An Axiomatic Semantics for RDF RDFSchema and DAMLOIL

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

What’s Inferable From Semantic Markup u Formal specification of intended meaning of semantic markup 4 RDF, RDF Schema, and DAML+OIL u Translation of semantic markup into first order logic (FOL) 4 Produces a logically equivalent ontology in FOL 4 FOL language is KIF (Knowledge Interchange Format) u Provides axioms that specify legal inferences and constraints [137] u Facilitates query answering and constraint checking 4 By traditional theorem provers and problem solvers 4 Provides specs for special purpose reasoners u 2 Provides basis for discussing language changes and extensions Knowledge Systems Laboratory, Stanford University

Translation Method 4 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” 4 Translate each RDF statement into a FOL sentence I. e. , into “(Property. Value P R V)” E. g. , “(Property. Value parent Joe John)” 4 Simplify property typing sentences using relation type I. e. , simplify “(Property. Value type R V)” to “(type R V)” E. g. , simplify “(Property. Value type Joe Person)” to “(type Joe Person)” 4 3 Add axioms from the semantics document Knowledge Systems Laboratory, Stanford University

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

RDF Schema Classes and Properties Classes [2] (2 axioms) Constraint. Resource Constraint. Property Properties [8] (18 axioms) sub. Class. Of sub. Property. Of see. Also is. Defined. By comment label range domain 5 Knowledge Systems Laboratory, Stanford University

DAML+OIL Classes Thing Nothing Disjoint Restriction Non. Negative. Integer Transitive. Property Unique. Property Unambiguous. Property List Ontology (10 classes; 14 axioms) 6 Knowledge Systems Laboratory, Stanford University

DAML+OIL Properties equivalent. To on. Property max. Cardinality. Q same. Class. As to. Class cardinality. Q same. Property. As has. Value inverse. Of disjoint. With has. Class first union. Of min. Cardinality rest disjoint. Union. Of max. Cardinality item intersection. Of cardinality version. Info complement. Of has. Class. Q imports one. Of min. Cardinality. Q (26 properties; 69 axioms) 7 Knowledge Systems Laboratory, Stanford University

Class Person and Person Joe u Persons are animals and a person’s parents are persons <rdfs: Class rdf: ID = "Person"> <rdfs: sub. Class. Of rdf: resource = "#Animal” /> <restricted. By> <Restriction> <on. Property resource = "#parent” /> <to. Class resource = "#Person” /> </Restriction> </restricted. By> </Class> u Joe is a person one of whose parents is John <Person ID = "Joe"> <parent resource = "#John” /> </Person> 8 Knowledge Systems Laboratory, Stanford University

Translation Into RDF Statements u Persons are animals (type Person Class) (sub. Class. Of Person Animal) u A person’s parents are persons (type R Restriction) (restricted. By Person R) (on. Property R parent) (to. Class R Person) u Joe is a person one of whose parents is John (type Joe Person) (parent Joe John) 9 Knowledge Systems Laboratory, Stanford University

Translation Into First Order Logic (type Person Class) (Type Person Class) (sub. Class. Of Person Animal) (Property. Value sub. Class. Of Person Animal) (type R Restriction) (Type R Restriction) (restricted. By Person R) (Property. Value restricted. By Person R) (on. Property R parent) (Property. Value on. Property R parent) (to. Class R Person) (Property. Value to. Class R Person) (type Joe Person) (Type Joe Person) (parent Joe John) (Property. Value Parent Joe John) 10 Knowledge Systems Laboratory, Stanford University

Primary Axiom For to. Class u If object R is a value of restricted. By for object C 1, and object P is a value of on. Property for R, and object C 2 is a value of to. Class for R, then for all objects I and V, if I is of type C 1 and V is a value of P for I, then V is type C 2. ” u (=> (and (Property. Value restricted. By ? c 1 ? r) (Property. Value on. Property ? r ? p) (Property. Value to. Class ? r ? c 2)) (forall (? i ? v) (=> (and (Type ? i ? c 1) (Property. Value ? p ? i ? v)) (Type ? v ? c 2)))) 11 Knowledge Systems Laboratory, Stanford University

Is John a Person? u From – 4 (Property. Value restricted. By Person R) 4 (Property. Value on. Property R parent) 4 (Property. Value to. Class R Person) 4 (=> (and (Property. Value restricted. By ? c 1 ? r) (Property. Value on. Property ? r ? p) (Property. Value to. Class ? r ? c 2)) (forall (? i ? v) (=> (and (Type ? i ? c 1) (Property. Value ? p ? i ? v)) (Type ? v ? c 2)))) u Infer – 4 (forall (? i ? v) (=> (and (Type ? i Person) (Property. Value parent ? i ? v)) (Type ? v Person))) 12 Knowledge Systems Laboratory, Stanford University

Is John a Person? u From – 4 (Type Joe Person) 4 (Property. Value Parent Joe John) 4 (forall (? i ? v) (=> (and (Type ? i Person) (Property. Value parent ? i ? v)) (Type ? v Person))) u Infer – 4 (Type John Person) u 13 “Yes”, John is a Person. Knowledge Systems Laboratory, Stanford University

Summary u Formal specification of intended meaning of semantic markup 4 RDF, RDF Schema, and DAML+OIL u Translation of semantic markup into first order logic (FOL) 4 Produces a logically equivalent ontology in KIF u Provides axioms that specify legal inferences and constraints u Facilitates query answering and constraint checking 4 By traditional theorem provers and problem solvers 4 Provides specs for special purpose reasoners u 14 Provides basis for discussing language changes and extensions Knowledge Systems Laboratory, Stanford University