RIF RDF and OWL Compatibility

W3C Editor's Draft 09 June 2008

This version:

http://www.w3.org/2005/rules/wg/draft/ED-rif-rdf-owl-20080609/

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-20080518/ (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May Be Superseded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 6 documents:

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

Please Comment By 2008-06-13

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) 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 OWL Web Ontology Language (OWL-Reference) are Web-based languages for representing and exchanging ontologies (i.e., data models). This document specifies how combinations of RIF BLD documents and RDF data and RDFS and OWL ontologies are 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. In the remainder, 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 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: one to many, rather than one-to-one. When a rule publisher A publishes its rules on the Web, it is hoped that there are several consumers that retrieve the RIF rules and RDF graphs from the Web, translate the RIF rules to their own rules language, 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 document, publishes the OWL ontology, and sends (or publishes) the RIF document, which includes a reference to the OWL ontology. A consumer of the rules retrieves the OWL ontology and translates the ontology and document into a combined ontology+rules description in its own rule extension of OWL.

A RIF document that refers to (imports) RDF graphs and/or RDFS/OWL ontologies, or any use of a RIF document with RDF graphs, is viewed as a combination of a document and a number of graphs and ontologies. This document specifies how, in such a combination, the document 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 (RIF-BLD) with the model theories of RDF (RDF-Semantics) and OWL (OWL-Semantics), respectively.

The notation of certain symbols, particularly IRIs and plain literals, in RIF is slightly different from the notation in RDF/OWL. This difference is illustrated in the Section Symbols in RIF Versus RDF/OWL.

The RDF semantics specification (RDF-Semantics) defines four notions of entailment for RDF graphs. The OWL semantics specification (OWL-Semantics) defines two notions of entailment for OWL ontologies, namely OWL Lite/DL and OWL Full. This document specifies the interaction between RIF and RDF/OWL for all six notions. The Section RDF Compatibility is concerned with the combination of RIF and RDF/RDFS. The combination of RIF and OWL is addressed in the Section OWL Compatibility. The semantics of the interaction between RIF and OWL DL is close in spirit to (SWRL).

RIF provides a mechanism for referring to (importing) RDF graphs and a means for specifying the context of this import, which corresponds to the intended entailment regime. The Section Importing RDF Graphs in RIF specifies how such import statements are used for representing RIF-RDF and RIF-OWL combinations.

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 documents, 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.

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

• All RIF statements are written using the RIF presentation syntax (RIF-BLD). Where possible, this document uses the shortcut syntax for IRIs and strings as defined in (RIF-DTB).
• 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#, xs 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#.

Where RDF/OWL has four kinds of constants: URI references (i.e., IRIs), plain literals without language tags, plain literals with language tags and typed literals (i.e., Unicode sequences with datatype IRIs) (RDF-Concepts), RIF has one kind of constants: Unicode sequences with symbol space IRIs (DTB).

Symbol spaces can be seen as groups of constants. Every datatype is a symbol space, but there are symbol spaces that are not datatypes. For example, the symbol space rif:iri groups all IRIs. The shortcut syntax for IRIs and strings (RIF-DTB), used throughout this document, corresponds with the syntax for IRIs and plain literals in (Turtle).

• IRIs, i.e., constants of the form "absolute-IRI"^^rif:iri may be written as <absolute-IRI> or 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.
• Strings, i.e., constants of the form "absolute-IRI"^^xs:string may be written as "absolute-IRI".

The correspondence between constant symbols in RDF graphs and RIF documents is explained in Table 1.

RIF does not have a notion corresponding to RDF blank nodes. RIF local symbols, written _symbolname, have some commonality with blank nodes; like the blank node label, the name of a local symbol is not exposed outside of the document. However, in contrast to blank nodes, which are essentially existentially quantified variables, RIF local symbols are constant symbols. Finally, variables in the bodies of RIF rules may be existentially quantified, and are thus similar to blank nodes; however, RIF BLD does not allow existentially quantified variables to occur in rule heads.

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]))

_:x ex:hasName "John" . 

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

Exists ?z ( And( ?z[rdf:type -> ex:nameBearer]  <http://a>[<http://p> -> ?z] ))

_:y rdf:type ex:nameBearer .
<http://a> <http://p> "John" . 

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. 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: xs:string, xs:decimal, xs:time, xs:date, xs: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.

Editor's Note: The list of required data types is to be replaced with a link.

Definition. Let T be the set of considered datatypes (cf. Section 5 of (RIF-BLD)), i.e., T includes at least all data types used in the combination under consideration and all datatypes required for RIF-BLD (RIF-DTB). A datatype map D is a conforming datatype map if it satisfies the following conditions:

1. The IRIs identifying all datatypes in T are in the domain of D.
2. D maps each IRI in its domain to the corresponding datatype in T.   ☐

Editor's Note: The terminology "considered datatype" might change if the terminology is changed in BLD.

The notion of well-typed literal loosely correspond with the notion of legal symbol 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) or
2. d is the IRI of a symbol space required by RIF BLD and s is in the lexical space of the symbol space.   ☐

Definition. A RIF-RDF combination is a pair < R,S>, where R is a RIF document 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.

Recall that Const is the set of constant symbols and Var is the set of variable symbols in RIF.

• 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.

For the purpose of the interpretation of imported documents, RIF BLD defines the notion of semantic multi-structures, which are nonempty sets {I¹, ..., Iⁿ} of semantic structures that are identical in all respects with the exception of the interpretation of local constants.

Given a semantic multi-structure I={I¹, ..., Iⁿ}, we use the symbol I to denote both the multi-structure and the common part of the individual structures I¹, ..., Iⁿ.

Definition. A common-rif-rdf-interpretation is a pair (I, I), where I is a semantic multi-structure and I 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 (union of the value spaces of all considered datatypes);
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>) 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.   ☐

Editor's Note: Make sure the concept of "considered datatype" is consistent with the terminology defined in BLD.

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_ind. 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_ind are included in LV (by definition, the value spaces of all considered datatypes are included in D_ind). 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 Document is empty. For example, consider the combination of an empty document 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>[<http://p> -> <http://b>]

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 an existentially closed RIF-BLD condition formula φ if TVal_I(φ)=t.   ☐

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

Forall ("a"="b")

does not have a model, since the symbols "a" and "b" are mapped to the (distinct) character strings "a" and "b", 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 an existentially closed RIF-BLD 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).   ☐

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 document is empty.

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 document 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, 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.

In this document, we are using OWL to refer to OWL 1. While OWL 2 is still in development it is unclear how RIF will interoperate with it. At the time of writing, we believe that with OWL2 the support for punning may be beneficial, and that there might be particular problems in using section 3.2.2.3, since the semantics of annotation properties might be different than in OWL 1.

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 condition formula φ is a 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 document R is a DL-Document 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-Document 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.

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: The above definition of DL-safeness is intended to identify a fragment of RIF-OWL DL combinations for which implementation is easier than full RIF-OWL DL. This definition should be considered AT RISK and may change based on implementation experience.

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 an existentially closed RIF-BLD condition formula 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 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.

A DL-semantic multi-structure is a nonempty set of DL-semantic structures {I¹, ..., Iⁿ} that are identical in all respects except that the mappings I_C¹, ..., I_Cⁿ might differ on the constants in Const that belong to the rif:local symbol space.   ☐

Given a DL-semantic multi-structure I={I¹, ..., Iⁿ}, we use the symbol I to denote both the multi-structure and the common part of the individual structures I¹, ..., Iⁿ.

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 DL-semantic multi-structure I is a model of a DL-Document 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 on 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 is a DL-semantic multi-structure and I 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(c) = set of all objects k in O such that I_truth(I_frame'(I_C(rdf:type))(k,I_C(<c>))) = t, for every class identifier c in V;
3. ER(p) = set of all pairs (k, l) in O × (O union LV) such that I_truth(I_frame'(I_C(<p>))( k, l ))) = t (true), for every data valued and individual valued property identifier p 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>) 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>]. 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> -> ?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 documents 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 an existentially closed RIF-BLD condition φ if TVal_I(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 an existentially closed DL-condition formula 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 members of 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; similarly 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 xs: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 document containing the following fact

ex:myiri[rdf:type -> xs: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 xs: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 document containing the following fact

ex:myiri[ex:hasChild -> "John"]

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

SubClassof(ex:A ex:B)

and a RIF document 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(p) = set of all pairs (k, l) in O × O such that I_truth(I_frame'(I_C(<p>))( k, l) ) = t (true), for every IRI p 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 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 an existentially closed RIF-BLD condition formula 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 document 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 document OWL-DL-annotation-entails the RIF condition formula ex:myOntology[ex:hasTitle -> "Example ontology"]; the combination does not OWL-DL-entail the formula.

In the previous sections, RIF-RDF Combinations and RIF-OWL combinations were defined in an abstract way, as pairs of documents and sets of RDF graphs/OWL ontologies. In addition, different semantics were specified based on the various RDF and OWL entailment regimes. RIF provides a mechanism for explicitly referring to (importing) RDF graphs from documents and specify the intended profile (entailment regime) through the use of Import statements.

This section specifies how RIF documents with such import statements are interpreted.

A RIF document contains a number of Import statements. Unary Import statements are used for importing RIF documents, and the interpretation of these statements is defined in (RIF-BLD). This section defines the interpretation of two-ary Import statements:

Import(t1 p1)
  ...
Import(tn pn)
  

Here, ti is an IRI constant of the form <absolute-IRI>, where absolute-IRI is the location of an RDF graph to be imported, and pi is an IRI constant denoting the profile to be used.

The profile determines which notions of model, satisfiability and entailment must be used. For example, if a RIF document R imports an RDF graph S with the profile RDFS, the notions of rdfs-model, rdfs-satisfiability, and rdfs-entailment should be used with the combination <R, {S}>.

In case several graphs are imported in a document, and these imports specify different profile, the highest of these profiles is used. For example, if a RIF document R imports an RDF graph S₁ with the profile RDF and an RDF graph S₂ with the profile OWL Full, the notions of OWL-Full-model, OWL-Full-satisfiability, and OWL-Full-entailment must be used with the combination <R, {S₁, S₂}>.

Finally, if a RIF document R imports an RDF graph S with the profile OWL DL, R must be a DL-Document, S must be the translation to RDF of an OWL DL ontology in abstract syntax form O, and the notions of OWL-DL-model, OWL-DL-satisfiability, and OWL-DL-entailment must be used with the combination <R, {O}>.

Let R be a RIF document such that

Import(<u1> <p1>)
  ...
Import(<un> <pn>)

are the two-ary import statements in R and all imported documents and let Profile be the set of profiles corresponding to the IRIs p1,...,pn.

If Profile contains only specific profiles, then:

• If Profile does not have a single highest profile, then the document must be rejected.

• If Profile contains only DL profiles and the RDF graphs accessible from the locations u1,...,un are OWL DL ontologies in graph form such that O₁,....,O_n are the same ontologies in abstract syntax form, then the combination C=<R,{O₁,....,O_n}> must be interpreted according to the highest among the profiles in Profile.

• Otherwise, the combination C=<R,{S₁,....,S_n}>, where S₁,....,S_n are RDF graphs accessible from the locations u1,...,un, must be interpreted according to the highest profile in Profile.

If Profile contains a generic profile, then the combination C=<R,{S₁,....,S_n}>, where S₁,....,S_n are RDF graphs accessible from the locations u1,...,un and C may be interpreted according to the highest among the specific profiles in Profile.

(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, 09 June 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-bld-20080609/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-bld/.

(RIF-DTB)

RIF Data Types and Built-Ins Axel Polleres, Harold Boley, Michael Kifer, eds. W3C Editor's Draft, 09 June 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-dtb-20080609/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-dtb/.

(XML-Schema2)

XML Schema Part 2: Datatypes, W3C Recommendation, 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.

(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/.

(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 Christian de Sainte Marie, eds. W3C Editor's Draft, 09 June 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-prd-20080609/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-prd/.

(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/.

(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.

(SWRL)

SWRL: A Semantic Web Rule Language Combining OWL and RuleML, I. Horrocks, P. F. Patel-Schneider, H. Boley, S. Tabet, B., and M. Dean, W3C Member Submission, 21 May 2004, http://www.w3.org/Submission/2004/SUBM-SWRL-20040521/. Latest version available at http://www.w3.org/Submission/SWRL/.

(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/.

Besides the namespace prefix is defined in the Overview, the following namespace prefix is used in this appendix: pred refers to the RIF namespace for built-in predicates http://www.w3.org/2007/rif-builtin-predicate# (RIF-DTB).

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.

   Forall ?x ?y (?x # ?y :- ?x[rdf:type -> ?y])

   Forall ?x ?y (?x[rdfs:subClassOf -> ?y] :- ?x ## ?y]) ))

Theorem A RIF-RDF combination <R,{S1,...,Sn}> is satisfiable iff there is a semantic multi-structure I that is a model of R, R^simple, tr_R(S1), ..., and tr_R(Sn)).

Proof. The theorem follows immediately from the following theorem and the observation that a combination (respectively, document) is satisfiable (respectively, has a model) if it does not entail the condition formula "a"="b".

Theorem A RIF-RDF combination C=<R,{S1,...,Sn}> simple-entails a generalized RDF graph T if and only if (R union tr_R(S1) union ... union tr_R(Sn)) entails tr_Q(T). C simple-entails an existentially closed RIF-BLD condition formula φ if and only if (R union R^simple union tr_R(S1) union ... union tr_R(Sn)) entails φ.

Editor's Note: Formulation of the entailment theorem is to be updated with a notion of merge of rule sets.

Proof. We prove both directions by contradiction: if the entailment does not hold on one side, we show that it also does not hold on the other.
In the proof we abbreviate (R union R^simple union tr_R(S1) union ... union tr_R(Sn)) with R'.

(=>) Assume R' does not entail φ. This means there is some semantic multi-structure I that is a model of R', but not of φ. Consider the pair interpretation (I, I), where I is defined as follows:

• IR=D_ind,
• IP is 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,
• LV=(union of the value spaces of all considered datatypes),
• 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,
• IS(i) = I_C(<i>) for every absolute IRI i in V_U, and
• IL((s, d)) = I_C(tr("s"^^d)) for every typed literal (s, d) in V_TL.

Clearly, (I, I) is a common-rif-rdf-interpretation: conditions 1-6 in the definition are satisfied by construction of I and conditions 7 and 8 are satisfied by condition 4 and by the fact that I is a model of R^simple.

Consider a graph Si in {S1,...,Sn}. Let x1,..., xm be the blank nodes in Si and let u1,..., um be the new IRIs that were obtained from the variables ?x1,..., ?xm through the skolemization in tr_R(Si). Now, let A be a mapping from blank nodes to elements in D_ind such that A(xj)=I_C(uj) for every blank node xj in Si. From the fact that I is a model of tr_R(Si) and by construction of I it follows that [I+A] satisfies Si, and so I satisfies Si.

We have that I is a model of R, by assumption. So, (I, I) satisfies C. Again, by assumption, I is not a model of φ. Therefore, C does not entail φ.

Assume now that R' does not entail tr_Q(T) and I is a model of R', but not of tr_Q(T). The common-rif-rdf-interpretation (I, I) is obtained in the same way as above, and so clearly satisfies C.
We proceed by contradiction. Assume I satisfies T. This means there is some mapping A from the blank nodes x1,...,xm in T to objects in D_ind such that [I+A] satisfies T.
Consider now the semantic multi-structure I*, which is the same as I, with the exception of the mapping I*_V on the variables ?x1,...,?xm, which is defined as follows: I*_V(?xj)=A(xj) for each blank node xj in S. By construction of I and since [I+A] satisfies T we can conclude that I* is a model of And(tr(t1)... tr(tm)), and so I is a model of tr_Q(T), violating the assumption that it is not. Therefore, (I, I) does not satisfy T and C does not entail φ.

(<=) Assume C does not entail φ. This means there is some common-rif-rdf-interpretation (I, I) that satisfies C such that I is not a model of φ.
Consider the semantic multi-structure I$, which is exactly the same I$, except for the mapping I$_C on new IRIs that were introduced in skolemization. The mapping of these new IRIs is defined as follows:
For each graph Si in {S1,...,Sn}, let x1,..., xm be the blank nodes in Si and let u1,..., um be the new IRIs that were obtained from the variables ?x1,..., ?xm through the skolemization in tr_R(Si). Now, since I satisfies Si, there must be a mapping A from blank nodes to elements in D_ind such that [I+A] satisfies Si. We define I$_C(uj)=A(xj) for every blank node xj in Si.

By assumption, I$ is a model of R (recall that I$ differs from I only on the new IRIs). Clearly, I$ is also a model of R^simple, by conditions 7, 8, and 4 in the definition of common-rif-rdf-interpretation.
From the fact that I satisfies Si and by construction of I$ it follows that I$ is a model of tr_R(Si). So, I$ is a model of R'. Since I is not a model of φ and φ does not contain any of the new IRIs, I$ is not the model of φ.
Therefore, R' does not entail φ.

Assume now that C does not entail T and (I, I) satisfies C, but I does not satisfy T. We obtain I$ from I in the same way as above, and so clearly satisfies R'. It can be shown analogous to the (=>) direction that if I$ is a model of tr_Q(T), then there is a blank node mapping A such that [I+A] satisfies T, and thus I satisfies S, violating the assumption that it does not. Therefore, I$ is not a model of tr_Q(T) and thus R' does not entail tr_Q(T).

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

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

   Forall ?x ("a"="b" :- And(?x[rdf:type -> rdf:XMLLiteral] ex:illxml(?x)))

Theorem A RIF-RDF combination <R,{S1,...,Sn}> is rdf-satisfiable iff there is a semantic multi-structure I that is a model of R, R^RDF, tr_R(S1), ..., and tr_R(Sn)).

Proof. The theorem follows immediately from the following theorem and the observation that a combination (respectively, document) is rdf-satisfiable (respectively, has a model) if it does not entail the condition formula "a"="b".

Theorem A RIF-RDF 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 rdf-entails an existentially closed RIF-BLD condition formula φ iff (R^RDF union R union tr_R(S1) union ... union tr_R(Sn)) entails φ.

Proof. In the proof we abbreviate (R union R^RDF union tr_R(S1) union ... union tr_R(Sn)) with R'.
The proof is then obtained from the proof of correspondence for simple entailment in the previous section with the following modifications: (*) in the (=>) direction we additionally need to show that I is an rdf-interpretation and (**) in the (<=) direction we need to slightly extend the definition of I$ to account for ex:illxml and show that I$ is a model of R^RDF.

(*) To show that I is an rdf-interpretation, we need to show that I satisfies the RDF axiomatic triples and the RDF semantic conditions.
Satisfaction of the axiomatic triples follows immediately from the inclusion of tr(t) in R^RDF for every RDF axiomatic triple t, the fact that I is a model of R^RDF, and construction of I. Consider the three RDF semantic conditions:

1	x is in IP if and only if <x, I(rdf:Property)> is in IEXT(I(rdf:type))
2	If "xxx"^^rdf:XMLLiteral is in V and xxx is a well-typed XML literal string, then IL("xxx"^^rdf:XMLLiteral) is the XML value of xxx; IL("xxx"^^rdf:XMLLiteral) is in LV; IEXT(I(rdf:type)) contains <IL("xxx"^^rdf:XMLLiteral), I(rdf:XMLLiteral)>
3	If "xxx"^^rdf:XMLLiteral is in V and xxx is an ill-typed XML literal string, then IL("xxx"^^rdf:XMLLiteral) is not in LV; IEXT(I(rdf:type)) does not contain <IL("xxx"^^rdf:XMLLiteral), I(rdf:XMLLiteral)>.

Satisfaction of condition 1 follows from satisfaction of the first rule in R^RDF in I and construction of I; specifically the second bullet.
Consider a well-typed XML literal "xxx"^^rdf:XMLLiteral. By the definition of satisfaction in RIF BLD, I_C("xxx"^^rdf:XMLLiteral) is the XML value of xxx, and is clearly in LV, by definition of I. The final part of condition 2 is satisfied by the second rule in R^RDF.
Satisfaction of condition 3 follows from the definition of LV and satisfaction of the third rule in R^RDF in I (if there were a non-well-typed XML literal in the class extension of rdf:XMLLiteral, then I would not be a model of this rule). This establishes the fact that I is an rdf-interpretation.

(**) Recall that, by assumption, ex:illxml is not used in R. Therefore, changing satisfaction of atomic formulas concerning ex:illxml does not affect satisfaction of R. We assume that k is a unique element, i.e., no other constant is mapped to k.
We define I$_C(k) as follows: For every non-well-typed literal of the form (s, rdf:XMLLiteral) such that I$_C(tr(s^^rdf:XMLLiteral))=l we define I_truth(I$_F(k)(l))=t; I$_truth(I$_F(k)(m))=f for any other object m in D_ind.

Consider R^RDF. Satisfaction of R^simple was established in the proof in the previous section. Satisfaction of the facts corresponding to the RDF axiomatic triples in I$ follows immediately from the definition of common-rif-rdf-interpretation and the fact that I is an rdf-interpretation, and thus satisfies all RDF axiomatic triples.
Since rdf:XMLLiteral is a required datatype, the set of non-well-typed XML literals is the same as the set of ill-typed XML literals. Satisfaction of the ex:illxml facts in R^RDF then follows immediately from the definition of I$. Satisfaction of the first, second, and third rule in R^RDF follow straightforwardly from the RDF semantic conditions 1, 2, and 3. This establishes the fact that I$ is a model of R^RDF.

Let T be the set of considered datatypes (cf. Section 5 of (RIF-BLD)), i.e., T includes at least all data types used in the combination under consideration and all datatypes required for RIF-BLD (RIF-DTB).

Editor's Note: The terminology "considered datatype" might change if the terminology is changed in BLD.

By (RIF-DTB), each datatype in T has an associated label DATATYPE (e.g., the label of xs:string is String) and a guard pred:isDATATYPE, which can be used to test whether a particular object is a value of the data type.

Editor's Note: Verify that these things are defined in the DTB document before publication.

     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 ("a"="b" :- And(?x[rdf:type -> rdfs:Literal] ex:illxml(?x)))

Theorem A RIF-RDF combination <R,{S1,...,Sn}> is rdfs-satisfiable if and only if there is a semantic multi-structure I that is a model of R, R^RDFS, tr_R(S1), ..., and tr_R(Sn)).

Proof. The theorem follows immediately from the following theorem and the observation that a combination (respectively, document) is rdfs-satisfiable (respectively, has a model) if it does not entail the condition formula "a"="b".

Theorem A RIF-RDF combination C=<R,{S1,...,Sn}> rdfs-entails a generalized RDF graph T if and only if (R^RDFS union R union tr_R(S1) union ... union tr_R(Sn)) entails tr_Q(T). C rdfs-entails an existentially closed RIF-BLD condition formula φ if and only if (R^RDFS union R union tr_R(S1) union ... union tr_R(Sn)) entails φ.

Proof. In the proof we abbreviate (R union R^RDFS union tr_R(S1) union ... union tr_R(Sn)) with R'.
The proof is then obtained from the proof of correspondence for RDF entailment in the previous section with the following modifications: (*) in the (=>) direction we need to slightly amend the definition of I to account for rdfs:Literal and show that I is an rdfs-interpretation and (**) in the (<=) direction we need to show that I$ is a model of R^RDFS.

(*) We amend the definition of I by changing the definition of LV to the following:

• LV=(union of the value spaces of all considered datatypes) union (set of all k in D_ind such that I_truth(I_frame(I_C(rdf:type))(k,I_C(rdfs:Literal)))=t).

Clearly, this change does not effect satisfaction of the RDF semantic conditions 1 and 2. To see that condition 3 is still satisfied, consider some ill typed XML literal t. Then, ex:illxml(tr(t)) is satisfied in I. If tr(t)[rdf:type -> rdfs:Literal] were to be satisfied as well, then, by the second last rule in the definition of R^RDFS, "a"="b" would be satisfied, which cannot be the case. Therefore, tr(t)[rdf:type -> rdfs:Literal] is not satisfied and thus IL(t) is not in ICEXT(rdfs:Literal). And, since IL(t) is not in the value space of any considered datatype, it is not in LV. To show that I is an rdfs-interpretation, we need to show that I satisfies the RDFS axiomatic triples and the RDF semantic conditions.
Satisfaction of the axiomatic triples follows immediately from the inclusion of tr(t) in R^RDFS for every RDFS axiomatic triple t, the fact that I is a model of R^RDFS, and construction of I. Consider the RDFS semantic conditions:

1	x is in ICEXT(y) if and only if <x,y> is in IEXT(I(rdf:type)) IC = ICEXT(I(rdfs:Class)) IR = ICEXT(I(rdfs:Resource)) LV = ICEXT(I(rdfs:Literal))
2	If <x,y> is in IEXT(I(rdfs:domain)) and <u,v> is in IEXT(x) then u is in ICEXT(y)
3	If <x,y> is in IEXT(I(rdfs:range)) and <u,v> is in IEXT(x) then v is in ICEXT(y)
4	IEXT(I(rdfs:subPropertyOf)) is transitive and reflexive on IP
5	If <x,y> is in IEXT(I(rdfs:subPropertyOf)) then x and y are in IP and IEXT(x) is a subset of IEXT(y)
6	If x is in IC then <x, I(rdfs:Resource)> is in IEXT(I(rdfs:subClassOf))
7	If <x,y> is in IEXT(I(rdfs:subClassOf)) then x and y are in IC and ICEXT(x) is a subset of ICEXT(y)
8	IEXT(I(rdfs:subClassOf)) is transitive and reflexive on IC
9	If x is in ICEXT(I(rdfs:ContainerMembershipProperty)) then: < x, I(rdfs:member)> is in IEXT(I(rdfs:subPropertyOf))
10	If x is in ICEXT(I(rdfs:Datatype)) then <x, I(rdfs:Literal)> is in IEXT(I(rdfs:subClassOf))

The first and second part of condition 1 are simply definitions of ICEXT and IC, respectively. Since I satisfies the first rule in the definition of R^RDFS it must be the case that every element k in D_ind is in ICEXT(I(rdfs:Resource)). Since IR=D_ind, it follows that IR = ICEXT(I(rdfs:Resource)). Clearly, every object in ICEXT(I(rdfs:Literal)) is in LV, by definition. Consider any value k in LV. By definition, either k is in the value space of some considered datatype or I_truth(I_frame(I_C(rdf:type))(k,I_C(rdfs:Literal)))=t. In the latter case, clearly k is in ICEXT(I(rdfs:Literal)). In the former case, k is in the value space of some datatype with some label D, and thus I_truth(I_F(I_C(pred:isD))(k))=t. By the last rule in R^RDFS, it must consequently be the case that I_truth(I_frame(I_C(rdf:type))(k,I_C(rdfs:Literal)))=t, and thus k is in ICEXT(I(rdfs:Literal)). So, I satisfies condition 1.

Satisfaction of conditions 2 through 10 in I follows immediately from satisfaction in I of the 2nd through the 12th rule in the definition of R^RDFS. This establishes the fact that I is an rdfs-interpretation.

(**) Consider R^RDFS. Satisfaction of R^RDF was established in the [[SWC#proof-rdf-entailment|proof] in the previous section. Satisfaction of the facts corresponding to the RDFS axiomatic triples in I$ follows immediately from the definition of common-rif-rdf-interpretation and the fact that I is an rdfs-interpretation, and thus satisfies all RDFS axiomatic triples.
Satisfaction of the 1st through the 12th, second, and third rule in R^RDFS follow straightforwardly from the RDFS semantic conditions 1 through 10. Satisfaction of the 13th rule follows from the fact that, given an ill-typed XML literal t, IL(t) is not in LV (by RDF semantic condition 3), ICEXT(rdfs:Literal)=LV, and the fact that the ex:illxml predicate is only true on ill-typed XML literals. Finally, satisfaction of the last rule in R^RDFS follows from the fact that ICEXT(rdfs:Literal)=LV, the definition of LV as a superset of the union of the value spaces of all datatypes, and the definition of the pred:isD predicates. This establishes the fact that I$ is a model of R^RDFS.

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.

OWL DLP restricts the OWL DL abstract syntax (OWL-Semantics), removing disjunction and extensional quantification from consequents of implications and removing negation and equality. The semantics of OWL DLP is the same as OWL DL.

Definition. An OWL DL ontology in abstract syntax form is an OWL DLP ontology if it respects the grammar below.   ☐

The basic syntax of ontologies and identifiers is the same as for OWL DL.

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

Facts are the same as for OWL DL, except that equality and inequality (SameIndividual and DifferentIndividual), as well as individuals without an identifier are not allowed.

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

type ::= Rdescription

The main restrictions posed by OWL DLP on the OWL DL syntax are on descriptions and axioms. Specifically, we need to distinguish between descriptions which are allowed on the right-hand side (Rdescription) and those allowed on the left-hand side (Ldescription) of subclass statements.

We start with descriptions that may be allowed on both sides

dataRange ::= datatypeID | 'rdfs:Literal'

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

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

We then proceed with the individual sides

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 ')'

Finally, we turn to axioms. We start with class axioms.

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 } )'

Property axioms in OWL DLP restrict those in OWL DL by disallowing functional and inverse functional properties, because these involve equality.

 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 DLP ontologies into a RIF DL-document. The embedding extends the embedding function tr. The embeddings of IRIs and literals is as defined in the Section Embedding Symbols.

Let T be the set of considered datatypes (cf. Section 5 of (RIF-BLD)), i.e., T includes at least all data types used in the combination under consideration and all datatypes required for RIF-BLD (RIF-DTB).

Editor's Note: The terminology "considered datatype" might change if the terminology is changed in BLD.

By (RIF-DTB), each datatype in T has an associated label DATATYPE (e.g., the label of xs:string is String) and positive and negative guards pred:isDATATYPE and pred:isNotDATATYPE, which can be used to test whether a particular object is (resp., is not) a value of the data type.

Editor's Note: Verify that these things are defined in the DTB document before publication.

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)

Besides the embedding in the previous table, we also need an axiomatization of some of the aspects of the OWL DL semantics, e.g., separation between individual and datatype domains. This axiomatization is defined relative to an OWL vocabulary V.

Theorem A RIF-OWL-DL-combination <R,{O1,...,On}>, where O1,...,On are OWL DLP ontologies with vocabulary V, is OWL-DL-satisfiable iff there is a semantic multi-structure I that is a model of tr(R union R^OWL-DL(V) union tr_O(tr_N(O1)) union ... union tr_O(tr_N(On))).

Proof. The theorem follows immediately from the following theorem and the observation that a combination (respectively, document) is OWL-DL-satisfiable (respectively, has a model) if it does not entail the condition formula "a"="b".

Theorem An OWL-DL-satisfiable RIF-OWL-DL-combination C=<R,{O1,...,On}>, where O1,...,On are OWL DLP ontologies with vocabulary V, OWL-DL-entails an existentially closed RIF-BLD condition formula φ iff tr(R union R^OWL-DL(V) union tr_O(tr_N(O1)) union ... union tr_O(tr_N(On))) entails φ.

Proof. The proof of the theorem consists of three steps. We first show
(*) Given an OWL DLP ontology O, O and tr_N(O) the same models. In the remainder we assume that all OWL DLP ontologies are normalized.
We then show
(**) C=<R,{O1,...,On}> OWL-DL-entails φ if and only if for every model of R union R^OWL-DL(V) union tr_O(tr_N(O1)) union ... union tr_O(tr_N(On)) holds that TVal_I(φ)=t.
Finally, we show
(***) tr(R union R^OWL-DL(V) union tr_O(tr_N(O1)) union ... union tr_O(tr_N(On))) entails φ if and only if for every model of R union R^OWL-DL(V) union tr_O(tr_N(O1)) union ... union tr_O(tr_N(On)) holds that TVal_I(φ)=t.

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.