OWL 2 Web Ontology Language
RDF-Based Semantics

W3C Recommendation 27 October 2009

This version:

http://www.w3.org/TR/2009/REC-owl2-rdf-based-semantics-20091027/

Latest version (series 2):

http://www.w3.org/TR/owl2-rdf-based-semantics/

Latest Recommendation:

http://www.w3.org/TR/owl-rdf-based-semantics

Previous version:

http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/ (color-coded diff)

Editors:

Michael Schneider, FZI Research Center for Information Technology

Contributors: (in alphabetical order)

Jeremy Carroll, HP (now at TopQuadrant)

Ivan Herman, W3C/ERCIM

Peter F. Patel-Schneider, Bell Labs Research, Alcatel-Lucent

Please refer to the errata for this document, which may include some normative corrections.

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

See also translations.

Copyright © 2009 W3C^® (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.

Abstract

Status of this Document

May Be Superseded

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

Summary of Changes

Please Send Comments

Please send any comments to public-owl-comments@w3.org (public archive). Although work on this document by the OWL Working Group is complete, comments may be addressed in the errata or in future revisions. Open discussion among developers is welcome at public-owl-dev@w3.org (public archive).

Endorsed By W3C

This document has been reviewed by W3C Members, by software developers, and by other W3C groups and interested parties, and is endorsed by the Director as a W3C Recommendation. It is a stable document and may be used as reference material or cited from another document. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.

Patents

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

1 Introduction (Informative)

This document defines the RDF-compatible model-theoretic semantics of OWL 2, referred to as the "OWL 2 RDF-Based Semantics". The OWL 2 RDF-Based Semantics gives a formal meaning to every RDF graph [RDF Concepts] and is fully compatible with the RDF Semantics specification [RDF Semantics]. The specification provided here is the successor to the original OWL 1 RDF-Compatible Semantics specification [OWL 1 RDF-Compatible Semantics].

Technically, the OWL 2 RDF-Based Semantics is defined as a semantic extension of "D-Entailment" (RDFS with datatype support), as specified in the RDF Semantics [RDF Semantics]. In other words, the meaning given to an RDF graph by the OWL 2 RDF-Based Semantics includes the meaning provided by the semantics of RDFS with datatypes, and additional meaning is specified for all the language constructs of OWL 2, such as Boolean connectives, sub property chains and qualified cardinality restrictions (see the OWL 2 Structural Specification [OWL 2 Specification] for further information on all the language constructs of OWL 2). The definition of the semantics for the extra constructs follows the design principles as applied to the RDF Semantics.

The content of this document is not meant to be self-contained but builds on top of the RDF Semantics document [RDF Semantics] by adding those aspects that are specific to OWL 2. Hence, the complete definition of the OWL 2 RDF-Based Semantics is given by the combination of both the RDF Semantics document and the document at hand. In particular, the terminology used in the RDF Semantics is reused here except for cases where a conflict exists with the rest of the OWL 2 specification.

The remainder of this section provides an overview of some of the distinguishing features of the OWL 2 RDF-Based Semantics and outlines the document's structure and content.

In Section 2, the syntax over which the OWL 2 RDF-Based Semantics is defined is the set of all RDF graphs [RDF Concepts]. The OWL 2 RDF-Based Semantics provides a precise formal meaning for every RDF graph. The language that is determined by RDF graphs being interpreted using the OWL 2 RDF-Based Semantics is called "OWL 2 Full". In this document, RDF graphs are also called "OWL 2 Full ontologies", or simply "ontologies", unless there is risk of confusion.

The OWL 2 RDF-Based Semantics interprets the RDF and RDFS vocabularies [RDF Semantics] and the OWL 2 RDF-Based vocabulary together with an extended set of datatypes and their constraining facets (see Section 3).

OWL 2 RDF-Based interpretations (Section 4) are defined on a universe (see Section 1.3 of the RDF Semantics specification [RDF Semantics] for an overview of the basic intuition of model-theoretic semantics). The universe is divided into parts, namely individuals, classes, and properties, which are identified with their RDF counterparts (see Figure 1). The part of individuals equals the whole universe. This means that all classes and properties are also individuals in their own right. Further, every name interpreted by an OWL 2 RDF-Based interpretation denotes an individual.

The three basic parts are divided into further parts as follows. The part of individuals subsumes the part of data values, which comprises the denotations of all literals. Also subsumed by the individuals is the part of ontologies. The part of classes subsumes the part of datatypes, which are classes consisting entirely of data values. Finally, the part of properties subsumes the parts of object properties, data properties, ontology properties and annotation properties. The part of object properties equals the whole part of properties, and therefore all other kinds of properties are also object properties.

For annotations properties note that annotations are not "semantic-free" under the OWL 2 RDF-Based Semantics. Just like every other triple or set of triples occurring in an RDF graph, an annotation is assigned a truth value by any given OWL 2 RDF-Based interpretation. Hence, although annotations are meant to be "semantically weak", i.e., their formal meaning does not significantly exceed that originating from the RDF Semantics specification, adding an annotation may still change the meaning of an ontology. A similar discussion holds for statements that are built from ontology properties, such as owl:imports, which are used to define relationships between two ontologies.

Every class represents a specific set of individuals, called the class extension of the class: an individual a is an instance of a class C, if a is a member of the class extension ICEXT(C). Since a class is itself an individual under the OWL 2 RDF-Based Semantics, classes are distinguished from their respective class extensions. This distinction allows, for example, that a class may be an instance of itself by being a member of its own class extension. Also, two classes may be equivalent by sharing the same class extension, although being different individuals, e.g., they do not need to share the same properties. Similarly, every property has an associated property extension that consists of pairs of individuals: an individual a₁ has a relationship to an individual a₂ with respect to a property p if the pair ( a₁ , a₂ ) is a member of the property extension IEXT(p). Again, properties are distinguished from their property extensions. In general, if there are no further constraints, an arbitrary extension may be associated with a given class or property, and two interpretations may associate distinct extensions with the same class or property.

Individuals may play different "roles". For example, an individual can be both a data property and an annotation property, since the different parts of the universe of an OWL 2 RDF-Based interpretation are not required to be mutually disjoint, or an individual can be both a class and a property by associating both a class extension and a property extension with it. In the latter case there will be no specific relationship between the class extension and the property extension of such an individual without further constraints. For example, the same individual can have an empty class extension while having a nonempty property extension.

The main part of the OWL 2 RDF-Based Semantics is Section 5, which specifies a formal meaning for all the OWL 2 language constructs by means of the OWL 2 RDF-Based semantic conditions. These semantic conditions extend all the semantic conditions given in the RDF Semantics [RDF Semantics]. The OWL 2 RDF-Based semantic conditions effectively determine which sets of RDF triples are assigned a specific meaning and what this meaning is. For example, semantic conditions exist that allow one to interpret the triple "C owl:disjointWith D" to mean that the denotations of the IRIs C and D have disjoint class extensions.

There is usually no need to provide localizing information (e.g., by means of "typing triples") for the IRIs occurring in an ontology. As for the RDF Semantics, the OWL 2 RDF-Based semantic conditions have been designed to ensure that the denotation of any IRI will be in the appropriate part of the universe. For example, the RDF triple "C owl:disjointWith D" is sufficient to deduce that the denotations of the IRIs C and D are actually classes. It is not necessary to explicitly add additional typing triples "C rdf:type rdfs:Class" and "D rdf:type rdfs:Class" to the ontology.

In the RDF Semantics, this kind of "automatic localization" was to some extent achieved by so called "axiomatic triples" [RDF Semantics], such as "rdf:type rdf:type rdf:Property" or "rdf:type rdfs:domain rdfs:Resource". However, there is no explicit normative collection of additional axiomatic triples for the OWL 2 RDF-Based Semantics; instead, the specific axiomatic aspects of the OWL 2 RDF-Based Semantics are determined by a subset of the OWL 2 RDF-Based semantic conditions. Section 6 discusses axiomatic triples in general and provides an example set of axiomatic triples that is compatible with the OWL 2 RDF-Based Semantics.

Section 7 compares the OWL 2 RDF-Based Semantics with the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. While the OWL 2 RDF-Based Semantics is based on the RDF Semantics specification [RDF Semantics], the OWL 2 Direct Semantics is a description logic style semantics. Several fundamental differences exist between the two semantics, but there is also a strong relationship basically stating that the OWL 2 RDF-Based Semantics is able to reflect all logical conclusions of the OWL 2 Direct Semantics. This means that the OWL 2 Direct Semantics can in a sense be regarded as a semantics subset of the OWL 2 RDF-Based Semantics. The precise relationship is given by the OWL 2 correspondence theorem.

Significant effort has been spent in keeping the design of the OWL 2 RDF-Based Semantics as close as possible to that of the original specification of the OWL 1 RDF-Compatible Semantics [OWL 1 RDF-Compatible Semantics]. While this aim was achieved to a large degree, the OWL 2 RDF-Based Semantics actually deviates from its predecessor in several aspects. In most cases, this is because of serious technical problems that would have arisen from a conservative semantic extension. One important change is that while so called "comprehension conditions" for the OWL 2 RDF-Based Semantics (see Section 8) still exist, these are not part of the normative set of semantic conditions anymore. The OWL 2 RDF-Based Semantics also corrects several errors of OWL 1. A list of differences between the two languages is given in Section 9.

The italicized keywords MUST, MUST NOT, SHOULD, SHOULD NOT, and MAY are used to specify normative features of OWL 2 documents and tools, and are interpreted as specified in RFC 2119 [RFC 2119].

2 Ontologies

This section determines the syntax for the OWL 2 RDF-Based Semantics, and gives an overview on typical content of ontologies for ontology management tasks.

2.1 Syntax

Following Sections 0.2 and 0.3 of the RDF Semantics specification [RDF Semantics], the OWL 2 RDF-Based Semantics is defined on every RDF graph (Section 6.2 of RDF Concepts [RDF Concepts]), i.e. on every set of RDF triples (Section 6.1 of RDF Concepts [RDF Concepts]).

In accordance with the rest of the OWL 2 specification (see Section 2.4 of the OWL 2 Structural Specification [OWL 2 Specification]), this document uses an extended notion of an RDF graph by allowing the RDF triples in an RDF graph to contain arbitrary IRIs ("Internationalized Resource Identifiers") according to RFC 3987 [RFC 3987]. In contrast, the RDF Semantics specification [RDF Semantics] is defined on RDF graphs containing URIs [RFC 2396]. This change is backward compatible with the RDF specification, since URIs are also IRIs.

Terminological note: The document at hand uses the term "IRI" in accordance with the rest of the OWL 2 specification (see Section 2.4 of the OWL 2 Structural Specification [OWL 2 Specification]), whereas the RDF Semantics specification [RDF Semantics] uses the term "URI reference". According to RFC 3987 [RFC 3987], the term "IRI" stands for an absolute resource identifier with optional fragment, which is what is being used throughout this document. In contrast, the term "IRI reference" additionally covers relative references, which are never used in this document.

Convention: In this document, IRIs are abbreviated in the way defined by Section 2.4 of the OWL 2 Structural Specification [OWL 2 Specification], i.e., the abbreviations consist of a prefix name and a local part, such as "prefix:localpart".

The definition of an RDF triple according to Section 6.1 of RDF Concepts [RDF Concepts] is restricted to cases where the subject of an RDF triple is an IRI or a blank node (Section 6.6 of RDF Concepts [RDF Concepts]), and where the predicate of an RDF triple is an IRI. As a consequence, the definition does not treat cases, where, for example, the subject of a triple is a literal (Section 6.5 of RDF Concepts [RDF Concepts]), as in "s" ex:p ex:o, or where the predicate of a triple is a blank node, as in ex:s _:p ex:o. In order to allow for interoperability with other existing and future technologies and tools, the document at hand does not explicitly forbid the use of generalized RDF graphs consisting of generalized RDF triples, which are defined to allow for IRIs, literals and blank nodes to occur in the subject, predicate and object position. Thus, an RDF graph MAY contain generalized RDF triples, but an implementation is not required to support generalized RDF graphs. Note that every RDF graph consisting entirely of RDF triples according to Section 6.1 of RDF Concepts [RDF Concepts] is also a generalized RDF graph.

Terminological notes: The term "OWL 2 Full" refers to the language that is determined by the set of all RDF graphs being interpreted using the OWL 2 RDF-Based Semantics. Further, in this document the term "OWL 2 Full ontology" (or simply "ontology", unless there is any risk of confusion) will be used interchangeably with the term "RDF graph".

2.2 Content of Ontologies (Informative)

While there do not exist any syntactic restrictions on the set of RDF graphs that can be interpreted by the OWL 2 RDF-Based Semantics, in practice an ontology will often contain certain kinds of constructs that are aimed to support ontology management tasks. Examples are ontology headers and ontology IRIs, as well as constructs that are about versioning, importing and annotating of ontologies, including the concept of incompatibility between ontologies.

These topics are outside the scope of this semantics specification. Section 3 of the OWL 2 Structural Specification [OWL 2 Specification] deals with these topics in detail, and can therefore be used as a guide on how to apply these constructs in OWL 2 Full ontologies accordingly. The mappings of all these constructs to their respective RDF encoding are defined in the OWL 2 RDF Mapping [OWL 2 RDF Mapping].

3 Vocabulary

This section specifies the OWL 2 RDF-Based vocabulary, and lists the names of the datatypes and facets used under the OWL 2 RDF-Based Semantics.

3.1 Standard Prefixes

Table 3.1 lists the standard prefix names and their prefix IRIs used in this document.

3.2 Vocabulary Terms

Table 3.2 lists the IRIs of the OWL 2 RDF-Based vocabulary, which is the set of vocabulary terms that are specific for the OWL 2 RDF-Based Semantics. This vocabulary extends the RDF and RDFS vocabularies as specified in Sections 3.1 and 4.1 of the RDF Semantics [RDF Semantics], respectively. Table 3.2 does not mention those IRIs that will be listed in Section 3.3 on datatype names or Section 3.4 on facet names.

Implementations are not required to support the IRI owl:onProperties, but MAY support it in order to realize n-ary dataranges with arity ≥ 2 (see Sections 7 and 8.4 of the OWL 2 Structural Specification [OWL 2 Specification] for further information).

Note: The use of the IRI owl:DataRange has been deprecated as of OWL 2. The IRI rdfs:Datatype SHOULD be used instead.

3.3 Datatype Names

Table 3.3 lists the IRIs of the datatypes used in the OWL 2 RDF-Based Semantics. The datatype rdf:XMLLiteral is described in Section 3.1 of the RDF Semantics [RDF Semantics]. All other datatypes are described in Section 4 of the OWL 2 Structural Specification [OWL 2 Specification]. The normative set of datatypes of the OWL 2 RDF-Based Semantics equals the set of datatypes described in Section 4 of the OWL 2 Structural Specification [OWL 2 Specification].

3.4 Facet Names

Table 3.4 lists the IRIs of the facets used in the OWL 2 RDF-Based Semantics. Each datatype listed in Section 3.3 has a (possibly empty) set of constraining facets. All facets are described in Section 4 of the OWL 2 Structural Specification [OWL 2 Specification] in the context of their respective datatypes. The normative set of facets of the OWL 2 RDF-Based Semantics equals the set of facets described in Section 4 of the OWL 2 Structural Specification [OWL 2 Specification].

In this specification, facets are used for defining datatype restrictions (see Section 5.7). For example, to refer to the set of all strings of length 5 one can restrict the datatype xsd:string (Section 3.3) by the facet xsd:length and the value 5.

4 Interpretations

The OWL 2 RDF-Based Semantics provides vocabulary interpretations and vocabulary entailment (see Section 2.1 of the RDF Semantics [RDF Semantics]) for the RDF and RDFS vocabularies and for the OWL 2 RDF-Based vocabulary. This section defines OWL 2 RDF-Based datatype maps and OWL 2 RDF-Based interpretations, and specifies what satisfaction of ontologies, consistency and entailment means under the OWL 2 RDF-Based Semantics. In addition, the so called "parts" of the universe of an OWL 2 RDF-Based interpretation are defined.

4.1 Datatype Maps

According to Section 5.1 of the RDF Semantics specification [RDF Semantics], a datatype d has the following components:

• LS(d), the lexical space of d, which is a set of lexical forms;
• VS(d), the value space of d, which is a set of data values;
• L2V(d), the lexical-to-value mapping of d, which maps lexical forms in LS(d) to data values in VS(d).

Terminological notes: The document at hand uses the term "data value" in accordance with the rest of the OWL 2 specification (see Section 4 of the OWL 2 Structural Specification [OWL 2 Specification]), whereas the RDF Semantics specification [RDF Semantics] uses the term "datatype value" instead. Further, the names "LS" and "VS", which stand for the lexical space and the value space of a datatype, respectively, are not used in the RDF Semantics specification, but have been introduced here for easier reference.

In this document, the basic definition of a datatype is extended to take facets into account. See Section 3.4 for information and an example on facets. Note that Section 5.1 of the RDF Semantics specification [RDF Semantics] explicitly permits that semantic extensions may impose more elaborate datatyping conditions than those listed above.

A datatype with facets d is a datatype that has the following additional components:

• FS(d), the facet space of d, which is a set of pairs of the form ( F , v ), where F is an IRI called the constraining facet and v is an arbitrary data value called the constraining value;
• F2V(d), the facet-to-value mapping of d, which maps each facet-value pair ( F , v ) in FS(d) to a subset of VS(d).

Note that it is not further specified what the nature of the denotation of a facet IRI is, i.e. it is only known that a facet IRI denotes some individual. Semantic extensions MAY impose further restrictions on the denotations of facets. In fact, Section 5.3 will define additional restrictions on facets.

Also note that for a datatype d and a facet-value pair ( F , v ) in FS(d) the value v is not required to be included in the value space VS(d) of d itself. For example, the datatype xsd:string (Section 3.3) has the facet xsd:length (Section 3.4), which takes nonnegative integers as its constraining values rather than strings.

In this document, it will always be assumed from now on that any datatype d is a datatype with facets. If the facet space FS(d) of a datatype d has not been explicitly defined, or if it is not derived from another datatype's facet space according to some well defined condition, then FS(d) is the empty set. Unless there is any risk of confusion, the term "datatype" will always refer to a datatype with facets.

Section 5.1 of the RDF Semantics specification [RDF Semantics] further defines a datatype map D to be a set of name-datatype pairs ( u , d ) consisting of an IRI u and a datatype d, such that no IRI appears twice in the set. As a consequence of what has been said before, in this document every datatype map D will entirely consist of datatypes with facets.

The following definition specifies what an OWL 2 RDF-Based datatype map is.

Note that Definition 4.1 does not prevent additional datatypes to be in an OWL 2 RDF-Based datatype map. For the special case of an OWL 2 RDF-Based datatype map D that exclusively contains the datatypes listed in Section 3.3, it is ensured that there are datatypes available for all the facet values, i.e., for every name-datatype pair ( u , d ) in D and for every facet-value pair ( F , v ) in FS(d) there exists a name-datatype pair ( u^* , d^* ) in D such that v is in VS(d^*).

4.2 Vocabulary Interpretations

From the RDF Semantics specification [RDF Semantics], let V be a set of literals and IRIs containing the RDF and RDFS vocabularies, and let D be a datatype map according to Section 5.1 of the RDF Semantics [RDF Semantics] (and accordingly Section 4.1). A D-interpretation I of V with respect to D is a tuple

IR is the universe of I, i.e., a nonempty set that contains at least the denotations of literals and IRIs in V. IP is a subset of IR, the properties of I. LV, the data values of I, is a subset of IR that contains at least the set of plain literals (see Section 6.5 of RDF Concepts [RDF Concepts]) in V, and the value spaces of each datatype of D. IEXT is used to associate properties with their property extension, and is a mapping from IP to the powerset of IR × IR. IS is a mapping from IRIs in V to their denotations in IR. In particular, IS(u) = d for any name-datatype pair ( u , d ) in D. IL is a mapping from typed literals "s"^^u in V to their denotations in IR, where IL("s"^^u) = L2V(d)(s), provided that d is a datatype of D, IS(u) = d, and s is in the lexical space LS(d); otherwise IL("s"^^u) is not in LV.

Convention: Following the practice introduced in Section 1.4 of the RDF Semantics [RDF Semantics], for a given interpretation I of a vocabulary V the notation "I(x)" will be used instead of "IL(x)" and "IS(x)" for the typed literals and IRIs x in V, respectively.

As detailed in the RDF Semantics [RDF Semantics], a D-interpretation has to meet all the semantic conditions for ground graphs and blank nodes, those for RDF interpretations and RDFS interpretations, and the "general semantic conditions for datatypes".

In this document, the basic definition of a D-interpretation is extended to take facets into account.

A D-interpretation with facets I is a D-interpretation for a datatype map D consisting entirely of datatypes with facets (Section 4.1), where I meets the following additional semantic conditions: for each name-datatype pair ( u , d ) in D and each facet-value pair ( F , v ) in the facet space FS(d)

• F is in the vocabulary V of I;
• a name-datatype pair ( u^* , d^* ) exists in D, such that v is in the value space VS(d^*).

In this document, it will always be assumed from now on that any D-interpretation I is a D-interpretation with facets. Unless there is any risk of confusion, the term "D-interpretation" will always refer to a D-interpretation with facets.

The following definition specifies what an OWL 2 RDF-Based interpretation is.

4.3 Satisfaction, Consistency and Entailment

The following definitions specify what it means for an RDF graph to be satisfied by a given OWL 2 RDF-Based interpretation, to be consistent under the OWL 2 RDF-Based Semantics, and to entail another RDF graph.

The notion of satisfaction under the OWL 2 RDF-Based Semantics is based on the notion of satisfaction for D-interpretations and Simple interpretations, as defined in the RDF Semantics [RDF Semantics]. In essence, in order to satisfy an RDF graph, an interpretation I has to satisfy all the triples in the graph, i.e., for a triple "s p o" it is necessary that the relationship ( I(s) , I(o) ) ∈ IEXT(I(p)) holds (special treatment exists for blank nodes, as detailed in Section 1.5 of the RDF Semantics [RDF Semantics]). In other words, the given graph has to be compatible with the specific form of the IEXT mapping of I. The distinguishing aspect of OWL 2 RDF-Based satisfaction is that an interpretation I needs to meet all the OWL 2 RDF-Based semantic conditions (see Section 5), which have a constraining effect on the possible forms an IEXT mapping can have.

4.4 Parts of the Universe

Table 4.1 defines the "parts" of the universe of a given OWL 2 RDF-Based interpretation I.

The second column tells the name of the part. The third column gives a definition of the part in terms of the mapping IEXT of I, and by referring to a particular term of the RDF, RDFS or OWL 2 RDF-Based vocabulary.

As an example, the part of all datatypes is named "IDC", and it is defined as the set of all individuals x for which the relationship "( x , I(rdfs:Datatype) ) ∈ IEXT(I(rdf:type))" holds. According to the semantics of rdf:type, as defined in Section 4.1 of the RDF Semantics [RDF Semantics], this means that the name "IDC" denotes the class extension (see Section 4.5) of I(rdfs:Datatype).

4.5 Class Extensions

The mapping ICEXT from IC to the powerset of IR, which associates classes with their class extension, is defined for every c ∈ IC as

5 Semantic Conditions

This section defines the semantic conditions of the OWL 2 RDF-Based Semantics. The semantic conditions presented here are basically only those for the specific constructs of OWL 2. The complete set of semantic conditions for the OWL 2 RDF-Based Semantics is the combination of the semantic conditions presented here and the semantic conditions for Simple Entailment, RDF, RDFS and D-Entailment, as specified in the RDF Semantics specification [RDF Semantics].

All semantic conditions in this section are defined with respect to an interpretation I. Section 5.1 specifies semantic conditions for the different parts of the universe of the interpretation being considered (compare Section 4.4). Section 5.2 and Section 5.3 list semantic conditions for the classes and the properties of the OWL 2 RDF-Based vocabulary. In the rest of this section, the OWL 2 RDF-Based semantic conditions for the different language constructs of OWL 2 are specified.

Conventions used in this Section

iff: Throughout this section the term "iff" is used as a shortform for "if and only if".

Conjunctive commas: A comma (",") separating two assertions in a semantic condition, as in "c ∈ IC , p ∈ IP", is read as a logical "and". Further, a comma separating two variables, as in "c, d ∈ IC", is used for abbreviating two comma separated assertions, "c ∈ IC , d ∈ IC" in this example.

Unscoped variables: If no explicit scope is given for a variable "x", as in "∀ x : …" or "{ x | … }", then "x" is unconstrained, which means x ∈ IR, i.e. "x" denotes an arbitrary individual in the universe.

Set cardinality: For a set S, an expression of the form "#S" means the number of elements in S.

Sequence expressions: An expression of the form "s sequence of a₁ , … , a_n ∈ S" means that "s" represents an RDF list of n ≥ 0 individuals a₁ , … , a_n, all of them being members of the set S. Precisely, s = I(rdf:nil) for n = 0; and for n > 0 there exist z₁ ∈ IR , … , z_n ∈ IR, such that

Note, as mentioned in Section 3.3.3 of the RDF Semantics [RDF Semantics], there are no semantic constraints that enforce "well-formed" sequence structures. So, for example, it is possible for a sequence head s to refer to more than one sequence.

Set names: The following names are used as convenient abbreviations for certain sets:

• ISEQ: The set of all sequences. This set equals the class extension of rdf:List, i.e., ISEQ := ICEXT(I(rdf:List)).
• INNI: The set of all nonnegative integers. This set equals the value space of the datatype xsd:nonNegativeInteger, i.e., INNI := ICEXT(I(xsd:nonNegativeInteger)), but is also subsumed by the value spaces of other numerical datatypes, such as xsd:integer.

Notes on the Form of Semantic Conditions (Informative)

One design goal of OWL 2 was to ensure an appropriate degree of alignment between the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics [OWL 2 Direct Semantics] under the different constraints the two semantics have to meet. The way this semantic alignment is described is via the OWL 2 correspondence theorem in Section 7.2. For this theorem to hold, the semantic conditions that treat the RDF encoding of OWL 2 axioms (compare Section 3.2.5 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping] and Section 9 of the OWL 2 Structural Specification [OWL 2 Specification]), such as inverse property axioms, must have the form of "iff" ("if-and-only-if") conditions. This means that these semantic conditions completely determine the semantics of the encoding of these constructs. On the other hand, the RDF encoding of OWL 2 expressions (compare Section 3.2.4 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping] and Sections 6 – 8 of the OWL 2 Structural Specification [OWL 2 Specification]), such as property restrictions, are treated by "if-then" conditions. These weaker semantic conditions for expressions are sufficient for the correspondence theorem to hold, so there is no necessity to define stronger "iff" conditions under the OWL 2 RDF-Based Semantics for these language constructs.

Special cases are the semantic conditions for Boolean connectives of classes and for enumerations. These language constructs build OWL 2 expressions. But for backward compatibility reasons there is also RDF encoding of axioms based on the vocabulary for these language constructs (see Table 18 in Section 3.2.5 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping]). For example, an RDF expression of the form

is mapped by the reverse RDF mapping to an OWL 2 axiom that states the equivalence of the class denoted by ex:c₁ with the union of the classes denoted by ex:c₂ and ex:c₃. In order to ensure that the correspondence theorem holds, and in accordance with the original OWL 1 RDF-Compatible Semantics specification [OWL 1 RDF-Compatible Semantics], the semantic conditions for the mentioned language constructs are therefore "iff" conditions.

Further, special treatment exists for OWL 2 axioms that have a multi-triple representation in RDF, where the different triples share a common "root node", such as the blank node "_:x" in the following example:

In essence, the semantic conditions for the encoding of these language constructs are "iff" conditions, as usual for axioms. However, in order to cope with the specific syntactic aspect of a "root node", the "iff" conditions of these language constructs have been split into two "if-then" conditions, where the "if-then" condition representing the right-to-left direction contains an additional premise having the form "∃ z ∈ IR". The purpose of this premise is to ensure the existence of an individual that is needed to satisfy the root node under the OWL 2 RDF-Based semantics. The language constructs in question are n-ary disjointness axioms in Section 5.10, and negative property assertions in Section 5.15.

The "if-then" semantic conditions in this section sometimes do not explicitly list all typing statements in their consequent that one might expect. For example, the semantic condition for owl:someValuesFrom restrictions in Section 5.6 does not list the statement "x ∈ ICEXT(I(owl:Restriction))" on its right hand side. Consequences are generally not mentioned, if they can already be deduced by other means. Often, these redundant consequences follow from the semantic conditions for vocabulary classes and vocabulary properties in Section 5.2 and Section 5.3, respectively, occasionally in connection with the semantic conditions for the parts of the universe in Section 5.1. In the example above, the omitted consequence can be obtained from the third column of the entry for owl:someValuesFrom in the table in Section 5.3, which determines that IEXT(I(owl:someValuesFrom)) ⊆ ICEXT(I(owl:Restriction)) × IC.

5.1 Semantic Conditions for the Parts of the Universe

Table 5.1 lists the semantic conditions for the parts of the universe of the OWL 2 RDF-Based interpretation being considered. Additional semantic conditions affecting these parts are given in Section 5.2.

The first column tells the name of the part, as defined in Section 4.4. The second column defines certain conditions on the part. In most cases, the column specifies for the part by which other part it is subsumed, and thus the position of the part in the "parts hierarchy" of the universe is narrowed down. The third column provides further information about the instances of those parts that consist of classes or properties. In general, if the part consists of classes, then for the class extensions of the member classes is specified by which part of the universe they are subsumed. If the part consists of properties, then the domains and ranges of the member properties are determined.

5.2 Semantic Conditions for the Vocabulary Classes

Table 5.2 lists the semantic conditions for the classes that have IRIs in the OWL 2 RDF-Based vocabulary. In addition, the table contains all those classes with IRIs in the RDF and RDFS vocabularies that represent parts of the universe of the OWL 2 RDF-Based interpretation being considered (Section 4.4). The semantic conditions for the remaining classes with names in the RDF and RDFS vocabularies can be found in the RDF Semantics specification [RDF Semantics].

The first column tells the IRI of the class. The second column defines of what particular kind a class is, i.e. whether it is a general class (a member of the part IC) or a datatype (a member of IDC). The third column specifies for the class extension of the class by which part of the universe (Section 4.4) it is subsumed: from an entry of the form "ICEXT(I(C)) ⊆ S", for a class IRI C and a set S, and given an RDF triple of the form "u rdf:type C", one can deduce that the relationship "I(u) ∈ S" holds. Note that some entries are of the form "ICEXT(I(C)) = S", which means that the class extension is exactly specified to be that set. See Section 5.1 for further semantic conditions on those classes that represent parts.

Not included in this table are the datatypes of the OWL 2 RDF-Based Semantics with IRIs listed in Section 3.3. For each such datatype IRI E, the following semantic conditions hold (as a consequence of the fact that E is a member of the datatype map of every OWL 2 RDF-Based interpretation according to Definition 4.2, and by the "general semantic conditions for datatypes" listed in Section 5.1 of the RDF Semantics [RDF Semantics]):

• I(E) ∈ IDC
• ICEXT(I(E)) ⊆ LV

5.3 Semantic Conditions for the Vocabulary Properties

Table 5.3 lists the semantic conditions for the properties that have IRIs in the OWL 2 RDF-Based vocabulary. In addition, the table contains all those properties with IRIs in the RDFS vocabulary that are specified to be annotation properties under the OWL 2 RDF-Based Semantics. The semantic conditions for the remaining properties with names in the RDF and RDFS vocabularies can be found in the RDF Semantics specification [RDF Semantics].

The first column tells the IRI of the property. The second column defines of what particular kind a property is, i.e. whether it is a general property (a member of the part IP), a datatype property (a member of IODP), an ontology property (a member of IOXP) or an annotation property (a member of IOAP). The third column specifies the domain and range of the property: from an entry of the form "IEXT(I(p)) ⊆ S₁ × S₂", for a property IRI p and sets S₁ and S₂, and given an RDF triple "s p o", one can deduce the relationships "I(s) ∈ S₁" and "I(o) ∈ S₂". Note that some entries are of the form "IEXT(I(p)) = S₁ × S₂", which means that the property extension is exactly specified to be the Cartesian product of the two sets.

Not included in this table are the facets of the OWL 2 RDF-Based Semantics with IRIs listed in Section 3.4, which are used to specify datatype restrictions (see Section 5.7). For each such facet IRI E, the following semantic conditions extend the basic semantics specification that has been given for datatypes with facets in Section 4.1:

• I(E) ∈ IODP
• IEXT(I(E)) ⊆ IR × LV

Implementations are not required to support the semantic condition for owl:onProperties, but MAY support it in order to realize n-ary dataranges with arity ≥ 2 (see Sections 7 and 8.4 of the OWL 2 Structural Specification [OWL 2 Specification] for further information).

Informative notes:

owl:topObjectProperty relates every two individuals in the universe with each other. Likewise, owl:topDataProperty relates every individual with every data value. Further, owl:bottomObjectProperty and owl:bottomDataProperty stand both for the empty relationship.

The ranges of the properties owl:deprecated and owl:hasSelf are not restricted in any form, and, in particular, they are not restricted to Boolean values. The actual object values of these properties do not have any intended meaning, but could as well have been defined to be of any other value. Therefore, the semantics given here are of a form that the values can be arbitrarily chosen without leading to any nontrivial semantic conclusions. It is, however, recommended to still use an object literal of the form "true"^^xsd:boolean in ontologies, in order to not get in conflict with the required usage of these properties in scenarios that ask for applying the reverse RDF mapping (compare Table 13 in Section 3.2.4 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping] for owl:hasSelf, and Section 5.5 of the OWL 2 Structural Specification [OWL 2 Specification] for owl:deprecated).

The range of the property owl:annotatedProperty is unrestricted, i.e. it is not specified as the set of properties. Annotations are meant to be "semantically weak", i.e. their formal meaning should not significantly exceed that originating from the RDF Semantics specification.

Several properties, such as owl:priorVersion, have been specified as both ontology properties and annotation properties, in order to be in line with both the original OWL 1 RDF-Compatible Semantics specification [OWL 1 RDF-Compatible Semantics] and the rest of the OWL 2 specification (see Section 5.5 of the OWL 2 Structural Specification [OWL 2 Specification]).

5.4 Semantic Conditions for Boolean Connectives

Table 5.4 lists the semantic conditions for Boolean connectives, including intersections, unions and complements of classes and datatypes. An intersection or a union of a collection of datatypes or a complement of a datatype is itself a datatype. While a complement of a class is created w.r.t. the whole universe, a datatype complement is created for a datatype w.r.t. the set of data values only.

Informative notes: Of the three pairs of semantic conditions in the table every first is an "iff" condition, since the corresponding OWL 2 language constructs are both class expressions and axioms. In contrast, the semantic condition on datatype complements is an "if-then" condition, since it only corresponds to a datarange expression. See the notes on the form of semantic conditions for further information. For the remaining semantic conditions that treat the cases of intersections and unions of datatypes it is sufficient to have "if-then" conditions, since stronger "iff" conditions would be redundant due to the more general "iff" conditions that already exist for classes. Note that the datatype related semantic conditions do not apply to empty sets, but one can still receive a datatype from an empty set by explicitly asserting the resulting class to be an instance of class rdfs:Datatype.

5.5 Semantic Conditions for Enumerations

Table 5.5 lists the semantic conditions for enumerations, i.e. classes that consist of an explicitly given finite set of instances. In particular, an enumeration entirely consisting of data values is a datatype.

Informative notes: The first semantic condition is an "iff" condition, since the corresponding OWL 2 language construct is both a class expression and an axiom. See the notes on the form of semantic conditions for further information. For the remaining semantic condition that treats the case of enumerations of data values it is sufficient to have an "if-then" condition, since a stronger "iff" condition would be redundant due to the more general "iff" condition that already exists for individuals. Note that the data value related semantic condition does not apply to empty sets, but one can still receive a datatype from an empty set by explicitly asserting the resulting class to be an instance of class rdfs:Datatype.

5.6 Semantic Conditions for Property Restrictions

Table 5.6 lists the semantic conditions for property restrictions.

Value restrictions require that some or all of the values of a certain property must be instances of a given class or data range, or that the property has a specifically defined value. By placing a self restriction on some given property one only considers those individuals that are reflexively related to themselves via this property. Cardinality restrictions determine how often a certain property is allowed to be applied to a given individual. Qualified cardinality restrictions are more specific than cardinality restrictions in that they determine the quantity of a property application with respect to a particular class or data range from which the property values are taken.

Implementations are not required to support the semantic conditions for owl:onProperties, but MAY support them in order to realize n-ary dataranges with arity ≥ 2 (see Sections 7 and 8.4 of the OWL 2 Structural Specification [OWL 2 Specification] for further information).

Informative notes: All the semantic conditions are "if-then" conditions, since the corresponding OWL 2 language constructs are class expressions. The "if-then" conditions generally only list those consequences on their right hand side that are specific for the respective condition, i.e. consequences that do not already follow by other means. See the notes on the form of semantic conditions for further information. Note that the semantic condition for self restrictions does not constrain the right hand side of a owl:hasSelf assertion to be the Boolean value "true"^^xsd:boolean. See Section 5.3 for an explanation.

5.7 Semantic Conditions for Datatype Restrictions

Table 5.7 lists the semantic conditions for datatype restrictions, which are used to define sub datatypes of existing datatypes by restricting the original datatype by means of a set of facet-value pairs. See Section 3.4 for information and an example on constraining facets.

Certain special cases exist: If no facet-value pair is applied to a given datatype, then the resulting datatype will be equivalent to the original datatype. Further, if a facet-value pair is applied to a datatype without being a member of the datatype's facet space, then the ontology cannot be satisfied and will therefore be inconsistent. In particular, a datatype restriction with one or more specified facet-value pairs will result in an inconsistent ontology, if applied to a datatype with an empty facet space.

The set IFS is defined by IFS(d) := { ( I(F) , v ) | ( F , v ) ∈ FS(d) } , where d is a datatype, F is the IRI of a constraining facet, and v is a constraining value of the facet. This set corresponds to the facet space FS(d), as defined in Section 4.1, but rather consists of pairs of the denotation of a facet and a value.

The mapping IF2V is defined by IF2V(d)(( I(F) , v )) := F2V(d)(( F , v )) , where d is a datatype, F is the IRI of a constraining facet, and v is a constraining value of the facet. This mapping corresponds to the facet-to-value mapping F2V(d), as defined in Section 4.1, resulting in the same subsets of the value space VS(d), but rather applies to pairs of the denotation of a facet and a value.

Informative notes: The semantic condition is an "if-then" condition, since the corresponding OWL 2 language construct is a datarange expression. The "if-then" condition only lists those consequences on its right hand side that are specific for the condition, i.e. consequences that do not already follow by other means. See the notes on the form of semantic conditions for further information.

5.8 Semantic Conditions for the RDFS Vocabulary

Table 5.8 extends the RDFS semantic conditions for subclass axioms, subproperty axioms, domain axioms and range axioms. The semantic conditions provided here are "iff" conditions, while the original semantic conditions, as specified in Section 4.1 of the RDF Semantics [RDF Semantics], are weaker "if-then" conditions. Only the additional semantic conditions are given here and the other conditions of RDF and RDFS are retained.

Informative notes: All the semantic conditions are "iff" conditions, since the corresponding OWL 2 language constructs are axioms. See the notes on the form of semantic conditions for further information.

5.9 Semantic Conditions for Equivalence and Disjointness

Table 5.9 lists the semantic conditions for specifying that two individuals are equal or different from each other, and that either two classes or two properties are equivalent or disjoint with each other, respectively. The property owl:equivalentClass is also used to formulate datatype definitions (see Section 9.4 of the OWL 2 Structural Specification [OWL 2 Specification] for information about datatype definitions). In addition, the table treats disjoint union axioms.

Informative notes: All the semantic conditions are "iff" conditions, since the corresponding OWL 2 language constructs are axioms. See the notes on the form of semantic conditions for further information.

5.10 Semantic Conditions for N-ary Disjointness

Table 5.10 lists the semantic conditions for specifying n-ary diversity and disjointness axioms, i.e. that several given individuals are mutually different from each other, and that several given classes or properties are mutually disjoint with each other, respectively.

Note that there are two alternative ways to specify owl:AllDifferent axioms, by using either the property owl:members that is used for all other constructs, too, or by applying the legacy property owl:distinctMembers. Both variants have an equivalent formal meaning.

Informative notes: The semantic conditions essentially represent "iff" conditions, since the corresponding OWL 2 language constructs are axioms. However, there are actually two semantic conditions for each language construct due to the multi-triple RDF encoding of these language constructs. The "if-then" conditions only list those consequences on their right hand side that are specific for the respective condition, i.e. consequences that do not already follow by other means. See the notes on the form of semantic conditions for further information.

5.11 Semantic Conditions for Sub Property Chains

Table 5.11 lists the semantic conditions for sub property chains, which allow for specifying complex property subsumption axioms.

As an example, one can define a sub property chain axiom that specifies the chain consisting of the property extensions of properties ex:hasFather and ex:hasBrother to be a sub relation of the extension of the property ex:hasUncle.

Informative notes: The semantic condition is an "iff" condition, since the corresponding OWL 2 language construct is an axiom. See the notes on the form of semantic conditions for further information. The semantics has been specified in a way such that a sub property chain axiom can be satisfied without requiring the existence of a property that has the property chain as its property extension.

5.12 Semantic Conditions for Inverse Properties

Table 5.12 lists the semantic conditions for inverse property axioms. The inverse of a given property is the corresponding property with subject and object swapped for each property assertion built from it.

Informative notes: The semantic condition is an "iff" condition, since the corresponding OWL 2 language construct is an axiom. See the notes on the form of semantic conditions for further information.

5.13 Semantic Conditions for Property Characteristics

Table 5.13 lists the semantic conditions for property characteristics.

If a property is functional, then at most one distinct value can be assigned to any given individual via this property. An inverse functional property can be regarded as a "key" property, i.e. no two different individuals can be assigned the same value via this property. A reflexive property relates every individual in the universe to itself, whereas an irreflexive property does not relate any individual with itself. If two individuals are related by a symmetric property, then this property also relates them reversely, while this is never the case for an asymmetric property. A transitive property that relates an individual a with an individual b, and the latter with an individual c, also relates a with c.

Informative notes: All the semantic conditions are "iff" conditions, since the corresponding OWL 2 language constructs are axioms. See the notes on the form of semantic conditions for further information.

5.14 Semantic Conditions for Keys

Table 5.14 lists the semantic conditions for Keys.

Keys provide an alternative to inverse functional properties (see Section 5.13). They allow for defining a property as a key local to a given class: the specified property will have the features of a key only for individuals being instances of the class, and no assumption is made about individuals for which membership of the class cannot be entailed. Further, it is possible to define "compound keys", i.e. several properties can be combined into a single key applicable to composite values. Note that keys are not functional by default under the OWL 2 RDF-Based Semantics.

Informative notes: The semantic condition is an "iff" condition, since the corresponding OWL 2 language construct is an axiom. See the notes on the form of semantic conditions for further information.

5.15 Semantic Conditions for Negative Property Assertions

Table 5.15 lists the semantic conditions for negative property assertions. They allow to state that two given individuals are not related by a given property.

The second form based on owl:targetValue is more specific than the first form based on owl:targetIndividual in that the second form is restricted to the case of negative data property assertions. Note that the second form will coerce the target value of a negative property assertion into a data value, due to the range defined for the property owl:targetValue in Section 5.3.

Informative notes: The semantic conditions essentially represent "iff" conditions, since the corresponding OWL 2 language constructs are axioms. However, there are actually two semantic conditions for each language construct, due to the multi-triple RDF encoding of these language constructs. The "if-then" conditions only list those consequences on their right hand side that are specific for the respective condition, i.e. consequences that do not already follow by other means. See the notes on the form of semantic conditions for further information.

6 Appendix: Axiomatic Triples (Informative)

The RDF Semantics specification [RDF Semantics] defines so called "axiomatic triples" as part of the semantics of RDF and RDFS. Unlike the RDF Semantics, the OWL 2 RDF-Based Semantics does not normatively specify any axiomatic triples, since one cannot expect to find a set of RDF triples that fully captures all "axiomatic aspects" of the OWL 2 RDF-Based Semantics. Furthermore, axiomatic triples for the OWL 2 RDF-Based Semantics could, in principle, contain arbitrarily complex class expressions, e.g. the union of several classes, and by this it becomes nonobvious which of several possible nonequivalent sets of axiomatic triples should be selected. However, the OWL 2 RDF-Based Semantics includes many semantic conditions that can in a sense be regarded as being "axiomatic", and thus can be considered a replacement for the missing axiomatic triples. After an overview on axiomatic triples for RDF and RDFS in Section 6.1, Sections 6.2 and 6.3 will discuss how the "axiomatic" semantic conditions of the OWL 2 RDF-Based Semantics relate to axiomatic triples. Based on this discussion, an explicit example set of axiomatic triples that is compatible with the OWL 2 RDF-Based Semantics will be provided in Section 6.4.

6.1 Axiomatic Triples in RDF

In RDF and RDFS [RDF Semantics], axiomatic triples are used to provide basic meaning for all the vocabulary terms of the two languages. This formal meaning is independent of any given RDF graph, and it even holds for vocabulary terms, which do not occur in a graph that is interpreted by an RDF or RDFS interpretation. As a consequence, all the axiomatic triples of RDF and RDFS are entailed by the empty graph, when being interpreted under the semantics of RDF or RDFS, respectively.

Examples of RDF and RDFS axiomatic triples are:

As shown by these examples, axiomatic triples are typically used by the RDF Semantics specification to determine the part of the universe to which the denotation of a vocabulary term belongs (1). In the case of a property, the domain (2) and range (3) is specified as well. Also, in some cases, hierarchical relationships between classes (4) or properties (5) of the vocabulary are determined.

Under the OWL 2 RDF-Based Semantics, all the axiomatic triples of RDF and RDFS could, in principle, be replaced by "axiomatic" semantic conditions that have neither premises nor bound variables. By applying the RDFS semantic conditions given in Section 5.8, the example axiomatic triples (1) – (5) can be equivalently restated as:

All the axiomatic triples of RDF and RDFS can be considered "simple" in the sense that they have in their object position only single terms from the RDF and RDFS vocabularies, and no complex class or property expressions appear there.

6.2 Axiomatic Triples for the Vocabulary Classes

The semantic conditions for vocabulary classes in Section 5.2 can be considered as corresponding to a set of axiomatic triples for the classes in the vocabulary of the OWL 2 RDF-Based Semantics.

First, for each IRI E occurring in the first column of Table 5.2, if the second column contains an entry of the form "I(E) ∈ S" for some set S, then this entry corresponds to an RDF triple of the form "E rdf:type C", where C is the IRI of a vocabulary class with ICEXT(I(C)) = S. In the table, S will always be either the part IC of all classes, or some sub part of IC. Hence, in a corresponding RDF triple the IRI C will be one of "rdfs:Class", "owl:Class" (S=IC in both cases) or "rdfs:Datatype" (S=IDC).

For example, for the IRI "owl:FunctionalProperty", the semantic condition

has the corresponding axiomatic triple

Further, for each IRI E in the first column of the table, if the third column contains an entry of the form "ICEXT(I(E)) ⊆ S" (or "ICEXT(I(E)) = S") for some set S, then this entry corresponds to an RDF triple of the form "E rdfs:subClassOf C" (or additionally "C rdfs:subClassOf E"), where C is the IRI of a vocabulary class with ICEXT(I(C)) = S. In each case, S will be one of the parts of the universe of I.

For example, the semantic condition

has the corresponding axiomatic triple

In addition, the semantic conditions for the parts of the universe in Table 5.1 of Section 5.1 have to be taken into account. In particular, if an entry in the second column of Table 5.1 is of the form "S₁ ⊆ S₂" for some sets S₁ and S₂, then this corresponds to an RDF triple of the form "C₁ owl:subClassOf C₂", where C₁ and C₂ are the IRIs of vocabulary classes with ICEXT(I(C₁)) = S₁ and ICEXT(I(C₂)) = S₂, respectively, according to Section 5.2.

Section 5.2 also specifies semantic conditions for all the datatypes of the OWL 2 RDF-Based Semantics, as listed in Section 3.3. For each datatype IRI E, such as E := "xsd:string", for the semantic conditions "I(E) ∈ IDC" and "ICEXT(I(E)) ⊆ LV" the corresponding axiomatic triples are of the form

In analogy to Section 6.1 for the RDF axiomatic triples, all the axiomatic triples for the vocabulary classes (including datatypes) can be considered "simple" in the sense that they will have in their object position only single terms from the RDF, RDFS and OWL 2 RDF-Based vocabularies (Section 3.2).

Note that some of the axiomatic triples obtained in this way already follow from the semantics of RDF and RDFS, as defined in the RDF Semantics [RDF Semantics].

6.3 Axiomatic Triples for the Vocabulary Properties

The semantic conditions for vocabulary properties in Section 5.3 can be considered as corresponding to a set of axiomatic triples for the properties in the vocabulary of the OWL 2 RDF-Based Semantics.

First, for each IRI E occurring in the first column of Table 5.3, if the second column contains an entry of the form "I(E) ∈ S" for some set S, then this entry corresponds to an RDF triple of the form "E rdf:type C", where C is the IRI of a vocabulary class with ICEXT(I(C)) = S. In the table, S will always be either the part IP of all properties, or some sub part of IP. Hence, in a corresponding RDF triple the IRI C will be one of "rdf:Property", "owl:ObjectProperty", (S=IP in both cases), "owl:DatatypeProperty" (S=IODP), "owl:OntologyProperty" (S=IOXP) or "owl:AnnotationProperty" (S=IOAP).

For example, for the IRI "owl:disjointWith", the semantic condition

has the corresponding axiomatic triple

Further, for each IRI E in the first column of the table, if the third column contains an entry of the form "IEXT(I(E)) ⊆ S₁ × S₂" for some sets S₁ and S₂, then this entry corresponds to RDF triples of the form "E rdfs:domain C₁" and "E rdfs:range C₂", where C₁ and C₂ are the IRIs of vocabulary classes with ICEXT(I(C₁)) = S₁ and ICEXT(I(C₂)) = S₂, respectively. Note that the sets S₁ and S₂ do not always correspond to any of the parts of the universe of I.

For example, the semantic condition

has the corresponding axiomatic triples

Exceptions are the semantic conditions "IEXT(I(owl:topObjectProperty)) = IR × IR" and "IEXT(I(owl:topDataProperty)) = IR × LV", since the exactly specified property extensions of these properties cannot be expressed solely by domain and range axiomatic triples. For example, the domain and range axiomatic triples for owl:sameAs are equal to those for owl:topObjectProperty, but the property extension of owl:sameAs is different from the property extension of owl:topObjectProperty.

Section 5.3 also specifies semantic conditions for all the facets of the OWL 2 RDF-Based Semantics, as listed in Section 3.4. For each facet IRI E, such as E := "xsd:length", for the semantic conditions "I(E) ∈ IODP" and "IEXT(I(E)) ⊆ IR × LV" the corresponding axiomatic triples are of the form

In analogy to Section 6.1 for the RDF axiomatic triples, all the axiomatic triples for the vocabulary properties (including facets) can be considered "simple" in the sense that they will have in their object position only single terms from the RDF, RDFS and OWL 2 RDF-Based vocabularies (Section 3.2).

6.4 A Set of Axiomatic Triples

This section provides a concrete example set of axiomatic triples based on the discussion in the Sections 6.2 and 6.3. The axiomatic triples are grouped by different tables for the classes and the properties of the OWL 2 RDF-Based vocabulary, for the datatypes and the facets of the OWL 2 RDF-Based Semantics, and for some of the classes and properties of the RDFS vocabulary. Note that this set of axiomatic triples is not meant to be free of redundancy.

7 Appendix: Relationship to the Direct Semantics (Informative)

This section compares the OWL 2 RDF-Based Semantics with the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. While the OWL 2 RDF-Based Semantics is based on the RDF Semantics specification [RDF Semantics], the OWL 2 Direct Semantics is a description logic style semantics. Several fundamental differences exist between the two semantics, but there is also a strong relationship basically stating that the OWL 2 RDF-Based Semantics is able to reflect all logical conclusions of the OWL 2 Direct Semantics. This means that the OWL 2 Direct Semantics can in a sense be regarded as a semantics subset of the OWL 2 RDF-Based Semantics.

Technically, the comparison will be performed by comparing the sets of entailments that hold for each of the two semantics, respectively. The definition of an OWL 2 RDF-Based entailment was given in Section 4.3 of this document, while the definition of an OWL 2 Direct entailment is provided in Section 2.5 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. In both cases, entailments are defined for pairs of ontologies, and such an ordered pair of two ontologies will be called an entailment query in this section.

Comparing the two semantics by means of entailments will only be meaningful if the entailment queries allow for applying both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics to them. In order to ensure this, the comparison will be restricted to entailment queries, for which the left-hand side and right-hand side ontologies are both OWL 2 DL ontologies in RDF graph form. These are RDF graphs that, by applying the reverse RDF mapping [OWL 2 RDF Mapping], can be transformed into corresponding OWL 2 DL ontologies in Functional Syntax form according to the functional style syntax defined in the OWL 2 Structural Specification [OWL 2 Specification], and which must further meet all the restrictions on OWL 2 DL ontologies that are specified in Section 3 of the OWL 2 Structural Specification [OWL 2 Specification]. In fact, these restrictions must be mutually met by both ontologies that occur in an entailment query, i.e. all these restrictions need to be satisfied as if the two ontologies would be part of a single ontology. Any entailment query that adheres to the conditions defined here will be called an OWL 2 DL entailment query.

Ideally, the relationship between the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics would be of the form that every OWL 2 DL entailment query that is an OWL 2 Direct entailment is also an OWL 2 RDF-Based entailment. However, this desirable relationship cannot hold in general due to a variety of differences that exist between the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics, as demonstrated in Section 7.1.

Fortunately, the problems resulting from these semantic differences can be overcome in a way that for every OWL 2 DL entailment query there is another one for which the desired entailment relationship indeed holds, and the new entailment query is semantically equivalent to the original entailment query under the OWL 2 Direct Semantics. This is the gist of the OWL 2 correspondence theorem, which will be presented in Section 7.2. The proof of this theorem, given in Section 7.3, will further demonstrate that such a substitute OWL 2 DL entailment query can always be algorithmically constructed by means of simple syntactic transformations.

7.1 Example on Semantic Differences

This section will show that differences exist between the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics, and it will be demonstrated how these semantic differences complicate a comparison of the two semantics in terms of entailments. An example OWL 2 DL entailment query will be given, which will happen to be an OWL 2 Direct entailment without being an OWL 2 RDF-Based entailment. The section will explain the different reasons and will provide a resolution of each of them. It will turn out that the example entailment query can be syntactically transformed into another OWL 2 DL entailment query that is both an OWL 2 Direct entailment and an OWL 2 RDF-Based entailment, while being semantically unchanged compared to the original entailment query under the OWL 2 Direct Semantics. This example will motivate the OWL 2 correspondence theorem in Section 7.2 and its proof in Section 7.3.

The example entailment query consists of the following pair ( G₁^* , G₂^* ) of RDF graphs:

Both G₁^* and G₂^* are OWL 2 DL ontologies in RDF graph form and can therefore be mapped by the reverse RDF mapping [OWL 2 RDF Mapping] to the following two OWL 2 DL ontologies in Functional Syntax form F(G₁^*) and F(G₂^*):

Note that F(G₁^*) and F(G₂^*) mutually meet the restrictions on OWL 2 DL ontologies as specified in Section 3 of the OWL 2 Structural Specification [OWL 2 Specification]. For example, none of the IRIs being declared as a class in F(G₁^*) is declared as a datatype in F(G₂^*), since this would not be allowed for an OWL 2 DL entailment query.

It follows that F(G₁^*) OWL 2 Direct entails F(G₂^*). To show this, only the axioms (4) of F(G₁^*) and (5) of F(G₂^*) have to be considered. None of the other statements in the two ontologies are relevant for this OWL 2 Direct entailment to hold, since they do not have a formal meaning under the OWL 2 Direct Semantics. However, it turns out that the RDF graph G₁^* does not OWL 2 RDF-Based entail G₂^*, for reasons discussed in detail now.

Reason 1: An Annotation in F(G₂^*). The ontology F(G₂^*) contains an annotation (6). The OWL 2 Direct Semantics does not give a formal meaning to annotations. In contrast, under the OWL 2 RDF-Based Semantics every RDF triple occurring in an RDF graph has a formal meaning, including the corresponding annotation triple (8) in G₂^*. Since this annotation triple only occurs in G₂^* but not in G₁^*, there will exist OWL 2 RDF-Based interpretations that satisfy G₁^* without satisfying triple (8) of G₂^*. Hence, G₁^* does not OWL 2 RDF-Based entail G₂^*.

Resolution of Reason 1. The annotation triple (8) in G₂^* will be removed, which will avoid requiring OWL 2 RDF-Based interpretations to interpret this triple. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since annotations are strictly optional in OWL 2 DL ontologies and may therefore be omitted. Also, this operation will not change the formal meaning of the ontologies under the OWL 2 Direct Semantics, since annotations do not have a formal meaning under this semantics.

Reason 2: An Entity Declaration exclusively in F(G₂^*). The ontology F(G₂^*) contains an entity declaration for the class IRI ex:c3 (4), for which there is no corresponding entity declaration in F(G₁^*). The OWL 2 Direct Semantics does not give a formal meaning to entity declarations, while the OWL 2 RDF-Based Semantics gives a formal meaning to the corresponding declaration statement (4) in G₂^*. The consequences are analog to those described for reason 1.

Resolution of Reason 2. The declaration statement (4) in G₂^* will be copied to G₁^*. An OWL 2 RDF-Based interpretation that satisfies the modified graph G₁^* will then also satisfy the declaration statement. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since the copied declaration statement is not in conflict with any of the other entity declarations in G₁^*. Also, this operation will not change the formal meaning of the ontologies under the OWL 2 Direct Semantics, since entity declarations do not have a formal meaning under this semantics.

Reason 3: Different Ontology IRIs in F(G₁^*) and F(G₂^*). The ontology IRIs for the two ontologies, given by (1) in F(G₁^*) and by (1) in F(G₂^*), differ from each other. The OWL 2 Direct Semantics does not give a formal meaning to ontology headers, while the OWL 2 RDF-Based Semantics gives a formal meaning to the corresponding header triples (1) in G₁^* and (1) in G₂^*. Since these header triples differ from each other, the consequences are analog to those described for reason 1.

Resolution of Reason 3. The IRI in the subject position of the header triple (1) in G₂^* is changed into a blank node. Due to the existential semantics of blank nodes under the OWL 2 RDF-Based Semantics the resulting triple will then be entailed by triple (1) in G₁^*. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since an ontology IRI is optional for an OWL 2 DL ontology. (Note, however, that it would have been an error to simply remove triple (1) from G₂^*, since an OWL 2 DL ontology is required to contain an ontology header.) Also, this operation will not change the formal meaning of the ontologies under the OWL 2 Direct Semantics, since ontology headers do not have a formal meaning under this semantics.

Reason 4: A Class Expression in F(G₂^*). Axiom (5) of F(G₂^*) contains a class expression that represents the union of the two classes denoted by ex:c2 and ex:c3. Within G₂^*, this class expression is represented by the triples (6) and (7), both having the blank node "_:x" in their respective subject position. The way the OWL 2 RDF-Based Semantics interprets these two triples differs from the way the OWL 2 Direct Semantics treats the class expression in axiom (5) of F(G₂^*).

The OWL 2 Direct Semantics treats classes as sets, i.e. subsets of the universe. Thus, the IRIs ex:c2 and ex:c3 in F(G₂^*) denote two sets, and the class expression in axiom (5) of F(G₂^*) therefore represents the set that consists of the union of these two sets.

The OWL 2 RDF-Based Semantics, on the other hand, treats classes as individuals, i.e. members of the universe. While every class under the OWL 2 RDF-Based Semantics represents a certain subset of the universe, namely its class extension, this set is actually distinguished from the class itself. For two given classes it is ensured under the OWL 2 RDF-Based Semantics, just as for the OWL 2 Direct Semantics, that the union of their class extensions will always exist as a subset of the universe. However, there is no guarantee that there will also exist an individual in the universe that has this set union as its class extension.

Under the OWL 2 RDF-Based Semantics, triple (7) of G₂^* essentially claims that a class exists being the union of two other classes. But since the existence of such a union class is not ensured by G₁^*, there will be OWL 2 RDF-Based interpretations that satisfy G₁^* without satisfying triple (7) of G₂^*. Hence, G₁^* does not OWL 2 RDF-Based entail G₂^*.

Resolution of Reason 4. The triples (6) and (7) of G₂^* are copied to G₁^* together with the new triple "_:x owl:equivalentClass _:x". In addition, for the IRI ex:c3, which only occurs in the union class expression but not in G₁^*, an entity declaration is added to G₁^* by the resolution of reason 2. If an OWL 2 RDF-Based interpretation satisfies the modified graph G₁^*, then the triples (6) and (7) of G₂^* will now be satisfied. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since the whole set of added triples validly encodes an OWL 2 axiom, and since none of the restrictions on OWL 2 DL ontologies is hurt. Also, this operation will not change the formal meaning of the ontologies under the OWL 2 Direct Semantics, since the added equivalence axiom is a tautology under this semantics.

Note that it would have been an error to simply copy the triples (6) and (7) of G₂^* to G₁^*, without also adding the new triple "_:x owl:equivalentClass _:x". This would have produced a class expression that has no connection to any axiom in the ontology. An OWL 2 DL ontology is basically a set of axioms and does not allow for the occurrence of "dangling" class expressions. This is the reason for actually "embedding" the class expression in an axiom. It would have also been wrong to use an arbitrary axiom for such an embedding, since it has to be ensured that the formal meaning of the original ontology does not change under the OWL 2 Direct Semantics. However, any tautological axiom that contains the original class expression would have been sufficient for this purpose as well.

Complete Resolution: The Transformed Entailment Query.

Combining the resolutions of all the above reasons leads to the following new pair of RDF graphs ( G₁ , G₂ ):

The following list reiterates the changes compared to the original RDF graphs G₁^* and G₂^*:

• Resolution of Reason 1 (Annotation): Triple (8) in G₂^* has been removed, i.e. there is no corresponding annotation triple in G₂.
• Resolution of Reason 2 (Entity Declaration): Triple (4) in G₂^* has been copied to G₁^*, becoming triple (4) in G₁.
• Resolution of Reason 3 (Ontology IRIs): The IRI in the subject position of triple (1) in G₂^* has been changed into a blank node, becoming triple (1) in G₂.
• Resolution of Reason 4 (Class Expression): Triples (6) and (7) in G₂^* have been copied to G₁^* together with the new triple "_:x owl:equivalentClass _:x", becoming triples (6), (7) and (8) in G₁.

G₁ and G₂ are again OWL 2 DL ontologies in RDF graph form and can be mapped to the following OWL 2 DL ontologies in Functional Syntax form F(G₁) and F(G₂), which again mutually meet the restrictions on OWL 2 DL ontologies:

As said earlier, all the applied changes preserve the formal meaning of the original OWL 2 DL ontologies under the OWL 2 Direct Semantics. Hence, it is still the case that F(G₁) OWL 2 Direct entails F(G₂). However, due to the syntactic transformation the situation has changed for the OWL 2 RDF-Based Semantics: it is now possible to show, by following the lines of argumentation for the resolutions of the different reasons given above, that G₁ OWL 2 RDF-Based entails G₂ as well.

7.2 Correspondence Theorem

This section presents the OWL 2 correspondence theorem, which compares the semantic expressivity of the OWL 2 RDF-Based Semantics with that of the OWL 2 Direct Semantics. The theorem basically states that the OWL 2 RDF-Based Semantics is able to reflect all the semantic conclusions of the OWL 2 Direct Semantics, where the notion of a "semantic conclusion" is technically expressed in terms of an entailment.

However, as discussed in Section 7.1, there exist semantic differences between the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics, which do not allow for stating that any OWL 2 DL entailment query that is an OWL 2 Direct entailment will always also be an OWL 2 RDF-Based entailment. Nevertheless, it can still be ensured that any given OWL 2 DL entailment query can be substituted by another OWL 2 DL entailment query in a way that for the substitute entailment query the desired relationship will really hold, while preserving the formal meaning compared to the original entailment query under the OWL 2 Direct Semantics.

In fact, the theorem only makes the seemingly weak assertion that such a substitute entailment query will always exist. But the actual proof for the theorem in Section 7.3 will be more concrete in that it will substitute each given OWL 2 DL entailment query with a variant that can be algorithmically constructed by applying a set of simple syntactic transformations to the original entailment query. One can get an idea of how this works from Section 7.1.

Technical Note on Corresponding Datatype Maps. A distinction exists between the format of an OWL 2 RDF-Based datatype map, as defined by Definition 4.1, and the format of an OWL 2 Direct datatype map, as defined in Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. It is, however, possible to translate between an OWL 2 RDF-Based datatype map D and the corresponding OWL 2 Direct datatype map F(D) in the following way:

For an OWL 2 RDF-Based datatype map D, the corresponding OWL 2 Direct datatype map F(D) := ( N_DT , N_LS , N_FS , ⋅ ^DT , ⋅ ^LS , ⋅ ^FS ) [OWL 2 Direct Semantics] is given by

• Datatype Names: N_DT is defined as the set of all IRIs u, for which there is a datatype d, such that ( u , d ) ∈ D.
• Lexical Space: For each datatype name u ∈ N_DT, set N_LS(u) := LS(d), where ( u , d ) ∈ D.
• Facet Space: For each datatype name u ∈ N_DT, set N_FS(u) := FS(d), where ( u , d ) ∈ D.
• Value Space: For each datatype name u ∈ N_DT, set (u) ^DT := VS(d), where ( u , d ) ∈ D.
• Lexical-to-Value Mapping: For each datatype name u ∈ N_DT and each lexical form a ∈ N_LS(u), set ( a , u ) ^LS := L2V(d)(a), where ( u , d ) ∈ D.
• Facet-to-Value Mapping: For each datatype name u ∈ N_DT and each facet-value pair ( F , v ) ∈ N_FS(u), set ( F , v ) ^FS := F2V(d)(( F , v )), where ( u , d ) ∈ D.

7.3 Proof for the Correspondence Theorem

This is the sketch of a proof for Theorem 7.1 (OWL 2 Correspondence Theorem) in Section 7.2. The proof sketch provides the basic line of argumentation for showing the theorem. However, for complexity reasons, some technical aspects of the theorem are only coarsely treated, and the proof sketch also refrains from considering the full amount of OWL 2 language constructs. For certain steps of the proof there are example calculations that focus only on a small fraction of language constructs, but which can be taken as a hint on how a complete proof taking into account every feature of the OWL 2 RDF-Based Semantics could be constructed in principle. A complete proof could make use of the observation that the definitions of the OWL 2 Direct Semantics and the OWL 2 RDF-Based Semantics, despite their technical differences as outlined in Section 7.1, are closely aligned with respect to the different language constructs of OWL 2.

The proof sketch will make use of an approach that will be called "balancing" throughout this section, and which will now be introduced. The basic idea is to substitute the original pair of RDF graphs in an OWL 2 DL entailment query by another entailment query having the same semantic characteristics under the OWL 2 Direct Semantics, but for which the technical differences between the two semantics specifications have no relevant consequences under the OWL 2 RDF-Based Semantics anymore. A concrete example for the application of this approach was given in Section 7.1.

Proof for the Balancing Lemma:

Let the graph pair ( G₁^* , G₂^* ) be a valid input. The resulting RDF graphs G₁ and G₂ are constructed as follows, starting from copies of G₁^* and G₂^*, respectively.

Since the initial versions of G₁ and G₂ are OWL 2 DL ontologies in RDF graph form, the canonical parsing process (CP) for computing the reverse RDF mapping, as described in Section 3 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping], can be applied. Based on CP, it is possible to identify within these graphs

• all entity types for every non-built-in IRI,
• all blank nodes that correspond to anonymous individuals, and
• all sub graphs that correspond to OWL 2 language constructs (ontology headers, declarations, expressions, axioms and annotations) as described in the OWL 2 Structural Specification [OWL 2 Specification].

Based on this observation, the following steps are performed:

1. Consistently substitute all blank nodes in G₂ such that G₁ and G₂ have no common blank nodes.
2. Apply CP to G₁ and G₂ (without changing these graphs) to identify the entity types of the IRIs, the anonymous individuals, and the sub graphs encoding OWL 2 language constructs.
3. For each sub graph g of G₂: remove g from G₂, if g is the RDF encoding of
   • an annotation, or
   • a deprecation statement, or
   • an annotation property axiom.
4. For the sub graph g of G₂ corresponding to the ontology header in F(G₂): substitute g in G₂ by a triple of the form "x rdf:type owl:Ontology", where x is a new blank node not yet used in G₂.
5. For each non-built-in IRI u in G₁ and G₂ and for each entity type T of u identified by CP: add to G₁ or G₂, respectively, the RDF triple "u rdf:type t", where t is the vocabulary class IRI corresponding to T.
6. For each plain or typed literal L in G₂: add to G₁ the RDF triple "o rdfs:comment L", where o is the IRI or blank node of the ontology header triple "o rdf:type owl:Ontology" in G₁.
7. For each sub graph g of G₂ that is the RDF encoding of an entity declaration: add g to G₁.
8. For each sub graph g of G₂ that is the RDF encoding of a property expression with root blank node x: add g to G₁ together with the RDF triple "x owl:equivalentProperty x".
9. For each sub graph g of G₂ that is the RDF encoding of a class expression with root blank node x: add g to G₁ together with the RDF triple "x owl:equivalentClass x".
10. For each sub graph g of G₂ that is the RDF encoding of a data range expression with root blank node x:
    • If g is part of a data property restriction expression, then nothing needs to be done, since the comprising restriction expression is covered by the treatment of class expressions, and therefore g occurs in G₁ as well.
    • Otherwise, add a declaration triple to G₁ for a new data property p that does not yet occur in G₁ and G₂. Then, the RDF encoding r of a universal data property restriction expression on property p is created for g. Let r have the new root blank node y. Add r to G₁ together with the RDF triple "y owl:equivalentClass y".
11. For each sub graph g of G₂ that is an RDF sequence with root blank node x, which does not occur in the RDF encoding of language constructs already treated by one of the earlier steps, i.e. g is part of the encoding of an axiom: create the RDF encoding r of an enumeration class expression with a new root blank node y having the main RDF triple "y owl:oneOf x". Then, add r to G₁ together with the RDF triple "y owl:equivalentClass y". Additionally, for every IRI u being a member of the RDF sequence, add to G₁ a typing triple "u rdf:type owl:NamedIndividual". If one of the sequence members is a blank node z that is the root node of some property expression or class expression e, then select a new IRI w not yet occurring in G₁, consistently replace z by w everywhere in r, add to G₁ the triple "w owl:equivalentProperty z" or "w owl:equivalentClass z", respectively, and add to G₁ the two triples "w rdf:type owl:NamedIndividual" and "w rdf:type t", where t is the vocabulary class IRI that represents the appropriate entity type of the expression e. No further treatment of e is needed, since e is treated by the earlier steps covering expressions.

In the following it is shown that all the claims of the balancing lemma hold.

A: Existence of a Terminating Algorithm. An algorithm exists for mapping the input graph pair ( G₁^* , G₂^* ) to the output graph pair ( G₁ , G₂ ), since CP (applied in step 2) is described in the form of an algorithm in the OWL 2 RDF Mapping [OWL 2 RDF Mapping], and since all other steps can obviously be performed algorithmically. The algorithm terminates, since CP terminates on arbitrary input graphs, and since all other steps can obviously be executed in finite time.

B: The Resulting RDF Graphs are OWL 2 DL Ontologies. The RDF graphs G₁ and G₂ are OWL 2 DL ontologies in RDF graph form that mutually meet the restrictions on OWL 2 DL ontologies, since the original RDF graphs G₁^* and G₂^* have this feature, and since each of the steps described above transforms a pair of RDF graphs with this feature again into a pair of RDF graphs with this feature, for the following reasons:

• The consistent substitution of blank nodes in step 1 does not change the structure of an OWL 2 DL ontology.
• The application of CP in step 2 does not change the graphs.
• Annotations, deprecation statements and annotation property axioms are optional information in an OWL 2 DL ontology and can therefore be omitted in step 3.
• The ontology header of an OWL 2 DL ontology does neither require the existence of an ontology IRI nor of any ontology properties, and so the substitution of the ontology header in step 4 is a valid operation.
• If an entity has some particular entity type for which there is no explicitly given entity declaration, then the entity declaration may be added, as done in step 5.
• It is allowed to add arbitrary annotations to the ontology header of an OWL 2 DL ontology, as done in step 6.
• Entity declarations may be copied from G₂ to G₁ in step 7 without conflict, since the original ontologies have been assumed to mutually meet the restrictions on OWL 2 DL ontologies regarding different entity declarations for the same IRI (e.g. that one IRI must not be the name of both an object property and a data property).
• Adding to G₁ an axiom that claims equivalence of some property expression (step 8) or class expression (step 9) with itself, where the expression already occurs in G₂, is an allowed operation, since the original ontologies are assumed to mutually meet the restrictions on OWL 2 DL ontologies concerning property and class expressions, and since no syntactic restrictions exist on this specific use of equivalence axioms.
• For the case of data ranges (step 10) it is sufficient to note that placing universal property restrictions on arbitrary (simple or complex) property expressions is allowed in OWL 2 DL. The rest of the argumentation follows the lines of the treatment of class expressions in step 9.
• For the treatment of RDF sequences in step 11: First, the enumeration class expressions being constructed from the RDF sequences are syntactically valid in OWL 2 DL, since all enumerated entries are IRIs by construction. Second, there is no restriction in OWL 2 DL disallowing axioms that claim equivalence of enumeration class expressions with themselves. Third, punning in OWL 2 DL allows a given non-built-in IRI of any entity type to be additionally declared as a named individual. Forth, there is no OWL 2 DL restriction forbidding to add an entity declaration for a new (i.e. not elsewhere used) IRI and to assert the denotation of this new IRI to be equivalent to some existing property or class expression. Hence, the resulting ontologies still mutually meet all syntactic restrictions on OWL 2 DL ontologies.

C: The Resulting Pair of RDF Graphs is Balanced. All the conditions of balanced pairs of RDF graphs are met by the pair ( G₁ , G₂ ) for the following reasons:

• Condition 1: It has already been shown in paragraph B that G₁ and G₂ mutually meet the restrictions on OWL 2 DL ontologies.
• Conditions 2.1 and 2.2 on nodes in G₁ and G₂ are met by steps 5 and 6, respectively.
• Condition 3 on ontology headers in G₂ is satisfied by step 4, always applying an anonymous ontology header.
• Conditions 4.1, 4.3 and 4.4 on annotations, deprecation statements and annotation property axioms in G₂, respectively, are all satisfied by step 3.
• Condition 4.2 on statements with ontology properties is implicitly satisfied by step 4, since the substitution of the ontology header in G₂ removes all existing statements with ontology properties.
• Condition 5.1 on entity declarations in G₂ being reflected in G₁ is satisfied by step 7.
• Conditions 5.2, 5.3 and 5.4 on property, class and data range expressions in G₂, respectively, being reflected in G₁ are met by steps 8, 9 and 10, respectively.
• Condition 5.5 on RDF sequences in G₂ being reflected in G₁ is satisfied by step 11.

D: The Resulting Ontologies are semantically equivalent with the Original Ontologies under the OWL 2 Direct Semantics. F(G₁) is semantically equivalent with F(G₁^*), since F(G₁) differs from F(G₁^*) only by (potentially):

• additional entity declarations (steps 5, 7 and 11), which have no formal meaning under the OWL 2 Direct Semantics;
• additional annotations (step 6), which have no formal meaning;
• additional tautological axioms (steps 8, 9, 10 and 11), which do not change the formal meaning;

F(G₂) is semantically equivalent with F(G₂^*), since F(G₂) differs from F(G₂^*) only by (potentially):

• differently labeled anonymous individuals (step 1), by which the formal meaning under the OWL 2 Direct Semantics keeps unchanged, since anonymous individuals are existentially interpreted;
• missing annotations, deprecation statements and annotation property axioms (step 3), which have no formal meaning;
• a modified ontology header (step 4), which has no formal meaning;
• additional entity declarations (step 5), which have no formal meaning.

End of Proof for the Balancing Lemma.

In the following, the correspondence theorem will be proven.

Assume that the premises of the correspondence theorem are true for a given pair ( G₁^* , G₂^* ) of RDF graphs. This allows for applying the balancing lemma, which provides the existence of corresponding RDF graphs G₁ and G₂ that are OWL 2 DL ontologies in RDF graph form, and which meet the definition of balanced graph pairs. Let F(G₁) and F(G₂) be the corresponding OWL 2 DL ontologies in Functional Syntax form. Then, the claimed relationship 1 of the correspondence theorem follows directly from relationship 1 of the balancing lemma and from condition 1 of the definition of balanced graph pairs. Further, the claimed relationships 2 and 3 of the correspondence theorem follow directly from the relationships 2 and 3 of the balancing lemma, respectively.

The rest of this proof will treat the claimed relationship 4 of the correspondence theorem, which states that if F(G₁) OWL 2 Direct entails F(G₂) with respect to F(D), then G₁ OWL 2 RDF-Based entails G₂ with respect to D. For this to see, an arbitrary OWL 2 RDF-Based interpretation I will be selected that OWL 2 RDF-Based satisfies G₁. For I, a closely corresponding OWL 2 Direct interpretation J will be constructed, and it will then be shown that J OWL 2 Direct satisfies F(G₁). Since it was assumed that F(G₁) OWL 2 Direct entails F(G₂), it will follow that J OWL 2 Direct satisfies F(G₂). Based on this result, it will then be possible to show that I also OWL 2 RDF-Based satisfies G₂. Since I was arbitrarily selected, this will mean that G₁ OWL 2 RDF-Based entails G₂.

Step 1: Selection of a Pair of Corresponding Interpretations.

Let F(G₁) OWL 2 Direct entail F(G₂) w.r.t. F(D), and let I be an OWL 2 RDF-Based interpretation of a vocabulary V^I w.r.t. D, such that I OWL 2 RDF-Based satisfies G₁.

Since the pair ( G₁ , G₂ ) is balanced, there exist entity declarations in F(G₁) for each entity type of every non-built-in IRI occurring in G₁: For each entity declaration of the form "Declaration(T(u))" in F(G₁), such that T is the entity type for some IRI u, a typing triple of the form "u rdf:type t" exists in G₁, where t is the vocabulary class IRI representing the part of the universe of I that corresponds to T. Since I OWL 2 RDF-Based satisfies G₁, all these declaration typing triples are OWL 2 RDF-Based satisfied by I, and thus all non-built-in IRIs in G₁ are instances of all their declared parts of the universe of I.

The vocabulary V^J := ( V^J_C , V^J_OP , V^J_DP , V^J_I , V^J_DT , V^J_LT , V^J_FA ) of the OWL 2 Direct interpretation J w.r.t. the datatype map F(D) is now constructed as follows.

• The set V^J_C of classes contains all IRIs in V^I that are declared as classes in F(G₁), together with all the required class names listed in Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics].
• The set V^J_OP of object properties contains all IRIs in V^I that are declared as object properties in F(G₁), together with all the required object property names listed in Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics].
• The set V^J_DP of data properties contains all IRIs in V^I that are declared as data properties in F(G₁), together with all the required data property names listed in Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics].
• The set V^J_I of individuals contains all IRIs in V^I that are declared as named individuals in F(G₁), and additionally all anonymous individuals occurring in F(G₁) and F(G₂).
• The set V^J_DT of datatypes is defined according to Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics] w.r.t. the datatype map F(D), together with all other IRIs in V^I that are declared as datatypes in F(G₁).
• The set V^J_LT of literals is defined according to Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics] w.r.t. the datatype map F(D).
• The set V^J_FA of facet-literal pairs is defined according to Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics] w.r.t. the datatype map F(D).

The OWL 2 Direct interpretation J := ( Δ_I , Δ_D , ⋅ ^C , ⋅ ^OP , ⋅ ^DP , ⋅ ^I , ⋅ ^DT , ⋅ ^LT , ⋅ ^FA ) is now defined as follows. The object and data domains of J are identified with the universe IR and the set of data values LV of I, respectively, i.e., Δ_I := IR and Δ_D := LV. The class interpretation function ⋅ ^C, the object property interpretation function ⋅ ^OP, the data property interpretation function ⋅ ^DP, the datatype interpretation function ⋅ ^DT, the literal interpretation function ⋅ ^LT, and the facet interpretation function ⋅ ^FA are defined according to Section 2.2 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. Specifically, for every non-built-in IRI u occurring in F(G₁):

• If u is declared as a class, then set u^C := ICEXT(I(u)), since G₁ contains the triple "u rdf:type owl:Class", i.e., I(u) ∈ IC.
• If u is declared as an object property, then set u^OP := IEXT(I(u)), since G₁ contains the triple "u rdf:type owl:ObjectProperty", i.e., I(u) ∈ IP.
• If u is declared as a data property, then set u^DP := IEXT(I(u)), since G₁ contains the triple "u rdf:type owl:DatatypeProperty", i.e., I(u) ∈ IODP.
• If u is declared as a named individual, then set u^I := I(u), since G₁ contains the triple "u rdf:type owl:NamedIndividual", i.e., I(u) ∈ IR.
• If u is declared as a datatype, then set u^DT := ICEXT(I(u)), since G₁ contains the triple "u rdf:type rdfs:Datatype", i.e., I(u) ∈ IDC.

Notes:

• A literal occurring in G₁ is mapped by the reverse RDF mapping to the same literal in F(G₁), and the formal meaning of a well-formed literal is analog for both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics.
• A blank node b occurring in G₁ that represents an anonymous individual is written as the same blank node b in F(G₁). Both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics treat anonymous individuals in an analog way as existential variables defined locally to a given ontology, i.e. some individual x exists in the universe to which all occurrences of b in the ontology can be mapped (see Section 1.5 in the RDF Semantics [RDF Semantics] for the precise definition on how blank nodes are treated under the OWL 2 RDF-Based Semantics). Hence, the same mapping from b to x can be used with both I and J.
• G₁ may also contain declarations for annotation properties. Since annotation properties have no formal meaning under the OWL 2 Direct Semantics, the OWL 2 Direct interpretation J does not treat them.
• With the above definition it is possible for J to have a nonseparated vocabulary according to Section 5.9 of the OWL 2 Structural Specification [OWL 2 Specification]. Since G₁ is an OWL 2 DL ontology in RDF graph form, it is allowed that the same IRI u may be declared as one or more of an individual name, either a class name or a datatype name, and either an object property name or a data property name. For the OWL 2 RDF-Based interpretation I, the IRI u will always denote the same individual in the universe IR, where I(u) may additionally have a class extension or a property extension, or both. For the OWL 2 Direct interpretation J, however, u will denote as an individual name an element of Δ_I, as a class name a subset of Δ_I, as a datatype name a subset of Δ_D, as an object property name a subset of Δ_I × Δ_I, and as a data property name a subset of Δ_I × Δ_D.

Step 2: Satisfaction of F(G₁) by the OWL 2 Direct Interpretation.

Based on the premise that I OWL 2 RDF-Based satisfies G₁, it has to be shown that J OWL 2 Direct satisfies F(G₁). For this to hold, it will be sufficient that J OWL 2 Direct satisfies every axiom A occurring in F(G₁). Let g_A be the sub graph of G₁ that is mapped to A by the reverse RDF mapping. The basic idea can roughly be described as follows:

Since I is an OWL 2 RDF-Based interpretation, all the OWL 2 RDF-Based semantic conditions are met by I. Due to the close alignment between the definitions in the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics, OWL 2 RDF-Based semantic conditions exist that semantically correspond to the definition of the interpretation of the axiom A. In particular, the antecedent of one of these semantic conditions will become true, if the RDF-encoding of A, i.e. the graph g_A, is satisfied (in the case of an "if-and-only-if" semantic condition this will generally be the left-to-right direction of that condition). Now, all the RDF triples in g_A are OWL 2 RDF-Based satisfied by I, since I OWL 2 RDF-Based satisfies G₁. Hence, the antecedent of the semantic condition becomes true, and therefore its consequent becomes true as well. This will reveal a certain semantic relationship that according to I holds between the denotations of the IRIs, literals and anonymous individuals occurring in g_A, which, roughly speaking, expresses the meaning of the OWL 2 axiom A. Because of the close semantic correspondence of the OWL 2 Direct interpretation J to I, the analog semantic relationship holds according to J between the denotations of the IRIs, literals and anonymous individuals occurring in A. This semantic relationship turns out to be compatible with the formal meaning of the axiom A as specified by the OWL 2 Direct Semantics, i.e. J satisfies A.

This basic idea is now demonstrated in more detail for a single example axiom A in F(G₁), which can be taken as a hint on how a complete proof taking into account every feature of the OWL 2 RDF-Based Semantics could be constructed in principle.

Since J OWL 2 Direct satisfies F(G₁), and since it has been postulated that F(G₁) OWL 2 Direct entails F(G₂), it follows that J OWL 2 Direct satisfies F(G₂).

Step 3: Satisfaction of G₂ by the OWL 2 RDF-Based Interpretation.

The last step will be to show that I OWL 2 RDF-Based satisfies G₂. For this to hold, I needs to OWL 2 RDF-Based satisfy every triple occurring in G₂. The basic idea can roughly be described as follows:

First: According to the "semantic conditions for ground graphs" in Section 1.4 of the RDF Semantics specification [RDF Semantics], all the IRIs and literals used in RDF triples in G₂ need to be in the vocabulary V^I of I. This is true for the following reason: Since the pair ( G₁ , G₂ ) is balanced, all IRIs and literals occurring in G₂ do also occur in G₁. Since I satisfies G₁, all IRIs and literals in G₁, including those in G₂, are contained in V^I due to the semantic conditions for ground graphs.

Second: If a set of RDF triples encodes an OWL 2 language construct that is not interpreted by the OWL 2 Direct Semantics, such as annotations, then G₂ should contain such a set of RDF triples only if they are also included in G₁. The reason is that with such triples there will, in general, exist OWL 2 RDF-Based interpretations only satisfying the graph G₁ but not G₂, which will render the pair ( G₁ , G₂ ) into a nonentailment (an exception are RDF triples that are true under every OWL 2 RDF-Based interpretation). Since the pair ( G₁ , G₂ ) is balanced, G₂ will not contain the RDF encoding for any annotations, statements with ontology properties, deprecation statements or annotation property axioms. Hence, there are no corresponding RDF triples that need to be satisfied by I.

Third: Since G₂ is an OWL 2 DL ontology in RDF graph form, the graph is partitioned by the reverse RDF mapping [OWL 2 RDF Mapping] into sub graphs corresponding to either ontology headers, entity declarations or axioms, where axioms may further consist of different kinds of expressions, such as Boolean class expressions. It has to be shown that all the triples in each such sub graph are OWL 2 RDF-Based satisfied by I.

For ontology headers: Let A be the ontology header of F(G₂) and let g_A be the corresponding sub graph of G₂. Since the pair ( G₁ , G₂ ) is balanced, g_A is encoded as a single RDF triple of the form "x rdf:type owl:Ontology", where x is either an IRI or a blank node. Since G₁ is an OWL 2 DL ontology in RDF graph form, G₁ also contains the encoding of an ontology header including a triple g₁ of the form "y rdf:type owl:Ontology", where y is either an IRI or a blank node. Since I OWL 2 RDF-Based satisfies G₁, g₁ is satisfied by I. If both y and x are IRIs, then, due to balancing, x equals y, and therefore g_A equals g₁, i.e. g_A is OWL 2 RDF-Based satisfied by I. Otherwise, balancing forces x to be a blank node, i.e. x is treated as an existential variable under the OWL 2 RDF-Based Semantics according to the "semantic conditions for blank nodes" [RDF Semantics]. From this observation, and from the premise that I satisfies g₁, it follows that g_A is OWL 2 RDF-Based satisfied by I.

For entity declarations: Let A be an entity declaration in F(G₂), and let g_A be the corresponding sub graph of G₂. Since the pair ( G₁ , G₂ ) is balanced, A occurs in F(G₁), and hence g_A is a sub graph of G₁. Since I OWL 2 RDF-Based satisfies G₁, I OWL 2 RDF-Based satisfies g_A.

For axioms: Let A be an axiom in F(G₂), and let g_A be the corresponding sub graph of G₂. Since I is an OWL 2 RDF-Based interpretation, all the OWL 2 RDF-Based semantic conditions are met by I. Due to the close alignment between the definitions in the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics, OWL 2 RDF-Based semantic conditions exist that semantically correspond to the definition of the interpretation of the axiom A. In particular, the consequent of one of these semantic conditions corresponds to the RDF-encoding of A, i.e. the graph g_A, except for declaration typing triples, for which satisfaction has already been shown (in the case of an "if-and-only-if" semantic condition this will generally be the right-to-left direction of that condition). Hence, in order to show that g_A is OWL 2 RDF-Based satisfied by I, it will be sufficient to show that the antecedent of this semantic condition is true. In general, several requirements have to be met to ensure this:

Requirement 1: The denotations of all the non-built-in IRIs in g_A have to be contained in the appropriate part of the universe of I. This can be shown as follows. For every non-built-in IRI u occurring in g_A, u also occurs in A. Since the pair ( G₁ , G₂ ) is balanced, there are entity declarations in F(G₂) for all the entity types of u, each being of the form D := "Declaration(T(u))" for some entity type T. From the reverse RDF mapping follows that for each such declaration D a typing triple d exists in G₂, being of the form d := "u rdf:type t", where t is the vocabulary class IRI representing the part of the universe of I that corresponds to the entity type T. It has already been shown that, for D being an entity declaration in F(G₂) and d being the corresponding sub graph in G₂, I OWL 2 RDF-Based satisfies d. Hence, I(u) is an individual contained in the appropriate part of the universe.

Requirement 2: For every expression E occurring in A, with the RDF encoding g_E in g_A, an individual has to exist in the universe of I that appropriately represents the denotation of E. Since I is an OWL 2 RDF-Based interpretation, all the OWL 2 RDF-Based semantic conditions are met by I. Due to the close alignment between the definitions in the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics, OWL 2 RDF-Based semantic conditions exist that semantically correspond to the definition of the interpretation of the expression E. In particular, the antecedent of one of these semantic conditions will become true, if the RDF-encoding of E, i.e. the graph g_E, is satisfied (in the case of an "if-and-only-if" semantic condition this will generally be the left-to-right direction of that condition). Now, since the pair ( G₁ , G₂ ) is balanced, g_E also occurs in G₁. So, since I OWL 2 RDF-Based satisfies G₁, g_E is OWL 2 RDF-Based satisfied by I. Hence, the antecedent of the semantic condition becomes true, and therefore its consequent becomes true as well. This will result in the existence of an individual with the required properties, when taking into account existential blank node semantics.

Requirement 3: A semantic relationship has to hold between the denotations of the IRIs, literals and anonymous individuals occurring in g_A with respect to I, which, roughly speaking, expresses the meaning of the OWL 2 axiom A. This is the case for the following reasons: First, the literals and anonymous individuals occurring in A and g_A, respectively, are interpreted in an analog way under the OWL 2 Direct Semantics and the OWL 2 RDF-Based Semantics. Second, it was assumed that the OWL 2 Direct interpretation J OWL 2 Direct satisfies A, and therefore a semantic relationship with the desired properties holds with respect to J. Third, J has been defined in close correspondence to I, so that for the semantic relationship expressed by J an analog semantic relationship holds with respect to I.

This basic idea is now demonstrated in more detail for a single example axiom A in F(G₂), which can be taken as a hint on how a complete proof taking into account every feature of the OWL 2 RDF-Based Semantics could be constructed in principle.

To conclude, for any OWL 2 RDF-Based interpretation I that OWL 2 RDF-Based satisfies G₁, I also OWL 2 RDF-Based satisfies G₂. Hence, G₁ OWL 2 RDF-Based entails G₂, and therefore relationship 4 of the correspondence theorem holds. Q.E.D.

8 Appendix: Comprehension Conditions (Informative)

The correspondence theorem in Section 7.2 shows that it is possible for the OWL 2 RDF-Based Semantics to reflect all the entailments of the OWL 2 Direct Semantics [OWL 2 Direct Semantics], provided that one allows for certain "harmless" syntactic transformations on the RDF graphs being considered. This makes numerous potentially desirable and useful entailments available that would otherwise be outside the scope of the OWL 2 RDF-Based Semantics, for the technical reasons discussed in Section 7.1. It seems natural to ask for similar entailments even when an entailment query does not consist of OWL 2 DL ontologies in RDF graph form. However, the correspondence theorem does not apply to such cases, and thus the OWL 2 Direct Semantics cannot be taken as a reference frame for "desirable" and "useful" entailments, or for when a graph transformation can be considered "harmless" or not.

As discussed in Section 7.1, a core obstacle for the correspondence theorem to hold was the RDF encoding of OWL 2 expressions, such as union class expressions, when they appear on the right hand side of an entailment query. Under the OWL 2 RDF-Based Semantics it is not generally ensured that an individual exists, which represents the denotation of such an expression. The "comprehension conditions" defined in this section are additional semantic conditions that provide the necessary individuals for every sequence, class and property expression. By this, the combination of the normative semantic conditions of the OWL 2 RDF-Based Semantics (Section 5) and the comprehension conditions can be regarded to "simulate" the semantic expressivity of the OWL 2 Direct Semantics on entailment queries consisting of arbitrary RDF graphs.

The combined semantics is, however, not primarily intended for use in actual implementations. The comprehension conditions add significantly to the complexity and expressivity of the basic semantics and, in fact, have proven to lead to formal inconsistency. But the combined semantics can still be seen as a generalized reference frame for "desirable" and "useful" entailments, and this can be used, for example, to evaluate methods that syntactically transform unrestricted entailment queries in order to receive additional entailments under the OWL 2 RDF-Based Semantics. Such a concrete method is, however, outside the scope of this specification.

Note: The conventions in the introduction of Section 5 ("Semantic Conditions") apply to the current section as well.

8.1 Comprehension Conditions for Sequences

Table 8.1 lists the comprehension conditions for sequences, i.e. RDF lists. These comprehension conditions provide the existence of sequences built from any finite combination of individuals contained in the universe.

8.2 Comprehension Conditions for Boolean Connectives

Table 8.2 lists the comprehension conditions for Boolean connectives (see Section 5.4 for the corresponding semantic conditions). These comprehension conditions provide the existence of complements for any class and datatype, and of intersections and unions built from any finite set of classes contained in the universe.

8.3 Comprehension Conditions for Enumerations

Table 8.3 lists the comprehension conditions for enumerations (see Section 5.5 for the corresponding semantic conditions). These comprehension conditions provide the existence of enumeration classes built from any finite set of individuals contained in the universe.

8.4 Comprehension Conditions for Property Restrictions

Table 8.4 lists the comprehension conditions for property restrictions (see Section 5.6 for the corresponding semantic conditions). These comprehension conditions provide the existence of cardinality restrictions on any property and for any nonnegative integer, as well as value restrictions on any property and on any class contained in the universe.

Note that the comprehension conditions for self restrictions constrains the right hand side of the produced owl:hasSelf assertions to be the Boolean value "true"^^xsd:boolean. This is in accordance with Table 13 in Section 3.2.4 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping].

Implementations are not required to support the comprehension conditions for owl:onProperties, but MAY support them in order to realize n-ary dataranges with arity ≥ 2 (see Sections 7 and 8.4 of the OWL 2 Structural Specification [OWL 2 Specification] for further information).