RIF RDF and OWL Compatibility

W3C Editor's Draft 22 February 2008

This version:

http://www.w3.org/2005/rules/wg/draft/ED-rif-rdf-owl-20080222/

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-20080219/ (color-coded diff)

Editors:

Jos de Bruijn, Free University of Bozen/Bolzano

---

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May Be Superseded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 5 documents:

1. RIF Basic Logic Dialect
2. RIF Framework for Logic Dialects
3. RIF Data Types and Built-Ins
4. RIF Use Cases and Requirements
5. RIF RDF and OWL Compatibility (this document)

Please Comment By 19 February 2008

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.

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

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. An individual who has actual knowledge of a patent which the individual believes contains Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.

The Rule Interchange Format (RIF), specifically the Basic Logic Dialect (BLD) [ RIF-BLD ], defines a means for the interchange of (logical) rules over the Web. Rules which are exchanged using RIF may refer to external data sources and may be based on certain data models which 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 the representation and exchange of ontologies (i.e., data models). This document specifies how combinations of RIF BLD Rule sets and RDF data and RDFS and OWL ontologies must be interpreted.

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 will address compatibility of these dialects with RDF and OWL as well. In the remainder of the document, when mentioning RIF, we mean 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 ], which is an XML-based format. RIF does not provide a format for exchanging RDF; 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.

This document does not, as yet, define whether or how RDF documents/graphs should be referred to from RIF rule sets. The specification of combinations in this document does not depend on (the existence of) this mechanism: it applies in case an RIF rule sets explicity points to (one or more) RDF documents, but also in case the references to the RDF document(s) are not interchanged using RIF, but using some other (out of bounds) mechanism.

A typical scenario for the use of RIF with RDF/OWL includes the exchange of rules which either use RDF data or an RDFS or OWL ontology. In terms of rule interchange the scenario is the following: interchange partner A has a rules language which is RDF/OWL-aware, i.e., it allows to use RDF data, it uses an RDFS or OWL ontology, or it extends RDF(S)/OWL. A sends its rules (using RIF), possibly with a reference to the appropriate RDF graph(s), to partner B. B can now translate the RIF rules into its own rules language, retrieve the RDF graph(s) (which is published most likely using RDF/XML), and process the rules in its own rule engine, which is also RDF/OWL-aware. The use case Vocabulary Mapping for Data Integration [ RIF-UCR ] is an example of the interchange of RIF rules which use RDF graphs.

A specialization of this use case is the publication of RIF rules which refer to RDF graphs (notice that publication is a specific kind of interchange). In such a scenario, a rule publisher A publishes its rules on the Web. There may be several consumers who retrieve the RIF rules and RDF graphs from the Web, and translate the RIF rules to their own rules languages. The use case Publishing Rules for Interlinked Metadata [ RIF-UCR ] is an example of the publication of RIF rules related to RDF graphs.

Another specialization of this use case 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 an RIF rule set, publishes the OWL ontology, and sends (or publishes) the RIF rule set, which includes a reference to the OWL ontology. The consumers of rules retrieves the OWL ontology, and translates the ontology and the rule set into a combined ontology+rules description in its own rules language which also extends OWL.

An RIF rule set which refers to RDF graphs, or any use of an RIF rule set with RDF graphs, is viewed as a combination of an RIF rule set and a number of RDF graphs. This document specifies how, in such a combination, the rule set interacts with the RDF graphs. With "interaction" we mean the conditions under which the combination is satisfiable, as well as the entailments defined for the combination. The interaction between RIF and RDF and 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.

The RDF semantics specification [ RDF-Semantics ] defines 4 notions of entailment for RDF graphs. At this stage it has not yet been decided which of these notions are of interest in RIF. Therefore, we specify the interaction between RIF and RDF for all 4 notions.

EDITOR'S NOTE: Currently, this document only defines how combinations of RIF rule sets and RDF/OWL should be interpreted; it does not suggest how references to RDF graphs are specified in RIF, nor does it specify which of the RDF entailment regimes (simple, RDF, RDFS, or D) should be used. Possible ways to refer to RDF graphs and RDFS/OWL ontologies include annotations in RIF rule sets 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: Embedding RDF Combinations describes how reasoning with combinations of RIF rules with RDF can be reduced to reasoning with RIF rule sets, which can be seen as a guide to describing how an RIF processor could be turned into an RDF-aware RIF processor. This reduction can be seen as a guide for interchange partners which do not have RDF-aware rule systems, but still want to be able to process RIF rules which refer to RDF graphs. In terms of the scenario above: if the interchange partner B does not have an RDF-aware rule system, but B can process RIF rules, then the appendix explains how the rule system could be used for processing combinations.

EDITOR'S NOTE: The future status of the appendix with the embedding is uncertain. The appendix is not about interchange, but rather about possible implementation, so it can be argued that it should not be included in this document. On the other hand, many think the appendix is useful. If we decide not to include it in this document, we might consider publishing it as a separate note (not recommendation-track document).

john brotherOf jack . 
jack parentOf mary . 

Forall ?x, ?y, ?z (?x["uncleOf"^^rif:iri -> ?z] :- 
     And( ?x["brotherOf"^^rif:iri -> ?y] ?y["parentOf"^^rif:iri -> ?z]))

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

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

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

From this combination we can derive the RIF condition formulas

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

as well as the RDF triples

_:y rdf:type 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. Combinations are pairs of RIF rule sets and sets 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, (correspond 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 graph of V is a set of generalized RDF triples s p o ., where s, p and o are names in V or blank nodes.

(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), we use the notion of datatype maps [ RDF-Semantics ].

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 considered. RIF BLD additionally requires the following datatypes to be considered: xsd:long, xsd:string, xsd:integer, xsd:decimal, xsd:time, xsd:dateTime, and rif:text; we call these datatypes the RIF-required datatypes. We define the notion of a conforming datatype map as a datatype map which 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 which is not an RIF-required 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 ].

We now define the notions of well- and ill-typed literals, which loosely correspond to the notions of well-formed and ill-formed 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: An RIF-RDF combination is a pair < R,S>, where R is a Rule set 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 4 (normative) kinds of interpretations, as well as corresponding notions of satisfiability and entailment:

• simple interpretations, which do not pose 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.

This distinction is reflected in the definitions of satisfaction and entailment in this section.

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 which 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, I_C, I_V, I_F, I_frame, I_SF, I_sub, I_isa, I₌, I_Truth>. We restrict our attention here to DTS, D, I_C, I_V, and I_slot. The other mappings which 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 non-empty set (the domain),
• I_C is a mapping from Const to D,
• I_V is a mapping from Var to D, and
• I_frame is (in BLD) a mapping from D to truth-valued functions of the form D × D → TV.

DEFINITION: A common 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 an 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 is a subset of D;

2. IP is a superset of the set of all k in D such that there exist a, b in D and I_frame(k)(a,b)=t (i.e. the truth value of I_frame(k)(a,b) is true);

3. (IR union IP) = D;

4. LV is a subset of IR and a superset of (D intersection (union of all value spaces DS));

5. IEXT(k) = the set of all pairs (a, b), with a, b in D, such that I_slot(k)(a,b)=t, for every k in D;

6. IS(i) = I_C("i"^^rif:iri) for every absolute IRI i in V_U;

7. IL((s, d)) = I_C("s"^^d) for every well-typed literal (s, d) in V_TL;

8. IEXT(IS(rdf:type)) is equal to the set of all pairs <a,b> in D × D such that I_isa(< a,b >)=t; and

9. IEXT(IS(rdfs:subClassOf)) is a superset of the set of all pairs <a,b> in D × D such that I_sub(< a,b >)=t.

Condition 1 ensures that all resources in an RDF interpretation correspond to elements in the RIF domain. Condition 2 ensures that the set of properties at least includes all elements which are used as properties in the RIF domain. Condition 3 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 4 ensures that all concrete values in D are included in LV. Condition 5 ensures that RDF triples are interpreted in the same way as frame formulas. Condition 6 ensures that IRIs are interpreted in the same way. Finally, condition 7 ensures that typed literals are interpreted in the same way. Note that no correspondences are defined for the mapping of names in RDF which are not symbols of RIF, e.g., ill-typed literals and RDF URI references which are not absolute IRIs. Condition 8 the ensures that typing in RDF and typing in RIF correspond, i.e. a rdf:type b is true iff a # b is true. Condition 9 the ensures that whenever an 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 6 and 7 is that IRIs of the form http://iri and typed literals of the form "http://iri"^^rif:iri which occur in an RDF graph are treated the same in RIF-RDF combinations, even if the RIF component is empty. For example, consider an RIF-RDF combination with an empty rule set and an RDF graph which contains the triple

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

This combination allows to derive, among other things, the following triples:

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

as well as the following frame formula:

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

DEFINITION: A common interpretation (I, I) simple-satisfies an RIF-RDF combination C=< R, S > if I satisfies 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 condition formula φ if I_truth(φ)=t.

Notice that not every combination is satisfiable. In fact, not every RIF rule set has a model. For example, the rule set 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 rdf-satisfies C if I is an rdf-interpretation; in this case (I, I) is called an rdf-model of C, and C is rdf-satisfiable.

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

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

DEFINITION: Given a conforming datatype map D, an 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 condition formula 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).

Two kinds of combinations of RIF rules with OWL ontologies are considered. The combination of RIF rules with the Full species of OWL is a straightforward extension of RIF-RDF compatibility (see the definition below), in which RDF triples correspond to RIF frame formulas. The combination of RIF rules with the DL and Lite species of OWL is slightly different; OWL classes and properties correspond to RIF unary and binary predicates, respectively. The discrepancy between the two kinds of combinations is overcome by interpreting frame formulas as unary and binary predicates and imposing certain restrictions on the use of variables in the rules.

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 explicitly considered in the remainder.

Syntactically speaking, OWL DL is a subset of OWL Full. The semantics of the DL and Full species are different, though [ 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, every OWL DL entailment is an OWL Full entailment.

Finally, the OWL DL RDF 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.

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.

Since RDF graphs and OWL Full ontologies cannot be distinguished, we use the notion of RIF-RDF combinations for the syntax of combinations of RIF rule sets with OWL Full ontologies.

For the combination of RIF rule sets with OWL DL ontologies we define the notion of RIF-OWL DL combinations based on the abstract syntax of OWL DL. We need to furthermore impose certain restrictions on the syntax of the rules. Specifically, we can only allow constant symbols in class and property positions.

DEFINITION: An RIF rule set R is a DL Rule set 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: An RIF-OWL DL combination is a pair < R,O>, where R is a DL Rule set 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.

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 extends 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-Full and RIF-OWL DL combinations, the semantics of RIF frame formulas is slightly altered in RIF-OWL DL combinations.

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 interpretation (I, I) OWL Full satisfies an RIF-RDF combination C=< R, S > if I satisfies R, I is an OWL Full interpretation, and I satisfies every RDF graph S in S; in this case (I, I) is called an OWL Full model of C, and C is OWL Full satisfiable.

DEFINITION: Given a conforming datatype map D, an 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 condition formula φ if every OWL Full model of C satisfies φ.

hasUncle(?x,?y) :- And(hasParent(?x,?y) hasBrother(?y,?z))

?x[hasUncle -> ?y] :- And(?x[hasParent -> ?y] ?y[hasBrother -> ?z])

We define a new truth valuation function for RIF formulas, which is the same as the truth valuation function defined in RIF-BLD, with the exception of frame formulas.

DEFINITION: Given an RIF semantic structure I = <TV, DTS, D, I_C, I_V, I_F, I_frame, I_SF, I_sub, I_isa, I₌, I_Truth>, the truth valuation function I_T-DL is obtained by modifying the truth valuation of frame formulas in I_Truth in the following way: I_T-DL (t [ rdf:type -> A ]) = I_R(A)(t) and I_T-DL (t₁ [ P -> t₂ ]) = I_R(P)(t₁, t₂).

DEFINITION: We say that I DL satisfies a rule Q then :- if, where Q is a quantification prefix for all the variables in the rule, if I^*_T-DL(then) ≥ I^*_T-DL(if) for every I^* that agrees with I everywhere except possibly on some variables mentioned in Q. I is a DL model of a rule set R if it DL satisfies every rule in the set.

DEFINITION: Given a conforming datatype map D, a common 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 an RIF 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. R=D;

2. LV is a subset of R and contains the value spaces of all data types in D;

3. EC(u) = set of all objects k in R such that I_R("u"^^rif:iri)(k) = t (true), for every IRI u in V;

4. ER(u) = set of all tuples ( k, l ) such that I_R("u"^^rif:iri)( k, l ) = t (true), for every data valued and individual valued property identifier u in V;

5. L((s, d)) = I_C("s"^^d) for every well-typed literal (s, d) in V;

6. S(i) = I_C("i"^^rif:iri) for every IRI i in V.

Condition 1 ensures that the domains of interpretation are the same. Condition 2 ensures that the set of literal values includes the value spaces of all considered datatypes. Condition 3 ensures that the interpretation (extension) of an OWL DL class u corresponds to the interpretation of the unary predicate with the same name in RIF. Condition 4 ensures that the interpretation (extension) of an OWL DL object or datatype property u corresponds to the interpretation of the binary predicates with the same name in RIF. Condition 5 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 6 ensures that individual identifiers in the OWL ontologies and the RIF rule sets are interpreted in the same way.

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

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

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

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

7. ER(u) = set of all tuples ( k, l ) such that I_R("u"^^rif:iri)( k, l ) = t (true), for every IRI u in V.

Condition 7 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) OWL DL annotation satisfies an RIF-OWL DL combination C=< R, O > if I satisfies R and I satisfies every OWL DL ontology in abstract syntax form O in O; in this case (I, I) is called an OWL DL annotation model of C, and C is OWL DL annotation satisfiable.

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

We illustrate the difference between the two kinds of OWL DL entailment 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 rule set which consists 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 rule set 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, H. Boley, M. Kifer (Editors), W3C Editor's Draft, http://www.w3.org/2005/rules/wg/wiki/BLD. Accessed on 2008-02-13T17:00 UTC.

[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/.

[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/.

For the embedding we use the concrete syntax of RIF and the N-Triples syntax for RDF.

Throughout this section the function tr is defined, which maps symbols, triples, and RDF graphs to RIF symbols, statements, and rule sets.

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 which include RDF graphs with RDF URI references which are not absolute IRIs.

The compact URIs used in the RIF rules in this section and the next are short for the complete URIs with the rif:iri datatype, e.g. rdf:type is short for "http://www.w3.org/1999/02/22-rdf-syntax-ns#type"^^rif:iri

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

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

Theorem A combination <R,{S1,...,Sn}> is rdf-satisfiable iff (R^RDF union R union tr_R(S1) union ... union tr_R(Sn)) has a model.

Theorem A combination C=<R,{S1,...,Sn}> rdf-entails a generalized RDF graph T iff (R^RDF union R union tr_R(S1) union ... union tr_R(Sn)) entails tr_Q(T). C simple-entails an RIF condition φ iff (R^RDF union R union tr_R(S1) union ... union tr_R(Sn)) entails φ.

    (Forall tr(s p o .)) for every <a href="http://www.w3.org/TR/rdf-mt/#RDFS_axiomatic_triples">RDFS axiomatic triple</a> s p o .) union

    (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 u[rdf:type -> rdfs:Datatype] | for every IRI in the domain of D) union

    (Forall "s"^^u[rdf:type -> "u"^^rif:iri] | for every well-typed literal (s , u ) in VTL) union

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

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. We note also that the notion of generalized RDF graphs is more liberal than the notion of RDF graphs used by SPARQL; generalized RDF graphs additionally allow blank nodes and literals in predicate positions.