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May Be Superseded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:

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.

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1 Introduction
2 Direct Model-Theoretic Semantics for OWL 2
3 Independence of the Semantics from the Datatype Map
4 Acknowledgments
5 References

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 = ( N_DT , N_LS , N_FS , ⋅ ^DT , ⋅ ^LS , ⋅ ^FS ) with the following components.

N_DT is a set of datatypes that does not contain the datatype rdfs:Literal.
N_LS is a function that assigns to each datatype DT ∈ N_DT a set N_LS(DT) of strings called lexical values. The set N_LS(DT) is called the lexical space of DT.
N_FS is a function that assigns to each datatype DT ∈ N_DT a set N_FS(DT) of pairs ⟨ F v ⟩, where F is a constraining facet and v is an arbitrary object called a value. The set N_FS(DT) is called the facet space of DT.
For each datatype DT ∈ N_DT, the interpretation function ⋅ ^DT assigns to DT a set (DT)^DT called the value space of DT.
For each datatype DT ∈ N_DT and each lexical value LV ∈ N_LS(DT), the interpretation function ⋅ ^LS assigns to the pair ⟨ LV DT ⟩ a data value (⟨ LV DT ⟩)^LS ∈ (DT)^DT.
For each datatype DT ∈ N_DT and each pair ⟨ F v ⟩ ∈ N_FS(DT), the interpretation function ⋅ ^FS assigns to ⟨ F v ⟩ a facet value (⟨ F v ⟩)^FS ⊆ (DT)^DT.

A vocabulary V = ( V_C , V_OP , V_DP , V_I , V_DT , V_LT , V_FA ) over a datatype map D is a 7-tuple consisting of the following elements:

V_C is a set of classes as defined in the OWL 2 Specification [OWL 2 Specification], containing at least the classes owl:Thing and owl:Nothing.
V_OP is a set of object properties as defined in the OWL 2 Specification [OWL 2 Specification], containing at least the object properties owl:topObjectProperty and owl:bottomObjectProperty.
V_DP is a set of data properties as defined in the OWL 2 Specification [OWL 2 Specification], containing at least the data properties owl:topDataProperty and owl:bttomDataProperty.
V_I is a set of individuals (named and anonymous) as defined in the OWL 2 Specification [OWL 2 Specification].
V_DT is the set of all datatypes of D extended with the datatype rdfs:Literal; that is, V_DT = N_DT ∪ { rdfs:Literal }.
V_LT is a set of literals LV^^DT for each datatype DT ∈ N_DT and each lexical value LV ∈ N_LS(DT).
V_FA is the set of pairs ⟨ F lt ⟩ for each constraining facet F, datatype DT ∈ N_DT, and literal lt ∈ V_LT such that ⟨ F (⟨ LV DT₁ ⟩)^LS ⟩ ∈ N_FS(DT), where LV is the lexical value of lt and DT₁ is the datatype of lt.

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

OP denotes an object property;
OPE denotes an object property expression;
DP denotes a data property;
DPE denotes a data property expression;
PE denotes an object property or a data property expression;
C denotes a class;
CE denotes a class expression;
DT denotes a datatype;
DR denotes a data range;
a denotes an individual (named or anonymous);
lt denotes a literal; and
F denotes a constraining facet.

2.2 Interpretations

Given a datatype map D and a vocabulary V over D, an interpretation Int = ( Δ_Int , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) for D and V is a 9-tuple with the following structure.

Δ_Int is a nonempty set called the object domain.
Δ_D is a nonempty set disjoint with Δ_Int called the data domain such that (DT)^DT ⊆ Δ_D for each datatype DT ∈ V_DT.
⋅ ^C is the class interpretation function that assigns to each class C ∈ V_C a subset (C)^C ⊆ Δ_Int such that
- (owl:Thing)^C = Δ_Int and
- (owl:Nothing)^C = ∅.
⋅ ^OP is the object property interpretation function that assigns to each object property OP ∈ V_OP a subset (OP)^OP ⊆ Δ_Int × Δ_Int such that
- (owl:topObjectProperty)^OP = Δ_Int × Δ_Int and
- (owl:bottomObjectProperty)^OP = ∅.
⋅ ^DP is the data property interpretation function that assigns to each data property DP ∈ V_DP a subset (DP)^DP ⊆ Δ_Int × Δ_D such that
- (owl:topDataProperty)^DP = Δ_Int × Δ_D and
- (owl:bottomDataProperty)^DP = ∅.
⋅ ^I is the individual interpretation function that assigns to each individual a ∈ V_I an element (a)^I ∈ Δ_Int.
⋅ ^DT is the datatype interpretation function that is the same as in D for all datatypes DT ∈ N_DT and is extended to rdfsLiteral by setting
- (rdfs:Literal)^DT = Δ_D.
⋅ ^LT is the literal interpretation function that is defined as (lt)^LT = (⟨ LV DT ⟩)^LS for each lt ∈ V_LT, where LV is the lexical value of lt and DT is the datatype of lt.
⋅ ^FA is the facet interpretation function that is defined as (⟨ F lt ⟩)^FA = (⟨ F (lt)^LT ⟩)^FS for each ⟨ F lt ⟩ ∈ V_FA.

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 Δ_D — that is, a set (DT)^DT ⊆ Δ_D. 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 Δ_D.

Table 3. Interpreting Data Ranges
Data Range	Interpretation ⋅ ^DT
IntersectionOf( DR₁ ... DR_n )	(DR₁)^DT ∩ ... ∩ (DR_n)^DT
UnionOf( DR₁ ... DR_n )	(DR₁)^DT ∪ ... ∪ (DR_n)^DT
ComplementOf( DR )	(Δ_D)ⁿ \ (DR)^DT where n is the arity of DR
OneOf( lt₁ ... lt_n )	{ (lt₁)^LT , ... , (lt_n)^LT }
DatatypeRestriction( DT F₁ lt₁ ... F_n lt_n )	(DT)^DT ∩ (⟨ F₁ lt₁ ⟩)^FA ∩ ... ∩ (⟨ F_n lt_n ⟩)^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 ... CE_n )	(CE₁)^C ∩ ... ∩ (CE_n)^C
UnionOf( CE₁ ... CE_n )	(CE₁)^C ∪ ... ∪ (CE_n)^C
ComplementOf( CE )	Δ_Int \ (CE)^C
OneOf( a₁ ... a_n )	{ (a₁)^I , ... , (a_n)^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 ... DPE_n DR )	{ x \| ∃ y₁, ... , y_n : ⟨ x , y_k ⟩ ∈ (DPE_k)^DP for each 1 ≤ k ≤ n and ⟨ y₁ , ... , y_n ⟩ ∈ (DR)^DT }
AllValuesFrom( DPE₁ ... DPE_n DR )	{ x \| ∀ y₁, ... , y_n : ⟨ x , y_k ⟩ ∈ (DPE_k)^DP for each 1 ≤ k ≤ n imply ⟨ y₁ , ... , y_n ⟩ ∈ (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 = ( Δ_Int , Δ_D , ⋅ ^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:

ISNAMED_O(x) = true for x ∈ Δ_Int 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 ... CE_n )	(CE_j)^C = (CE_k)^C for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n
DisjointClasses( CE₁ ... CE_n )	(CE_j)^C ∩ (CE_k)^C = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k
DisjointUnion( C CE₁ ... CE_n )	(C)^C = (CE₁)^C ∪ ... ∪ (CE_n)^C and (CE_j)^C ∩ (CE_k)^C = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k

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 ... OPE_n ) OPE )	∀ y₀ , ... , y_n : ⟨ y₀ , y₁ ⟩ ∈ (OPE₁)^OP and ... and ⟨ y_n-1 , y_n ⟩ ∈ (OPE_n)^OP imply ⟨ y₀ , y_n ⟩ ∈ (OPE)^OP
EquivalentProperties( OPE₁ ... OPE_n )	(OPE_j)^OP = (OPE_k)^OP for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n
DisjointProperties( OPE₁ ... OPE_n )	(OPE_j)^OP ∩ (OPE_k)^OP = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k
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 : x ∈ Δ_Int implies ⟨ x , x ⟩ ∈ (OPE)^OP
IrreflexiveProperty( OPE )	∀ x : x ∈ Δ_Int 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 ... DPE_n )	(DPE_j)^DP = (DPE_k)^DP for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n
DisjointProperties( DPE₁ ... DPE_n )	(DPE_j)^DP ∩ (DPE_k)^DP = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k
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 ... PE_n )	∀ x , y , z₁ , ... , z_n : if ISNAMED_O(x) and ISNAMED_O(y) and ISNAMED_O(z₁) and ... and ISNAMED_O(z_n) and x ∈ (CE)^C and y ∈ (CE)^C and for each 1 ≤ i ≤ n, if PE_i is an object property, then ⟨ x , z_i ⟩ ∈ (PE_i)^OP and ⟨ y , z_i ⟩ ∈ (PE_i)^OP, and if PE_i is a data property, then ⟨ x , z_i ⟩ ∈ (PE_i)^DP and ⟨ y , z_i ⟩ ∈ (PE_i)^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 ... a_n )	(a_j)^I = (a_k)^I for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n
DifferentIndividuals( a₁ ... a_n )	(a_j)^I ≠ (a_k)^I for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k
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 = ( Δ_Int , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) is a model of an OWL 2 ontology O if an interpretation Int₁ = ( Δ_Int , Δ_D , ⋅ ^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 , ... , x_n , y₁ , ... , y_m : [ A₁ ∧ ... ∧ A_k ]

where each A_i 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 x_j, and u a literal or some variable y_j.

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 = ( Δ_Int , Δ_D , ⋅ ^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 = ( Δ_Int , Δ_D , ⋅ ^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 = ( Δ_Int , Δ_D , ⋅ ^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:

One can interpret an OWL 2 ontology O by considering only the datatypes explicitly occurring in O.
When referring to various reasoning problems, the datatype map D need not be given explicitly, as it is sufficient to consider an implicit datatype map containing only the datatypes from the given ontology.
OWL 2 reasoners can provide datatypes not explicitly mentioned in this specification without fear that this will change the semantics of OWL 2 ontologies not using these datatypes.

Theorem DS1. Let O₁ and O₂ be OWL 2 ontologies over a vocabulary V and D = ( N_DT , N_LS , N_FS , ⋅ ^DT , ⋅ ^LS , ⋅ ^FS ) a datatype map such that each datatype mentioned in O₁ and O₂ is either rdfs:Literal or it occurs in N_DT. Furthermore, let D' = ( N_DT' , N_LS' , N_FS' , ⋅ ^DT ' , ⋅ ^LS ' , ⋅ ^FS ' ) be a datatype map such that N_DT ⊆ N_DT', N_LS(DT) = N_LS'(DT), and N_FS(DT) = N_FS'(DT) for each DT ∈ N_DT, 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 = ( Δ_Int , Δ_D , ⋅ ^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' = ( Δ_Int , Δ_D' , ⋅ ^C ' , ⋅ ^OP , ⋅ ^DP ' , ⋅ ^I , ⋅ ^DT ' , ⋅ ^LT ' , ⋅ ^FA ' ) be an interpretation such that

Δ_D' is obtained by extending Δ_D with the value space of all datatypes in N_DT' \ N_DT,
⋅ ^C ' coincides with ⋅ ^C on all classes, and
⋅ ^DP ' coincides with ⋅ ^DP on all data properties apart from owl:topDataProperty.

Clearly, ComplementOf( DR )^DT ⊆ ComplementOf( 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).

OWL 2 Web Ontology Language:Direct Semantics