W3C

OWL 2 Web Ontology Language
Direct Semantics (Second Edition)

W3C Recommendation 11 December 2012

This version:
http://www.w3.org/TR/2012/REC-owl2-direct-semantics-20121211/
Latest version (series 2):
http://www.w3.org/TR/owl2-direct-semantics/
Latest Recommendation:
http://www.w3.org/TR/owl-direct-semantics
Previous version:
http://www.w3.org/TR/2012/PER-owl2-direct-semantics-20121018/
Editors:
Boris Motik, University of Oxford
Peter F. Patel-Schneider, Nuance Communications
Bernardo Cuenca Grau, University of Oxford
Contributors: (in alphabetical order)
Ian Horrocks, University of Oxford
Bijan Parsia, University of Manchester
Uli Sattler, University of Manchester

Please refer to the errata for this document, which may include some normative corrections.

A color-coded version of this document showing changes made since the previous version is also available.

This document is also available in these non-normative formats: PDF version.

See also translations.


Abstract

The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the Semantic Web with formally defined meaning. OWL 2 ontologies provide classes, properties, individuals, and data values and are stored as Semantic Web documents. OWL 2 ontologies can be used along with information written in RDF, and OWL 2 ontologies themselves are primarily exchanged as RDF documents. The OWL 2 Document Overview describes the overall state of OWL 2, and should be read before other OWL 2 documents.

This document provides the direct model-theoretic semantics for OWL 2, which is compatible with the description logic SROIQ. Furthermore, this document defines the most common inference problems for OWL 2.

Status of this Document

May Be Superseded

This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

Summary of Changes

There have been no substantive changes since the previous version. For details on the minor changes see the change log and color-coded diff.

Please Send Comments

Please send any comments to public-owl-comments@w3.org (public archive). Although work on this document by the OWL Working Group is complete, comments may be addressed in the errata or in future revisions. Open discussion among developers is welcome at public-owl-dev@w3.org (public archive).

Endorsed By W3C

This document has been reviewed by W3C Members, by software developers, and by other W3C groups and interested parties, and is endorsed by the Director as a W3C Recommendation. It is a stable document and may be used as reference material or cited from another document. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.

Patents

This document was produced by a group operating under the 5 February 2004 W3C Patent Policy. W3C maintains a public list of any patent disclosures made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent.


Table of Contents


1 Introduction

This document defines the direct model-theoretic semantics of OWL 2. The semantics given here is strongly related to the semantics of description logics [Description Logics] and it extends the semantics of the description logic SROIQ [SROIQ]. As the definition of SROIQ does not provide for datatypes and punning, the semantics of OWL 2 is defined directly on the constructs of the structural specification of OWL 2 [OWL 2 Specification] instead of by reference to SROIQ. For the constructs available in SROIQ, the semantics of SROIQ trivially corresponds to the one defined in this document.

Since each OWL 1 DL ontology is an OWL 2 ontology, this document also provides a direct semantics for OWL 1 Lite and OWL 1 DL ontologies; this semantics is equivalent to the direct model-theoretic semantics of OWL 1 Lite and OWL 1 DL [OWL 1 Semantics and Abstract Syntax]. Furthermore, this document also provides the direct model-theoretic semantics for the OWL 2 profiles [OWL 2 Profiles].

The semantics is defined for OWL 2 axioms and ontologies, which should be understood as instances of the structural specification [OWL 2 Specification]. Parts of the structural specification are written in this document using the functional-style syntax.

OWL 2 allows ontologies, anonymous individuals, and axioms to be annotated; furthermore, annotations themselves can contain additional annotations. All these types of annotations, however, have no semantic meaning in OWL 2 and are ignored in this document. OWL 2 declarations are used only to disambiguate class expressions from data ranges and object property from data property expressions in the functional-style syntax; therefore, they are not mentioned explicitly in this document.

2 Direct Model-Theoretic Semantics for OWL 2

This section specifies the direct model-theoretic semantics of OWL 2 ontologies.

2.1 Vocabulary

A datatype map, formalizing datatype maps from the OWL 2 Specification [OWL 2 Specification], is a 6-tuple D = ( NDT , NLS , NFS , ⋅ DT , ⋅ LS , ⋅ FS ) with the following components:

The set of datatypes NDT of a datatype map D is not required to contain all datatypes from the OWL 2 datatype map; this allows one to talk about subsets of the OWL 2 datatype map, which may be necessary for the various profiles of OWL 2. If, however, D contains a datatype DT from the OWL 2 datatype map, then NLS(DT), NFS(DT), (DT)DT, ( LV , DT )LS for each LVNLS(DT), and ( F , v )FS for each ( F , v ) ∈ NFS(DT) are required to coincide with the definitions for DT in the OWL 2 datatype map.

A vocabulary V = ( VC , VOP , VDP , VI , VDT , VLT , VFA ) over a datatype map D is a 7-tuple consisting of the following elements:

Given a vocabulary V, the following conventions are used in this document to denote different syntactic parts of OWL 2 ontologies:

2.2 Interpretations

Given a datatype map D and a vocabulary V over D, an interpretation I = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) for D and V is a 10-tuple with the following structure:

The following sections define the extensions of ⋅ OP, ⋅ DT, and ⋅ C to object property expressions, data ranges, and class expressions.

2.2.1 Object Property Expressions

The object property interpretation function ⋅ OP is extended to object property expressions as shown in Table 1.

Table 1. Interpreting Object Property Expressions
Object Property Expression Interpretation ⋅ OP
ObjectInverseOf( OP ) { ( x , y ) | ( y , x ) ∈ (OP)OP }

2.2.2 Data Ranges

The datatype interpretation function ⋅ DT is extended to data ranges as shown in Table 3. All datatypes in OWL 2 are unary, so each datatype DT is interpreted as a unary relation over ΔD — that is, as a set (DT)DTΔD. OWL 2 currently does not define data ranges of arity more than one; however, by allowing for n-ary data ranges, the syntax of OWL 2 provides a "hook" allowing implementations to introduce extensions such as comparisons and arithmetic. An n-ary data range DR is interpreted as an n-ary relation (DR)DT over ΔD — that is, as a set (DT)DTD)n

Table 3. Interpreting Data Ranges
Data Range Interpretation ⋅ DT
DataIntersectionOf( DR1 ... DRn ) (DR1)DT ∩ ... ∩ (DRn)DT
DataUnionOf( DR1 ... DRn ) (DR1)DT ∪ ... ∪ (DRn)DT
DataComplementOf( DR ) D)n \ (DR)DT where n is the arity of DR
DataOneOf( lt1 ... ltn ) { (lt1)LT , ... , (ltn)LT }
DatatypeRestriction( DT F1 lt1 ... Fn ltn ) (DT)DT ∩ ( F1 , lt1 )FA ∩ ... ∩ ( Fn , ltn )FA

2.2.3 Class Expressions

The class interpretation function ⋅ C is extended to class expressions as shown in Table 4. For S a set, #S denotes the number of elements in S.

Table 4. Interpreting Class Expressions
Class Expression Interpretation ⋅ C
ObjectIntersectionOf( CE1 ... CEn ) (CE1)C ∩ ... ∩ (CEn)C
ObjectUnionOf( CE1 ... CEn ) (CE1)C ∪ ... ∪ (CEn)C
ObjectComplementOf( CE ) ΔI \ (CE)C
ObjectOneOf( a1 ... an ) { (a1)I , ... , (an)I }
ObjectSomeValuesFrom( OPE CE ) { x | ∃ y : ( x, y ) ∈ (OPE)OP and y(CE)C }
ObjectAllValuesFrom( OPE CE ) { x | ∀ y : ( x, y ) ∈ (OPE)OP implies y(CE)C }
ObjectHasValue( OPE a ) { x | ( x , (a)I ) ∈ (OPE)OP }
ObjectHasSelf( OPE ) { x | ( x , x ) ∈ (OPE)OP }
ObjectMinCardinality( n OPE ) { x | #{ y | ( x , y ) ∈ (OPE)OP } ≥ n }
ObjectMaxCardinality( n OPE ) { x | #{ y | ( x , y ) ∈ (OPE)OP } ≤ n }
ObjectExactCardinality( n OPE ) { x | #{ y | ( x , y ) ∈ (OPE)OP } = n }
ObjectMinCardinality( n OPE CE ) { x | #{ y | ( x , y ) ∈ (OPE)OP and y(CE)C } ≥ n }
ObjectMaxCardinality( n OPE CE ) { x | #{ y | ( x , y ) ∈ (OPE)OP and y(CE)C } ≤ n }
ObjectExactCardinality( n OPE CE ) { x | #{ y | ( x , y ) ∈ (OPE)OP and y(CE)C } = n }
DataSomeValuesFrom( DPE1 ... DPEn DR ) { x | ∃ y1, ... , yn : ( x , yk ) ∈ (DPEk)DP for each 1 ≤ kn and ( y1 , ... , yn ) ∈ (DR)DT }
DataAllValuesFrom( DPE1 ... DPEn DR ) { x | ∀ y1, ... , yn : ( x , yk ) ∈ (DPEk)DP for each 1 ≤ kn imply ( y1 , ... , yn ) ∈ (DR)DT }
DataHasValue( DPE lt ) { x | ( x , (lt)LT ) ∈ (DPE)DP }
DataMinCardinality( n DPE ) { x | #{ y | ( x , y ) ∈ (DPE)DP} ≥ n }
DataMaxCardinality( n DPE ) { x | #{ y | ( x , y ) ∈ (DPE)DP } ≤ n }
DataExactCardinality( n DPE ) { x | #{ y | ( x , y ) ∈ (DPE)DP } = n }
DataMinCardinality( n DPE DR ) { x | #{ y | ( x , y ) ∈ (DPE)DP and y(DR)DT } ≥ n }
DataMaxCardinality( n DPE DR ) { x | #{ y | ( x , y ) ∈ (DPE)DP and y(DR)DT } ≤ n }
DataExactCardinality( n DPE DR ) { x | #{ y | ( x , y ) ∈ (DPE)DP and y(DR)DT } = n }

2.3 Satisfaction in an Interpretation

An axiom or an ontology is satisfied in an interpretation I = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) if the appropriate condition from the following sections holds.

2.3.1 Class Expression Axioms

Satisfaction of OWL 2 class expression axioms in I is defined as shown in Table 5.

Table 5. Satisfaction of Class Expression Axioms in an Interpretation
Axiom Condition
SubClassOf( CE1 CE2 ) (CE1)C(CE2)C
EquivalentClasses( CE1 ... CEn ) (CEj)C = (CEk)C for each 1 ≤ jn and each 1 ≤ kn
DisjointClasses( CE1 ... CEn ) (CEj)C(CEk)C = ∅ for each 1 ≤ jn and each 1 ≤ kn such that jk
DisjointUnion( C CE1 ... CEn ) (C)C = (CE1)C ∪ ... ∪ (CEn)C and
(CEj)C(CEk)C = ∅ for each 1 ≤ jn and each 1 ≤ kn such that jk

2.3.2 Object Property Expression Axioms

Satisfaction of OWL 2 object property expression axioms in I is defined as shown in Table 6.

Table 6. Satisfaction of Object Property Expression Axioms in an Interpretation
Axiom Condition
SubObjectPropertyOf( OPE1 OPE2 ) (OPE1)OP(OPE2)OP
SubObjectPropertyOf( ObjectPropertyChain( OPE1 ... OPEn ) OPE ) y0 , ... , yn : ( y0 , y1 ) ∈ (OPE1)OP and ... and ( yn-1 , yn ) ∈ (OPEn)OP imply ( y0 , yn ) ∈ (OPE)OP
EquivalentObjectProperties( OPE1 ... OPEn ) (OPEj)OP = (OPEk)OP for each 1 ≤ jn and each 1 ≤ kn
DisjointObjectProperties( OPE1 ... OPEn ) (OPEj)OP(OPEk)OP = ∅ for each 1 ≤ jn and each 1 ≤ kn such that jk
ObjectPropertyDomain( OPE CE ) x , y : ( x , y ) ∈ (OPE)OP implies x(CE)C
ObjectPropertyRange( OPE CE ) x , y : ( x , y ) ∈ (OPE)OP implies y(CE)C
InverseObjectProperties( OPE1 OPE2 ) (OPE1)OP = { ( x , y ) | ( y , x ) ∈ (OPE2)OP }
FunctionalObjectProperty( OPE ) x , y1 , y2 : ( x , y1 ) ∈ (OPE)OP and ( x , y2 ) ∈ (OPE)OP imply y1 = y2
InverseFunctionalObjectProperty( OPE ) x1 , x2 , y : ( x1 , y ) ∈ (OPE)OP and ( x2 , y ) ∈ (OPE)OP imply x1 = x2
ReflexiveObjectProperty( OPE ) x : xΔI implies ( x , x ) ∈ (OPE)OP
IrreflexiveObjectProperty( OPE ) x : xΔI implies ( x , x ) ∉ (OPE)OP
SymmetricObjectProperty( OPE ) x , y : ( x , y ) ∈ (OPE)OP implies ( y , x ) ∈ (OPE)OP
AsymmetricObjectProperty( OPE ) x , y : ( x , y ) ∈ (OPE)OP implies ( y , x ) ∉ (OPE)OP
TransitiveObjectProperty( OPE ) x , y , z : ( x , y ) ∈ (OPE)OP and ( y , z ) ∈ (OPE)OP imply ( x , z ) ∈ (OPE)OP

2.3.3 Data Property Expression Axioms

Satisfaction of OWL 2 data property expression axioms in I is defined as shown in Table 7.

Table 7. Satisfaction of Data Property Expression Axioms in an Interpretation
Axiom Condition
SubDataPropertyOf( DPE1 DPE2 ) (DPE1)DP(DPE2)DP
EquivalentDataProperties( DPE1 ... DPEn ) (DPEj)DP = (DPEk)DP for each 1 ≤ jn and each 1 ≤ kn
DisjointDataProperties( DPE1 ... DPEn ) (DPEj)DP(DPEk)DP = ∅ for each 1 ≤ jn and each 1 ≤ kn such that jk
DataPropertyDomain( DPE CE ) x , y : ( x , y ) ∈ (DPE)DP implies x(CE)C
DataPropertyRange( DPE DR ) x , y : ( x , y ) ∈ (DPE)DP implies y(DR)DT
FunctionalDataProperty( DPE ) x , y1 , y2 : ( x , y1 ) ∈ (DPE)DP and ( x , y2 ) ∈ (DPE)DP imply y1 = y2

2.3.4 Datatype Definitions

Satisfaction of datatype definitions in I is defined as shown in Table 8.

Table 8. Satisfaction of Datatype Definitions in an Interpretation
Axiom Condition
DatatypeDefinition( DT DR ) (DT)DT = (DR)DT

2.3.5 Keys

Satisfaction of keys in I is defined as shown in Table 9.

Table 9. Satisfaction of Keys in an Interpretation
Axiom Condition
HasKey( CE ( OPE1 ... OPEm ) ( DPE1 ... DPEn ) ) x , y , z1 , ... , zm , w1 , ... , wn :
    if x(CE)C and xNAMED and
       y(CE)C and yNAMED and
       ( x , zi ) ∈ (OPEi)OP and ( y , zi ) ∈ (OPEi)OP and ziNAMED for each 1 ≤ im and
       ( x , wj ) ∈ (DPEj)DP and ( y , wj ) ∈ (DPEj)DP for each 1 ≤ jn
    then x = y

2.3.6 Assertions

Satisfaction of OWL 2 assertions in I is defined as shown in Table 10.

Table 10. Satisfaction of Assertions in an Interpretation
Axiom Condition
SameIndividual( a1 ... an ) (aj)I = (ak)I for each 1 ≤ jn and each 1 ≤ kn
DifferentIndividuals( a1 ... an ) (aj)I(ak)I for each 1 ≤ jn and each 1 ≤ kn such that jk
ClassAssertion( CE a ) (a)I(CE)C
ObjectPropertyAssertion( OPE a1 a2 ) ( (a1)I , (a2)I ) ∈ (OPE)OP
NegativeObjectPropertyAssertion( OPE a1 a2 ) ( (a1)I , (a2)I ) ∉ (OPE)OP
DataPropertyAssertion( DPE a lt ) ( (a)I , (lt)LT ) ∈ (DPE)DP
NegativeDataPropertyAssertion( DPE a lt ) ( (a)I , (lt)LT ) ∉ (DPE)DP

2.3.7 Ontologies

An OWL 2 ontology O is satisfied in an interpretation I if all axioms in the axiom closure of O (with anonymous individuals standardized apart as described in Section 5.6.2 of the OWL 2 Specification [OWL 2 Specification]) are satisfied in I.

2.4 Models

Given a datatype map D, an interpretation I = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) for D is a model of an OWL 2 ontology O w.r.t. D if an interpretation J = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ J , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) for D exists such that ⋅ J coincides with ⋅ I on all named individuals and J satisfies O.

Thus, an interpretation I satisfying O is also a model of O. In contrast, a model I of O may not satisfy O directly; however, by modifying the interpretation of anonymous individuals, I can always be coerced into an interpretation J that satisfies O.

2.5 Inference Problems

Let D be a datatype map and V a vocabulary over D. Furthermore, let O and O1 be OWL 2 ontologies, CE, CE1, and CE2 class expressions, and a a named individual, such that all of them refer only to the vocabulary elements in V. Furthermore, variables are symbols that are not contained in V. Finally, a Boolean conjunctive query Q is a closed formula of the form

∃ x1 , ... , xn , y1 , ... , ym : [ A1 ∧ ... ∧ Ak ]

where each Ai is an atom of the form C(s), OP(s,t), or DP(s,u) with C a class, OP an object property, DP a data property, s and t individuals or some variable xj, and u a literal or some variable yj.

The following inference problems are often considered in practice.

Ontology Consistency: O is consistent (or satisfiable) w.r.t. D if a model of O w.r.t. D and V exists.

Ontology Entailment: O entails O1 w.r.t. D if every model of O w.r.t. D and V is also a model of O1 w.r.t. D and V.

Ontology Equivalence: O and O1 are equivalent w.r.t. D if O entails O1 w.r.t. D and O1 entails O w.r.t. D.

Ontology Equisatisfiability: O and O1 are equisatisfiable w.r.t. D if O is satisfiable w.r.t. D if and only if O1 is satisfiable w.r.t D.

Class Expression Satisfiability: CE is satisfiable w.r.t. O and D if a model I = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) of O w.r.t. D and V exists such that (CE)C ≠ ∅.

Class Expression Subsumption: CE1 is subsumed by a class expression CE2 w.r.t. O and D if (CE1)C(CE2)C for each model I = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) of O w.r.t. D and V.

Instance Checking: a is an instance of CE w.r.t. O and D if (a)I(CE)C for each model I = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) of O w.r.t. D and V.

Boolean Conjunctive Query Answering: Q is an answer w.r.t. O and D if Q is true in each model of O w.r.t. D and V according to the standard definitions of first-order logic.

In order to ensure that ontology entailment, class expression satisfiability, class expression subsumption, and instance checking are decidable, the following restriction w.r.t. O needs to be satisfied:

Each class expression of type MinObjectCardinality, MaxObjectCardinality, ExactObjectCardinality, and ObjectHasSelf that occurs in O1, CE, CE1, and CE2 can contain only object property expressions that are simple in the axiom closure Ax of O.

For ontology equivalence to be decidable, O1 needs to satisfy this restriction w.r.t. O and vice versa. These restrictions are analogous to the first condition from Section 11.2 of the OWL 2 Specification [OWL 2 Specification].

3 Independence of the Direct Semantics from the Datatype Map in OWL 2 DL (Informative)

OWL 2 DL has been defined so that the consequences of an OWL 2 DL ontology O do not depend on the choice of a datatype map, as long as the datatype map chosen contains all the datatypes occurring in O. This statement is made precise by the following theorem, and it has several useful consequences:

Theorem DS1. Let O1 and O2 be OWL 2 DL ontologies over a vocabulary V and D = ( NDT , NLS , NFS , ⋅ DT , ⋅ LS , ⋅ FS ) a datatype map such that each datatype mentioned in O1 and O2 is rdfs:Literal, a datatype defined in the respective ontology, or it occurs in NDT. Furthermore, let D' = ( NDT' , NLS' , NFS' , ⋅ DT ' , ⋅ LS ' , ⋅ FS ' ) be a datatype map such that NDTNDT', NLS(DT) = NLS'(DT), and NFS(DT) = NFS'(DT) for each DTNDT, and ⋅ DT ', ⋅ LS ', and ⋅ FS ' are extensions of ⋅ DT, ⋅ LS, and ⋅ FS, respectively. Then, O1 entails O2 w.r.t. D if and only if O1 entails O2 w.r.t. D'.

Proof. Without loss of generality, one can assume O1 and O2 to be in negation-normal form [Description Logics]. Furthermore, since datatype definitions in O1 and O2 are acyclic, one can assume that each defined datatype has been recursively replaced with its definition; thus, all datatypes in O1 and O2 are from NDT ∪ { rdfs:Literal }. The claim of the theorem is equivalent to the following statement: an interpretation I w.r.t. D and V exists such that O1 is and O2 is not satisfied in I if and only if an interpretation I' w.r.t. D' and V exists such that O1 is and O2 is not satisfied in I'. The (⇐) direction is trivial since each interpretation I w.r.t. D' and V is also an interpretation w.r.t. D and V. For the (⇒) direction, assume that an interpretation I = ( ΔI , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA , NAMED ) w.r.t. D and V exists such that O1 is and O2 is not satisfied in I. Let I' = ( ΔI , ΔD' , ⋅ C ' , ⋅ OP , ⋅ DP ' , ⋅ I , ⋅ DT ' , ⋅ LT ' , ⋅ FA ' , NAMED ) be an interpretation such that

Clearly, DataComplementOf( DR )DTDataComplementOf( DR )DT ' for each data range DR that is either a datatype, a datatype restriction, or an enumerated data range. The owl:topDataProperty property can occur in O1 and O2 only in tautologies. The interpretation of all other data properties is the same in I and I', so (CE)C = (CE)C ' for each class expression CE occurring in O1 and O2. Therefore, O1 is and O2 is not satisfied in I'. QED

4 Appendix: Change Log (Informative)

4.1 Changes Since Recommendation

This section summarizes the changes to this document since the Recommendation of 27 October, 2009.

4.2 Changes Since Proposed Recommendation

No changes have been made to this document since the Proposed Recommendation of 22 September, 2009.

4.3 Changes Since Candidate Recommendation

This section summarizes the changes to this document since the Candidate Recommendation of 11 June, 2009.

4.4 Changes Since Last Call

This section summarizes the changes to this document since the Last Call Working Draft of 21 April, 2009.

5 Acknowledgments

The starting point for the development of OWL 2 was the OWL1.1 member submission, itself a result of user and developer feedback, and in particular of information gathered during the OWL Experiences and Directions (OWLED) Workshop series. The working group also considered postponed issues from the WebOnt Working Group.

This document has been produced by the OWL Working Group (see below), and its contents reflect extensive discussions within the Working Group as a whole. The editors extend special thanks to Markus Krötzsch (FZI), Michael Schneider (FZI) and Thomas Schneider (University of Manchester) for their thorough reviews.

The regular attendees at meetings of the OWL Working Group at the time of publication of this document were: Jie Bao (RPI), Diego Calvanese (Free University of Bozen-Bolzano), Bernardo Cuenca Grau (Oxford University Computing Laboratory), Martin Dzbor (Open University), Achille Fokoue (IBM Corporation), Christine Golbreich (Université de Versailles St-Quentin and LIRMM), Sandro Hawke (W3C/MIT), Ivan Herman (W3C/ERCIM), Rinke Hoekstra (University of Amsterdam), Ian Horrocks (Oxford University Computing Laboratory), Elisa Kendall (Sandpiper Software), Markus Krötzsch (FZI), Carsten Lutz (Universität Bremen), Deborah L. McGuinness (RPI), Boris Motik (Oxford University Computing Laboratory), Jeff Pan (University of Aberdeen), Bijan Parsia (University of Manchester), Peter F. Patel-Schneider (Bell Labs Research, Alcatel-Lucent), Sebastian Rudolph (FZI), Alan Ruttenberg (Science Commons), Uli Sattler (University of Manchester), Michael Schneider (FZI), Mike Smith (Clark & Parsia), Evan Wallace (NIST), Zhe Wu (Oracle Corporation), and Antoine Zimmermann (DERI Galway). We would also like to thank past members of the working group: Jeremy Carroll, Jim Hendler, and Vipul Kashyap.

6 References

6.1 Normative References

[OWL 2 Specification]
OWL 2 Web Ontology Language: Structural Specification and Functional-Style Syntax (Second Edition) Boris Motik, Peter F. Patel-Schneider, Bijan Parsia, eds. W3C Recommendation, 11 December 2012, http://www.w3.org/TR/2012/REC-owl2-syntax-20121211/. Latest version available at http://www.w3.org/TR/owl2-syntax/.

6.2 Nonnormative References

[Description Logics]
The Description Logic Handbook: Theory, Implementation, and Applications, second edition. Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi, and Peter F. Patel-Schneider, eds. Cambridge University Press, 2007. Also see the Description Logics Home Page.
[OWL 1 Semantics and Abstract Syntax]
OWL Web Ontology Language: Semantics and Abstract Syntax. Peter F. Patel-Schneider, Patrick Hayes, and Ian Horrocks, eds. W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-semantics-20040210/. Latest version available at http://www.w3.org/TR/owl-semantics/.
[OWL 2 Profiles]
OWL 2 Web Ontology Language: Profiles (Second Edition) Boris Motik, Bernardo Cuenca Grau, Ian Horrocks, Zhe Wu, Achille Fokoue, Carsten Lutz, eds. W3C Recommendation, 11 December 2012, http://www.w3.org/TR/2012/REC-owl2-profiles-20121211/. Latest version available at http://www.w3.org/TR/owl2-profiles/.
[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.