OWL Full Semantics -- RDF-Compatible Model-Theoretic Semantics by Peter F. Patel-Schneider, Patrick Hayes and Ian Horrocks W3C Recommendation, Presented by Jie Bao RPI Sept 11, 2008 Part 2 of RDF/OWL Semantics Tutorial

Disclaimer The semantics and inference rules about RDFS Plus /RDFS 3.0 are rolely Jie Bao’s own and do not reflect the positions of either W3C (or any of its working group) or any of the RDFS Plus /RDF 3.0 proposals (citation on the page RDFS Plus: a Rule Subset of OWL ).RDFS Plus: a Rule Subset of OWL 2

A Layer Cake of Languages OWL2 OWL (RDFS Plus) RDF(S) You Are Here 3

Not Covered in the Talk Datatype Annotation Ontology house keeping (e.g., imports) OWL comprehension conditions 4

Outline Review of RDF Semantics OWL Overview RDFS 3.0 Semantics OWL Full Universe OWL Full Interpretation Conditions 5

RDF(S) Vocabulary RDFRDFS rdf:type rdf:Property rdfs:domain rdfs:range rdfs:Resource rdfs:Class rdfs:subClassOf rdfs:subPropertyOf … others (rectification, annotation, literal, collection, container) 6

RDFS Interpretation IP IR IEXT V IS IR x IR IC ICEXT vocabulary rdfs:Class rdfs:Resource rdf:Property extension of classes extension of properties 7

Outline Review of RDF Semantics OWL Overview RDFS Plus Semantics OWL Full Universe OWL Full Interpretation Conditions 8

OWL Family OWL Full OWL DL (SHOIN(D)) OWL Lite (SHIF(D)) RDFS Plus (or RDFS 3.0) 9

From RDF to OWL 2 Full RDF RDFS RDFS+ OWL Full OWL 2 RL OWL 2 Full Covered next time 10

OWL Extensions to RDFS Constructing classes: – e.g.,     Constructing properties: – e.g., inverseOf Property characteristics: – e.g., transitive, functional, symmetric Mapping – Equality, non-equality (between classes, properties, ind.) 11

Direct MT Sem. vs RDF MT Sem. Direct Model-Theoretical Semantics – For OWL DL (thus also OWL Lite) – Simpler than the RDF MT Semantics – Corresponds to the semantics of DL SHOIN(D) – Decidability guaranteed RDF-Compatible Model-Theoretical Semantics – For OWL Full (thus also OWL DL and OWL Lite) – Extends RDFS Semantics 12

Outline Review of RDF Semantics OWL Overview RDFS Plus Semantics OWL Full Universe OWL Full Interpretation Conditions 13

RDFS Plus: a Rule Subset of OWL Design intuition: Scalable, easier to implement using rule inference RDFS Plus / OWL Prime / RDFS 3.0 – Dean Allemang, James Hendler. Semantic Web for the Working Ontologist, Chapter 7Semantic Web for the Working Ontologist – Oracle: OWL Prime Related proposals – AllegroGraph RDFS++: – OWL 2 RL

RDFS Plus Vocabulary EqualityProperty Characteristics owl:equivalentClass, owl:equivalentProperty, owl:sameAs owl:inverseOf owl:TransitiveProperty, owl:SymmetricProperty, owl:FuncionalProperty, owl:InverseFunctionalProperty owl:ObjectProperty, owl:DatatypeProperty + RDFS vocabulary 15

RDFS Plus Semantics If E isthen owl:ObjectProperty IS(E) ∈ IC and IEXT (IS (E))=IOOP ⊆ IEXT(IP) owl:DatatypeProperty IS(E) ∈ IC and IEXT (IS (E))=IODP ⊆ IEXT(IP) If E is then ∈ IEXT (IS (E)) iff owl:equivalentClass x,y ∈ IC and ICEXT(x)=ICEXT(y) owl:equivalentProperty x,y ∈ IOOP ∪ IODP and IEXT (x) = IEXT (y) owl:sameAsx = y 16

RDFS Plus Semantics If E is then c ∈ ICEXT (IS (E)) iff owl:TransitiveProperty, ∈ IEXT (c) implies ∈ IEXT (c) and c ∈ IOOP owl:SymmetricProperty ∈ IEXT (c) implies ∈ IEXT (c) and c ∈ IOOP owl:FuncionalProperty, ∈ IEXT (c) implies y 1 = y 2 and c ∈ IOOP ∪ IODP owl:InverseFunctionalProperty, ∈ IEXT (c) implies x 1 = x 2 and c ∈ IOOP If E is then ∈ IEXT (IS(E)) iff owl:inverseOf x,y ∈ IOOP and ∈ IEXT (x) iff ∈ IEXT (y) 17

RDFS Plus Semantics Extensional Semantic Conditions ∈ IEXT(IS(rdfs:subClassOf)) Iff* c, d ∈ IC, ICEXT(c) ⊆ ICEXT(d) ∈ IEXT(IS(rdfs:subPropertyOf)) p, q ∈ IP, IEXT(p) ⊆ IEXT(q) ∈ IEXT(IS(rdfs:domain)) p ∈ IP, c ∈ IC, ∈ IEXT(p) → x ∈ ICEXT(c) ∈ IEXT(IS(rdfs:range)) p ∈ IP, c ∈ IC, ∈ IEXT(p) → y ∈ ICEXT(c) * By default, RDFS uses “only if”, OWL 1 Full and OWL 2 Full uses “iff” 18

Inference Rules Complete rule set is in backup slides Ifthen (?x, owl:sameAs, ?y)(?y, owl:sameAs, ?x) (?c 1, owl:equivalentClass, ?c 2 ) (?x, rdf:type, ?c 1 ) (?x, rdf:type, ?c 2 ) (?p, rdf:type, owl:FunctionalProperty) (?x, ?p, ?y 1 ) T(?x, ?p, ?y 2 ) (?y 1, owl:sameAs, ?y 2 ) (?p 1, owl:inverseOf, ?p 2 ) (?x, ?p 1, ?y)(?y, ?p 2, ?x) (?p, rdfs:domain, ?c) (?x, ?p, ?y)(?x, rdf:type, ?c) Some examples: 19

Outline Review of RDF Semantics OWL Overview RDFS Plus Semantics OWL Full Universe OWL Full Interpretation Conditions 20

OWL Vocabulary ClassesClass Construction owl:Class owl:Thing owl:Nothing owl:complementOf owl:intersectionOf owl:unionOf Boolean owl:Restriction owl:onProperty owl:allValuesFrom owl:someValuesFrom owl:hasValue qualification Non-equality owl:differentFrom owl:disjointWith owl:AllDifferent owl:distinctMembers owl:cardinality owl:minCardinality owl:maxCardinality cardinality owl:oneOf + RDFS Plus vocabulary 21

Recall: RDFS Interpretation IP IR IEXT V IS IR x IR IC ICEXT vocabulary rdfs:Class rdfs:Resource rdf:Property extension of classes extension of properties 22

OWL Full Interpretation IP IR IEXT V IS IR x IR IC ICEXT vocabulary rdfs:Class =owl:Class rdfs:Resource =owl:Thing rdf:Property = {owl:ObjectProperty, owl:DatatypeProperty, owl:AnnotationProperty, owl:OntologyProperty} extension of classes extension of properties 23

OWL Full vs OWL DL OWL-DLOWL Full Relation to RDFS universe owl:Thing <=rdfs:Resource owl:Class <= rdfs:Class P <= rdf:Property owl:Thing = rdfs:Resource owl:Class = rdfs:Class P = rdf:Property Pairwise Disjointness YesNo DecidabilityYesNo P is the union of owl:ObjectProperty, owl:DatatypeProperty, owl:AnnotationProperty, and owl:OntologyProperty Note: in OWL Full, an element can be an individual (owl:Thing element), a class (owl:Class element) and an property (P element) at the same time. 24

True or False? In OWL Full owl:Thing rdfs:subClassOfowl:Class owl:Classrdfs:subClassOf owl:Thing owl:Thingrdf:type owl:Class owl:Classrdf:typeowl:Class rdf:Propertyrdf:type owl:Class Refer: OWL RDF Schema: Thing and Class: 09/threads.html#00004http://ontolog.cim3.net/forum/ontolog-forum/ /threads.html#

Outline Review of RDF Semantics OWL Overview RDFS Plus Semantics OWL Full Universe OWL Full Interpretation Conditions 26

OWL Classes and Properties If E is then IS (E) ∈ ICEXT(IS (E))=and owl:ClassICIOCIOC=IC owl:ThingIOCIOT IOT=IR and IOT ≠ ∅ owl:NothingIOC{} If E is then if e ∈ ICEXT(IS (E)) then Note owl:Class ICEXT (e) ⊆ IOT Instances of OWL classes are OWL individuals. owl:ObjectPropertyIEXT (e) ⊆ IOT×IOT Values for individual-valued properties are OWL individuals. 27

Boolean Operations and Enumeration If E is then ∈ IEXT(IS (E)) iff owl:complementOf x,y ∈ IOC and ICEXT(x)=IOT-ICEXT(y) owl:unionOf x ∈ IOC and y is a sequence of y 1,…y n over IOC and ICEXT(x) = ICEXT(y 1 ) ∪ … ∪ ICEXT(y n ) owl:intersectionOf x ∈ IOC and y is a sequence of y 1,…y n over IOC and ICEXT(x) = ICEXT(y 1 ) ∩…∩ ICEXT(y n ) owl:oneOf x ∈ IC and y is a sequence of y 1,…y n over IOT or over ILV and ICEXT(x) = {y 1,..., y n } If E isand then if ∈ IEXT(IS (E)) then owl:oneOf l is a sequence of y 1,…y n over IOT x ∈ IOC 28

Restriction (Anonymous Class) If E is then IS(E) ∈ ICEXT(IS(E))=and owl:RestrictionICIORIOR ⊆ IOC If E is and ∈ IEXT(IS(E))) ∧ ∈ IEXT(IS(owl:onProperty))) then x ∈ IOR, y ∈ IOC, p ∈ IOOP, and ICEXT(x) = owl:allValuesFrom {u ∈ IOT | ∈ IEXT(p) implies v ∈ ICEXT(y) } owl:someValuesFrom {u ∈ IOT | ∃ ∈ IEXT(p) such that v ∈ ICEXT(y) } then x ∈ IOR, y ∈ IOT, p ∈ IOOP, and ICEXT(x) = owl:hasValue {u ∈ IOT | ∈ IEXT(p) } then x ∈ IOR, y is a non-negative integer, p ∈ IOOP, and ICEXT(x) = owl:minCardinality {u ∈ IOT | card({v ∈ IOT : ∈ IEXT(p)}) ≥ y } owl:maxCardinality, owl:cardinality defined similarly Note: Content on this page is simplified by omitting datatype properties 29

Non-equality If E is then ∈ IEXT (IS(E)) iff owl:disjointWith x,y ∈ IOC and ICEXT(x)∩ICEXT(y)={} owl:differentFromx ≠ y More: Comprehension conditions (which require the existence of appropriate OWL descriptions and data ranges ) – not covered 30

Conclusions RDFS Plus A scalable rule subset of OWL Full, with MT semantics Equality + Property Characteristics Has extensional semantic conditions (while RDFS has not) OWL Full Extends RDFS Plus, with MT semantics OWL Full universe = RDFS universe – rdfs:Class = owl:Class ; rdfs:Resource = owl:Thing; owl:ObjectProperty <= rdf:Property No distinction between classes, properties and individuals Next talk: OWL 2Full 31

Further Reading Ian Horrocks, Peter F. Patel-Schneider, Frank van Harmelen - From SHIQ and RDF to OWL: the making of a Web Ontology Language. In J. Web Sem. 1(1):7-26, 2003.(URL)URL Turner, David; Carroll, Jeremy J. Comparing OWL Semantics. Technical Reports HPL HP Lab, (URL)URL 32

Backup 33

Other OWL Vocabulary owl:DatatypeProperty, owl:DataRange owl:Ontology owl:imports, owl:priorVersion, owl:backwardCompatibleWith, and owl:incompatibleWith, owl:versionInfo owl:OntologyProperty owl:DeprecatedClass, owl:DeprecatedProperty owl:AnnotationProperty 34

Exercise Prove tautology in RDFS: – rdfs:subPropertyOf rdfs:subPropertyOf rdfs:subPropertyOf – rdfs:domain rdfs:domain rdf:Property – rdfs:doman rdfs:range rdf:Class – rdf:Property rdf:type rdfs:Class Prove tautology in OWL Full: – owl:sameAs owl:sameAs owl:sameAs 35

RDFS Plus Rules (1) Ifthen (?s, ?p, ?o) (?s, owl:sameAs, ?s) (?p, owl:sameAs, ?p) (?o, owl:sameAs, ?o) (?x, owl:sameAs, ?y)(?y, owl:sameAs, ?x) (?x, owl:sameAs, ?y) (?y, owl:sameAs, ?z) (?x, owl:sameAs, ?z) (?s, owl:sameAs, ?s‘) (?s, ?p, ?o)(?s', ?p, ?o) (?p, owl:sameAs, ?p‘) (?s, ?p, ?o)(?s, ?p', ?o) (?o, owl:sameAs, ?o‘) (?s, ?p, ?o)(?s, ?p, ?o') d Equality rules 36

RDFS Plus Rules (2) Ifthen (?c 1, owl:equivalentClass, ?c 2 ) (?x, rdf:type, ?c 1 ) (?x, rdf:type, ?c 2 ) (?c 1, owl:equivalentClass, ?c 2 ) (?x, rdf:type, ?c 2 ) (?x, rdf:type, ?c 1 ) (?c1, owl:equivalentClass, ?c2) (?c1, rdfs:subClassOf, ?c2) (?c2, rdfs:subClassOf, ?c1) (?p1, owl:equivalentProperty, ?p2) (?p1, rdfs:subPropertyOf, ?p2) (?p2, rdfs:subPropertyOf, ?p1) (?p 1, owl:equivalentProperty, ?p 2 ) (?x, ?p 1, ?y) (?x, ?p 2, ?y) (?p 1, owl:equivalentProperty, ?p 2 ) (?x, ?p 2, ?y) (?x, ?p 1, ?y) Equality rules 37

RDFS Plus Rules (3) Ifthen (?p, rdf:type, owl:FunctionalProperty) (?x, ?p, ?y 1 ) T(?x, ?p, ?y 2 ) (?y 1, owl:sameAs, ?y 2 ) (?p, rdf:type, owl:InverseFunctionalProperty) (?x 1, ?p, ?y) T(?x 2, ?p, ?y) (?x 1, owl:sameAs, ?x 2 ) (?p, rdf:type, owl:SymmetricProperty) (?x, ?p, ?y) (?y, ?p, ?x) (?p, rdf:type, owl:TransitiveProperty) (?x, ?p, ?y) (?y, ?p, ?z) (?x, ?p, ?z) (?p 1, owl:inverseOf, ?p 2 ) (?x, ?p 1, ?y)(?y, ?p 2, ?x) (?p 1, owl:inverseOf, ?p 2 ) (?x, ?p 2, ?y)(?y, ?p 1, ?x) Property characteristic rules 38

RDFS Plus Rules (4) Ifthen (?c, rdf:type, owl:Class) (?c, rdfs:subClassOf, ?c) (?c, owl:equivalentClasses, ?c) (?p, rdf:type, owl:ObjectProperty) (?p, rdfs:subPropertyOf, ?p) (?p, owl:equivalentProperty, ?p) (?p, rdf:type, owl:DatatypeProperty) (?p, rdfs:subPropertyOf, ?p) (?p, owl:equivalentProperty, ?p) OWL Class and Property Declaration 39

RDFS Plus Rules (5) Ifthen (?x, ?p, ?y) (?p, rdf:type rdf:Property) (?x, rdf:type rdfs:Resource) (?y, rdf:type rdfs:Resource) (?p, rdf:type rdf:Property)(?p, rdfs:subPropertyOf ?p) (?c, rdf:type rdfs:Class) (?c, rdfs:subClassOf rdfs:Resource) (?c, rdfs:subClassOf ?c) (?p1, rdfs:subPropertyOf, ?p2) (?x, ?p1, ?y)(?x, ?p2, ?y) (?c1, rdfs:subClassOf, ?c2) (?x, rdf:type, ?c1)(?x, rdf:type, ?c2) (?c1, rdfs:subClassOf, ?c2) (?c2, rdfs:subClassOf, ?c3) (?c1, rdfs:subClassOf, ?c3) (?p1, rdfs:subPropertyOf, ?p2) (?p2, rdfs:subPropertyOf, ?p3) (?p1, rdfs:subPropertyOf, ?p3) RDFS Rules 40

RDFS Plus Rules (6) Ifthen (?p, rdfs:domain, ?c) (?x, ?p, ?y)(?x, rdf:type, ?c) (?p, rdfs:range, ?c) (?x, ?p, ?y)(?y, rdf:type, ?c) Rules due to Extensional Semantic Conditions (?p, rdfs:domain, ?c1) (?c1, rdfs:subClassOf, ?c2)(?p, rdfs:domain, ?c2) (?p2, rdfs:domain, ?c) (?p1, rdfs:subPropertyOf, ?p2)(?p1, rdfs:domain, ?c) (?p, rdfs:range, ?c1) (?c1, rdfs:subClassOf, ?c2)(?p, rdfs:range, ?c2) (?p2, rdfs:range, ?c) (?p1, rdfs:subPropertyOf, ?p2)(?p1, rdfs:range, ?c) RDFS Rules (domain & range) 41