Profiles-v2
__NUMBEREDHEADINGS__
- Document title:
- OWL 2 Web Ontology Language
Profiles (Second Edition)
- Authors
- Bernardo Cuenca Grau, Oxford University
- Boris Motik, Oxford University
- Zhe Wu, Oracle
- Achille Fokoue, IBM
- Carsten Lutz, Dresden University of Technology
- 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
- This document is an evolution of the OWL 1.1 Web Ontology Language: Tractable Fragments document that forms part of the OWL 1.1 Web Ontology Language W3C Member Submission.
Copyright © 2008-2009 W3C^{®} (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.
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 of OWL 2, each of them achieving efficiency in a different way and useful in different application scenarios:
- The OWL 2 EL profile enables polynomial time algorithms for the reasoning tasks consistency, classification, and instance checking. Dedicated reasoning algorithms for this profile are available and have been demonstrated to be implementable in a highly scalable way.
- In the OWL 2 QL profile, conjunctive query answering can be implemented using conventional relational database systems (after rewriting the query). In particular, this requires that conjunctive query answering is in LogSpace regarding data complexity. As in OWL 2 EL, there are polynomial time algorithms for consistency, classification, and instance checking.
- The OWL RL profile enables the implementation of reasoning algorithms by forward chaining rules applied to triples of the RDF serialization. In particular, such rule approaches can be used to compute consistency, classification, and instance checking in polynomial time. Conjunctive query answering in OWL RL can be implemented by extending a standard relational database system with rules.
The profiles OWL 2 EL and QL are defined by placing restrictions on the syntax of OWL 2 DL.
Syntactic restrictions can be specified by modifying the grammar of the functional-style syntax [OWL 2 Specification], and (possibly) giving additional non-structural restrictions. In this document, the modified grammars are specified in two ways. In each section defining a profile, only the difference to the full grammar is given; that is, only the productions that differ from [OWL 2 Specification] are presented and the productions that are the same as in [OWL 2 Specification] are not repeated. In order to make this document self-contained, the full grammar for each of the profiles is given in the Appendix.
The reasoning tasks mentioned in the description of the profiles are defined as follows.
- consistency: given an ontology, decide whether it is contradictory (i.e., has any models);
- classification: given an ontology, compute the subclass relation between all its classes;
- instance checking: given an individual a and a class C in an ontology, decide whether a is an instance of C;
- conjunctive query answering: given an ontology O and a conjunctive query q, return all tuples of individuals in O that match q.
It will be worth having a larger introduction. It's worth mentioning that there are many possible profiles that might usefully be the focus of a community, as well as that implementations of the profiles might support profile *families* (OWL 2 QL really comes to mind there).
Longer intro should mention each of the three profiles and a one-liner rationale for each in language that is as non-technical as possible. Could also mention: OWL DL as another sensible implementation profile; that users/implementators MAY use any other profile that they find helpful; deprecate OWL Lite, see ISSUE-107
Apart from the ones specified here, there are many other possible profiles of OWL 2. Although we don't specifically document OWL lite [OWL 1 Reference] in this document, it is the intention of the working group that all OWL Lite ontologies will be 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.
2 OWL 2 EL
The OWL 2 EL profile [EL++,EL++ Update] is designed as a maximal subset of OWL 2 that
- captures the expressive power used by many large-scale ontologies and
- for which the following reasoning problems can be decided in polynomial time: satisfiability, subsumption, classification, and instance checking.
A main design principle of OWL 2 EL is to focus on the class constructors ObjectIntersectionOf and ObjectSomeValuesFrom, but to provide ObjectAllValuesFrom only in the form of range restrictions. Many biomedical ontologies, such as SNOMED CT, fall within this profile.
2.1 Feature Overview
OWL 2 EL provides the following features:
- 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)
- class inclusion (SubClassOf)
- class equivalence (EquivalentClasses)
- class disjointness (DisjointClasses)
- object property inclusion (SubObjectPropertyOf), possibly involving 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 (KeyFor)
- a subset of the datatypes of OWL 2
The following features of OWL 2 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 and DisjointUnion)
- 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 object properties (FunctionalObjectProperty)
- 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
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:
- rdfs:Literal
- xsd:decimal
- xsd:integer
- xsd:nonNegativeInteger
- xsd:dateTime
- xsd:date
- xsd:string
- xsd:normalizedString
- xsd:anyURI
- xsd:token
- xsd:Name
- xsd:NCName
- xsd:hexBinary
- xsd:base64Binary
- owl:internationalizedString
The following predefined OWL 2 datatypes cannot be used in OWL 2 EL: xsd:double, xsd:float, xsd:long, xsd:int, xsd:short, xsd:byte, xsd:nonPositiveInteger, xsd:positiveInteger, xsd:negativeInteger, xsd:unsignedLong, xsd:unsignedInt, xsd:unsignedShort, xsd:unsignedByte, xsd:time, xsd:gYear, xsd:gMonth, xsd:gDay, xsd:gYearMonth, and xsd:gMonthDay.
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 OWL 2 [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 [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 [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 [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. 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 OP_{1} if object properties OP_{i}, 2 ≤ i ≤ k, exist such that Ax contains all of the following axioms:
- 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 SubPropertyOf axiom in which in the first item of the previous definition does not contain a property chain. Range restrictions on reflexive and transitive roles are generally allowed, unless they are used in axioms that are explicitly forbidden using the previous definition.
3 OWL 2 QL
OWL 2 QL is a syntactic profile of OWL 2 that admits sound and complete reasoning in LOGSPACE with respect to the size of the data (assertions). OWL 2 QL includes most of the main features of conceptual models such as UML class diagrams and ER diagrams.
Several variants of DL-Lite have been described in the literature. The variant presented here is called DL-Lite_{A}: it is one of the most expressive variants of DL-Lite, and it contains the intersection between RDFS and OWL 2. In order to achieve the desired complexity of query answering, certain restrictions are imposed on the interaction between subproperty and functionality axioms.
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 can be used.
3.1 Feature Overview
The following constructs can be used to define subclass expressions in SubClassOf axioms:
- classes
- existential quantification (ObjectSomeValuesFrom) where the class is limited to owl:Thing
The following constructs can be used to define superclass expressions in SubClassOf axioms:
- classes
- existential quantification to a class (ObjectSomeValuesFrom)
- negation (ObjectComplementOf)
- intersection (ObjectIntersectionOf)
All axioms in OWL 2 QL are constrained in a way that is compliant with these restrictions. Thus, OWL 2 QL supports the following axioms:
- subclass axioms (SubClassOf)
- class expression equivalence (EquivalentClasses)
- class expression disjointness (DisjointClasses)
- functional object properties (FunctionalObjectProperty)
- 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)
- keys (KeyFor)
The following features of OWL 2 are not supported in OWL 2 QL:
- existential quantification to a class expression or a data range (ObjectSomeValuesFrom and DataSomeValuesFrom 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)
- property inclusions (SubObjectPropertyOf involving property chains)
- transitive properties (TransitiveObjectProperty)
- reflexive properties (ReflexiveObjectProperty)
- irreflexive properties (IrreflexiveObjectProperty)
- asymmetric properties (AsymmetricObjectProperty)
- inverse-functional properties (InverseFunctionalObjectProperty)
3.2 Profile Specification
The productions for OWL 2 QL are defined in the following sections. The expressive power of OWL 2 QL is such that the global restriction on axioms defined in Section 11 of [OWL 2 Specification] are vacuously satisfied in every DL-lite ontology.
3.2.1 Entities
OWL 2 QL supports all OWL 2 entities, including all predefined classesand properties. The following datatypes are supported in OWL 2 QL:
- rdfs:Literal
- xsd:decimal
- xsd:integer
- xsd:nonNegativeInteger
- xsd:positiveInteger
- xsd:dateTime
- xsd:date
- xsd:string
- xsd:normalizedString
- xsd:anyURI
- xsd:token
- xsd:Name
- xsd:NCName
- xsd:hexBinary
- xsd:base64Binary
- owl:internationalizedString
The following predefined OWL 2 datatypes cannot be used in OWL 2 QL: xsd:double, xsd:float, xsd:long, xsd:int, xsd:short, xsd:byte, xsd:nonPositiveInteger, xsd:negativeInteger, xsd:unsignedLong, xsd:unsignedInt, xsd:unsignedShort, xsd:unsignedByte, xsd:time, xsd:gYear, xsd:gMonth, xsd:gDay, xsd:gYearMonth, and xsd:gMonthDay.
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 in the antecedents of implications; such class expressions can, for example, occur as subclass expressions in SubClassOf axioms. The superClassExpression production defines the classes that can occur in the consequents of implications; such class expressions can, for example, 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. Different OWL 2 class axioms from the structural specification [OWL 2 Specification] that refer to the ClassExpression production, however, are redefined and restricted them to appropriate forms of class expressions.
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 transitive, asymmetric, reflexive and irreflexive object properties, and it redefines the object property domain and range axioms to use the appropriate class expressions. Finally, OWL 2 QL allows for functional object properties.
ObjectPropertyDomain := 'PropertyDomain' '(' { annotation } ObjectPropertyExpression superClass ')'
ObjectPropertyRange := 'PropertyRange' '(' { annotation } ObjectPropertyExpression superClass ')'
SubObjectPropertyOf := 'SubPropertyOf' '(' { annotation } ObjectPropertyExpression ObjectPropertyExpression ')'
ObjectPropertyAxiom :=
SubObjectPropertyOf | EquivalentObjectProperties |
DisjointObjectProperties | InverseObjectProperties |
ObjectPropertyDomain | ObjectPropertyRange |
SymmetricObjectProperty | FunctionalObjectProperty
OWL 2 QL provides the same axioms about data properties as OWL 2 [OWL 2 Specification], with the exception that the object property domain axioms are redefined to use the appropriate class expressions.
DataPropertyDomain := 'PropertyDomain' '(' { annotation } DataPropertyExpression superClass ')'
OWL 2 QL disallows negative object property assertions and equality axioms. Furthermore, class assertions in OWL 2 QL can involve only atomic classes. Equality and 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 in OWL 2, with the difference that the new productions are used for various types of axioms.
3.2.6 Global Restrictions
The global restrictions on axioms from Section 11 of the structural specification [OWL 2 Specification] are not relevant in OWL 2 QL because it allows for neither number restrictions nor property chains. OWL 2 QL, however, introduces a different set of global restrictions that the axiom closure Ax of an OWL 2 QL ontology O must satisfy.
- Ax must axiomatize the Unique Name Assumption (UNA): each pair of individuals i_{1} and i_{2} occurring in some axiom in Ax must also occur in Ax in an axiom of the form DifferentIndividuals( ... i_{1} ... i_{2} ... ).
- No object property OP is allowed to occur in Ax in both a FunctionalProperty or a KeyFor axiom and in a superproperty expression in SubPropertyOf axiom.
- No data property DP is allowed to occur in Ax in both a FunctionalProperty or a KeyFor axiom and in a superproperty expression in SubPropertyOf axiom.
4 OWL 2 RL
OWL 2 RL is a profile of OWL 2 that allows for scalable reasoning using rule-based technologies.
An important design goal for OWL 2 RL is flexibility. On the one hand, OWL 2 RL can accommodate OWL 2 DL applications that can trade the full expressivity of the language for efficiency; on the other hand, OWL 2 RL can also accommodate RDF(S) applications that need some added expressivity from OWL. This is achieved by providing both:
- A partial axiomatisation of the OWL 2 RDF-Based Semantics in the form of first-order implications (see Section 4.2). The axiomatisation can be used as the basis for practical inference implementations that use rule-based technologies and operate directly on RDF triples.
- A syntactically defined subset of OWL 2 for which such rule based implementations can provide computational guarantees (see Section 4.3), in particular completeness for certain forms of query answering (see Conformance).
4.1 Feature Overview
OWL 2 RL is an expressive profile in that it allows for all the constructs of OWL 2. By restricting the way in which these constructors are used, however, it is still possible to implement effective reasoning systems using relatively simple technology. These restrictions are designed so as to avoid the need to infer the existence of individuals not explicitly present in the knowledge base. This is achieved mainly by using the constructs of OWL 2 in the subclass and superclass expressions of SubClassOf axioms only according to the patterns shown in Table 1.
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) |
Ontologies satisfying these (and a few other) restrictions are called Simple OWL 2 RL ontologies (see Section 4.3). For OWL 2 RL ontologies that do not satisfy these restrictions, implementations based on the axiomatisation provided in Section 4.2 will still compute valid entailments, but they can no longer be guaranteed to compute all such entailments (see Conformance).
4.2 Reasoning in OWL 2 RL and RDF Graphs using Rules
This section presents a partial axiomatisation of the OWL 2 RDF-Based Semantics in the form of first-order (material) implications. This provides a useful starting point for practical implementation using rule-based technologies.
The implications are given as universally quantified first-order implications over a ternary predicate T. This predicate represents RDF triples; thus, T(s, p, o) represents a RDF triple with the subject s, predicate p, and the object o. Variables in the implications are preceded with the question mark. The propositional symbol false is a special symbol denoting contradiction: if it is derived, then the RDF graph is inconsistent.
Many conditions contain atoms that must 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 following conjunction of triples, 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 axiomatisation is split into several tables for easier navigation. Table 3 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.
It would be good if all the rows in the tables were numbered so that we can refer to them easily, even if it's only in the markup. Consider trying to discuss a particular mapping rule in the RDF mapping.
Please add real names (not consequtive numbers, which are hostages to fortune) for each rule, and add HTML anchors
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 |
Table 4 specifies the semantic conditions on 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, ?y_{1}) T(?x, ?p, ?y_{2}) |
T(?y_{1}, owl:sameAs, ?y_{2}) |
T(?p, rdf:type, owl:InverseFunctionalProperty) T(?x_{1}, ?p, ?y) T(?x_{2}, ?p, ?y) |
T(?x_{1}, owl:sameAs, ?x_{2}) |
T(?p, rdf:type, owl:ReflexiveProperty) T(?x, ?y, ?z) |
T(?x, ?p, ?x) T(?y, ?p, ?y) T(?z, ?p, ?z) |
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(?p_{1}, rdfs:subPropertyOf, ?p_{2}) T(?x, ?p_{1}, ?y) |
T(?x, ?p_{2}, ?y) |
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}) |
T(?p_{1}, owl:equivalentProperty, ?p_{2}) T(?x, ?p_{1}, ?y) |
T(?x, ?p_{2}, ?y) |
T(?p_{1}, owl:equivalentProperty, ?p_{2}) T(?x, ?p_{2}, ?y) |
T(?x, ?p_{1}, ?y) |
T(?p_{1}, owl:propertyDisjointWith, ?p_{2}) T(?x, ?p_{1}, ?y) T(?x, ?p_{2}, ?y) |
false |
T(?p_{1}, owl:inverseOf, ?p_{2}) T(?x, ?p_{1}, ?y) |
T(?y, ?p_{2}, ?x) |
T(?p_{1}, owl:inverseOf, ?p_{2}) T(?x, ?p_{2}, ?y) |
T(?y, ?p_{1}, ?x) |
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 5 specifies the semantic conditions on classes.
If | then | |
---|---|---|
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) | |
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}) | |
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 |
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, ?p, ?y) T(?u, rdf:type, ?x) |
false | |
T(?x, owl:maxCardinality, "1"^^xsd:nonNegativeInteger) T(?x, owl:onProperty, ?p) T(?u, ?p, ?y_{1}) T(?u, ?p, ?y_{2}) T(?u, rdf:type, ?x) |
T(?y_{1}, owl:sameAs, ?y_{2}) | |
T(?c, owl:oneOf, ?x) LIST[?x, ?y_{1}, ..., ?y_{n}] |
T(?y_{i}, rdf:type, ?c) | for each 1 ≤ i ≤ n |
Table 6 specifies the semantic conditions on class axioms.
If | then |
---|---|
T(?c_{1}, rdfs:subClassOf, ?c_{2}) T(?x, rdf:type, ?c_{1}) |
T(?x, rdf:type, ?c_{2}) |
T(?c_{1}, owl:equivalentClass, ?c_{2}) T(?x, rdf:type, ?c_{1}) |
T(?x, rdf:type, ?c_{2}) |
T(?c_{1}, owl:equivalentClass, ?c_{2}) T(?x, rdf:type, ?c_{2}) |
T(?x, rdf:type, ?c_{1}) |
T(?c_{1}, owl:disjointClasses, ?c_{2}) T(?x, rdf:type, ?c_{1}) T(?x, rdf:type, ?c_{2}) |
false |
Table 7 specifies the semantic restrictions on the vocabulary used to define the schema.
If | then |
---|---|
T(?c, rdf:type, owl:Class) | T(?c, rdfs:subClassOf, ?c) T(?c, owl:equivalentClasses, ?c) |
T(?c_{1}, rdfs:subClassOf, ?c_{2}) T(?c_{2}, rdfs:subClassOf, ?c_{3}) |
T(?c_{1}, rdfs:subClassOf, ?c_{3}) |
T(?c_{1}, owl:equivalentClass, ?c_{2}) | T(?c_{1}, rdfs:subClassOf, ?c_{2}) T(?c_{2}, rdfs:subClassOf, ?c_{1}) |
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(?p_{1}, rdfs:subPropertyOf, ?p_{2}) T(?p_{2}, rdfs:subPropertyOf, ?p_{3}) |
T(?p_{1}, rdfs:subPropertyOf, ?p_{3}) |
T(?p_{1}, owl:equivalentProperty, ?p_{2}) | T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) T(?p_{2}, rdfs:subPropertyOf, ?p_{1}) |
T(?p, rdfs:domain, ?c_{1}) T(?c_{1}, rdfs:subClassOf, ?c_{2}) |
T(?p, rdfs:domain, ?c_{2}) |
T(?p_{2}, rdfs:domain, ?c) T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) |
T(?p_{1}, rdfs:domain, ?c) |
T(?p, rdfs:range, ?c_{1}) T(?c_{1}, rdfs:subClassOf, ?c_{2}) |
T(?p, rdfs:range, ?c_{2}) |
T(?p_{2}, rdfs:range, ?c) T(?p_{1}, rdfs:subPropertyOf, ?p_{2}) |
T(?p_{1}, rdfs:range, ?c) |
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}) |
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}) |
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}) |
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}) |
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}) |
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}) |
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) |
Should soemthing be specified about literals
4.3 Simple OWL 2 RL Ontologies
Simple OWL 2 RL Ontologies are syntactically defined. As with OWL 2 QL, the definition is not only in terms of a set of supported constructs, but it also restricts the places in which these constructs can be used. The idea is based on Description Logic Programs [DLP] — a logic obtained by intersecting description logics with rule-based languages.
4.3.1 Entities
OWL 2 RL does not impose any restrictions on OWL 2 entities; hence, all entities from [OWL 2 Specification] are supported. OWL 2 RL also supports all well-known entities of OWL 2, apart from owl:TopObjectProperty, owl:BottomObjectProperty, owl:TopDataProperty, and owl:BottomDataProperty built-in properties.
4.3.2 Property Expressions
Property expressions in OWL 2 RL are identical to the property expressions in OWL 2 [OWL 2 Specification].
4.3.3 Class Expressions
There are three types of class expressions in OWL 2 RL. The subClassExpression production defines the class expressions that can occur in the antecedents of implications; such class expressions can, for example, occur as subclass expressions in a SubClassOf axiom. The superClassExpressions production defines the classes that can occur in the consequents of implications; such class expressions can, for example, occur as superclass expressions in a SubClassOf axiom. Finally, the equivClassExpressions production defines the classes that can occur in an EquivalentClasses axiom.
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' '(' subClassExpression 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 ')'
4.3.4 Data Ranges
In OWL 2 RL, data ranges are restricted to supported datatypes.
DataRange := Datatype
4.3.5 Axioms
OWL 2 RL redefines all of [OWL 2 Specification] that refer to ClassExpression. 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 property domain and range axioms are restricted to the appropriate form of class expressions.
ObjectPropertyDomain := 'PropertyDomain' '(' { annotation } ObjectPropertyExpression superClassExpression ')'
ObjectPropertyRange := 'PropertyRange' '(' { annotation } ObjectPropertyExpression superClassExpression ')'
DataPropertyDomain := 'PropertyDomain' '(' { annotation } DataPropertyExpression superClassExpression ')'
OWL 2 RL restricts the positive assertions to a particular type of classes, 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
Keys are redefined in OWL 2 RL to allow for correct type of class expression in the axiom.
KeyFor := 'KeyFor' '(' { annotation } ObjectPropertyExpression | DataPropertyExpression { ObjectPropertyExpression | DataPropertyExpression } subClassExpression ')'
All other axioms in OWL 2 RL are defined as in OWL 2.
4.4 Relationship between OWL 2 RL and OWL 2 Full
Let AXIOMS be a set containing all the implications listed in Section 4.2; let O be a simple OWL 2 RL ontology in 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 let F be a set of assertions of the following form:
- ClassAssertion( C a ) where C is a class
- PropertyAssertion( P a_{1} a_{2} ) where P is an object property
- PropertyAssertion( P a v ) where P is a data property
- SameIndividual( a_{1} ... a_{n} )
- DifferentIndividuals( a_{1} ... a_{n} )
Furthermore, let RDF(O) and RDF(F) be the translations of O and F into RDF graphs as specified in [OWL 2 RDF Mapping], in which triples are represented using the T predicate. Then, the following relationship between consequences in OWL 2 RL and OWL 2 Full holds:
F is a consequence of O under the OWL 2 RDF-Based semantics [OWL 2 RDF-Based Semantics] if and only if RDF(F) is a consequence of RDF(O) ∪ AXIOMS under the standard first-order semantics.
The main result of this section seems intuitive, but do we have a formal proof?
5 Computational Properties
This section describes the computational complexity of important reasoning problems in the described profiles.
Note that in languages that are propositionally closed (i.e., that provide, either implicitly or explicitly, conjunction, union and negation of class descriptions), such as OWL 2, the problems of ontology consistency, class expression satisfiability, class expression subsumption and instance checking can be reduced to each other in polynomial time. However, none of the described profiles is propositionally closed, and these reasoning problems may thus have different complexity and require different algorithmic solutions.
This section describes the computational complexity of the most relevant reasoning problems of the languages defined in this document. The reasoning problems considered here are the following:
- Ontology Consistency: Check whether a given ontology has at least one model.
- Class Expression Satisfiability: Given an ontology O and a class expression CE, verify whether there is a model of O in which the interpretation of CE is not empty.
- Class Expression Subsumption: Given an ontology O and class expressions CE_{1} and CE_{2}, verify whether the interpretation of CE_{1} is a subset of the interpretation of CE_{2} in every model of O
- Instance Checking: Given an ontology, an individual a and a class expression CE, verify whether a is an instance of CE in every model of the ontology.
- Conjunctive Query Answering: Given an ontology O and a Boolean conjunctive query Q (that is, a closed, existentially quantified conjunction of atoms in which class and property expressions occur as unary and binary atoms), check whether Q is true in every model of O
When evaluating the 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 and the facts. In the case of conjunctive query answering, the combined complexity also includes the size of the query.
Table 8 summarizes the known complexity results for OWL 2, 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.
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* | |
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 | |
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 | |
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 |
In Table 8, I wonder if we should add references to papers establishing complexity results for various profiles.
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 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 and Boris Motik, eds., 2008.
- [OWL 2 RDF-Based Semantics]
- OWL 2 Web Ontology Language: RDF-Based Semantics. Michael Schneider, 2008.
- [OWL 1 Reference]
- OWL Web Ontology Language Reference. Mike Dean and Guus Screiber, eds., 2004.
- [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.
- [XML Schema Datatypes]
- XML Schema Part 2: Datatypes Second Edition. Paul V. Biron and Ashok Malhotra, eds. W3C Recommendation 28 October 2004.
7 Appendix: Complete Grammars for Profiles
7.1 OWL 2 EL
7.2 OWL 2 QL
7.3 OWL 2 RL
7.4 Appendix: OWL-R Full
OWL-R Full is defined by weakening the semantic conditions on an interpretation from OWL 2 Full. Section 4.3.1 illustrates the principles according to which this weakening has been derived; these have strongly been inspired by [pD*]. The language is then formally defined in Section 4.3.2 by axiomatizing the weakened semantic conditions using first-order implications. The latter section provides a useful starting point for practical implementation using rule-based technologies.
HP notes the following design choice concerning expressivity vs ease-of-implementation with respect to: (a) equality and (b) b-node introduction. We have found both of these features add to expressivity and usefulness of a rule based approach to OWL Full reasoning, and both lead to performance issues. We note that the OWL-R definition includes (a) and not (b). That trade-off seems appropriate to us but this is a choice that should be highlighted during public review.
7.4.1 Weakening the Semantic Conditions of OWL 2 Full
This section presents the principles according to which the [OWL 2 Full Semantics] is weakened in OWL-R Full. To make discussion easier to understand, the following paragraph first summarizes the basic definitions used in [OWL 2 Full Semantics].
A datatype map D is a partial mapping from URI references to datatypes that maps datatype URIs to the appropriate [XML Schema Datatypes]. Given a datatype map, an interpretation in OWL 2 Full is defined as follows. For V a set of URI references and literals containing the reserved vocabulary of RDF, RDFS, and OWL, and for D a datatype map, a D-interpretation of V is a tuple I = ( R_{I} , P_{I} , EXT_{I} , S_{I} , L_{I} , LV_{I} ). R_{I} is the domain of discourse or universe — that is, a nonempty set that contains the denotations of URI references and literals in V. P_{I} is a subset of R_{I} consisting of the properties of I. EXT_{I} is used to give meaning to properties, and is a mapping from P_{I} to P(R_{I} × R_{I}). S_{I} is a mapping from URI references in V to their denotations in R_{I}. L_{I} is a mapping from typed literals in V to their denotations in R_{I}. LV_{I} is a subset of R_{I} that contains at least the set of all Unicode strings, the set of pairs of Unicode strings and language tags, and the value spaces for each datatype in D. The set of all classes in R_{I} is C_{I}, and the mapping CEXT_{I} from C_{I} to P(R_{I}) is defined as CEXT_{I}(c) = { x ∈ R_{I} | < x , c > ∈ EXT_{I}(S_{I}(rdf:type)) }. CEXT_{I}(c) maps a class c to its extension. A D-interpretations must meet several other conditions, as detailed in the OWL 2 Full semantics. Finally, the following important sets are used in the definitions of the semantic conditions of OWL 2 Full. IOOP denotes the set of OWL object properties and IODP denotes the set of OWL datatype properties; both are subsets of P_{I}. IOC, a subset of C_{I}, denotes the set of OWL classes, and IDC is the set of OWL datatypes. IOR represents the set of OWL restrictions. IOT is the set of OWL individuals.
In OWL-R Full, the weakening of the OWL 2 Full semantic conditions on an interpretation is mainly done by weakening some equivalences in the OWL Full semantics to implications. For example, the semantics of the owl:someValuesFrom restriction is defined in OWL 2 Full using the following restrictions on the RDF interpretation:
If | < x , y > ∈ EXT_{I}(S_{I}(owl:someValuesFrom)) ∧ < x , p > ∈ EXT_{I}(S_{I}(owl:onProperty)) | then | x ∈ IOR, y ∈ IOC ∪ IDC, p ∈ IOOP ∪ IODP, and CEXT_{I}(x) = { u ∈ IOT | ∃ < u , v > ∈ EXT_{I}(p) such that v ∈ CEXT_{I}(y) } |
---|
In a simplified form, these conditions can be understood as the following two implications:
If | < x , y > ∈ EXT_{I}(S_{I}(owl:someValuesFrom)) ∧ < x , p > ∈ EXT_{I}(S_{I}(owl:onProperty)) ∧ < u , v > ∈ EXT_{I}(p) ∧ < v , y > ∈ EXT_{I}(S_{I}(rdf:type)) |
then | < u , x > ∈ EXT_{I}(S_{I}(rdf:type)). |
---|---|---|---|
If | < x , y > ∈ EXT_{I}(S_{I}(owl:someValuesFrom)) ∧ < x , p > ∈ EXT_{I}(S_{I}(owl:onProperty)) ∧ < u , x > ∈ EXT_{I}(S_{I}(rdf:type)) |
then | ∃ v such that < u , v > ∈ EXT_{I}(p) ∧ < v , y > ∈ EXT_{I}(S_{I}(rdf:type)). |
The first implication captures the notion of existential restrictions occurring in the antecedents of implications, while the second implication captures the notion of existential restrictions occurring in the consequents of implications. In OWL-R Full, the second implication is discarded. Note the parallel with OWL-R DL, where syntactic restrictions prevent existential restrictions occurring in the consequents of implications.
Next, the restrictions that define OWL-R Full are listed. Instead of repeating all the intricate definitions of OWL Full, this section just specifies the difference to the definitions in the OWL Full document. For readers less familiar with OWL Full semantics, the next section provides a more self-contained axiomatization of OWL-R.
It would be good if tables in the OWL Full semantics document were numbered so that we can refer to them easily from this document.
Similarly, I think numbering all the rows would be helpful too, even if it's only in the markup. Consider trying to discuss a particular mapping rule in the RDF mapping.
- The conditions in the table defining the characteristics of OWL classes, datatypes, and properties are changed as follows:
- The conditions defining the sets IOT, LV_{I}, and IX are dropped.
- The if-and-only-if in the definition of the semantics of owl:FunctionalProperty, owl:InverseFunctionalProperty, owl:SymmetricProperty, and owl:TransitiveProperty is changed into only-if.
- In the table defining the semantics of rdfs:subClassOf, rdfs:subPropertyOf, rdfs:domain, and rdfs:range, the if-and-only-if condition in the third column of the table header is changed into only-if.
- In the table defining the characteristics of OWL vocabulary related to equivalence, the if-and-only-if condition in the second column of the table header is changed into only-if.
- In the table defining the conditions on OWL vocabulary related to boolean combinations and sets, the conditions for owl:complementOf is dropped, and the conditions for owl:oneOf and owl:unionOf are changed into only-if. The condition for owl:intersectionOf is left unchanged.
- The table defining the conditions on OWL restrictions is modified as follows.
- The condifion for owl:allValuesFrom is changed into CEXT_{I}(x) ⊆ { u ∈ IOT | < u , v > ∈ EXT_{I}(p) implies v ∈ CEXT_{I}(y) }.
- The condifion for owl:someValuesFrom is changed into CEXT_{I}(x) ⊇ {u ∈ IOT | < u , v > ∈ EXT_{I}(p) implies v ∈ CEXT_{I}(y) }.
- The condition for owl:hasValue is let unchanged.
- The conditions for cardinality restrictions are changed such that y must be 1, and CEXT_{I}(x) = [...] is changed into CEXT_{I}(x) ⊆ [...].
- The comprehension principles are dropped.