2008 09-04

1. RDF Semantics
by Patrick Hayes
W3C Recommendation
http://www.w3.org/TR/rdf-mt/
Presented by Jie Bao
RPI
Sept 4, 2008
Part 1 of RDF/OWL Semantics Tutorial
http://tw.rpi.edu/wiki/index.php/RDF_and_OWL_Semantics

2. A Layer Cake of Languages
OWL2
OWL
(RDFS 3.0)
You
RDF(S) Are
Here

3. Outline
• What is Semantics?
• RDF: Syntax
• RDF Graph and Simple Entailment
• RDF Interpretation
• RDFS Interpretation

4. What is Semantics
Semant Inferen
Syntax Logic
ics ce
Merriam-Webster: the study of meanings
Wikipedia: the study of meaning in communication.

5. What is Semantics?
• Intensional Meaning
– TW Students are Students with affiliation to the
Tetherless World Group
• Extensional Meaning
– TW Students are the set {Jiao, Ankesh, Jesse,…}

6. Model Theory
Used to link intensional
meaning and extensional
meaning
“Model theory assumes that the
language refers to a 'world', and
Alfred Tarski describes the minimal conditions that
1901-1983 a world must satisfy in order to
Picure source: wikipedia
assign an appropriate meaning for
every expression in the language.”
--RDF Semantics

7. Model: an Example
Expression:
TW Students are Students with affiliation to the
Tetherless World Group
A Model:
…

8. A Few Jargons
• An interpretation is a world with each symbol and each
Interpretation expression assigned an extension.
• An model of a logic theory is an interpretation of the
Model theory that satisfies all constraints specified by the theory
• A logic theory is consistent if it has a model.
Consistency
• A symbol or expression x is satisfiable w.r.t. a logic theory
Satisfiability K if there is a model of K with x’s extension not empty.
• A logic theory K entails another logical theory K’ if every
Entailment model of K is a model of K’

9. Outline
• What is Semantics?
• RDF: Syntax
• RDF Graph and Simple Entailment
• RDF Interpretation
• RDFS Interpretation

10. RDF Family
RDFS RDFS Interpretation
Vocabulary
RDF
Vocabulary RDF Interpretation
RDF Graph
Simple Interpretation
Syntax Semantics

11. Not Covered in the Talk
• Blank Node (b-Node)
• Literals (Datatypes)
• Containers
• Collections
• Reification
• Annotation
• Entailment rules (rule inference)

12. RDF: Triple and Graph
• Triple: (subject, property, object)
– UB × U × UBL (Url, Blank node, Literal)
– e.g., (Jim, is-a, Professor)
– e.g., (Jim, has-surname, “Hendler”) – not covered
– e.g.,(Jim, has-pet, _:x) – not covered
is-a
Professor
Jim has-surname “Hendler”
has-pet
• Graph: A set of triples

13. Outline
• What is Semantics?
• RDF: Syntax
• RDF Graph and Simple Entailment
• RDF Interpretation
• RDFS Interpretation

14. Simple Interpretation
A simple interpretation I of a vocabulary V is defined by:
1. A non-empty set IR of resources, called the domain or universe of I.
2. A set IP, called the set of properties of I.
3. A mapping IEXT from IP into the powerset of IR x IR i.e. the set of sets of
pairs <x,y> with x and y in IR .
4. A mapping IS from URI references in V into (IR union IP)
5. A mapping IL from typed literals in V into IR.
6. A distinguished subset LV of IR, called the set of literal values, which
contains all the plain literals in V
We do not consider RDF vocabulary (e.g., rdf:type), yet.

15. Simple Interpretation
V
IS
IP IR
IEXT

16. Simple Interpretation Example
V={a, b, c}
Picture courtesy of “RDF Semantics”(Figure 1)

17. Simple Semantic Conditions
• if E is a URI reference in V then I(E) = IS(E)
• if E is a ground triple s p o. then I(E) = true if s, p and o are in
V, I(p) is in IP and <I(s),I(o)> is in IEXT(I(p)) otherwise I(E)=
false.
• if E is a ground RDF graph then I(E) = false if I(E') = false for
some triple E' in E, otherwise I(E) =true
• if E is a plain literal "aaa" in V then I(E) = aaa
• if E is a plain literal "aaa"@ttt in V then I(E) = <aaa, ttt>
• if E is a typed literal in V then I(E) = IL(E)
• If E is a blank node and A(E) is defined then [I+A](E) = A(E)
• If E is an RDF graph then I(E) = true if [I+A'](E) = true for some
mapping A' from blank(E) to IR, otherwise I(E)= false

18. Note to Simple Interpreation
• IP may not be in IR
• A property (an element in IP) and its extension
(mapping by IEXT) are separated.
– Thus avoids paradox like the barber paradox (A
barber shaves only those men who do not shave themselves.)

19. Outline
• What is Semantics?
• RDF: Syntax
• RDF Graph and Simple Entailment
• RDF Interpretation
• RDFS Interpretation

20. RDF Vocabulary (rdfV)
• rdf:type rdf:Property
• rdf:XMLLiteral rdf:nil rdf:List rdf:Statement
rdf:subject rdf:predicate rdf:object rdf:first
rdf:rest rdf:Seq rdf:Bag rdf:Alt rdf:_1 rdf:_2 ...
rdf:value

21. RDF Semantic Conditions
• x is in IP if and only if <x, I(rdf:Property)> is in
IEXT(I(rdf:type))
– Thus, RDF properties (IP) must be resources (IR) in
the universe.
– (rdf:type rdf:type rdf:Property ) is always true
• More conditions for literals

22. RDF Interpretation Example
Vocabulary: rdfV + V={a,b,c}
Picture courtesy of “RDF Semantics”(Figure 2)

23. Outline
• What is Semantics?
• RDF: Syntax
• RDF Graph and Simple Entailment
• RDF Interpretation
• RDFS Interpretation

24. RDFS Vocabulary (rdfsV)
• rdfs:domain rdfs:range rdfs:Resource
• rdfs:Class rdfs:subClassOf rdfs:subPropertyOf
• rdfs:Literal rdfs:Datatype
• rdfs:member rdfs:Container
rdfs:ContainerMembershipProperty
• rdfs:comment rdfs:seeAlso rdfs:isDefinedBy
rdfs:label

25. RDFS Semantic Conditions
On classes
• x is in ICEXT(y) if and only if <x,y> is in IEXT(I(rdf:type))
– IC = ICEXT(I(rdfs:Class))
– IR = ICEXT(I(rdfs:Resource))
– LV = ICEXT(I(rdfs:Literal))
• If x is in IC then <x, I(rdfs:Resource)> is in
IEXT(I(rdfs:subClassOf))
• If <x,y> is in IEXT(I(rdfs:subClassOf)) then x and y are in IC and
ICEXT(x) is a subset of ICEXT(y)
• IEXT(I(rdfs:subClassOf)) is transitive and reflexive on IC

26. RDFS Semantic Conditions
On properties
• If <x,y> is in IEXT(I(rdfs:domain)) and <u,v> is in
IEXT(x) then u is in ICEXT(y)
• If <x,y> is in IEXT(I(rdfs:range)) and <u,v> is in IEXT(x)
then v is in ICEXT(y)
• IEXT(I(rdfs:subPropertyOf)) is transitive and reflexive
on IP
• If <x,y> is in IEXT(I(rdfs:subPropertyOf)) then x and y
are in IP and IEXT(x) is a subset of IEXT(y)
More for container and literals

27. RDFS Axiomatic triples
Domains
• rdf:type rdfs:domain rdfs:Resource .
rdfs:domain rdfs:domain rdf:Property .
rdfs:range rdfs:domain rdf:Property .
rdfs:subPropertyOf rdfs:domain rdf:Property .
rdfs:subClassOf rdfs:domain rdfs:Class .

28. RDFS Axiomatic triples
Ranges
• rdf:type rdfs:range rdfs:Class .
rdfs:domain rdfs:range rdfs:Class .
rdfs:range rdfs:range rdfs:Class .
rdfs:subPropertyOf rdfs:range rdf:Property .
rdfs:subClassOf rdfs:range rdfs:Class .
More for container, reification, literal, and
annotation…

29. RDFS-Valid Triples
• rdfs:Resource rdf:type rdfs:Class .
• rdfs:Class rdf:type rdfs:Class .
• rdf:Property rdf:type rdfs:Class .
• rdfs:domain rdf:type rdf:Property .
rdfs:range rdf:type rdf:Property .
rdfs:subPropertyOf rdf:type rdf:Property .
rdfs:subClassOf rdf:type rdf:Property.

30. Conclusions
• Model Theory gives semantics to RDF(S)
• RDF and RDFS vocabularies pose semantic
constraints on interpretations
– RDF: type, Property
– RDFS: domain, range, Resource, Class, subClassOf
subPropertyOf
• Will see OWL 1 and OWL 2 extensions to
RDF(S) in the future

31. More on RDF Semantics
• Herman J. ter Horst - Completeness, decidability and
complexity of entailment for RDF Schema and a
semantic extension involving the OWL vocabulary. In
J. Web Sem. 3(2-3):79-115, 2005.
• Jos de Bruijn, Stijn Heymans - Logical Foundations of
(e)RDF(S): Complexity and Reasoning. In ISWC/ASWC
pp. 86-99, 2007.
• Jeff Z. Pan, Ian Horrocks - RDFS(FA) and RDF MT: Two
Semantics for RDFS. In International Semantic Web
Conference pp. 30-46, 2003.