Introduction to OWL 2 Knowledge Technologies
Knowledge Technologies
Manolis Koubarakis
1
A
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2
Knowledge Technologies
Manolis Koubarakis
2
Acknowledgement
•
This presentation is based on the OWL 2
Web Ontology Language Structural
Specification and Functional-Style Syntax
available at
http://www.w3.org/TR/owl2-
syntax/
•
Much of the material in this presentation is
verbatim from the above specification.
Knowledge Technologies
Manolis Koubarakis
3
Outline
•
Features of OWL 2
•
Structural Specification
•
Functional Syntax
•
Other Syntaxes
•
Examples
•
Semantics of OWL 2
•
OWL 2 Profiles
Knowledge Technologies
Manolis Koubarakis
4
The Semantic Web “Layer Cake”
Knowledge Technologies
Manolis Koubarakis
5
OWL 2 Basics
•
OWL 2 is the current version of the
W
e
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a
W3C recommendation as of
October 2009.
•
The previous version of OWL (OWL 1) became
a W3C recommendation in 2004.
•
All W3C documents about OWL 2 can be found
at
http://www.w3.org/TR/2009/REC-owl2-
overview-20091027/
.
Knowledge Technologies
Manolis Koubarakis
6
The Structure of OWL 2
Knowledge Technologies
Manolis Koubarakis
7
OWL 2 Basics (cont’d)
•
OWL 2 is a language for writing ontologies
for the Web.
•
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Knowledge Technologies
Manolis Koubarakis
8
OWL 2 Terminology
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Knowledge Technologies
Manolis Koubarakis
9
OWL 2 Terminology (cont’d)
•
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Knowledge Technologies
Manolis Koubarakis
10
Annotations
•
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E
xample
:
A
class can be given a human-readable label
that provides a more descriptive name for the class.
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Knowledge Technologies
Manolis Koubarakis
11
IRIs
•
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Knowledge Technologies
Manolis Koubarakis
12
The Structure of an Ontology
Knowledge Technologies
Manolis Koubarakis
13
Ontology IRI and Version IRIs
•
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•
Example:
–
Ontology IRI:
<http://www.example.com/my>
–
Version IRIs:
<
http://www.example.com/my
/1.0>,
<
http://www.example.com/my
/2.0>,
…
•
An ontology without an ontology IRI
must not
contain a version IRI.
•
Ontology IRIs and version IRIs should satisfy various uniqueness constraints that
OWL 2 tools should check, for detecting possible problems.
Knowledge Technologies
Manolis Koubarakis
14
Ontology Document
•
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Example: The document of the current version of an
ontology should always be accessible via the
ontology IRI and the current version IRI.
Knowledge Technologies
Manolis Koubarakis
15
Imports
•
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Knowledge Technologies
Manolis Koubarakis
16
OWL 2 Syntaxes
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s
e
r
s
t
o
r
e
a
d
.
•
T
h
e
O
W
L
X
M
L
s
y
n
t
a
x
:
t
h
i
s
i
s
a
n
X
M
L
s
y
n
t
a
x
f
o
r
O
W
L
d
e
f
i
n
e
d
b
y
a
n
X
M
L
s
c
h
e
m
a
.
•
OWL Syntax Converter:
•
http://owl.cs.manchester.ac.uk/tools/webapps/owl-syntax-converter/
Knowledge Technologies
Manolis Koubarakis
17
BNF Grammar for the Functional
Syntax of OWL 2
ontologyDocument := { prefixDeclaration } OntologyprefixDeclaration :=
'Prefix' '('prefixName
‘
:
=' fullIRI
')'
Ontology :=
'Ontology' '('[ ontologyIRI [ versionIRI ] ]
directlyImportsDocuments
ontologyAnnotations
axioms
')'
ontologyIRI := IRI
versionIRI := IRI
directlyImportsDocuments := { 'Import' '('IRI
')'
}
axioms := { Axiom }Knowledge Technologies
Manolis Koubarakis
18
Example
Prefix(
ex
:=<http://www.example.com/ontology1#>)
Prefix(owl:=<http://www.w3.org/2002/07/owl#>)
Ontology(<http://www.example.com/ontology1>
Import(<http://www.example.com/ontology2>)
Annotation(rdfs:label "An example
ontology
")
SubClassOf(
ex
:
Person
owl:Thing)
SubClassOf(
ex
:
Male
ex:Person
)
SubClassOf(
ex
:
Female
ex:Person
)
)
Knowledge Technologies
Manolis Koubarakis
19
Things One Can Define in OWL 2
Knowledge Technologies
Manolis Koubarakis
20
Classes
•
C
l
a
s
s
e
s
(
e
.
g
.
,
a
:
F
e
m
a
l
e
)
r
e
p
r
e
s
e
n
t
s
e
t
s
o
f
i
n
d
i
v
i
d
u
a
l
s
.
•
Built-in classes:
–
owl:Thing
, which
represents the set of all
individuals.
–
owl:Nothing
, which
represents the empty
set.
Knowledge Technologies
Manolis Koubarakis
21
Things One Can Define in OWL 2
(cont’d)
Knowledge Technologies
Manolis Koubarakis
22
Object Properties
•
O
b
j
e
c
t
p
r
o
p
e
r
t
i
e
s
(
e
.
g
.
,
a
:
p
a
r
e
n
t
O
f
)
c
o
n
n
e
c
t
p
a
i
r
s
o
f
i
n
d
i
v
i
d
u
a
l
s
.
•
Built-in object properties:
–
owl:topObjectProperty
,
which
connects
all possible pairs of individuals.
–
owl:bottomObjectProperty
, which
does
not
connect any pair of individuals.
Knowledge Technologies
Manolis Koubarakis
23
Object Property Expressions
•
O
b
j
e
c
t
p
r
o
p
e
r
t
i
e
s
c
a
n
b
e
u
s
e
d
t
o
f
o
r
m
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
24
Inverse Object Propert
y
Expressions
•
A
n
i
n
v
e
r
s
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
I
n
v
e
r
s
e
O
f
(
P
)
c
o
n
n
e
c
t
s
a
n
i
n
d
i
v
i
d
u
a
l
I
1
w
i
t
h
I
2
i
f
a
n
d
o
n
l
y
i
f
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
P
c
o
n
n
e
c
t
s
I
2
w
i
t
h
I
1
.
•
Example: If an ontology contains the axiom
ObjectPropertyAssertion(a:fatherOf a:Peter a:Stewie)
then the ontology entails
ObjectPropertyAssertion(
ObjectInverseOf(
a:fatherOf
)
a:Stewie
a:Peter)
P
ObjectInverseOf(P)
I2
I1
Knowledge Technologies
Manolis Koubarakis
25
Things One Can Define in OWL 2
(cont’d)
Knowledge Technologies
Manolis Koubarakis
26
Data Properties
•
D
a
t
a
p
r
o
p
e
r
t
i
e
s
(
e
.
g
.
,
a
:
h
a
s
A
g
e
)
c
o
n
n
e
c
t
i
n
d
i
v
i
d
u
a
l
s
w
i
t
h
l
i
t
e
r
a
l
s
.
•
Built-in properties:
–
owl:topDataProperty
,
which
connects all
possible individuals with all literals.
–
owl:bottomDataProperty
,
which
does not
connect any individual with a literal.
Knowledge Technologies
Manolis Koubarakis
27
Data Property Expressions
•
T
h
e
o
n
l
y
a
l
l
o
w
e
d
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
i
s
a
d
a
t
a
p
r
o
p
e
r
t
y
.
Knowledge Technologies
Manolis Koubarakis
28
Things One Can Define in OWL 2
(cont’d)
Knowledge Technologies
Manolis Koubarakis
29
Annotation Properties
•
A
n
n
o
t
a
t
i
o
n
p
r
o
p
e
r
t
i
e
s
c
a
n
b
e
u
s
e
d
t
o
p
r
o
v
i
d
e
a
n
a
n
n
o
t
a
t
i
o
n
f
o
r
a
n
o
n
t
o
l
o
g
y
,
a
x
i
o
m
,
o
r
a
n
I
R
I
.
•
U
s
e
r
s
c
a
n
d
e
f
i
n
e
t
h
e
i
r
o
w
n
a
n
n
o
t
a
t
i
o
n
p
r
o
p
e
r
t
i
e
s
(
w
e
w
i
l
l
s
e
e
h
o
w
l
a
t
e
r
o
n
)
o
r
u
s
e
t
h
e
a
v
a
i
l
a
b
l
e
b
u
i
l
t
-
i
n
a
n
n
o
t
a
t
i
o
n
p
r
o
p
e
r
t
i
e
s
:
–
rdfs:label
,
rdfs:comment
,
rdfs:see
Also,
rdfs:isDefinedBy
–
owl:deprecated
,
owl:versionInfo
,
owl:priorVersion
,
o
wl:backwardCompatibleWith
,
owl:incompatibleWith
Knowledge Technologies
Manolis Koubarakis
30
Things One Can Define in OWL 2
(cont’d)
Knowledge Technologies
Manolis Koubarakis
31
Individuals
•
I
n
d
i
v
i
d
u
a
l
s
r
e
p
r
e
s
e
n
t
a
c
t
u
a
l
o
b
j
e
c
t
s
f
r
o
m
t
h
e
d
o
m
a
i
n
.
•
There are two types of individuals:
–
N
a
m
e
d
i
n
d
i
v
i
d
u
a
l
s
a
r
e
g
i
v
e
n
a
n
e
x
p
l
i
c
i
t
n
a
m
e
(
a
n
I
R
I
e
.
g
.
,
a
:
P
e
t
e
r
)
t
h
a
t
c
a
n
b
e
u
s
e
d
i
n
a
n
y
o
n
t
o
l
o
g
y
t
o
r
e
f
e
r
t
o
t
h
e
s
a
m
e
o
b
j
e
c
t
.
–
A
n
o
n
y
m
o
u
s
i
n
d
i
v
i
d
u
a
l
s
d
o
n
o
t
h
a
v
e
a
g
l
o
b
a
l
n
a
m
e
.
T
h
e
y
c
a
n
b
e
d
e
f
i
n
e
d
u
s
i
n
g
a
n
a
m
e
(
e
.
g
.
,
_
:
s
o
m
e
b
o
d
y
)
l
o
c
a
l
t
o
t
h
e
o
n
t
o
l
o
g
y
t
h
e
y
a
r
e
c
o
n
t
a
i
n
e
d
i
n
.
T
h
e
y
a
r
e
l
i
k
e
b
l
a
n
k
n
o
d
e
s
i
n
R
D
F
.
Knowledge Technologies
Manolis Koubarakis
32
Things One Can Define in OWL 2
(cont’d)
Knowledge Technologies
Manolis Koubarakis
33
Things One Can Define in OWL 2
(cont’d)
Knowledge Technologies
Manolis Koubarakis
34
Datatypes
•
D
a
t
a
t
y
p
e
s
a
r
e
e
n
t
i
t
i
e
s
t
h
a
t
r
e
p
r
e
s
e
n
t
s
e
t
s
o
f
d
a
t
a
v
a
l
u
e
s
.
•
OWL 2 offers a rich set of data types: decimal numbers, integers, floating
point numbers, rationals, reals, strings, binary data, IRIs and time instants.
•
In most cases, these data types are taken from XML schema. From RDF
and RDFS, we have
rdf:XMLLitera
l,
rdf:PlainLiteral
and
rdfs:Literal
.
•
rdfs:Literal
contains all the elements of other data types.
•
There are also the OWL datatypes
owl:real
and
owl:rational.
•
F
o
r
m
a
l
l
y
,
t
h
e
d
a
t
a
t
y
p
e
s
s
u
p
p
o
r
t
e
d
a
r
e
s
p
e
c
i
f
i
e
d
i
n
t
h
e
O
W
L
2
d
a
t
a
t
y
p
e
m
a
p
.
Knowledge Technologies
Manolis Koubarakis
35
Datatypes (cont’d)
•
In a datatype map, e
ach datatype is identified by an IRI and is defined by
the following components:
–
T
h
e
v
a
l
u
e
s
p
a
c
e
i
s
t
h
e
s
e
t
o
f
v
a
l
u
e
s
o
f
t
h
e
d
a
t
a
t
y
p
e
.
E
l
e
m
e
n
t
s
o
f
t
h
e
v
a
l
u
e
s
p
a
c
e
a
r
e
c
a
l
l
e
d
d
a
t
a
v
a
l
u
e
s
.
–
T
h
e
l
e
x
i
c
a
l
s
p
a
c
e
i
s
a
s
e
t
o
f
s
t
r
i
n
g
s
t
h
a
t
c
a
n
b
e
u
s
e
d
t
o
r
e
f
e
r
t
o
d
a
t
a
v
a
l
u
e
s
.
E
a
c
h
m
e
m
b
e
r
o
f
t
h
e
l
e
x
i
c
a
l
s
p
a
c
e
i
s
c
a
l
l
e
d
a
l
e
x
i
c
a
l
f
o
r
m
,
a
n
d
i
t
i
s
m
a
p
p
e
d
t
o
a
p
a
r
t
i
c
u
l
a
r
d
a
t
a
v
a
l
u
e
.
–
T
h
e
f
a
c
e
t
s
p
a
c
e
i
s
a
s
e
t
o
f
p
a
i
r
s
o
f
t
h
e
f
o
r
m
(
F
,
v
)
w
h
e
r
e
F
i
s
a
n
I
R
I
c
a
l
l
e
d
a
c
o
n
s
t
r
a
i
n
i
n
g
f
a
c
e
t
,
a
n
d
v
i
s
a
n
a
r
b
i
t
r
a
r
y
d
a
t
a
v
a
l
u
e
c
a
l
l
e
d
t
h
e
c
o
n
s
t
r
a
i
n
i
n
g
v
a
l
u
e
.
E
a
c
h
s
u
c
h
p
a
i
r
i
s
m
a
p
p
e
d
t
o
a
s
u
b
s
e
t
o
f
t
h
e
v
a
l
u
e
s
p
a
c
e
o
f
t
h
e
d
a
t
a
t
y
p
e
.
Knowledge Technologies
Manolis Koubarakis
36
Facet Space
•
F
o
r
t
h
e
X
M
L
S
c
h
e
m
a
d
a
t
a
t
y
p
e
s
x
s
d
:
d
o
u
b
l
e
,
x
s
d
:
f
l
o
a
t
,
a
n
d
x
s
d
:
d
e
c
i
m
a
l
,
t
h
e
c
o
n
s
t
r
a
i
n
i
n
g
f
a
c
e
t
s
a
l
l
o
w
e
d
a
r
e
:
x
s
d
:
m
i
n
I
n
c
l
u
s
i
v
e
,
x
s
d
:
m
a
x
I
n
c
l
u
s
i
v
e
,
x
s
d
:
m
i
n
E
x
c
l
u
s
i
v
e
a
n
d
x
s
d
:
m
a
x
E
x
c
l
u
s
i
v
e
.
•
Example: The pair
(xsd:minInclusive,v)
of the facet space
denotes
the set of all numbers
x
from the value space of
the
datatype
such that
x=v
or
x>v
.
•
Similarly for other datatypes.
•
W
e
w
i
l
l
s
e
e
l
a
t
e
r
h
o
w
c
o
n
s
t
r
a
i
n
i
n
g
f
a
c
e
t
s
c
a
n
b
e
u
s
e
d
t
o
d
e
f
i
n
e
d
a
t
a
r
a
n
g
e
s
.
Knowledge Technologies
Manolis Koubarakis
37
Literals
•
L
i
t
e
r
a
l
s
r
e
p
r
e
s
e
n
t
d
a
t
a
v
a
l
u
e
s
s
u
c
h
a
s
p
a
r
t
i
c
u
l
a
r
s
t
r
i
n
g
s
o
r
i
n
t
e
g
e
r
s
.
T
h
e
y
a
r
e
a
n
a
l
o
g
o
u
s
t
o
R
D
F
l
i
t
e
r
a
l
s
.
•
Examples:
–
"
1
"
^
^
x
s
d
:
i
n
t
e
g
e
r
(
t
y
p
e
d
l
i
t
e
r
a
l
)
–
"
F
a
m
i
l
y
G
u
y
"
(
p
l
a
i
n
l
i
t
e
r
a
l
,
a
n
a
b
b
r
e
v
i
a
t
i
o
n
f
o
r
"
F
a
m
i
l
y
G
u
y
"
^
^
r
d
f
:
P
l
a
i
n
L
i
t
e
r
a
l
)
–
"
P
a
d
r
e
d
e
f
a
m
i
l
i
a
"
@
e
s
(
p
l
a
i
n
l
i
t
e
r
a
l
w
i
t
h
l
a
n
g
u
a
g
e
t
a
g
,
a
n
a
b
b
r
e
v
i
a
t
i
o
n
f
o
r
"
P
a
d
r
e
d
e
f
a
m
i
l
i
a
@
e
s
"
^
^
r
d
f
:
P
l
a
i
n
L
i
t
e
r
a
l
)
Knowledge Technologies
Manolis Koubarakis
38
Things One Can Define in OWL 2
(cont’d)
Knowledge Technologies
Manolis Koubarakis
39
Data Ranges
•
D
a
t
a
r
a
n
g
e
s
r
e
p
r
e
s
e
n
t
s
e
t
s
o
f
t
u
p
l
e
s
o
f
l
i
t
e
r
a
l
s
.
T
h
e
y
a
r
e
d
e
f
i
n
e
d
u
s
i
n
g
d
a
t
a
t
y
p
e
s
a
n
d
c
o
n
s
t
r
a
i
n
i
n
g
f
a
c
e
t
s
.
•
Examples:
–
The set of integers greater than 10.
–
The set of strings that contain “good” as a substring.
–
The set of
(x,y)
such that
x
and
y
are integers and
x < y
.
•
E
a
c
h
d
a
t
a
r
a
n
g
e
i
s
a
s
s
o
c
i
a
t
e
d
w
i
t
h
a
p
o
s
i
t
i
v
e
a
r
i
t
y
,
w
h
i
c
h
d
e
t
e
r
m
i
n
e
s
t
h
e
s
i
z
e
o
f
i
t
s
t
u
p
l
e
s
.
•
Datatypes are themselves data ranges of arity 1.
•
D
a
t
a
r
a
n
g
e
s
a
r
e
u
s
e
d
i
n
r
e
s
t
r
i
c
t
i
o
n
s
o
n
d
a
t
a
p
r
o
p
e
r
t
i
e
s
,
a
s
w
e
w
i
l
l
s
e
e
l
a
t
e
r
w
h
e
n
w
e
d
e
f
i
n
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
40
Data Ranges
Knowledge Technologies
Manolis Koubarakis
41
BNF for Data Ranges
DataRange :=
Datatype |
DataIntersectionOf |
DataUnionOf |
DataComplementOf |
DataOneOf |
DatatypeRestriction
DataIntersectionOf :=
'DataIntersectionOf' '('DataRange DataRange
{ DataRange } ')'
DataUnionOf :=
'DataUnionOf' '(' DataRange DataRange { DataRange }')'
DataComplementOf :=
'DataComplementOf' '('DataRange
')'
DataOneOf :=
'DataOneOf' '(' Literal { Literal } ')'
Knowledge Technologies
Manolis Koubarakis
42
Examples
DataIntersectionOf(xsd:nonNegativeInteger
xsd:nonPositiveInteger)
DataUnionOf(xsd:string xsd:integer)
DataComplementOf(xsd:positiveInteger)
DataOneOf("Peter" "John
")
Knowledge Technologies
Manolis Koubarakis
43
Datatype Restrictions
DatatypeRestriction :=
'DatatypeRestriction' '('Datatype constrainingFacet
restrictionValue
{ constrainingFacet restrictionValue } ')
’
constrainingFacet := IRI
restrictionValue := Literal
Knowledge Technologies
Manolis Koubarakis
44
Datatype Restrictions
•
A
d
a
t
a
t
y
p
e
r
e
s
t
r
i
c
t
i
o
n
D
a
t
a
t
y
p
e
R
e
s
t
r
i
c
t
i
o
n
(
D
T
F
1
l
t
1
.
.
.
F
n
l
t
n
)
c
o
n
s
i
s
t
s
o
f
a
u
n
a
r
y
d
a
t
a
t
y
p
e
D
T
a
n
d
n
p
a
i
r
s
(
F
i
,
l
t
i
)
w
h
e
r
e
F
i
i
s
a
c
o
n
s
t
r
a
i
n
i
n
g
f
a
c
e
t
o
f
D
T
a
n
d
l
t
i
a
l
i
t
e
r
a
l
v
a
l
u
e
.
•
T
h
e
d
a
t
a
r
a
n
g
e
r
e
p
r
e
s
e
n
t
e
d
b
y
a
d
a
t
a
t
y
p
e
r
e
s
t
r
i
c
t
i
o
n
i
s
u
n
a
r
y
a
n
d
i
s
o
b
t
a
i
n
e
d
b
y
r
e
s
t
r
i
c
t
i
n
g
t
h
e
v
a
l
u
e
s
p
a
c
e
o
f
D
T
a
c
c
o
r
d
i
n
g
t
o
t
h
e
c
o
n
j
u
n
c
t
i
o
n
o
f
a
l
l
(
F
i
,
l
t
i
)
.
•
O
b
s
e
r
v
a
t
i
o
n
:
T
h
u
s
,
a
l
t
h
o
u
g
h
t
h
e
d
e
f
i
n
i
t
i
o
n
o
f
d
a
t
a
r
a
n
g
e
s
p
e
a
k
s
o
f
t
u
p
l
e
s
o
f
a
n
y
a
r
i
t
y
,
t
h
e
s
y
n
t
a
x
d
e
f
i
n
e
d
a
l
l
o
w
s
o
n
l
y
u
n
a
r
y
d
a
t
a
r
a
n
g
e
s
.
Knowledge Technologies
Manolis Koubarakis
45
Example
•
The following data
type restriction
represents
the
set of
integers 5, 6, 7, 8,
and 9:
DatatypeRestriction(xsd:integer
xsd:minInclusive "5"^^xsd:integer
xsd:maxExclusive "10"^^xsd:integer)
Knowledge Technologies
Manolis Koubarakis
46
Things One Can Define in OWL 2
(cont’d)
Knowledge Technologies
Manolis Koubarakis
47
Class Expressions
•
C
l
a
s
s
n
a
m
e
s
a
n
d
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
c
a
n
b
e
u
s
e
d
t
o
c
o
n
s
t
r
u
c
t
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
.
•
T
h
e
s
e
a
r
e
e
s
s
e
n
t
i
a
l
l
y
t
h
e
c
o
m
p
l
e
x
c
o
n
c
e
p
t
s
o
r
d
e
s
c
r
i
p
t
i
o
n
s
t
h
a
t
w
e
c
a
n
d
e
f
i
n
e
i
n
D
L
s
.
•
C
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
r
e
p
r
e
s
e
n
t
s
e
t
s
o
f
i
n
d
i
v
i
d
u
a
l
s
b
y
f
o
r
m
a
l
l
y
s
p
e
c
i
f
y
i
n
g
c
o
n
d
i
t
i
o
n
s
o
n
t
h
e
i
n
d
i
v
i
d
u
a
l
s
'
p
r
o
p
e
r
t
i
e
s
;
i
n
d
i
v
i
d
u
a
l
s
s
a
t
i
s
f
y
i
n
g
t
h
e
s
e
c
o
n
d
i
t
i
o
n
s
a
r
e
s
a
i
d
t
o
b
e
i
n
s
t
a
n
c
e
s
o
f
t
h
e
r
e
s
p
e
c
t
i
v
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
48
Ways to Form Class Expressions
•
Class expressions can be formed by:
–
A
p
p
l
y
i
n
g
t
h
e
s
t
a
n
d
a
r
d
B
o
o
l
e
a
n
c
o
n
n
e
c
t
i
v
e
s
t
o
s
i
m
p
l
e
r
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
o
r
b
y
e
n
u
m
e
r
a
t
i
n
g
t
h
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
b
e
l
o
n
g
t
o
a
n
e
x
p
r
e
s
s
i
o
n
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
49
Boolean Connectives and Enumeration
of Individuals
Knowledge Technologies
Manolis Koubarakis
50
Intersection Class Expression
s
•
A
n
i
n
t
e
r
s
e
c
t
i
o
n
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
I
n
t
e
r
s
e
c
t
i
o
n
O
f
(
C
E
1
.
.
.
C
E
n
)
c
o
n
t
a
i
n
s
a
l
l
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
i
n
s
t
a
n
c
e
s
o
f
a
l
l
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
C
E
i
f
o
r
1
≤
i
≤
n
.
•
Example:
ObjectIntersectionOf(a:Dog a:CanTalk)
Knowledge Technologies
Manolis Koubarakis
51
Union Class Expression
s
•
A
u
n
i
o
n
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
U
n
i
o
n
O
f
(
C
E
1
.
.
.
C
E
n
)
c
o
n
t
a
i
n
s
a
l
l
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
i
n
s
t
a
n
c
e
s
o
f
a
t
l
e
a
s
t
o
n
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
i
f
o
r
1≤i≤n
.
•
Example:
ObjectUnionOf(a:Man a:Woman)
Knowledge Technologies
Manolis Koubarakis
52
Complement Class Expressions
•
A
c
o
m
p
l
e
m
e
n
t
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
C
o
m
p
l
e
m
e
n
t
O
f
(
C
E
)
c
o
n
t
a
i
n
s
a
l
l
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
n
o
t
i
n
s
t
a
n
c
e
s
o
f
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
.
•
Example:
ObjectComplementOf(a:Man)
Knowledge Technologies
Manolis Koubarakis
53
Example Inference
•
From
DisjointClasses(a:Man a:Woman)
ClassAssertion(a:Woman a:Lois)
we can infer
ClassAssertion(
ObjectComplementOf(a:Man)
a:Lois)
…and in DL
•
From
Woman
isSubsumedBy
¬Man
Woman(LOIS)
we can infer
¬ Man(LOIS)
Knowledge Technologies
Manolis Koubarakis
54
Knowledge Technologies
Manolis Koubarakis
55
E
n
u
m
e
r
a
t
i
o
n
o
f
I
n
d
i
v
i
d
u
a
l
s
•
A
n
e
n
u
m
e
r
a
t
i
o
n
o
f
i
n
d
i
v
i
d
u
a
l
s
O
b
j
e
c
t
O
n
e
O
f
(
a
1
.
.
.
a
n
)
c
o
n
t
a
i
n
s
e
x
a
c
t
l
y
t
h
e
i
n
d
i
v
i
d
u
a
l
s
a
i
w
i
t
h
1
≤
i
≤
n
.
•
Example:
ObjectOneOf(a:Peter a:Lois
a:Stewie a:Meg a:Chris a:Brian)
Knowledge Technologies
Manolis Koubarakis
56
Example Inference
•
From
EquivalentClasses(a:GriffinFamilyMember
ObjectOneOf(a:Peter a:Lois a:Stewie a:Meg
a:Chris a:Brian))
DifferentIndividuals(a:Quagmire a:Peter a:Lois
a:Stewie a:Meg a:Chris a:Brian)
we can infer
ClassAssertion(
ObjectComplementOf(a:GriffinFamilyMember)
a:Quagmire
)
Knowledge Technologies
Manolis Koubarakis
57
Example Inference (con’td)
•
From
ClassAssertion(a:GriffinFamilyMember a:Peter)
ClassAssertion(a:GriffinFamilyMember a:Lois)
ClassAssertion(a:GriffinFamilyMember a:Stewie)
ClassAssertion(a:GriffinFamilyMember a:Meg)
ClassAssertion(a:GriffinFamilyMember a:Chris)
ClassAssertion(a:GriffinFamilyMember a:Brian)
DifferentIndividuals(a:Quagmire a:Peter a:Lois a:Stewie
a:Meg a:Chris a:Brian)
Can we infer this:
ClassAssertion(
ObjectComplementOf(a:GriffinFamilyMember) a:Quagmire
)
?
Knowledge Technologies
Manolis Koubarakis
58
Ways to Form Class Expressions
(cont’d)
•
Class expressions can be formed by:
–
A
p
p
l
y
i
n
g
t
h
e
s
t
a
n
d
a
r
d
B
o
o
l
e
a
n
c
o
n
n
e
c
t
i
v
e
s
t
o
s
i
m
p
l
e
r
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
o
r
b
y
e
n
u
m
e
r
a
t
i
n
g
t
h
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
b
e
l
o
n
g
t
o
a
n
e
x
p
r
e
s
s
i
o
n
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
59
Object Property Restrictions
Knowledge Technologies
Manolis Koubarakis
60
Existential Quantification
•
A
n
e
x
i
s
t
e
n
t
i
a
l
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
S
o
m
e
V
a
l
u
e
s
F
r
o
m
(
O
P
E
C
E
)
c
o
n
s
i
s
t
s
o
f
a
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
a
n
d
a
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
a
n
i
n
d
i
v
i
d
u
a
l
t
h
a
t
i
s
a
n
i
n
s
t
a
n
c
e
o
f
C
E
.
•
Example:
ObjectSomeValuesFrom(a:fatherOf a:Man)
•
If
OPE is simple,
the above
class expression
is
equivalent with
the class expression
ObjectMinCardinality(1 OPE CE)
Knowledge Technologies
Manolis Koubarakis
61
Example Inference
•
From
ObjectPropertyAssertion(a:fatherOf
a:Peter a:Stewie)
ClassAssertion(a:Man a:Stewie)
we can infer
ClassAssertion(
ObjectSomeValuesFrom(a:fatherOf
a:Man)
a:Peter)
Peter
Stewie
Man
fatherOf
Knowledge Technologies
Manolis Koubarakis
62
Univers
al Quantification
•
A
u
n
i
v
e
r
s
a
l
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
A
l
l
V
a
l
u
e
s
F
r
o
m
(
O
P
E
C
E
)
c
o
n
s
i
s
t
s
o
f
a
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
a
n
d
a
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
O
P
E
o
n
l
y
t
o
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
i
n
s
t
a
n
c
e
s
o
f
C
E
.
•
Example:
Object
All
ValuesFrom(a:fatherOf a:Man)
•
If
OPE is simple,
the above
class expression
is
equivalent with
the class expression
ObjectM
ax
Cardinality(0 OPE ObjectComplementOf(CE))
Knowledge Technologies
Manolis Koubarakis
63
Example Inference
•
From
ObjectPropertyAssertion(a:hasPet a:Peter a:Brian)
ClassAssertion(a:Dog a:Brian)
ClassAssertion(
ObjectMaxCardinality(1 a:hasPet) a:Peter)
we can infer
ClassAssertion(
Object
All
ValuesFrom(a:
hasPet
a:
Dog
)
a:Peter)
Knowledge Technologies
Manolis Koubarakis
64
I
ndividual Value Restriction
•
A
n
i
n
d
i
v
i
d
u
a
l
v
a
l
u
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
H
a
s
V
a
l
u
e
(
O
P
E
a
)
c
o
n
s
i
s
t
s
o
f
a
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
a
n
d
a
n
i
n
d
i
v
i
d
u
a
l
a
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
a
.
•
Example:
ObjectHasValue(a:fatherOf a:Stewie)
•
The above
class expression
is equivalent to
the class
expression
ObjectSomeValuesFrom(OPE ObjectOneOf(a)).
Knowledge Technologies
Manolis Koubarakis
65
Example Inference
•
From
ObjectPropertyAssertion(a:fatherOf
a:Peter a:Stewie)
we can infer
ClassAssertion(
ObjectHasValue(a:fatherOf a:Stewie)
a:Peter)
Knowledge Technologies
Manolis Koubarakis
66
Self-Restriction
•
A
s
e
l
f
-
r
e
s
t
r
i
c
t
i
o
n
O
b
j
e
c
t
H
a
s
S
e
l
f
(
O
P
E
)
c
o
n
s
i
s
t
s
o
f
a
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
t
h
e
m
s
e
l
v
e
s
.
•
Example:
ObjectHasSelf(a:likes)
Peter
likes
Knowledge Technologies
Manolis Koubarakis
67
Example Inference
•
From
ObjectPropertyAssertion(a:likes
a:Peter a:Peter)
we can infer
ClassAssertion(
ObjectHasSelf(a:likes)
a:Peter)
Knowledge Technologies
Manolis Koubarakis
68
Ways to Form Class Expressions
(cont’d)
•
Class expressions can be formed by:
–
A
p
p
l
y
i
n
g
t
h
e
s
t
a
n
d
a
r
d
B
o
o
l
e
a
n
c
o
n
n
e
c
t
i
v
e
s
t
o
s
i
m
p
l
e
r
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
o
r
b
y
e
n
u
m
e
r
a
t
i
n
g
t
h
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
b
e
l
o
n
g
t
o
a
n
e
x
p
r
e
s
s
i
o
n
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
69
Object Property Cardinality
Restrictions
•
Object property c
ardinality restrictions
are distinguished
into:
–
Q
u
a
l
i
f
i
e
d
:
a
p
p
l
y
o
n
l
y
t
o
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
a
n
d
a
r
e
i
n
s
t
a
n
c
e
s
o
f
t
h
e
q
u
a
l
i
f
y
i
n
g
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
.
(e.g. >3hasChild.Male)
–
U
n
q
u
a
l
i
f
i
e
d
:
a
p
p
l
y
t
o
a
l
l
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
(
t
h
i
s
i
s
e
q
u
i
v
a
l
e
n
t
t
o
t
h
e
q
u
a
l
i
f
i
e
d
c
a
s
e
w
i
t
h
t
h
e
q
u
a
l
i
f
y
i
n
g
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
e
q
u
a
l
t
o
o
w
l
:
T
h
i
n
g
)
(
e
.
g
.
>
3
h
a
s
C
h
i
l
d
)
.
Knowledge Technologies
Manolis Koubarakis
70
Object Property Cardinality
Restrictions
Knowledge Technologies
Manolis Koubarakis
71
Minimum Cardinality
•
A
m
i
n
i
m
u
m
c
a
r
d
i
n
a
l
i
t
y
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
M
i
n
C
a
r
d
i
n
a
l
i
t
y
(
n
O
P
E
C
E
)
c
o
n
s
i
s
t
s
o
f
a
n
o
n
n
e
g
a
t
i
v
e
i
n
t
e
g
e
r
n
,
a
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
,
a
n
d
a
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
a
t
l
e
a
s
t
n
d
i
f
f
e
r
e
n
t
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
i
n
s
t
a
n
c
e
s
o
f
C
E
.
I
f
C
E
i
s
m
i
s
s
i
n
g
,
i
t
i
s
t
a
k
e
n
t
o
b
e
o
w
l
:
T
h
i
n
g
.
•
Example:
ObjectMinCardinality(2 a:fatherOf a:Man)
Knowledge Technologies
Manolis Koubarakis
72
Example Inference
•
From
ObjectPropertyAssertion(a:fatherOf a:Peter a:Stewie)
ClassAssertion(a:Man a:Stewie)
ObjectPropertyAssertion(a:fatherOf a:Peter a:Chris)
ClassAssertion(a:Man a:Chris)
DifferentIndividuals(a:Chris a:Stewie)
we can infer
ClassAssertion(
ObjectMinCardinality(2 a:fatherOf a:Man)
a:Peter)
Knowledge Technologies
Manolis Koubarakis
73
Maximum Cardinality
•
A
m
a
x
i
m
u
m
c
a
r
d
i
n
a
l
i
t
y
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
M
a
x
C
a
r
d
i
n
a
l
i
t
y
(
n
O
P
E
C
E
)
c
o
n
s
i
s
t
s
o
f
a
n
o
n
n
e
g
a
t
i
v
e
i
n
t
e
g
e
r
n
,
a
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
,
a
n
d
a
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
a
t
m
o
s
t
n
d
i
f
f
e
r
e
n
t
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
i
n
s
t
a
n
c
e
s
o
f
C
E
.
I
f
C
E
i
s
m
i
s
s
i
n
g
,
i
t
i
s
t
a
k
e
n
t
o
b
e
o
w
l
:
T
h
i
n
g
.
•
Example:
ObjectMaxCardinality(2 a:hasPet)
Knowledge Technologies
Manolis Koubarakis
74
Example Inference
•
From
ObjectPropertyAssertion(a:hasPet
a:Peter a:Brian)
ClassAssertion(ObjectMaxCardinality(1
a:hasPet) a:Peter)
we can infer
ClassAssertion(
ObjectM
ax
Cardinality(2
a:hasPet
)
a:Peter)
Knowledge Technologies
Manolis Koubarakis
75
Example Inference
•
From
ObjectPropertyAssertion(a:hasDaughter
a:Peter a:Meg)
ObjectPropertyAssertion(a:hasDaughter
a:Peter a:Megan)
ClassAssertion(ObjectMaxCardinality(1
a:hasDaughter) a:Peter)
we can infer
SameIndividual(a:Meg a:Megan)
Knowledge Technologies
Manolis Koubarakis
76
Exact Cardinality
•
A
n
e
x
a
c
t
c
a
r
d
i
n
a
l
i
t
y
e
x
p
r
e
s
s
i
o
n
O
b
j
e
c
t
E
x
a
c
t
C
a
r
d
i
n
a
l
i
t
y
(
n
O
P
E
C
E
)
c
o
n
s
i
s
t
s
o
f
a
n
o
n
n
e
g
a
t
i
v
e
i
n
t
e
g
e
r
n
,
a
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
,
a
n
d
a
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
e
x
a
c
t
l
y
n
d
i
f
f
e
r
e
n
t
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
i
n
s
t
a
n
c
e
s
o
f
C
E
.
•
Example:
ObjectExactCardinality(1 a:hasPet a:Dog)
•
The above
expression is equivalent to
ObjectIntersectionOf(
ObjectMinCardinality
(
n OPE CE)
ObjectMaxCardinality(n OPE CE)).
Knowledge Technologies
Manolis Koubarakis
77
Example Inference
•
From
ObjectPropertyAssertion(a:hasPet a:Peter a:Brian)
ClassAssertion(a:Dog a:Brian)
ClassAssertion(
ObjectAllValuesFrom(a:hasPet
ObjectUnionOf(ObjectOneOf(a:Brian)
ObjectComplementOf(a:Dog)))
a:Peter)
we can infer
ClassAssertion(ObjectExactCardinality(1 a:hasPet
a:Dog) a:Peter)
Knowledge Technologies
Manolis Koubarakis
78
Ways to Form Class Expressions
(cont’d)
•
Class expressions can be formed by:
–
A
p
p
l
y
i
n
g
t
h
e
s
t
a
n
d
a
r
d
B
o
o
l
e
a
n
c
o
n
n
e
c
t
i
v
e
s
t
o
s
i
m
p
l
e
r
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
o
r
b
y
e
n
u
m
e
r
a
t
i
n
g
t
h
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
b
e
l
o
n
g
t
o
a
n
e
x
p
r
e
s
s
i
o
n
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
79
Data Property Restrictions
•
Data property restrictions
are similar to the restrictions on object property
expressions
.
•
T
h
e
m
a
i
n
d
i
f
f
e
r
e
n
c
e
i
s
t
h
a
t
t
h
e
e
x
p
r
e
s
s
i
o
n
s
f
o
r
e
x
i
s
t
e
n
t
i
a
l
a
n
d
u
n
i
v
e
r
s
a
l
q
u
a
n
t
i
f
i
c
a
t
i
o
n
a
l
l
o
w
f
o
r
n
-
a
r
y
d
a
t
a
r
a
n
g
e
s
.
•
G
i
v
e
n
t
h
e
s
y
n
t
a
x
f
o
r
d
a
t
a
r
a
n
g
e
s
g
i
v
e
n
e
a
r
l
i
e
r
,
o
n
l
y
u
n
a
r
y
d
a
t
a
r
a
n
g
e
s
a
r
e
s
u
p
p
o
r
t
e
d
.
•
H
o
w
e
v
e
r
,
t
h
e
s
p
e
c
i
f
i
c
a
t
i
o
n
a
p
r
o
v
i
d
e
t
h
e
s
y
n
t
a
c
t
i
c
c
o
n
s
t
r
u
c
t
s
n
e
e
d
e
d
t
o
h
a
v
e
n
-
a
r
y
d
a
t
a
r
a
n
g
e
s
e
.
g
.
,
s
e
t
s
o
f
r
e
c
t
a
n
g
l
e
s
d
e
f
i
n
e
d
b
y
a
p
p
r
o
p
r
i
a
t
e
g
e
o
m
e
t
r
i
c
c
o
n
s
t
r
a
i
n
t
s
.
•
The
“
Data Range Extension: Linear Equations
” W3C
note proposes an
extension to OWL 2 for defining
n-ary
data ranges in terms of linear
(in)equations with rational coefficients.
See
http://www.w3.org/TR/owl2-dr-
linear/
.
Knowledge Technologies
Manolis Koubarakis
80
Data Property Restrictions
Knowledge Technologies
Manolis Koubarakis
81
Existential Quantification
•
A
n
e
x
i
s
t
e
n
t
i
a
l
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
D
a
t
a
S
o
m
e
V
a
l
u
e
s
F
r
o
m
(
D
P
E
1
.
.
.
D
P
E
n
D
R
)
c
o
n
s
i
s
t
s
o
f
n
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
D
P
E
i
,
1
≤
i
≤
n
,
a
n
d
a
d
a
t
a
r
a
n
g
e
D
R
w
h
o
s
e
a
r
i
t
y
m
u
s
t
b
e
n
.
•
Such a class expression contains all those individuals that are connected by
DPEi
to literals
lti,1≤i≤n
,
such that the tuple
(lt1 ,...,ltn)
is in
DR
.
•
Example:
DataSomeValuesFrom(a:hasAge
DatatypeRestriction(xsd:integer xsd:maxExclusive
"20"^^xsd:integer))
•
A class expression of the form
DataSomeValuesFrom(DPE DR)
is
equivalent to
the class expression
DataMinCardinality(1 DPE D
R
).
Knowledge Technologies
Manolis Koubarakis
82
Example Inference
•
From
DataPropertyAssertion(a:hasAge a:Meg
"17"^^xsd:integer)
we can infer
ClassAssertion(
DataSomeValuesFrom(a:hasAge
DatatypeRestriction(xsd:integer
xsd:maxExclusive "20"^^xsd:integer))
a:Meg)
Knowledge Technologies
Manolis Koubarakis
83
Universal Quantification
•
A
u
n
i
v
e
r
s
a
l
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
D
a
t
a
A
l
l
V
a
l
u
e
s
F
r
o
m
(
D
P
E
1
.
.
.
D
P
E
n
D
R
)
c
o
n
s
i
s
t
s
o
f
n
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
D
P
E
i
,
1
≤
i
≤
n
,
a
n
d
a
d
a
t
a
r
a
n
g
e
D
R
w
h
o
s
e
a
r
i
t
y
m
u
s
t
b
e
n
.
•
Such a class expression contains all those individuals that are connected by
DPEi
only to literals
lti,1≤i≤n,
such that each tuple
(lt1,...,ltn)
is
in
DR
.
•
Example:
DataAllValuesFrom(a:hasZIP xsd:integer)
•
A class expression of the form
DataAllValuesFrom(DPE DR)
can be
seen as a syntactic shortcut for the class expression
DataMaxCardinality(0 DPE DataComplementOf(DR)).
Knowledge Technologies
Manolis Koubarakis
84
Example Inference
•
From
DataPropertyAssertion(a:hasZIP
_:
a1
"02903"^^xsd:integer)
FunctionalDataProperty(a:hasZIP)
we can infer
ClassAssertion(
DataAllValuesFrom(a:hasZIP xsd:integer)
_:
a1)
Knowledge Technologies
Manolis Koubarakis
85
Literal Value Restriction
•
A
l
i
t
e
r
a
l
v
a
l
u
e
c
l
a
s
s
r
e
s
t
r
i
c
t
i
o
n
D
a
t
a
H
a
s
V
a
l
u
e
(
D
P
E
l
t
)
c
o
n
s
i
s
t
s
o
f
a
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
D
P
E
a
n
d
a
l
i
t
e
r
a
l
l
t
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
D
P
E
t
o
l
t
.
•
Example:
DataHasValue(a:hasAge "17"^^xsd:integer)
•
Each such class expression
is equivalent to
the class
expression
DataSomeValuesFrom(DPE DataOneOf
(
lt)).
Knowledge Technologies
Manolis Koubarakis
86
Ways to Form Class Expressions
(cont’d)
•
Class expressions can be formed by:
–
A
p
p
l
y
i
n
g
t
h
e
s
t
a
n
d
a
r
d
B
o
o
l
e
a
n
c
o
n
n
e
c
t
i
v
e
s
t
o
s
i
m
p
l
e
r
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
o
r
b
y
e
n
u
m
e
r
a
t
i
n
g
t
h
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
b
e
l
o
n
g
t
o
a
n
e
x
p
r
e
s
s
i
o
n
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
–
P
l
a
c
i
n
g
r
e
s
t
r
i
c
t
i
o
n
s
o
n
t
h
e
c
a
r
d
i
n
a
l
i
t
y
o
f
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
87
Data Property Cardinality
Restrictions
•
Data property c
ardinality restrictions can
be
distinguished into:
–
Q
u
a
l
i
f
i
e
d
:
t
h
e
y
o
n
l
y
a
p
p
l
y
t
o
l
i
t
e
r
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
t
h
e
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
a
n
d
a
r
e
i
n
t
h
e
q
u
a
l
i
f
y
i
n
g
d
a
t
a
r
a
n
g
e
.
–
U
n
q
u
a
l
i
f
i
e
d
:
t
h
e
y
a
p
p
l
y
t
o
a
l
l
l
i
t
e
r
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
t
h
e
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
.
T
h
i
s
i
s
e
q
u
i
v
a
l
e
n
t
t
o
t
h
e
q
u
a
l
i
f
i
e
d
c
a
s
e
w
i
t
h
t
h
e
q
u
a
l
i
f
y
i
n
g
d
a
t
a
r
a
n
g
e
e
q
u
a
l
t
o
r
d
f
s
:
L
i
t
e
r
a
l
.
Knowledge Technologies
Manolis Koubarakis
88
Minimum Cardinality
•
A
m
i
n
i
m
u
m
c
a
r
d
i
n
a
l
i
t
y
e
x
p
r
e
s
s
i
o
n
D
a
t
a
M
i
n
C
a
r
d
i
n
a
l
i
t
y
(
n
D
P
E
D
R
)
c
o
n
s
i
s
t
s
o
f
a
n
o
n
n
e
g
a
t
i
v
e
i
n
t
e
g
e
r
n
,
a
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
D
P
E
,
a
n
d
a
u
n
a
r
y
d
a
t
a
r
a
n
g
e
D
R
,
a
n
d
i
t
c
o
n
t
a
i
n
s
a
l
l
t
h
o
s
e
i
n
d
i
v
i
d
u
a
l
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
b
y
D
P
E
t
o
a
t
l
e
a
s
t
n
d
i
f
f
e
r
e
n
t
l
i
t
e
r
a
l
s
i
n
D
R
.
I
f
D
R
i
s
n
o
t
p
r
e
s
e
n
t
,
i
t
i
s
t
a
k
e
n
t
o
b
e
r
d
f
s
:
L
i
t
e
r
a
l
.
•
Example:
DataMinCardinality(2 a:hasName)
•
There are similar definitions for
DataM
ax
Cardinality(n DPE DR)
and
Data
Exact
Cardinality(n DPE DR)
.
Knowledge Technologies
Manolis Koubarakis
89
Example Inference
•
From
DataPropertyAssertion(a:hasName a:Meg
"Meg Griffin")
DataPropertyAssertion(a:hasName a:Meg
"Megan Griffin")
we can infer
ClassAssertion(
DataMinCardinality(2 a:hasName)
a:Meg)
Knowledge Technologies
Manolis Koubarakis
90
M
aximum/Exact
Cardinality
•
Defined similarly.
Knowledge Technologies
Manolis Koubarakis
91
What Have we Achieved so far?
•
W
e
h
a
v
e
e
x
p
l
a
i
n
e
d
w
h
a
t
t
h
e
“
t
h
i
n
g
s
”
t
h
a
t
o
n
e
c
a
n
d
e
f
i
n
e
i
n
O
W
L
2
a
r
e
.
•
N
o
w
l
e
t
u
s
s
e
e
h
o
w
t
o
u
s
e
t
h
e
s
e
“
t
h
i
n
g
s
”
t
o
r
e
p
r
e
s
e
n
t
k
n
o
w
l
e
d
g
e
a
b
o
u
t
a
d
o
m
a
i
n
.
•
I
n
O
W
L
2
k
n
o
w
l
e
d
g
e
i
s
r
e
p
r
e
s
e
n
t
e
d
b
y
a
x
i
o
m
s
:
s
t
a
t
e
m
e
n
t
s
t
h
a
t
s
a
y
w
h
a
t
i
s
t
r
u
e
i
n
t
h
e
d
o
m
a
i
n
o
f
i
n
t
e
r
e
s
t
.
Knowledge Technologies
Manolis Koubarakis
92
Axioms
Knowledge Technologies
Manolis Koubarakis
93
Class Expression Axioms
Knowledge Technologies
Manolis Koubarakis
94
Subclass Axioms
•
A
s
u
b
c
l
a
s
s
a
x
i
o
m
S
u
b
C
l
a
s
s
O
f
(
C
E
1
C
E
2
)
s
t
a
t
e
s
t
h
a
t
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
1
i
s
a
s
u
b
c
l
a
s
s
o
f
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
2
.
•
Example:
SubClassOf(a:Child a:Person)
•
The properties known from RDFS for
SubClassOf
hold
here as well:
–
Reflexivity
–
Transitivity
–
If
x
is an instance of class
A
and class
A
is a subclass of class
B
,
then
x
is an instance of
B
as well.
Knowledge Technologies
Manolis Koubarakis
95
Example Inferences
•
From
SubClassOf(a:Baby a:Child)
SubClassOf(a:Child a:Person)
ClassAssertion(a:Baby a:Stewie)
we can infer
SubClassOf(a:Baby a:Person)
ClassAssertion(a:
Child
a:Stewie)
ClassAssertion(a:
Person
a:Stewie)
Knowledge Technologies
Manolis Koubarakis
96
Example Inferences
•
From
SubClassOf(a:PersonWithChild
ObjectSomeValuesFrom(a:hasChild ObjectUnionOf(a:Boy
a:Girl)))
(PersonWithChild SubClassOf hasChild some (Boy or
Girl))
SubClassOf(a:Boy a:Child)
SubClassOf(a:Girl a:Child)
SubClassOf(ObjectSomeValuesFrom(a:hasChild a:Child)
a:Parent)
(hasChild some Child SubClassOf Parent)
we can infer
SubClassOf(a:PersonWithChild a:Parent)
Knowledge Technologies
Manolis Koubarakis
97
Equivalent Classes
•
A
n
e
q
u
i
v
a
l
e
n
t
c
l
a
s
s
e
s
a
x
i
o
m
E
q
u
i
v
a
l
e
n
t
C
l
a
s
s
e
s
(
C
E
1
.
.
.
C
E
n
)
s
t
a
t
e
s
t
h
a
t
a
l
l
o
f
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
C
E
i
,
1
≤
i
≤
n
,
a
r
e
s
e
m
a
n
t
i
c
a
l
l
y
e
q
u
i
v
a
l
e
n
t
t
o
e
a
c
h
o
t
h
e
r
.
•
Example:
EquivalentClasses(a:Boy
ObjectIntersectionOf(a:Child a:M
ale)
)
•
An axiom
EquivalentClasses(CE1 CE2)
is
equivalent to the
conjunction of the
following two axioms:
SubClassOf(CE1 CE2)
SubClassOf(CE2 CE1)
Knowledge Technologies
Manolis Koubarakis
98
Example Inferences
•
From
EquivalentClasses(a:Boy
ObjectIntersectionOf(a:Child a:Male))
ClassAssertion(a:Child a:Chris)
ClassAssertion(a:Male a:Chris)
we can infer
ClassAssertion(a:Boy a:Chris)
Knowledge Technologies
Manolis Koubarakis
99
Example Inferences
•
From
EquivalentClasses(a:MongrelOwner
ObjectSomeValuesFrom(a:hasPet a:Mongrel))
(
MongrelOwner ≡ hasPet some Mongrel
)
EquivalentClasses(a:DogOwner ObjectSomeValuesFrom(a:hasPet
a:Dog))
(
DogOwner ≡ hasPet some Dog
)
SubClassOf(a:Mongrel a:Dog)
(Mongrel SubClassOf Dog)
ClassAssertion(a:MongrelOwner a:Peter)
we can infer
SubClassOf(a:MongrelOwner a:DogOwner)
ClassAssertion(a:DogOwner a:Peter)
Knowledge Technologies
Manolis Koubarakis
100
Disjoint Classes
•
A
d
i
s
j
o
i
n
t
c
l
a
s
s
e
s
a
x
i
o
m
D
i
s
j
o
i
n
t
C
l
a
s
s
e
s
(
C
E
1
.
.
.
C
E
n
)
s
t
a
t
e
s
t
h
a
t
a
l
l
o
f
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
C
E
i
,
1
≤
i
≤
n
,
a
r
e
p
a
i
r
w
i
s
e
d
i
s
j
o
i
n
t
.
•
Example:
DisjointClasses(a:Boy a:Girl)
•
An axiom
DisjointClasses(CE1 CE2)
is equivalent
to the following axiom:
SubClassOf(CE1 ObjectComplementOf(CE
2
))
Knowledge Technologies
Manolis Koubarakis
101
Disjoint Union of Classes
•
A
d
i
s
j
o
i
n
t
u
n
i
o
n
a
x
i
o
m
D
i
s
j
o
i
n
t
U
n
i
o
n
(
C
C
E
1
.
.
.
C
E
n
)
s
t
a
t
e
s
t
h
a
t
a
c
l
a
s
s
C
i
s
a
d
i
s
j
o
i
n
t
u
n
i
o
n
o
f
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
s
C
E
i
,
1
≤
i
≤
n
,
a
l
l
o
f
w
h
i
c
h
a
r
e
p
a
i
r
w
i
s
e
d
i
s
j
o
i
n
t
.
•
S
u
c
h
a
x
i
o
m
s
a
r
e
s
o
m
e
t
i
m
e
s
r
e
f
e
r
r
e
d
t
o
a
s
c
o
v
e
r
i
n
g
a
x
i
o
m
s
,
a
s
t
h
e
y
s
t
a
t
e
t
h
a
t
t
h
e
e
x
t
e
n
s
i
o
n
s
o
f
a
l
l
C
E
i
e
x
a
c
t
l
y
c
o
v
e
r
t
h
e
e
x
t
e
n
s
i
o
n
o
f
C
.
•
Example:
DisjointUnion(a:Child a:Boy a:Girl)
•
Each such axiom
is equivalent to the conjunction of
the following two
axioms:
EquivalentClasses(C ObjectUnionOf(CE1 ... CEn))
DisjointClasses
(
CE1 ... CEn)
Boy
Girl
Child
Knowledge Technologies
Manolis Koubarakis
102
Example Inferences
•
From
DisjointUnion(a:Child a:Boy a:Girl)
ClassAssertion(a:Child a:Stewie)
ClassAssertion(ObjectComplementOf(a:Girl)
a:Stewie)
we can infer
ClassAssertion(a:Boy a:Stewie)
Knowledge Technologies
Manolis Koubarakis
103
Axioms (cont’d)
Knowledge Technologies
Manolis Koubarakis
104
Object Property Axioms
•
O
W
L
2
p
r
o
v
i
d
e
s
a
x
i
o
m
s
t
h
a
t
c
a
n
b
e
u
s
e
d
t
o
c
h
a
r
a
c
t
e
r
i
z
e
a
n
d
e
s
t
a
b
l
i
s
h
r
e
l
a
t
i
o
n
s
h
i
p
s
b
e
t
w
e
e
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
105
Object Property Axioms
Knowledge Technologies
Manolis Koubarakis
106
O
b
j
e
c
t
S
u
b
p
r
o
p
e
r
t
y
A
x
i
o
m
s
•
Object subproperty axioms are analogous to subclass
axioms
.
•
T
h
e
b
a
s
i
c
f
o
r
m
o
f
a
n
o
b
j
e
c
t
s
u
b
p
r
o
p
e
r
t
y
a
x
i
o
m
i
s
S
u
b
O
b
j
e
c
t
P
r
o
p
e
r
t
y
O
f
(
O
P
E
1
O
P
E
2
)
.
•
This axiom states that the object property expression
OPE1
is a subproperty of the object property expression
OPE2
— that is, if an individual
x
is connected by
OPE1
to an individual
y
, then
x
is also connected by
OPE2
to
y
.
•
SubObjectPropertyOf
is a reflexive and transitive
relation.
Knowledge Technologies
Manolis Koubarakis
107
Example Inferences
•
From
SubObjectPropertyOf(a:hasDog a:hasPet)
ObjectPropertyAssertion(a:hasDog a:Peter
a:Brian)
we can infer
ObjectPropertyAssertion(a:hasPet a:Peter
a:Brian)
Knowledge Technologies
Manolis Koubarakis
108
O
b
j
e
c
t
S
u
b
p
r
o
p
e
r
t
y
A
x
i
o
m
s
:
I
n
c
l
u
s
i
o
n
s
w
i
t
h
P
r
o
p
e
r
t
y
C
h
a
i
n
s
•
If
OPE1, …, OPEn
are object properties then
O
P
E
1
…
O
P
E
n
i
s
c
a
l
l
e
d
a
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
c
h
a
i
n
.
•
The more complex form
of object subproperty axioms
is
SubObjectPropertyOf(
ObjectPropertyChain(OPE1 ... OPEn) OPE).
•
This axiom states that, if an individual
x
1
is connected by a
sequence of object property expressions
OPE1, ..., OPEn
with
an individual
x
n
, then
x
1
is also connected with
x
n
by the object
property expression
OPE
.
•
T
h
e
s
e
a
x
i
o
m
s
a
r
e
k
n
o
w
n
a
s
c
o
m
p
l
e
x
r
o
l
e
i
n
c
l
u
s
i
o
n
s
i
n
t
h
e
D
L
l
i
t
e
r
a
t
u
r
e
.
x
1
x
2
x
3
x
n
R
1
R
2
R
3
R
N
R
Knowledge Technologies
Manolis Koubarakis
109
Example Inferences
•
From
SubObjectPropertyOf(
ObjectPropertyChain(a:hasMother a:hasSister)
a:hasAunt)
ObjectPropertyAssertion(a:hasMother a:Stewie a:Lois)
ObjectPropertyAssertion(a:hasSister a:Lois
a:Carol)
we can infer
ObjectPropertyAssertion(a:hasAunt a:Stewie a:Carol)
x
1
x
2
x
3
hasMother
hasAunt
hasSister
Knowledge Technologies
Manolis Koubarakis
110
Equivalent Object Properties
•
A
n
e
q
u
i
v
a
l
e
n
t
o
b
j
e
c
t
p
r
o
p
e
r
t
i
e
s
a
x
i
o
m
E
q
u
i
v
a
l
e
n
t
O
b
j
e
c
t
P
r
o
p
e
r
t
i
e
s
(
O
P
E
1
.
.
.
O
P
E
n
)
s
t
a
t
e
s
t
h
a
t
a
l
l
o
f
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
O
P
E
i
,
1
≤
i
≤
n
,
a
r
e
s
e
m
a
n
t
i
c
a
l
l
y
e
q
u
i
v
a
l
e
n
t
t
o
e
a
c
h
o
t
h
e
r
.
•
The axiom
EquivalentObjectProperties(OPE1
OPE2)
is equivalent to the following two axioms:
SubObjectPropertyOf(OPE1 OPE2)
SubObjectPropertyOf(OPE2 OPE1)
Knowledge Technologies
Manolis Koubarakis
111
Example Inferences
•
From
EquivalentObjectProperties(a:hasBrother a:hasMaleSibling)
ObjectPropertyAssertion(a:hasBrother a:Chris a:Stewie)
ObjectPropertyAssertion(a:hasMaleSibling a:Stewie a:Chris)
we can infer
ObjectPropertyAssertion(a:hasMaleSibling a:Chris a:Stewie)
ObjectPropertyAssertion(a:hasBrother a:Stewie a:Chris)
Knowledge Technologies
Manolis Koubarakis
112
Disjoint Object Properties
•
A
d
i
s
j
o
i
n
t
o
b
j
e
c
t
p
r
o
p
e
r
t
i
e
s
a
x
i
o
m
D
i
s
j
o
i
n
t
O
b
j
e
c
t
P
r
o
p
e
r
t
i
e
s
(
O
P
E
1
.
.
.
O
P
E
n
)
s
t
a
t
e
s
t
h
a
t
a
l
l
o
f
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
O
P
E
i
,
1
≤
i
≤
n
,
a
r
e
p
a
i
r
w
i
s
e
d
i
s
j
o
i
n
t
.
•
Example:
DisjointObjectProperties(a:hasFather
a:hasMother)
Knowledge Technologies
Manolis Koubarakis
113
Inverse Object Properties
•
A
n
i
n
v
e
r
s
e
o
b
j
e
c
t
p
r
o
p
e
r
t
i
e
s
a
x
i
o
m
I
n
v
e
r
s
e
O
b
j
e
c
t
P
r
o
p
e
r
t
i
e
s
(
O
P
E
1
O
P
E
2
)
s
t
a
t
e
s
t
h
a
t
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
1
i
s
a
n
i
n
v
e
r
s
e
o
f
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
2
.
•
Each such axiom
is equivalent with
the following:
EquivalentObjectProperties(OPE1
ObjectInverseOf(OPE2))
Knowledge Technologies
Manolis Koubarakis
114
Example Inferences
•
From
InverseObjectProperties(a:hasFather a:fatherOf)
ObjectPropertyAssertion(a:hasFather a:Stewie
a:Peter)
ObjectPropertyAssertion(a:fatherOf a:Peter a:Chris)
we can infer
ObjectPropertyAssertion(a:fatherOf a:Peter a:Stewie)
ObjectPropertyAssertion(a:hasFather a:Chris a:Peter
)
Knowledge Technologies
Manolis Koubarakis
115
Object Property Domain Axioms
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
d
o
m
a
i
n
a
x
i
o
m
O
b
j
e
c
t
P
r
o
p
e
r
t
y
D
o
m
a
i
n
(
O
P
E
C
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
d
o
m
a
i
n
o
f
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
—
t
h
a
t
i
s
,
i
f
a
n
i
n
d
i
v
i
d
u
a
l
x
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
w
i
t
h
s
o
m
e
o
t
h
e
r
i
n
d
i
v
i
d
u
a
l
,
t
h
e
n
x
i
s
a
n
i
n
s
t
a
n
c
e
o
f
C
E
.
•
Each such axiom
is equivalent to
the following axiom:
SubClassOf(ObjectSomeValuesFrom(OPE
owl:Thing) CE)
Knowledge Technologies
Manolis Koubarakis
116
Example Inferences
•
From
ObjectPropertyDomain(a:hasDog a:Person)
ObjectPropertyAssertion(a:hasDog a:Peter
a:Brian)
we can infer
ClassAssertion(a:Person a:Peter)
Knowledge Technologies
Manolis Koubarakis
117
Object Property Range
Axioms
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
r
a
n
g
e
a
x
i
o
m
O
b
j
e
c
t
P
r
o
p
e
r
t
y
R
a
n
g
e
(
O
P
E
C
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
r
a
n
g
e
o
f
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
—
t
h
a
t
i
s
,
i
f
s
o
m
e
i
n
d
i
v
i
d
u
a
l
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
w
i
t
h
a
n
i
n
d
i
v
i
d
u
a
l
x
,
t
h
e
n
x
i
s
a
n
i
n
s
t
a
n
c
e
o
f
C
E
.
•
Each such axiom
is equivalent to
the following axiom:
SubClassOf(owl:Thing ObjectAllValuesFrom(OPE
CE))
Knowledge Technologies
Manolis Koubarakis
118
Example Inferences
•
From
ObjectPropertyRange(a:hasDog a:Dog)
ObjectPropertyAssertion(a:hasDog
a:Peter a:Brian)
we can infer
ClassAssertion(a:Dog a:Brian)
Knowledge Technologies
Manolis Koubarakis
119
Object Property Axioms
(cont’d)
Knowledge Technologies
Manolis Koubarakis
120
Functional Object Properties
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
f
u
n
c
t
i
o
n
a
l
i
t
y
a
x
i
o
m
F
u
n
c
t
i
o
n
a
l
O
b
j
e
c
t
P
r
o
p
e
r
t
y
(
O
P
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
f
u
n
c
t
i
o
n
a
l
—
t
h
a
t
i
s
,
f
o
r
e
a
c
h
i
n
d
i
v
i
d
u
a
l
x
,
t
h
e
r
e
c
a
n
b
e
a
t
m
o
s
t
o
n
e
d
i
s
t
i
n
c
t
i
n
d
i
v
i
d
u
a
l
y
s
u
c
h
t
h
a
t
x
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
y
.
•
Each such axiom
is equivalent to
the following
axiom:
SubClassOf(owl:Thing
ObjectMaxCardinality(1 OPE))
Knowledge Technologies
Manolis Koubarakis
121
Example Inferences
•
From
FunctionalObjectProperty(a:hasFather)
ObjectPropertyAssertion(a:hasFather a:Stewie
a:Peter)
ObjectPropertyAssertion(a:hasFather a:Stewie
a:Peter_Griffin)
What do we infer?
Knowledge Technologies
Manolis Koubarakis
122
Example Inferences
•
From
FunctionalObjectProperty(a:hasFather)
ObjectPropertyAssertion(a:hasFather a:Stewie
a:Peter)
ObjectPropertyAssertion(a:hasFather a:Stewie
a:Peter_Griffin)
What do we infer?
SameIndividual(a:Peter a:Peter_Griffin
)
Knowledge Technologies
Manolis Koubarakis
123
Inverse-Functional Object
Properties
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
i
n
v
e
r
s
e
f
u
n
c
t
i
o
n
a
l
i
t
y
a
x
i
o
m
I
n
v
e
r
s
e
F
u
n
c
t
i
o
n
a
l
O
b
j
e
c
t
P
r
o
p
e
r
t
y
(
O
P
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
i
n
v
e
r
s
e
-
f
u
n
c
t
i
o
n
a
l
—
t
h
a
t
i
s
,
f
o
r
e
a
c
h
i
n
d
i
v
i
d
u
a
l
x
,
t
h
e
r
e
c
a
n
b
e
a
t
m
o
s
t
o
n
e
i
n
d
i
v
i
d
u
a
l
y
s
u
c
h
t
h
a
t
y
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
w
i
t
h
x
.
•
Each such axiom
is equivalent to
the following axiom:
SubClassOf(owl:Thing
ObjectMaxCardinality(1
ObjectInverseOf(OPE)))
What do we infer?
Knowledge Technologies
Manolis Koubarakis
124
Example Inferences
•
From
InverseFunctionalObjectProperty(a:fatherOf)
ObjectPropertyAssertion(a:fatherOf a:Peter a:Stewie)
ObjectPropertyAssertion(a:fatherOf a:Peter_Griffin
a:Stewie)
What do we infer?
Knowledge Technologies
Manolis Koubarakis
125
Example Inferences
•
From
InverseFunctionalObjectProperty(a:fatherOf)
ObjectPropertyAssertion(a:fatherOf a:Peter a:Stewie)
ObjectPropertyAssertion(a:fatherOf a:Peter_Griffin
a:Stewie)
What do we infer?
SameIndividual(a:Peter a:Peter_Griffin
)
Knowledge Technologies
Manolis Koubarakis
126
Reflexive Object Properties
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
r
e
f
l
e
x
i
v
i
t
y
a
x
i
o
m
R
e
f
l
e
x
i
v
e
O
b
j
e
c
t
P
r
o
p
e
r
t
y
(
O
P
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
r
e
f
l
e
x
i
v
e
—
t
h
a
t
i
s
,
e
a
c
h
i
n
d
i
v
i
d
u
a
l
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
i
t
s
e
l
f
.
•
Each such axiom
is equivalent
to
the following
axiom:
SubClassOf(owl:Thing ObjectHasSelf(
OPE))
Knowledge Technologies
Manolis Koubarakis
127
Example Inferences
•
From
ReflexiveObjectProperty(a:knows)
we can infer
ObjectPropertyAssertion(a:knows
a:Peter a:Peter)
Knowledge Technologies
Manolis Koubarakis
128
Irreflexive Object Properties
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
i
r
r
e
f
l
e
x
i
v
i
t
y
a
x
i
o
m
I
r
r
e
f
l
e
x
i
v
e
O
b
j
e
c
t
P
r
o
p
e
r
t
y
(
O
P
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
i
r
r
e
f
l
e
x
i
v
e
—
t
h
a
t
i
s
,
n
o
i
n
d
i
v
i
d
u
a
l
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
i
t
s
e
l
f
.
•
Each such axiom
is equivalent to
the following
axiom:
SubClassOf(ObjectHasSelf(OPE)
owl:Nothing)
Knowledge Technologies
Manolis Koubarakis
129
Symmetric Object Properties
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
s
y
m
m
e
t
r
y
a
x
i
o
m
S
y
m
m
e
t
r
i
c
O
b
j
e
c
t
P
r
o
p
e
r
t
y
(
O
P
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
s
y
m
m
e
t
r
i
c
—
t
h
a
t
i
s
,
i
f
a
n
i
n
d
i
v
i
d
u
a
l
x
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
a
n
i
n
d
i
v
i
d
u
a
l
y
,
t
h
e
n
y
i
s
a
l
s
o
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
x
.
•
Example:
SymmetricObjectProperty(a:friend)
•
Each such axiom
is equivalent to
the following axiom:
SubObjectPropertyOf(OPE
ObjectInverseOf(OPE))
Knowledge Technologies
Manolis Koubarakis
130
Asymmetric Object Properties
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
a
s
y
m
m
e
t
r
y
a
x
i
o
m
A
s
y
m
m
e
t
r
i
c
O
b
j
e
c
t
P
r
o
p
e
r
t
y
(
O
P
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
a
s
y
m
m
e
t
r
i
c
—
t
h
a
t
i
s
,
i
f
a
n
i
n
d
i
v
i
d
u
a
l
x
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
a
n
i
n
d
i
v
i
d
u
a
l
y
,
t
h
e
n
y
c
a
n
n
o
t
b
e
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
x
.
•
Example
AsymmetricObjectProperty(a:parentOf)
Knowledge Technologies
Manolis Koubarakis
131
Transitive Object Properties
•
A
n
o
b
j
e
c
t
p
r
o
p
e
r
t
y
t
r
a
n
s
i
t
i
v
i
t
y
a
x
i
o
m
T
r
a
n
s
i
t
i
v
e
O
b
j
e
c
t
P
r
o
p
e
r
t
y
(
O
P
E
)
s
t
a
t
e
s
t
h
a
t
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
i
s
t
r
a
n
s
i
t
i
v
e
—
t
h
a
t
i
s
,
i
f
a
n
i
n
d
i
v
i
d
u
a
l
x
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
a
n
i
n
d
i
v
i
d
u
a
l
y
t
h
a
t
i
s
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
a
n
i
n
d
i
v
i
d
u
a
l
z
,
t
h
e
n
x
i
s
a
l
s
o
c
o
n
n
e
c
t
e
d
b
y
O
P
E
t
o
z
.
•
Each such axiom
is equivalent to
the following axiom:
SubObjectPropertyOf(ObjectPropertyChain(OPE
OPE) OPE)
Knowledge Technologies
Manolis Koubarakis
132
Example Inferences
•
From
TransitiveObjectProperty(a:ancestorOf)
ObjectPropertyAssertion(a:ancestorOf a:Carter
a:Lois)
ObjectPropertyAssertion(a:ancestorOf a:Lois a:Meg)
we can infer
ObjectPropertyAssertion(a:ancestorOf a:Carter a:Meg)
Knowledge Technologies
Manolis Koubarakis
133
Axioms (cont’d)
Knowledge Technologies
Manolis Koubarakis
134
Data Property Axioms
Knowledge Technologies
Manolis Koubarakis
135
Data Property Axioms (cont’d)
•
OWL 2 also provides for data property
axioms. Their structure
and semantics
is
similar to
the corresponding
object
property axioms
.
•
We will not present data property axioms
in detail. We will only give some examples.
Knowledge Technologies
Manolis Koubarakis
136
Examples
•
From
SubDataPropertyOf(a:hasLastName a:hasName)
DataPropertyAssertion(a:hasLastName a:Peter
"Griffin")
we can infer
DataPropertyAssertion(a:hasName a:Peter
"Griffin")
Knowledge Technologies
Manolis Koubarakis
137
Examples (cont’d)
•
The ontology
FunctionalDataProperty(a:hasAge)
DataPropertyAssertion(a:hasAge a:Meg
"17"^^xsd:integer)
DataPropertyAssertion(a:hasAge a:Meg
"17.0"^^xsd:decimal)
DataPropertyAssertion(a:hasAge a:Meg "+17"^^xsd:int)
is consistent because the different age literals given map to the same
value.
Knowledge Technologies
Manolis Koubarakis
138
Examples (cont’d)
•
The ontology
FunctionalDataProperty(a:numberOfChildren)
DataPropertyAssertion(a:numberOfChildren
a:Meg "+0"^^xsd:float)
DataPropertyAssertion(a:numberOfChildren
a:Meg "-0"^^xsd:float)
is unsatisfiable because l
iterals
"+0"^^xsd:float
and
"-0"^^xsd:float
are mapped to distinct data values
+0
and
-0
in the value space of
xs
d
:float
; these data
values are equal, but not identical.
Knowledge Technologies
Manolis Koubarakis
139
Axioms (cont’d)
Knowledge Technologies
Manolis Koubarakis
140
Datatype Definitions
•
A
d
a
t
a
t
y
p
e
d
e
f
i
n
i
t
i
o
n
D
a
t
a
t
y
p
e
D
e
f
i
n
i
t
i
o
n
(
D
T
D
R
)
d
e
f
i
n
e
s
a
n
e
w
d
a
t
a
t
y
p
e
D
T
a
s
b
e
i
n
g
s
e
m
a
n
t
i
c
a
l
l
y
e
q
u
i
v
a
l
e
n
t
t
o
t
h
e
d
a
t
a
r
a
n
g
e
D
R
;
t
h
e
l
a
t
t
e
r
m
u
s
t
b
e
a
u
n
a
r
y
d
a
t
a
r
a
n
g
e
.
•
T
h
e
d
a
t
a
t
y
p
e
s
d
e
f
i
n
e
d
b
y
d
a
t
a
t
y
p
e
d
e
f
i
n
i
t
i
o
n
a
x
i
o
m
s
s
u
p
p
o
r
t
n
o
f
a
c
e
t
s
s
o
t
h
e
y
m
u
s
t
n
o
t
o
c
c
u
r
i
n
d
a
t
a
t
y
p
e
r
e
s
t
r
i
c
t
i
o
n
s
.
Knowledge Technologies
Manolis Koubarakis
141
Example
DatatypeDefinition(a:SSN
DatatypeRestriction(xsd:string
xsd:pattern "[0-9]{3}-[0-9]{2}-[0-9]{4}"))DataPropertyRange(a:hasSSN
a:SSN)
Knowledge Technologies
Manolis Koubarakis
142
Axioms (cont’d)
Knowledge Technologies
Manolis Koubarakis
143
Keys
•
A
k
e
y
a
x
i
o
m
HasKey(CE (OPE1 ... OPEm) (DPE1 ... DPEn))
s
t
a
t
e
s
t
h
a
t
e
a
c
h
n
a
m
e
d
i
n
s
t
a
n
c
e
o
f
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
i
s
u
n
i
q
u
e
l
y
i
d
e
n
t
i
f
i
e
d
b
y
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
O
P
E
i
a
n
d
/
o
r
t
h
e
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
D
P
E
j
.
•
I
n
t
h
i
s
c
a
s
e
,
n
o
t
w
o
d
i
s
t
i
n
c
t
,
n
a
m
e
d
i
n
s
t
a
n
c
e
s
o
f
C
E
c
a
n
c
o
i
n
c
i
d
e
o
n
t
h
e
v
a
l
u
e
s
o
f
a
l
l
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
O
P
E
i
a
n
d
a
l
l
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
s
D
P
E
j
.
•
A key axiom of the form
HasKey(owl:Thing (OPE) ())
is similar to the
axiom
InverseFunctionalObjectProperty(OPE).
Their main
difference is that the former axiom is applicable only to individuals that are
explicitly named in an ontology, while the latter axiom is also applicable to
unnamed individuals.
Knowledge Technologies
Manolis Koubarakis
144
Example Inferences
•
From
HasKey(owl:Thing () ( a:hasSSN))
DataPropertyAssertion(a:hasSSN a:Peter "123-45-
6789")
DataPropertyAssertion(a:hasSSN a:Peter_Griffin "123-
45-6789")
we can infer
SameIndividual(a:Peter a:Peter_Griffin)
Knowledge Technologies
Manolis Koubarakis
145
Example Inferences
•
From
HasKey(a:GriffinFamilyMember () (a:hasName))
DataPropertyAssertion(a:hasName a:Peter "Peter")
ClassAssertion(a:GriffinFamilyMember a:Peter)
DataPropertyAssertion(a:hasName a:Peter_Griffin "Peter")
ClassAssertion(a:GriffinFamilyMember a:Peter_Griffin)
DataPropertyAssertion(a:hasName a:StPeter "Peter")
we can infer
SameIndividual(a:Peter a:Peter_Griffin)
Knowledge Technologies
Manolis Koubarakis
146
Example
•
The ontology
HasKey(a:GriffinFamilyMember () (a:hasName))
DataPropertyAssertion(a:hasName a:Peter "Peter")
DataPropertyAssertion(a:hasName a:Peter "Kichwa-
Tembo")
ClassAssertion(a:GriffinFamilyMember a:Peter)
is consistent because a
key axiom does not make all the properties
used in it functional.
Knowledge Technologies
Manolis Koubarakis
147
Axioms (cont’d)
Knowledge Technologies
Manolis Koubarakis
148
Declarations
•
In
an OWL 2 ontology
,
the entities (individuals,
classes, properties) used
can be, and
sometimes even needs to be, declared
.
•
D
e
c
l
a
r
a
t
i
o
n
s
a
r
e
n
o
n
l
o
g
i
c
a
l
a
x
i
o
m
s
.
T
h
e
y
h
a
v
e
n
o
s
e
m
a
n
t
i
c
s
b
u
t
c
a
n
a
l
l
o
w
O
W
L
2
t
o
o
l
s
t
o
c
a
t
c
h
e
r
r
o
r
s
.
•
D
e
c
l
a
r
a
t
i
o
n
s
a
r
e
o
p
t
i
o
n
a
l
.
B
u
t
i
n
O
W
L
D
L
c
l
a
s
s
e
s
,
d
a
t
a
t
y
p
e
s
a
n
d
p
r
o
p
e
r
t
i
e
s
o
f
v
a
r
i
o
u
s
k
i
n
d
s
n
e
e
d
t
o
b
e
d
e
c
l
a
r
e
d
a
s
s
u
c
h
.
Knowledge Technologies
Manolis Koubarakis
149
BNF for Entity Declarations
Declaration :=
'Declaration' '('axiomAnnotations
Entity
')‘
Entity :=
'Class' '('Class
')'
|
'Datatype' '('Datatype
')'
|
'ObjectProperty' '('ObjectProperty
')'
|
'DataProperty' '('DataProperty
')'
|
'AnnotationProperty' '('AnnotationProperty
')'
|
'NamedIndividual' '('NamedIndividual
')'
Knowledge Technologies
Manolis Koubarakis
150
Example
Declaration(Class(a:Person))
Declaration(NamedIndividual(a:Peter))
ClassAssertion(a:Person a:Peter)
Knowledge Technologies
Manolis Koubarakis
151
Axioms (cont’d)
Knowledge Technologies
Manolis Koubarakis
152
A
s
s
e
r
t
i
o
n
s
•
O
W
L
2
s
u
p
p
o
r
t
s
a
r
i
c
h
s
e
t
o
f
a
x
i
o
m
s
f
o
r
s
t
a
t
i
n
g
a
s
s
e
r
t
i
o
n
s
a
b
o
u
t
i
n
d
i
v
i
d
u
a
l
s
:
–
Individual equality
–
Individual inequality
–
Class assertion
–
Positive object property assertion
–
Negative object property assertion
–
Positive data property assertion
–
Negative data property assertion
•
A
s
s
e
r
t
i
o
n
s
a
r
e
o
f
t
e
n
a
l
s
o
c
a
l
l
e
d
f
a
c
t
s
.
T
h
e
y
a
r
e
p
a
r
t
o
f
t
h
e
A
B
o
x
i
n
D
L
s
.
Knowledge Technologies
Manolis Koubarakis
153
Individual Equality
Axiom
•
A
n
i
n
d
i
v
i
d
u
a
l
e
q
u
a
l
i
t
y
a
x
i
o
m
S
a
m
e
I
n
d
i
v
i
d
u
a
l
(
a
1
.
.
.
a
n
)
s
t
a
t
e
s
t
h
a
t
a
l
l
o
f
t
h
e
i
n
d
i
v
i
d
u
a
l
s
a
i
,
1
≤
i
≤
n
,
a
r
e
e
q
u
a
l
t
o
e
a
c
h
o
t
h
e
r
.
Knowledge Technologies
Manolis Koubarakis
154
Example Inference
•
From
SameIndividual(a:Meg a:Megan)
ObjectPropertyAssertion(a:hasBrother a:Meg
a:Stewie)
we can infer
ObjectPropertyAssertion(a:hasBrother a:Megan
a:Stewie)
Knowledge Technologies
Manolis Koubarakis
155
Individual Inequality
Axiom
•
A
n
i
n
d
i
v
i
d
u
a
l
i
n
e
q
u
a
l
i
t
y
a
x
i
o
m
D
i
f
f
e
r
e
n
t
I
n
d
i
v
i
d
u
a
l
s
(
a
1
.
.
.
a
n
)
s
t
a
t
e
s
t
h
a
t
a
l
l
o
f
t
h
e
i
n
d
i
v
i
d
u
a
l
s
a
i
,
1
≤
i
≤
n
,
a
r
e
d
i
f
f
e
r
e
n
t
f
r
o
m
e
a
c
h
o
t
h
e
r
.
•
Example:
DifferentIndividuals(a:Peter a:Meg
a:Chris a:Stewie)
Knowledge Technologies
Manolis Koubarakis
156
Class Assertions
•
A
c
l
a
s
s
a
s
s
e
r
t
i
o
n
C
l
a
s
s
A
s
s
e
r
t
i
o
n
(
C
E
a
)
s
t
a
t
e
s
t
h
a
t
t
h
e
i
n
d
i
v
i
d
u
a
l
a
i
s
a
n
i
n
s
t
a
n
c
e
o
f
t
h
e
c
l
a
s
s
e
x
p
r
e
s
s
i
o
n
C
E
.
•
Example:
ClassAssertion(a:Dog a:Brian)
Knowledge Technologies
Manolis Koubarakis
157
Object Property Assertions
•
A
p
o
s
i
t
i
v
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
a
s
s
e
r
t
i
o
n
O
b
j
e
c
t
P
r
o
p
e
r
t
y
A
s
s
e
r
t
i
o
n
(
O
P
E
a
1
a
2
)
s
t
a
t
e
s
t
h
a
t
t
h
e
i
n
d
i
v
i
d
u
a
l
a
1
i
s
c
o
n
n
e
c
t
e
d
b
y
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
t
o
t
h
e
i
n
d
i
v
i
d
u
a
l
a
2
.
•
A
n
e
g
a
t
i
v
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
a
s
s
e
r
t
i
o
n
N
e
g
a
t
i
v
e
O
b
j
e
c
t
P
r
o
p
e
r
t
y
A
s
s
e
r
t
i
o
n
(
O
P
E
a
1
a
2
)
s
t
a
t
e
s
t
h
a
t
t
h
e
i
n
d
i
v
i
d
u
a
l
a
1
i
s
n
o
t
c
o
n
n
e
c
t
e
d
b
y
t
h
e
o
b
j
e
c
t
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
O
P
E
t
o
t
h
e
i
n
d
i
v
i
d
u
a
l
a
2
.
Knowledge Technologies
Manolis Koubarakis
158
Examples
ObjectPropertyAssertion(a:hasDog
a:Peter a:Brian)
NegativeObjectPropertyAssertion(
a:hasSon a:Peter a:Meg)
Knowledge Technologies
Manolis Koubarakis
159
Data Property Assertions
•
A
p
o
s
i
t
i
v
e
d
a
t
a
p
r
o
p
e
r
t
y
a
s
s
e
r
t
i
o
n
D
a
t
a
P
r
o
p
e
r
t
y
A
s
s
e
r
t
i
o
n
(
D
P
E
a
l
t
)
s
t
a
t
e
s
t
h
a
t
t
h
e
i
n
d
i
v
i
d
u
a
l
a
i
s
c
o
n
n
e
c
t
e
d
b
y
t
h
e
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
D
P
E
t
o
t
h
e
l
i
t
e
r
a
l
l
t
.
•
A
n
e
g
a
t
i
v
e
d
a
t
a
p
r
o
p
e
r
t
y
a
s
s
e
r
t
i
o
n
N
e
g
a
t
i
v
e
D
a
t
a
P
r
o
p
e
r
t
y
A
s
s
e
r
t
i
o
n
(
D
P
E
a
l
t
)
s
t
a
t
e
s
t
h
a
t
t
h
e
i
n
d
i
v
i
d
u
a
l
a
i
s
n
o
t
c
o
n
n
e
c
t
e
d
b
y
t
h
e
d
a
t
a
p
r
o
p
e
r
t
y
e
x
p
r
e
s
s
i
o
n
D
P
E
t
o
t
h
e
l
i
t
e
r
a
l
l
t
.
Knowledge Technologies
Manolis Koubarakis
160
Example Inference
•
From
DataPropertyAssertion(a:hasAge a:Meg "17"^^xsd:integer)
SubClassOf(
DataSomeValuesFrom(a:hasAge
DatatypeRestriction(xsd:integer
xsd:minInclusive "13"^^xsd:integer
xsd:maxInclusive "19"^^xsd:integer
)
)
a:Teenager
)
we can infer
ClassAssertion(a:Teenager a:Meg)
Knowledge Technologies
Manolis Koubarakis
161
Annotations
•
O
W
L
2
a
p
p
l
i
c
a
t
i
o
n
s
o
f
t
e
n
n
e
e
d
w
a
y
s
t
o
a
s
s
o
c
i
a
t
e
a
d
d
i
t
i
o
n
a
l
i
n
f
o
r
m
a
t
i
o
n
w
i
t
h
o
n
t
o
l
o
g
i
e
s
,
e
n
t
i
t
i
e
s
,
a
n
d
a
x
i
o
m
s
.
T
o
t
h
i
s
e
n
d
,
O
W
L
2
p
r
o
v
i
d
e
s
f
o
r
a
n
n
o
t
a
t
i
o
n
s
o
n
o
n
t
o
l
o
g
i
e
s
,
a
x
i
o
m
s
,
a
n
d
e
n
t
i
t
i
e
s
.
•
A
n
n
o
t
a
t
i
o
n
s
a
r
e
f
i
r
s
t
-
c
l
a
s
s
c
i
t
i
z
e
n
s
i
n
O
W
L
2
;
t
h
e
i
r
s
t
r
u
c
t
u
r
e
i
s
i
n
d
e
p
e
n
d
e
n
t
o
f
t
h
e
u
n
d
e
r
l
y
i
n
g
s
y
n
t
a
x
a
n
d
t
h
e
y
a
r
e
d
i
f
f
e
r
e
n
t
t
h
a
n
c
o
m
m
e
n
t
s
t
h
a
t
a
s
y
n
t
a
x
(
e
.
g
.
,
O
W
L
X
M
L
)
m
i
g
h
t
a
l
l
o
w
.
•
Annotations have no formal semantics, thus they do not
participate in the meaning of an ontology (under the
OWL 2 direct semantics).
Knowledge Technologies
Manolis Koubarakis
162
Axioms (cont’d)
Knowledge Technologies
Manolis Koubarakis
163
Annotation of Entities and
Anonymous Individuals
•
T
he
axiom
AnnotationAssertion
(
AP as av
)
states
that the annotation subject
as
is annotated with the
annotation property
AP
(user defined or built-in)
and the
annotation value
av
.
•
a
s
c
a
n
b
e
a
n
e
n
t
i
t
y
(
i
.
e
.
,
i
n
d
i
v
i
d
u
a
l
,
c
l
a
s
s
o
r
p
r
o
p
e
r
t
y
)
o
r
a
n
a
n
o
n
y
m
o
u
s
i
n
d
i
v
i
d
u
a
l
.
•
Example:
AnnotationAssertion(rdfs:label a:Person
"Represents the set of all people.")
Knowledge Technologies
Manolis Koubarakis
164
Annotations of Axioms, Annotations
and Ontologies
•
OWL 2 also provides the construct
Annotation
({A} AP v)where
AP
is an
annotation property
(user defined or built-in)
,
v
is
a literal, an IRI, or an anonymous individual and
{A}are 0 or more annotations.
•
T
h
e
a
b
o
v
e
c
o
n
s
t
r
u
c
t
c
a
n
b
e
u
s
e
d
f
o
r
a
n
n
o
t
a
t
i
o
n
s
o
f
a
x
i
o
m
s
a
n
d
o
n
t
o
l
o
g
i
e
s
.
I
t
c
a
n
a
l
s
o
b
e
u
s
e
d
f
o
r
a
n
n
o
t
a
t
i
o
n
s
o
f
a
n
n
o
t
a
t
i
o
n
s
t
h
e
m
s
e
l
v
e
s
.
Knowledge Technologies
Manolis Koubarakis
165
Examples
SubClassOf(
Annotation(rdfs:comment "
Persons
are humans
.")
a:Person
a
:
Human
)
Knowledge Technologies
Manolis Koubarakis
166
Examples (cont’d)
Prefix(
ex
:=<http://www.example.com/ontology1#>)
Prefix(owl:=<http://www.w3.org/2002/07/owl#>)
Ontology(<http://www.example.com/ontology1>
Import(<http://www.example.com/ontology2>)
Annotation(rdfs:label "An example
ontology
")
SubClassOf(
ex
:Child owl:Thing)
)
Knowledge Technologies
Manolis Koubarakis
167
Annotation Properties
•
Various annotation properties can be defined by users
(e.g., an integer ID in the Foundational Model of
Anatomy ontology; see
http://sig.biostr.washington.edu/projects/fm/AboutFM.htm
l
).
•
To help users in their modeling, OWL 2 also offers the
constructs:
–
SubAnnotationPropertyOf(AP1 AP2)
states that the
annotation property
AP1
is a subproperty of the annotation
property
AP2
.
–
AnnotationPropertyDomain(AP U)
states that the domain
of the annotation property
AP
is the IRI
U
.
–
AnnotationPropertyRange(AP U)
states that the range of
the annotation property
AP
is the IRI
U
.
Knowledge Technologies
Manolis Koubarakis
168
Metamodeling
•
O
W
L
2
e
n
a
b
l
e
s
m
e
t
a
m
o
d
e
l
i
n
g
b
y
a
l
l
o
w
i
n
g
t
h
e
s
a
m
e
I
R
I
I
t
o
r
e
f
e
r
t
o
m
o
r
e
t
h
a
n
o
n
e
t
y
p
e
o
f
e
n
t
i
t
y
(
e
.
g
.
,
a
n
i
n
d
i
v
i
d
u
a
l
a
n
d
a
c
l
a
s
s
)
.
T
h
i
s
i
s
c
a
l
l
e
d
“
p
u
n
n
i
n
g
”
i
n
t
h
e
l
i
t
e
r
a
t
u
r
e
.
•
Example:
ClassAssertion(
a
:Father
a
:John)
ClassAssertion(
a
:SocialRole
a
:Father)
•
In the above example, IRI
a:Father
is first used as a class and
then as an individual.
•
T
h
e
d
i
r
e
c
t
m
o
d
e
l
-
t
h
e
o
r
e
t
i
c
s
e
m
a
n
t
i
c
s
o
f
O
W
L
2
a
c
c
o
m
m
o
d
a
t
e
s
t
h
i
s
b
y
u
n
d
e
r
s
t
a
n
d
i
n
g
t
h
e
c
l
a
s
s
a
:
F
a
t
h
e
r
a
n
d
t
h
e
i
n
d
i
v
i
d
u
a
l
a
:
F
a
t
h
e
r
a
s
t
w
o
d
i
f
f
e
r
e
n
t
v
i
e
w
s
o
n
t
h
e
s
a
m
e
I
R
I
,
i
.
e
.
t
h
e
y
a
r
e
i
n
t
e
r
p
r
e
t
e
d
s
e
m
a
n
t
i
c
a
l
l
y
a
s
i
f
t
h
e
y
w
e
r
e
d
i
s
t
i
n
c
t
.
Knowledge Technologies
Manolis Koubarakis
169
Semantics
•
There are two alternative ways of assigning meaning to
ontologies in OWL 2
:
–
T
h
e
d
i
r
e
c
t
m
o
d
e
l
-
t
h
e
o
r
e
t
i
c
s
e
m
a
n
t
i
c
s
.
T
h
i
s
p
r
o
v
i
d
e
s
a
m
e
a
n
i
n
g
f
o
r
O
W
L
2
i
n
a
D
L
s
t
y
l
e
b
y
u
n
d
e
r
s
t
a
n
d
i
n
g
O
W
L
2
c
o
n
s
t
r
u
c
t
s
a
s
c
o
n
s
t
r
u
c
t
s
o
f
t
h
e
D
L
S
R
O
I
Q
(
w
i
t
h
t
h
e
e
x
c
e
p
t
i
o
n
o
f
d
a
t
a
t
y
p
e
s
a
n
d
p
u
n
n
i
n
g
t
h
a
t
a
r
e
n
o
t
i
n
c
l
u
d
e
d
i
n
S
R
O
I
Q
b
u
t
s
t
i
l
l
c
o
v
e
r
e
d
b
y
t
h
i
s
s
e
m
a
n
t
i
c
s
)
.
S
e
e
h
t
t
p
:
/
/
w
w
w
.
w
3
.
o
r
g
/
T
R
/
o
w
l
2
-
d
i
r
e
c
t
-
s
e
m
a
n
t
i
c
s
/
.
Knowledge Technologies
Manolis Koubarakis
170
Semantics
•
There are two alternative ways of assigning meaning to
ontologies in OWL 2
:
–
T
h
e
R
D
F
-
b
a
s
e
d
s
e
m
a
n
t
i
c
s
.
T
h
i
s
i
s
a
n
e
x
t
e
n
s
i
o
n
o
f
t
h
e
s
e
m
a
n
t
i
c
s
o
f
R
D
F
S
(
D
-
e
n
t
a
i
l
m
e
n
t
i
n
p
a
r
t
i
c
u
l
a
r
)
a
n
d
i
s
b
a
s
e
d
o
n
v
i
e
w
i
n
g
O
W
L
2
o
n
t
o
l
o
g
i
e
s
a
s
R
D
F
g
r
a
p
h
s
.
F
o
r
t
h
e
e
x
a
c
t
r
e
l
a
t
i
o
n
s
h
i
p
o
f
t
h
e
t
w
o
s
e
m
a
n
t
i
c
s
,
s
e
e
h
t
t
p
:
/
/
w
w
w
.
w
3
.
o
r
g
/
T
R
/
o
w
l
2
-
r
d
f
-
b
a
s
e
d
-
s
e
m
a
n
t
i
c
s
/
.
Knowledge Technologies
Manolis Koubarakis
171
OWL 2 DL and OWL 2 Full
•
I
n
f
o
r
m
a
l
l
y
,
t
h
e
n
o
t
i
o
n
"
O
W
L
2
D
L
"
i
s
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D
L
.
•
See the OWL 2 Structural Specification and Functional-
Style Syntax for details.
Knowledge Technologies
Manolis Koubarakis
172
Examples of Additional Conditions
in OWL 2 DL
•
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)
.
Knowledge Technologies
Manolis Koubarakis
173
Axiom Closure of an Ontology
•
T
h
e
i
m
p
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.
Knowledge Technologies
Manolis Koubarakis
174
Example (not OWL 2 DL!)
Prefix(ex:=<http://www.example.com/ontology1#>)
Prefix(owl:=<http://www.w3.org/2002/07/owl#>)
Ontology(<http://www.example.com/ontology1>
SubObjectPropertyOf(
ObjectPropertyChain(ex:hasFather ex:hasBrother)
ex:hasUncle)
EquivalentClasses(ex:PersonWithThreeUncles
ObjectExactCardinality(3 ex:hasUncle ex:Person))
)
N
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.
Knowledge Technologies
Manolis Koubarakis
175
OWL 2 DL and OWL 2 Full (cont’d)
•
W
e
c
a
n
t
h
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n
k
o
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2
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a
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d
O
W
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2
F
u
l
l
i
n
t
w
o
w
a
y
s
:
–
OWL 2 DL
is
a syntactically restricted version of OWL
2 Full
.
OWL 2 Full is undecidable
while
OWL 2 DL
is
not. T
here are several production quality reasoners
that cover the entire OWL 2 DL language
(e.g., Pellet
and HermiT).
–
OWL 2 Full
is an
extension of RDFS. As such, the
RDF-Based Semantics for OWL 2 Full follows the
RDFS semantics and general syntactic philosophy
(i.e., everything is a triple and the language is fully
reflective).
Knowledge Technologies
Manolis Koubarakis
176
OWL 2 Profiles
•
I
n
a
d
d
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t
i
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n
t
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2
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p
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t
i
e
s
a
l
r
e
a
d
y
.
•
The
OWL 2 Profiles document provides a clear template for
specifying additional profiles.
Knowledge Technologies
Manolis Koubarakis
177
OWL 2 EL
•
The OWL 2 EL profile is a subset of OWL 2 that
–
i
s
p
a
r
t
i
c
u
l
a
r
l
y
s
u
i
t
a
b
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a
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g
o
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t
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s
t
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a
t
d
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v
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y
l
a
r
g
e
n
u
m
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s
o
f
c
l
a
s
s
e
s
a
n
d
/
o
r
p
r
o
p
e
r
t
i
e
s
,
–
captures the expressive power used by many such ontologies, and
–
f
o
r
w
h
i
c
h
o
n
t
o
l
o
g
y
c
o
n
s
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t
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n
c
y
,
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l
a
s
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e
x
p
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s
s
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o
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s
u
b
s
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m
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t
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n
,
a
n
d
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s
t
a
n
c
e
c
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k
i
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g
c
a
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d
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c
i
d
e
d
i
n
p
o
l
y
n
o
m
i
a
l
t
i
m
e
.
•
E
xample
:
OWL 2 EL
is
sufficient to express the very large biomedical ontology
SNOMED CT
. The specialized reasoner ELK can classify all 400,000 classes of this
ontology in 5 seconds using 4 cores (
http://www.cs.ox.ac.uk/isg/tools/ELK/
).
•
The acronym EL comes from the fact that the profile is based on the DL family of
languages EL. See the relevant paper
–
Pushing the EL Envelope
.
Franz Baader, Sebastian Brandt, and Carsten Lutz. In
Proc. of the 19th Joint Int. Conf. on Artificial Intelligence (IJCAI 2005), 2005
.
Available from
http://lat.inf.tu-
dresden.de/research/papers/2005/BaaderBrandtLutz-IJCAI-05.pdf
Knowledge Technologies
Manolis Koubarakis
178
OWL 2 EL Specification
•
T
y
p
e
s
o
f
c
l
a
s
s
r
e
s
t
r
i
c
t
i
o
n
s
a
l
l
o
w
e
d
:
–
existential quantification to a class expression
(
ObjectSomeValuesFrom
) or a data range
(
DataSomeValuesFrom
)
–
existential quantification to an individual
(
ObjectHasValue
) or a literal (
DataHasValue
)
–
self-restriction (
ObjectHasSelf
)
–
e
n
u
m
e
r
a
t
i
o
n
s
i
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(
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)
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l
e
l
i
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e
r
a
l
(
D
a
t
a
O
n
e
O
f
)
–
intersection of classes (
ObjectIntersectionOf
)
and data ranges (
DataIntersectionOf
)
Knowledge Technologies
Manolis Koubarakis
179
OWL 2 EL Specification (cont’d)
•
T
y
p
e
s
o
f
a
x
i
o
m
s
a
l
l
o
w
e
d
:
–
class inclusion (
SubClassOf
)
–
class equivalence (
EquivalentClasses
)
–
class disjointness (
DisjointClasses
)
–
object property inclusion (
SubObjectPropertyOf
)
with or without property chains, and data property
inclusion (
SubDataPropertyOf
)
–
property equivalence
(
EquivalentObjectProperties
and
EquivalentDataProperties
)
Knowledge Technologies
Manolis Koubarakis
180
OWL 2 EL Specification (cont’d)
–
transitive object properties (
TransitiveObjectProperty
)
–
reflexive object properties (
ReflexiveObjectProperty
)
–
domain restrictions (
ObjectPropertyDomain
and
DataPropertyDomain
)
–
range restrictions (
ObjectPropertyRange
and
DataPropertyRange
)
–
assertions (
SameIndividual, DifferentIndividuals,
ClassAssertion, ObjectPropertyAssertion,
DataPropertyAssertion,
NegativeObjectPropertyAssertion,
and
NegativeDataPropertyAssertion
)
–
functional data properties (
FunctionalDataProperty
)
–
keys (
HasKey
)
Knowledge Technologies
Manolis Koubarakis
181
OWL 2 EL Specification (cont’d)
•
C
o
n
s
t
r
u
c
t
s
n
o
t
s
u
p
p
o
r
t
e
d
:
–
universal quantification to a class expression
(
ObjectAllValuesFrom
) or a data range
(
DataAllValuesFrom
)
–
cardinality restrictions (
ObjectMaxCardinality
,
ObjectMinCardinality
,
ObjectExactCardinality
,
DataMaxCardinality
,
DataMinCardinality
, and
DataExactCardinality
)
–
disjunction (
ObjectUnionOf
,
DisjointUnion
, and
DataUnionOf
)
–
class negation (
ObjectComplementOf
)
–
enumerations involving more than one individual (
ObjectOneOf
and
DataOneOf
)
Knowledge Technologies
Manolis Koubarakis
182
OWL 2 EL Specification (cont’d)
–
disjoint properties (
DisjointObjectProperties
and
DisjointDataProperties
)
–
irreflexive object properties
(
IrreflexiveObjectProperty
)
–
inverse object properties
(
InverseObjectProperties
)
–
functional and inverse-functional object properties
(
FunctionalObjectProperty
and
InverseFunctionalObjectProperty
)
–
symmetric object properties
(
SymmetricObjectProperty
)
–
asymmetric object properties
(
AsymmetricObjectProperty
)
Knowledge Technologies
Manolis Koubarakis
183
OWL 2 QL
•
T
h
e
O
W
L
2
Q
L
p
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o
f
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i
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2
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l
a
t
i
o
n
a
l
d
a
t
a
b
a
s
e
s
.
•
I
n
t
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i
s
p
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f
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l
e
s
o
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n
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OBDA
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186
OWL Syntaxes (cont’d)
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Knowledge Technologies
Manolis Koubarakis
187
Example
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Jack is a person but not a parent.
Knowledge Technologies
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188
Functional-Style Syntax
ClassAssertion(
ObjectIntersectionOf(:Person
ObjectComplementOf(:Parent))
:
Jack
)
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189
RDF/XML Syntax
<rdf:Description rdf:about="Jack">
<rdf:type>
<owl:Class>
<owl:intersectionOf
rdf:parseType="Collection">
<owl:Class rdf:about="Person"/>
<owl:Class>
<owl:complementOf rdf:resource="Parent"/>
</owl:Class>
</owl:intersectionOf>
</owl:Class>
</rdf:type>
</rdf:Description>
Knowledge Technologies
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190
Turtle Syntax
:Jack rdf:type [
rdf:type owl:Class;
owl:intersectionOf (
:Person
[ rdf:type owl:Class;
owl:complementOf :Parent
]
)
] .
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191
Manchester Syntax
Individual: Jack
Types: Person and not Parent
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192
OWL/XML Syntax
<ClassAssertion>
<ObjectIntersectionOf>
<Class IRI="Person"/>
<ObjectComplementOf>
<Class IRI="Parent"/>
</ObjectComplementOf>
</ObjectIntersectionOf>
<NamedIndividual IRI="Jack"/>
</ClassAssertion>
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Readings
•
The document
http://www.w3.org/TR/2009/REC-owl2-overview-
20091027/
gives an overview of the OWL 2 specification of the W3C
OWL Working Group. In the documents referenced there, you will
find all the information that you may need.
•
You should read at least the Primer (
http://www.w3.org/TR/owl2-
primer/
) and Structural Specification and Functional Style Syntax
(
http://www.w3.org/TR/owl2-syntax/
) .
•
The following survey paper introduces OWL 2, explains its
relationship with DL
SROIQ,
and discusses various OWL tools and
applications:
–
Ian Horrocks and Peter F. Patel-Schneider. KR and Reasoning on the
Semantic Web: OWL. In Handbook of Semantic Web Technologies,
chapter 9. Springer, 2010. Available from
http://www.cs.ox.ac.uk/people/ian.horrocks/Publications/download/2010/
HoPa10a.pdf
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194
Readings (cont’d)
•
The DL
SROIQ
on which OWL 2 is based is fully described in the
paper
–
The Even More Irresistible SROIQ. Ian Horrocks, Oliver Kutz, and Uli
Sattler. In Proc. of the 10th Int. Conf. on Principles of Knowledge
Representation and Reasoning (KR 2006). AAAI Press, 2006. Available
from
http://www.cs.manchester.ac.uk/~sattler/publications/sroiq-TR.pdf
.
Explore the basics of OWL 2, the Web Ontology Language, through a presentation by Manolis Koubarakis. Learn about OWL 2's structural specification, functional syntax, semantics, and profiles. Delve into the terminology of OWL 2, understanding individuals, classes, properties, expressions, and axioms. Discover how OWL 2 is used to represent knowledge on the web, building upon concepts from description logics.
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Presentation Transcript
An Introduction to OWL 2 Knowledge Technologies Manolis Koubarakis 1
Acknowledgement This presentation is based on the OWL 2 Web Ontology Language Structural Specification and Functional-Style Syntax available at http://www.w3.org/TR/owl2- syntax/ Much of the material in this presentation is verbatim from the above specification. Knowledge Technologies Manolis Koubarakis 2
Outline Features of OWL 2 Structural Specification Functional Syntax Other Syntaxes Examples Semantics of OWL 2 OWL 2 Profiles Knowledge Technologies Manolis Koubarakis 3
The Semantic Web Layer Cake Knowledge Technologies Manolis Koubarakis 4
OWL 2 Basics OWL 2 is the current version of the Web Ontology Language and a W3C recommendation as of October 2009. The previous version of OWL (OWL 1) became a W3C recommendation in 2004. All W3C documents about OWL 2 can be found at http://www.w3.org/TR/2009/REC-owl2- overview-20091027/ . Knowledge Technologies Manolis Koubarakis 5
The Structure of OWL 2 Knowledge Technologies Manolis Koubarakis 6
OWL 2 Basics (contd) OWL 2 is a language for writing ontologies for the Web. It is based on well-known concepts and results from description logics. Like DLs, OWL 2 is a language for representing knowledge about things, groups of things, and relations between things. Knowledge Technologies Manolis Koubarakis 7
OWL 2 Terminology The things or objects about which knowledge is represented (e.g., John, Mary) are called individuals. Groups of things (e.g., female) are called classes. Relations between things (e.g., married) are called properties. Individuals, classes and properties are called entities. Knowledge Technologies Manolis Koubarakis 8
OWL 2 Terminology (contd) As in DLs, entities can be combined using constructors to form complex descriptions called expressions. To represent knowledge in OWL (like in any other KR language), we make statements. These statements are called axioms. Knowledge Technologies Manolis Koubarakis 9
Annotations Entities, expressions and axioms form the logical part of OWL 2. They can be given a precise semantics and inferences can be drawn from them. In addition, entities, axioms, and ontologies can be annotated. Example: A class can be given a human-readable label that provides a more descriptive name for the class. Annotations have no effect on the logical aspects of an ontology. For the purposes of the OWL 2 semantics, annotations are treated as not being present. Knowledge Technologies Manolis Koubarakis 10
IRIs Ontologies and their elements are identified using International Resource Identifiers (IRIs). In OWL 2, an IRI can be written in full or it can be abbreviated as prefix:lname as in XML qualified names where prefix is a namespace and lname is the local name with respect to the namespace. Knowledge Technologies Manolis Koubarakis 11
The Structure of an Ontology Knowledge Technologies Manolis Koubarakis 12
Ontology IRI and Version IRIs An ontology may have an ontology IRI, which is used to identify it. If an ontology has an ontology IRI, the ontology may additionally have a version IRI, which is used to identify the version of the ontology. The version IRI may, but need not be equal to the ontology IRI. An ontology series is identified using an ontology IRI, and each version in the series is assigned a different version IRI. Only one version of the ontology is the current one. Example: Ontology IRI: <http://www.example.com/my> Version IRIs: <http://www.example.com/my/1.0>, <http://www.example.com/my/2.0>, An ontology without an ontology IRI must not contain a version IRI. Ontology IRIs and version IRIs should satisfy various uniqueness constraints that OWL 2 tools should check, for detecting possible problems. Knowledge Technologies Manolis Koubarakis 13
Ontology Document Each ontology is associated with an ontology document which physically contains the ontology stored in a particular way (e.g., a text file). An ontology document should be accessible via the IRIs determined by the rules defined in the W3C specification. Example: The document of the current version of an ontology should always be accessible via the ontology IRI and the current version IRI. Knowledge Technologies Manolis Koubarakis 14
Imports An OWL 2 ontology can import (directly or indirectly) other ontologies in order to gain access to their entities, expressions and axioms, thus providing the basic facility for ontology modularization. Example: an ontology of sensors can import a geospatial ontology to specify the location of sensors. Knowledge Technologies Manolis Koubarakis 15
OWL 2 Syntaxes The Functional-Style syntax. This syntax is designed to be easier for specification purposes and to provide a foundation for the implementation of OWL 2 tools such as APIs and reasoners. This is the syntax we will use in this presentation. The RDF/XML syntax: this is just RDF/XML, with a particular translation for the OWL constructs. Here one can use other popular syntaxes for RDF, e.g., Turtle syntax. The Manchester syntax: this is a frame-based syntax that is designed to be easier for users to read. The OWL XML syntax: this is an XML syntax for OWL defined by an XML schema. OWL Syntax Converter: http://owl.cs.manchester.ac.uk/tools/webapps/owl-syntax-converter/ Knowledge Technologies Manolis Koubarakis 16
BNF Grammar for the Functional Syntax of OWL 2 ontologyDocument := { prefixDeclaration } Ontology prefixDeclaration := 'Prefix' '(' prefixName :=' fullIRI ')' Ontology := 'Ontology' '(' [ ontologyIRI [ versionIRI ] ] directlyImportsDocuments ontologyAnnotations axioms ')' ontologyIRI := IRI versionIRI := IRI directlyImportsDocuments := { 'Import' '(' IRI ')' } axioms := { Axiom } Knowledge Technologies Manolis Koubarakis 17
Example Prefix(ex:=<http://www.example.com/ontology1#>) Prefix(owl:=<http://www.w3.org/2002/07/owl#>) Ontology(<http://www.example.com/ontology1> Import(<http://www.example.com/ontology2>) Annotation(rdfs:label "An example ontology") SubClassOf(ex:Person owl:Thing) SubClassOf(ex:Male ex:Person) SubClassOf(ex:Female ex:Person) ) Knowledge Technologies Manolis Koubarakis 18
Things One Can Define in OWL 2 Knowledge Technologies Manolis Koubarakis 19
Classes Classes (e.g., a:Female) represent sets of individuals. Built-in classes: owl:Thing, which represents the set of all individuals. owl:Nothing, which represents the empty set. Knowledge Technologies Manolis Koubarakis 20
Things One Can Define in OWL 2 (cont d) Knowledge Technologies Manolis Koubarakis 21
Object Properties Object properties (e.g., a:parentOf) connect pairs of individuals. Built-in object properties: owl:topObjectProperty, which connects all possible pairs of individuals. owl:bottomObjectProperty, which does not connect any pair of individuals. Knowledge Technologies Manolis Koubarakis 22
Object Property Expressions Object properties can be used to form object property expressions. Knowledge Technologies Manolis Koubarakis 23
Inverse Object Property Expressions An inverse object property expression ObjectInverseOf(P) connects an individual I1 with I2 if and only if the object property P connects I2 with I1. P I2 I1 ObjectInverseOf(P) Example: If an ontology contains the axiom ObjectPropertyAssertion(a:fatherOf a:Peter a:Stewie) then the ontology entails ObjectPropertyAssertion(ObjectInverseOf(a:fatherOf) a:Stewie a:Peter) Knowledge Technologies Manolis Koubarakis 24
Things One Can Define in OWL 2 (cont d) Knowledge Technologies Manolis Koubarakis 25
Data Properties Data properties (e.g., a:hasAge) connect individuals with literals. Built-in properties: owl:topDataProperty, which connects all possible individuals with all literals. owl:bottomDataProperty, which does not connect any individual with a literal. Knowledge Technologies Manolis Koubarakis 26
Data Property Expressions The only allowed data property expression is a data property. Knowledge Technologies Manolis Koubarakis 27
Things One Can Define in OWL 2 (cont d) Knowledge Technologies Manolis Koubarakis 28
Annotation Properties Annotation properties can be used to provide an annotation for an ontology, axiom, or an IRI. Users can define their own annotation properties (we will see how later on) or use the available built-in annotation properties: rdfs:label, rdfs:comment, rdfs:seeAlso, rdfs:isDefinedBy owl:deprecated, owl:versionInfo, owl:priorVersion, owl:backwardCompatibleWith, owl:incompatibleWith Knowledge Technologies Manolis Koubarakis 29
Things One Can Define in OWL 2 (cont d) Knowledge Technologies Manolis Koubarakis 30
Individuals Individuals represent actual objects from the domain. There are two types of individuals: Named individuals are given an explicit name (an IRI e.g., a:Peter) that can be used in any ontology to refer to the same object. Anonymous individuals do not have a global name. They can be defined using a name (e.g., _:somebody) local to the ontology they are contained in. They are like blank nodes in RDF. Knowledge Technologies Manolis Koubarakis 31
Things One Can Define in OWL 2 (cont d) Knowledge Technologies Manolis Koubarakis 32
Things One Can Define in OWL 2 (cont d) Knowledge Technologies Manolis Koubarakis 33
Datatypes Datatypes are entities that represent sets of data values. OWL 2 offers a rich set of data types: decimal numbers, integers, floating point numbers, rationals, reals, strings, binary data, IRIs and time instants. In most cases, these data types are taken from XML schema. From RDF and RDFS, we have rdf:XMLLiteral, rdf:PlainLiteral and rdfs:Literal. rdfs:Literal contains all the elements of other data types. There are also the OWL datatypes owl:real and owl:rational. Formally, the data types supported are specified in the OWL 2 datatype map. Knowledge Technologies Manolis Koubarakis 34
Datatypes (contd) In a datatype map, each datatype is identified by an IRI and is defined by the following components: The value space is the set of values of the datatype. Elements of the value space are called data values. The lexical space is a set of strings that can be used to refer to data values. Each member of the lexical space is called a lexical form, and it is mapped to a particular data value. The facet space is a set of pairs of the form (F,v) where F is an IRI called a constraining facet, and v is an arbitrary data value called the constraining value. Each such pair is mapped to a subset of the value space of the datatype. Knowledge Technologies Manolis Koubarakis 35
Facet Space For the XML Schema datatypes xsd:double, xsd:float, and xsd:decimal, the constraining facets allowed are: xsd:minInclusive, xsd:maxInclusive, xsd:minExclusive and xsd:maxExclusive. Example: The pair(xsd:minInclusive,v) of the facet space denotes the set of all numbers x from the value space of the datatype such that x=v or x>v. Similarly for other datatypes. We will see later how constraining facets can be used to define data ranges. Knowledge Technologies Manolis Koubarakis 36
Literals Literals represent data values such as particular strings or integers. They are analogous to RDF literals. Examples: "1"^^xsd:integer (typed literal) "Family Guy" (plain literal, an abbreviation for "Family Guy"^^rdf:PlainLiteral) "Padre de familia"@es (plain literal with language tag, an abbreviation for "Padre de familia@es"^^rdf:PlainLiteral) Knowledge Technologies Manolis Koubarakis 37
Things One Can Define in OWL 2 (cont d) Knowledge Technologies Manolis Koubarakis 38
Data Ranges Data ranges represent sets of tuples of literals. They are defined using datatypes and constraining facets. Examples: The set of integers greater than 10. The set of strings that contain good as a substring. The set of (x,y) such that x and y are integers and x < y. Each data range is associated with a positive arity, which determines the size of its tuples. Datatypes are themselves data ranges of arity 1. Data ranges are used in restrictions on data properties, as we will see later when we define class expressions. Knowledge Technologies Manolis Koubarakis 39
Data Ranges Knowledge Technologies Manolis Koubarakis 40
BNF for Data Ranges DataRange := Datatype | DataIntersectionOf | DataUnionOf | DataComplementOf | DataOneOf | DatatypeRestriction DataIntersectionOf := 'DataIntersectionOf' '(' DataRange DataRange { DataRange } ')' DataUnionOf := 'DataUnionOf' '(' DataRange DataRange { DataRange } ')' DataComplementOf := 'DataComplementOf' '(' DataRange ')' DataOneOf := 'DataOneOf' '(' Literal { Literal } ')' Knowledge Technologies Manolis Koubarakis 41
Examples DataIntersectionOf(xsd:nonNegativeInteger xsd:nonPositiveInteger) DataUnionOf(xsd:string xsd:integer) DataComplementOf(xsd:positiveInteger) DataOneOf("Peter" "John") Knowledge Technologies Manolis Koubarakis 42
Datatype Restrictions DatatypeRestriction := 'DatatypeRestriction' '(' Datatype constrainingFacet restrictionValue { constrainingFacet restrictionValue } ') constrainingFacet := IRI restrictionValue := Literal Knowledge Technologies Manolis Koubarakis 43
Datatype Restrictions A datatype restrictionDatatypeRestriction(DT F1 lt1 ... Fn ltn) consists of a unary datatype DT and n pairs(Fi,lti) where Fi is a constraining facet of DT and lti a literal value. The data range represented by a datatype restriction is unary and is obtained by restricting the value space of DT according to the conjunction of all (Fi,lti). Observation: Thus, although the definition of data range speaks of tuples of any arity, the syntax defined allows only unary data ranges. Knowledge Technologies Manolis Koubarakis 44
Example The following data type restriction represents the set of integers 5, 6, 7, 8, and 9: DatatypeRestriction(xsd:integer xsd:minInclusive "5"^^xsd:integer xsd:maxExclusive "10"^^xsd:integer) Knowledge Technologies Manolis Koubarakis 45
Things One Can Define in OWL 2 (cont d) Knowledge Technologies Manolis Koubarakis 46
Class Expressions Class names and property expressions can be used to construct class expressions. These are essentially the complex concepts or descriptions that we can define in DLs. Class expressions represent sets of individuals by formally specifying conditions on the individuals' properties; individuals satisfying these conditions are said to be instances of the respective class expressions. Knowledge Technologies Manolis Koubarakis 47
Ways to Form Class Expressions Class expressions can be formed by: Applying the standard Boolean connectives to simpler class expressions or by enumerating the individuals that belong to an expression. Placing restrictions on object property expressions. Placing restrictions on the cardinality of object property expressions. Placing restrictions on data property expressions. Placing restrictions on the cardinality of data property expressions. Knowledge Technologies Manolis Koubarakis 48
Boolean Connectives and Enumeration of Individuals Knowledge Technologies Manolis Koubarakis 49
Intersection Class Expressions An intersection class expression ObjectIntersectionOf(CE1 ... CEn) contains all individuals that are instances of all class expressions CEi for 1 i n. Example: ObjectIntersectionOf(a:Dog a:CanTalk) Knowledge Technologies Manolis Koubarakis 50
