W3C

OWL 2 Web Ontology Language:
Profiles

W3C Editor's Draft 29 September 2008

This version:
http://www.w3.org/2007/OWL/draft/ED-owl2-profiles-20080929/
Latest editor's draft:
http://www.w3.org/2007/OWL/draft/owl2-profiles/
Previous version:
http://www.w3.org/2007/OWL/draft/ED-owl2-profiles-20080922/ (color-coded diff)
Authors:
Bernardo Cuenca Grau, Oxford University
Boris Motik, Oxford University
Zhe Wu, Oracle
Achille Fokoue, IBM
Carsten Lutz, Dresden University of Technology


Abstract

OWL 2 extends the W3C OWL Web Ontology Language with a small but useful set of features that have been requested by users, for which effective reasoning algorithms are now available, and that OWL tool developers are willing to support. The new features include extra syntactic sugar, additional property and qualified cardinality constructors, extended datatype support, simple metamodelling, and extended annotations.
This document provides a specification of several profiles of OWL 2 which can be more simply and/or efficiently implemented. In logic, profiles are often called fragments. Most profiles are defined by placing restrictions on the syntax of OWL 2. These restrictions have been 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May Be Superseded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. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

This document is being published as one of a set of 7 documents:

  1. Structural Specification and Functional-Style Syntax
  2. Direct Semantics
  3. RDF-Based Semantics
  4. Mapping to RDF Graphs
  5. XML Serialization
  6. Profiles (this document)
  7. Conformance and Test Cases

Please Comment By ASAP

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

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

Patents

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Contents


1 Introduction

The purpose of an OWL 2 profile is to provide a trimmed down version of OWL 2 that trades expressive power for efficiency of reasoning. In logic, a profile is usually called a fragment or a sublanguage. This document describes three important profiles, each of which achieves efficiency in a different way and is useful in different application scenarios.

The choice of profile will depend on the structure of the ontologies used in the application and on the reasoning tasks to be performed, for example (ontology) consistency, (class) satisfiability, (class) subsumption, classification and conjunctive query answering. Precise definitions of these tasks can be found in Section 5.

OWL 2 profiles are defined by placing restrictions on the OWL 2 syntax. 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, the modified grammars are specified in two ways. In each profile definition, only the difference with respect 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-style syntax are not repeated. In order to make this document self-contained, the full grammar for each of the profiles is given in the Appendix.

An ontology in any profile can be written into a document by using any of the syntaxes 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. Although we don't specifically document OWL lite [OWL 1 Reference] in this document, all OWL Lite ontologies are OWL 2 DL ontologies and so OWL Lite can be viewed as a profile of OWL 2. OWL 1 DL can also be viewed as a profile of OWL 2.

The italicized keywords MUST, MUST NOT, SHOULD, SHOULD NOT, and MAY specify certain aspects of the normative behavior of OWL 2 tools, and are interpreted as specified in RFC 2119 [RFC 2119].

2 OWL 2 EL

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

OWL 2 EL provides class constructors that are sufficient to express many complex ontologies, such as the biomedical 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. In particular, the following types of class restrictions are supported:

OWL 2 EL supports the following axioms, all of which are restricted to the allowed set of class expressions.

The following features of OWL 2 are not supported in OWL 2 EL:

2.2 Profile Specification

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

2.2.1 Entities

The entities of OWL 2 EL are exactly as in OWL 2. Furthermore, OWL 2 EL supports the owl:Thing and owl:Nothing predefined classes, as well as the predefined object and data properties owl:TopObjectProperty, owl:BottomObjectProperty, owl:TopDataProperty, and owl:BottomDataProperty. Finally, it supports the following datatypes:

The set of supported datatypes has been designed such that the intersection of the value spaces of any set of these datatypes is finite, which is necessary to obtain the desired computational properties [EL++]. Consequently, the following predefined OWL 2 datatypes MUST NOT be used in OWL 2 EL: owl:realPlus, 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.

2.2.2 Property Expressions

Inverse properties are not supported in OWL 2 EL, so object property expressions are restricted to named properties. Data property expressions are defined in the same way as in OWL 2.

ObjectPropertyExpression:= ObjectProperty

2.2.3 Class Expressions

In order to allow for efficient reasoning, OWL 2 EL restricts the set of supported class expressions to ObjectIntersectionOf, ObjectSomeValuesFrom, ObjectExistsSelf, ObjectHasValue, DataSomeValuesFrom, DataHasValue, and objectOneOf containing a single individual.

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

The class expressions are as defined in the same way as in the structural specification [OWL 2 Specification], with the exception of the objectOneOf class expression, which in OWL 2 EL admits only a single individual.

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

2.2.4 Data Ranges

A data range expression is restricted in OWL 2 EL to the predefined datatypes admitted in OWL 2 EL and to enumerated datatypes consisting of a single literal.

DataRange:= Datatype | DataOneOf
DataOneOf:= 'OneOf' '(' Literal ')'

2.2.5 Axioms

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

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

OWL 2 EL provides the same axioms about data properties as OWL 2 apart from DisjointDataProperty. These axioms are defined in the same way as in the structural specification [OWL 2 Specification].

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

The assertions in OWL 2 EL, as well as all other axioms, are the same as in OWL 2, with the difference that class object property expressions are 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 structural specification [OWL 2 Specification] with an additional condition [EL++ Update]. In order to define this condition, the following notion is used.

Let CE be a class expression. We say that Ax imposes a range restriction to CE on an object property OP1 if Ax contains the following axioms, where k 1 is an integer and OPi are object properties:

The axiom closure Ax of an OWL 2 EL ontology MUST obey the restrictions described in Section 11 of the structural specification [OWL 2 Specification] and, in addition, if

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

This additional restriction is vacuously true for each SubObjectPropertyOf axiom in which in the first item of the previous definition does not contain a property chain. There are no additional restrictions for range restrictions on reflexive and transitive roles; that is, a range restriction can be placed on a reflexive and/or transitive role provided that it satisfies the afore-mentioned restriction.

3 OWL 2 QL

The OWL 2 QL profile admits sound and complete reasoning in LOGSPACE with respect to the size of the data (assertions). OWL 2 QL includes many of the main features of conceptual models such as UML class diagrams and ER diagrams.

OWL 2 QL is based on the DL-Lite family of description logics. Several variants of DL-Lite have been described in the literature [DL-Lite]. OWL 2 QL is based on DL-LiteR an expressive DL containing the intersection of RDFS and OWL 2. DL-LiteR does not require the unique name assumption (UNA), since making this assumption would have no impact on the semantic consequences of a DL-LiteR ontology. More expressive variants of DL-Lite, such as DL-LiteA, extend DL-LiteR with functional properties, and these 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 interaction between properties used in different types of axiom. Basing OWL 2 QL on DL-LiteR avoids practical problems involved in the explicit axiomatization of UNA. Other variants of DL-Lite can also be supported on top of OWL 2 QL, but may require additional restrictions on the structure of ontologies [DL-Lite].

3.1 Feature Overview

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

Table 1. Syntactic Restrictions on Class Expressions in OWL 2 QL
Subclass Expressions Superclass Expressions
a class
existential quantification (ObjectSomeValuesFrom)
where the class is limited to owl:Thing
a class
existential quantification to a class (ObjectSomeValuesFrom)
negation (ObjectComplementOf)
intersection (ObjectIntersectionOf)

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

The following features of OWL 2 are not supported in OWL 2 QL:

3.2 Profile Specification

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

3.2.1 Entities

OWL 2 QL supports all OWL 2 entities, including all predefined classes and properties. Furthermore, the following datatypes are supported in OWL 2 QL:

The set of supported datatypes has been designed such that the intersection of the value spaces of any set of these datatypes is finite, which is necessary to obtain the desired computational properties. Consequently, the following predefined OWL 2 datatypes MUST NOT be used in OWL 2 QL: owl:realPlus, 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.

3.2.2 Property Expressions

OWL 2 QL object and data property expressions are the same as in OWL 2.

3.2.3 Class Expressions

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

subClassExpression:=
Class |
'SomeValuesFrom' '(' ObjectPropertyExpression owl:Thing ')'
superClassExpression:=
Class |
'SomeValuesFrom' '(' ObjectPropertyExpression Class ')'
'ComplementOf' '(' subClassExpression ')' |
'IntersectionOf' '(' superClassExpression superClassExpression { superClassExpression } ')'

3.2.4 Data Ranges

A data range expression is restricted in OWL 2 QL to the predefined datatypes.

DataRange:= Datatype

3.2.5 Axioms

The class axioms of OWL 2 QL are the same as in OWL 2, with the exception that DisjointUnion is disallowed; however, all axioms that refer to the ClassExpression production are redefined so as to use subClassExpression and/or superClassExpression as appropriate.

SubClassOf:= 'SubClassOf' '(' { annotation } subClassExpression superClassExpression ')'
EquivalentClasses:= 'EquivalentClasses' '(' { annotation } subClassExpression subClassExpression { subClassExpression } ')'
DisjointClasses:= 'DisjointClasses' '(' { annotation } subClassExpression subClassExpression { subClassExpression } ')'
ClassAxiom:= SubClassOf | EquivalentClasses | DisjointClasses

OWL 2 QL disallows the use of property chains in property inclusion axioms; however, simple property inclusions are supported. Furthermore, OWL 2 QL disallows the use of functional, transitive, asymmetric, reflexive and irreflexive object properties, and it restricts the class expressions in object property domain and range axioms to superClassExpression.

ObjectPropertyDomain:= 'PropertyDomain' '(' { annotation } ObjectPropertyExpression superClassExpression ')'
ObjectPropertyRange:= 'PropertyRange' '(' { annotation } ObjectPropertyExpression superClassExpression ')'
SubObjectPropertyOf:= 'SubPropertyOf' '(' { annotation } ObjectPropertyExpression ObjectPropertyExpression ')'
ObjectPropertyAxiom:=
SubObjectPropertyOf | EquivalentObjectProperties |
DisjointObjectProperties | InverseObjectProperties |
ObjectPropertyDomain | ObjectPropertyRange |
SymmetricObjectProperty

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

DataPropertyDomain:= 'PropertyDomain' '(' { annotation } DataPropertyExpression superClassExpression ')'
DataPropertyAxiom:=
SubDataPropertyOf | EquivalentDataProperties | DisjointDataProperties |
DataPropertyDomain | DataPropertyRange

OWL 2 QL disallows negative object property assertions and equality axioms. Furthermore, class assertions in OWL 2 QL can involve only atomic classes. Inequality axioms and property assertions are the same as in OWL 2.

ClassAssertion:= 'ClassAssertion' '(' { annotation } Class Individual ')'
Assertion:= DifferentIndividuals | ClassAssertion | ObjectPropertyAssertion | DataPropertyAssertion

Finally, the axioms in OWL 2 QL are the same as those in OWL 2, with the exception that keys are not allowed.

Axiom:= Declaration | ClassAxiom | ObjectPropertyAxiom | DataPropertyAxiom | Assertion | EntityAnnotation | AnonymousIndividualAnnotation

4 OWL 2 RL

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

For ontologies satisfying the syntactic constraints described in Section 4.2, a suitable rule-based implementation will have desirable computational properties; for example, it can return all and only the correct answers to certain kinds of query (see Section 4.4 and Conformance). Such an implementation can also be used with arbitrary RDF graphs. In this case, however, these properties no longer hold in particular, it is no longer possible to guarantee that all correct answers can be returned.

4.1 Feature Overview

Restricting the way in which OWL 2 constructs are used makes it possible to implement reasoning systems using rule-based reasoning engines, while still providing desirable computational guarantees. These restrictions are designed so as to avoid the need to infer the existence of individuals not explicitly present in the knowledge base, and to avoid the need for nondeterministic reasoning. This is achieved by restricting the use of OWL 2 constructs to certain syntactic positions. For example in SubClassOf axioms, the constructs in the subclass and superclass expressions must follow the usage patterns shown in Table 2.

Table 2. Syntactic Restrictions on Class Expressions in OWL 2 RL
Subclass Expressions Superclass Expressions
a class
a nominal class (OneOf)
intersection of class expressions (ObjectIntersectionOf)
union of class expressions (ObjectUnionOf)
existential quantification to a class expressions (ObjectSomeValuesFrom)
existential quantification to an individual (ObjectHasValue)
a class
intersection of classes (ObjectIntersectionOf)
universal quantification to a class expressions (ObjectAllValuesFrom)
at-most 1 cardinality restrictions (ObjectMaxCardinality 1)
existential quantification to an individual (ObjectHasValue)

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), reflexive object property axioms (ReflexiveObjectProperty), and negative object and data property assertions (NegativeObjectPropertyAssertion and NegativeDataPropertyAssertion).

Implementations based on the partial axiomatization (presented in Section 4.3) can also be used with arbitrary RDF graphs, but in this case it is no longer possible to provide the above mentioned computational guarantees. Such implementations will, however, still produce only correct entailments (see Conformance).

4.2 Profile Specification

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

4.2.1 Entities

The entities of OWL 2 RL are exactly as in OWL 2. Furthermore, OWL 2 RL supports the owl:Thing and owl:Nothing predefined classes; however, it does not support the predefined object and data properties owl:TopObjectProperty, owl:BottomObjectProperty, owl:TopDataProperty, and owl:BottomDataProperty. Finally, it supports the following datatypes:

The set of supported datatypes has been designed such that the intersection of the value spaces of any set of these datatypes is finite, which is necessary to obtain the desired computational properties. Consequently, the following predefined OWL 2 datatypes MUST NOT be used in OWL 2 RL: owl:realPlus, 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.

4.2.2 Property Expressions

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

4.2.3 Class Expressions

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

zeroOrOne := '0' | '1'
subClassExpression:=
Class other than owl:Thing |
'OneOf' '(' Individual { Individual } ')'
'IntersectionOf' '(' subClassExpression subClassExpression { subClassExpression } ')' |
'UnionOf' '(' subClassExpression subClassExpression { subClassExpression } ')' |
'SomeValuesFrom' '(' ObjectPropertyExpression subClassExpression ')' |
'SomeValuesFrom' '(' DataPropertyExpression { DataPropertyExpression } DataRange ')' |
'HasValue' '(' ObjectPropertyExpression Individual ')' |
'HasValue' '(' DataPropertyExpression Literal ')'
superClassExpression:=
Class |
'IntersectionOf' '(' superClassExpression superClassExpression { superClassExpression } ')' |
'AllValuesFrom' '(' ObjectPropertyExpression superClassExpression ')' |
'AllValuesFrom' '(' DataPropertyExpression { DataPropertyExpression } DataRange ')' |
'MaxCardinality' '(' zeroOrOne ObjectPropertyExpression [ subClassExpression ] ')' |
'MaxCardinality' '(' zeroOrOne DataPropertyExpression [ DataRange ] ')' |
'HasValue' '(' ObjectPropertyExpression Individual ')' |
'HasValue' '(' DataPropertyExpression Literal ')'
equivClassExpression:=
Class other than owl:Thing |
'IntersectionOf' '(' equivClassExpression equivClassExpression { equivClassExpression } ')' |
'HasValue' '(' ObjectPropertyExpression Individual ')' |
'HasValue' '(' DataPropertyExpression Literal ')'

4.2.4 Data Ranges

A data range expression is restricted in OWL 2 RL to the predefined datatypes admitted in OWL 2 RL.

DataRange:= Datatype

4.2.5 Axioms

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

ClassAxiom:= SubClassOf | EquivalentClasses | DisjointClasses
SubClassOf:= 'SubClassOf' '(' { annotation } subClassExpression superClassExpression ')'
EquivalentClasses:= 'EquivalentClasses' '(' { annotation } equivClassExpression equivClassExpression { equivClassExpression } ')'
DisjointClasses:= 'DisjointClasses' '(' { annotation } subClassExpression subClassExpression { subClassExpression } ')'

OWL 2 RL axioms about property expressions are as in OWL 2, the only difference being that class expressions in property domain and range axioms are restricted to superClassExpression.

ObjectPropertyDomain:= 'PropertyDomain' '(' { annotation } ObjectPropertyExpression superClassExpression ')'
ObjectPropertyRange:= 'PropertyRange' '(' { annotation } ObjectPropertyExpression superClassExpression ')'
DataPropertyDomain:= 'PropertyDomain' '(' { annotation } DataPropertyExpression superClassExpression ')'

OWL 2 RL restricts class expressions in positive assertions to superClassExpression, and it disallows negative property assertions. Equality and inequality between individuals and positive assertions are the same as in OWL 2.

ClassAssertion:= 'ClassAssertion' '(' { annotation } Individual superClassExpression ')'
Assertion:= SameIndividual | DifferentIndividuals | ClassAssertion | ObjectPropertyAssertion | DataPropertyAssertion

OWL 2 RL restricts class expressions in keys to subClassExpression.

HasKey:= 'HasKey' '(' { annotation } subClassExpression ObjectPropertyExpression | DataPropertyExpression { ObjectPropertyExpression | DataPropertyExpression } ')'

Axioms about properties are redefined in OWL 2 RL to disallow the reflexive properties.

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

All other axioms in OWL 2 RL are defined as in OWL 2.

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 in the form of first-order (material) implications; we will call this axiomatization the OWL 2 RL/RDF rules. These rules provide a useful starting point for practical implementation using rule-based technologies.

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 allowed in all positions (similar to the partial generalization in pD* [pD*] and to generalized RDF triples in RIF [RIF]); thus, T(s, p, o) represents a generalized RDF triple with the subject s, predicate p, and the object o. Variables in the implications are preceded with a question mark. The propositional symbol false is a special symbol denoting contradiction: if it is derived, then the initial RDF graph was inconsistent.

Many conditions contain atoms that match to the list construct of RDF. In order to simplify the presentation 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 anywhere where 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. Table 4 axiomatizes the semantics of equality. In particular, it defines the equality relation on resources owl:sameAs as being reflexive, symmetric, and transitive, and it axiomatizes the standard replacement properties of equality for it.

Editor's Note: When the rule set is finalized, rows in the tables will be numbered/named and given HTML anchors.


Table 4. The Semantics of Equality
If then
T(?s,?p,?o)
T(?s, owl:sameAs,?s)
T(?p, owl:sameAs,?p)
T(?o, owl:sameAs,?o)
T(?x, owl:sameAs,?y) T(?y, owl:sameAs,?x)
T(?x, owl:sameAs,?y)
T(?y, owl:sameAs,?z)
T(?x, owl:sameAs,?z)
T(?s, owl:sameAs,?s')
T(?s,?p,?o)
T(?s',?p,?o)
T(?p, owl:sameAs,?p')
T(?s,?p,?o)
T(?s,?p',?o)
T(?o, owl:sameAs,?o')
T(?s,?p,?o)
T(?s,?p,?o')
T(?x, owl:sameAs,?y)
T(?x, owl:differentFrom,?y)
false
T(?yi, owl:sameAs,?yj)
T(?x, rdf:type, owl:AllDifferent)
LIST[?x,?y1, ...,?yn]
false for each 1 i < j n

Table 5 specifies the semantic conditions on axioms about properties.

Table 5. The Semantics of Axioms about Properties
If then
T(?p, rdfs:domain,?c)
T(?x,?p,?y)
T(?x, rdf:type,?c)
T(?p, rdfs:range,?c)
T(?x,?p,?y)
T(?y, rdf:type,?c)
T(?p, rdf:type, owl:FunctionalProperty)
T(?x,?p,?y1)
T(?x,?p,?y2)
T(?y1, owl:sameAs,?y2)
T(?p, rdf:type, owl:InverseFunctionalProperty)
T(?x1,?p,?y)
T(?x2,?p,?y)
T(?x1, owl:sameAs,?x2)
T(?p, rdf:type, owl:IrreflexiveProperty)
T(?x,?p,?x)
false
T(?p, rdf:type, owl:SymmetricProperty)
T(?x,?p,?y)
T(?y,?p,?x)
T(?p, rdf:type, owl:AsymmetricProperty)
T(?x,?p,?y)
T(?y,?p,?x)
false
T(?p, rdf:type, owl:TransitiveProperty)
T(?x,?p,?y)
T(?y,?p,?z)
T(?x,?p,?z)
T(?p1, rdfs:subPropertyOf,?p2)
T(?x,?p1,?y)
T(?x,?p2,?y)
T(?sc, owl:propertyChain,?x)
LIST[?x,?p1, ...,?pn]
T(?sc, rdfs:subPropertyOf,?p)
T(?u1,?p1,?u2)
T(?u2,?p2,?u3)
...
T(?un,?pn,?un+1)
T(?u1,?p,?un+1)
T(?p1, owl:equivalentProperty,?p2)
T(?x,?p1,?y)
T(?x,?p2,?y)
T(?p1, owl:equivalentProperty,?p2)
T(?x,?p2,?y)
T(?x,?p1,?y)
T(?p1, owl:propertyDisjointWith,?p2)
T(?x,?p1,?y)
T(?x,?p2,?y)
false
T(?z, rdf:type, owl:AllDisjointProperties)
LIST[?z,?p1, ...,?pn]
T(?x,?pi,?y)
T(?x,?pj,?y)
false for each 1 i < j n
T(?p1, owl:inverseOf,?p2)
T(?x,?p1,?y)
T(?y,?p2,?x)
T(?p1, owl:inverseOf,?p2)
T(?x,?p2,?y)
T(?y,?p1,?x)
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)

Table 6 specifies the semantic conditions on classes.

Table 6. The Semantics of Classes
If then
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)
T(?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)
T(?c, owl:unionOf,?x)
LIST[?x,?c1, ...,?cn]
T(?y, rdf:type,?ci)
T(?y, rdf:type,?c) for each 1 i n
T(?x, owl:someValuesFrom,?y)
T(?x, owl:onProperty,?p)
T(?u,?p,?v)
T(?v, rdf:type,?y)
T(?u, rdf:type,?x)
T(?x, owl:allValuesFrom,?y)
T(?x, owl:onProperty,?p)
T(?u, rdf:type,?x)
T(?u,?p,?v)
T(?v, rdf:type,?y)
T(?x, owl:hasValue,?y)
T(?x, owl:onProperty,?p)
T(?u, rdf:type,?x)
T(?u,?p,?y)
T(?x, owl:hasValue,?y)
T(?x, owl:onProperty,?p)
T(?u,?p,?y)
T(?u, rdf:type,?x)
T(?x, owl:maxCardinality, "0"^^xsd:nonNegativeInteger)
T(?x, owl:onProperty,?p)
T(?u, rdf:type,?x)
T(?u,?p,?y)
false
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)
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
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)
T(?y1, owl:sameAs,?y2)
T(?c, owl:oneOf,?x)
LIST[?x,?y1, ...,?yn]
T(?yi, rdf:type,?c) for each 1 i n

Table 7 specifies the semantic conditions on class axioms.

Table 7. The Semantics of Class Axioms
If then
T(?c1, rdfs:subClassOf,?c2)
T(?x, rdf:type,?c1)
T(?x, rdf:type,?c2)
T(?c1, owl:equivalentClass,?c2)
T(?x, rdf:type,?c1)
T(?x, rdf:type,?c2)
T(?c1, owl:equivalentClass,?c2)
T(?x, rdf:type,?c2)
T(?x, rdf:type,?c1)
T(?c1, owl:disjointWith,?c2)
T(?x, rdf:type,?c1)
T(?x, rdf:type,?c2)
false
T(?y, rdf:type, owl:AllDisjointClasses)
LIST[?y,?c1, ...,?cn]
T(?x, rdf:type,?ci)
T(?x, rdf:type,?cj)
false for each 1 i < j n

Table 8 specifies the semantics of datatype literals.

Table 8. The Semantics of Datatype Literals
If then
true T(lt, rdf:type, dt) for each literal lt and each datatype dt supported in OWL 2 RL
such that the data value of lt is contained in the value space of dt
true T(lt1, owl:sameAs, lt2) for all literals lt1 and lt2 with the same data value
true T(lt1, owl:differentFrom, lt2) for all literals lt1 and lt2 with different data values
T(lt, rdf:type, dt) false for each literal lt and each datatype dt supported in OWL 2 RL
such that the data value of lt is not contained 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 Vocabulary
If then
T(?c, rdf:type, owl:Class) T(?c, rdfs:subClassOf,?c)
T(?c, owl:equivalentClass,?c)
T(?c1, rdfs:subClassOf,?c2)
T(?c2, rdfs:subClassOf,?c3)
T(?c1, rdfs:subClassOf,?c3)
T(?c1, owl:equivalentClass,?c2) T(?c1, rdfs:subClassOf,?c2)
T(?c2, rdfs:subClassOf,?c1)
T(?p, rdf:type, owl:ObjectProperty) T(?p, rdfs:subPropertyOf,?p)
T(?p, owl:equivalentProperty,?p)
T(?p, rdf:type, owl:DatatypeProperty) T(?p, rdfs:subPropertyOf,?p)
T(?p, owl:equivalentProperty,?p)
T(?p1, rdfs:subPropertyOf,?p2)
T(?p2, rdfs:subPropertyOf,?p3)
T(?p1, rdfs:subPropertyOf,?p3)
T(?p1, owl:equivalentProperty,?p2) T(?p1, rdfs:subPropertyOf,?p2)
T(?p2, rdfs:subPropertyOf,?p1)
T(?p, rdfs:domain,?c1)
T(?c1, rdfs:subClassOf,?c2)
T(?p, rdfs:domain,?c2)
T(?p2, rdfs:domain,?c)
T(?p1, rdfs:subPropertyOf,?p2)
T(?p1, rdfs:domain,?c)
T(?p, rdfs:range,?c1)
T(?c1, rdfs:subClassOf,?c2)
T(?p, rdfs:range,?c2)
T(?p2, rdfs:range,?c)
T(?p1, rdfs:subPropertyOf,?p2)
T(?p1, rdfs:range,?c)
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)
T(?c1, rdfs:subClassOf,?c2)
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)
T(?c1, rdfs:subClassOf,?c2)
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)
T(?c1, rdfs:subClassOf,?c2)
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)
T(?c1, rdfs:subClassOf,?c2)
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)
T(?c2, rdfs:subClassOf,?c1)
T(?c, owl:intersectionOf,?x)
LIST[?x,?c1, ...,?cn]
T(?c, rdfs:subClassOf,?c1)
T(?c, rdfs:subClassOf,?c2)
...
T(?c, rdfs:subClassOf,?cn)
T(?c, owl:unionOf,?x)
LIST[?x,?c1, ...,?cn]
T(?c1, rdfs:subClassOf,?c)
T(?c2, rdfs:subClassOf,?c)
...
T(?cn, rdfs:subClassOf,?c)

OWL 2 RL/RDF rules include neither the axiomatic triples and entailment rules of RDF and RDFS [RDF Semantics] nor the axiomatic triples for the relevant OWL vocabulary [OWL 2 RDF-Based Semantics], as these might cause performance problems in practice. An OWL 2 RL/RDF implementation MAY include these triples and entailment rules as necessary without invalidating the conformance requirements for OWL 2 RL [Conformance].

Theorem 1. Let R be the OWL 2 RL/RDF rules as defined above; and let O1 and O2 be OWL 2 RL ontologies in both of which no URI is used for more than one type of entity (i.e., no URIs is used both as, say, a class and an individual), and where all axioms in O2 are assertions of the following form with a, a1, ..., an named individuals:

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

Editor's Note: ToDo: Add proof sketch and/or pointers into the literature.

5 Computational Properties

This section describes the computational complexity of the most relevant reasoning problems of the languages defined in this document. The reasoning problems considered here ontology consistency, class expression satisfiability, class expression subsumption, instance checking, and (Boolean) conjunctive query answering [OWL 2 Semantics]. When evaluating complexity, the following parameters will be considered:

Table 10 summarizes the known complexity results for OWL 2 DL, OWL 1 DL, OWL 2 EL, OWL 2 QL, and OWL 2 RL. Whenever the complexity for a given problem is described as "Open", * denotes that the problem's decidability is still an open question; if * is omitted, then the problem is known to be decidable but precise complexity bounds have not yet been established.

Table 10. Complexity of the Profiles
Language Reasoning Problems Taxonomic Complexity Data Complexity Query Complexity Combined Complexity
OWL 2 DL Ontology Consistency, Class Expression Satisfiability,
Class Expression Subsumption, Instance Checking
2NEXPTIME-complete Open
(NP-Hard)
Not Applicable 2NEXPTIME-complete
Conjunctive Query Answering Open* Open* Open* Open*
OWL 1 DL Ontology Consistency, Class Expression Satisfiability,
Class Expression Subsumption, Instance Checking
NEXPTIME-complete Open
(NP-Hard)
Not Applicable NEXPTIME-complete
Conjunctive Query Answering Open* Open* Open* Open*
OWL 2 EL Ontology Consistency, Class Expression Satisfiability,
Class Expression Subsumption, Instance Checking
PTIME-complete PTIME-complete Not Applicable PTIME-complete
Conjunctive Query Answering PTIME-complete PTIME-complete NP-complete PSPACE-complete
OWL 2 QL Ontology Consistency, Class Expression Satisfiability,
Class Expression Subsumption, Instance Checking,
In PTIME In LOGSPACE Not Applicable In PTIME
Conjunctive Query Answering In PTIME In LOGSPACE NP-complete NP-complete
OWL 2 RL Ontology Consistency, Class Expression Satisfiability,
Class Expression Subsumption, Instance Checking
PTIME-complete PTIME-complete Not Applicable PTIME-complete
Conjunctive Query Answering PTIME-complete PTIME-complete NP-complete NP-complete

6 References

[OWL 2 Specification]
OWL 2 Web Ontology Language:Structural Specification and Functional-Style Syntax Boris Motik, Peter F. Patel-Schneider, Ian Horrocks. W3C Editor's Draft, 29 September 2008, http://www.w3.org/2007/OWL/draft/ED-owl2-syntax-20080929/. Latest version available at http://www.w3.org/2007/OWL/draft/owl2-syntax/.
[OWL 2 Semantics]
OWL 2 Web Ontology Language: Model-Theoretic Semantics. Bernardo Cuenca Grau and Boris Motik, eds., 2008.
[OWL 2 RDF Mapping]
OWL 2 Web Ontology Language:Mapping to RDF Graphs Bernardo Cuenca Grau, Boris Motik, Peter F. Patel-Schneider. W3C Editor's Draft, 29 September 2008, http://www.w3.org/2007/OWL/draft/ED-owl2-mapping-to-rdf-20080929/. Latest version available at http://www.w3.org/2007/OWL/draft/owl2-mapping-to-rdf/.
[OWL 2 RDF-Based Semantics]
OWL 2 Web Ontology Language:RDF-Based Semantics Michael Schneider. W3C Editor's Draft, 29 September 2008, http://www.w3.org/2007/OWL/draft/ED-owl2-rdf-based-semantics-20080929/. Latest version available at http://www.w3.org/2007/OWL/draft/owl2-rdf-based-semantics/.
[OWL 1 Reference]
OWL Web Ontology Language Reference. Mike Dean and Guus Screiber, eds., 2004.
[Conformance]
Conformance. Details to be filled in when the document is finished.
[RDF Semantics]
RDF Semantics. Patrick Hayes, ed., 2004.
[DL Handbook]
The Description Logic Handbook. Franz Baader, Diego Calvanese, Deborah L. McGuinness, Daniele Nardi, Peter F. Patel-Schneider, eds., 2007
[EL++]
Pushing the EL Envelope. Franz Baader, Sebastian Brandt, and Carsten Lutz. In Proc. of the 19th Joint Int. Conf. on Artificial Intelligence (IJCAI 2005), 2005.
[EL++ Update]
Pushing the EL Envelope Further. Franz Baader, Sebastian Brandt, and Carsten Lutz. In Proc. of the Washington DC workshop on OWL: Experiences and Directions (OWLED08DC), 2008.
[DL-Lite]
Tractable Reasoning and Efficient Query Answering in Description Logics: The DL-Lite Family. Diego Calvanese, Giuseppe de Giacomo, Domenico Lembo, Maurizio Lenzerini, Riccardo Rosati. J. of Automated Reasoning 39(3):385429, 2007.
[Complexity]
Complexity Results and Practical Algorithms for Logics in Knowledge Representation. Stephan Tobies. Ph.D Dissertation, 2002
[DLP]
Description Logic Programs: Combining Logic Programs with Description Logic. Benjamin N. Grosof, Ian Horrocks, Raphael Volz, and Stefan Decker. in Proc. of the 12th Int. World Wide Web Conference (WWW 2003), Budapest, Hungary, 2003. pp.: 4857
[pD*]
Completeness, decidability and complexity of entailment for RDF Schema and a semantic extension involving the OWL vocabulary. Herman J. ter Horst. J. of Web Semantics 3(23):79115, 2005.
[RIF]
RIF RDF and OWL Compatibility. Jos de Bruijn, ed. W3C Working Draft 30 July 2008
[XML Schema Datatypes]
XML Schema Part 2: Datatypes Second Edition. Paul V. Biron and Ashok Malhotra, eds. W3C Recommendation 28 October 2004.
[RFC 2119]
RFC 2119: Key words for use in RFCs to Indicate Requirement Levels. Network Working Group, S. Bradner. Internet Best Current Practice, March 1997.
[SNOMED CT]
Systematized Nomenclature of Medicine Clinical Terms. International Health Terminology Standards Development Organization (IHTSDO).

7 Appendix: Complete Grammars for Profiles

Editor's Note: This appendix will contain the full grammars of each of the profiles. The grammar will be completed when the technical work on each of the profiles has been finished.

7.1 OWL 2 EL

7.2 OWL 2 QL

7.3 OWL 2 RL