Difference between revisions of "Direct Semantics"
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 class="name"  HasKey( CE ( OPE<sub>1</sub> ... OPE<sub>m</sub> ) ( DPE<sub>1</sub> ... DPE<sub>n</sub> ) )   class="name"  HasKey( CE ( OPE<sub>1</sub> ... OPE<sub>m</sub> ) ( DPE<sub>1</sub> ... DPE<sub>n</sub> ) )  
−   style="textalign: left"  ∀ ''x'' , ''y'' , ''z<sub>1</sub>'' , ... , ''z<sub>m</sub>'' , ''w<sub>1</sub>'' , ... , ''w<sub>n</sub>'' :<br /> if ''x'' ∈ ''(CE)<sup>C</sup>'' and ''  +   style="textalign: left"  ∀ ''x'' , ''y'' , ''z<sub>1</sub>'' , ... , ''z<sub>m</sub>'' , ''w<sub>1</sub>'' , ... , ''w<sub>n</sub>'' :<br /> if ''x'' ∈ ''(CE)<sup>C</sup>'' and ''x'' ∈ ''NAMED'' and<br /> ''y'' ∈ ''(CE)<sup>C</sup>'' and ''y'' ∈ ''NAMED'' and<br /> ( ''x'' , ''z<sub>i</sub>'' ) ∈ ''(OPE<sub>i</sub>)<sup>OP</sup>'' and ( ''y'' , ''z<sub>i</sub>'' ) ∈ ''(OPE<sub>i</sub>)<sup>OP</sup>'' and ''z<sub>i</sub>'' ∈ ''NAMED'' for each 1 ≤ ''i'' ≤ ''m'' and<br /> ( ''x'' , ''w<sub>j</sub>'' ) ∈ ''(DPE<sub>j</sub>)<sup>DP</sup>'' and ( ''y'' , ''w<sub>j</sub>'' ) ∈ ''(DPE<sub>j</sub>)<sup>DP</sup>'' for each 1 ≤ ''j'' ≤ ''n''<br /> then x = y 
    
}  } 
Revision as of 17:10, 15 April 2010
__NUMBEREDHEADINGS__
 Document title:
 OWL 2 Web Ontology Language
Direct Semantics (Second Edition)
 Editors
 Boris Motik, Oxford University Computing Laboratory
 Peter F. PatelSchneider, Bell Labs Research, AlcatelLucent
 Bernardo Cuenca Grau, Oxford University Computing Laboratory
 Contributors (alphabetical order)
 Ian Horrocks, Oxford University Computing Laboratory
 Bijan Parsia, University of Manchester
 Uli Sattler, University of Manchester
 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 modeltheoretic 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
Copyright © 20082009 W3C^{®} (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.
Contents
 1 Introduction
 2 Direct ModelTheoretic Semantics for OWL 2
 3 Independence of the Direct Semantics from the Datatype Map in OWL 2 DL (Informative)
 4 Appendix: Change Log (Informative)
 5 Acknowledgments
 6 References
1 Introduction
This document defines the direct modeltheoretic 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 modeltheoretic semantics of OWL 1 Lite and OWL 1 DL [OWL 1 Semantics and Abstract Syntax]. Furthermore, this document also provides the direct modeltheoretic 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 functionalstyle 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 functionalstyle syntax; therefore, they are not mentioned explicitly in this document.
2 Direct ModelTheoretic Semantics for OWL 2
This section specifies the direct modeltheoretic semantics of OWL 2 ontologies.
2.1 Vocabulary
A datatype map, formalizing datatype maps from the OWL 2 Specification [OWL 2 Specification], is a 6tuple D = ( N_{DT} , N_{LS} , N_{FS} , ⋅ ^{DT} , ⋅ ^{LS} , ⋅ ^{FS} ) with the following components:
 N_{DT} is a set of datatypes (more precisely, names 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 forms. 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 data value called the constraining 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 form 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 ) the set ( F , v )^{FS} ⊆ (DT)^{DT}.
The set of datatypes N_{DT} 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 N_{LS}(DT), N_{FS}(DT), (DT)^{DT}, ( LV , DT )^{LS} for each LV ∈ N_{LS}(DT), and ( F , v )^{FS} for each ( F , v ) ∈ N_{FS}(DT) are required to coincide with the definitions for DT in the OWL 2 datatype map.
A vocabulary V = ( V_{C} , V_{OP} , V_{DP} , V_{I} , V_{DT} , V_{LT} , V_{FA} ) over a datatype map D is a 7tuple 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:bottomDataProperty.
 V_{I} is a set of individuals (named and anonymous) as defined in the OWL 2 Specification [OWL 2 Specification].
 V_{DT} is a set containing all datatypes of D, the datatype rdfs:Literal, and possibly other datatypes; that is, N_{DT} ∪ { rdfs:Literal } ⊆ V_{DT}.
 V_{LT} is a set of literals LV^^DT for each datatype DT ∈ N_{DT} and each lexical form 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 , DT_{1} )^{LS} ) ∈ N_{FS}(DT), where LV is the lexical form of lt and DT_{1} 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;
 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 I = ( Δ_{I} , Δ_{D} , ⋅ ^{C} , ⋅ ^{OP} , ⋅ ^{DP} , ⋅ ^{I} , ⋅ ^{DT} , ⋅ ^{LT} , ⋅ ^{FA} , NAMED ) for D and V is a 10tuple with the following structure:
 Δ_{I} is a nonempty set called the object domain.
 Δ_{D} is a nonempty set disjoint with Δ_{I} 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_{I} such that
 (owl:Thing)^{C} = Δ_{I} 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_{I} × Δ_{I} such that
 (owl:topObjectProperty)^{OP} = Δ_{I} × Δ_{I} 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_{I} × Δ_{D} such that
 (owl:topDataProperty)^{DP} = Δ_{I} × 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_{I}.
 ⋅ ^{DT} is the datatype interpretation function that assigns to each datatype DT ∈ V_{DT} a subset (DT)^{DT} ⊆ Δ_{D} such that
 ⋅ ^{DT} is the same as in D for each datatype DT ∈ N_{DT}, and
 (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 form 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}.
 NAMED is a subset of Δ_{I} such that (a)^{I} ∈ NAMED for each named individual a ∈ V_{I}.
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.
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 Δ_{D} — that is, as a set (DT)^{DT} ⊆ Δ_{D}. OWL 2 currently does not define data ranges of arity more than one; however, by allowing for nary data ranges, the syntax of OWL 2 provides a "hook" allowing implementations to introduce extensions such as comparisons and arithmetic. An nary data range DR is interpreted as an nary relation (DR)^{DT} over Δ_{D} — that is, as a set (DT)^{DT} ⊆ (Δ_{D})^{n}
Data Range  Interpretation ⋅ ^{DT} 

DataIntersectionOf( DR_{1} ... DR_{n} )  (DR_{1})^{DT} ∩ ... ∩ (DR_{n})^{DT} 
DataUnionOf( DR_{1} ... DR_{n} )  (DR_{1})^{DT} ∪ ... ∪ (DR_{n})^{DT} 
DataComplementOf( DR )  (Δ_{D})^{n} \ (DR)^{DT} where n is the arity of DR 
DataOneOf( lt_{1} ... lt_{n} )  { (lt_{1})^{LT} , ... , (lt_{n})^{LT} } 
DatatypeRestriction( DT F_{1} lt_{1} ... F_{n} lt_{n} )  (DT)^{DT} ∩ ( F_{1} , lt_{1} )^{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.
Class Expression  Interpretation ⋅ ^{C} 

ObjectIntersectionOf( CE_{1} ... CE_{n} )  (CE_{1})^{C} ∩ ... ∩ (CE_{n})^{C} 
ObjectUnionOf( CE_{1} ... CE_{n} )  (CE_{1})^{C} ∪ ... ∪ (CE_{n})^{C} 
ObjectComplementOf( CE )  Δ_{I} \ (CE)^{C} 
ObjectOneOf( a_{1} ... a_{n} )  { (a_{1})^{I} , ... , (a_{n})^{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( DPE_{1} ... DPE_{n} DR )  { x  ∃ y_{1}, ... , y_{n} : ( x , y_{k} ) ∈ (DPE_{k})^{DP} for each 1 ≤ k ≤ n and ( y_{1} , ... , y_{n} ) ∈ (DR)^{DT} } 
DataAllValuesFrom( DPE_{1} ... DPE_{n} DR )  { x  ∀ y_{1}, ... , y_{n} : ( x , y_{k} ) ∈ (DPE_{k})^{DP} for each 1 ≤ k ≤ n imply ( y_{1} , ... , y_{n} ) ∈ (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 interpretation I = ( Δ_{I} , Δ_{D} , ⋅ ^{C} , ⋅ ^{OP} , ⋅ ^{DP} , ⋅ ^{I} , ⋅ ^{DT} , ⋅ ^{LT} , ⋅ ^{FA} , NAMED ) satisfies an the axiom 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.
Axiom  Condition 

SubClassOf( CE_{1} CE_{2} )  (CE_{1})^{C} ⊆ (CE_{2})^{C} 
EquivalentClasses( CE_{1} ... CE_{n} )  (CE_{j})^{C} = (CE_{k})^{C} for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n 
DisjointClasses( CE_{1} ... CE_{n} )  (CE_{j})^{C} ∩ (CE_{k})^{C} = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
DisjointUnion( C CE_{1} ... CE_{n} )  (C)^{C} = (CE_{1})^{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 I is defined as shown in Table 6.
Axiom  Condition 

SubObjectPropertyOf( OPE_{1} OPE_{2} )  (OPE_{1})^{OP} ⊆ (OPE_{2})^{OP} 
SubObjectPropertyOf( ObjectPropertyChain( OPE_{1} ... OPE_{n} ) OPE )  ∀ y_{0} , ... , y_{n} : ( y_{0} , y_{1} ) ∈ (OPE_{1})^{OP} and ... and ( y_{n1} , y_{n} ) ∈ (OPE_{n})^{OP} imply ( y_{0} , y_{n} ) ∈ (OPE)^{OP} 
EquivalentObjectProperties( OPE_{1} ... OPE_{n} )  (OPE_{j})^{OP} = (OPE_{k})^{OP} for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n 
DisjointObjectProperties( OPE_{1} ... OPE_{n} )  (OPE_{j})^{OP} ∩ (OPE_{k})^{OP} = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
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( OPE_{1} OPE_{2} )  (OPE_{1})^{OP} = { ( x , y )  ( y , x ) ∈ (OPE_{2})^{OP} } 
FunctionalObjectProperty( OPE )  ∀ x , y_{1} , y_{2} : ( x , y_{1} ) ∈ (OPE)^{OP} and ( x , y_{2} ) ∈ (OPE)^{OP} imply y_{1} = y_{2} 
InverseFunctionalObjectProperty( OPE )  ∀ x_{1} , x_{2} , y : ( x_{1} , y ) ∈ (OPE)^{OP} and ( x_{2} , y ) ∈ (OPE)^{OP} imply x_{1} = x_{2} 
ReflexiveObjectProperty( OPE )  ∀ x : x ∈ Δ_{I} implies ( x , x ) ∈ (OPE)^{OP} 
IrreflexiveObjectProperty( OPE )  ∀ x : x ∈ Δ_{I} 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.
Axiom  Condition 

SubDataPropertyOf( DPE_{1} DPE_{2} )  (DPE_{1})^{DP} ⊆ (DPE_{2})^{DP} 
EquivalentDataProperties( DPE_{1} ... DPE_{n} )  (DPE_{j})^{DP} = (DPE_{k})^{DP} for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n 
DisjointDataProperties( DPE_{1} ... DPE_{n} )  (DPE_{j})^{DP} ∩ (DPE_{k})^{DP} = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
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 , y_{1} , y_{2} : ( x , y_{1} ) ∈ (DPE)^{DP} and ( x , y_{2} ) ∈ (DPE)^{DP} imply y_{1} = y_{2} 
2.3.4 Datatype Definitions
Satisfaction of datatype definitions in I is defined as shown in Table 8.
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.
Axiom  Condition 

HasKey( CE ( OPE_{1} ... OPE_{m} ) ( DPE_{1} ... DPE_{n} ) )  ∀ x , y , z_{1} , ... , z_{m} , w_{1} , ... , w_{n} : if x ∈ (CE)^{C} and x ∈ NAMED and y ∈ (CE)^{C} and y ∈ NAMED and ( x , z_{i} ) ∈ (OPE_{i})^{OP} and ( y , z_{i} ) ∈ (OPE_{i})^{OP} and z_{i} ∈ NAMED for each 1 ≤ i ≤ m and ( x , w_{j} ) ∈ (DPE_{j})^{DP} and ( y , w_{j} ) ∈ (DPE_{j})^{DP} for each 1 ≤ j ≤ n then x = y 
2.3.6 Assertions
Satisfaction of OWL 2 assertions in I is defined as shown in Table 10.
Axiom  Condition 

SameIndividual( a_{1} ... a_{n} )  (a_{j})^{I} = (a_{k})^{I} for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n 
DifferentIndividuals( a_{1} ... 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} 
ObjectPropertyAssertion( OPE a_{1} a_{2} )  ( (a_{1})^{I} , (a_{2})^{I} ) ∈ (OPE)^{OP} 
NegativeObjectPropertyAssertion( OPE a_{1} a_{2} )  ( (a_{1})^{I} , (a_{2})^{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 interpretation I satisfies an OWL 2 ontology O 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 = ( Δ_{I} , Δ_{D} , ⋅ ^{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 = ( Δ_{I} , Δ_{D} , ⋅ ^{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 O_{1} be OWL 2 ontologies, CE, CE_{1}, and CE_{2} 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
∃ x_{1} , ... , x_{n} , y_{1} , ... , y_{m} : [ A_{1} ∧ ... ∧ 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 O_{1} w.r.t. D if every model of O w.r.t. D and V is also a model of O_{1} w.r.t. D and V.
Ontology Equivalence: O and O_{1} are equivalent w.r.t. D if O entails O_{1} w.r.t. D and O_{1} entails O w.r.t. D.
Ontology Equisatisfiability: O and O_{1} are equisatisfiable w.r.t. D if O is satisfiable w.r.t. D if and only if O_{1} is satisfiable w.r.t D.
Class Expression Satisfiability: CE is satisfiable w.r.t. O and D if a model I = ( Δ_{I} , Δ_{D} , ⋅ ^{C} , ⋅ ^{OP} , ⋅ ^{DP} , ⋅ ^{I} , ⋅ ^{DT} , ⋅ ^{LT} , ⋅ ^{FA} , NAMED ) of O w.r.t. D and V exists such that (CE)^{C} ≠ ∅.
Class Expression Subsumption: CE_{1} is subsumed by a class expression CE_{2} w.r.t. O and D if (CE_{1})^{C} ⊆ (CE_{2})^{C} for each model I = ( Δ_{I} , Δ_{D} , ⋅ ^{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 = ( Δ_{I} , Δ_{D} , ⋅ ^{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 firstorder 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 O_{1}, CE, CE_{1}, and CE_{2} can contain only object property expressions that are simple in the axiom closure Ax of O.
For ontology equivalence to be decidable, O_{1} 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:
 One can apply the direct semantics to an OWL 2 DL 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 DL reasoners can provide datatypes not explicitly mentioned in this specification without fear that this will change the meaning of OWL 2 DL ontologies not using these datatypes.
Theorem DS1. Let O_{1} and O_{2} be OWL 2 DL 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 O_{1} and O_{2} is rdfs:Literal, a datatype defined in the respective ontology, 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, O_{1} entails O_{2} w.r.t. D if and only if O_{1} entails O_{2} w.r.t. D'.
Proof. Without loss of generality, one can assume O_{1} and O_{2} to be in negationnormal form [Description Logics]. Furthermore, since datatype definitions in O_{1} and O_{2} are acyclic, one can assume that each defined datatype has been recursively replaced with its definition; thus, all datatypes in O_{1} and O_{2} are from N_{DT} ∪ { 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 O_{1} is and O_{2} is not satisfied in I if and only if an interpretation I' w.r.t. D' and V exists such that O_{1} is and O_{2} 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 = ( Δ_{I} , Δ_{D} , ⋅ ^{C} , ⋅ ^{OP} , ⋅ ^{DP} , ⋅ ^{I} , ⋅ ^{DT} , ⋅ ^{LT} , ⋅ ^{FA} , NAMED ) w.r.t. D and V exists such that O_{1} is and O_{2} is not satisfied in I. Let I' = ( Δ_{I} , Δ_{D}' , ⋅ ^{C '} , ⋅ ^{OP} , ⋅ ^{DP '} , ⋅ ^{I} , ⋅ ^{DT '} , ⋅ ^{LT '} , ⋅ ^{FA '} , NAMED ) 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, DataComplementOf( DR )^{DT} ⊆ DataComplementOf( 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 O_{1} and O_{2} 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 O_{1} and O_{2}. Therefore, O_{1} is and O_{2} is not satisfied in I'. QED
4 Appendix: Change Log (Informative)
4.1 Changes Since Proposed Recommendation
No changes have been made to this document since the Proposed Recommendation of 22 September, 2009.
4.2 Changes Since Candidate Recommendation
This section summarizes the changes to this document since the Candidate Recommendation of 11 June, 2009.
 An editorial comment was added to clarify the role played by the OWL 2 datatype map.
4.3 Changes Since Last Call
This section summarizes the changes to this document since the Last Call Working Draft of 21 April, 2009.
 Some minor editorial changes were made.
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 BozenBolzano), Bernardo Cuenca Grau (Oxford University Computing Laboratory), Martin Dzbor (Open University), Achille Fokoue (IBM Corporation), Christine Golbreich (Université de Versailles StQuentin 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. PatelSchneider (Bell Labs Research, AlcatelLucent), 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 FunctionalStyle Syntax. Boris Motik, Peter F. PatelSchneider, and Bijan Parsia, eds., 2009.
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. PatelSchneider, 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. PatelSchneider, Patrick Hayes, and Ian Horrocks, eds. W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/RECowlsemantics20040210/. Latest version available at http://www.w3.org/TR/owlsemantics/.
 [OWL 2 Profiles]
 OWL 2 Web Ontology Language: Profiles. Boris Motik, Bernardo Cuenca Grau, Ian Horrocks, Zhe Wu, Achille Fokoue, and Carsten Lutz, eds., 2009.
 [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.