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Revision as of 18:47, 19 November 2008
__NUMBEREDHEADINGS__
- Document title:
- OWL 2 Web Ontology Language
Profiles (Second Edition)
- Editors
- Boris Motik, Oxford University
- Bernardo Cuenca Grau, Oxford University
- Ian Horrocks, Oxford University
- Zhe Wu, Oracle
- Achille Fokoue, IBM
- Carsten Lutz, University of Bremen
- Contributors
- Diego Calvanese, Free University of Bolzano
- Giuseppe De Giacomo, University of Rome “La Sapienza”
- Bijan Parsia, The University of Manchester
- Peter F. Patel-Schneider, Bell Labs Research, Alcatel-Lucent
- Note: The complete list of contributors is being compiled and will be included in the next draft.
- Abstract
- The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the Semantic Web with formally defined meaning. OWL 2 ontologies provide classes, properties, individuals, and data values and are stored as Semantic Web documents. OWL 2 ontologies can be used along with information written in RDF, and OWL 2 ontologies themselves are primarily exchanged as RDF documents. The OWL 2 Document Overview describes the overall state of OWL 2, and should be read before other OWL 2 documents.
This document provides 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
Copyright © 2008-2009 W3C^{®} (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.
Contents
1 Introduction
An OWL 2 profile (commonly called a fragment or a sublanguage in computational logic) is a trimmed down version of OWL 2 that trades some expressive power for the efficiency of reasoning. This document describes three profiles of OWL 2, each of which achieves efficiency in a different way and is useful in different application scenarios. The choice of which profile to use in practice will depend on the structure of the ontologies and the reasoning tasks at hand.
- OWL 2 EL is particularly useful in applications employing ontologies that contain very large numbers of properties and/or classes. This profile captures the expressive power used by many such ontologies and is a subset of OWL 2 for which the basic reasoning problems can be performed in time that is polynomial with respect to the size of the ontology [EL++]. Dedicated reasoning algorithms for this profile are available and have been demonstrated to be implementable in a highly scalable way.
- OWL 2 QL is aimed at applications that use very large volumes of instance data, and where query answering is the most important reasoning task. In OWL 2 QL, conjunctive query answering can be implemented using conventional relational database systems. Using a suitable reasoning technique, sound and complete conjunctive query answering can be performed in LOGSPACE with respect to the size of the data (assertions). As in OWL 2 EL, polynomial time algorithms can be used to implement the ontology consistency and class expression subsumption reasoning problems. The expressive power of the profile is necessarily quite limited, although it does include most of the main features of conceptual models such as UML class diagrams and ER diagrams.
- OWL 2 RL is aimed at applications that require scalable reasoning without sacrificing too much expressive power. It is designed to accommodate OWL 2 applications that can trade the full expressivity of the language for efficiency, as well as RDF(S) applications that need some added expressivity. OWL 2 RL reasoning systems can be implemented using rule-based reasoning engines. The ontology consistency, class expression satisfiability, class expression subsumption, instance checking, and conjunctive query answering problems can be solved in time that is polynomial with respect to the size of the ontology.
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. Furthermore, 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. This document does not list OWL Lite [OWL 1 Reference]; however, all OWL Lite ontologies are OWL 2 ontologies, so OWL Lite can be 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 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
- captures the expressive power used by many large-scale ontologies and
- for which ontology satisfiability, class expression subsumption, and instance checking can be decided in polynomial time.
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:
- existential quantification to a class expression (ObjectSomeValuesFrom) or a data range (DataSomeValuesFrom)
- existential quantification to an individual (ObjectHasValue) or a literal (DataHasValue)
- self-restriction (ObjectExistsSelf)
- 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 class expressions.
- 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 (DatAllaValuesFrom)
- cardinality restrictions (ObjectMaxCardinality, ObjectMinCardinality, ObjectExactCardinality, DataMaxCardinality, DataMinCardinality, and DataExactCardinality)
- 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 2 Specification], and OWL 2 EL supports all predefined classes and properties. Furthermore, OWL 2 EL supports the following datatypes:
- rdf:text
- rdfs:Literal
- owl:real
- xsd:decimal
- xsd:integer
- xsd:nonNegativeInteger
- xsd:string
- xsd:normalizedString
- xsd:token
- xsd:Name
- xsd:NCName
- xsd:NMTOKEN
- xsd:hexBinary
- xsd:base64Binary
- xsd:anyURI
- owl:dateTime
The set of supported datatypes has been designed such that the intersection of the value spaces of any set of these datatypes either empty or infinite, which is necessary to obtain the desired computational properties [EL++]. Consequently, the following 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 the structural specification [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 class expressions to ObjectIntersectionOf, ObjectSomeValuesFrom, ObjectExistsSelf, ObjectHasValue, DataSomeValuesFrom, DataHasValue, and ObjectOneOf containing a single individual.
ClassExpression :=
Class | ObjectIntersectionOf | ObjectOneOf |
ObjectSomeValuesFrom | ObjectExistsSelf | ObjectHasValue |
DataSomeValuesFrom | DataHasValue
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, intersections of data ranges, and to enumerated datatypes consisting of a single literal.
DataRange := Datatype | DataIntersectionOf | DataOneOf
DataOneOf := 'OneOf' '(' Literal ')'
2.2.5 Axioms
The class axioms of OWL 2 EL are the same as in the structural specification [OWL 2 Specification], 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 the structural specification [OWL 2 Specification] apart from DisjointDataProperty.
DataPropertyAxiom :=
SubDataPropertyOf |
EquivalentDataProperties |
DataPropertyDomain |
DataPropertyRange |
FunctionalDataProperty
The assertions in OWL 2 EL, as well as all other axioms, are the same as in the structural specification [OWL 2 Specification], 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.
The set of axioms Ax imposes a range restriction to a class expression CE on an object property OP_{1} if Ax contains the following axioms, where k ≥ 1 is an integer and OP_{i} are object properties:
SubPropertyOf( OP_{1} OP_{2})
...
SubPropertyOf( OP_{k-1} OP_{k} )
PropertyRange( OP_{k} CE )
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
- Ax contains SubPropertyOf( PropertyChain( OP_{1} ... OP_{n} ) OP ) and
- Ax imposes a range restriction to some class expression CE on OP
then Ax MUST impose a range restriction to CE on OP_{n}.
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 previously 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), while providing many of the 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. It is based on the DL-Lite family of description logics. Several variants of DL-Lite have been described in the literature [DL-Lite], and DL-Lite_{R} provides the logical underpinning for OWL 2 QL. DL-Lite_{R} does not require the unique name assumption (UNA), since making this assumption would have no impact on the semantic consequences of a DL-Lite_{R} ontology. More expressive variants of DL-Lite, such as DL-Lite_{A}, extend DL-Lite_{R} 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-Lite_{R} 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.
Subclass Expressions | Superclass Expressions |
---|---|
a class existential quantification (ObjectSomeValuesFrom) where the class is limited to owl:Thing existential quantification to a data range (DataSomeValuesFrom) |
a class existential quantification to a class (ObjectSomeValuesFrom) existential quantification to a data range (DataSomeValuesFrom) negation (ObjectComplementOf) intersection (ObjectIntersectionOf) |
OWL 2 QL supports the following axioms, constrained so as to be compliant with the mentioned restrictions 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)
- assertions other than the equality assertions (DifferentIndividuals, ClassAssertion, ObjectPropertyAssertion, and DataPropertyAssertion)
The following constructs are not supported in OWL 2 QL:
- existential quantification to a class expression or a data range (ObjectSomeValuesFrom in the subclass position)
- self-restriction (ObjectExistsSelf)
- existential quantification to an individual or a literal (ObjectHasValue, DataHasValue)
- nominals (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)
- reflexive properties (ReflexiveObjectProperty)
- irreflexive properties (IrreflexiveObjectProperty)
- asymmetric properties (AsymmetricObjectProperty)
- keys (HasKey)
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
Entities are defined in OWL 2 QL in the same way as in the structural specification [OWL 2 Specification], and OWL 2 QL supports all predefined classes and properties. Furthermore, OWL 2 QL supports the following datatypes:
- rdf:text
- rdfs:Literal
- owl:real
- xsd:decimal
- xsd:integer
- xsd:nonNegativeInteger
- xsd:string
- xsd:normalizedString
- xsd:token
- xsd:Name
- xsd:NCName
- xsd:NMTOKEN
- xsd:hexBinary
- xsd:base64Binary
- xsd:anyURI
- owl:dateTime
The set of supported datatypes has been designed such that the intersection of the value spaces of any set of these datatypes is either empty or infinite, which is necessary to obtain the desired computational properties. Consequently, the following 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 the structural specification [OWL 2 Specification].
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 ')' |
'SomeValuesFrom' '(' DataPropertyExpression DataRange ')'
superClassExpression :=
Class |
'SomeValuesFrom' '(' ObjectPropertyExpression Class ')' |
'SomeValuesFrom' '(' DataPropertyExpression DataRange ')' |
'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 and the intersection 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 2 Specification], 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' '(' axiomAnnotations subClassExpression superClassExpression ')'
EquivalentClasses := 'EquivalentClasses' '(' axiomAnnotations subClassExpression subClassExpression { subClassExpression } ')'
DisjointClasses := 'DisjointClasses' '(' axiomAnnotations 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' '(' axiomAnnotations ObjectPropertyExpression superClassExpression ')'
ObjectPropertyRange := 'PropertyRange' '(' axiomAnnotations ObjectPropertyExpression superClassExpression ')'
SubObjectPropertyOf := 'SubPropertyOf' '(' axiomAnnotations 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' '(' axiomAnnotations 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 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 2 Specification], with the exception that keys are not allowed.
Axiom := Declaration | ClassAxiom | ObjectPropertyAxiom | DataPropertyAxiom | Assertion | AnnotationAxiom
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 4.3 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 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 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.
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
Entities are defined in OWL 2 RL in the same way as in the structural specification [OWL 2 Specification]. OWL 2 RL supports the the predefined classes owl:Nothing and owl:Thing, but the usage of the latter class is restricted by the grammar of OWL 2 RL. Furthermore, OWL 2 RL does not support the predefined object and data properties owl:topObjectProperty, owl:bottomObjectProperty, owl:topDataProperty, and owl:bottomDataProperty. Finally, OWL 2 RL supports the following datatypes:
- rdf:text
- rdfs:Literal
- owl:real
- xsd:decimal
- xsd:integer
- xsd:nonNegativeInteger
- xsd:string
- xsd:normalizedString
- xsd:token
- xsd:Name
- xsd:NCName
- xsd:NMTOKEN
- xsd:hexBinary
- xsd:base64Binary
- xsd:anyURI
- owl:dateTime
The set of supported datatypes has been designed such that the intersection of the value spaces of any set of these datatypes is either empty or infinite, which is necessary to obtain the desired computational properties. Consequently, the following 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' '(' ObjectPropertyExpression owl:Thing ')' |
'SomeValuesFrom' '(' DataPropertyExpression { DataPropertyExpression } DataRange ')' |
'HasValue' '(' ObjectPropertyExpression Individual ')' |
'HasValue' '(' DataPropertyExpression Literal ')'
superClassExpression :=
Class other than owl:Thing |
'IntersectionOf' '(' superClassExpression superClassExpression { superClassExpression } ')' |
'AllValuesFrom' '(' ObjectPropertyExpression superClassExpression ')' |
'AllValuesFrom' '(' DataPropertyExpression { DataPropertyExpression } DataRange ')' |
'MaxCardinality' '(' zeroOrOne ObjectPropertyExpression [ subClassExpression ] ')' |
'MaxCardinality' '(' zeroOrOne ObjectPropertyExpression owl:Thing ')' |
'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 and the intersection of data ranges.
DataRange := Datatype | DataIntersectionOf
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' '(' axiomAnnotations subClassExpression superClassExpression ')'
EquivalentClasses := 'EquivalentClasses' '(' axiomAnnotations equivClassExpression equivClassExpression { equivClassExpression } ')'
DisjointClasses := 'DisjointClasses' '(' axiomAnnotations subClassExpression subClassExpression { subClassExpression } ')'
OWL 2 RL axioms about property expressions are as in the structural specification [OWL 2 Specification], the only difference being that class expressions in property domain and range axioms are restricted to superClassExpression.
ObjectPropertyDomain := 'PropertyDomain' '(' axiomAnnotations ObjectPropertyExpression superClassExpression ')'
ObjectPropertyRange := 'PropertyRange' '(' axiomAnnotations ObjectPropertyExpression superClassExpression ')'
DataPropertyDomain := 'PropertyDomain' '(' axiomAnnotations 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 the structural specification [OWL 2 Specification].
ClassAssertion := 'ClassAssertion' '(' axiomAnnotations Individual superClassExpression ')'
Assertion := SameIndividual | DifferentIndividuals | ClassAssertion | ObjectPropertyAssertion | DataPropertyAssertion
OWL 2 RL restricts class expressions in keys to subClassExpression.
HasKey := 'HasKey' '(' axiomAnnotations 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 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 in the form of first-order (material) implications; this axiomatization is called 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, e_{1}, ..., e_{n}] is used as an abbreviation for the conjunction of triples shown in Table 3, where z_{2}, ..., z_{n} are fresh variables that do not occur anywhere where the abbreviation is used.
T(h, rdf:first, e_{1}) | T(h, rdf:rest, z_{2}) |
T(z_{2}, rdf:first, e_{2}) | T(z_{2}, rdf:rest, z_{3}) |
... | ... |
T(z_{n}, rdf:first, e_{n}) | T(z_{n}, rdf:rest, rdf:nil) |
The axiomatization is split into several tables for easier navigation. Each rule is given a short unique name.
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.
If | 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(?y_{i}, owl:sameAs, ?y_{j}) T(?x, rdf:type, owl:AllDifferent) LIST[?x, ?y_{1}, ..., ?y_{n}] |
false | for each 1 ≤ i < j ≤ n |
Table 5 specifies the semantic conditions on axioms about properties.
If | then | ||
---|---|---|---|
prp-ap | true | T(ap, rdf:type, owl:AnnotationProperty) | for each built-in annotation property of OWL 2 RL |
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, ?y_{1}) T(?x, ?p, ?y_{2}) |
T(?y_{1}, owl:sameAs, ?y_{2}) | |
prp-ifp | T(?p, rdf:type, owl:InverseFunctionalProperty) T(?x_{1}, ?p, ?y) T(?x_{2}, ?p, ?y) |
T(?x_{1}, owl:sameAs, ?x_{2}) | |
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(?p_{1}, rdfs:subPropertyOf, ?p_{2}) T(?x, ?p_{1}, ?y) |
T(?x, ?p_{2}, ?y) | |
prp-spo2 | T(?sc, owl:propertyChain, ?x) LIST[?x, ?p_{1}, ..., ?p_{n}] T(?sc, rdfs:subPropertyOf, ?p) T(?u_{1}, ?p_{1}, ?u_{2}) T(?u_{2}, ?p_{2}, ?u_{3}) ... T(?u_{n}, ?p_{n}, ?u_{n+1}) |
T(?u_{1}, ?p, ?u_{n+1}) | |
prp-eqp1 | T(?p_{1}, owl:equivalentProperty, ?p_{2}) T(?x, ?p_{1}, ?y) |
T(?x, ?p_{2}, ?y) | |
prp-eqp2 | T(?p_{1}, owl:equivalentProperty, ?p_{2}) T(?x, ?p_{2}, ?y) |
T(?x, ?p_{1}, ?y) | |
prp-pdw | T(?p_{1}, owl:propertyDisjointWith, ?p_{2}) T(?x, ?p_{1}, ?y) T(?x, ?p_{2}, ?y) |
false | |
prp-adp | T(?z, rdf:type, owl:AllDisjointProperties) LIST[?z, ?p_{1}, ..., ?p_{n}] T(?x, ?p_{i}, ?y) T(?x, ?p_{j}, ?y) |
false | for each 1 ≤ i < j ≤ n |
prp-inv1 | T(?p_{1}, owl:inverseOf, ?p_{2}) T(?x, ?p_{1}, ?y) |
T(?y, ?p_{2}, ?x) | |
prp-inv2 | T(?p_{1}, owl:inverseOf, ?p_{2}) T(?x, ?p_{2}, ?y) |
T(?y, ?p_{1}, ?x) | |
prp-key | T(?c, owl:hasKey, ?u) LIST[?u, ?p_{1}, ..., ?p_{n}] T(?x, rdf:type, ?c) T(?x, ?p_{1}, ?z_{1}) ... T(?x, ?p_{n}, ?z_{n}) T(?y, rdf:type, ?c) T(?y, ?p_{1}, ?z_{1}) ... T(?y, ?p_{n}, ?z_{n}) |
T(?x, owl:sameAs, ?y) |
Table 6 specifies the semantic conditions on classes.
If | then | ||
---|---|---|---|
cls-thing | true | T(owl:Thing, rdf:type, owl:Class) | |
cls-nothing1 | true | 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, ?c_{1}, ..., ?c_{n}] T(?y, rdf:type, ?c_{1}) T(?y, rdf:type, ?c_{2}) ... T(?y, rdf:type, ?c_{n}) |
T(?y, rdf:type, ?c) | |
cls-int2 | T(?c, owl:intersectionOf, ?x) LIST[?x, ?c_{1}, ..., ?c_{n}] T(?y, rdf:type, ?c) |
T(?y, rdf:type, ?c_{1}) T(?y, rdf:type, ?c_{2}) ... T(?y, rdf:type, ?c_{n}) | |
cls-uni | T(?c, owl:unionOf, ?x) LIST[?x, ?c_{1}, ..., ?c_{n}] T(?y, rdf:type, ?c_{i}) |
T(?y, rdf:type, ?c) | for each 1 ≤ i ≤ n |
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-svf2 | T(?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-hv1 | T(?x, owl:hasValue, ?y) T(?x, owl:onProperty, ?p) T(?u, rdf:type, ?x) |
T(?u, ?p, ?y) | |
cls-hv2 | T(?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, ?y_{1}) T(?u, ?p, ?y_{2}) |
T(?y_{1}, owl:sameAs, ?y_{2}) | |
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, ?y_{1}) T(?y_{1}, rdf:type, ?c) T(?u, ?p, ?y_{2}) T(?y_{2}, rdf:type, ?c) |
T(?y_{1}, owl:sameAs, ?y_{2}) | |
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, ?y_{1}) T(?u, ?p, ?y_{2}) |
T(?y_{1}, owl:sameAs, ?y_{2}) | |
cls-oo | T(?c, owl:oneOf, ?x) LIST[?x, ?y_{1}, ..., ?y_{n}] |
T(?y_{i}, rdf:type, ?c) | for each 1 ≤ i ≤ n |
Table 7 specifies the semantic conditions on class axioms.
If | then | ||
---|---|---|---|
cax-sco | T(?c_{1}, rdfs:subClassOf, ?c_{2}) T(?x, rdf:type, ?c_{1}) |
T(?x, rdf:type, ?c_{2}) | |
cax-eqc1 | T(?c_{1}, owl:equivalentClass, ?c_{2}) T(?x, rdf:type, ?c_{1}) |
T(?x, rdf:type, ?c_{2}) | |
cax-eqc2 | T(?c_{1}, owl:equivalentClass, ?c_{2}) T(?x, rdf:type, ?c_{2}) |
T(?x, rdf:type, ?c_{1}) | |
cax-dw | T(?c_{1}, owl:disjointWith, ?c_{2}) T(?x, rdf:type, ?c_{1}) T(?x, rdf:type, ?c_{2}) |
false | |
cax-adc | T(?y, rdf:type, owl:AllDisjointClasses) LIST[?y, ?c_{1}, ..., ?c_{n}] T(?x, rdf:type, ?c_{i}) T(?x, rdf:type, ?c_{j}) |
false | for each 1 ≤ i < j ≤ n |
Table 8 specifies the semantics of datatypes.
If | then | ||
---|---|---|---|
dt-type1 | true | T(dt, rdf:type, rdfs:Datatype) | for each datatype dt supported in OWL 2 RL |
dt-type2 | 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 |
dt-eq | true | T(lt_{1}, owl:sameAs, lt_{2}) | for all literals lt_{1} and lt_{2} with the same data value |
dt-diff | true | T(lt_{1}, owl:differentFrom, lt_{2}) | for all literals lt_{1} and lt_{2} with different data values |
dt-not-type | 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.
If | then | |
---|---|---|
scm-cls | T(?c, rdf:type, owl:Class) | T(?c, rdfs:subClassOf, ?c) T(?c, owl:equivalentClass, ?c) T(?c, rdsf:subClassOf, owl:Thing) T(owl:Nothing, rdsf:subClassOf, ?c) |
scm-sco | T(?c_{1}, rdfs:subClassOf, ?c_{2}) T(?c_{2}, rdfs:subClassOf, ?c_{3}) |
T(?c_{1}, rdfs:subClassOf, ?c_{3}) |
scm-eqc | T(?c_{1}, owl:equivalentClass, ?c_{2}) | T(?c_{1}, rdfs:subClassOf, ?c_{2}) T(?c_{2}, rdfs:subClassOf, ?c_{1}) |
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(?p_{1}, rdfs:subPropertyOf, ?p_{2}) T(?p_{2}, rdfs:subPropertyOf, ?p_{3}) |
T(?p_{1}, rdfs:subPropertyOf, ?p_{3}) |
scm-eqp | T(?p_{1}, owl:equivalentProperty, ?p_{2}) | T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) T(?p_{2}, rdfs:subPropertyOf, ?p_{1}) |
scm-dom1 | T(?p, rdfs:domain, ?c_{1}) T(?c_{1}, rdfs:subClassOf, ?c_{2}) |
T(?p, rdfs:domain, ?c_{2}) |
scm-dom2 | T(?p_{2}, rdfs:domain, ?c) T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) |
T(?p_{1}, rdfs:domain, ?c) |
scm-rng1 | T(?p, rdfs:range, ?c_{1}) T(?c_{1}, rdfs:subClassOf, ?c_{2}) |
T(?p, rdfs:range, ?c_{2}) |
scm-rng2 | T(?p_{2}, rdfs:range, ?c) T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) |
T(?p_{1}, rdfs:range, ?c) |
scm-hv | T(?c_{1}, owl:hasValue, ?i) T(?c_{1}, owl:onProperty, ?p_{1}) T(?c_{2}, owl:hasValue, ?i) T(?c_{2}, owl:onProperty, ?p_{2}) T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) |
T(?c_{1}, rdfs:subClassOf, ?c_{2}) |
scm-svf1 | T(?c_{1}, owl:someValuesFrom, ?y_{1}) T(?c_{1}, owl:onProperty, ?p) T(?c_{2}, owl:someValuesFrom, ?y_{2}) T(?c_{2}, owl:onProperty, ?p) T(?y_{1}, rdfs:subClassOf, ?y_{2}) |
T(?c_{1}, rdfs:subClassOf, ?c_{2}) |
scm-svf2 | T(?c_{1}, owl:someValuesFrom, ?y) T(?c_{1}, owl:onProperty, ?p_{1}) T(?c_{2}, owl:someValuesFrom, ?y) T(?c_{2}, owl:onProperty, ?p_{2}) T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) |
T(?c_{1}, rdfs:subClassOf, ?c_{2}) |
scm-avf1 | T(?c_{1}, owl:allValuesFrom, ?y_{1}) T(?c_{1}, owl:onProperty, ?p) T(?c_{2}, owl:allValuesFrom, ?y_{2}) T(?c_{2}, owl:onProperty, ?p) T(?y_{1}, rdfs:subClassOf, ?y_{2}) |
T(?c_{1}, rdfs:subClassOf, ?c_{2}) |
scm-avf2 | T(?c_{1}, owl:allValuesFrom, ?y) T(?c_{1}, owl:onProperty, ?p_{1}) T(?c_{2}, owl:allValuesFrom, ?y) T(?c_{2}, owl:onProperty, ?p_{2}) T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) |
T(?c_{2}, rdfs:subClassOf, ?c_{1}) |
scm-int | T(?c, owl:intersectionOf, ?x) LIST[?x, ?c_{1}, ..., ?c_{n}] |
T(?c, rdfs:subClassOf, ?c_{1}) T(?c, rdfs:subClassOf, ?c_{2}) ... T(?c, rdfs:subClassOf, ?c_{n}) |
scm-uni | T(?c, owl:unionOf, ?x) LIST[?x, ?c_{1}, ..., ?c_{n}] |
T(?c_{1}, rdfs:subClassOf, ?c) T(?c_{2}, rdfs:subClassOf, ?c) ... T(?c_{n}, 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. Furthermore, let O_{1} and O_{2} 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 O_{2} are assertions of the following form with a, a_{1}, ..., a_{n} named individuals:
- ClassAssertion( C a ) where C is a class
- PropertyAssertion( OP a_{1} a_{2} ) where OP is an object property
- PropertyAssertion( DP a v ) where DP is a data property
- SameIndividual( a_{1} ... a_{n} )
- DifferentIndividuals( a_{1} ... a_{n} )
Furthermore, let RDF(O_{1}) and RDF(O_{2}) be translations of O_{1} and O_{2}, respetively, into RDF graphs as specified in the OWL 2 Mapping to RDF Graphs [OWL 2 RDF Mapping]; and let FO(RDF(O_{1})) and FO(RDF(O_{2})) 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, O_{1} entails O_{2} under the OWL 2 RDF-Based semantics [OWL 2 RDF-Based Semantics] if and only if FO(RDF(O_{1})) ∪ R entails FO(RDF(O_{2})) under the standard first-order semantics.
Proof Sketch. Without loss of generality, it can be assumed that all axioms in O_{1} are fully normalized — that is, that all class expressions in the axioms are of depth at most one. Let DLP(O_{1}) be the set of rules obtained by translating O_{1} into a set of rules as in Description Logic Programs [DLP].
Consider now each assertion A ∈ O_{2} that is entailed by DLP(O_{1}) (or, equivalently, by O_{1}). Let dt be a derivation tree for A from DLP(O_{1}). By examining the set of OWL 2 RL constructs, it is possible to see that each such tree can be transformed to a derivation tree dt' for RDF(A) from RDF(O_{1}). Each assertion B occurring in dt is of the form as specified in the theorem. The tree dt' can, roughly speaking, be obtained from dt by replacing each assertion B with RDF(B) and by replacing each rule from DLP(O_{1}) with a corresponding rule from Tables 3–8. Consequently, RDF(O_{1}) entails RDF(A).
Since no URI in O_{1} is used as both an individual and a class or a property, RDF(O_{1}) does not entail a triple of the form T(a:i1, owl:sameAs, a:i2) where either a:i1 or a:i2 is used in O_{1} as a class or a property. This allows one to transform a derivation tree for RDF(A) from RDF(O_{1}) to a derivation tree for A from DLP(O_{1}) in a way that is analogous to the previous case. QED
5 Computational Properties
This section describes the computational complexity of the most relevant reasoning problems of the languages defined in this document. For an introduction to computational complexity, please refer to a textbook on complexity such as [Papadimitriou]. The reasoning problems considered here ontology consistency, class expression satisfiability, class expression subsumption, instance checking, and (Boolean) conjunctive query answering [OWL 2 Direct Semantics]. When evaluating complexity, the following parameters will be considered:
- Data Complexity: the complexity measured with respect to the total size of the assertions in the ontology
- Taxonomic Complexity: the complexity measured with respect to the total size of the axioms in the ontology
- Query Complexity: the complexity measured with respect to the total size of the query
- Combined Complexity: the complexity measured with respect to both the size of the axioms, the size of the assertions, and, in the case of conjunctive query answering, the size of the query as well
Table 10 summarizes the known complexity results for OWL 2 under both RDF and the direct semantics, OWL 2 EL, OWL 2 QL, OWL 2 RL, and OWL 1 DL. The meaning of the entries is as follows:
- Decidability open means that it is not known whether this reasoning problem is decidable at all.
- Decidable, but complexity open means that decidability of this reasoning problem is known, but not its exact computational complexity. If available, known lower bounds are given in parenthesis; for example, (NP-Hard) means that this problem is at least as hard as any other problem in NP.
- X-complete for X one of the complexity classes explained below indicates that tight complexity bounds are known — that is, the problem is known to be both in the complexity class X (i.e., an algorithm is known that only uses time/space in X) and hard for X (i.e., it is at least as hard as any other problem in X). The following is a brief sketch of the classes used in this table, from the most complex one down to the simplest ones.
- 2NEXPTIME is the class of problems solvable by a nondeterministic algorithm in time that is at most double exponential in the size of the input (i.e., roughly 2^{2n}, for n the size of the input).
- NEXPTIME is the class of problems solvable by a nondeterministic algorithm in time that is at most exponential in the size of the input (i.e., roughly 2^{n}, for n the size of the input).
- PSPACE is the class of problems solvable by a nondeterministic algorithm using space that is at most polynomial in the size of the input (i.e., roughly n^{c}, for n the size of the input and c a constant).
- NP is the class of problems solvable by a nondeterministic algorithm using time that is at most polynomial in the size of the input (i.e., roughly n^{c}, for n the size of the input and c a constant).
- PTIME is the class of problems solvable by a deterministic algorithm using time that is at most polynomial in the size of the input (i.e., roughly n^{c}, for n the size of the input and c a constant). PTIME is often referred to as tractable, whereas the problems in the classes above are often referred to as intractable.
- LOGSPACE is the class of problems solvable by a deterministic algorithm using space that is at most logarithmic in the size of the input (i.e., roughly log(n), for n the size of the input and c a constant).
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, or what 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 by this class on all input problems.
Language | Reasoning Problems | Taxonomic Complexity | Data Complexity | Query Complexity | Combined Complexity |
---|---|---|---|---|---|
OWL 2 RDF-Based Semantics |
Ontology Consistency, Class Expression Satisfiability, Class Expression Subsumption, Instance Checking, Conjunctive Query Answering |
Undecidable | Undecidable | Undecidable | Undecidable |
OWL 2 Direct Semantics |
Ontology Consistency, Class Expression Satisfiability, Class Expression Subsumption, Instance Checking |
2NEXPTIME-complete (NEXPTIME if property hierarchies are bounded) | Decidable, but complexity open (NP-Hard) |
Not Applicable | 2NEXPTIME-complete (NEXPTIME if property hierarchies are bounded) |
Conjunctive Query Answering | Decidability open | Decidability open | Decidability open | Decidability 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 | |
OWL 1 DL | Ontology Consistency, Class Expression Satisfiability, Class Expression Subsumption, Instance Checking |
NEXPTIME-complete | Decidable, but complexity open (NP-Hard) |
Not Applicable | NEXPTIME-complete |
Conjunctive Query Answering | Decidability open | Decidability open | Decidability open | Decidability open |
6 References
- [OWL 2 Specification]
- OWL 2 Web Ontology Language: Structural Specification and Functional-Style Syntax. Peter F. Patel-Schneider, Ian Horrocks, and Boris Motik, eds., 2008.
- [OWL 2 Direct Semantics]
- OWL 2 Web Ontology Language: Direct 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 and Boris Motik, eds., 2008.
- [OWL 2 RDF-Based Semantics]
- OWL 2 RDF-Based Semantics. Michael Schneider, ed., 2008.
- [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):385–429, 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.: 48–57
- [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(2–3):79–115, 2005.
- [RIF]
- RIF RDF and OWL Compatibility. Jos de Bruijn, ed. W3C Working Draft 30 July 2008
- [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).