OWL Full Semantics -- RDF-Compatible Model-Theoretic Semantics by Peter F. Patel-Schneider, Patrick...

OWL Full Semantics-- RDF-Compatible Model-Theoretic Semantics

by Peter F. Patel-Schneider, Patrick Hayes and Ian HorrocksW3C Recommendation, 2004

http://www.w3.org/TR/owl-semantics/rdfs.html

Presented by Jie BaoRPI

Sept 11, 2008

Part 2 of RDF/OWL Semantics Tutorialhttp://tw.rpi.edu/wiki/index.php/

RDF_and_OWL_Semantics

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 ).

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A Layer Cake of Languages

You Are

Here

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Not Covered in the Talk

• Datatype• Annotation• Ontology house keeping (e.g., imports)• OWL comprehension conditions

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Outline

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

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RDF(S) Vocabulary

RDF RDFSrdf:type rdf:Property

rdfs:domain rdfs:range rdfs:Resource rdfs:Class rdfs:subClassOf rdfs:subPropertyOf

… others (rectification, annotation, literal, collection, container)

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RDFS Interpretation

IP

IRIEXT

V

IS

IR x IR

IC

ICEXT

vocabulary

rdfs:Class

rdfs:Resource

rdf:Property

extension of classes

extension of properties7

Outline

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

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OWL Family

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From RDF to OWL 2 Full

RDF

RDFS

RDFS+

OWL FullOWL 2 RL

OWL 2 Full

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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.)

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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

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Outline


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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 7– Oracle: OWL Prime http://www.w3.org/2007/OWL/wiki/OracleOwlPrime

• Related proposals– AllegroGraph RDFS++:

http://agraph.franz.com/support/learning/Overview-of-RDFS++.lhtml– OWL 2 RL http://www.w3.org/2007/OWL/wiki/Profiles#OWL_2_RL

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RDFS Plus Vocabulary

Equality Property Characteristicsowl:equivalentClass, owl:equivalentProperty, owl:sameAs

owl:inverseOfowl:TransitiveProperty, owl:SymmetricProperty, owl:FuncionalProperty, owl:InverseFunctionalPropertyowl:ObjectProperty, owl:DatatypeProperty

+ RDFS vocabulary

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RDFS Plus Semantics

If E is then

owl:ObjectPropertyIS(E) IC and IEXT∈ (IS (E))=IOOP

IEXT(IP)⊆

owl:DatatypePropertyIS(E) IC and IEXT∈ (IS (E))=IODP

IEXT(IP)⊆

If E is then <x,y> 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:sameAs x = y 16

RDFS Plus Semantics

If E is then c ICEXT∈ (IS (E)) iff

owl:TransitiveProperty <x,y>, <y,z> IEXT∈ (c) implies <x,z> IEXT∈ (c)

and c IOOP∈

owl:SymmetricProperty<x,y> IEXT∈ (c) implies <y, x> IEXT∈ (c)

and c IOOP∈

owl:FuncionalProperty <x,y1>, <x,y2> IEXT∈ (c) implies y1 = y2

and c IOOP IODP∈ ∪

owl:InverseFunctionalProperty

<x1,y>, <x2,y> IEXT∈ (c) implies x1 = x2

and c IOOP∈

If E is then <x,y> IEXT (IS(E)) iff∈

owl:inverseOfx,y IOOP and ∈ <u,v> IEXT∈ (x) iff <v,u> IEXT∈

(y)

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RDFS Plus Semantics

Extensional Semantic Conditions

<c,d> IEXT(IS(rdfs:subClassOf)) ∈

Iff*

c, d IC,∈ICEXT(c) ICEXT(d) ⊆

<p,q> ∈IEXT(IS(rdfs:subPropertyOf))

p, q IP,∈IEXT(p) IEXT(q) ⊆

<p,c> IEXT(IS(rdfs:domain)) ∈p IP, c IC,∈ ∈<x,y> IEXT(p) → x ∈ ∈ICEXT(c)

<p,c> IEXT(IS(rdfs:range)) ∈p IP, c IC,∈ ∈<x,y> IEXT(p) → y ∈ ∈ICEXT(c) * By default, RDFS uses “only if”, OWL 1 Full and OWL 2 Full uses “iff”

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Inference Rules

Complete rule set is in backup slides

If then (?x, owl:sameAs, ?y) (?y, owl:sameAs, ?x)

(?c1, owl:equivalentClass, ?c2)

(?x, rdf:type, ?c1) (?x, rdf:type, ?c2)

(?p, rdf:type, owl:FunctionalProperty)(?x, ?p, ?y1) T(?x, ?p, ?y2)

(?y1, owl:sameAs, ?y2)

(?p1, owl:inverseOf, ?p2) (?x, ?p1, ?y) (?y, ?p2, ?x)

(?p, rdfs:domain, ?c) (?x, ?p, ?y) (?x, rdf:type, ?c)

Some examples:

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Outline


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OWL Vocabulary

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Recall: RDFS Interpretation

IP

IRIEXT

V

IS

IR x IR

IC

ICEXT

vocabulary

rdfs:Class

rdfs:Resource

rdf:Property



OWL Full Interpretation

IP

IRIEXT

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}



OWL Full vs OWL DLOWL-DL OWL Full

Relation to RDFS universe

owl:Thing <=rdfs:Resourceowl:Class <= rdfs:ClassP <= rdf:Property

owl:Thing = rdfs:Resourceowl:Class = rdfs:ClassP = rdf:Property

Pairwise Disjointness

Yes No

Decidability Yes No

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.

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True or False?In OWL Full• owl:Thing rdfs:subClassOf owl:Class• owl:Class rdfs:subClassOf owl:Thing• owl:Thing rdf:type owl:Class• owl:Class rdf:type owl:Class• rdf:Property rdf:type owl:Class

Refer: • OWL RDF Schema: http://www.w3.org/2002/07/owl • Thing and Class:

http://ontolog.cim3.net/forum/ontolog-forum/2008-09/threads.html#00004

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Outline


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OWL Classes and Properties

If E isthen

IS (E)∈ ICEXT(IS (E))= and

owl:Class IC IOC IOC=IC

owl:Thing IOC IOT IOT=IR and IOT ≠ ∅

owl:Nothing IOC {}

If E is then if e ICEXT(IS ∈

(E)) then Note

owl:Class ICEXT (e) IOT⊆Instances of OWL classes are OWL individuals.

owl:ObjectProperty IEXT (e) IOT×IOT⊆Values for individual-valued properties are OWL individuals.

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Boolean Operations and Enumeration

If E is then <x,y> IEXT(IS∈ (E)) iffowl:complementOf x,y IOC and ICEXT(x)=IOT-ICEXT(y)∈

owl:unionOfx IOC and y is a sequence of y∈ 1,…yn over IOC

and ICEXT(x) = ICEXT(y1) … ICEXT(y∪ ∪ n)

owl:intersectionOfx IOC and y is a sequence of y∈ 1,…yn over IOC

and ICEXT(x) = ICEXT(y1) ∩…∩ ICEXT(yn)

owl:oneOfx IC and y is a sequence of y∈ 1,…yn over IOT or

over ILV and ICEXT(x) = {y1,..., yn} If E is and then if <x,l> IEXT(IS∈ (E)) then

owl:oneOfl is a sequence of y1,…yn over IOT

x IOC ∈

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Restriction (Anonymous Class)If E is

thenIS(E)∈ ICEXT(IS(E))= and

owl:Restriction IC IOR IOR IOC⊆

If E is and<x,y> IEXT(IS(E))) ∈ ∧

<x,p> IEXT(IS(owl:onProperty)))∈then x IOR, y IOC, p IOOP, and ICEXT(x) =∈ ∈ ∈

owl:allValuesFrom {u IOT | <u,v> IEXT(p) implies v ICEXT(y) }∈ ∈ ∈

owl:someValuesFrom {u IOT | <u,v> IEXT(p) such that v ICEXT(y) } ∈ ∃ ∈ ∈

then x IOR, y IOT, p IOOP, and ICEXT(x) =∈ ∈ ∈

owl:hasValue {u IOT | <u, y> IEXT(p) }∈ ∈

then x IOR, y is a non-negative integer, ∈p IOOP, and ICEXT(x) =∈

owl:minCardinality {u IOT | card({v IOT : <u,v> IEXT(p)}) ≥ y }∈ ∈ ∈

owl:maxCardinality, owl:cardinality defined similarlyNote: Content on this page is simplified by omitting datatype properties

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Non-equalityIf E is then <x,y> IEXT∈ (IS(E)) iff

owl:disjointWith x,y IOC and ICEXT(x)∩ICEXT(y)={}∈

owl:differentFrom x ≠ y

More: Comprehension conditions (which require the existence of appropriate OWL descriptions and data ranges ) – not covered

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ConclusionsRDFS 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

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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)

• Turner, David; Carroll, Jeremy J. Comparing OWL Semantics. Technical Reports HPL-2007-146. HP Lab, 2007. (URL)

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Backup

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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

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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

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RDFS Plus Rules (1)

If then

(?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

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RDFS Plus Rules (2)If then





(?c1, owl:equivalentClass, ?c2) (?c1, rdfs:subClassOf, ?c2)(?c2, rdfs:subClassOf, ?c1)

(?p1, owl:equivalentProperty, ?p2) (?p1, rdfs:subPropertyOf, ?p2)(?p2, rdfs:subPropertyOf, ?p1)

(?p1, owl:equivalentProperty, ?p2) (?x, ?

p1, ?y) (?x, ?p2, ?y)

(?p1, owl:equivalentProperty, ?p2) (?x, ?

p2, ?y) (?x, ?p1, ?y)

Equality rules 37


(?p, rdf:type, owl:FunctionalProperty)(?x, ?p, ?y1) T(?x, ?p, ?y2)

(?y1, owl:sameAs, ?y2)

(?p, rdf:type, owl:InverseFunctionalProperty)(?x1, ?p, ?y) T(?x2, ?p, ?y)

(?x1, owl:sameAs, ?x2)

(?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)



Property characteristic rules

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RDFS Plus Rules (4)

If then

(?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

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RDFS Plus Rules (5)

If then

(?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 Rules40


(?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)

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OWL Full Semantics -- RDF-Compatible Model-Theoretic Semantics by Peter F. Patel-Schneider, Patrick...

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