OWL 1.1 extends the W3C OWL Web Ontology Language with a small but useful set of features that have been requested by users, for which effective reasoning algorithms are now available, and that OWL tool developers are willing to support. The new features include extra syntactic sugar, additional property and qualified cardinality constructors, extended datatype support, simple metamodeling, and extended annotations. This document provides a model-theoretic semantics for OWL 1.1.
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1 Introduction
This document defines the formal semantics of OWL 1.1. The semantics given here
follows the principles for defining the semantics of description logics
[Description Logics] and is
compatible with the description logic SROIQ presented in
[SROIQ]. Unfortunately, the definition of
SROIQ given in [SROIQ] does not provide
for datatypes and metamodeling. Therefore, the semantics of OWL 1.1 is defined in a direct
model-theoretic way, by interpreting the constructs of the functional-style syntax from
[OWL 1.1 Specification].
For the constructs available in SROIQ, the semantics of SROIQ
trivially corresponds to the one defined in this document.
OWL 1.1 does not have an RDF-compatible semantics. Ontologies expressed in OWL RDF
are given semantics by converting then into the functional-style syntax and interpreting
the result as specified in this document.
OWL 1.1 allows for annotations of ontologies and ontology axioms. Annotations,
however, have no semantic meaning in OWL 1.1 and are ignored in this document.
Since OWL 1.1 is an extension of OWL DL, this document also provides a formal semantics
for OWL DL and it is equivalent to the definition given in
[OWL Abstract Syntax and Semantics].
2 Model-Theoretic Semantics for OWL 1.1
A vocabulary (or signature)
V = ( NC , NPo , NPd , NI , ND , NV )
is a 6-tuple where
- NC is a set of OWL classes,
- NPo is a set of object properties,
- NPd is a set of data properties,
- NI is a set of individuals, and
- ND is a set of datatypes each associated with a positive integer datatype arity,
- NV is a set of well-formed constants.
Since OWL 1.1 allows punning [Metamodeling]
in the signature, we do not require the sets NC , NPo , NPd ,
NI , ND , and NV to be pair-wise disjoint. Thus,
the same name can be used in an ontology to denote a class, a datatype,
a property (object or data), an individual, and a constant. The set ND
is defined as it is because a datatype is defined by its name and the arity, and such a
definition allows one to reuse the same name with different arities.
The semantics of OWL 1.1 is defined with respect to a concrete domain,
which is a tuple D = ( ΔD , .D ) where
- ΔD is a fixed set called the data domain,
- .D assigns to each constant v ∈ NV an element vD of ΔD, and
- .D assigns to each datatype d ∈ ND with arity n an n-ary relation dD over ΔD.
The set of datatypes ND in each OWL 1.1 vocabulary
must include a unary datatype rdfs:Literal interpreted as ΔD;
furthermore, it must also include the following unary datatypes:
xsd:string,
xsd:boolean,
xsd:decimal,
xsd:float,
xsd:double,
xsd:dateTime,
xsd:time,
xsd:date,
xsd:gYearMonth,
xsd:gYear,
xsd:gMonthDay,
xsd:gDay,
xsd:gMonth,
xsd:hexBinary,
xsd:base64Binary,
xsd:anyURI,
xsd:normalizedString,
xsd:token,
xsd:language,
xsd:NMTOKEN,
xsd:Name,
xsd:NCName,
xsd:integer,
xsd:nonPositiveInteger,
xsd:negativeInteger,
xsd:long,
xsd:int,
xsd:short,
xsd:byte,
xsd:nonNegativeInteger,
xsd:unsignedLong,
xsd:unsignedInt,
xsd:unsignedShort,
xsd:unsignedByte, and
xsd:positiveInteger.
These datatypes, as well as the well-formed constants from NV, are interpreted as specified in
[XML Schema Datatypes].
The set ΔD is a fixed set that must be large enough; that is,
it must contain the extension of each datatype from ND and, apart from that, an
infinite number of other objects. Such a definition is ambiguous, as it does not uniquely single
out a particular set ΔD; however, the choice of the actual set is not
actually relevant for the definition of the semantics, as long as it contains the interpretations
of all datatypes that one can "reasonably think of." This allows the implementations to
support datatypes other than the ones mentioned in the previous paragraphs without affecting
the semantics.
Given a vocabulary V and a concrete domain D, an interpretation
I = ( ΔI , .Ic , .Ipo , .Ipd , .Ii ) is a 5-tuple where
- ΔI is a nonempty set disjoint with ΔD, called the object domain;
- .Ic is the class interpretation function that assigns to each OWL class A ∈ NIc a subset AIc of ΔI ;
- .Ipo is the object property interpretation function that assigns to each object property R ∈ NPo a subset RIpo of ΔI x ΔI ;
- .Ipd is the data property interpretation function that assigns to each data property U ∈ NPd a subset UIpd of ΔI x ΔD ;
- .Ii is the individual interpretation function that assigns to each individual a ∈ NI an element aIi from ΔI.
We extend the object interpretation function .Ipo to object property expressions as shown in Table 1.
Table 1. Interpreting Property Expressions
| Property |
Interpretation |
| InverseObjectProperty(R) |
{ ( x , y ) | ( y , x ) ∈ RIpo } |
We extend the interpretation function .D to data ranges as shown in Table 2.
Table 2. Interpreting Data Ranges
| Data Range |
Interpretation |
| DataOneOf(v1 ... vn) |
{ v1D , ... , vnD } |
| DataComplementOf(DR) |
( ΔD )n \ DRD where n is the arity of DR |
| DatatypeRestriction(DR f v) |
the n-ary relation over ΔD obtained by applying the facet f with value v
to the data range DR as specified in [XML Schema Datatypes] |
We extend the class interpretation function .Ic to classes as shown in Table 3.
With #S we denote the number of elements in a set S.
Table 3. Interpreting Classes
| Class |
Interpretation |
| owl:Thing |
ΔI |
| owl:Nothing |
empty set |
| ObjectComplementOf(C) |
ΔI \ CIc |
| ObjectIntersectionOf(C1 ... Cn) |
C1Ic ∩ ... ∩ CnIc |
| ObjectUnionOf(C1 ... Cn) |
C1Ic ∪ ... ∪ CnIc |
| ObjectOneOf(a1 ... an) |
{ a1Ii , ... , anIi } |
| ObjectSomeValuesFrom(R C) |
{ x | ∃ y : ( x, y ) ∈ RIpo and y ∈ CIc } |
| ObjectAllValuesFrom(R C) |
{ x | ∀ y : ( x, y ) ∈ RIpo implies y ∈ CIc } |
| ObjectHasValue(R a) |
{ x | ( x, aIi ) ∈ RIpo } |
| ObjectExistsSelf(R) |
{ x | ( x, x ) ∈ RIpo } |
| ObjectMinCardinality(n R C) |
{ x | #{ y | ( x, y ) ∈ RIpo and y ∈ CIc } ≥ n } |
| ObjectMaxCardinality(n R C) |
{ x | #{ y | ( x, y ) ∈ RIpo and y ∈ CIc } ≤ n } |
| ObjectExactCardinality(n R C) |
{ x | #{ y | ( x, y ) ∈ RIpo and y ∈ CIc } = n } |
| DatatSomeValuesFrom(U1 ... Un DR) |
{ x | ∃ y1, ..., yn : ( x, yk ) ∈ UkIpd for each 1 ≤ k ≤ n and ( y1, ..., yn ) ∈ DRD } |
| DatatAllValuesFrom(U1 ... Un DR) |
{ x | ∀ y1, ..., yn : ( x, yk ) ∈ UkIpd for each 1 ≤ k ≤ n implies ( y1, ..., yn ) ∈ DRD } |
| DataHasValue(U v) |
{ x | ( x, vD ) ∈ UIpd } |
| DataMinCardinality(n U DR) |
{ x | #{ y | ( x, y ) ∈ UIpd and y ∈ DRD } ≥ n } |
| DataMaxnCardinality(n U DR) |
{ x | #{ y | ( x, y ) ∈ UIpd and y ∈ DRD } ≤ n } |
| DataExactCardinality(n U DR) |
{ x | #{ y | ( x, y ) ∈ UIpd and y ∈ DRD } = n } |
Satisfaction of OWL 1.1 axioms in an interpretation I is defined as shown in Table 4.
With o we denote the composition of binary relations.
Table 4. Satisfaction of Axioms in an Interpretation
| Axiom |
Condition |
| SubClassOf(C D) |
CIc ⊆ DIc |
| EquivalentClasses(C1 ... Cn) |
CjIc = CkIc for each 1 ≤ j , k ≤ n |
| DisjointClasses(C1 ... Cn) |
CjIc ∩ CkIc is empty for each 1 ≤ j , k ≤ n and j ≠ k |
| DisjointUnion(A C1 ... Cn) |
AIc = C1Ic ∪ ... ∪ CnIc and CjIc ∩ CkIc is empty for each 1 ≤ j , k ≤ n and j ≠ k |
| SubObjectPropertyOf(R S) |
RIpo ⊆ SIpo |
| SubObjectPropertyOf(SubObjectPropertyChain(R1 ... Rn) S) |
R1Ipo o ... o RnIpo ⊆ SIpo |
| EquivalentObjectProperties(R1 ... Rn) |
RjIpo = RkIpo for each 1 ≤ j , k ≤ n |
| DisjointObjectProperties(R1 ... Rn) |
RjIpo ∩ RkIpo is empty for each 1 ≤ j , k ≤ n and j ≠ k |
| ObjectPropertyDomain(R C) |
{ x | ∃ y : (x , y ) ∈ RIpo } ⊆ CIc |
| ObjectPropertyRange(R C) |
{ y | ∃ x : (x , y ) ∈ RIpo } ⊆ CIc |
| InverseObjectProperties(R S) |
RIpo = { ( x , y ) | ( y , x ) ∈ SIpo } |
| FunctionalObjectProperty(R) |
( x , y1 ) ∈ RIpo and ( x , y2 ) ∈ RIpo imply y1 = y2 |
| InverseFunctionalObjectProperty(R) |
( x1 , y ) ∈ RIpo and ( x2 , y ) ∈ RIpo imply x1 = x2 |
| ReflexiveObjectProperty(R) |
x ∈ ΔI implies ( x , x ) ∈ RIpo |
| IrreflexiveObjectProperty(R) |
x ∈ ΔI implies ( x , x ) is not in RIpo |
| SymmetricObjectProperty(R) |
( x , y ) ∈ RIpo implies ( y , x ) ∈ RIpo |
| AntisymmetricObjectProperty(R) |
( x , y ) ∈ RIpo implies ( y , x ) is not in RIpo |
| TransitiveObjectProperty(R) |
RIpo o RIpo ⊆ RIpo |
| SubDataPropertyOf(U V) |
UIpd ⊆ VIpd |
| EquivalentDataProperties(U1 ... Un) |
UjIpd = UkIpd for each 1 ≤ j , k ≤ n |
| DisjointDataProperties(U1 ... Un) |
UjIpd ∩ UkIpd is empty for each 1 ≤ j , k ≤ n and j ≠ k |
| DataPropertyDomain(U C) |
{ x | ∃ y : (x , y ) ∈ UIpd } ⊆ CIc |
| DataPropertyRange(U DR) |
{ y | ∃ x : (x , y ) ∈ UIpd } ⊆ DRD |
| FunctionalDataProperty(U) |
( x , y1 ) ∈ UIpd and ( x , y2 ) ∈ UIpd imply y1 = y2 |
| SameIndividual(a1 ... an) |
ajIi = akIi for each 1 ≤ j , k ≤ n |
| DifferentIndividuals(a1 ... an) |
ajIi ≠ akIi for each 1 ≤ j , k ≤ n and j ≠ k |
| ClassAssertion(a C) |
aIi ∈ CIc |
| ObjectPropertyAssertion(R a b) |
( aIi , bIi ) ∈ RIpo |
| NegativeObjectPropertyAssertion(R a b) |
( aIi , bIi ) is not in RIpo |
| DataPropertyAssertion(U a v) |
( aIi , vD ) ∈ UIpd |
| NegativeDataPropertyAssertion(U a v) |
( aIi , vD ) is not in UIpd |
Let O be an OWL 1.1 ontology with vocabulary V.
O is consistent if an interpretation I exists that satisfies all the axioms of the
axiom closure of O (the axiom closure of O is defined in
[OWL 1.1 Specification]);
such I is then called a model of O.
A class C is satisfiable w.r.t. O if there is a model I of
O such that CIc is not empty. O
entails an OWL 1.1 ontology O' with vocabulary V if every model
of O is also a model of O'; furthermore, O and O' are
equivalent if O entails O' and O' entails O.
References
- [Description Logics]
-
The Description Logic Handbook.
Franz Baader, Diego Calvanese, Deborah McGuinness, Daniele Nardi, Peter Patel-Schneider, editors.
Cambridge University Press, 2003; and Description Logics Home Page.
- [Metamodeling]
-
On the Properties of Metamodeling in OWL.
Boris Motik. In Proceedings of ISWC-2005
- [OWL 1.1 Specification]
-
OWL 1.1 Web Ontology Language: Structural Specification and Functional-Style Syntax.
Peter F. Patel-Schneider, Ian Horrocks, and Boris Motik, eds., 2006.
- [OWL Abstract Syntax and Semantics]
-
OWL Web Ontology Language: Semantics and Abstract Syntax.
Peter F. Patel-Schneider, Pat Hayes, and Ian Horrocks, Editors, W3C Recommendation, 10 February 2004.
- [SROIQ]
-
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.
- [XML Schema Datatypes]
-
XML Schema Part 2: Datatypes Second Edition.
Paul V. Biron and Ashok Malhotra, eds.
W3C Recommendation 28 October 2004.