W3C

OWL 2 Web Ontology Language:
Direct Semantics

W3C Working Draft 02 December 2008

This version:
http://www.w3.org/TR/2008/WD-owl2-semantics-20081202/
Latest version:
http://www.w3.org/TR/owl2-semantics/
Previous version:
http://www.w3.org/TR/2008/WD-owl2-semantics-20081008/
Editors:
Boris Motik, Oxford University
Peter F. Patel-Schneider, Bell Labs Research, Alcatel-Lucent
Bernardo Cuenca Grau, Oxford University
Contributors:
Ian Horrocks, Oxford University
Bijan Parsia, University of Manchester
Uli Sattler, University of Manchester

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



Abstract

OWL 2 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 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May Be Superseded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/.

This document is being published as one of a set of 11 documents:

  1. Structural Specification and Functional-Style Syntax
  2. Direct Semantics (this document)
  3. RDF-Based Semantics
  4. Conformance and Test Cases
  5. Mapping to RDF Graphs
  6. XML Serialization
  7. Profiles
  8. Quick Reference Guide
  9. New Features and Rationale
  10. Manchester Syntax
  11. rdf:text: A Datatype for Internationalized Text

Last Call

The Working Group believes it has completed its design work for the technologies specified this document, so this is a "Last Call" draft. The design is not expected to change significantly, going forward, and now is the key time for external review, before the implementation phase.

Summary of Changes

This document has been updated to keep in sync with the Syntax document. The most significant update is in the formal definition of the datatype map.

Please Comment By 23 January 2009

The OWL Working Group seeks public feedback on these Working Drafts. Please send your comments to public-owl-comments@w3.org (public archive). If possible, please offer specific changes to the text that would address your concern. You may also wish to check the Wiki Version of this document for internal-review comments and changes being drafted which may address your concerns.

No Endorsement

Publication as a Working Draft does not imply endorsement by the W3C Membership. This is a draft document and may be updated, replaced or obsoleted by other documents at any time. It is inappropriate to cite this document as other than work in progress.

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. An individual who has actual knowledge of a patent which the individual believes contains Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.


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 is compatible with 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 OWL 2 is an extension of OWL DL, this document also provides a direct semantics for OWL Lite and OWL DL; this semantics is equivalent to the official semantics of OWL Lite and OWL DL [OWL Abstract Syntax and Semantics]. Furthermore, this document also provides the direct model-theoretic semantics for the OWL 2 profiles [OWL 2 Profiles].

The semantics is defined for an 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 for annotations of ontologies, anonymous individuals, axioms, and other annotations. Annotations of all these types, 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 is a 6-tuple D = ( NDT , NLS , NFS , ⋅ DT , ⋅ LS , ⋅ FS ) with the following components.

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 Int = ( ΔInt , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA ) for D and V is a 9-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
InverseOf( 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, a set (DT)DTΔD. Data ranges, however, can be n-ary, as this allows implementations to extend OWL 2 with built-in operations such as comparisons or arithmetic. An n-ary data range DR is interpreted as an n-ary relation (DR)DT over ΔD.

Table 3. Interpreting Data Ranges
Data Range Interpretation ⋅ DT
IntersectionOf( DR1 ... DRn ) (DR1)DT ∩ ... ∩ (DRn)DT
UnionOf( DR1 ... DRn ) (DR1)DT ∪ ... ∪ (DRn)DT
ComplementOf( DR ) D)n \ (DR)DT where n is the arity of DR
OneOf( 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
IntersectionOf( CE1 ... CEn ) (CE1)C ∩ ... ∩ (CEn)C
UnionOf( CE1 ... CEn ) (CE1)C ∪ ... ∪ (CEn)C
ComplementOf( CE ) ΔInt \ (CE)C
OneOf( a1 ... an ) { (a1)I , ... , (an)I }
SomeValuesFrom( OPE CE ) { x | ∃ y : ⟨ x, y ⟩ ∈ (OPE)OP and y(CE)C }
AllValuesFrom( OPE CE ) { x | ∀ y : ⟨ x, y ⟩ ∈ (OPE)OP implies y(CE)C }
HasValue( OPE a ) { x | ⟨ x , (a)I ⟩ ∈ (OPE)OP }
HasSelf( OPE ) { x | ⟨ x , x ⟩ ∈ (OPE)OP }
MinCardinality( n OPE ) { x | #{ y | ⟨ x , y ⟩ ∈ (OPE)OP } ≥ n }
MaxCardinality( n OPE ) { x | #{ y | ⟨ x , y ⟩ ∈ (OPE)OP } ≤ n }
ExactCardinality( n OPE ) { x | #{ y | ⟨ x , y ⟩ ∈ (OPE)OP } = n }
MinCardinality( n OPE CE ) { x | #{ y | ⟨ x , y ⟩ ∈ (OPE)OP and y(CE)C } ≥ n }
MaxCardinality( n OPE CE ) { x | #{ y | ⟨ x , y ⟩ ∈ (OPE)OP and y(CE)C } ≤ n }
ExactCardinality( n OPE CE ) { x | #{ y | ⟨ x , y ⟩ ∈ (OPE)OP and y(CE)C } = n }
SomeValuesFrom( DPE1 ... DPEn DR ) { x | ∃ y1, ... , yn : ⟨ x , yk ⟩ ∈ (DPEk)DP for each 1 ≤ kn and ⟨ y1 , ... , yn ⟩ ∈ (DR)DT }
AllValuesFrom( DPE1 ... DPEn DR ) { x | ∀ y1, ... , yn : ⟨ x , yk ⟩ ∈ (DPEk)DP for each 1 ≤ kn imply ⟨ y1 , ... , yn ⟩ ∈ (DR)DT }
HasValue( DPE lt ) { x | ⟨ x , (lt)LT ⟩ ∈ (DPE)DP }
MinCardinality( n DPE ) { x | #{ y | ⟨ x , y ⟩ ∈ (DPE)DP} ≥ n }
MaxCardinality( n DPE ) { x | #{ y | ⟨ x , y ⟩ ∈ (DPE)DP } ≤ n }
ExactCardinality( n DPE ) { x | #{ y | ⟨ x , y ⟩ ∈ (DPE)DP } = n }
MinCardinality( n DPE DR ) { x | #{ y | ⟨ x , y ⟩ ∈ (DPE)DP and y(DR)DT } ≥ n }
MaxCardinality( n DPE DR ) { x | #{ y | ⟨ x , y ⟩ ∈ (DPE)DP and y(DR)DT } ≤ n }
ExactCardinality( n DPE DR ) { x | #{ y | ⟨ x , y ⟩ ∈ (DPE)DP and y(DR)DT } = n }

2.3 Satisfaction in an Interpretation

An interpretation Int = ( ΔInt , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA ) satisfies an axiom w.r.t. an ontology O if the axiom satisfies appropriate conditions listed in the following sections. Satisfaction of axioms in Int is defined w.r.t. O because satisfaction of key axioms uses the following function:

ISNAMEDO(x) = true for xΔInt if and only if (a)I = x for some named individual a occurring in the axiom closure of O

2.3.1 Class Expression Axioms

Satisfaction of OWL 2 class expression axioms in Int w.r.t. O 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 Int w.r.t. O is defined as shown in Table 6.

Table 6. Satisfaction of Object Property Expression Axioms in an Interpretation
Axiom Condition
SubPropertyOf( OPE1 OPE2 ) (OPE1)OP(OPE2)OP
SubPropertyOf( PropertyChain( OPE1 ... OPEn ) OPE ) y0 , ... , yn : ⟨ y0 , y1 ⟩ ∈ (OPE1)OP and ... and ⟨ yn-1 , yn ⟩ ∈ (OPEn)OP imply ⟨ y0 , yn ⟩ ∈ (OPE)OP
EquivalentProperties( OPE1 ... OPEn ) (OPEj)OP = (OPEk)OP for each 1 ≤ jn and each 1 ≤ kn
DisjointProperties( OPE1 ... OPEn ) (OPEj)OP(OPEk)OP = ∅ for each 1 ≤ jn and each 1 ≤ kn such that jk
PropertyDomain( OPE CE ) x , y : ⟨ x , y ⟩ ∈ (OPE)OP implies x(CE)C
PropertyRange( OPE CE ) x , y : ⟨ x , y ⟩ ∈ (OPE)OP implies y(CE)C
InverseProperties( OPE1 OPE2 ) (OPE1)OP = { ⟨ x , y ⟩ | ⟨ y , x ⟩ ∈ (OPE2)OP }
FunctionalProperty( OPE ) x , y1 , y2 : ⟨ x , y1 ⟩ ∈ (OPE)OP and ⟨ x , y2 ⟩ ∈ (OPE)OP imply y1 = y2
InverseFunctionalProperty( OPE ) x1 , x2 , y : ⟨ x1 , y ⟩ ∈ (OPE)OP and ⟨ x2 , y ⟩ ∈ (OPE)OP imply x1 = x2
ReflexiveProperty( OPE ) x : xΔInt implies ⟨ x , x ⟩ ∈ (OPE)OP
IrreflexiveProperty( OPE ) x : xΔInt implies ⟨ x , x ⟩ ∉ (OPE)OP
SymmetricProperty( OPE ) x , y : ⟨ x , y ⟩ ∈ (OPE)OP implies ⟨ y , x ⟩ ∈ (OPE)OP
AsymmetricProperty( OPE ) x , y : ⟨ x , y ⟩ ∈ (OPE)OP implies ⟨ y , x ⟩ ∉ (OPE)OP
TransitiveProperty( 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 Int w.r.t. O is defined as shown in Table 7.

Table 7. Satisfaction of Data Property Expression Axioms in an Interpretation
Axiom Condition
SubPropertyOf( DPE1 DPE2 ) (DPE1)DP(DPE2)DP
EquivalentProperties( DPE1 ... DPEn ) (DPEj)DP = (DPEk)DP for each 1 ≤ jn and each 1 ≤ kn
DisjointProperties( DPE1 ... DPEn ) (DPEj)DP(DPEk)DP = ∅ for each 1 ≤ jn and each 1 ≤ kn such that jk
PropertyDomain( DPE CE ) x , y : ⟨ x , y ⟩ ∈ (DPE)DP implies x(CE)C
PropertyRange( DPE DR ) x , y : ⟨ x , y ⟩ ∈ (DPE)DP implies y(DR)DT
FunctionalProperty( DPE ) x , y1 , y2 : ⟨ x , y1 ⟩ ∈ (DPE)DP and ⟨ x , y2 ⟩ ∈ (DPE)DP imply y1 = y2

2.3.4 Keys

Satisfaction of keys in Int w.r.t. O is defined as shown in Table 8.

Table 8. Satisfaction of Keys in an Interpretation
Axiom Condition
HasKey( CE PE1 ... PEn ) x , y , z1 , ... , zn  :
    if ISNAMEDO(x) and ISNAMEDO(y) and ISNAMEDO(z1) and ... and ISNAMEDO(zn) and x(CE)C and y(CE)C and
       for each 1 ≤ in,
          if PEi is an object property, then ⟨ x , zi ⟩ ∈ (PEi)OP and ⟨ y , zi ⟩ ∈ (PEi)OP, and
          if PEi is a data property, then ⟨ x , zi ⟩ ∈ (PEi)DP and ⟨ y , zi ⟩ ∈ (PEi)DP
    then x = y

2.3.5 Assertions

Satisfaction of OWL 2 assertions in Int w.r.t. O is defined as shown in Table 9.

Table 9. 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
PropertyAssertion( OPE a1 a2 ) (a1)I , (a2)I ⟩ ∈ (OPE)OP
NegativePropertyAssertion( OPE a1 a2 ) (a1)I , (a2)I ⟩ ∉ (OPE)OP
PropertyAssertion( DPE a lt ) (a)I , (lt)LT ⟩ ∈ (DPE)DP
NegativePropertyAssertion( DPE a lt ) (a)I , (lt)LT ⟩ ∉ (DPE)DP

2.3.6 Ontologies

Int satisfies an OWL 2 ontology O if all axioms in the axiom closure of O (with anonymous individuals renamed apart as described in Section 5.6.2 of the OWL 2 Specification [OWL 2 Specification]) are satisfied in Int w.r.t. O.

2.4 Models

An interpretation Int = ( ΔInt , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA ) is a model of an OWL 2 ontology O if an interpretation Int1 = ( ΔInt , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I1 , ⋅ DT , ⋅ LT , ⋅ FA ) exists such that ⋅ I1 coincides with ⋅ I on all named individuals and Int1 satisfies O.

Thus, an interpretation Int satisfying O is also a model of O. In contrast, a model Int of O may not satisfy O directly; however, by modifying the interpretation of anonymous individuals, Int can always be coerced into an interpretation Int1 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. 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 Int = ( ΔInt , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA ) 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 Int = ( ΔInt , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA ) 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 Int = ( ΔInt , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA ) 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.

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 Semantics from the Datatype Map

The semantics of OWL 2 has been defined in such a way that the semantics of an OWL 2 ontology O does 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, which has several useful consequences:

Theorem DS1. Let O1 and O2 be OWL 2 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 either rdfs:Literal 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]. The claim of the theorem is equivalent to the following statement: an interpretation Int w.r.t. D and V exists such that O1 is and O2 is not satisfied in Int if and only if an interpretation Int' w.r.t. D' and V exists such that O1 is and O2 is not satisfied in Int'. The (⇐) direction is trivial since each interpretation Int w.r.t. D' and V is also an interpretation w.r.t. D and V. For the (⇒) direction, assume that an interpretation Int = ( ΔInt , ΔD , ⋅ C , ⋅ OP , ⋅ DP , ⋅ I , ⋅ DT , ⋅ LT , ⋅ FA ) w.r.t. D and V exists such that O1 is and O2 is not satisfied in Int. Let Int' = ( ΔInt , ΔD' , ⋅ C ' , ⋅ OP , ⋅ DP ' , ⋅ I , ⋅ DT ' , ⋅ LT ' , ⋅ FA ' ) be an interpretation such that

Clearly, ComplementOf( DR )DTComplementOf( DR )DT ' for each data range DR that is 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 Int and Int', so (CE)C = (CE)C ' for each class expression CE occurring in O1 and O2. Therefore, O1 is and O2 is not satisfied in Int'. QED

4 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 is the product of the OWL Working Group (see below) whose members deserve recognition for their time and commitment. 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), Martin Dzbor (Open University), Achille Fokoue (IBM Corporation), Christine Golbreich (Université de Versailles St-Quentin), Sandro Hawke (W3C/MIT), Ivan Herman (W3C/ERCIM), Rinke Hoekstra (University of Amsterdam), Ian Horrocks (Oxford University), Elisa Kendall (Sandpiper Software), Markus Krötzsch (FZI), Carsten Lutz (Universität Bremen), Boris Motik (Oxford University), Jeff Pan (University of Aberdeen), Bijan Parsia (University of Manchester), Peter F. Patel-Schneider (Bell Labs Research, Alcatel-Lucent), Alan Ruttenberg (Science Commons), Uli Sattler (University of Manchester), Michael Schneider (FZI), Mike Smith (Clark & Parsia), Evan Wallace (NIST), and Zhe Wu (Oracle Corporation). We would also like to thank past members of the working group: Jeremy Carroll, Jim Hendler and Vipul Kashyap.

5 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.
[OWL 2 Specification]
OWL 2 Web Ontology Language:Structural Specification and Functional-Style Syntax Boris Motik, Peter F. Patel-Schneider, Bijan Parsia, eds. W3C Working Draft, 02 December 2008, http://www.w3.org/TR/2008/WD-owl2-syntax-20081202/. Latest version available at http://www.w3.org/TR/owl2-syntax/.
[OWL 2 Profiles]
OWL 2 Web Ontology Language:Structural Specification and Functional-Style Syntax Boris Motik, Peter F. Patel-Schneider, Bijan Parsia, eds. W3C Working Draft, 02 December 2008, http://www.w3.org/TR/2008/WD-owl2-syntax-20081202/. Latest version available at http://www.w3.org/TR/owl2-syntax/.
[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.
[RFC-4646]
RFC 4646 - Tags for Identifying Languages. M. Phillips and A. Davis. IETF, September 2006, http://www.ietf.org/rfc/rfc4646.txt. Latest version is available as BCP 47, (details).