OWL 2 Web Ontology Language Profiles (Second Edition) · 12/11/2012 · The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the Semantic Web with formally

OWL 2 Web Ontology LanguageProfiles (Second Edition)W3C Recommendation 11 December 2012

This version:http://www.w3.org/TR/2012/REC-owl2-profiles-20121211/

Latest version (series 2):http://www.w3.org/TR/owl2-profiles/

Latest Recommendation:http://www.w3.org/TR/owl-profiles

Previous version:http://www.w3.org/TR/2012/PER-owl2-profiles-20121018/

Editors:Boris Motik, University of OxfordBernardo Cuenca Grau, University of OxfordIan Horrocks, University of OxfordZhe Wu, OracleAchille Fokoue, IBM CorporationCarsten Lutz, University of Bremen

Contributors: (in alphabetical order)Diego Calvanese, Free University of Bozen-BolzanoJeremy Carroll, HP (now at TopQuadrant)Giuseppe De Giacomo, Sapienza Università di RomaJim Hendler, Rensselaer Polytechnic InstituteIvan Herman, W3C/ERCIMBijan Parsia, University of ManchesterPeter F. Patel-Schneider, Nuance CommunicationsAlan Ruttenberg, Science Commons (Creative Commons)Uli Sattler, University of ManchesterMichael Schneider, FZI Research Center for Information Technology

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 alsoavailable.

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

See also translations.

Copyright © 2012 W3C® (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark anddocument use rules apply.

OWL 2 Web Ontology Language Profiles (Second Edition) W3C Recommendation 11 December 2012

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Abstract

The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the SemanticWeb 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 usedalong with information written in RDF, and OWL 2 ontologies themselves are primarilyexchanged as RDF documents. The OWL 2 Document Overview describes the overall state ofOWL 2, and should be read before other OWL 2 documents.

This document provides a specification of several profiles of OWL 2 which can be more simplyand/or efficiently implemented. In logic, profiles are often called fragments. Most profiles aredefined by placing restrictions on the structure of OWL 2 ontologies. These restrictions havebeen specified by modifying the productions of the functional-style syntax.

Status of this Document

May Be Superseded

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

Summary of Changes

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

Please Send Comments

Please send any comments to [email protected] (public archive). Although work onthis document by the OWL Working Group is complete, comments may be addressed in theerrata or in future revisions. Open discussion among developers is welcome at [email protected] (public archive).

Endorsed By W3C

This document has been reviewed by W3C Members, by software developers, and by other W3Cgroups and interested parties, and is endorsed by the Director as a W3C Recommendation. It isa 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 topromote its widespread deployment. This enhances the functionality and interoperability of theWeb.

Patents

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



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Table of Contents

• 1 Introduction• 2 OWL 2 EL

◦ 2.1 Feature Overview◦ 2.2 Profile Specification

▪ 2.2.1 Entities▪ 2.2.2 Property Expressions▪ 2.2.3 Class Expressions▪ 2.2.4 Data Ranges▪ 2.2.5 Axioms▪ 2.2.6 Global Restrictions

• 3 OWL 2 QL◦ 3.1 Feature Overview◦ 3.2 Profile Specification

▪ 3.2.1 Entities▪ 3.2.2 Property Expressions▪ 3.2.3 Class Expressions▪ 3.2.4 Data Ranges▪ 3.2.5 Axioms

• 4 OWL 2 RL◦ 4.1 Feature Overview◦ 4.2 Profile Specification

▪ 4.2.1 Entities▪ 4.2.2 Property Expressions▪ 4.2.3 Class Expressions▪ 4.2.4 Data Ranges▪ 4.2.5 Axioms

◦ 4.3 Reasoning in OWL 2 RL and RDF Graphs using Rules• 5 Computational Properties• 6 Appendix: Complete Grammars for Profiles

◦ 6.1 OWL 2 EL◦ 6.2 OWL 2 QL◦ 6.3 OWL 2 RL

• 7 Appendix: Change Log (Informative)◦ 7.1 Changes Since Recommendation◦ 7.2 Changes Since Proposed Recommendation◦ 7.3 Changes Since Candidate Recommendation◦ 7.4 Changes Since Last Call

• 8 Acknowledgments• 9 References

◦ 9.1 Normative References◦ 9.2 Nonnormative References

1 Introduction

An OWL 2 profile (commonly called a fragment or a sublanguage in computational logic) is atrimmed down version of OWL 2 that trades some expressive power for the efficiency ofreasoning. This document describes three profiles of OWL 2, each of which achieves efficiency ina different way and is useful in different application scenarios. The profiles are independent ofeach other, so (prospective) users can skip over the descriptions of profiles that are not ofinterest to them. The choice of which profile to use in practice will depend on the structure of



the ontologies and the reasoning tasks at hand (see Section 10 of the OWL 2 Primer [OWL 2Primer] for more help in understanding and selecting profiles).

• OWL 2 EL is particularly useful in applications employing ontologies that contain verylarge numbers of properties and/or classes. This profile captures the expressive powerused by many such ontologies and is a subset of OWL 2 for which the basic reasoningproblems can be performed in time that is polynomial with respect to the size of theontology [EL++] (see Section 5 for more information on computational complexity).Dedicated reasoning algorithms for this profile are available and have beendemonstrated to be implementable in a highly scalable way. The EL acronym reflects theprofile's basis in the EL family of description logics [EL++], logics that provide onlyExistential quantification.

• OWL 2 QL is aimed at applications that use very large volumes of instance data, andwhere query answering is the most important reasoning task. In OWL 2 QL, conjunctivequery answering can be implemented using conventional relational database systems.Using a suitable reasoning technique, sound and complete conjunctive query answeringcan be performed in LOGSPACE with respect to the size of the data (assertions). As inOWL 2 EL, polynomial time algorithms can be used to implement the ontologyconsistency and class expression subsumption reasoning problems. The expressivepower of the profile is necessarily quite limited, although it does include most of themain features of conceptual models such as UML class diagrams and ER diagrams. TheQL acronym reflects the fact that query answering in this profile can be implemented byrewriting queries into a standard relational Query Language.

• OWL 2 RL is aimed at applications that require scalable reasoning without sacrificingtoo much expressive power. It is designed to accommodate OWL 2 applications that cantrade the full expressivity of the language for efficiency, as well as RDF(S) applicationsthat need some added expressivity. OWL 2 RL reasoning systems can be implementedusing rule-based reasoning engines. The ontology consistency, class expressionsatisfiability, class expression subsumption, instance checking, and conjunctive queryanswering problems can be solved in time that is polynomial with respect to the size ofthe ontology. The RL acronym reflects the fact that reasoning in this profile can beimplemented using a standard Rule Language.

OWL 2 profiles are defined by placing restrictions on the structure of OWL 2 ontologies.Syntactic restrictions can be specified by modifying the grammar of the functional-style syntax[OWL 2 Specification] and possibly giving additional global restrictions. In this document, themodified grammars are specified in two ways. In each profile definition, only the difference withrespect to the full grammar is given; that is, only the productions that differ from the functional-style syntax are presented, while the productions that are the same as in the functional-stylesyntax are not repeated. Furthermore, the full grammar for each of the profiles is given in theAppendix. Note that none of the profiles is a subset of another.

An ontology in any profile can be written into an ontology document by using any of thesyntaxes of OWL 2.

Apart from the ones specified here, there are many other possible profiles of OWL 2 — there are,for example, a whole family of profiles that extend OWL 2 QL. This document does not list OWLLite [OWL 1 Reference]; however, all OWL Lite ontologies are OWL 2 ontologies, so OWL Lite canbe viewed as a profile of OWL 2. Similarly, OWL 1 DL can also be viewed as a profile of OWL 2.

The italicized keywords must, must not, should, should not, and may are used to specifynormative features of OWL 2 documents and tools, and are interpreted as specified in RFC 2119[RFC 2119].



http://www.w3.org/TR/2012/REC-owl2-primer-20121211/#OWL_2_Profiles

2 OWL 2 EL

The OWL 2 EL profile [EL++,EL++ Update] is designed as a subset of OWL 2 that

• is particularly suitable for applications employing ontologies that define very largenumbers of classes and/or properties,

• captures the expressive power used by many such ontologies, and• for which ontology consistency, class expression subsumption, and instance checking

can be decided in polynomial time.

For example, OWL 2 EL provides class constructors that are sufficient to express the very largebiomedical ontology SNOMED CT [SNOMED CT].

2.1 Feature Overview

OWL 2 EL places restrictions on the type of class restrictions that can be used in axioms. Inparticular, the following types of class restrictions are supported:

• existential quantification to a class expression (ObjectSomeValuesFrom) or a datarange (DataSomeValuesFrom)

• existential quantification to an individual (ObjectHasValue) or a literal (DataHasValue)• self-restriction (ObjectHasSelf)• enumerations involving a single individual (ObjectOneOf) or a single literal

(DataOneOf)• intersection of classes (ObjectIntersectionOf) and data ranges (DataIntersectionOf)

OWL 2 EL supports the following axioms, all of which are restricted to the allowed set of classexpressions:

• class inclusion (SubClassOf)• class equivalence (EquivalentClasses)• class disjointness (DisjointClasses)• object property inclusion (SubObjectPropertyOf) with or without property chains, and

data property inclusion (SubDataPropertyOf)• property equivalence (EquivalentObjectProperties and EquivalentDataProperties),• transitive object properties (TransitiveObjectProperty)• reflexive object properties (ReflexiveObjectProperty)• domain restrictions (ObjectPropertyDomain and DataPropertyDomain)• range restrictions (ObjectPropertyRange and DataPropertyRange)• assertions (SameIndividual, DifferentIndividuals, ClassAssertion,

ObjectPropertyAssertion, DataPropertyAssertion,NegativeObjectPropertyAssertion, and NegativeDataPropertyAssertion)

• functional data properties (FunctionalDataProperty)• keys (HasKey)

The following constructs are not supported in OWL 2 EL:

• universal quantification to a class expression (ObjectAllValuesFrom) or a data range(DataAllValuesFrom)

• cardinality restrictions (ObjectMaxCardinality, ObjectMinCardinality,ObjectExactCardinality, DataMaxCardinality, DataMinCardinality, andDataExactCardinality)

• disjunction (ObjectUnionOf, DisjointUnion, and DataUnionOf)• class negation (ObjectComplementOf)• enumerations involving more than one individual (ObjectOneOf and DataOneOf)• disjoint properties (DisjointObjectProperties and DisjointDataProperties)



• irreflexive object properties (IrreflexiveObjectProperty)• inverse object properties (InverseObjectProperties)• functional and inverse-functional object properties (FunctionalObjectProperty and

InverseFunctionalObjectProperty)• symmetric object properties (SymmetricObjectProperty)• asymmetric object properties (AsymmetricObjectProperty)

2.2 Profile Specification

The following sections specify the structure of OWL 2 EL ontologies.

2.2.1 Entities

Entities are defined in OWL 2 EL in the same way as in the structural specification [OWL 2Specification], and OWL 2 EL supports all predefined classes and properties. Furthermore, OWL 2EL supports the following datatypes:

• rdf:PlainLiteral• rdf:XMLLiteral• rdfs:Literal• owl:real• owl:rational• xsd:decimal• xsd:integer• xsd:nonNegativeInteger• xsd:string• xsd:normalizedString• xsd:token• xsd:Name• xsd:NCName• xsd:NMTOKEN• xsd:hexBinary• xsd:base64Binary• xsd:anyURI• xsd:dateTime• xsd:dateTimeStamp

The set of supported datatypes has been designed such that the intersection of the valuespaces of any set of these datatypes is either empty or infinite, which is necessary to obtain thedesired computational properties [EL++]. Consequently, the following datatypes must not beused in OWL 2 EL: xsd:double, xsd:float, xsd:nonPositiveInteger, xsd:positiveInteger,xsd:negativeInteger, xsd:long, xsd:int, xsd:short, xsd:byte, xsd:unsignedLong, xsd:unsignedInt,xsd:unsignedShort, xsd:unsignedByte, xsd:language, and xsd:boolean.

Finally, OWL 2 EL does not support anonymous individuals.

Individual := NamedIndividual



2.2.2 Property Expressions

Inverse properties are not supported in OWL 2 EL, so object property expressions are restrictedto named properties. Data property expressions are defined in the same way as in the structuralspecification [OWL 2 Specification].

ObjectPropertyExpression := ObjectProperty

2.2.3 Class Expressions

In order to allow for efficient reasoning, OWL 2 EL restricts the set of supported classexpressions to ObjectIntersectionOf, ObjectSomeValuesFrom, ObjectHasSelf,ObjectHasValue, DataSomeValuesFrom, DataHasValue, and ObjectOneOf containing a singleindividual.

ClassExpression :=Class | ObjectIntersectionOf | ObjectOneOf |ObjectSomeValuesFrom | ObjectHasValue | ObjectHasSelf |DataSomeValuesFrom | DataHasValue

ObjectOneOf := 'ObjectOneOf' '(' Individual ')'

2.2.4 Data Ranges

A data range expression is restricted in OWL 2 EL to the predefined datatypes admitted in OWL2 EL, intersections of data ranges, and to enumerations of literals consisting of a single literal.

DataRange := Datatype | DataIntersectionOf | DataOneOfDataOneOf := 'DataOneOf' '(' Literal ')'

2.2.5 Axioms

The class axioms of OWL 2 EL are the same as in the structural specification [OWL 2Specification], with the exception that DisjointUnion is disallowed. Different class axioms aredefined in the same way as in the structural specification [OWL 2 Specification], with thedifference that they use the new definition of ClassExpression.

ClassAxiom := SubClassOf | EquivalentClasses | DisjointClasses

OWL 2 EL supports the following object property axioms, which are defined in the same way asin the structural specification [OWL 2 Specification], with the difference that they use the newdefinition of ObjectPropertyExpression.



ObjectPropertyAxiom :=EquivalentObjectProperties | SubObjectPropertyOf |ObjectPropertyDomain | ObjectPropertyRange |ReflexiveObjectProperty | TransitiveObjectProperty

OWL 2 EL provides the same axioms about data properties as the structural specification [OWL 2Specification] apart from DisjointDataProperties.

DataPropertyAxiom :=SubDataPropertyOf | EquivalentDataProperties |DataPropertyDomain | DataPropertyRange | FunctionalDataProperty

The assertions in OWL 2 EL, as well as all other axioms, are the same as in the structuralspecification [OWL 2 Specification], with the difference that class object property expressionsare restricted as defined in the previous sections.

2.2.6 Global Restrictions

OWL 2 EL extends the global restrictions on axioms from Section 11 of the structuralspecification [OWL 2 Specification] with an additional condition [EL++ Update]. In order todefine this condition, the following notion is used.

The set of axioms Ax imposes a range restriction to a class expression CE on an object propertyOP1 if Ax contains the following axioms, where k ≥ 1 is an integer and OPi are object properties:

SubObjectPropertyOf( OP1 OP2)...SubObjectPropertyOf( OPk-1 OPk )ObjectPropertyRange( OPk CE )

The axiom closure Ax of an OWL 2 EL ontology must obey the restrictions described in Section11 of the structural specification [OWL 2 Specification] and, in addition, if

• Ax contains SubObjectPropertyOf( ObjectPropertyChain( OP1 ... OPn ) OP ) and• Ax imposes a range restriction to some class expression CE on OP

then Ax must impose a range restriction to CE on OPn.

This additional restriction is vacuously true for each SubObjectPropertyOf axiom in which inthe first item of the previous definition does not contain a property chain. There are noadditional restrictions for range restrictions on reflexive and transitive roles — that is, a rangerestriction can be placed on a reflexive and/or transitive role provided that it satisfies thepreviously mentioned restriction.

3 OWL 2 QL

The OWL 2 QL profile is designed so that sound and complete query answering is in LOGSPACE(more precisely, in AC0) with respect to the size of the data (assertions), while providing many ofthe main features necessary to express conceptual models such as UML class diagrams and ER



diagrams. In particular, this profile contains the intersection of RDFS and OWL 2 DL. It isdesigned so that data (assertions) that is stored in a standard relational database system can bequeried through an ontology via a simple rewriting mechanism, i.e., by rewriting the query intoan SQL query that is then answered by the RDBMS system, without any changes to the data.

OWL 2 QL is based on the DL-Lite family of description logics [DL-Lite]. Several variants of DL-Lite have been described in the literature, and DL-LiteR provides the logical underpinning forOWL 2 QL. DL-LiteR does not require the unique name assumption (UNA), since making thisassumption would have no impact on the semantic consequences of a DL-LiteR ontology. Moreexpressive variants of DL-Lite, such as DL-LiteA, extend DL-LiteR with functional properties, andthese can also be extended with keys; however, for query answering to remain in LOGSPACE,these extensions require UNA and need to impose certain global restrictions on the interactionbetween properties used in different types of axiom. Basing OWL 2 QL on DL-LiteR avoidspractical problems involved in the explicit axiomatization of UNA. Other variants of DL-Lite canalso be supported on top of OWL 2 QL, but may require additional restrictions on the structure ofontologies.


OWL 2 QL is defined not only in terms of the set of supported constructs, but it also restricts theplaces in which these constructs are allowed to occur. The allowed usage of constructs in classexpressions is summarized in Table 1.

Table 1. Syntactic Restrictions on Class Expressions in OWL 2 QLSubclass Expressions Superclass Expressions

a classexistential quantification(ObjectSomeValuesFrom)

where the class is limited to owl:Thingexistential quantification to a data range(DataSomeValuesFrom)

a classintersection (ObjectIntersectionOf)negation (ObjectComplementOf)existential quantification to a class(ObjectSomeValuesFrom)existential quantification to a data range(DataSomeValuesFrom)

OWL 2 QL supports the following axioms, constrained so as to be compliant with the mentionedrestrictions on class expressions:

• subclass axioms (SubClassOf)• class expression equivalence (EquivalentClasses)• class expression disjointness (DisjointClasses)• inverse object properties (InverseObjectProperties)• property inclusion (SubObjectPropertyOf not involving property chains and

SubDataPropertyOf)• property equivalence (EquivalentObjectProperties and EquivalentDataProperties)• property domain (ObjectPropertyDomain and DataPropertyDomain)• property range (ObjectPropertyRange and DataPropertyRange)• disjoint properties (DisjointObjectProperties and DisjointDataProperties)• symmetric properties (SymmetricObjectProperty)• reflexive properties (ReflexiveObjectProperty)• irreflexive properties (IrreflexiveObjectProperty)• asymmetric properties (AsymmetricObjectProperty)• assertions other than individual equality assertions and negative property assertions

(DifferentIndividuals, ClassAssertion, ObjectPropertyAssertion, andDataPropertyAssertion)



The following constructs are not supported in OWL 2 QL:

• existential quantification to a class expression or a data range (ObjectSomeValuesFromand DataSomeValuesFrom) in the subclass position

• self-restriction (ObjectHasSelf)• existential quantification to an individual or a literal (ObjectHasValue, DataHasValue)• enumeration of individuals and literals (ObjectOneOf, DataOneOf)• universal quantification to a class expression or a data range (ObjectAllValuesFrom,

DataAllValuesFrom)• cardinality restrictions (ObjectMaxCardinality, ObjectMinCardinality,

ObjectExactCardinality, DataMaxCardinality, DataMinCardinality,DataExactCardinality)

• disjunction (ObjectUnionOf, DisjointUnion, and DataUnionOf)• property inclusions (SubObjectPropertyOf) involving property chains• functional and inverse-functional properties (FunctionalObjectProperty,

InverseFunctionalObjectProperty, and FunctionalDataProperty)• transitive properties (TransitiveObjectProperty)• keys (HasKey)• individual equality assertions and negative property assertions

OWL 2 QL does not support individual equality assertions (SameIndividual): adding suchaxioms to OWL 2 QL would increase the data complexity of query answering, so that it is nolonger first order rewritable, which means that query answering could not be implementeddirectly using relational database technologies. However, an ontology O that includes individualequality assertions, but is otherwise OWL 2 QL, could be handled by computing thereflexive–symmetric–transitive closure of the equality (SameIndividual) relation in O (thisrequires answering recursive queries and can be implemented in LOGSPACE w.r.t. the size ofdata) [DL-Lite-bool], and then using this relation in query answering procedures to simulateindividual equality reasoning [Automated Reasoning].


The productions for OWL 2 QL are defined in the following sections. Note that each OWL 2 QLontology must satisfy the global restrictions on axioms defined in Section 11 of the structuralspecification [OWL 2 Specification].

3.2.1 Entities

Entities are defined in OWL 2 QL in the same way as in the structural specification [OWL 2Specification], and OWL 2 QL supports all predefined classes and properties. Furthermore, OWL2 QL supports the following datatypes:

• rdf:PlainLiteral• rdf:XMLLiteral• rdfs:Literal• owl:real• owl:rational• xsd:decimal• xsd:integer• xsd:nonNegativeInteger• xsd:string• xsd:normalizedString• xsd:token• xsd:Name• xsd:NCName• xsd:NMTOKEN



• xsd:hexBinary• xsd:base64Binary• xsd:anyURI• xsd:dateTime• xsd:dateTimeStamp

The set of supported datatypes has been designed such that the intersection of the valuespaces of any set of these datatypes is either empty or infinite, which is necessary to obtain thedesired computational properties. Consequently, the following datatypes must not be used inOWL 2 QL: xsd:double, xsd:float, xsd:nonPositiveInteger, xsd:positiveInteger,xsd:negativeInteger, xsd:long, xsd:int, xsd:short, xsd:byte, xsd:unsignedLong, xsd:unsignedInt,xsd:unsignedShort, xsd:unsignedByte, xsd:language, and xsd:boolean.

Finally, OWL 2 QL does not support anonymous individuals.



OWL 2 QL object and data property expressions are the same as in the structural specification[OWL 2 Specification].


In OWL 2 QL, there are two types of class expressions. The subClassExpression productiondefines the class expressions that can occur as subclass expressions in SubClassOf axioms, andthe superClassExpression production defines the classes that can occur as superclassexpressions in SubClassOf axioms.

subClassExpression :=Class |subObjectSomeValuesFrom | DataSomeValuesFrom

subObjectSomeValuesFrom := 'ObjectSomeValuesFrom' '(' ObjectPropertyExpressionowl:Thing ')'

superClassExpression :=Class |superObjectIntersectionOf | superObjectComplementOf |superObjectSomeValuesFrom | DataSomeValuesFrom

superObjectIntersectionOf := 'ObjectIntersectionOf' '(' superClassExpressionsuperClassExpression { superClassExpression } ')'superObjectComplementOf := 'ObjectComplementOf' '(' subClassExpression ')'superObjectSomeValuesFrom := 'ObjectSomeValuesFrom' '('ObjectPropertyExpression Class ')'

3.2.4 Data Ranges

A data range expression is restricted in OWL 2 QL to the predefined datatypes and theintersection of data ranges.



DataRange := Datatype | DataIntersectionOf

3.2.5 Axioms

The class axioms of OWL 2 QL are the same as in the structural specification [OWL 2Specification], with the exception that DisjointUnion is disallowed; however, all axioms thatrefer to the ClassExpression production are redefined so as to use subClassExpression and/orsuperClassExpression as appropriate.

SubClassOf := 'SubClassOf' '(' axiomAnnotations subClassExpressionsuperClassExpression ')'EquivalentClasses := 'EquivalentClasses' '(' axiomAnnotations subClassExpressionsubClassExpression { subClassExpression } ')'DisjointClasses := 'DisjointClasses' '(' axiomAnnotations subClassExpressionsubClassExpression { subClassExpression } ')'ClassAxiom := SubClassOf | EquivalentClasses | DisjointClasses

OWL 2 QL disallows the use of property chains in property inclusion axioms; however, simpleproperty inclusions are supported. Furthermore, OWL 2 QL disallows the use of functional andtransitive object properties, and it restricts the class expressions in object property domain andrange axioms to superClassExpression.

ObjectPropertyDomain := 'ObjectPropertyDomain' '(' axiomAnnotationsObjectPropertyExpression superClassExpression ')'ObjectPropertyRange := 'ObjectPropertyRange' '(' axiomAnnotationsObjectPropertyExpression superClassExpression ')'SubObjectPropertyOf := 'SubObjectPropertyOf' '(' axiomAnnotationsObjectPropertyExpression ObjectPropertyExpression ')'ObjectPropertyAxiom :=

SubObjectPropertyOf | EquivalentObjectProperties |DisjointObjectProperties | InverseObjectProperties |ObjectPropertyDomain | ObjectPropertyRange |ReflexiveObjectProperty |SymmetricObjectProperty | AsymmetricObjectProperty

OWL 2 QL disallows functional data property axioms, and it restricts the class expressions indata property domain axioms to superClassExpression.

DataPropertyDomain := 'DataPropertyDomain' '(' axiomAnnotationsDataPropertyExpression superClassExpression ')'DataPropertyAxiom :=

SubDataPropertyOf | EquivalentDataProperties | DisjointDataProperties |DataPropertyDomain | DataPropertyRange



OWL 2 QL disallows negative object property assertions and individual equality axioms.Furthermore, class assertions in OWL 2 QL can involve only atomic classes. Inequality axiomsand property assertions are the same as in the structural specification [OWL 2 Specification].

ClassAssertion := 'ClassAssertion' '(' axiomAnnotations Class Individual ')'Assertion := DifferentIndividuals | ClassAssertion | ObjectPropertyAssertion |DataPropertyAssertion

Finally, the axioms in OWL 2 QL are the same as those in the structural specification [OWL 2Specification], with the exception that key axioms are not allowed.

Axiom := Declaration | ClassAxiom | ObjectPropertyAxiom | DataPropertyAxiom |DatatypeDefinition | Assertion | AnnotationAxiom

4 OWL 2 RL

The OWL 2 RL profile is aimed at applications that require scalable reasoning without sacrificingtoo much expressive power. It is designed to accommodate both OWL 2 applications that cantrade the full expressivity of the language for efficiency, and RDF(S) applications that needsome added expressivity from OWL 2. This is achieved by defining a syntactic subset of OWL 2which is amenable to implementation using rule-based technologies (see Section 4.2), andpresenting a partial axiomatization of the OWL 2 RDF-Based Semantics [OWL 2 RDF-BasedSemantics] in the form of first-order implications that can be used as the basis for such animplementation (see Section 4.3). The design of OWL 2 RL was inspired by Description LogicPrograms [DLP] and pD* [pD*].

For ontologies satisfying the syntactic constraints described in Section 4.2, a suitable rule-basedimplementation (e.g., one based on the partial axiomatization presented in Section 4.3) willhave desirable computational properties; for example, it can return all and only the correctanswers to certain kinds of query (see Theorem PR1 and OWL 2 Conformance [OWL 2Conformance]). Such an implementation can also be used with arbitrary RDF graphs. In thiscase, however, these properties no longer hold — in particular, it is no longer possible toguarantee that all correct answers can be returned, for example if the RDF graph uses the built-in vocabulary in unusual ways. Such an implementation will, however, still produce only correctentailments (see OWL 2 Conformance [OWL 2 Conformance]).


Restricting the way in which constructs are used makes it possible to implement reasoningsystems using rule-based reasoning engines, while still providing desirable computationalguarantees. These restrictions are designed so as to avoid the need to infer the existence ofindividuals not explicitly present in the knowledge base, and to avoid the need fornondeterministic reasoning. This is achieved by restricting the use of constructs to certainsyntactic positions. For example in SubClassOf axioms, the constructs in the subclass andsuperclass expressions must follow the usage patterns shown in Table 2.

Table 2. Syntactic Restrictions on Class Expressions in OWL 2 RLSubclass Expressions Superclass Expressions



a class other than owl:Thingan enumeration of individuals(ObjectOneOf)intersection of class expressions(ObjectIntersectionOf)union of class expressions(ObjectUnionOf)existential quantification to a classexpression (ObjectSomeValuesFrom)existential quantification to a data range(DataSomeValuesFrom)existential quantification to an individual(ObjectHasValue)existential quantification to a literal(DataHasValue)

a class other than owl:Thingintersection of classes (ObjectIntersectionOf)negation (ObjectComplementOf)universal quantification to a class expression(ObjectAllValuesFrom)existential quantification to an individual(ObjectHasValue)at-most 0/1 cardinality restriction to a classexpression (ObjectMaxCardinality 0/1)universal quantification to a data range(DataAllValuesFrom)existential quantification to a literal(DataHasValue)at-most 0/1 cardinality restriction to a datarange (DataMaxCardinality 0/1)

All axioms in OWL 2 RL are constrained in a way that is compliant with these restrictions. Thus,OWL 2 RL supports all axioms of OWL 2 apart from disjoint unions of classes (DisjointUnion)and reflexive object property axioms (ReflexiveObjectProperty).


The productions for OWL 2 RL are defined in the following sections. OWL 2 RL is defined not onlyin terms of the set of supported constructs, but it also restricts the places in which theseconstructs can be used. Note that each OWL 2 RL ontology must satisfy the global restrictionson axioms defined in Section 11 of the structural specification [OWL 2 Specification].

4.2.1 Entities

Entities are defined in OWL 2 RL in the same way as in the structural specification [OWL 2Specification]. OWL 2 RL supports the predefined classes owl:Nothing and owl:Thing, but theusage of the latter class is restricted by the grammar of OWL 2 RL. Furthermore, OWL 2 RL doesnot support the predefined object and data properties owl:topObjectProperty,owl:bottomObjectProperty, owl:topDataProperty, and owl:bottomDataProperty. Finally, OWL 2 RLsupports the following datatypes:

• rdf:PlainLiteral• rdf:XMLLiteral• rdfs:Literal• xsd:decimal• xsd:integer• xsd:nonNegativeInteger• xsd:nonPositiveInteger• xsd:positiveInteger• xsd:negativeInteger• xsd:long• xsd:int• xsd:short• xsd:byte• xsd:unsignedLong• xsd:unsignedInt• xsd:unsignedShort• xsd:unsignedByte• xsd:float• xsd:double



• xsd:string• xsd:normalizedString• xsd:token• xsd:language• xsd:Name• xsd:NCName• xsd:NMTOKEN• xsd:boolean• xsd:hexBinary• xsd:base64Binary• xsd:anyURI• xsd:dateTime• xsd:dateTimeStamp

The set of supported datatypes has been designed to allow for an implementation in rulesystems. The owl:real and owl:rational datatypes must not be used in OWL 2 RL.


Property expressions in OWL 2 RL are identical to the property expressions in the structuralspecification [OWL 2 Specification].


There are three types of class expressions in OWL 2 RL. The subClassExpression productiondefines the class expressions that can occur as subclass expressions in SubClassOf axioms; thesuperClassExpression production defines the class expressions that can occur as superclassexpressions in SubClassOf axioms; and the equivClassExpressions production defines theclasses that can occur in EquivalentClasses axioms.

zeroOrOne := '0' | '1'

subClassExpression :=Class other than owl:Thing |subObjectIntersectionOf | subObjectUnionOf | ObjectOneOf |subObjectSomeValuesFrom | ObjectHasValue |DataSomeValuesFrom | DataHasValue

subObjectIntersectionOf := 'ObjectIntersectionOf' '(' subClassExpressionsubClassExpression { subClassExpression } ')'subObjectUnionOf := 'ObjectUnionOf' '(' subClassExpression subClassExpression {subClassExpression } ')'subObjectSomeValuesFrom :=

'ObjectSomeValuesFrom' '(' ObjectPropertyExpression subClassExpression ')' |'ObjectSomeValuesFrom' '(' ObjectPropertyExpression owl:Thing ')'

superClassExpression :=Class other than owl:Thing |superObjectIntersectionOf | superObjectComplementOf |superObjectAllValuesFrom | ObjectHasValue | superObjectMaxCardinality |DataAllValuesFrom | DataHasValue | superDataMaxCardinality

superObjectIntersectionOf := 'ObjectIntersectionOf' '(' superClassExpressionsuperClassExpression { superClassExpression } ')'superObjectComplementOf := 'ObjectComplementOf' '(' subClassExpression ')'



superObjectAllValuesFrom := 'ObjectAllValuesFrom' '(' ObjectPropertyExpressionsuperClassExpression ')'superObjectMaxCardinality :=

'ObjectMaxCardinality' '(' zeroOrOne ObjectPropertyExpression [subClassExpression ] ')' |

'ObjectMaxCardinality' '(' zeroOrOne ObjectPropertyExpression owl:Thing ')'superDataMaxCardinality := 'DataMaxCardinality' '(' zeroOrOneDataPropertyExpression [ DataRange ] ')'

equivClassExpression :=Class other than owl:Thing |equivObjectIntersectionOf |ObjectHasValue |DataHasValue

equivObjectIntersectionOf := 'ObjectIntersectionOf' '(' equivClassExpressionequivClassExpression { equivClassExpression } ')'

4.2.4 Data Ranges

A data range expression is restricted in OWL 2 RL to the predefined datatypes admitted in OWL2 RL and the intersection of data ranges.


4.2.5 Axioms

OWL 2 RL redefines all axioms of the structural specification [OWL 2 Specification] that refer toclass expressions. In particular, it restricts various class axioms to use the appropriate form ofclass expressions (i.e., one of subClassExpression, superClassExpression, orequivClassExpression), and it disallows the DisjointUnion axiom.

ClassAxiom := SubClassOf | EquivalentClasses | DisjointClassesSubClassOf := 'SubClassOf' '(' axiomAnnotations subClassExpressionsuperClassExpression ')'EquivalentClasses := 'EquivalentClasses' '(' axiomAnnotationsequivClassExpression equivClassExpression { equivClassExpression } ')'DisjointClasses := 'DisjointClasses' '(' axiomAnnotations subClassExpressionsubClassExpression { subClassExpression } ')'

OWL 2 RL axioms about property expressions are as in the structural specification [OWL 2Specification], the only differences being that class expressions in property domain and rangeaxioms are restricted to superClassExpression, and that the use of reflexive properties isdisallowed.



ObjectPropertyDomain := 'ObjectPropertyDomain' '(' axiomAnnotationsObjectPropertyExpression superClassExpression ')'ObjectPropertyRange := 'ObjectPropertyRange' '(' axiomAnnotationsObjectPropertyExpression superClassExpression ')'DataPropertyDomain := 'DataPropertyDomain' '(' axiomAnnotationsDataPropertyExpression superClassExpression ')'ObjectPropertyAxiom :=

SubObjectPropertyOf | EquivalentObjectProperties |DisjointObjectProperties | InverseObjectProperties |ObjectPropertyDomain | ObjectPropertyRange |FunctionalObjectProperty | InverseFunctionalObjectProperty |IrreflexiveObjectProperty |SymmetricObjectProperty | AsymmetricObjectPropertyTransitiveObjectProperty

OWL 2 RL restricts class expressions in positive assertions to superClassExpression. All otherassertions are the same as in the structural specification [OWL 2 Specification].

ClassAssertion := 'ClassAssertion' '(' axiomAnnotations superClassExpressionIndividual ')'

OWL 2 RL restricts class expressions in keys to subClassExpression.

HasKey := 'HasKey' '(' axiomAnnotations subClassExpression '(' {ObjectPropertyExpression } ')' '(' { DataPropertyExpression } ')' ')'

All other axioms in OWL 2 RL are defined as in the structural specification [OWL 2 Specification].

4.3 Reasoning in OWL 2 RL and RDF Graphs using Rules

This section presents a partial axiomatization of the OWL 2 RDF-Based Semantics [OWL 2 RDF-Based Semantics] in the form of first-order implications; this axiomatization is called the OWL 2RL/RDF rules. These rules provide a useful starting point for practical implementation using rule-based technologies such as logic programming [Logic Programming, Lloyd].

The rules are given as universally quantified first-order implications over a ternary predicate T.This predicate represents a generalization of RDF triples in which bnodes and literals are allowedin all positions (similar to the partial generalization in pD* [pD*] and to generalized RDF triplesin RIF [RIF RDF & OWL]); thus, T(s, p, o) represents a generalized RDF triple with the subjects, predicate p, and the object o. Variables in the implications are preceded with a question mark.The rules that have empty "if" parts should be understood as being always applicable. Thepropositional symbol false is a special symbol denoting contradiction: if it is derived, then theinitial RDF graph was inconsistent. The set of rules listed in this section is not minimal, ascertain rules are implied by other ones; this was done to make the definition of the semanticconsequences of each piece of OWL 2 vocabulary self-contained.

Many conditions contain atoms that match to the list construct of RDF. In order to simplify thepresentation of the rules, LIST[h, e1, ..., en] is used as an abbreviation for the conjunction



of triples shown in Table 3, where z2, ..., zn are fresh variables that do not occur anywherewhere the abbreviation is used.

Table 3. Expansion of LIST[h, e1, ..., en]

T(h, rdf:first, e1) T(h, rdf:rest, z2)

T(z2, rdf:first, e2) T(z2, rdf:rest, z3)

... ...

T(zn, rdf:first, en) T(zn, rdf:rest, rdf:nil)

The axiomatization is split into several tables for easier navigation. Each rule is given a shortunique name. The rows of several tables specify rules that need to be instantiated for eachcombination of indices given in the right-most column.

Table 4 axiomatizes the semantics of equality. In particular, it defines the equality relationowl:sameAs as being reflexive, symmetric, and transitive, and it axiomatizes the standardreplacement properties of equality for it.

Table 4. The Semantics of EqualityIf then

eq-ref T(?s, ?p, ?o)

T(?s,owl:sameAs, ?s)T(?p,owl:sameAs, ?p)T(?o,owl:sameAs, ?o)

eq-sym T(?x, owl:sameAs, ?y) T(?y,owl:sameAs, ?x)

eq-trans

T(?x, owl:sameAs, ?y)T(?y, owl:sameAs, ?z)

T(?x,owl:sameAs, ?z)

eq-rep-s

T(?s, owl:sameAs, ?s')T(?s, ?p, ?o) T(?s', ?p, ?o)

eq-rep-p

T(?p, owl:sameAs, ?p')T(?s, ?p, ?o) T(?s, ?p', ?o)

eq-rep-o

T(?o, owl:sameAs, ?o')T(?s, ?p, ?o) T(?s, ?p, ?o')

eq-diff1

T(?x, owl:sameAs, ?y)T(?x, owl:differentFrom, ?y) false

eq-diff2

T(?x, rdf:type,owl:AllDifferent)T(?x, owl:members, ?y)LIST[?y, ?z1, ..., ?zn]T(?zi, owl:sameAs, ?zj)

false for each 1 ≤ i < j ≤n

eq-diff3

T(?x, rdf:type,owl:AllDifferent) false for each 1 ≤ i < j ≤

n



T(?x, owl:distinctMembers, ?y)LIST[?y, ?z1, ..., ?zn]T(?zi, owl:sameAs, ?zj)

Table 5 specifies the semantic conditions on axioms about properties.

Table 5. The Semantics of Axioms about PropertiesIf then

prp-ap

T(ap, rdf:type,owl:AnnotationProperty)

for each built-inannotationproperty of OWL 2RL

prp-dom

T(?p, rdfs:domain, ?c)T(?x, ?p, ?y) T(?x, rdf:type, ?c)

prp-rng

T(?p, rdfs:range, ?c)T(?x, ?p, ?y) T(?y, rdf:type, ?c)

prp-fp

T(?p, rdf:type,owl:FunctionalProperty)T(?x, ?p, ?y1)T(?x, ?p, ?y2)

T(?y1, owl:sameAs, ?y2)

prp-ifp

T(?p, rdf:type,owl:InverseFunctionalProperty)T(?x1, ?p, ?y)T(?x2, ?p, ?y)

T(?x1, owl:sameAs, ?x2)

prp-irp

T(?p, rdf:type,owl:IrreflexiveProperty)T(?x, ?p, ?x)

false

prp-symp

T(?p, rdf:type,owl:SymmetricProperty)T(?x, ?p, ?y)

T(?y, ?p, ?x)

prp-asyp

T(?p, rdf:type,owl:AsymmetricProperty)T(?x, ?p, ?y)T(?y, ?p, ?x)

false

prp-trp

T(?p, rdf:type,owl:TransitiveProperty)T(?x, ?p, ?y)T(?y, ?p, ?z)

T(?x, ?p, ?z)

prp-spo1

T(?p1,rdfs:subPropertyOf, ?p2)T(?x, ?p1, ?y)

T(?x, ?p2, ?y)

prp-spo2

T(?p,owl:propertyChainAxiom, ?x)LIST[?x, ?p1, ..., ?pn]T(?u1, ?p1, ?u2)

T(?u1, ?p, ?un+1)



T(?u2, ?p2, ?u3)...T(?un, ?pn, ?un+1)

prp-eqp1

T(?p1,owl:equivalentProperty, ?p2)T(?x, ?p1, ?y)

T(?x, ?p2, ?y)

prp-eqp2

T(?p1,owl:equivalentProperty, ?p2)T(?x, ?p2, ?y)

T(?x, ?p1, ?y)

prp-pdw

T(?p1,owl:propertyDisjointWith, ?p2)T(?x, ?p1, ?y)T(?x, ?p2, ?y)

false

prp-adp

T(?x, rdf:type,owl:AllDisjointProperties)T(?x, owl:members, ?y)LIST[?y, ?p1, ..., ?pn]T(?u, ?pi, ?v)T(?u, ?pj, ?v)

false for each 1 ≤ i < j≤ n

prp-inv1

T(?p1, owl:inverseOf, ?p2)T(?x, ?p1, ?y) T(?y, ?p2, ?x)

prp-inv2

T(?p1, owl:inverseOf, ?p2)T(?x, ?p2, ?y) T(?y, ?p1, ?x)

prp-key

T(?c, owl:hasKey, ?u)LIST[?u, ?p1, ..., ?pn]T(?x, rdf:type, ?c)T(?x, ?p1, ?z1)...T(?x, ?pn, ?zn)T(?y, rdf:type, ?c)T(?y, ?p1, ?z1)...T(?y, ?pn, ?zn)

T(?x, owl:sameAs, ?y)

prp-npa1

T(?x,owl:sourceIndividual, ?i1)T(?x,owl:assertionProperty, ?p)T(?x,owl:targetIndividual, ?i2)T(?i1, ?p, ?i2)

false

prp-npa2

T(?x,owl:sourceIndividual, ?i)T(?x,owl:assertionProperty, ?p)T(?x, owl:targetValue, ?lt)T(?i, ?p, ?lt)

false



Table 6 specifies the semantic conditions on classes.

Table 6. The Semantics of ClassesIf then

cls-thing

T(owl:Thing,rdf:type, owl:Class)

cls-nothing1

T(owl:Nothing,rdf:type, owl:Class)

cls-nothing2 T(?x, rdf:type, owl:Nothing) false

cls-int1

T(?c, owl:intersectionOf, ?x)LIST[?x, ?c1, ..., ?cn]T(?y, rdf:type, ?c1)T(?y, rdf:type, ?c2)...T(?y, rdf:type, ?cn)

T(?y, rdf:type, ?c)

cls-int2T(?c, owl:intersectionOf, ?x)LIST[?x, ?c1, ..., ?cn]T(?y, rdf:type, ?c)

T(?y, rdf:type, ?c1)T(?y, rdf:type, ?c2)...T(?y, rdf:type, ?cn)

cls-uniT(?c, owl:unionOf, ?x)LIST[?x, ?c1, ..., ?cn]T(?y, rdf:type, ?ci)

T(?y, rdf:type, ?c) for each1 ≤ i ≤ n

cls-comT(?c1, owl:complementOf, ?c2)T(?x, rdf:type, ?c1)T(?x, rdf:type, ?c2)

false

cls-svf1

T(?x, owl:someValuesFrom, ?y)T(?x, owl:onProperty, ?p)T(?u, ?p, ?v)T(?v, rdf:type, ?y)

T(?u, rdf:type, ?x)

cls-svf2T(?x, owl:someValuesFrom, owl:Thing)T(?x, owl:onProperty, ?p)T(?u, ?p, ?v)

T(?u, rdf:type, ?x)

cls-avf

T(?x, owl:allValuesFrom, ?y)T(?x, owl:onProperty, ?p)T(?u, rdf:type, ?x)T(?u, ?p, ?v)

T(?v, rdf:type, ?y)

cls-hv1T(?x, owl:hasValue, ?y)T(?x, owl:onProperty, ?p)T(?u, rdf:type, ?x)

T(?u, ?p, ?y)

cls-hv2T(?x, owl:hasValue, ?y)T(?x, owl:onProperty, ?p)T(?u, ?p, ?y)

T(?u, rdf:type, ?x)



cls-maxc1

T(?x, owl:maxCardinality,"0"^^xsd:nonNegativeInteger)T(?x, owl:onProperty, ?p)T(?u, rdf:type, ?x)T(?u, ?p, ?y)

false

cls-maxc2

T(?x, owl:maxCardinality,"1"^^xsd:nonNegativeInteger)T(?x, owl:onProperty, ?p)T(?u, rdf:type, ?x)T(?u, ?p, ?y1)T(?u, ?p, ?y2)

T(?y1,owl:sameAs, ?y2)

cls-maxqc1

T(?x, owl:maxQualifiedCardinality,"0"^^xsd:nonNegativeInteger)T(?x, owl:onProperty, ?p)T(?x, owl:onClass, ?c)T(?u, rdf:type, ?x)T(?u, ?p, ?y)T(?y, rdf:type, ?c)

false

cls-maxqc2

T(?x, owl:maxQualifiedCardinality,"0"^^xsd:nonNegativeInteger)T(?x, owl:onProperty, ?p)T(?x, owl:onClass, owl:Thing)T(?u, rdf:type, ?x)T(?u, ?p, ?y)

false

cls-maxqc3

T(?x, owl:maxQualifiedCardinality,"1"^^xsd:nonNegativeInteger)T(?x, owl:onProperty, ?p)T(?x, owl:onClass, ?c)T(?u, rdf:type, ?x)T(?u, ?p, ?y1)T(?y1, rdf:type, ?c)T(?u, ?p, ?y2)T(?y2, rdf:type, ?c)


cls-maxqc4

T(?x, owl:maxQualifiedCardinality,"1"^^xsd:nonNegativeInteger)T(?x, owl:onProperty, ?p)T(?x, owl:onClass, owl:Thing)T(?u, rdf:type, ?x)T(?u, ?p, ?y1)T(?u, ?p, ?y2)


cls-oo T(?c, owl:oneOf, ?x)LIST[?x, ?y1, ..., ?yn]

T(?y1, rdf:type, ?c)...T(?yn, rdf:type, ?c)

Table 7 specifies the semantic conditions on class axioms.

Table 7. The Semantics of Class AxiomsIf then



cax-sco

T(?c1, rdfs:subClassOf, ?c2)T(?x, rdf:type, ?c1)

T(?x,rdf:type, ?c2)

cax-eqc1

T(?c1, owl:equivalentClass, ?c2)T(?x, rdf:type, ?c1)

T(?x,rdf:type, ?c2)

cax-eqc2

T(?c1, owl:equivalentClass, ?c2)T(?x, rdf:type, ?c2)

T(?x,rdf:type, ?c1)

cax-dwT(?c1, owl:disjointWith, ?c2)T(?x, rdf:type, ?c1)T(?x, rdf:type, ?c2)

false

cax-adc

T(?x, rdf:type,owl:AllDisjointClasses)T(?x, owl:members, ?y)LIST[?y, ?c1, ..., ?cn]T(?z, rdf:type, ?ci)T(?z, rdf:type, ?cj)

false for each 1 ≤ i < j≤ n

Table 8 specifies the semantics of datatypes.

Table 8. The Semantics of DatatypesIf then

dt-type1

T(dt, rdf:type,rdfs:Datatype)

for each datatype dt supported in OWL 2RL

dt-type2

T(lt, rdf:type,dt)

for each literal lt and each datatype dtsupported in OWL 2 RLsuch that the data value of lt is containedin the value space of dt

dt-eq T(lt1, owl:sameAs,lt2)

for all literals lt1 and lt2 with the samedata value

dt-diff

T(lt1,owl:differentFrom,lt2)

for all literals lt1 and lt2 with differentdata values

dt-not-type

T(lt,rdf:type,dt)

false

for each literal lt and each datatype dtsupported in OWL 2 RLsuch that the data value of lt is notcontained in the value space of dt

Table 9 specifies the semantic restrictions on the vocabulary used to define the schema.

Table 9. The Semantics of Schema VocabularyIf then

scm-cls T(?c, rdf:type, owl:Class)

T(?c, rdfs:subClassOf, ?c)T(?c, owl:equivalentClass, ?c)T(?c, rdfs:subClassOf,owl:Thing)



T(owl:Nothing,rdfs:subClassOf, ?c)

scm-sco T(?c1, rdfs:subClassOf, ?c2)T(?c2, rdfs:subClassOf, ?c3)

T(?c1, rdfs:subClassOf, ?c3)

scm-eqc1 T(?c1, owl:equivalentClass, ?c2)

T(?c1, rdfs:subClassOf, ?c2)T(?c2, rdfs:subClassOf, ?c1)

scm-eqc2

T(?c1, rdfs:subClassOf, ?c2)T(?c2, rdfs:subClassOf, ?c1)

T(?c1, owl:equivalentClass, ?c2)

scm-op T(?p, rdf:type,owl:ObjectProperty)

T(?p, rdfs:subPropertyOf, ?p)T(?p,owl:equivalentProperty, ?p)

scm-dp T(?p, rdf:type,owl:DatatypeProperty)

T(?p, rdfs:subPropertyOf, ?p)T(?p,owl:equivalentProperty, ?p)

scm-spo T(?p1, rdfs:subPropertyOf, ?p2)T(?p2, rdfs:subPropertyOf, ?p3)

T(?p1, rdfs:subPropertyOf, ?p3)

scm-eqp1 T(?p1, owl:equivalentProperty, ?p2)

T(?p1, rdfs:subPropertyOf, ?p2)T(?p2, rdfs:subPropertyOf, ?p1)

scm-eqp2

T(?p1, rdfs:subPropertyOf, ?p2)T(?p2, rdfs:subPropertyOf, ?p1)

T(?p1,owl:equivalentProperty, ?p2)

scm-dom1

T(?p, rdfs:domain, ?c1)T(?c1, rdfs:subClassOf, ?c2)

T(?p, rdfs:domain, ?c2)

scm-dom2

T(?p2, rdfs:domain, ?c)T(?p1, rdfs:subPropertyOf, ?p2)

T(?p1, rdfs:domain, ?c)

scm-rng1

T(?p, rdfs:range, ?c1)T(?c1, rdfs:subClassOf, ?c2)

T(?p, rdfs:range, ?c2)

scm-rng2

T(?p2, rdfs:range, ?c)T(?p1, rdfs:subPropertyOf, ?p2)

T(?p1, rdfs:range, ?c)

scm-hv

T(?c1, owl:hasValue, ?i)T(?c1, owl:onProperty, ?p1)T(?c2, owl:hasValue, ?i)T(?c2, owl:onProperty, ?p2)T(?p1, rdfs:subPropertyOf, ?p2)


scm-svf1

T(?c1, owl:someValuesFrom, ?y1)T(?c1, owl:onProperty, ?p)T(?c2, owl:someValuesFrom, ?y2)T(?c2, owl:onProperty, ?p)T(?y1, rdfs:subClassOf, ?y2)


scm-svf2

T(?c1, owl:someValuesFrom, ?y)T(?c1, owl:onProperty, ?p1)T(?c2, owl:someValuesFrom, ?y)




T(?c2, owl:onProperty, ?p2)T(?p1, rdfs:subPropertyOf, ?p2)

scm-avf1

T(?c1, owl:allValuesFrom, ?y1)T(?c1, owl:onProperty, ?p)T(?c2, owl:allValuesFrom, ?y2)T(?c2, owl:onProperty, ?p)T(?y1, rdfs:subClassOf, ?y2)


scm-avf2

T(?c1, owl:allValuesFrom, ?y)T(?c1, owl:onProperty, ?p1)T(?c2, owl:allValuesFrom, ?y)T(?c2, owl:onProperty, ?p2)T(?p1, rdfs:subPropertyOf, ?p2)


scm-int T(?c, owl:intersectionOf, ?x)LIST[?x, ?c1, ..., ?cn]

T(?c, rdfs:subClassOf, ?c1)T(?c, rdfs:subClassOf, ?c2)...T(?c, rdfs:subClassOf, ?cn)

scm-uni T(?c, owl:unionOf, ?x)LIST[?x, ?c1, ..., ?cn]

T(?c1, rdfs:subClassOf, ?c)T(?c2, rdfs:subClassOf, ?c)...T(?cn, rdfs:subClassOf, ?c)

In order to avoid potential performance problems in practice, OWL 2 RL/RDF rules do not includethe axiomatic triples of RDF and RDFS (i.e., those triples that must be satisfied by, respectively,every RDF and RDFS interpretation) [RDF Semantics] and the relevant OWL vocabulary [OWL 2RDF-Based Semantics]; moreover, OWL 2 RL/RDF rules include most, but not all of theentailment rules of RDFS [RDF Semantics]. An OWL 2 RL/RDF implementation may include thesetriples and entailment rules as necessary without invalidating the conformance requirements forOWL 2 RL [OWL 2 Conformance].

Theorem PR1. Let R be the OWL 2 RL/RDF rules as defined above. Furthermore, let O1 and O2be OWL 2 RL ontologies satisfying the following properties:

• neither O1 nor O2 contains a IRI that is used for more than one type of entity (i.e., noIRIs is used both as, say, a class and an individual);

• O1 does not contain SubAnnotationPropertyOf, AnnotationPropertyDomain, andAnnotationPropertyRange axioms; and

• each axiom in O2 is an assertion of the form as specified below, for a, a1, ..., an namedindividuals:

◦ ClassAssertion( C a ) where C is a class,◦ ObjectPropertyAssertion( OP a1 a2 ) where OP is an object property,◦ DataPropertyAssertion( DP a v ) where DP is a data property, or◦ SameIndividual( a1 ... an ).

Furthermore, let RDF(O1) and RDF(O2) be translations of O1 and O2, respectively, into RDFgraphs as specified in the OWL 2 Mapping to RDF Graphs [OWL 2 RDF Mapping]; and letFO(RDF(O1)) and FO(RDF(O2)) be the translation of these graphs into first-order theories inwhich triples are represented using the T predicate — that is, T(s, p, o) represents an RDFtriple with the subject s, predicate p, and the object o. Then, O1 entails O2 under the OWL 2Direct Semantics [OWL 2 Direct Semantics] if and only if FO(RDF(O1)) ∪ R entails FO(RDF(O2))under the standard first-order semantics.



Proof Sketch. Without loss of generality, it can be assumed that all axioms in O1 are fullynormalized — that is, that all class expressions in the axioms are of depth at most one. LetDLP(O1) be the set of rules obtained by translating O1 into a set of rules as in Description LogicPrograms [DLP].

Consider now each assertion A ∈ O2 that is entailed by DLP(O1) (or, equivalently, by O1). Let dtbe a derivation tree for A from DLP(O1). By examining the set of OWL 2 RL constructs, it ispossible to see that each such tree can be transformed to a derivation tree dt' for FO(RDF(A))from FO(RDF(O1)) ∪ R. Each assertion B occurring in dt is of the form as specified in thetheorem. The tree dt' can, roughly speaking, be obtained from dt by replacing each assertion Bwith FO(RDF(B)) and by replacing each rule from DLP(O1) with a corresponding rule from Tables3–8. Consequently, FO(RDF(O1)) ∪ R entails FO(RDF(A)).

Since no IRI in O1 is used as both an individual and a class or a property, FO(RDF(O1)) ∪ R doesnot entail a triple of the form T(a:i1, owl:sameAs, a:i2) where either a:i1 or a:i2 is used inO1 as a class or a property. This allows one to transform a derivation tree for FO(RDF(A)) fromFO(RDF(O1)) ∪ R to a derivation tree for A from DLP(O1) in a way that is analogous to theprevious case. QED

5 Computational Properties

This section describes the computational complexity of the most relevant reasoning problems ofthe languages defined in this document. For an introduction to computational complexity, pleaserefer to a textbook on complexity such as [Papadimitriou]. The reasoning problems consideredhere are ontology consistency, class expression satisfiability, class expression subsumption,instance checking, and (Boolean) conjunctive query answering [OWL 2 Direct Semantics]. Whenevaluating complexity, the following complexity measures are considered:

• Data Complexity: the complexity measured with respect to the total size of theassertions in the ontology

• Taxonomic Complexity: the complexity measured with respect to the total size of theaxioms (without assertions) in the ontology

• Query Complexity: the complexity measured with respect to the total size of theconjunctive query (defined only for conjunctive query answering)

• Combined Complexity: the complexity measured with respect to the total size of theaxioms and the assertions in the ontology, the conjunctive query (in the case ofconjunctive query answering), and the expression(s) being checked (in the case of classexpression satisfiability, class expression subsumption, or instance checking)

Table 10 summarizes the known complexity results for OWL 2 under both RDF and the directsemantics, OWL 2 EL, OWL 2 QL, OWL 2 RL, and OWL 1 DL. The meaning of the entries is asfollows:

• Decidability open means that it is not known whether this reasoning problem isdecidable at all.

• Decidable, but complexity open means that decidability of this reasoning problem isknown, but not its exact computational complexity. If available, known lower bounds aregiven in parenthesis; for example, (NP-Hard) means that this problem is at least ashard as any other problem in NP.

• X-complete for X one of the complexity classes explained below indicates that tightcomplexity bounds are known — that is, the problem is known to be both in thecomplexity class X (i.e., an algorithm is known that only uses time/space in X) and hardfor X (i.e., it is at least as hard as any other problem in X). The following is a brief sketchof the classes used in this table, from the most complex one down to the simplest ones.



◦ N2EXPTIME is the class of problems solvable by a nondeterministic algorithm intime that is at most double exponential in the size of the input (i.e., roughly 22n

,for n the size of the input).

◦ NEXPTIME is the class of problems solvable by a nondeterministic algorithm intime that is at most exponential in the size of the input (i.e., roughly 2n, for nthe size of the input).

◦ PSPACE is the class of problems solvable by a deterministic algorithm usingspace that is at most polynomial in the size of the input (i.e., roughly nc, for nthe size of the input and c a constant).

◦ NP is the class of problems solvable by a nondeterministic algorithm using timethat is at most polynomial in the size of the input (i.e., roughly nc, for n the sizeof the input and c a constant).

◦ PTIME is the class of problems solvable by a deterministic algorithm using timethat is at most polynomial in the size of the input (i.e., roughly nc, for n the sizeof the input and c a constant). PTIME is often referred to as tractable, whereasthe problems in the classes above are often referred to as intractable.

◦ LOGSPACE is the class of problems solvable by a deterministic algorithm usingspace that is at most logarithmic in the size of the input (i.e., roughly log(n), forn the size of the input and c a constant). NLOGSPACE is the nondeterministicversion of this class.

◦ AC0 is a proper subclass of LOGSPACE and defined not via Turing Machines, butvia circuits: AC0 is the class of problems definable using a family of circuits ofconstant depth and polynomial size, which can be generated by a deterministicTuring machine in logarithmic time (in the size of the input). Intuitively, AC0

allows us to use polynomially many processors but the run-time must beconstant. A typical example of an AC0 problem is the evaluation of first-orderqueries over databases (or model checking of first-order sentences over finitemodels), where only the database (first-order model) is regarded as the inputand the query (first-order sentence) is assumed to be fixed. The undirectedgraph reachability problem is known to be in LogSpace, but not in AC0.

The results below refer to the worst-case complexity of these reasoning problems and, as such,do not say that implemented algorithms necessarily run in this class on all input problems, orwhat space/time they use on some/typical/certain kind of problems. For X-complete problems,these results only say that a reasoning algorithm cannot use less time/space than indicated bythis class on all input problems.

Table 10. Complexity of the Profiles

Language ReasoningProblems

TaxonomicComplexity

DataComplexity

QueryComplexity

CombinedComplexity

OWL 2RDF-BasedSemantics

OntologyConsistency,ClassExpressionSatisfiability,ClassExpressionSubsumption,InstanceChecking,ConjunctiveQueryAnswering

Undecidable Undecidable Undecidable Undecidable



OntologyConsistency,ClassExpressionSatisfiability,ClassExpressionSubsumption,InstanceChecking

N2EXPTIME-complete

(NEXPTIME-complete if the

propertyhierarchy can betranslated into a

polynomially-sized

nondeterministicfinite

automaton)

Decidable,but

complexityopen

(NP-Hard)

NotApplicable

N2EXPTIME-complete

(NEXPTIME-complete if the

propertyhierarchy can betranslated into a

polynomially-sized


automaton)

OWL 2Direct

Semantics

ConjunctiveQueryAnswering

Decidabilityopen

Decidabilityopen

Decidabilityopen

Decidabilityopen


PTIME-complete PTIME-complete

NotApplicable PTIME-complete

OWL 2 EL


in EXPTIME(PTIME-completeif the property

hierarchy can betranslated into a

polynomially-sized


automaton)

PTIME-complete

NP-complete

in EXPTIME(PSPACE-

complete if theproperty

hierarchy can betranslated into a

polynomially-sized


automaton)


NLogSpace-complete In AC0 Not

ApplicableNLogSpace-

completeOWL 2 QL


NLogSpace-complete In AC0 NP-

complete NP-complete

OWL 2 RL OntologyConsistency PTIME-complete PTIME-

completeNot

Applicable PTIME-complete



ClassExpressionSatisfiability,ClassExpressionSubsumption,InstanceChecking


NotApplicable

co-NP-complete(PTIME-completefor atomic class

expressions)



NP-complete NP-complete


NEXPTIME-complete

Decidable,but

complexityopen

(NP-Hard)

NotApplicable

NEXPTIME-complete

OWL 1 DL


Decidabilityopen

Decidabilityopen

Decidabilityopen

Decidabilityopen

6 Appendix: Complete Grammars for Profiles

This appendix contains the grammars for all three profiles of OWL 2.

6.1 OWL 2 EL

The grammar of OWL 2 EL consists of the general definitions from Section 13.1 of the OWL 2Specification [OWL 2 Specification], as well as the following productions.

Class := IRI

Datatype := IRI

ObjectProperty := IRI

DataProperty := IRI

AnnotationProperty := IRI


NamedIndividual := IRI

Literal := typedLiteral | stringLiteralNoLanguage | stringLiteralWithLanguage



http://www.w3.org/TR/2012/REC-owl2-syntax-20121211/#General_Definitions

typedLiteral := lexicalForm '^^' DatatypelexicalForm := quotedStringstringLiteralNoLanguage := quotedStringstringLiteralWithLanguage := quotedString languageTag

ObjectPropertyExpression := ObjectProperty

DataPropertyExpression := DataProperty

DataRange := Datatype | DataIntersectionOf | DataOneOf

DataIntersectionOf := 'DataIntersectionOf' '(' DataRange DataRange {DataRange } ')'

DataOneOf := 'DataOneOf' '(' Literal ')'

ClassExpression :=Class | ObjectIntersectionOf | ObjectOneOf |ObjectSomeValuesFrom | ObjectHasValue | ObjectHasSelf |DataSomeValuesFrom | DataHasValue

ObjectIntersectionOf := 'ObjectIntersectionOf' '(' ClassExpressionClassExpression { ClassExpression } ')'

ObjectOneOf := 'ObjectOneOf' '(' Individual ')'

ObjectSomeValuesFrom := 'ObjectSomeValuesFrom' '(' ObjectPropertyExpressionClassExpression ')'

ObjectHasValue := 'ObjectHasValue' '(' ObjectPropertyExpression Individual ')'

ObjectHasSelf := 'ObjectHasSelf' '(' ObjectPropertyExpression ')'

DataSomeValuesFrom := 'DataSomeValuesFrom' '(' DataPropertyExpression {DataPropertyExpression } DataRange ')'

DataHasValue := 'DataHasValue' '(' DataPropertyExpression Literal ')'

Axiom := Declaration | ClassAxiom | ObjectPropertyAxiom | DataPropertyAxiom |DatatypeDefinition | HasKey | Assertion | AnnotationAxiom

ClassAxiom := SubClassOf | EquivalentClasses | DisjointClassesSubClassOf := 'SubClassOf' '(' axiomAnnotations subClassExpressionsuperClassExpression ')'



subClassExpression := ClassExpressionsuperClassExpression := ClassExpression

EquivalentClasses := 'EquivalentClasses' '(' axiomAnnotations ClassExpressionClassExpression { ClassExpression } ')'

DisjointClasses := 'DisjointClasses' '(' axiomAnnotations ClassExpressionClassExpression { ClassExpression } ')'

ObjectPropertyAxiom :=EquivalentObjectProperties | SubObjectPropertyOf |ObjectPropertyDomain | ObjectPropertyRange |ReflexiveObjectProperty | TransitiveObjectProperty

SubObjectPropertyOf := 'SubObjectPropertyOf' '(' axiomAnnotationssubObjectPropertyExpression superObjectPropertyExpression ')'subObjectPropertyExpression := ObjectPropertyExpression |propertyExpressionChainpropertyExpressionChain := 'ObjectPropertyChain' '(' ObjectPropertyExpressionObjectPropertyExpression { ObjectPropertyExpression } ')'superObjectPropertyExpression := ObjectPropertyExpression

EquivalentObjectProperties := 'EquivalentObjectProperties' '(' axiomAnnotationsObjectPropertyExpression ObjectPropertyExpression { ObjectPropertyExpression }')'

ObjectPropertyDomain := 'ObjectPropertyDomain' '(' axiomAnnotationsObjectPropertyExpression ClassExpression ')'

ObjectPropertyRange := 'ObjectPropertyRange' '(' axiomAnnotationsObjectPropertyExpression ClassExpression ')'

ReflexiveObjectProperty := 'ReflexiveObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

TransitiveObjectProperty := 'TransitiveObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

DataPropertyAxiom :=SubDataPropertyOf | EquivalentDataProperties |DataPropertyDomain | DataPropertyRange | FunctionalDataProperty

SubDataPropertyOf := 'SubDataPropertyOf' '(' axiomAnnotationssubDataPropertyExpression superDataPropertyExpression ')'subDataPropertyExpression := DataPropertyExpressionsuperDataPropertyExpression := DataPropertyExpression

EquivalentDataProperties := 'EquivalentDataProperties' '(' axiomAnnotationsDataPropertyExpression DataPropertyExpression { DataPropertyExpression } ')'



DataPropertyDomain := 'DataPropertyDomain' '(' axiomAnnotationsDataPropertyExpression ClassExpression ')'

DataPropertyRange := 'DataPropertyRange' '(' axiomAnnotationsDataPropertyExpression DataRange ')'

FunctionalDataProperty := 'FunctionalDataProperty' '(' axiomAnnotationsDataPropertyExpression ')'

DatatypeDefinition := 'DatatypeDefinition' '(' axiomAnnotations DatatypeDataRange ')'

HasKey := 'HasKey' '(' axiomAnnotations ClassExpression '(' {ObjectPropertyExpression } ')' '(' { DataPropertyExpression } ')' ')'

Assertion :=SameIndividual | DifferentIndividuals | ClassAssertion |ObjectPropertyAssertion | NegativeObjectPropertyAssertion |DataPropertyAssertion | NegativeDataPropertyAssertion

sourceIndividual := IndividualtargetIndividual := IndividualtargetValue := Literal

SameIndividual := 'SameIndividual' '(' axiomAnnotations Individual Individual {Individual } ')'

DifferentIndividuals := 'DifferentIndividuals' '(' axiomAnnotations IndividualIndividual { Individual } ')'

ClassAssertion := 'ClassAssertion' '(' axiomAnnotations ClassExpressionIndividual ')'

ObjectPropertyAssertion := 'ObjectPropertyAssertion' '(' axiomAnnotationsObjectPropertyExpression sourceIndividual targetIndividual ')'

NegativeObjectPropertyAssertion := 'NegativeObjectPropertyAssertion' '('axiomAnnotations ObjectPropertyExpression sourceIndividual targetIndividual ')'

DataPropertyAssertion := 'DataPropertyAssertion' '(' axiomAnnotationsDataPropertyExpression sourceIndividual targetValue ')'

NegativeDataPropertyAssertion := 'NegativeDataPropertyAssertion' '('axiomAnnotations DataPropertyExpression sourceIndividual targetValue ')'



6.2 OWL 2 QL

The grammar of OWL 2 QL consists of the general definitions from Section 13.1 of the OWL 2Specification [OWL 2 Specification], as well as the following productions.

Class := IRI

Datatype := IRI


DataProperty := IRI




Literal := typedLiteral | stringLiteralNoLanguage | stringLiteralWithLanguagetypedLiteral := lexicalForm '^^' DatatypelexicalForm := quotedStringstringLiteralNoLanguage := quotedStringstringLiteralWithLanguage := quotedString languageTag

ObjectPropertyExpression := ObjectProperty | InverseObjectProperty

InverseObjectProperty := 'ObjectInverseOf' '(' ObjectProperty ')'




subClassExpression :=Class |subObjectSomeValuesFrom | DataSomeValuesFrom

subObjectSomeValuesFrom := 'ObjectSomeValuesFrom' '(' ObjectPropertyExpressionowl:Thing ')'

superClassExpression :=Class |superObjectIntersectionOf | superObjectComplementOf |superObjectSomeValuesFrom | DataSomeValuesFrom




superObjectIntersectionOf := 'ObjectIntersectionOf' '(' superClassExpressionsuperClassExpression { superClassExpression } ')'

superObjectComplementOf := 'ObjectComplementOf' '(' subClassExpression ')'

superObjectSomeValuesFrom := 'ObjectSomeValuesFrom' '('ObjectPropertyExpression Class ')'

DataSomeValuesFrom := 'DataSomeValuesFrom' '(' DataPropertyExpressionDataRange ')'

Axiom := Declaration | ClassAxiom | ObjectPropertyAxiom | DataPropertyAxiom |DatatypeDefinition | Assertion | AnnotationAxiom


SubClassOf := 'SubClassOf' '(' axiomAnnotations subClassExpressionsuperClassExpression ')'

EquivalentClasses := 'EquivalentClasses' '(' axiomAnnotations subClassExpressionsubClassExpression { subClassExpression } ')'

DisjointClasses := 'DisjointClasses' '(' axiomAnnotations subClassExpressionsubClassExpression { subClassExpression } ')'

ObjectPropertyAxiom :=SubObjectPropertyOf | EquivalentObjectProperties |DisjointObjectProperties | InverseObjectProperties |ObjectPropertyDomain | ObjectPropertyRange |ReflexiveObjectProperty |SymmetricObjectProperty | AsymmetricObjectProperty

SubObjectPropertyOf := 'SubObjectPropertyOf' '(' axiomAnnotationsObjectPropertyExpression ObjectPropertyExpression ')'


DisjointObjectProperties := 'DisjointObjectProperties' '(' axiomAnnotationsObjectPropertyExpression ObjectPropertyExpression { ObjectPropertyExpression }')'

InverseObjectProperties := 'InverseObjectProperties' '(' axiomAnnotationsObjectPropertyExpression ObjectPropertyExpression ')'

ObjectPropertyDomain := 'ObjectPropertyDomain' '(' axiomAnnotations



ObjectPropertyExpression superClassExpression ')'

ObjectPropertyRange := 'ObjectPropertyRange' '(' axiomAnnotationsObjectPropertyExpression superClassExpression ')'

ReflexiveObjectProperty := 'ReflexiveObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

SymmetricObjectProperty := 'SymmetricObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

AsymmetricObjectProperty := 'AsymmetricObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

DataPropertyAxiom :=SubDataPropertyOf | EquivalentDataProperties | DisjointDataProperties |DataPropertyDomain | DataPropertyRange



DisjointDataProperties := 'DisjointDataProperties' '(' axiomAnnotationsDataPropertyExpression DataPropertyExpression { DataPropertyExpression } ')'

DataPropertyDomain := 'DataPropertyDomain' '(' axiomAnnotationsDataPropertyExpression superClassExpression ')'



Assertion := DifferentIndividuals | ClassAssertion | ObjectPropertyAssertion |DataPropertyAssertion





ClassAssertion := 'ClassAssertion' '(' axiomAnnotations Class Individual ')'



6.3 OWL 2 RL

The grammar of OWL 2 RL consists of the general definitions from Section 13.1 of the OWL 2Specification [OWL 2 Specification], as well as the following productions.

Class := IRI

Datatype := IRI


DataProperty := IRI


Individual := NamedIndividual | AnonymousIndividual


AnonymousIndividual := nodeID

Literal := typedLiteral | stringLiteralNoLanguage | stringLiteralWithLanguagetypedLiteral := lexicalForm '^^' DatatypelexicalForm := quotedStringstringLiteralNoLanguage := quotedStringstringLiteralWithLanguage := quotedString languageTag

ObjectPropertyExpression := ObjectProperty | InverseObjectProperty

InverseObjectProperty := 'ObjectInverseOf' '(' ObjectProperty ')'







zeroOrOne := '0' | '1'

subClassExpression :=Class other than owl:Thing |subObjectIntersectionOf | subObjectUnionOf | ObjectOneOf |subObjectSomeValuesFrom | ObjectHasValue |DataSomeValuesFrom | DataHasValue

subObjectIntersectionOf := 'ObjectIntersectionOf' '(' subClassExpressionsubClassExpression { subClassExpression } ')'

subObjectUnionOf := 'ObjectUnionOf' '(' subClassExpression subClassExpression {subClassExpression } ')'

subObjectSomeValuesFrom :='ObjectSomeValuesFrom' '(' ObjectPropertyExpression subClassExpression ')' |'ObjectSomeValuesFrom' '(' ObjectPropertyExpression owl:Thing ')'

superClassExpression :=Class other than owl:Thing |superObjectIntersectionOf | superComplementOf |superObjectAllValuesFrom | ObjectHasValue | superObjectMaxCardinality |DataAllValuesFrom | DataHasValue | superDataMaxCardinality

superObjectIntersectionOf := 'ObjectIntersectionOf' '(' superClassExpressionsuperClassExpression { superClassExpression } ')'

superObjectComplementOf := 'ObjectComplementOf' '(' subClassExpression ')'

superObjectAllValuesFrom := 'ObjectAllValuesFrom' '(' ObjectPropertyExpressionsuperClassExpression ')'

superObjectMaxCardinality :='ObjectMaxCardinality' '(' zeroOrOne ObjectPropertyExpression [

subClassExpression ] ')' |'ObjectMaxCardinality' '(' zeroOrOne ObjectPropertyExpression owl:Thing ')'

superDataMaxCardinality := 'DataMaxCardinality' '(' zeroOrOneDataPropertyExpression [ DataRange ] ')' |

equivClassExpression :=Class other than owl:Thing |equivObjectIntersectionOf |ObjectHasValue |DataHasValue

equivObjectIntersectionOf := 'ObjectIntersectionOf' '(' equivClassExpressionequivClassExpression { equivClassExpression } ')'

ObjectOneOf := 'ObjectOneOf' '(' Individual { Individual }')'

ObjectHasValue := 'ObjectHasValue' '(' ObjectPropertyExpression Individual ')'



DataSomeValuesFrom := 'DataSomeValuesFrom' '(' DataPropertyExpression {DataPropertyExpression } DataRange ')'

DataAllValuesFrom := 'DataAllValuesFrom' '(' DataPropertyExpression {DataPropertyExpression } DataRange ')'

DataHasValue := 'DataHasValue' '(' DataPropertyExpression Literal ')'

Axiom := Declaration | ClassAxiom | ObjectPropertyAxiom | DataPropertyAxiom |DatatypeDefinition | HasKey | Assertion | AnnotationAxiom


SubClassOf := 'SubClassOf' '(' axiomAnnotations subClassExpressionsuperClassExpression ')'

EquivalentClasses := 'EquivalentClasses' '(' axiomAnnotationsequivClassExpression equivClassExpression { equivClassExpression } ')'

DisjointClasses := 'DisjointClasses' '(' axiomAnnotations subClassExpressionsubClassExpression { subClassExpression } ')'

ObjectPropertyAxiom :=SubObjectPropertyOf | EquivalentObjectProperties |DisjointObjectProperties | InverseObjectProperties |ObjectPropertyDomain | ObjectPropertyRange |FunctionalObjectProperty | InverseFunctionalObjectProperty |IrreflexiveObjectProperty |SymmetricObjectProperty | AsymmetricObjectPropertyTransitiveObjectProperty

SubObjectPropertyOf := 'SubObjectPropertyOf' '(' axiomAnnotationssubObjectPropertyExpression superObjectPropertyExpression ')'subObjectPropertyExpression := ObjectPropertyExpression |propertyExpressionChainpropertyExpressionChain := 'ObjectPropertyChain' '(' ObjectPropertyExpressionObjectPropertyExpression { ObjectPropertyExpression } ')'superObjectPropertyExpression := ObjectPropertyExpression


DisjointObjectProperties := 'DisjointObjectProperties' '(' axiomAnnotationsObjectPropertyExpression ObjectPropertyExpression { ObjectPropertyExpression }')'

InverseObjectProperties := 'InverseObjectProperties' '(' axiomAnnotations



ObjectPropertyExpression ObjectPropertyExpression ')'

ObjectPropertyDomain := 'ObjectPropertyDomain' '(' axiomAnnotationsObjectPropertyExpression superClassExpression ')'

ObjectPropertyRange := 'ObjectPropertyRange' '(' axiomAnnotationsObjectPropertyExpression superClassExpression ')'

FunctionalObjectProperty := 'FunctionalObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

InverseFunctionalObjectProperty := 'InverseFunctionalObjectProperty' '('axiomAnnotations ObjectPropertyExpression ')'

IrreflexiveObjectProperty := 'IrreflexiveObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

SymmetricObjectProperty := 'SymmetricObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

AsymmetricObjectProperty := 'AsymmetricObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

TransitiveObjectProperty := 'TransitiveObjectProperty' '(' axiomAnnotationsObjectPropertyExpression ')'

DataPropertyAxiom :=SubDataPropertyOf | EquivalentDataProperties | DisjointDataProperties |DataPropertyDomain | DataPropertyRange | FunctionalDataProperty



DisjointDataProperties := 'DisjointDataProperties' '(' axiomAnnotationsDataPropertyExpression DataPropertyExpression { DataPropertyExpression } ')'

DataPropertyDomain := 'DataPropertyDomain' '(' axiomAnnotationsDataPropertyExpression superClassExpression ')'


FunctionalDataProperty := 'FunctionalDataProperty' '(' axiomAnnotationsDataPropertyExpression ')'




HasKey := 'HasKey' '(' axiomAnnotations subClassExpression '(' {ObjectPropertyExpression } ')' '(' { DataPropertyExpression } ')' ')'

Assertion :=SameIndividual | DifferentIndividuals | ClassAssertion |ObjectPropertyAssertion | NegativeObjectPropertyAssertion |DataPropertyAssertion | NegativeDataPropertyAssertion


SameIndividual := 'SameIndividual' '(' axiomAnnotations Individual Individual {Individual } ')'


ClassAssertion := 'ClassAssertion' '(' axiomAnnotations superClassExpressionIndividual ')'


NegativeObjectPropertyAssertion := 'NegativeObjectPropertyAssertion' '('axiomAnnotations ObjectPropertyExpression sourceIndividual targetIndividual ')'


NegativeDataPropertyAssertion := 'NegativeDataPropertyAssertion' '('axiomAnnotations DataPropertyExpression sourceIndividual targetValue ')'

7 Appendix: Change Log (Informative)

7.1 Changes Since Recommendation

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

• With the publication of the XML Schema Definition Language (XSD) 1.1 Part 2: DatatypesRecommendation of 5 April 2012, the elements of OWL 2 which are based on XSD 1.1are now considered required, and the note detailing the optional dependency on theXSD 1.1 Candidate Recommendation of 30 April, 2009 has been removed from the"Status of this Document" section.





http://www.w3.org/TR/2012/REC-xmlschema11-2-20120405/

http://www.w3.org/TR/2009/CR-xmlschema11-2-20090430/

• The description of the computational properties of the various profiles in Section 5 hasbeen edited to improve clarity and precision.

• A redundant grammar production for ReflexiveObjectProperty was removed fromSection 6.3.

• Minor typographical errors were corrected as detailed on the OWL 2 Errata page.

7.2 Changes Since Proposed Recommendation

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

7.3 Changes Since Candidate Recommendation

This section summarizes the changes to this document since the Candidate Recommendation of11 June, 2009.

• The "Features At Risk" warning w.r.t. the owl:rational and rdf:XMLLiteral datatypes wasremoved: implementation support has been adequately demonstrated, and the featuresare no longer considered at risk (see Resolution 5 and Resolution 6, 05 August 2009).

• A note on the origin of the profile names was added, and it was pointed out that none ofthe profiles is a subset of another.

• A citation of Lloyd's Foundations of Logic Programming was added.• Some minor editorial changes were made.

7.4 Changes Since Last Call

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

• One grammar production was fixed in order to align it with the grammar in StructuralSpecification and Functional-Style Syntax

• The name of rdf:text was changed to rdf:PlainLiteral.• A reference to logic programming was added.• Some minor editorial changes were made.

8 Acknowledgments

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

This document has been produced by the OWL Working Group (see below), and its contentsreflect extensive discussions within the Working Group as a whole. The editors extend specialthanks to Jie Bao (RPI), Jim Hendler (RPI) and Jeff Pan (University of Aberdeen) for their thoroughreviews.

The regular attendees at meetings of the OWL Working Group at the time of publication of thisdocument were: Jie Bao (RPI), Diego Calvanese (Free University of Bozen-Bolzano), BernardoCuenca Grau (Oxford University Computing Laboratory), Martin Dzbor (Open University), AchilleFokoue (IBM Corporation), Christine Golbreich (Université de Versailles St-Quentin and LIRMM),Sandro Hawke (W3C/MIT), Ivan Herman (W3C/ERCIM), Rinke Hoekstra (University ofAmsterdam), Ian Horrocks (Oxford University Computing Laboratory), Elisa Kendall (SandpiperSoftware), Markus Krötzsch (FZI), Carsten Lutz (Universität Bremen), Deborah L. McGuinness



http://www.w3.org/2007/OWL/wiki/Errata

http://www.w3.org/TR/2009/PR-owl2-profiles-20090922/

http://www.w3.org/TR/2009/PR-owl2-profiles-20090922/

http://www.w3.org/TR/2009/CR-owl2-profiles-20090611/

http://www.w3.org/TR/2009/CR-owl2-profiles-20090611/

http://www.w3.org/2007/OWL/meeting/2009-08-05#resolution_5

http://www.w3.org/2007/OWL/meeting/2009-08-05#resolution_6

http://www.w3.org/TR/2009/WD-owl2-profiles-20090421/

http://www.w3.org/TR/2009/WD-owl2-profiles-20090421/

http://www.w3.org/Submission/2006/10/

http://www.webont.org/owled/

http://www.webont.org/owled/

http://www.w3.org/2001/sw/WebOnt/webont-issues.html

http://www.w3.org/2004/OWL/

(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 ofManchester), 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 pastmembers of the working group: Jeremy Carroll, Jim Hendler, and Vipul Kashyap.

9 References

9.1 Normative References

[OWL 2 Conformance]OWL 2 Web Ontology Language: Conformance (Second Edition) Michael Smith, IanHorrocks, Markus Krötzsch, Birte Glimm, eds. W3C Recommendation, 11 December 2012,http://www.w3.org/TR/2012/REC-owl2-conformance-20121211/. Latest version available athttp://www.w3.org/TR/owl2-conformance/.

[OWL 2 Direct Semantics]OWL 2 Web Ontology Language: Direct Semantics (Second Edition) Boris Motik, Peter F.Patel-Schneider, Bernardo Cuenca Grau, eds. W3C Recommendation, 11 December 2012,http://www.w3.org/TR/2012/REC-owl2-direct-semantics-20121211/. Latest version availableat http://www.w3.org/TR/owl2-direct-semantics/.

[OWL 2 RDF Mapping]OWL 2 Web Ontology Language: Mapping to RDF Graphs (Second Edition) Peter F. Patel-Schneider, Boris Motik, eds. W3C Recommendation, 11 December 2012, http://www.w3.org/TR/2012/REC-owl2-mapping-to-rdf-20121211/. Latest version available athttp://www.w3.org/TR/owl2-mapping-to-rdf/.

[OWL 2 RDF-Based Semantics]OWL 2 Web Ontology Language: RDF-Based Semantics (Second Edition) Michael Schneider,editor. W3C Recommendation, 11 December 2012, http://www.w3.org/TR/2012/REC-owl2-rdf-based-semantics-20121211/. Latest version available at http://www.w3.org/TR/owl2-rdf-based-semantics/.

[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. W3CRecommendation, 11 December 2012, http://www.w3.org/TR/2012/REC-owl2-syntax-20121211/. Latest version available at http://www.w3.org/TR/owl2-syntax/.

[RFC 2119]RFC 2119: Key words for use in RFCs to Indicate Requirement Levels. Network WorkingGroup, S. Bradner. IETF, March 1997, http://www.ietf.org/rfc/rfc2119.txt

9.2 Nonnormative References

[Complexity]Complexity Results and Practical Algorithms for Logics in Knowledge Representation.Stephan Tobies. Ph.D Dissertation, 2002

[DL-Lite]Tractable Reasoning and Efficient Query Answering in Description Logics: The DL-LiteFamily. Diego Calvanese, Giuseppe de Giacomo, Domenico Lembo, Maurizio Lenzerini,Riccardo Rosati. J. of Automated Reasoning 39(3):385–429, 2007

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OWL 2 Web Ontology Language Profiles (Second Edition) · 12/11/2012 · The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the Semantic Web with formally