RIF RDF and OWL Compatibility

W3C Editor's Draft 10 April 2008

This version:

http://www.w3.org/2005/rules/wg/draft/ED-rif-rdf-owl-20080410/

Latest editor's draft:

http://www.w3.org/2005/rules/wg/draft/rif-rdf-owl/

Previous version:

http://www.w3.org/2005/rules/wg/draft/ED-rif-rdf-owl-20080222/ (color-coded diff)

Editors:

Jos de Bruijn, Free University of Bozen/Bolzano

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Copyright © 2008 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May Be Superseded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/.

Set of Documents

This document is being published as one of a set of 3 documents:

1. RIF Basic Logic Dialect
2. RIF Framework for Logic Dialects
3. RIF RDF and OWL Compatibility (this document)

Note for Working Group

This draft is ready for Working Group Review. A publication decision is scheduled for 15 April.

Please Comment By 2008-04-14

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

No Endorsement

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

Patents

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The Rule Interchange Format (RIF), specifically the Basic Logic Dialect (BLD) (RIF-BLD), is a format for interchanging logical rules over the Web. Rules that are exchanged using RIF may refer to external data sources and may be based on data models that are represented using a language different from RIF. The Resource Description Framework RDF (RDF-Concepts) is a Web-based language for the representation and exchange of data; RDF Schema (RDFS) (RDF-Schema) and the Web Ontology Language OWL (OWL-Reference) are Web-based languages for representing and exchanging ontologies (i.e., data models). This document specifies how combinations of RIF BLD Rulesets and RDF data and RDFS and OWL ontologies must be interpreted; i.e., it specifies how RIF interoperates with RDF/OWL.

The RIF working group plans to develop further dialects besides BLD, most notably a dialect based on Production Rules (RIF-PRD); these dialects are not necessarily extensions of BLD. Future versions of this document may address compatibility of these dialects with RDF and OWL as well. In the remainder of the document, RIF is understood to refer to RIF BLD (RIF-BLD).

RDF data and RDFS and OWL ontologies are represented using RDF graphs. Several syntaxes have been proposed for the exchange of RDF graphs, the normative syntax being RDF/XML (RDF-Syntax). RIF does not provide a format for exchanging RDF graphs, since this would be a duplication. Instead, it is assumed that RDF graphs are exchanged using RDF/XML, or any other syntax that can be used for representing or exchanging RDF graphs.

A typical scenario for the use of RIF with RDF/OWL is the exchange of rules that either use RDF data or an RDFS or OWL ontology: an interchange partner A has a rules language that is RDF/OWL-aware, i.e., it supports the use of RDF data, it uses an RDFS or OWL ontology, or it extends RDF(S)/OWL. A sends its rules using RIF, possibly with references to the appropriate RDF graph(s), to partner B. B receives the rules and retrieves the referenced RDF graph(s) (published as, e.g., RDF/XML (RDF-SYNTAX)). The rules are translated to the internal rules language of B and are processed, together with the RDF graphs, using the RDF/OWL-aware rule engine of B. The use case Vocabulary Mapping for Data Integration (RIF-UCR) is an example of the interchange of RIF rules that use RDF data and RDFS ontologies.

A specialization of this scenario is the publication of RIF rules that refer to RDF graphs: publication is a special kind of interchange. A rule publisher A publishes its rules on the Web. There may be several consumers that retrieve the RIF rules and RDF graphs from the Web, translate the RIF rules to their own rules languages, and process them together with the RDF graphs in their own rules engine. The use case Publishing Rules for Interlinked Metadata (RIF-UCR) illustrates the publication scenario.

Another specialization of the exchange scenario is the interchange of rule extensions to OWL (RIF-UCR). The intention of the rule publisher in this scenario is to extend an OWL ontology with rules: interchange partner A has a rules language that extends OWL. A splits its ontology+rules description into a separate OWL ontology and a RIF ruleset, publishes the OWL ontology, and sends (or publishes) the RIF ruleset, which includes a reference to the OWL ontology. A consumer of the rules retrieves the OWL ontology and translates the ontology and ruleset into a combined ontology+rules description in its own rule extension of OWL.

A RIF ruleset that refers to RDF graphs and/or RDFS/OWL ontologies, or any use of a RIF ruleset with RDF graphs, is viewed as a combination of a ruleset and a number of graphs and ontologies. This document specifies how, in such a combination, the ruleset and the graphs and ontologies interoperate in a technical sense, i.e., the conditions under which the combination is satisfiable (i.e., consistent), as well as the entailments (i.e., logical consequences) of the combination. The interaction between RIF and RDF/OWL is realized by connecting the model theory of RIF (specified in (RIF-BLD)) with the model theories of RDF (specified in (RDF-Semantics)) and OWL (specified in (OWL-Semantics)), respectively.

Throughout this document the following conventions are used when writing RIF and RDF statements in examples and definitions.

• RIF constants in the symbol space rif:iri, i.e., constants that are absolute IRIs, are abbreviated. Specifically, constants of the form "absolute-IRI"^^rif:iri are written as compact IRIs (CURIE), i.e., as prefix:localname, where prefix is understood to refer to an IRI namespace-IRI, and prefix:localname stands for the IRI (absolute-IRI) obtained by concatenating namespace-IRI and localname.
• RDF triples are written using the Turtle syntax (Turtle): for the purposes of this document, triples are written as s p o, where s, p, o are IRIs delimited with '<' and '>', compact IRIs prefix:localname, or typed literals "literal"^^datatype-IRI.
• The following namespace prefixes are used throughout this document: ex refers to the example namespace http://example.org/example#, xsd refers to the XML schema namespace http://www.w3.org/2001/XMLSchema#, rdf refers to the RDF namespace http://www.w3.org/1999/02/22-rdf-syntax-ns#, rdfs refers to the RDFS namespace http://www.w3.org/2000/01/rdf-schema#, owl refers to the OWL namespace http://www.w3.org/2002/07/owl#, and rif refers to the RIF namespace http://www.w3.org/2007/rif#.

The RDF semantics specification (RDF-Semantics) defines four notions of entailment for RDF graphs. At this stage it has not been decided which of these notions are of interest in RIF. The OWL semantics specification (OWL-Semantics) defines two notions of entailment for OWL ontologies, namely OWL Lite/DL and OWL Full; both notions are of interest in RIF. This document specifies the interaction between RIF and RDF/OWL for all six notions.

EDITOR'S NOTE: Currently, this document only defines how combinations of RIF rulesets and RDF/OWL should be interpreted; it does not suggest how references to RDF graphs and OWL ontologies are specified in RIF, nor does it specify which of the RDF/OWL entailment regimes (simple, RDF, RDFS, D, OWL DL, OWL Full) should be used. Finally, there are two notions of interoperation with OWL DL: one that does not consider annotation properties, and one that does it is an open issue whether this distinction should be reflected in the syntax. Possible ways to refer to RDF graphs and RDFS/OWL ontologies include annotations in RIF rulesets and extensions of the syntax of RIF. Note that no agreement has yet been reached on this issue, and that especially the issue of the specification of entailment regimes is controversial (see http://lists.w3.org/Archives/Public/public-rif-wg/2007Jul/0030.html and the ensuing thread). See the Annotations page for a proposal for extending RIF with annotations.

The #Appendix: Embeddings (Informative) describes how reasoning with combinations of RIF rules with RDF and a subset of OWL DL can be reduced to reasoning with RIF rulesets, which can be seen as a guide to describing how a RIF processor could be turned into an RDF/OWL-aware RIF processor. This reduction can be seen as a guide for interchange partners that do not have RDF-aware rule systems, but want to be able to process RIF rules that refer to RDF graphs. In terms of the aforementioned scenario: if the interchange partner B does not have an RDF/OWL-aware rule system, but B can process RIF rules, then the appendix explains how B's rule system could be used for processing RIF-RDF.

ex:john ex:brotherOf ex:jack . 
ex:jack ex:parentOf ex:mary . 

Forall ?x, ?y, ?z (?x[ex:uncleOf -> ?z] :- 
     And(?x[ex:brotherOf -> ?y] ?y[ex:parentOf -> ?z]))

Typed literals in RDF may be ill-typed, which means that the literal string is not part of the lexical space of the datatype under consideration. Examples of such ill-typed literals are "abc"^^xsd:integer, "2"^^xsd:boolean, and "<non-valid-XML"^^rdf:XMLLiteral. Ill-typed literals are not expected to be used very often. However, as the RDF recommendation (RDF-Concepts) allows creating RDF graphs with ill-typed literals, their occurrence cannot be completely ruled out.

Rules that include ill-typed symbols are not legal RIF rules, so there are no RIF symbols that correspond to ill-typed literals. As with blank nodes, variables do range over objects denoted by such literals. The following example illustrates the interaction between RDF and RIF in the face of ill-typed literals and blank nodes.

Consider a combination of an RDF graph that contains the triple

_:x ex:hasName "a"^^xsd:integer . 

saying that there is some blank node that has a name, which is an ill-typed literal, and a RIF ruleset that contains the rules

Forall ?x, ?y ( ?x[rdf:type -> ex:nameBearer] :- ?x[ex:hasName -> ?y] )
Forall ?x, ?y ( "http://a"^^rif:iri["http://p"^^rif:iri -> ?y] :- ?x[ex:hasName -> ?y] )

which say that whenever there is a some x that has some name y, then x is of type ex:nameBearer and http://a has a property http://p with value y.

From this combination the following RIF condition formula can be derived:

Exists ?z ( And( ?z[rdf:type -> ex:nameBearer]  "http://a"^^rif:iri["http://p"^^rif:iri -> ?z] ))

as can the following RDF triples:

_:y rdf:type ex:nameBearer .
<http://a> <http://p> "a"^^xsd:integer . 

However, "http://a"^^rif:iri["http://p"^^rif:iri -> "a"^^xsd:integer] cannot be derived, because it is not a well-formed RIF formula, due to the fact that "a" is not an integer; it is not in the lexical space of the datatype xsd:integer.

This remainder of this section formally defines combinations of RIF rules with RDF graphs and the semantics of such combinations. A combination consists of a RIF ruleset and a set of RDF graphs. The semantics of combinations is defined in terms of combined models, which are pairs of RIF and RDF interpretations. The interaction between the two interpretations is defined through a number of conditions. Entailment is defined as model inclusion, as usual.

An RDF vocabulary V consists of the following sets of names:

• absolute IRIs V_U, (corresponds to the Concepts and Abstract Syntax term "RDF URI references"; see the (End note on RDF URI references))

• plain literals V_PL (i.e., character strings with an optional language tag), and

• typed literals V_TL (i.e., pairs of character strings and datatype IRIs).

The syntax of the names in these sets is defined in RDF Concepts and Abstract Syntax (RDF-Concepts). Besides these names, there is an infinite set of blank nodes, which is disjoint from the sets of literals and IRIs.

DEFINITION: Given an RDF vocabulary V, a generalized RDF triple of V is a statement of the form s p o, where s, p and o are names in V or blank nodes.

DEFINITION: Given an RDF vocabulary V, a generalized RDF graph is a set of generalized RDF triples of V.

(See the (End note on generalized RDF graphs))

Even though RDF allows the use of arbitrary datatype IRIs in typed literals, not all such datatype IRIs are recognized in the semantics. In fact, simple entailment does not recognize any datatype and RDF and RDFS entailment recognize only the datatype rdf:XMLLiteral. Furthermore, RDF allows expressing typed literals for which the literal string is not in the lexical space of the datatype; such literals are called ill-typed literals. RIF, in contrast, does not allow ill-typed literals in the syntax. To facilitate discussing datatypes, and specifically datatypes supported in specific contexts (required for D-entailment), the notion of datatype maps (RDF-Semantics) is used.

A datatype map is a partial mapping from IRIs to datatypes.

RDFS, specifically D-entailment, allows the use of arbitrary datatype maps, as long as the rdf:XMLLiteral datatype is included in the map. RIF BLD additionally requires the following datatypes to be included: xsd:string, xsd:decimal, xsd:time, xsd:date, xsd:dateTime, and rif:text; these datatypes are the RIF-required datatypes. A conforming datatype map is a datatype map that recognizes at least the RIF-required datatypes.

DEFINITION: A datatype map D is a conforming datatype map if it satisfies the following conditions:

1. No RIF-supported symbol space that is not a datatype (these are rif:local and rif:iri in RIF BLD) is in the domain of D.
2. The IRIs identifying all RIF-required datatypes are in the domain of D.
3. D maps IRIs identifying XML schema datatypes to the respective data types (XML-SCHEMA2), rdf:XMLLiteral to the rdf:XMLLiteral datatype (RDF-Concepts), and rif:text to the rif:text primitive datatype (RIF-BLD).

The notions of well- and ill-typed literals loosely correspond to the notions of legal and illegal symbols in RIF:

DEFINITION: Given a conforming datatype map D, a typed literal (s, d) is a well-typed literal if

1. d is in the domain of D and s is in the lexical space of D(d),
2. d is the IRI of a symbol space supported by RIF BLD and s is in the lexical space of the symbol space, or
3. d is not in the domain of D and does not identify a symbol space supported by RIF.

Otherwise (s, d) is an ill-typed literal.

DEFINITION: A RIF-RDF-combination is a pair < R,S>, where R is a RIF ruleset and S is a set of generalized RDF graphs of a vocabulary V.

When clear from the context, RIF-RDF-combinations are referred to simply as combinations.

The RDF Semantics document (RDF-Semantics) defines four normative kinds of interpretations, as well as corresponding notions of satisfiability and entailment:

• simple interpretations, which do not impose any conditions on the RDF and RDFS vocabularies,
• RDF interpretations, which impose additional conditions on the RDF vocabulary,
• RDFS interpretations, which impose additional conditions on the RDFS vocabulary, and
• D-interpretations, which impose additional conditions on the treatment of datatypes, relative to a datatype map D.

Those four types of interpretations are reflected in the definitions of satisfaction and entailment in this section.

The correspondence between RIF semantic structures (interpretations) and RDF interpretations is defined through a number of conditions that ensure the correspondence in the interpretation of names (i.e., IRIs and literals) and formulas, i.e., the correspondence between RDF triples of the form s p o and RIF frames of the form s'[p' -> o'], where s', p', and o' are RIF symbols corresponding to the RDF symbols s, p, and o, respectively (cf. Table 1).

As defined in (RDF-Semantics), a simple interpretation of a vocabulary V is a tuple I=< IR, IP, IEXT, IS, IL, LV >, where

• IR is a non-empty set of resources (the domain),
• IP is a set of properties,
• IEXT is an extension function, which is a mapping from IP into the power set of IR × IR,
• IS is a mapping from IRIs in V into (IR union IP),
• IL is a mapping from typed literals in V into IR, and
• LV is the set of literal values, which is a subset of IR, and includes all plain literals in V.

Rdf-, rdfs-, and D-interpretations are simple interpretations that satisfy certain conditions:

• A simple interpretation I of a vocabulary V is an rdf-interpretation if V includes the RDF vocabulary and I satisfies the rdf axiomatic triples and the rdf semantic conditions.
• An rdf-interpretation I of a vocabulary V is an rdfs-interpretation if V includes the RDFS vocabulary and I satisfies the rdfs axiomatic triples and the rdfs semantic conditions.
• Given a datatype map D, an rdfs-interpretation I of a vocabulary V is a D-interpretation if V includes the IRIs in the domain of D and I satisfies the general semantic conditions for datatypes for every pair <d, D(d)> such that d is in the domain of D.

As defined in (RIF-BLD), a semantic structure is a tuple of the form I = <TV, DTS, D, D_ind, D_func, I_C, I_V, I_F, I_frame, I_SF, I_sub, I_isa, I₌, I_external, I_truth>. The specification of RIF-RDF compatibility is only concerned with DTS, D, I_C, I_V, I_frame, I_sub, I_isa, and I_truth. The other mappings that are parts of a semantic structure are not used in the definition of combinations.

• DTS is the set of datatypes, which have associated datatype identifiers,
• D is a set (the domain),
• D_ind is a non-empty subset of D,
• I_C is a mapping from individual identifiers in Const to D_ind and from function symbols in Const to D,
• I_V is a mapping from Var to D_ind,
• I_frame is a mapping from D_ind to functions of the form SetOfFiniteBags(D_ind × D_ind) → D,
• I_sub is a mapping from D_ind × D_ind to D,
• I_isa is a mapping from D_ind × D_ind to D, and
• I_truth is a mapping from D to TV.

DEFINITION: A common-rif-rdf-interpretation is a pair (I, I), where I = <TV, DTS, D, D_ind, D_func, I_C, I_V, I_F, I_frame, I_SF, I_sub, I_isa, I₌, I_external, I_truth> is a RIF semantic structure and I=<IR, IP, IEXT, IS, IL, LV> is an RDF interpretation of a vocabulary V, such that the following conditions hold:

1. (IR union IP) = D_ind;
2. IP is a superset of the set of all k in D_ind such that there exist some a, b in D_ind and I_truth(I_frame(k)(a,b))=t;
3. LV is a superset of (D_ind intersection (union of the value spaces of all datatypes in DTS));
4. IEXT(k) = the set of all pairs (a, b), with a, b, and k in D_ind, such that I_truth(I_frame(k)(a,b))=t;
5. IS(i) = I_C("i"^^rif:iri) for every absolute IRI i in V_U;
6. IL((s, d)) = I_C("s"^^d) for every well-typed literal (s, d) in V_TL;
7. IEXT(IS(rdf:type)) is equal to the set of all pairs <a,b> in D_ind × D_ind such that I_truth(I_isa(< a,b >))=t; and
8. IEXT(IS(rdfs:subClassOf)) is a superset of the set of all pairs <a,b> in D_ind × D_ind such that I_truth(I_sub(< a,b >))=t.

Condition 1 ensures that the combination of resources and properties corresponds exactly to the RIF domain; note that if I is an rdf-, rdfs-, or D-interpretation, IP is a subset of IR, and thus IR=D. Condition 2 ensures that the set of RDF properties at least includes all elements that are used as properties in frames in the RIF domain. Condition 3 ensures that all concrete values in D are included in LV. Condition 4 ensures that RDF triples are interpreted in the same way as frame formulas. Condition 5 ensures that IRIs are interpreted in the same way. Condition 6 ensures that typed literals are interpreted in the same way. Note that no correspondences are defined for the mapping of names in RDF that are not symbols of RIF, e.g., ill-typed literals and RDF URI references that are not absolute IRIs. Condition 7 ensures that typing in RDF and typing in RIF correspond, i.e., a rdf:type b is true iff a # b is true. Finally, condition 8 ensures that whenever a RIF subclass statement holds, the corresponding RDF subclass statement holds as well, i.e., a rdfs:subClassOf b is true if a ## b is true.

One consequence of conditions 5 and 6 is that IRIs of the form http://iri and typed literals of the form "http://iri"^^rif:iri that occur in an RDF graph are treated the same in RIF-RDF-combinations, even if the RIF Ruleset is empty. For example, consider the combination of an empty ruleset and an RDF graph that contains the triple

<http://a> <http://p> "http://b"^^rif:iri .
 

This combination allows the derivation of, among other things, the following triple:

<http://a> <http://p> <http://b> .

as well as the following frame formula:

"http://a"^^rif:iri ["http://p"^^rif:iri -> "http://b"^^rif:iri]

DEFINITION: A common-rif-rdf-interpretation (I, I) satisfies a RIF-RDF-combination C=< R, S > if I is a model of R and I satisfies every RDF graph S in S; in this case (I, I) is called a simple model, or model, of C, and C is satisfiable. (I, I) satisfies a generalized RDF graph S if I satisfies S. (I, I) satisfies a closed RIF-BLD condition φ if TVal_I((φ)=t.

Notice that not every combination is satisfiable. In fact, not every RIF ruleset has a model. For example, the ruleset consisting of the rule

Forall ("1"^^xsd:integer="2"^^xsd:integer)

does not have a model, since the symbols "1"^^xsd:integer and "2"^^xsd:integer are mapped to the (distinct) numbers 1 and 2, respectively, in every semantic structure.

Rdf-, rdfs-, and D-satisfiability are defined through additional restrictions on I:

DEFINITION: A model (I, I) of a combination C is an rdf-model of C if I is an rdf-interpretation; in this case C is rdf-satisfiable.

DEFINITION: A model (I, I) of a combination C is an rdfs-model of C if I is an rdfs-interpretation; in this case C is rdfs-satisfiable.

DEFINITION: Given a conforming datatype map D, a model (I, I) of a combination C is a D-model of C if I is a D-interpretation; in this case C is D-satisfiable.

DEFINITION: Given a conforming datatype map D, a RIF-RDF-combination C D-entails a generalized RDF graph S if every D-model of C satisfies S. Likewise, C D-entails a closed RIF-BLD condition φ if every D-model of C satisfies φ.

The other notions of entailment are defined analogously:

DEFINITION: A combination C simple-entails S (resp., φ) if every simple model of C satisfies S (resp., φ).

DEFINITION: A combination C rdf-entails S (resp., φ) if every rdf-model of C satisfies S (resp., φ).

DEFINITION: A combination C rdfs-entails S (resp., φ) if every rdfs-model of C satisfies S (resp., φ).

OWL (OWL-Reference) specifies three increasingly expressive species, namely Lite, DL, and Full. OWL Lite is a syntactic subset of OWL DL, but the semantics is the same (OWL-Semantics). Since every OWL Lite ontology is an OWL DL ontology, the Lite species is not considered separately in this document.

Syntactically speaking, OWL DL is a subset of OWL Full, but the semantics of the DL and Full species are different (OWL-Semantics). While OWL DL has an abstract syntax with a direct model-theoretic semantics, the semantics of OWL Full is an extension of the semantics of RDFS, and is defined on the RDF syntax of OWL. Consequently, the OWL Full semantics does not extend the OWL DL semantics; however, all derivations sanctioned by the OWL DL semantics are sanctioned by the OWL Full semantics.

Finally, the OWL DL RDF syntax, which is based on the OWL abstract syntax, does not extend the RDF syntax, but rather restricts it: every OWL DL ontology is an RDF graph, but not every RDF graph is an OWL DL ontology. OWL Full and RDF have the same syntax: every RDF graph is an OWL Full ontology and vice versa. This syntactical difference is reflected in the definition of RIF-OWL compatibility: combinations of RIF with OWL DL are based on the OWL abstract syntax, whereas combinations with OWL Full are based on the RDF syntax.

Since the OWL Full syntax is the same as the RDF syntax and the OWL Full semantics is an extension of the RDF semantics, the definition of RIF-OWL Full compatibility is a straightforward extension of RIF-RDF compatibility. Defining RIF-OWL DL compatibility in the same way would entail losing certain semantic properties of OWL DL. One of the main reasons for this is the difference in the way classes and properties are interpreted in OWL Full and OWL DL. In the Full species, classes and properties are both interpreted as objects in the domain of interpretation, which are then associated with subsets of and binary relations over the domain of interpretation using rdf:type and the extension function IEXT, as in RDF. In the DL species, classes and properties are directly interpreted as subsets of and binary relations over the domain. The latter is a key property of Description Logic semantics that enables the use of Description Logic reasoning techniques for processing OWL DL descriptions. Defining RIF-OWL DL compatibility as an extension of RIF-RDF compatibility would define a correspondence between OWL DL statements and RIF frame formulas. Since RIF frame formulas are interpreted using an extension function, the same way as in RDF, defining the correspondence between them and OWL DL statements would change the semantics of OWL statements, even if the RIF ruleset is empty. Consider, for example, an OWL DL ontology with a class membership statement

a rdf:type C .

This statement says that the set denoted by C contains at least one element that is denoted by a. The corresponding RIF frame formula is

a[rdf:type -> C]

The terms a, rdf:type, and C are all interpreted as elements in the individual domain, and the pair of elements denoted by a and C is in the extension of the element denoted by rdf:type.

This semantic discrepancy has practical implications in terms of entailments. Consider, for example, an OWL DL ontology with two class membership statements

a rdf:type C .
D rdf:type owl:Class .

and a RIF ruleset

Forall ?x ?y ?x=?y 

which says that every element is the same as every other element (note that such statements can also be written in OWL using owl:Thing and owl:hasValue). From the naïve combination of the two one can derive C=D, and indeed

a rdf:type D .

This derivation is not sanctioned by the OWL DL semantics, because even if every element is the same as every other element, the class D might be interpreted as the empty set.

A RIF-OWL combination that is faithful to the OWL DL semantics requires interpreting classes and properties as sets and binary relations, respectively, suggesting that correspondence could be defined with unary and binary predicates. It is, however, also desirable that there be uniform syntax for the RIF component of both OWL DL and RDF/OWL Full combinations, because one may not know at time of writing the rules which type of inference will be used. Consider, for example, an RDF graph S with the following statement

a rdf:type C .

and a RIF ruleset with the rule

Forall ?x ?x[rdf:type -> D] :- ?x[rdf:type -> C]

The combination of the two, according to the specification of [#RDF_Compatibility], allows deriving

a rdf:type D .

Now, the RDF graph S is also an OWL DL ontology. Therefore, one would expect the triple to be derived by RIF-OWL DL combinations as well.

To ensure that the RIF-OWL DL combination is faithful to the OWL DL semantics and to enable using the same, or similar, rules with both OWL DL and RDF/OWL Full, the interpretation of frame formulas s[p -> o] in the RIF-OWL DL combinations is slightly different from their interpretation in RIF BLD and syntactical restrictions are imposed on the use of variables and function terms and frame formulas.

Note that the abstract syntax form of OWL DL allows so-called punning (this is not allowed in the RDF syntax), i.e., the same IRI may be used in an individual position, a property position, and a class position; the interpretation of the IRI depends on its context. Since combinations of RIF and OWL DL are based on the abstract syntax of OWL DL, punning may also be used in these combinations. This paves the way towards combination with OWL 1.1, which is envisioned to allow punning in all its syntaxes.

EDITOR'S NOTE: The semantics of RIF-OWL DL combinations is similar in spirit to the Semantic Web Rule Language proposal. However, a reference to SWRL from the above text does not seem appropriate.

Since RDF graphs and OWL Full ontologies cannot be distinguished, the syntax of RIF-OWL Full combinations is the same as the syntax of RIF-RDF-combinations.

The syntax of OWL ontologies in RIF-OWL DL combinations is specified by the abstract syntax of OWL DL. Certain restrictions are imposed on the syntax of the RIF rules in combinations with OWL DL. Specifically, the only terms allowed in class and property positions in frame formulas are constant symbols.

DEFINITION: A RIF-BLD condition φ is a RIF DL-condition if for every frame formula a[b -> c] in φ it holds that b is a constant and if b = rdf:type, then c is a constant.

DEFINITION: A RIF-BLD ruleset R is a DL-Ruleset if for every frame formula a[b -> c] in every rule of R it holds that b is a constant and if b = rdf:type, then c is a constant.

DEFINITION: A RIF-OWL-DL-combination is a pair < R,O>, where R is a DL-Ruleset and O is a set of OWL DL ontologies in abstract syntax form of a vocabulary V.

When clear from the context, RIF-OWL-DL-combinations are referred to simply as combinations.

In the literature, several restrictions on the use of variables in combinations of rules and Description Logics have been identified (Motik05, Rosati06) for the purpose of decidable reasoning. These restrictions are specified for RIF-OWL-DL combinations.

Given a set of OWL DL ontologies in abstract syntax form O, a variable ?x in a RIF rule Q then :- if is DL-safe if it occurs in an atomic formula in if that is not of the form s[P -> o] or s[rdf:type -> A], where P or A, respectively, occurs in one of the ontologies in O. A RIF rule Q then :- if is DL-safe, given O if every variable that occurs in then :- if is DL-safe. A RIF rule Q then :- if is weakly DL-safe, given O if every variable that occurs in then is DL-safe and every variable in if that is not DL-safe occurs only in atomic formulas in if that are of the form s[P -> o] or s[rdf:type -> A], where P or A, respectively, occurs in one of the ontologies in O.

EDITOR'S NOTE: it is not strictly necessary to disallow disjunctions in the definition, but it would make the definition a lot more complex. It would require defining the disjunctive normal form of a condition formula and define safeness with respect to each disjunct. Given that the safeness restriction is meant for implementation purposes, and that converting rules to disjunctive normal form is extremely expensive, it is probably a reasonable restriction to disallow disjunction.

DEFINITION: A RIF-OWL-DL-combination < R,O> is DL-safe if every rule in R is DL-safe, given O. A RIF-OWL-DL-combination < R,O> is weakly DL-safe if every rule in R is weakly DL-safe, given O.

EDITOR'S NOTE: Do we want additional safeness restrictions to ensure that variables do not cross the abstract-concrete domain boundary?

The semantics of RIF-OWL Full combinations is a straightforward extension of the #Semantics of RIF-RDF Combinations.

The semantics of RIF-OWL-DL-combinations cannot straightforwardly extend the semantics of RIF RDF combinations, because OWL DL does not extend the RDF semantics. In order to keep the syntax of the rules uniform between RIF-OWL Full and RIF-OWL-DL-combinations, the semantics of RIF frame formulas is slightly altered in RIF-OWL-DL-combinations.

A D-interpretation I is an OWL Full interpretation if it interprets the OWL vocabulary and it satisfies the conditions in the sections 5.2 and 5.3 in (OWL Semantics).

The semantics of RIF-OWL Full combinations is a straightforward extension of the semantics of RIF-RDF-combinations. It is based on the same notion of common-interpretations, but defines additional notions of satisfiability and entailment.

DEFINITION: Given a conforming datatype map D, a common-rif-rdf-interpretation (I, I) is an OWL-Full-model of a RIF-RDF-combination C=< R, S > if I is a model of R, I is an OWL Full interpretation, and I satisfies every RDF graph S in S; in this case C is OWL-Full-satisfiable.

DEFINITION: Given a conforming datatype map D, a RIF-RDF-combination C OWL-Full-entails a generalized RDF graph S if every OWL-Full-model of C satisfies S. Likewise, C OWL-Full-entails a closed RIF-BLD condition φ if every OWL-Full-model of C satisfies φ.

The modification of the semantics of RIF frame formulas is achieved by modifying the mapping function for frame formulas (I_frame), and leaving the RIF BLD semantics (RIF-BLD) otherwise unchanged.

Namely, frame formulas of the form s[rdf:type -> o] are interpreted as membership of s in the set denoted by o and frame formulas of the form s[p -> o], where p is not rdf:type, as membership of the pair (s, o) in the binary relation denoted by p.

DEFINITION: A RIF DL-semantic structure is a tuple I = <TV, DTS, D, D_ind, D_func, I_C, I_V, I_F, I_frame', I_SF, I_sub, I_isa, I₌, I_external, I_truth>, where I_frame' is a mapping from D_ind to total functions of the form SetOfFiniteFrame'Bags(D × D) → D, such that for each pair (a, b) in SetOfFiniteFrame'Bags(D × D) holds that if a≠I_C(rdf:type), then b in D_ind; all other elements of the structure are defined as in RIF semantic structures.

We define I(o[a₁->v₁ ... a_k->v_k]) = I_frame(I(o))({<I(a₁),I(v₁)>, ..., <I(a_n),I(v_n)>}). The truth valuation function TVal_I is then defined as in RIF BLD.

DEFINITION: A RIF DL-semantic structure I is a model of a DL-Ruleset R if TVal_I(R)=t.

As defined in (OWL-Semantics), an abstract OWL interpretation with respect to a datatype map D, with vocabulary V is a tuple I=< R, EC, ER, L, S, LV >, where

• R is a non-empty set of resources (the domain); O is a non-empty subset of R, disjoint from LV,
• EC is a mapping from class and datatype identifiers to subsets of O and LV, respectively,
• ER is a mapping from property identifiers to subsets of R × R,
• L is a mapping from typed literals in V into LV,
• S is a mapping from IRIs in V into R, and
• LV is the set of literal values, which is a subset of R, and includes all plain literals in V.

The OWL semantics imposes a number of further restrictions on the mapping functions as well as the set of resources R, to achieve a separation of the interpretation of class, datatype, ontology property, datatype property, annotation property, and ontology property identifiers.

DEFINITION: Given a conforming datatype map D, a common-rif-dl-interpretation is a pair (I, I), where I = <TV, DTS, D, I_C, I_V, I_F, I_frame', I_SF, I_sub, I_isa, I₌, I_Truth> is a RIF DL-semantic structure and I=<R, EC, ER, L, S, LV> is an abstract OWL interpretation with respect to D of a vocabulary V, such that the following conditions hold

1. (O union LV)=D_ind;
2. EC(u) = set of all objects k in O such that I_truth(I_frame'(I_C(rdf:type))(k,I_C("u"^^rif:iri))) = t, for every class identifier u in V;
3. ER(u) = set of all pairs (k, l) in O × (O union LV) such that I_truth(I_frame'(I_C("u"^^rif:iri))( k, l ))) = t (true), for every data valued and individual valued property identifier u in V;
4. L((s, d)) = I_C("s"^^d) for every well-typed literal (s, d) in V;
5. S(i) = I_C("i"^^rif:iri) for every individual identifier i in V.

Condition 1 ensures that the relevant parts of the domains of interpretation are the same. Condition 2 ensures that the interpretation (extension) of an OWL DL class u corresponds to the interpretation of frames of the form ?x[rdf:type -> "u"^^rif:iri]. Condition 3 ensures that the interpretation (extension) of an OWL DL object or datatype property u corresponds to to the interpretation of frames of the form ?x["u"^^rif:iri -> ?y]. Condition 4 ensures that typed literals of the form (s, d) in OWL DL are interpreted in the same way as constants of the form "s"^^d in RIF. Finally, condition 5 ensures that individual identifiers in the OWL ontologies and the RIF rulesets are interpreted in the same way.

Using the definition of common-rif-dl-interpretation, satisfaction, models, and entailment are defined in the usual way:

DEFINITION: Given a conforming datatype map D, a common-rif-dl-interpretation (I, I) is an OWL-DL-model of a RIF-OWL-DL-combination C=< R, O > if I is a model of R and I satisfies every OWL DL ontology in abstract syntax form O in O; in this case C is OWL-DL-satisfiable. (I, I) is an OWL-DL-model of an OWL DL ontology in abstract syntax form O if I satisfies O. (I, I) is an OWL-DL-model of a closed RIF-BLD condition φ if I_truth(φ)=t.

DEFINITION: Given a conforming datatype map D, a RIF-OWL-DL-combination C OWL-DL-entails an OWL DL ontology in abstract syntax form O if every OWL-DL-model of C is an OWL-DL-model of O. Likewise, C OWL-DL-entails a closed RIF DL-condition φ if every OWL-DL-model of C is an OWL-DL-model of φ.

Recall that in an abstract OWL interpretation I the sets O, which is used for interpreting individuals, and LV, which is used for interpreting literals (data values), are disjoint and that EC maps class identifiers to subsets of O and datatype identifiers to subsets of LV. The disjointness entails that data values cannot be member sof a class and individuals cannot be members of a datatype.

In RIF, variable quantification ranges over D_ind. So, the same variable may be assigned to an abstract individual or a concrete data value. Additionally, RIF constants (e.g., IRIs) denoting individuals can be written in place of a data value, such as the value of a data-valued property or in datatype membership statements; similar for constants denoting data values. Such statements cannot be satisfied in any common-rif-dl-interpretation, due to the constraints on the EC and ER functions. The following example illustrates several such statements.

Consider the datatype xsd:string and a RIF-OWL DL combination consisting of the set containing only the OWL DL ontology

ex:myiri rdf:type ex:A .

and a RIF ruleset containing the following fact

ex:myiri[rdf:type -> xsd:string]

This combination is not OWL-DL-satisfiable, because ex:myiri is an individual identifier and S maps individual identifiers to elements in O, which is disjoint from the elements in the datatype xsd:string.

Consider a RIF-OWL DL combination consisting of the set containing only the OWL DL ontology

ex:hasChild rdf:type owl:ObjectProperty .

and a RIF ruleset containing the following fact

ex:myiri[ex:hasChild -> "John"^^xsd:string]

This combination is not OWL-DL-satisfiable, because ex:hasChild is an object property, and values of object properties may not be concrete data values.

Consider a RIF-OWL DL combination consisting of the OWL DL ontology

ex:A rdfs:subClassOf ex:B

and a RIF ruleset containing the following rule

Forall ?x ?x[rdf:type -> ex:A]

This combination is not OWL-DL-satisfiable, because the rule requires every element, including every concrete data value, to be a member of the class ex:A. However, the mapping EC in any abstract OWL interpretation requires every member of ex:A to be an element of O, and concrete data values may not be members of O.

DEFINITION: Given a conforming datatype map D, a common-rif-dl-interpretation (I, I) is a common-DL-annotation-interpretation if the following condition holds

  6. ER(u) = set of all pairs (k, l) in O × O such that I_truth(I_frame'(I_C("u"^^rif:iri))( k, l) ) = t (true), for every IRI u in V.

Condition 6, which strengthens condition 3, ensures that the interpretation of all properties (also annotation and ontology properties) in the OWL DL ontologies corresponds with their interpretation in the RIF rules.

DEFINITION: Given a conforming datatype map D, a common-DL-annotation-interpretation (I, I) is an OWL-DL-annotation-model of a RIF-OWL-DL-combination C=< R, O > if I is a DL-model of R and I satisfies every OWL DL ontology in abstract syntax form O in O; in this case C is OWL-DL-annotation-satisfiable.

DEFINITION: Given a conforming datatype map D, a RIF-RDF-combination C OWL-DL-annotation-entails an OWL DL ontology in abstract syntax form O if every OWL-DL-annotation-model of C is an OWL-DL-model of O. Likewise, C OWL-DL-annotation-entails a closed RIF-BLD condition φ if every OWL-DL-annotation-model of C is an OWL-DL-model of φ.

The difference between the two kinds of OWL DL entailment can be illustrated using an example. Consider the following OWL DL ontology in abstract syntax form

Ontology (ex:myOntology
  Annotation(dc:title "Example ontology"))

which defines an ontology with a single annotation (title). Consider also a ruleset consisting of the following rule:

Forall ?x, ?y ( ?x[ex:hasTitle -> ?y] :- ?x[dc:title -> ?y])

which says that whenever something has a dc:title, it has the same ex:hasTitle.

The combination of the ontology and the ruleset OWL-DL-annotation-entails the RIF condition formula ex:myOntology[ex:hasTitle -> "Example ontology"^^xsd:string]; the combination does not OWL-DL-entail the formula.

[OWL-Semantics]

OWL Web Ontology Language Semantics and Abstract Syntax, P. F. Patel-Schneider, P. Hayes, I. Horrocks, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-semantics-20040210/. Latest version available at http://www.w3.org/TR/owl-semantics/.

[RDF-CONCEPTS]

Resource Description Framework (RDF): Concepts and Abstract Syntax, G. Klyne, J. Carroll (Editors), W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/. Latest version available at http://www.w3.org/TR/rdf-concepts/.

[RDF-SEMANTICS]

RDF Semantics, P. Hayes, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-mt-20040210/. Latest version available at http://www.w3.org/TR/rdf-mt/.

[RIF-BLD]

RIF Basic Logic Dialect Harold Boley, Michael Kifer, eds. W3C Editor's Draft, 10 April 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-bld-20080410/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-bld/.

[XML-SCHEMA2]

XML Schema Part 2: Datatypes, W3C Recommendation, World Wide Web Consortium, 2 May 2001. This version is http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/. Latest version available at http://www.w3.org/TR/xmlschema-2/.

[CURIE]

CURIE Syntax 1.0, M. Birbeck and S. McCarron, W3C Working Draft, 26 November 2007, http://www.w3.org/TR/2007/WD-curie-20071126/. Latest version available at http://www.w3.org/TR/curie/.

[DLP]

Description Logic Programs: Combining Logic Programs with Description Logics, B. Grosof, R. Volz, I. Horrocks, S. Decker. In Proc. of the 12th International World Wide Web Conference (WWW 2003), 2003.

[Motik05]

Query Answering for OWL-DL with rules, B. Motik, U. Sattler, R. Studer, Journal of Web Semantics 3(1): 41-60, 2005.

[RDF-Schema]

RDF Vocabulary Description Language 1.0: RDF Schema, D. Brickley, R.V. Guha, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-schema-20040210/. Latest version available at http://www.w3.org/TR/rdf-schema/.

[RDF-SYNTAX]

RDF/XML Syntax Specification (Revised), D. Beckett, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-syntax-grammar-20040210/. Latest version available at http://www.w3.org/TR/rdf-syntax-grammar/.

[RFC-3066]

RFC 3066 - Tags for the Identification of Languages, H. Alvestrand, IETF, January 2001. This document is http://www.isi.edu/in-notes/rfc3066.txt.

[RIF-PRD]

RIF Production Rule dialect, C. de Sainte Marie (Editors), Editor's Draft. Latest version available at http://www.w3.org/2005/rules/wg/wiki/PRdialect.

[RIF-UCR]

RIF Use Cases and Requirements, A. Ginsberg, D. Hirtle, F. !McCabe, P.-L. Patranjan (Editors), W3C Working Draft, 10 July 2006, http://www.w3.org/TR/2006/WD-rif-ucr-20060710/. Latest version available at http://www.w3.org/TR/rif-ucr/.

[OWL-Reference]

OWL Web Ontology Language Reference, M. Dean, G. Schreiber, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-ref-20040210/. Latest version available at http://www.w3.org/TR/owl-ref/.

[Rosati06]

DL+log: Tight Integration of Description Logics and Disjunctive Datalog, R. Rosati, Proceedings of the 10th International Conference on Principles of Knowledge Representation and Reasoning, pp. 68-78, 2005.

[Turtle]

Turtle - Terse RDF Triple Language, D. Beckett and T. Berners-Lee, W3C Team Submission, 14 January 2008, http://www.w3.org/TeamSubmission/2008/SUBM-turtle-20080114/. Latest version available at http://www.w3.org/TeamSubmission/turtle/.

Given a combination C=< R,S>, the function tr maps RDF symbols of a vocabulary V and a set of blank nodes B to RIF symbols, as defined in following table.

The embedding is not defined for combinations that include RDF graphs with RDF URI references that are not absolute IRIs.

   Forall ?x ?x[rdf:type -> rdf:Property] :- Exists ?y,?z (?y[?x -> ?z]),

   Forall ?x ?x[rdf:type -> rdf:XMLLiteral] :- wellxml(?x),

   Forall ?x "1"^^xsd:integer="2"^^xsd:integer :- And(?x[rdf:type -> rdf:XMLLiteral] illxml(?x))

     Forall ?x ?x[rdf:type -> rdfs:Resource],

     Forall ?u,?v,?x,?y ?u[rdf:type -> ?y] :- And(?x[rdfs:domain -> ?y] ?u[?x -> ?v]),

     Forall ?u,?v,?x,?y ?v[rdf:type -> ?y] :- And(?x[rdfs:range -> ?y] ?u[?x -> ?v]),

     Forall ?x ?x[rdfs:subPropertyOf -> ?x] :- ?x[rdf:type -> rdf:Property],

     Forall ?x,?y,?z ?x[rdfs:subPropertyOf -> ?z] :- And (?x[rdfs:subPropertyOf -> ?y] ?y[rdfs:subPropertyOf -> ?z]),

     Forall ?x,?y,?z1,?z2 ?z1[?y -> ?z2] :- And (?x[rdfs:subPropertyOf -> ?y] ?z1[?x -> ?z2]),

     Forall ?x ?x[rdfs:subClassOf -> rdfs:Resource] :- ?x[rdf:type -> rdfs:Class],

     Forall ?x,?y,?z ?z[rdf:type -> ?y] :- And (?x[rdfs:subClassOf -> ?y] ?z[rdf:type -> ?x]),

     Forall ?x ?x[rdfs:subClassOf -> ?x] :- ?x[rdf:type -> rdfs:Class],

     Forall ?x,?y,?z ?x[rdfs:subClassOf -> ?z] :- And (?x[rdfs:subClassOf -> ?y] ?y[rdfs:subClassOf -> ?z]),

     Forall ?x ?x[rdfs:subPropertyOf -> rdfs:member] :- ?x[rdf:type -> rdfs:ContainerMembershipProperty],

     Forall ?x ?x[rdfs:subClassOf -> rdfs:Literal] :- ?x[rdf:type -> rdfs:Datatype],

     Forall ?x ?x[rdf:type -> rdfs:Literal] :- lit(?x),

     Forall ?x "1"^^xsd:integer="2"^^xsd:integer :- And(?x[rdf:type -> rdfs:Literal] illxml(?x))

In this section we therefore consider a subset of OWL DL in RIF-OWL DL combinations. We define OWL DLP, which is inspired by so-called Description Logic programs (DLP), and define how reasoning with RIF-OWL DLP combinations can be reduced to reasoning with RIF.

ontology ::= 'Ontology(' [ ontologyID ] { directive } ')'
directive ::= 'Annotation(' ontologyPropertyID ontologyID ')'
        | 'Annotation(' annotationPropertyID URIreference ')'
        | 'Annotation(' annotationPropertyID dataLiteral ')'
        | 'Annotation(' annotationPropertyID individual ')'
        | axiom
        | fact

datatypeID ::= URIreference
classID ::= URIreference
individualID ::= URIreference
ontologyID ::= URIreference
datavaluedPropertyID ::= URIreference
individualvaluedPropertyID ::= URIreference
annotationPropertyID ::= URIreference
ontologyPropertyID ::= URIreference

dataLiteral ::= typedLiteral | plainLiteral
typedLiteral ::= lexicalForm^^URIreference
plainLiteral ::= lexicalForm | lexicalForm@languageTag
lexicalForm ::= as in RDF, a unicode string in normal form C
languageTag ::= as in RDF, an XML language tag

fact ::= individual 
individual ::= 'Individual(' individualID { annotation } 
  { 'type(' type ')' } { value } ')'
value ::= 'value(' individualvaluedPropertyID individualID ')'
       | 'value(' individualvaluedPropertyID  individual ')'
       | 'value(' datavaluedPropertyID  dataLiteral ')'

type ::= Rdescription

dataRange ::= datatypeID | 'rdfs:Literal'

description ::= classID
           | restriction
           | 'intersectionOf(' { description } ')'

restriction ::= 'restriction(' datavaluedPropertyID dataRestrictionComponent 
  { dataRestrictionComponent } ')'
           | 'restriction(' individualvaluedPropertyID individualRestrictionComponent 
  { individualRestrictionComponent } ')'
dataRestrictionComponent ::= 'value(' dataLiteral ')'
individualRestrictionComponent ::= 'value(' individualID ')'

Ldescription ::= description
           | Lrestriction
           | 'unionOf(' { Ldescription } ')'
           | 'intersectionOf(' { Ldescription } ')'
           | 'oneOf(' { individualID } ')'

Lrestriction ::= 'restriction(' datavaluedPropertyID LdataRestrictionComponent 
  { LdataRestrictionComponent } ')'
           | 'restriction(' individualvaluedPropertyID LindividualRestrictionComponent 
  { LindividualRestrictionComponent } ')'
LdataRestrictionComponent ::= 'someValuesFrom(' dataRange ')'
           | 'value(' dataLiteral ')'
LindividualRestrictionComponent ::= 'someValuesFrom(' description ')'
           | 'value(' individualID ')'

Rdescription ::= description
           | Rrestriction
           | 'intersectionOf(' { Rdescription } ')'

Rrestriction ::= 'restriction(' datavaluedPropertyID RdataRestrictionComponent 
  { RdataRestrictionComponent } ')'
           | 'restriction(' individualvaluedPropertyID RindividualRestrictionComponent 
  { RindividualRestrictionComponent } ')'
RdataRestrictionComponent ::= 'allValuesFrom(' dataRange ')'
           | 'value(' dataLiteral ')'
RindividualRestrictionComponent ::= 'allValuesFrom(' description ')'
           | 'value(' individualID ')'

axiom ::= 'Class(' classID  ['Deprecated'] 'complete' { annotation } { description } ')'
axiom ::= 'Class(' classID  ['Deprecated'] 'partial' { annotation } { Rdescription } ')'

axiom ::= 'DisjointClasses(' Ldescription Ldescription { Ldescription } ')'
       | 'EquivalentClasses(' description { description } ')'
       | 'SubClassOf(' Ldescription Rdescription ')'

axiom ::= 'Datatype(' datatypeID ['Deprecated']  { annotation } )'

 axiom ::= 'DatatypeProperty(' datavaluedPropertyID ['Deprecated'] { annotation } 
               { 'super(' datavaluedPropertyID ')'}
               { 'domain(' description ')' } { 'range(' dataRange ')' } ')'
       | 'ObjectProperty(' individualvaluedPropertyID ['Deprecated'] { annotation } 
               { 'super(' individualvaluedPropertyID ')' }
               [ 'inverseOf(' individualvaluedPropertyID ')' ] [ 'Symmetric' ] 
               [ 'Transitive' ]
               { 'domain(' description ')' } { 'range(' description ')' } ')'
       | 'AnnotationProperty(' annotationPropertyID { annotation } ')'
       | 'OntologyProperty(' ontologyPropertyID { annotation } ')'

 axiom ::= 'EquivalentProperties(' datavaluedPropertyID datavaluedPropertyID  { datavaluedPropertyID } ')'
       | 'SubPropertyOf(' datavaluedPropertyID  datavaluedPropertyID ')'
       | 'EquivalentProperties(' individualvaluedPropertyID individualvaluedPropertyID  { individualvaluedPropertyID } ')'
       | 'SubPropertyOf(' individualvaluedPropertyID  individualvaluedPropertyID ')'

Ontology( [ ontologyID ] 
 directive1
 ... 
 directiven )

...

Individual( individualID 
  annotation1 
  ...
  annotationn 
  type1
  ...
  typem
  value1
  ...
  valuek )

  ...

Individual( individualID value1 )
  ...
Individual( individualID valuek )

Individual( individualID 
  type(intersectionOf(
   description1
   ...
   descriptionn
  ))

  ...

Individual( individualID type(X))

Individual( individualID type(X))

Individual( individualID 
 type(restriction(propertyID 
 allValuesFrom(X))))

SubClassOf( oneOf(individualID) 
  restriction(propertyID allValuesFrom(X)))

Class( classID  [Deprecated] 
 complete 
  annotation1 
  ...
  annotationn 
  description1
  ...
  descriptionm )

EquivalentClasses(classID  
 intersectionOf(description1
  ...
  descriptionm )

Class( classID  [Deprecated] 
 partial
  annotation1 
  ...
  annotationn 
  description1
  ...
  descriptionm )

SubClassOf(classID  
 intersectionOf(description1
  ...
  descriptionm )

DisjointClasses(
  description1
  ...
  descriptionm )

 ...

 ...

EquivalentClasses(
  description1
  ...
  descriptionm )

  ...

SubClassOf(description X)

SubClassOf(description X)

SubClassOf(description 
  ...intersectionOf(
    description1 
    ...
    descriptionn
  )...)

...

DatatypeProperty( propertyID [ Deprecated ]
  annotation1 
  ...
  annotationn 
  super(superproperty1)
  ...
  super(superpropertym)
  domain(domaindescription1)
  ...
  domain(domaindescriptionj)
  range(rangedescription1)
  ...
  range(rangedescriptionk) )

SubPropertyOf(propertyID superproperty1)
  ... 
SubPropertyOf(propertyID superpropertym)

  ...

  ...

ObjectProperty( propertyID [ Deprecated ]
  annotation1 
  ...
  annotationn 
  super(superproperty1)
  ...
  super(superpropertym)
  [ inverseOf( inversePropertyID ) ] 
  [ Symmetric ] 
  [ Transitive ]
  domain(domaindescription1)
  ...
  domain(domaindescriptionl)
  range(rangedescription1)
  ...
  range(rangedescriptionk) )

SubPropertyOf(propertyID superproperty1)
  ... 
SubPropertyOf(propertyID superpropertym)

  ...

  ...

ObjectProperty( propertyID
  [ inverseOf( inversePropertyID ) ] )
ObjectProperty( propertyID [ Symmetric ] )
ObjectProperty( propertyID [ Transitive ] )

EquivalentProperties(
  property1
  ...
  propertym )

  ...

We now proceed with the embedding of normalized OWL DL ontologies into a RIF DL-ruleset. The embedding extends the embedding function tr. The embeddings of IRIs and literals is as defined in the Section #Embedding_Symbols.

EDITOR'S NOTE: This embedding assumes that for a given datatype identifier D, there is unary built-in predicate isD, called the "positive guard" for D, which is always interpreted as the value space of the datatype denoted by D and there is a built-in isNotD, called the "negative guard" for D, which is always interpreted as the complement of the value space of the datatype denoted by D.

directive1 
 ... 
directiven

...

Individual( individualID type(A) )

Individual( individualID 
  type(restriction(propertyID value(b))) )

Individual( individualID 
  value(propertyID b) )

SubPropertyOf(property1 property2)

ObjectProperty(propertyID)

ObjectProperty(property1
  inverseOf(property2) )

ObjectProperty(propertyID
  Symmetric )

ObjectProperty(propertyID
  Transitive )

SubClassOf(description1 description2)

Theorem A RIF-OWL-DL-combination <R,{O₁,...,O_n}>, where O₁,...,O_n are OWL DLP ontologies, is OWL-DL-satisfiable iff tr(R union tr_O(tr_N(O₁)) union ... union tr_O(tr_N(O_n))) has a model.

Theorem An OWL-DL-satisfiable RIF-OWL-DL-combination C=<R,{O₁,...,O_n}>, where O₁,...,O_n are OWL DLP ontologies, OWL-DL-entails a closed RIF condition φ iff tr(R union tr_O(tr_N(O₁)) union ... union tr_O(tr_N(O_n))) entails φ.

Generalized RDF graphs: Standard RDF graphs, as defined in (RDF-Concepts), do not allow the use of literals in subject and predicate positions and blank nodes in predicate positions. The RDF Core working group has listed two issues questioning the restrictions that literals may not occur in subject and blank nodes may not occur in predicate positions in triples. Anticipating lifting of these restrictions in a possible future version of RDF, we use the more liberal notion of generalized RDF graph. We note that the definitions of interpretations, models, and entailment in the RDF semantics document (RDF-Semantics) also apply to such generalized RDF graphs.

We note that every standard RDF graph is a generalized RDF graph. Therefore, our definition of combinations applies to standard RDF graphs as well.