Joe Person)” • Add axioms from the semantics document

4. RDF Classes and Properties Properties [6] (12 axioms) type subject predicate object value _1, _2, _3, … Classes [10] (18 axioms) Resource Property Class Literal Statement Container Bag Seq Alt ContainerMembershipProperty

5. RDF Schema Classes and Properties Classes [2] (2 axioms) ConstraintResource ConstraintProperty Properties [8] (18 axioms) subClassOf subPropertyOf seeAlso isDefinedBy comment label range domain

6. DAML+OIL Classes Thing Nothing Disjoint Restriction NonNegativeInteger TransitiveProperty UniqueProperty UnambiguousProperty List Ontology (10 classes; 14 axioms)

7. DAML+OIL Properties equivalentTo sameClassAs samePropertyAs disjointWith unionOf disjointUnionOf intersectionOf complementOf oneOf onProperty toClass hasValue hasClass minCardinality maxCardinality cardinality hasClassQ minCardinalityQ maxCardinalityQ cardinalityQ inverseOf first rest item versionInfo imports (26 properties; 69 axioms)

8. Class Person and Person Joe • Persons are animals and a person’s parents are persons <rdfs:Class rdf:ID = "Person"> <rdfs:subClassOf rdf:resource = "#Animal” /> <restrictedBy> <Restriction> <onProperty resource = "#parent” /> <toClass resource = "#Person” /> </Restriction> </restrictedBy> </Class> • Joe is a person one of whose parents is John <Person ID = "Joe"> <parent resource = "#John” /> </Person>

9. Translation Into RDF Statements • Persons are animals (type Person Class) (subClassOf Person Animal) • A person’s parents are persons (type R Restriction) (restrictedBy Person R) (onProperty R parent) (toClass R Person) • Joe is a person one of whose parents is John (type Joe Person) (parent Joe John)

10. Translation Into First Order Logic (type Person Class)  (Type Person Class) (subClassOf Person Animal)  (PropertyValue subClassOf Person Animal) (type R Restriction)  (Type R Restriction) (restrictedBy Person R)  (PropertyValue restrictedBy Person R) (onProperty R parent)  (PropertyValue onProperty R parent) (toClass R Person)  (PropertyValue toClass R Person) (type Joe Person)  (Type Joe Person) (parent Joe John)  (PropertyValue Parent Joe John)

11. Primary Axiom For toClass • If object R is a value of restrictedBy for object C1, and object P is a value of onProperty for R, and object C2 is a value of toClass for R, then for all objects I and V, if I is of type C1 and V is a value of P for I, then V is type C2.” • (=> (and (PropertyValue restrictedBy ?c1 ?r) (PropertyValue onProperty ?r ?p) (PropertyValue toClass ?r ?c2)) (forall (?i ?v) (=> (and (Type ?i ?c1) (PropertyValue ?p ?i ?v)) (Type ?v ?c2))))

12. Is John a Person? • From – • (PropertyValue restrictedBy Person R) • (PropertyValue onProperty R parent) • (PropertyValue toClass R Person) • (=> (and (PropertyValue restrictedBy ?c1 ?r) (PropertyValue onProperty ?r ?p) (PropertyValue toClass ?r ?c2)) (forall (?i ?v) (=> (and (Type ?i ?c1) (PropertyValue ?p ?i ?v)) (Type ?v ?c2)))) • Infer – • (forall (?i ?v) (=> (and (Type ?i Person) (PropertyValue parent ?i ?v)) (Type ?v Person)))

13. Is John a Person? • From – • (Type Joe Person) • (PropertyValue Parent Joe John) • (forall (?i ?v) (=> (and (Type ?i Person) (PropertyValue parent ?i ?v)) (Type ?v Person))) • Infer – • (Type John Person) • “Yes”, John is a Person.

14. Summary • Formal specification of intended meaning of semantic markup • RDF, RDF Schema, and DAML+OIL • Translation of semantic markup into first order logic (FOL) • Produces a logically equivalent ontology in KIF • Provides axioms that specify legal inferences and constraints • Facilitates query answering and constraint checking • By traditional theorem provers and problem solvers • Provides specs for special purpose reasoners • Provides basis for discussing language changes and extensions