2008 09-06 owl full

1. OWL Full Semantics
-- RDF-Compatible Model-Theoretic Semantics
by Peter F. Patel-Schneider, Patrick Hayes and Ian Horrocks
W3C Recommendation, 2004
http://www.w3.org/TR/owl-semantics/rdfs.html
Presented by Jie Bao
RPI
Sept 11, 2008
Part 2 of RDF/OWL Semantics Tutorial
http://tw.rpi.edu/wiki/index.php/RDF_and_OWL_Semantics

2. 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 ).
2

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

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

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

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

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

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

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

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

11. 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

12. 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

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

14. 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
14

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

16. RDFS Plus Semantics
If E is then
IS(E) ∈IC and IEXT (IS (E))=IOOP
owl:ObjectProperty
⊆IEXT(IP)
IS(E) ∈IC and IEXT (IS (E))=IODP
owl:DatatypeProperty
⊆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

17. RDFS Plus Semantics
If E is ∈
then c∈ICEXT (IS (E)) iff
∈ ∈
<x,y>, <y,z>∈IEXT (c) implies <x,z>∈IEXT (c)
owl:TransitiveProperty
and c ∈IOOP
<x,y> ∈ IEXT (c) implies <y, x>∈IEXT (c)
∈
owl:SymmetricProperty
and c ∈IOOP
<x,y1>, <x,y2> ∈ IEXT (c) implies y1 = y2
owl:FuncionalProperty
and c∈IOOP ∪ IODP
∈
<x ,y>, <x2,y>∈IEXT (c) implies x1 = x2
owl:InverseFunctionalProperty 1
and c∈IOOP
If E is ∈
then <x,y>∈IEXT (IS(E)) iff
owl:inverseOf x,y∈IOOP and <u,v>∈IEXT (x) iff <v,u>∈IEXT (y)
∈ ∈
17

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

19. Inference Rules
Some examples:
If then
(?x, owl:sameAs, ?y) (?y, owl:sameAs, ?x)
(?c1, owl:equivalentClass, ?c2)
(?x, rdf:type, ?c2)
(?x, rdf:type, ?c1)
(?p, rdf:type, owl:FunctionalProperty)
(?y1, owl:sameAs, ?y2)
(?x, ?p, ?y1) T(?x, ?p, ?y2)
(?p1, owl:inverseOf, ?p2) (?x, ?p1, ?y) (?y, ?p2, ?x)
(?p, rdfs:domain, ?c) (?x, ?p, ?y) (?x, rdf:type, ?c)
Complete rule set is in backup slides
19

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

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

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

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

24. OWL Full vs OWL DL
OWL-DL OWL Full
Relation to owl:Thing <=rdfs:Resource owl:Thing = rdfs:Resource
RDFS universe owl:Class <= rdfs:Class owl:Class = rdfs:Class
P <= rdf:Property P = rdf:Property
Pairwise Yes No
Disjointness
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.
24

25. 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
25

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

27. OWL Classes and Properties
then
If E is
∈
IS (E)∈ ICEXT(IS (E))= and
owl:Class IC IOC IOC=IC
owl:Thing IOC IOT IOT=IR and IOT ≠ ∅
owl:Nothing IOC {}
∈
then if e∈ICEXT(IS
If E is Note
(E)) then
Instances of OWL classes are OWL
owl:Class ICEXT (e)⊆IOT
individuals.
Values for individual-valued
owl:ObjectProperty IEXT (e)⊆IOT×IOT
properties are OWL individuals.
27

28. Boolean Operations and Enumeration
If E is ∈
then <x,y>∈IEXT(IS (E)) iff
owl:complementOf x,y∈ IOC and ICEXT(x)=IOT-ICEXT(y)
x∈IOC and y is a sequence of y1,…yn over IOC
owl:unionOf
and ICEXT(x) = ICEXT(y1) ∪…∪ ICEXT(yn)
x∈IOC and y is a sequence of y1,…yn over IOC
owl:intersectionOf
and ICEXT(x) = ICEXT(y1) ∩…∩ ICEXT(yn)
x∈IC and y is a sequence of y1,…yn over IOT or
owl:oneOf
over ILV and ICEXT(x) = {y1,..., yn}
If E is and ∈
then if <x,l>∈IEXT(IS (E)) then
l is a sequence of y1,…yn
owl:oneOf x∈IOC
over IOT
28

29. Restriction (Anonymous Class)
then
If E is
IS(E)∈ ICEXT(IS(E))= and
owl:Restriction IC IOR IOR⊆IOC
If E is and
<x,y>∈IEXT(IS(E))) ∧
∈ then x∈IOR, y∈IOC, p∈IOOP, and ICEXT(x) =
∈ ∈ ∈
<x,p>∈IEXT(IS(owl:onProperty)))
∈
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 similarly
Note: Content on this page is simplified by omitting datatype properties
29

30. Non-equality
If 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
30

31. 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

32. 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)
32

33. Backup
33

34. 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

35. 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

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

37. RDFS Plus Rules (2)
If then
(?c1, owl:equivalentClass, ?c2)
(?x, rdf:type, ?c2)
(?x, rdf:type, ?c1)
(?c1, owl:equivalentClass, ?c2)
(?x, rdf:type, ?c1)
(?x, rdf:type, ?c2)
(?c1, rdfs:subClassOf, ?c2)
(?c1, owl:equivalentClass, ?c2)
(?c2, rdfs:subClassOf, ?c1)
(?p1, rdfs:subPropertyOf, ?p2)
(?p1, owl:equivalentProperty, ?p2)
(?p2, rdfs:subPropertyOf, ?p1)
(?p1, owl:equivalentProperty, ?p2)
(?x, ?p2, ?y)
(?x, ?p1, ?y)
(?p1, owl:equivalentProperty, ?p2)
(?x, ?p1, ?y)
(?x, ?p2, ?y)
Equality rules 37

38. RDFS Plus Rules (3)
If then
(?p, rdf:type, owl:FunctionalProperty)
(?y1, owl:sameAs, ?y2)
(?x, ?p, ?y1) T(?x, ?p, ?y2)
(?p, rdf:type, owl:InverseFunctionalProperty)
(?x1, owl:sameAs, ?x2)
(?x1, ?p, ?y) T(?x2, ?p, ?y)
(?p, rdf:type, owl:SymmetricProperty)
(?y, ?p, ?x)
(?x, ?p, ?y)
(?p, rdf:type, owl:TransitiveProperty)
(?x, ?p, ?z)
(?x, ?p, ?y) (?y, ?p, ?z)
(?p1, owl:inverseOf, ?p2) (?x, ?p1, ?y) (?y, ?p2, ?x)
(?p1, owl:inverseOf, ?p2) (?x, ?p2, ?y) (?y, ?p1, ?x)
Property characteristic rules
38

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

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

41. RDFS Plus Rules (6)
If then
(?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