W3C W3C Member Submission

OWL 1.1 Web Ontology Language
Model-Theoretic Semantics

W3C Member Submission 19 December 2006

This version:
http://www.w3.org/Submission/2006/SUBM-owl11-semantics-20061219/
Latest version:
http://www.w3.org/Submission/owl11-semantics/
Authors:
Bernardo Cuenca Grau, The University of Manchester
Boris Motik, The University of Manchester
Contributors:
Ian Horrocks, The University of Manchester
Bijan Parsia, The University of Manchester
Peter F. Patel-Schneider, Bell Labs Research, Lucent Technologies
Ulrike Sattler, The University of Manchester

Abstract

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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Table of Contents


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

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

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

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 yCIc }
ObjectAllValuesFrom(R C) { x | ∀ y : ( x, y )RIpo implies yCIc }
ObjectHasValue(R a) { x | ( x, aIi )RIpo }
ObjectExistsSelf(R) { x | ( x, x )RIpo }
ObjectMinCardinality(n R C) { x | #{ y | ( x, y )RIpo and yCIc } ≥ n }
ObjectMaxCardinality(n R C) { x | #{ y | ( x, y )RIpo and yCIc } ≤ n }
ObjectExactCardinality(n R C) { x | #{ y | ( x, y )RIpo and yCIc } = n }
DatatSomeValuesFrom(U1 ... Un DR) { x | ∃ y1, ..., yn : ( x, yk )UkIpd for each 1 ≤ kn and ( y1, ..., yn )DRD }
DatatAllValuesFrom(U1 ... Un DR) { x | ∀ y1, ..., yn : ( x, yk )UkIpd for each 1 ≤ kn implies ( y1, ..., yn )DRD }
DataHasValue(U v) { x | ( x, vD )UIpd }
DataMinCardinality(n U DR) { x | #{ y | ( x, y )UIpd and yDRD } ≥ n }
DataMaxnCardinality(n U DR) { x | #{ y | ( x, y )UIpd and yDRD } ≤ n }
DataExactCardinality(n U DR) { x | #{ y | ( x, y )UIpd and yDRD } = 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) CIcDIc
EquivalentClasses(C1 ... Cn) CjIc = CkIc for each 1 ≤ j , kn
DisjointClasses(C1 ... Cn) CjIcCkIc is empty for each 1 ≤ j , kn and jk
DisjointUnion(A C1 ... Cn) AIc = C1Ic ∪ ... ∪ CnIc and CjIcCkIc is empty for each 1 ≤ j , kn and jk
SubObjectPropertyOf(R S) RIpoSIpo
SubObjectPropertyOf(SubObjectPropertyChain(R1 ... Rn) S) R1Ipo o ... o RnIpoSIpo
EquivalentObjectProperties(R1 ... Rn) RjIpo = RkIpo for each 1 ≤ j , kn
DisjointObjectProperties(R1 ... Rn) RjIpoRkIpo is empty for each 1 ≤ j , kn and jk
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) UIpdVIpd
EquivalentDataProperties(U1 ... Un) UjIpd = UkIpd for each 1 ≤ j , kn
DisjointDataProperties(U1 ... Un) UjIpdUkIpd is empty for each 1 ≤ j , kn and jk
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 , kn
DifferentIndividuals(a1 ... an) ajIiakIi for each 1 ≤ j , kn and jk
ClassAssertion(a C) aIiCIc
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.