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=== Ontology Annotations ===
 
=== Ontology Annotations ===
Apart from a set of axioms, an OWL 2 ontology contains a set of [[#Annotations |annotations]]. These annotations can be used to associate information with an ontology, such as the name of the ontology creator or the version of the ontology. OWL 2 provides several well-known ontology annotations, as specified in [[#Built-in_Annotation_Properties|Section 9.3]].
+
Apart from a set of axioms, an OWL 2 ontology contains a set of [[#Annotations |annotations]]. These annotations can be used to associate information with an ontology, such as the name of the ontology creator or the version of the ontology. OWL 2 provides several well-known ontology annotations, as specified in [[#Built-in_Annotation_Properties|Section 9.3]]. The usage of these particular annotation properties on entities other than ontologies is discouraged.
  
 
=== Imports ===
 
=== Imports ===

Revision as of 21:52, 3 April 2008

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Document title:
OWL 2 Web Ontology Language
Structural Specification and Functional-Style Syntax (Second Edition)
Authors
Boris Motik, Oxford University
Peter F. Patel-Schneider, Bell Labs Research, Alcatel-Lucent
Ian Horrocks, Oxford University
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 defines a functional-style syntax for OWL 2, and provides an informal discussion of the meaning of the additional constructs. As well, an informational structural specification of OWL 2 ontologies is provided.
Status of this Document
This document is an evolution of the OWL 1.1 Web Ontology Language: Structural Specification and Functional-Style Syntax document that forms part of the OWL 1.1 Web Ontology Language W3C Member Submission.

Copyright © 2006-2007 by the Authors. This document is available under the W3C Document License. See the W3C Intellectual Rights Notice and Legal Disclaimers for additional information.

1 Introduction

In this specification, OWL ontologies are defined in ways intended to be, on the one hand, largely independant of the concrete exchange syntaxes, and yet, on the other, sufficient for the clear specification of those syntaxes. There are several reasons for this stemming from the fact that some exchange syntaxes (e.g., the canonical RDF/XML) make it difficult to relate OWL to the canonical underlying formalisms (i.e., description logics or first order logic). Defining the semantics directly in terms of such syntaxes would make it much more difficult to understand OWL by making the large literature available on the underlying formalisms more inaccessible. Similarly, in order to assure decidability (or, in some profiles, polynomial decidability), certain "non-structural" restrictions on axioms need to be imposed. It is very difficult to clearly and correctly impose these directly on triple structures. Finally, a number of OWL tools, APIs, and concrete syntaxes work at a non-triple level. Given the need to clearly map OWL into terms that make sense for these cases, and the difficulty of several specification tasks when approached from the RDF perspective, OWL is defined first in terms of non-triple structural form.

In order to determine the particular concrete syntax for some bit of OWL, one should consult the corresponding mapping tables in the corresponding documents. The primary structural specification is in terms of UML diagrams given in this document, with a canonical functional syntax which instantiates the UML.

Editor's Note: Are we going to make this true: "For reader convenience, you may choose to hide the diagrams, or the grammar, or both."

Conceptually, OWL ontologies can be broken down into the following categories of parts:

  • entities --- classes, properties, individuals, etc.; syntactically, these are the sorts of things which are named by URIs; they can be thought of as primitive terms or names;
  • expressions --- these are descriptive characterizations of entities; e.g., a class expression defines a class, typically in terms of features of its possible instances, i.e., descriptions of the sort of thing which are members of that class; obviously, expressions are parameterized by the sort of entity they characterize, thus we have class expressions, or property expressions; different sorts of expression have richer or poorer support in OWL; for example, OWL has a rather rich class expression functionality but comparatively poor property expressions;
  • axioms --- these make statements, i.e., true or false assertions that relate entities or expressions with other entities or expressions.

These three categories form the logical part of OWL ontologies. Additionally, entities, axioms, and ontologies may be annotated. Annotations have no effect on the logical aspects of an ontology, though they may be significant to tools or critical for applications. In many cases, the point of the logical portion of an ontology is to support the exploitation of the annotation part. The significance of annotations on tools or applications is not defined by the OWL specifications.

Finally, there is one non-logical construct with a pre-defined meaning, owl:imports, which allows distinct ontologies to be combined.

In concrete syntaxes, there may be additional syntactic features such as mechanisms for abbreviating URIs.

2 Basic Definitions

Editor's Note: See Issue-4 (syntax reordering) and Issue-82 (Metamodel diagrams).

The UML notation used in this document is restricted to a very limited subset of UML class diagram notation. The names of abstract classes (that is, the classes that are not intended to be instantiated) are written in italic.

The grammar of OWL 2 is presented in the standard BNF notation. Nonterminal symbols are written in bold (e.g., owlClassURI), terminal symbols are written in single quotes (e.g. 'ObjectPropertyRange'), zero or more instances of a symbol is denoted with curly braces (e.g., { description }), alternative productions are denoted with the vertical bar (e.g., fact | declaration), and zero or one instances of a symbol are denoted with square brackets (e.g., [ description ]).

2.1 Associations and Structural Equivalence

Many associations between components of OWL 2 ontologies are of one-to-many type; for example, an ObjectUnionOf description contains a set of disjuncts. Usually, it is important to know whether the components in the association are ordered and whether repetitions are allowed. This is made clear by attaching the following UML stereotypes to associations between components:

  • The <<set>> stereotype denotes that the associated components are unordered and that repetitions are not allowed.
  • The <<list>> stereotype denotes that the associated components are ordered and that repetitions are allowed.

To make this definition precise, it is necessary to say when two ontology components are considered to be the same. This is captured by the notion of structural equivalence, defined as follows. Components o1 and o2 are structurally equivalent if the following conditions hold:

  • If o1 and o2 are atomic values, such as strings, integers, or IRIs (URIs), they are structurally equivalent if they are equal using equality for their atomic type, i.e., they are the same string, integer, or IRI.
  • If o1 and o2 are sets, they are structurally equivalent if each element of o1 is structurally equivalent to some element of o2 and vice versa.
  • If o1 and o2 are lists, they are structurally equivalent if they contain the same number of elements and each element of o1 is structurally equivalent to the element of o2 with the same index.
  • If o1 and o2 are complex components composed of other components, they are structurally equivalent if
    • both o1 and o2 are of the same type,
    • each component of o1 is structurally equivalent to the corresponding component of o2, and
    • each association of o1 is structurally equivalent to the corresponding association of o2.

For example, the description ObjectUnionOf( Person Animal ) is structurally equivalent to description ObjectUnionOf( Animal Person ) because the order of the elements in a set is not important. Note that structural equivalence is not a semantic notion, as it is based only on comparing object structures defined in this document. For example, ObjectUnionOf( Person ObjectComplementOf( Person ) ) is not structurally equivalent to owl:Thing even though it is semantically equivalent to it.

Although the <<set>> stereotype is widely used in the specification, ontology files written in one of the linear syntaxes (e.g., XML or RDF/XML) are not expected to be duplicate free. Defining the structure of the language using sets, however, facilitates the specification of APIs for manipulating OWL 2 ontologies programmatically; furthermore, it provides the basis for the definition of complex operations on OWL 2 ontologies, such as retraction of axioms.

2.2 URIs, Namespaces, and Integers

Ontologies and their elements are identified using International Resource Identifiers (IRIs) [RFC-3987]. OWL 1 uses Uniform Resource Locators (URIs) to identify objects. We use the term 'URI' in OWL 2 as well, to avoid introducing new terminology.

For readability, IRIs can be abbreviated using CURIEs [CURIE], whose syntax is compatibly defined as follows.

curie := [ [ prefix ] ':' ] reference
prefix := NCName
reference := irelative-ref
irelative-ref := as defined in [RFC-3987]
NCName := as defined in [XML Namespaces]

The syntax of full and abbreviated IRIs in OWL 2 is defined as follows.

Full-IRI := '<' IRI as defined in [RFC-3987] '>'
Abbreviated-IRI := curie
URI := Full-IRI | Abbreviated-IRI

Abbreviated IRIs are turned into full IRIs by looking up the value matched to the prefix production in the namespace definitions associated with an ontology and concatenting the associated Full-IRI value with the value matched to reference in the Abbreviated-IRI production. The result must be a valid IRI. Abbreviated IRIs with no prefix are handled by the namespace definition with no prefix.

Some grammar productions use numbers, which are defined as follows:

zero := '0'
nonZero := '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
digit := zero | nonZero
postiveInteger := nonzero { digit }
nonNegativeInteger := zero | positiveInteger

Editor's Note: The actual namespaces used in the specification are subject to discussion and might change in future.

The following standard namespace prefixes are used throughout this specification:

Namespace prefix Namespace
rdf http://www.w3.org/1999/02/22-rdf-syntax-ns#
rdfs http://www.w3.org/2000/01/rdf-schema#
xsd http://www.w3.org/2001/XMLSchema#
owl http://www.w3.org/2002/07/owl#
owl2 http://www.w3.org/2006/12/owl2#
owl2xml http://www.w3.org/2006/12/owl2-xml#

3 Ontologies

Editor's Note: See Issue-21 (import-target-match) and Issue-24 (1-version-allowed-policy).

Ontologies are the basic structure in OWL. They primarily consist of a set of assertions about the world and may include assertions from other, imported, ontologies.

OWL Ontologies
Figure 1. OWL Ontologies


3.1 Syntax

The syntax for OWL 2 ontology files is defined as follows:

ontologyFile := { namespace } ontology
namespace := 'Namespace' '(' [ prefix ] '=' Full-IRI ')'
ontology := 'Ontology' '(' [ ontologyURI ] { import } { annotation } { axiom } ')'
ontologyURI := URI
import := 'Import' '(' URI ')'
axiom := classAxiom | objectPropertyAxiom | dataPropertyAxiom | fact | declaration | entityAnnotation

The namespace production defines an abbreviation for namespaces in a document. In each document, only one namespace declaration can exist for a given prefix. These prefixes are then used to expand abbreviated IRIs as specified in Section 2.2.


3.2 Ontology URIs

Each ontology can have an ontology URI. If present, this URI must be unique, and it need not be equal to the physical location of the ontology file. For example, a file for an ontology with a URI http://www.my.domain.com/example need not be physically stored in that location. A specification of a mechanism for physically locating an ontology from its ontology URI is not in scope of this specification.

3.3 Axioms

The main component of an OWL 2 ontology is the set of axioms that it contains. Note that this definition does not allow repetitions of structurally equivalent axioms in an ontology. OWL 2 ontology files are, however, not expected to enforce this and tools can simply eliminate duplicates.

3.4 Ontology Annotations

Apart from a set of axioms, an OWL 2 ontology contains a set of annotations. These annotations can be used to associate information with an ontology, such as the name of the ontology creator or the version of the ontology. OWL 2 provides several well-known ontology annotations, as specified in Section 9.3. The usage of these particular annotation properties on entities other than ontologies is discouraged.

3.5 Imports

In OWL 1, owl:imports was a special annotation URI, which denotes that an ontology imports another ontology. In OWL 2, imports are not ontology annotations, but are a separate primitive, as discussed next; the owl:imports annotation property has no built-in meaning.

Each ontology contains a possibly empty set of import constructs. An ontology O directly imports an ontology O' if O contains an import construct whose value is the ontology URI of O'. The relation imports is defined as a transitive closure of the relation directly imports. The axiom closure of an ontology O is the smallest set containing all the axioms of O and of all ontologies that O imports. Intuitively, an import construct states that, when reasoning with an ontology O, one should consider not only the axioms of O, but the entire axiom closure of O.

3.6 Example

Ontology(http://example.org/ontology1 
        Import(http://example.org/ontology2)
        Label("The example")
        SubClassOf(Human Animal)
)

This example ontology has a URI (given on the first line), one imports statement, an ontology annotation which associates a label with the ontology, and a single (subclass) axiom.

4 Entities

Entities are the fundamental building blocks of OWL 2 ontologies. The description of entities (e.g., classes, properties, and individuals) and their relations is the canonical purpose of an ontology.

NOTE: For readers with a logic background, entities are the elements of the
signature of an OWL ontology.

The Hierarchy of Entities in OWL 2
Figure 2. The Hierarchy of Entities in OWL 2

4.1 Syntax

All entities may have an associated URI. The syntax for writing entity URIs in OWL 2 is as follows:

datatypeURI := URI
owlClassURI := URI
objectPropertyURI := URI
dataPropertyURI := URI
annotationPropertyURI := URI
individualURI := URI

Entities are written in the following way:

entity := datatype | owlClass | objectProperty | dataProperty | annotationProperty | individual
datatype := 'Datatype' '(' datatypeURI ')'
owlClass := 'OWLClass' '(' owlClassURI ')'
objectProperty := 'ObjectProperty' '(' objectPropertyURI ')'
dataProperty := 'DataProperty' '(' dataPropertyURI ')'
annotationProperty := 'AnnotationProperty' '(' annotationPropertyURI ')'
individual := 'Individual' '(' individualURI ')'


OWL 2 uses constants to describe atomic values, such as strings or integers. The quoting mechanism in strings is a subset of the one used in N-triples [RDF Test Cases].

string := '"' a Unicode string in normal form C with double quotes and backslashes replaced by the double quote or backslash preceeded by a backslash '"'
languageTag := a language tag as specified in [RFC-4646]
untypedConstant := string [ '@' languageTag ]
typedConstant := string '^^' datatypeURI
constant := typedConstant | untypedConstant


Note that entities, like expressions, only appear in an ontology as part of an axiom or annotation. It is not unreasonable to think of entities as atomic expressions.

Just like ontologies, entities can also have annotations associated with them. The mechanism for annotating entities is described in more detail in Section 9.2.

4.2 Predefined Entities

OWL 2 defines several well-known entities that have the predefined semantics. These entities are identified by the following predefined URIs:

  • The class with URI owl:Thing is the set of all objects. (In DL literature this is often called the top concept.)
  • The class with URI owl:Nothing is the empty set of objects. (In DL literature this is often called the bottom concept.)
  • The unary datatype with URI rdfs:Literal is the set of all concrete objects.
  • The datatypes with URIs as mentioned in [OWL 2 Semantics].

4.3 Example

5 Class Expressions

Editor's Note: See Issue-65 (excess vocab).

OWL 2 provides an expressive language for forming class expressions.Readers familiar with model theory may which to consult the Semantics Document.

5.1 Propositional Connectives

Editor's Note: Would like to have more class expression specific stuff, similar to what's in the first sentence of the next section. BJP

Class expressions can be thought of as descriptions of sets of individuals, or as descriptions which are true (or false) of particular individuals. These descriptions can be combined in ways that produce set theoretical combination of their underlying sets (or that change the set of conditions which must be true of an instance of the combined expression). OWL 2 contains the standard boolean connectives, and (objectIntersectionOf), not (objectComplementOf), and or (objectUnionOf).

Propositional Connectives for the Formation of Descriptions
Figure 3. Propositional Connectives for the Formation of Descriptions

The propositional connectives are presented in Figure 3. The description construct objectUnionOf forms a disjunction of a set of descriptions, objectIntersectionOf is a conjunction of a set of descriptions, objectComplementOf is a negation of a description, and objectOneOf is a descrption that contains exactly the objects denoted by the set of specified individuals.

5.1.1 Syntax

objectUnionOf := 'ObjectUnionOf' '(' description description { description } ')'
objectIntersectionOf := 'ObjectIntersectionOf' '(' description description { description } ')'
objectComplementOf := 'ObjectComplementOf' '(' description ')'
objectOneOf := 'ObjectOneOf' '(' individualURI { individualURI }')'

5.1.2 Example

5.2 Object Value Restrictions

OWL 2 also allows descriptions to be defined by means of restrictions on object properties, as shown in Figure 4.

OWL 2 Descriptions Defined by Restriction on Object Properties
Figure 4. OWL 2 Classes Defined by Restriction on Object Properties

The construct objectAllValuesFrom denotes the set of objects that are connected via the given object property only to instances of the given description, objectSomeValuesFrom denotes the set of objects that are connected via the given object property to at least one instance of the given description, objectExistsSelf denotes the set of objects that are connected to themselves via the given object property, and objectHasValue denotes the set of objects that are connected via the given object property to the object denoted by the given individual.

Finally, OWL 2 descriptions can be defined by restricting the cardinality of associations between objects, as shown in Figure 5.

OWL 2 Descriptions Defined by Restricting Object Property Cardinalities
Figure 5. OWL 2 Descrptions Defined by Restricting Object Property Cardinalities

Cardinality restrictions can be qualified or unqualified, depending on whether there is a restriction on the description of the connected individual; an unqualified cardinality restriction is equivalent to a qualified one where the restricting description is owl:Thing. The construct objectMinCardinality denotes the set of objects that are connected via the given object property to at least the given number of instances of the given description, the construct objectMaxCardinality denotes the set of objects that are connected via the given object property to at most the given number of instances of the given description, and the description objectExactCardinality denotes the set of objects that are connected via the given object property to exactly the given number of instances of the given description.

5.2.1 Syntax

objectAllValuesFrom := 'ObjectAllValuesFrom' '(' objectPropertyExpression description ')'
objectSomeValuesFrom := 'ObjectSomeValuesFrom' '(' objectPropertyExpression description ')'
objectExistsSelf := 'ObjectExistsSelf' '(' objectPropertyExpression ')'
objectHasValue := 'ObjectHasValue' '(' objectPropertyExpression individualURI ')'

cardinality := nonNegativeInteger
objectMinCardinality := 'ObjectMinCardinality' '(' cardinality objectPropertyExpression [ description ] ')'
objectMaxCardinality := 'ObjectMaxCardinality' '(' cardinality objectPropertyExpression [ description ] ')'
objectExactCardinality := 'ObjectExactCardinality' '(' cardinality objectPropertyExpression [ description ] ')'

5.2.2 Example

5.3 Data Value Restrictions

OWL 2 also allows for the definition of descriptions by stating restrictions on data properties, as shown in Figure 6.

OWL 2 Descriptions Defined by Restriction on Data Properties
Figure 6. OWL 2 Descriptions Defined by Restriction on Data Properties

The notable distinction with respect to object property restrictions is that dataAllValuesFrom and dataSomeValuesFrom restrictions take a list of data property expressions, and not just a single property expression. This is in order to support descriptions such as "objects whose width is greater than their height", where the values of width and height are specified using two data properties. In such definitions, the arity of the given data range must be equal to the number of the given data properties.

Figure 7 shows the restrictions that can be built by stating cardinality restrictions on data properties. The arity of data range must be one.

OWL 2 Descriptions Defined by Restriction on Data Properties
Figure 7. OWL 2 Descriptions Defined by Restriction on Data Properties

5.3.1 Syntax

dataAllValuesFrom := 'DataAllValuesFrom' '(' dataPropertyExpression { dataPropertyExpression } dataRange ')'
dataSomeValuesFrom := 'DataSomeValuesFrom' '(' dataPropertyExpression { dataPropertyExpression } dataRange ')'
dataHasValue := 'DataHasValue' '(' dataPropertyExpression constant ')'
dataMinCardinality := 'DataMinCardinality' '(' cardinality dataPropertyExpression [ dataRange ] ')'
dataMaxCardinality := 'DataMaxCardinality' '(' cardinality dataPropertyExpression [ dataRange ] ')'
dataExactCardinality := 'DataExactCardinality' '(' cardinality dataPropertyExpression [ dataRange ] ')'

5.3.2 Example

5.4 Unified Grammar for Expressions

The following grammar production integrates all types of descriptions in OWL 2:

description := owlClassURI | objectUnionOf | objectIntersectionOf | objectComplementOf | objectOneOf |
    objectAllValuesFrom | objectSomeValuesFrom | objectExistsSelf | objectHasValue |
    objectMinCardinality | objectMaxCardinality | objectExactCardinality |
    dataAllValuesFrom | dataSomeValuesFrom | dataHasValue |
    dataMinCardinality | dataMaxCardinality | dataExactCardinality

6 Object Property Expressions

Object properties can be combined into more complex expressions, as show in Figure 8.

Object Property Expressions
Figure 8. Object Property Expressions

The only non-atomic object property expressions in OWL 2 are inverse property expressions.

For symmetry, OWL 2 also allows for data property expressions, as shown in Figure 9; the only type of data property expressions are, however, data properties. The grammar for data property expressions is as follows:

Data Property Expressions
Figure 9. Data Property Expressions

6.1 Syntax

inverseObjectProperty := 'InverseObjectProperty' '(' objectPropertyExpression ')'
objectPropertyExpression := objectPropertyURI | inverseObjectProperty

dataPropertyExpression := dataPropertyURI

6.2 Example

7 Data Range Expressions

Editor's Note: See Issue-5 (n-ary datatypes), Issue-25 (values-class), Issue-31 (XSD defined datatypes), Issue-53 (Linear inequality between two data types), and Issue-71 (datarange language range).

OWL 2 provides several ways to define a range over data values, as shown in Figure 10.

Review comment from MikeSmith 06:44, 26 November 2007 (EST)
Text addressing ISSUE-31: "Canonical URI for externally defined datatypes" is provided on the discussion page and proposed for inclusion here.

Data Ranges of OWL 2
Figure 10. Data Ranges of OWL 2

A datatype is a fundamental type of data range that is defined by a URI. Each datatype URI is associated with a predefined arity (note that the same datatype URI cannot be used with different arities). The list of the datatypes supported in OWL 2 is given in [OWL 2 Semantics]; furthermore, this list can be extended by implementations as needed. The meaning of OWL 2 ontologies containing a datatype URI not supported by an implementation is not defined by this specification; implementations are allowed to signal an error in this case.

Complex data ranges can be constructed from the simpler ones using the dataComplementOf constructor, which takes a data range and returns its complement (with the same arity), Furthermore, data ranges (of arity one) consisting exactly of the specified set of constants can be formed using the dataOneOf constructor. Finally, the datatypeRestriction constructor creates a data range by applying one or more facet restriction to a datatype. A facet restriction consists of a constant restriction value and a facet type that is applied to the data range in question. The following facet types are supported in OWL 2: length, minLength, maxLength, pattern, minInclusive, minExclusive, maxInclusive, maxExclusive, totalDigits, and fractionDigits.

7.1 Facet-Datatype Compatibility

Not all datatypes are compatible with all facets; the allowed combinations are listed in Table 1. The semantics of the facets is defined in the XML Schema Datatypes Specification [XML Schema Datatypes].

Table 1. Compatibility Between Datatypes and Facets
Datatype Allowed Facets
xsd:boolean pattern
xsd:decimal, xsd:double, xsd:float,
xsd:long, xsd:integer, xsd:int, xsd:short, xsd:byte,
xsd:nonPositiveInteger, xsd:nonNegativeInteger,
xsd:positiveInteger, xsd:negativeInteger,
xsd:unsignedLong, xsd:unsignedInt,
xsd:unsignedShort, xsd:unsignedByte,
xsd:dateTime, xsd:time, xsd:date,
xsd:gYear, xsd:gMonth, xsd:gDay,
xsd:gYearMonth, xsd:gMonthDay
minInclusive, minExclusive,
maxInclusive, maxExclusive,
totalDigits, fractionDigits, pattern
xsd:string, xsd:normalizedString, xsd:anyURI,
xsd:token, xsd:language, xsd:NMTOKEN,
xsd:Name, xsd:NCName,
xsd:hexBinary, xsd:base64Binary
length, minLength, maxLength, pattern


7.2 Syntax

dataComplementOf := 'DataComplementOf' '(' dataRange ')'
dataOneOf := 'DataOneOf' '(' constant { constant } ')'
datatypeFacet :=
    'length' | 'minLength' | 'maxLength' | 'pattern' |
    'minInclusive' | 'minExclusive' | 'maxInclusive' | 'maxExclusive' |
    'totalDigits' | 'fractionDigits'
restrictionValue := constant
datatypeRestriction := 'DatatypeRestriction' '(' datatypeURI datatypeFacet restrictionValue { datatypeFacet restrictionValue } ')'
dataRange := datatypeURI | dataComplementOf | dataOneOf | datatypeRestriction

7.3 Example

8 Axioms

This section lists the types of axiom that can be stated in OWL 2. As already mentioned, an axiom may contain an arbitrary number of annotations; furthermore, although annotations do not affect the semantics of an axiom, they are taken into account in the definition of structural equivalence.

8.1 Class Axioms

Editor's Note: Need to replace "descriptions" with "expressions" methinks.

The class axioms of OWL 2 are shown in Figure 11.

The Class Axioms of OWL 2
Figure 11. The Class Axioms of OWL 2

The subClassOf axiom states that one description is a subclass of another description. The equivalentClasses axiom takes a set of descriptions and states that they are all equivalent. The disjointClasses axiom takes a set of descriptions and states that all descriptions from the set are pair-wise disjoint. Finally, the disjointUnion axiom defines a class as a union of descriptions, all of which are pair-wise disjoint.

8.1.1 Syntax

subClass := description
superClass := description
subClassOf := 'SubClassOf' '(' { annotation } subClass superClass ')'
equivalentClasses := 'EquivalentClasses' '(' { annotation } description description { description } ')'
disjointClasses := 'DisjointClasses' '(' { annotation } description description { description } ')'
disjointUnion := 'DisjointUnion' '(' { annotation } owlClassURI description description { description } ')'
classAxiom := subClassOf | equivalentClasses | disjointClasses | disjointUnion

8.1.2 Example

8.2 Object Property Axioms

Editor's Note: See Issue-22 (role-rule-sugar) and Issue-83 (Property Chain Axioms).

OWL 2 provides for several different kinds of object property axioms. For clarity these are presented in two separate diagrams, the first of which is Figure 12.

Object Property Axioms, Part I
Figure 12. Object Property Axioms, Part I

The equivalentObjectProperties axiom takes a set of object properties and states that they are all equivalent, and the disjointObjectProperties axiom takes a set of object properties and states that all properties from the set are pair-wise disjoint. Furthermore, objectPropertyDomain and objectPropertyRange specify the domain and the range description, respectively, of an object property. Finally, inverseObjectProperties axiomatizes two properties to be inverse of each other.

In addition, OWL 2 provides for axioms that allow the assertion of various characteristics of an object property, as specified in Figure 13.

Axioms Defining Characteristics of Object Properties, Part II
Figure 13. Axioms Defining Characteristics of Object Properties, Part II

Each of these axioms takes an object property and asserts that the property has a certain characteristic, such as being functional or transitive.

8.2.1 Syntax

subObjectPropertyExpression := objectPropertyExpression | 'SubObjectPropertyChain' '(' objectPropertyExpression objectPropertyExpression { objectPropertyExpression } ')'
subObjectPropertyOf := 'SubObjectPropertyOf' '(' { annotation } subObjectPropertyExpression objectPropertyExpression ')'
equivalentObjectProperties := 'EquivalentObjectProperties' '(' { annotation } objectPropertyExpression objectPropertyExpression { objectPropertyExpression } ')'
disjointObjectProperties := 'DisjointObjectProperties' '(' { annotation } objectPropertyExpression objectPropertyExpression { objectPropertyExpression } ')'
objectPropertyDomain := 'ObjectPropertyDomain' '(' { annotation } objectPropertyExpression description ')'
objectPropertyRange := 'ObjectPropertyRange' '(' { annotation } objectPropertyExpression description ')'
inverseObjectProperties := 'InverseObjectProperties' '(' { annotation } objectPropertyExpression objectPropertyExpression ')'

functionalObjectProperty := 'FunctionalObjectProperty' '(' { annotation } objectPropertyExpression ')'
inverseFunctionalObjectProperty := 'InverseFunctionalObjectProperty' '(' { annotation } objectPropertyExpression ')'
reflexiveObjectProperty := 'ReflexiveObjectProperty' '(' { annotation } objectPropertyExpression ')'
irreflexiveObjectProperty := 'IrreflexiveObjectProperty' '(' { annotation } objectPropertyExpression ')'
symmetricObjectProperty := 'SymmetricObjectProperty' '(' { annotation } objectPropertyExpression ')'
asymmetricObjectProperty := 'AsymmetricObjectProperty' '(' { annotation } objectPropertyExpression ')'
transitiveObjectProperty := 'TransitiveObjectProperty' '(' { annotation } objectPropertyExpression ')'

Editor's Note: I'm unsure what to do with the unifying grammar profiles

objectPropertyAxiom :=
    subObjectPropertyOf | equivalentObjectProperties |
    disjointObjectProperties | inverseObjectProperties |
    objectPropertyDomain | objectPropertyRange |
    functionalObjectProperty | inverseFunctionalObjectProperty |
    reflexiveObjectProperty | irreflexiveObjectProperty |
    symmetricObjectProperty | asymmetricObjectProperty |
    transitiveObjectProperty

8.2.2 Example

8.3 Data Property Axioms

Editor's Note: See Issue-8 (dataproperty chains).

Data property axioms are similar to object property axioms, and are shown in Figure 14.

Data Property Axioms of OWL 2
Figure 14. Data Property Axioms of OWL 2

Note that the arity of the data range used in a dataPropertyRange axiom must be one. The axioms are described by the following grammar:

8.3.1 Syntax

subDataPropertyOf := 'SubDataPropertyOf' '(' { annotation } dataPropertyExpression dataPropertyExpression ')'
equivalentDataProperties := 'EquivalentDataProperties' '(' { annotation } dataPropertyExpression dataPropertyExpression { dataPropertyExpression } ')'
disjointDataProperties := 'DisjointDataProperties' '(' { annotation } dataPropertyExpression dataPropertyExpression { dataPropertyExpression } ')'
dataPropertyDomain := 'DataPropertyDomain' '(' { annotation } dataPropertyExpression description ')'
dataPropertyRange := 'DataPropertyRange' '(' { annotation } dataPropertyExpression dataRange ')'
functionalDataProperty := 'FunctionalDataProperty' '(' { annotation } dataPropertyExpression ')'

Editor's Note: I'm unsure what to do with the unifying grammar profiles

The following grammar production merges all productions for data property axioms:

dataPropertyAxiom :=
    subDataPropertyOf | equivalentDataProperties | disjointDataProperties |
    dataPropertyDomain | dataPropertyRange | functionalDataProperty

8.3.2 Example

8.4 Facts

Editor's Note: See Issue-3 (anonymous individuals) and Issue-23 (scoped-names).

OWL 2 supports a rich set of axioms for stating facts. Figure 15 shows the facts that can be stated about individuals and descriptions.

Facts
Figure 15. Facts

The sameIndividual axiom states that each of the individuals from a given set denotes the same object, whereas the differentIndividuals axiom states that each of the individuals from a given set denotes a different object. The classAssertion axiom states that the object denoted by the given individual is an instance of the given description.

The facts about object properties are shown in Figure 16.

Object Property Assertions
Figure 16. Object Property Assertions

The objectPropertyAssertion states that the objects denoted by the given individuals are connected by the given property, whereas the negativeObjectPropertyAssertion states that the objects denoted by the given individuals are not connected by the given property.

The structure of axioms asserting facts about data properties is similar and is shown in Figure 17.

Data Property Assertions
Figure 17. Data Property Assertions

The dataPropertyAssertion states that the value of a data property for an object denoted by the given individual is the given constant, whereas the negativeDataPropertyAssertion states the opposite.

8.4.1 Syntax

fact := sameIndividual | differentIndividuals | classAssertion |
    objectPropertyAssertion | negativeObjectPropertyAssertion |
    dataPropertyAssertion | negativeDataPropertyAssertion

sameIndividual := 'SameIndividual' '(' { annotation } individualURI individualURI { individualURI } ')'
differentIndividuals := 'DifferentIndividuals' '(' { annotation } individualURI individualURI { individualURI } ')'
classAssertion := 'ClassAssertion' '(' { annotation } individualURI description ')'

sourceIndividualURI := individualURI
targetIndividualURI := individualURI
objectPropertyAssertion := 'ObjectPropertyAssertion' '(' { annotation } objectPropertyExpression sourceIndividualURI targetIndividualURI ')'
negativeObjectPropertyAssertion := 'NegativeObjectPropertyAssertion' '(' { annotation } objectPropertyExpression sourceIndividualURI targetIndividualURI ')'

targetValue := constant
dataPropertyAssertion := 'DataPropertyAssertion' '(' { annotation } dataPropertyExpression sourceIndividualURI targetValue ')'
negativeDataPropertyAssertion := 'NegativeDataPropertyAssertion' '(' { annotation } dataPropertyExpression sourceIndividualURI targetValue ')'

8.4.2 Example

9 Annotations

Ontologies can contain information which does not affect the logical meaning of the ontology (or any part thereof). Some such information is entirely implicit from the OWL point of view: for example, a user may want to physically group axioms together which "go together" in some way; however, since an OWL ontology contains a set of axioms, this grouping has no logical significance and certain tools (including structure oriented OWL editors) may ignore it. Similarly, certain serializations may have a comment facility (e.g., RDF/XML and the OWL XML serializations both support XML comments) which is not preserved at the structural level. Users who use these serialization specific mechanisms must take care when manipulating their ontologies using structure oriented tools and, in general, such use is not recommended.

One could use OWL entities and axioms which are not intended to model the subject domain, but, instead, to provide extra-modeling information. For example, it is common to use the Dublin Core vocabulary to represent who originally coined ("created") a class. One could represent this information, using punning, at the logical level. However, this mixes domain modeling with ontology record keeping which can make the ontology more difficult to understand and use.

To solve the problems with these alternatives, OWL 2 provides an annotation system which allows users to associate information with entities, ontologies, and axioms in a structurally significant way which, nevertheless, does not affect the logical meaning of the ontology. For example, the axiom

SubClassOf( Human Animal )

is not structurally equivalent to the axiom

SubClassOf( Comment("Humans are a type of animals.") Human Animal)

even though the semantics of the two axioms are equivalent.

Annotations can be thought as a form of expression which may be inserted into axioms or appear in ontology structures. To associate an annotation expression with an entity, one uses a special kind of axiom. An annotation consists of an annotation property that specifies the type of annotation and a value for the annotation. OWL 2 allows for two kinds of annotation values, as shown in Figure 18.

  • Annotation values can be constants. Note that these need not be just strings; rather, any OWL 2 constant can be used. For example, one can create an annotation whose value is a URI formatted according to the XML Schema xsd:anyURI type specification.
  • Annotation values can be ontology entities. Such annotations make it clearer that the value is not just some constant, but an entity from this or some other ontology.

Annotations in OWL 2
Figure 18. Annotations in OWL 2

9.1 Syntax

explicitAnnotationByConstant := 'Annotation' '(' annotationPropertyURI constant ')'
annotationByConstant := explicitAnnotationByConstant | labelAnnotation | commentAnnotation
annotationByEntity := 'Annotation' '(' annotationPropertyURI entity ')'
annotation := annotationByConstant | annotationByEntity

9.2 Entity Annotations

Editor's Note: See Issue-16 (entity annotations).

Often, it is desirable to annotate entities in an ontology; such an annotation might, for example, specify a "human-friendly" label or comment. OWL 2 provides entity annotations for this purpose; their structure is shown in Figure 19.

Entity Annotations in OWL 2
Figure 19. Entity Annotations in OWL 2

Note that an entity annotation axiom provides for two types of annotation -- one for the axiom itself and one for the entity. It is important to distinguish these two types of annotation: the first one refers to the axiom (e.g., says who has asserted it), whereas the second one refers to the entity itself (e.g., provides a human-friendly label). The grammar for entity annotations is as follows:

annotationForAxiom := annotation
annotationForEntity := annotation
entityAnnotation := 'EntityAnnotation' '(' { annotationForAxiom } entity annotationForEntity { annotationForEntity } ')'

Note that the production for the entityAnnotation nonterminal requires an entity and not a URI. Thus, an OWL class should be annotated as follows:

EntityAnnotation(OWLClass(Person) Comment("The set of all humans."))

This is so that the type of the entity being annotated can easily be determined from the syntactic form of the entity annotation axiom.

9.3 Built-in Annotation Properties

Annotation properties with the common URIs rdfs:label and rdfs:comment are abbreviated as follows:

labelAnnotation := 'Label' '(' constant ')'
commentAnnotation := 'Comment' '(' constant ')'

OWL 2 provides several well-known ontology annotations. These have no meaning in the model-theoretic semantics; however, they may be used by software to manage ontology version.

  • An owl:priorVersion annotation can have an individual as a value. The URI of this individual points to the prior version of the containing ontology.
  • An owl:backwardCompatibleWith annotation can have an individual as a value. The URI of this individual points to the prior version of the containing ontology, and that prior version is compatible with this ontology.
  • An owl:incompatibleWith annotation can have an individual as a value. The URI of this individual points to the prior version of the containing ontology, and that prior version is incompatible with this ontology.

10 Nonstructural Restrictions on Axioms

As explained in [SROIQ], to obtain a decidable language, the axiom closure Ax of each OWL 2 ontology must obey certain nonstructural restrictions, as defined next. In this section, we assume that all object property expressions in Ax are either object properties or inverses of an object property. This can be ensured by replacing all expressions InverseObjectProperty( InverseObjectProperty(PE) ) with PE.

For an object property expression PE, the inverse property expression INV(PE) is defined as follows:

  • if PE is an object property OP, then INV(PE) = InverseObjectProperty(OP);
  • if PE is of the form InverseObjectProperty(OP) for OP an object property, then INV(PE) = OP.

An object property expression PE is composite in Ax if Ax contains an axiom of the form

  • SubObjectPropertyOf(SubObjectPropertyChain(PE1 ... PEn) PE) with n > 1, or
  • SubObjectPropertyOf(SubObjectPropertyChain(PE1 ... PEn) INV(PE)) with n > 1, or
  • TransitiveObjectProperty(PE), or
  • TransitiveObjectProperty(INV(PE)).

The object property hierarchy relation → is the smallest relation on object property expressions for which the following conditions hold (AB means that → holds for A and B):

  • if Ax contains an axiom SubObjectPropertyOf(PE1 PE2), then PE1PE2 holds; and
  • if Ax contains an axiom EquivalentObjectProperties(PE1 PE2), then PE1PE2 and PE2PE1 hold; and
  • if Ax contains an axiom InverseObjectProperties(PE1 PE2), then PE1INV(PE2) and INV(PE2)PE1 hold; and
  • if Ax contains an axiom SymmetricObjectProperty(PE), then PEINV(PE) holds; and
  • if PE1PE2 holds, then INV(PE1)INV(PE2) holds as well.

The relation →* is the reflexive-transitive closure of →. An object property expression PE is simple in Ax if, for each object property expression PE' such that PE'* PE holds, PE' is not composite.

The axioms Ax satisfy the nonstructural restrictions of OWL 2 if the following three conditions hold:

  • Each class in Ax of the following form contains only simple object properties:
    • ObjectMinCardinality, ObjectMaxCardinality, ObjectExactCardinality, and ObjectExistsSelf.
  • Each axiom in Ax of the following form contains only simple object properties:
    • FunctionalObjectProperty, InverseFunctionalObjectProperty, IrreflexiveObjectProperty, AsymmetricObjectProperty, and DisjointObjectProperties.
  • A strict partial order < on the object property expressions must exist that fulfills the following conditions:
    • If x < y holds, then y →* x does not hold;
    • Each axiom in Ax of the form SubObjectPropertyOf(SUB PE) where SUB is of the form SubObjectPropertyChain(PE1 ... PEn) with n ≥ 2 fulfils the following conditions:
      • n = 2 and PE1 = PE2 = PE, or
      • PEi < PE for each 1 ≤ i ≤ n, or
      • PE1 = PE and PEi < PE for each 2 ≤ i ≤ n, or
      • PEn = PE and PEi < PE for each 1 ≤ i ≤ n-1.

11 Declarations and Structural Consistency

Editor's Note: See Issue-19 (declarations-p) and Issue-20 (annotate-declarations?).

In addition to axioms, OWL 2 also provides declarations, which can be used to check the structural consistency of an ontology. In OWL 2, it is possible to use a URI for an entity without explicitly stating that an entity with such a name exists. For example, the following is a correct OWL 2 ontology:

Ontology(<http://www.my.domain.com/example>
     SubClassOf( Human Animal )
)

In this example, the classes Human and Animal are used without explicitly stating that they exist. This is desirable in many situations, since it allows "anyone to say anything about anything," which is consistent with the vision of the Semantic Web. This approach can, however, make it difficult to detect structural errors in ontologies; a typographical error in a class name, for example, simply introduces a new class with a different name. This issue can be addressed by using declarations to check the structural integrity of an ontology. We say that an entity is declared in an ontology O if either O or some ontology O' imported by O contains a declaration axiom for the entity. Note that an ontology may contain multiple declarations for the same entity, and that imported ontologies may do so as well. An ontology O is structurally consistent if each entity occurring in an axiom from the axiom closure of O is declared in O. Note that, by this definition, the check applies also the entities used in annotations; that is, these entities must be properly declared as well for an ontology to be structurally consistent. Note that, although structurally consistency can be a very useful feature in some applications, OWL 2 does not require ontologies to be structurally consistent. Thus, an ontology can be used even if it does not contain any declarations. As shown in Figure 20, a declaration is a kind of axiom; this is in order to allow an ontology to be treated as simply a set axioms. A declaration, however, imposes no constraints on the model-theoretic interpretation of an ontology (in the sense of [OWL 2 Semantics]).

Entity Declarations in OWL 2
Figure 20. Entity Declarations in OWL 2

The grammar for declarations is as follows:

declaration := 'Declaration' '(' { annotation } entity ')'

Note that the production for the declaration nonterminal requires an entity and not a URI. Thus, an OWL class should be declared as follows:

Declaration(OWLClass(Person))

This is so that the type of the entity being declared can easily be determined from the syntactic form of the declaration.

12 Appendix: Differences from OWL 1 Abstract Syntax

OWL 2 departs in its conceptual design and in syntax from OWL 1 Abstract Syntax. In this section we summarize the differences and explain the rationale behind the changes.

12.1 Dropping the Frame-Like Syntax

OWL 1 provides a frame-like syntax that allows several aspects of a class, property or individual to be defined in a single axiom. For example, one can write the following axiom in OWL 1:

ObjectProperty(partOf inverseOf(containedIn) inverseFunctional transitive
     Label("Specifies that an object is a part of another object.")
)

This type of axiom may cause problems in practice. In the first place, it bundles many different features of the given object into a single axiom. While this may be convenient when ontologies are being manipulated by hand, it is not convenient for manipulating them programmatically. In fact, most implementations of OWL 1 break such axioms apart into several "atomic" axioms, each dealing with only a single feature of the object. However, this may cause problems with round-tripping, as the structure of the ontology may be destroyed in the process. In the second place, this type of axiom is often misinterpreted as a declaration and unique "definition" of the given object. In OWL 1, however, objects may be used without being the subject of any such axiom, and there may be many such axioms relating to the same object. Finally, OWL 1 does not provide means to annotate axioms, which has proved to be quite useful in practice. These problems are addressed in OWL 2 in several ways. First, the frame-like notation has been dropped in favor of a more fine-grained structure of axioms, where each axiom describes just one feature of the given object. Second, OWL 2 provides explicit declarations, and an explicit definition of the notion of structural consistency. Finally, all axioms in OWL 2 can be annotated, and entity annotation axioms provide means for that. For example, the above mentioned OWL 1 axiom can be represented in OWL 2 as follows:

Declaration(ObjectProperty(partOf))
EntityAnnotation(ObjectProperty(partOf)
     Label("Specifies that an object is a part of another object.")
)
EquivalentObjectProperties(partOf InverseObjectProperty(containedIn))
InverseFunctionalObjectProperty(partOf)
TransitiveObjectProperty(Comment("The partOf property is transitive.") partOf)

Although OWL 2 is more verbose, this should not be a problem given that most OWL ontologies are created using ontology engineering tools. Moreover, such tools are free to present the information to the user in a more intuitive (possibly frame-like) way.

12.2 Inverse Property Expressions

In OWL 1, all properties are atomic, but it is possible to assert that one object property is the inverse of another. For example, one can assert the following axiom in OWL 1:

ObjectProperty(hasPart inverse isPartOf)

In OWL 2, property expressions such as InverseObjectProperty(hasPart) can be used in class expressions, which avoids the need to give a name to every inverse property. If desired, however, names can still be given to inverse properties. For example, the following OWL 2 axiom asserts that isPartOf is the inverse of hasPart, and is thus equivalent to the mentioned OWL 1 axiom:

EquivalentObjectProperties(InverseObjectProperty(hasPart) isPartOf)

12.3 Separating the Vocabulary for Names

Editor's Note: See Issue-17 (role punning), Issue-18 (property typing), Issue-65 (excess vocab) and Issue-69 (punning).

OWL 1 relies on the separation of names between concepts, individuals, and object and data properties, but the type of a given name may not be obvious from the context in which it is used. Consider, for example, the following OWL 1 axiom:

SubClassOf(Lion restriction(eats someValuesFrom(Animal)))

From this axiom alone, it is not clear how to interpret the class restriction(eats someValuesFrom Animal): one possibility is to treat eats as an object property and Animal as a class, and the other one is to treat eats as a data property and Animal as a datatype. Examining other axioms in the ontology (or in an imported ontology) may make it possible to disambiguate this one, but this is not required in OWL 1: an ontology containing only the mentioned axiom is a correct OWL 1 ontology in which the intended usage of eats and Animal cannot be disambiguated. Hence, parsing an OWL 1 ontology requires not only two passes through the ontology being parsed, but also possibly parsing imported ontologies as well; this makes parser implementation complex and error prone. This problem is addressed in OWL 2 by explicitly typing each usage of a name. For example, the above mentioned axiom can be represented in OWL 2 as follows:

SubClassOf(Lion ObjectSomeValuesFrom(eats Animal))

This makes it completely clear that eats is intended to be interpreted as an object property and Animal as a class.


13 Index

Syntax Category
annotation
asymmetricObjectProperty Axiom
axiom Ontology
classAssertion Axiom
classAxiom Ontology
constant
dataAllValuesFrom Description
dataComplementOf Description
dataExactCardinality Description
dataHasValue Description
dataMaxCardinality Description
dataMinCardinality Description
dataOneOf Description
dataPropertyAssertion Axiom
dataPropertyAxiom Ontology
dataPropertyDomain Axiom
dataPropertyExpression
dataPropertyRange Axiom
dataSomeValuesFrom Description
dataRange Datarange
datatypeFacet Datarange
datatypeRestriction Datarange
declaration Axiom
description Description
differentIndividuals Axiom
disjointClasses Axiom
disjointDataProperties Axiom
disjointObjectProperties Axiom
disjointUnion Axiom
entity
entityAnnotation Axiom
equivalentClasses Axiom
equivalentDataProperties Axiom
equivalentObjectProperties Axiom
fact Ontology
functionalDataProperty Axiom
functionalObjectProperty Axiom
inverseFunctionalObjectProperty Axiom
inverseObjectProperties Axiom
inverseObjectProperty
irreflexiveObjectProperty Axiom
Namespace Ontology
negativeDataPropertyAssertion Axiom
negativeObjectPropertyAssertion Axiom
objectAllValuesFrom Description
objectComplementOf Description
objectExactCardinality Description
objectExistsSelf Description
objectHasValue Description
objectIntersectionOf Description
objectMaxCardinality Description
objectMinCardinality Description
objectOneOf Description
objectPropertyAssertion Axiom
objectPropertyAxiom Ontology
objectPropertyDomain Axiom
objectPropertyExpression
objectPropertyRange Axiom
objectSomeValuesFrom Description
objectUnionOf Description
reflexiveObjectProperty Axiom
sameIndividual Axiom
subClassOf Axiom
subDataPropertyOf Axiom
subObjectPropertyOf Axiom
symmetricObjectProperty Axiom
transitiveObjectProperty Axiom
ontology Ontology
URI

14 References

[OWL 2 Semantics]
OWL 2 Web Ontology Language: Model-Theoretic Semantics. Bernardo Cuenca Grau and Boris Motik, eds., 2006.
[SROIQ]
The Even More Irresistible SROIQ. Ian Horrocks, Oliver Kutz, and Uli Sattler. In Proc. of the 10th Int. Conf. on Principles of Knowledge Representation and Reasoning (KR 2006). AAAI Press, 2006.
[XML Namespaces]
Namespaces in XML 1.0 (Second Edition). Tim Bray, Dave Hollander, Andrew Layman, and Richard Tobin, eds. W3C Recommendation 16 August 2006.
[XML Schema Datatypes]
XML Schema Part 2: Datatypes Second Edition. Paul V. Biron and Ashok Malhotra, eds. W3C Recommendation 28 October 2004.
[RDF Syntax]
RDF/XML Syntax Specification (Revised). Dave Beckett, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/rdf-syntax-grammar/.
[RFC-4646]
RFC 4646 - Tags for Identifying Languages. M. Phillips and A. Davis. IETF, September 2006, http://www.ietf.org/rfc/rfc4646.txt. Latest version is available as BCP 47, (details) .
[RFC-3987]
RFC 3987 - Internationalized Resource Identifiers (IRIs). M. Duerst, M. Suignard. IETF, January 2005, http://www.ietf.org/rfc/rfc3987.txt.
[OWL Semantics and Abstract Syntax]
OWL Web Ontology Language: Semantics and Abstract Syntax Peter F. Patel-Schneider, Patrick Hayes, and Ian Horrocks, eds. W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-semantics-20040210/. Latest version available at http://www.w3.org/TR/owl-semantics/.
[CURIE]
CURIE Syntax 1.0: A syntax for expressing Compact URIs. M. Birbeck, S. McCarron, Editors, W3C Working Draft, 26 November 2007, http://www.w3.org/TR/2007/WD-curie-20071126/.
[RDF Test Cases]
RDF Test Cases. Jan Grant and Dave Beckett, Editors, W3C Recommendation 10 February 2004, http://www.w3.org/TR/rdf-testcases/.