RIF RDF and OWL Compatibility

W3C Editor's Draft 17 July 2008

This version:

http://www.w3.org/2005/rules/wg/draft/ED-rif-rdf-owl-20080717/

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-20080709/ (color-coded diff)

F2F10 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May Be Superseded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 Datatypes and Built-Ins 1.0

Please Comment By 2008-07-21

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.

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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, RDFS, and 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; 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 and/or RDFS or OWL ontologies: 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 in RIF, particularly IRIs and plain literals, is slightly different from the notation in RDF/OWL. These differences are 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 and OWL 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 blank nodes _:x, IRIs delimited with '<' and '>', compact IRIs prefix:localname, plain literals without language tags "literal", plain literals with language tags "literal"@lang, 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 [RIF-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 correspondence between constant symbols in RDF graphs and RIF documents is explained in Table 1.

The shortcut syntax for IRIs and strings [RIF-DTB], used throughout this document, corresponds with the syntax for IRIs and plain literals in [Turtle], a commonly used syntax for RDF.

• 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 "my string"^^xs:string may be written as "my string".

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. In many applications and deployment scenarios, this difference may be inconsequential. However the results will differ when an RDF graph is used in a non-assertional context, such as in a query pattern.

Finally, variables in the bodies of RIF rules or in query patterns 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:named] :- ?x[ex:hasName -> ?y] )
Forall ?x ?y ( <http://a>[<http://p> -> ?y] :- ?x[ex:hasName -> ?y] )

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

_:y rdf:type ex:named .
<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).

In addition, there is an infinite set of blank nodes, which is disjoint from the sets of names. See RDF Concepts and Abstract Syntax [RDF-Concepts] for precise definitions of these concepts.

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)

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.

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 rdf:XMLLiteral is in the domain of the map. RIF BLD requires a number of additional datatypes to be included; these are the RIF-required datatypes [RIF-DTB].

When checking consistency of a combination < R,S> or entailment of a graph S or RIF formula φ by a combination < R,S>, the set of considered datatypes is the union of the set of RIF-required datatypesand the sets of datatypes used in R, the documents imported into R, and φ (when considering entailment of φ).

Definition. Let T be a set of datatypes. A datatype map D is conforming with T if it satisfies the following conditions:

1. Every IRI identifying a datatypes in T is in the domain of D.
2. D maps each IRI in its domain to the corresponding datatype in T.   ☐

Note that it follows from the definition that every datatype used in the RIF document in the combination or the entailed RIF formula (when considering entailment questions) is included in any datatype map conforming to the set of considered datatypes. There may be datatypes used in an RDF graph in the combination that are not included in such a datatype map.

Definition. Given a 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.   ☐

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 interpretation of the RDF vocabulary,
• RDFS interpretations, which impose additional conditions on the interpretation of the RDF and RDFS vocabularies, 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. the Section Symbols in RIF Versus RDF/OWL).

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,
• D_func is a non-empty subset of D,
• I_C is a mapping from constants to D such that constants in individual position are mapped to D_ind and constants in function positions are mapped to D_func,
• 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, ..., I_n} 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, ..., I_n}, we use the symbol I to denote both the multi-structure and the common part of the individual structures I₁, ..., I_n.

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(a)(k,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(a)(k,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.   ☐

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. Similarly for plain literals without language tags of the form "mystring" and typed literals of the form "mystring"^^xs:string. For example, consider the combination of an empty document and an RDF graph that contains the triples

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

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

<http://a> <http://p> <http://b> .
<http://a> <http://p> "abc"^^xs:string .

as well as the following frame formulas:

<http://a>[<http://p> -> <http://b>]
<http://a>[<http://p> -> "abc"]

Definition. A common-rif-rdf-interpretation (I, I) satisfies a RIF-RDF combination C=< R, S > if I is a model of R and I satisfies every RDF graph S in S; in this case (I, I) is called a simple-model, or model, of C, and C is satisfiable. (I, I) satisfies a generalized RDF graph S if I satisfies S. (I, I) satisfies a 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 fact

"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. Let (I, I) be a model of a combination C and let D be a datatype map conforming with the set of datatypes in I. (I, I) is a D-model of C if I is a D-interpretation; in this case C is D-satisfiable.   ☐

Definition. Let C be a RIF-RDF combination, let S be a generalized RDF graph, let φ be a condition formula, and let D be a datatype map conforming with the set of considered datatypes. C D-entails S if every D-model of C satisfies S. Likewise, C D-entails 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. This is a key property of Description Logic semantics and 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 were 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 a 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 and function terms in 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 as specified in [OWL-Semantics], as opposed to OWL 2, which is currently in the process of being defined by the OWL working group. (See the End note on OWL 2)

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. A DL-frame formula is a frame formula a[b₁ -> c₁ ... b_n -> c_n] such that n≥1 and for every b_i, with 1≤i≤n, it holds that b_i is a constant and if b_i = rdf:type, then c_i is a constant.

Definition. A condition formula φ is a DL-condition if every frame formula in φ is a DL-frame formula.   ☐

Definition. A RIF-BLD document formula R is a RIF-BLD DL-document formula if every frame formula in R is a DL-frame formula.   ☐

Definition. A RIF-OWL-DL-combination is a pair < R,O>, where R is a RIF-BLD DL-document formula and O is a set of OWL DL ontologies in abstract syntax form of an OWL 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 effective reasoning. This section specifies such safeness restrictions 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 H :- B is DL-safe if it occurs in an atomic formula in B that is not of the form s[P -> o] or s[rdf:type -> A], where P or A occurs in one of the ontologies in O. A RIF rule Q H :- B is DL-safe, given O if every variable that occurs in H :- B is DL-safe. A RIF rule Q H :- B is weakly DL-safe, given O if every variable that occurs in H is DL-safe.

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.   ☐

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. Let (I, I) be a common-rif-rdf-interpretation that is a model of a RIF-RDF combination C=< R, S > and let D be a datatype map conforming with the set of datatypes in I. (I, I) is an owl-full-model of C if I is an OWL Full interpretation with respect to D; in this case C is OWL-Full-satisfiable with respect to D.   ☐

Definition. Let C be a RIF-RDF combination, let S be a generalized RDF graph, let φ be a condition formula, and let D be a datatype map conforming with the set of considered datatypes. C owl-full-entails S with respect to D if every owl-full-model of C satisfies S. Likewise, C owl-full-entails φ with respect to D 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 SetOfFiniteBags(D × D) → D, such that for each pair (a, b) in SetOfFiniteBags(D × D) it 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, ..., I_n} that are identical in all respects except that the mappings I_1C, ..., I_nC might differ on the constants in Const that belong to the rif:local symbol space.   ☐

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

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 RIF-BLD DL-document formula 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); there is a non-empty subset of R, called O, which is 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 datatype map D, a common-rif-owl-dl-interpretation with respect to D 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. D is conforming with the datatypes in I;
2. (O union LV)=D_ind;
3. EC(c) = set of all objects k such that I_truth(I_frame'(k)({(I_C(rdf:type),I_C(<c>))})) = t, for every class or datatype identifier c in V;
4. ER(p) = set of all pairs (k, l) such that I_truth(I_frame'(k)({(I_C(<p>), l)})) = t (true), for every data valued and individual valued property identifier p 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>) for every individual identifier i in V.   ☐

Condition 2 ensures that the relevant parts of the domains of interpretation are the same. Condition 3 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 4 ensures that the interpretation (extension) of an OWL DL object or datatype property u corresponds to the interpretation of frames of the form ?x[<u> -> ?y]. 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 documents are interpreted in the same way.

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

Definition. A common-rif-owl-dl-interpretation (I, I) with respect to a datatype map D 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 with respect to D. (I, I) is an owl-dl-model of an OWL DL ontology in abstract syntax form O if I satisfies O. (I, I) is an owl-dl-model of a DL-condition formula φ if TVal_I(φ)=t.   ☐

Definition. Let C be a RIF-OWL-DL-combination, let O be an OWL DL ontology in abstract syntax form, let φ be a DL-condition formula, and let D be a datatype map conforming with the set of considered datatypes. C owl-dl-entails O with respect to D if every common-rif-owl-dl-interpretation with respect to D that is an owl-dl-model of C is an owl-dl-model of O. Likewise, C owl-dl-entails φ with respect to D if every common-rif-owl-dl-interpretation with respect to D that is an owl-dl-model of C is an owl-dl-model of φ.   ☐

Recall that in an abstract OWL interpretation I the sets O and LV, which are used for interpreting, respectively, 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-owl-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 cannot be members of O.

Note that the above definition of RIF-OWL DL compatibility does not consider ontology and annotation properties, in contrast to the definition of compatibility of RIF with OWL Full, where there is no clear distinction between annotation and ontology properties and other kinds of properties. Therefore, it is not possible to "access" or use the values of these properties in the RIF rules. This limitation is overcome in the following definition. It is envisioned that the user will choose whether annotation and ontology properties are to be considered. It might be the case that OWL 2, the successor of OWL currently under development, will not define the semantics for annotation and ontology properties in the same way as OWL; therefore, the below definition may not extend to cover OWL 2.

Definition. Given a datatype map D, a common-rif-owl-dl-interpretation (I, I) is a common-dl-annotation-interpretation with respect to D if the following condition holds

  7. ER(p) = set of all pairs (k, l) in O × O such that I_truth(I_frame'(k)({(I_C(<p>), l)}) ) = t (true), for every IRI p in V.   ☐

Condition 7, 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. A common-dl-annotation-interpretation with respect to a datatype map D (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. Let C be a RIF-OWL-DL-combination, let O be an OWL DL ontology in abstract syntax form, let φ be a DL-condition formula, and let D be a datatype map conforming with the set of considered datatypes. C owl-dl-annotation-entails O with respect to D if every common-rif-owl-dl-interpretation with respect to D that is an owl-dl-annotation-model of C is an owl-dl-model of O. Likewise, C owl-dl-annotation-entails φ with respect to D if every common-rif-owl-dl-interpretation with respect to D that is an 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 consisting of a RIF document and a set 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 specifying 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 must be used with the combination <R, {S}>.

Profiles are ordered as defined later in this section. If several graphs are imported in a document, and these imports specify different profiles, 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 RIF-BLD DL-document formula, 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,...,u_n 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,...,u_n, 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,...,u_n, may be interpreted according to the highest among the specific profiles in Profile.

The following members of the joint RIF-OWL task force have contributed to the OWL Compatibility section in this document: Mike Dean, Peter F. Patel-Schneider, and Ulrike Sattler.

The regular attendees at meetings of the Rule Interchange Format (RIF) Working Group at the time of the publication were: Adrian Paschke (REWERSE), Axel Polleres (DERI), Chris Welty (IBM), Christian de Sainte Marie (ILOG), Dave Reynolds (HP), Gary Hallmark (ORACLE), Harold Boley (NRC), Hassan Aït-Kaci (ILOG), Igor Mozetic (JFI), John Hall (OMG), Jos de Bruijn (FUB), Leora Morgenstern (IBM), Michael Kifer (Stony Brook), Mike Dean (BBN), Sandro Hawke (W3C/MIT), and Stella Mitchell (IBM).

[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. Carrol, 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, 17 July 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-bld-20080717/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-bld/.

[RIF-DTB]

RIF Datatypes and Built-Ins 1.0 Axel Polleres, Harold Boley, Michael Kifer, eds. W3C Editor's Draft, 17 July 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-dtb-20080717/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-dtb/.

[CURIE]

CURIE Syntax 1.0, M. Birbeck, S. McCarron, W3C Working Draft, 6 May 2008, 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, 17 July 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-prd-20080717/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-prd/.

[RIF-UCR]

RIF Use Cases and Requirements Adrian Paschke, David Hirtle, Allen Ginsberg, Paula-Lavinia Patranjan, Frank McCabe, eds. W3C Editor's Draft, 17 July 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-ucr-20080717/. Latest version available at http://www.w3.org/2005/rules/wg/draft/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., 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, 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/.

[XML-Schema2]

XML Schema Part 2: Datatypes Second Edition, P. V. Biron, A. Malhotra, ditors, W3C Recommendation, 28 October 2004, http://www.w3.org/TR/2004/REC-xmlschema-2-20041028/. Latest version available at http://www.w3.org/TR/xmlschema-2/.

RIF-RDF combinations can be embedded into RIF documents in a fairly straightforward way, thereby demonstrating how a RIF-compliant translator without native support for RDF can process RIF-RDF combinations. RIF-OWL combinations cannot be embedded in RIF, in the general case. However, there is a subset of OWL DL, the so-called DLP subset [DLP], for which RIF-OWL DL combinations that can be embedded.

The embeddings are defined using the embedding function tr, which maps symbols, triples, RDF graphs, and OWL DL ontologies in abstract syntax form to RIF symbols, statements, and documents, respectively.

Besides the namespace prefixes 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].

To facilitate the definition of the embeddings we define the notion of a merge of RIF formulas.

Definition. Let R={R₁,...,R_n} be a set of document, group, and rule formulas, such that there are no prefix or base directives or relative IRIs in R and directive₁₁, ..., directive_nm are all the import directives occurring in document formulas in R. The merge of R, denoted merge (R), is defined as Document(directive₁₁ ... directive_nm Group(R*₁ ... R*_n)), where R*_i is obtained from R_i in the following way:

• if R_i is a document formula of the form Document(directive_i1 ... directive_im Γ), then R*_i=Γ;
• if R_i is a group formula, then R*_i=R_i;
• and if R_i is a RIF-BLD rule, then R*_i=Group(R_i).   ☐

Note that the requirement that no prefix or based directives or relative IRIs are included in any of the formulas to be merged is not a real limitation, since compact IRIs can be rewritten to absolutes IRIs, as can relative IRIs by exploiting a base directive or the location of the document.

Editor's Note: We note here is that the embeddings in this appendix use equality, which is a feature of RIF BLD that is at risk. However, equality is not a crucial feature for the embeddings; removing equality from embedded combinations is fairly straightforward.

The embedding is not defined for combinations that include infinite RDF graphs and for combinations that include RDF graphs with RDF URI references that are not absolute IRIs (see the End note on RDF URI references) or plain literals that are not in the lexical space of the xs:string datatype [XML-Schema2]. In addition, for the embedding of RDFS entailment, each datatype must have an associated guard predicate.

In the remainder of this section we first define the embedding of symbols, triples, and graphs, after which we define the axiomatization of simple, RDF, and RDFS entailment of combinations and, finally, demonstrate faithfulness of the embedding.

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.

In the table, the mapping tr' is an injective function that maps typed literals to new constants in the rif:local symbol space, where a new constant is a constant that is not used in the documents or its vicinity (e.g., entailed formula or entailing combination). It "generates" a new constant from a typed literal.

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

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

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

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. We first consider condition formulas (the second part of the theorem), after which we consider graphs (the first part of the theorem).
In the proof we abbreviate merge({R, R^simple, tr_R(S₁), ..., tr_R(S_n)}) 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 (I, I), where I is the interpretation 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 S_i in {S₁,...,S_n}. Let x₁,..., x_m be the blank nodes in S_i and let u₁,..., u_m be the new IRIs that were obtained from the variables ?x₁,..., ?x_m through the skolemization in tr_R(S_i), i.e., u_i=sk(?x_i). Now, let A be a mapping from blank nodes to elements in D_ind such that A(x_j)=I_C(u_j) for every blank node x_j in S_i. From the fact that I is a model of tr_R(S_i) and by construction of I it follows that [I+A] satisfies S_i (see Section 1.5 of [RDF-Semantics])), and so I satisfies S_i.

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), which means there is a semantic multi-structure I that 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,...,x_m 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,...,?x_m, which is defined as follows: I*_V(?x_j)=A(x_j) for each blank node x_j 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(t_m)), 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 T.

(<=) 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 S_i in {S₁,...,S_n}, let x₁,..., x_m be the blank nodes in S_i and let u₁,..., u_m be the new IRIs that were obtained from the variables ?x₁,..., ?x_m through the skolemization in tr_R(S_i). Now, since I satisfies S_i, there must be a mapping A from blank nodes to elements in D_ind such that [I+A] satisfies S_i. We define I'_C(u_j)=A(x_j) for every blank node x_j in S_i.

By assumption, I' is a model of R (recall that I' differs from I only on the new IRIs, which are not in R). 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 S_i and by construction of I' it follows that I' is a model of tr_R(S_i). 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, which means there is a common-rif-owl-dl-interpretation (I, I) that 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).   ☐

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

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

Let T be the set of considered datatypes. We assume that each datatype in T has an associated capitalized short name Datatype (e.g., the short name 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 datatype; see [RIF-DTB] for definitions of guards for the RIF-required datatypes.

   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)))

Here, inconsistencies may occur if non-well-typed XML literals, axiomatized using the ex:illxml predicate, are in the class extension of rdf:XMLLiteral. If this situation occurs, "a"="b" is derived, which is an inconsistency in RIF.

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

Proof. In the proof we abbreviate merge({R^RDF, R, tr_R(S₁), ..., tr_R(S_n)}) 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 extend I to ensure it satisfies the RDF axiomatic triples and 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.

(*) For any positive integer j such that rdf:_j does not occur in the context of C, I and I are extended such that IS(rdf:_j)=I_C(rdf:_j)=I_C(rdf:_m) (see the definition of finite-axiomatic triples above for the definition of m). Clearly, this does not affect satisfaction of R' or non-satisfaction of φ, respectively tr_Q(T). 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 finite-axiomatic triple t, the fact that I is a model of R^RDF, and construction of I, and the extension of I to satisfy the infinite axiomatic triples. 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 (a) IL("xxx"^^rdf:XMLLiteral) is the XML value of xxx; (b) IL("xxx"^^rdf:XMLLiteral) is in LV; (c) 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 (a) IL("xxx"^^rdf:XMLLiteral) is not in LV; (b) 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 (condition 2a), and is clearly in LV (condition 2b), by definition of I. Condition 2c is satisfied by satisfaction of the second rule in R^RDF in I.
Satisfaction of 3a follows from satisfaction of the fourth rule in R^RDF and the definition of LV. (3b) follows from satisfaction of the third rule in R^RDF (3b) 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 I'_C(ex:illxml)=k is a unique element, i.e., no other constant is mapped to k.
We define I'_F(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 each 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.
Satisfaction of the ex:illxml facts in R^RDF 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.   ☐

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

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

• every RDFS axiomatic triple not of the form rdf:_i ... ... is an RDFS finite-axiomatic triple,
• rdf:_m rdf:type rdfs:ContainerMembershipProperty, rdf:_m rdfs:domain rdfs:Resource, and rdf:_m rdfs:range rdfs:Resource, for some positive integer m such that rdf:_m does not occur in the context of C, are RDFS finite-axiomatic triples, and
• if rdf:_j occurs in the context of C, for some positive integer j, then rdf:_j rdf:type rdfs:ContainerMembershipProperty, rdf:_j rdfs:domain rdfs:Resource, and rdf:_j rdfs:range rdfs:Resource are RDFS finite-axiomatic triples.

The set of considered datatypes T is defined as before.

     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 C=<R,{S₁,...,S_n}> rdfs-entails a generalized RDF graph T if and only if merge({R, R^RDFS, tr_R(S₁), ..., tr_R(S_n)}) entails tr_Q(T). C rdfs-entails a condition formula φ if and only if merge({R, R^RDFS, tr_R(S₁), ..., tr_R(S_n)}) entails φ.

Proof. In the proof we abbreviate merge({R, R^RDFS, tr_R(S₁), ..., tr_R(S_n)}) 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 axiomatic triples and the semantic conditions 1 and 2. To see that condition 3 is still satisfied, consider some non-well-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 finite-axiomatic triple t, the fact that I is a model of R^RDFS, construction of I, and the extension of I in the proof of the RDF entailment embedding. Consider the RDFS semantic conditions:

1	(a) x is in ICEXT(y) if and only if <x,y> is in IEXT(I(rdf:type)) (b) IC = ICEXT(I(rdfs:Class)) (c) IR = ICEXT(I(rdfs:Resource)) (d) 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))

Conditions 1a and 1b 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)), establishing 1c. 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)). This establishes satisfaction of condition 1d in I.

Satisfaction in I of conditions 2 through 10 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 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 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.   ☐

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

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

It is known that expressive Description Logic languages such as OWL DL cannot be straightforwardly embedded into typical rules languages such as RIF BLD [RIF-BLD], because of features such as disjunction and negation.

In this section we 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.

The embedding of RIF-OWL DL combinations is not defined for combinations that include infinite OWL ontologies and for combinations that include ontologies with RDF URI references that are not absolute IRIs or plain literals that are not in the lexical space of the xs:string datatype. In addition, each datatype used in the combination must have associated positive and negative guard predicates [DTB].

OWL DLP restricts the OWL DL abstract syntax [OWL-Semantics], disallowing disjunction and extensional quantification in consequents of implications, as well as negation and equality. The semantics of OWL DLP is that of OWL DL.

The syntax is defined through an EBNF grammar, which is derived from the grammar of the OWL abstract syntax [OWL-Semantics]. Any OWL DL ontology in abstract syntax form that conforms to this grammar is an OWL DLP ontology.

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

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

datatypeID ::= absolute-IRI
classID ::= absolute-IRI
individualID ::= absolute-IRI
ontologyID ::= absolute-IRI
datavaluedPropertyID ::= absolute-IRI
individualvaluedPropertyID ::= absolute-IRI
annotationPropertyID ::= absolute-IRI
ontologyPropertyID ::= absolute-IRI

dataLiteral ::= typedLiteral | plainLiteral
typedLiteral ::= lexicalForm^^absolute-IRI
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, OWL DLP distinguishes between descriptions that 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(' Ldescription ')'
           | 'value(' individualID ')'

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

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

Definition. An OWL DL ontology in abstract syntax form is an OWL DLP ontology if it conforms with the grammar above.   ☐

Recall that the semantics of frame formulas in DL-document formulas is different from the semantics of frame formulas in RIF documents. Nonetheless, DL-document formulas can be embedded into RIF documents, by translating frame formulas to predicate formulas. The mapping tr is the identity mapping on all RIF formulas, with the exception of frame formulas, as defined in the following table.

In the table, the mapping tr' is an injective function that maps constants to new constants, i.e., constants that are not used in the original document or its vicinity (e.g., entailed or entailing formula). It "generates" a new constant from an existing one.

For the purpose of making statements about this embedding, we define a notion of entailment for DL-document formulas.

Definition. A RIF-BLD DL-document formula R dl-entails a DL-condition φ if for every dl-semantic multi-structure I that is a model of R it holds that TVal_I(φ)=t.   ☐

The following lemma establishes faithfulness with respect to entailment of the embedding.

RIF-BLD DL-document formula Lemma A RIF-BLD DL-document formula R dl-entails a DL-condition φ if and only if tr(R) entails tr(φ).

Proof. We prove both directions by contradiction: if the entailment does not hold on the one side, we show that it also does not hold on the other.

(=>) Assume tr(R) does not entail tr(φ). This means there is some semantic multi-structure 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> that is a model of tr(R), but not of tr(φ).
Consider the dl-semantic multi-structure 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>, with I*_C and I*_frame' defined as follows: Let t be an element in D such that I_truth(t)=t and let f in D be such that I_truth(f)=f.

• for every constant c' used as unary or binary predicate symbol in tr(R) or tr(φ) such that c'=tr'(c) for some constant c, I*_C( c')=I_C(c); I*_C(c*)=I*_C(c*) for every other constant c*;
• for every constant c' used as unary predicate symbol in tr(R) or tr(φ) such that c'=tr'(c) for some constant c, and every object k in D_ind, if I_truth(I_F(I_C(c'))(k))=t, I*_frame'(k)((I_C(rdf:type), I_C(c))=t,
• for every constant b' used as binary predicate symbol in tr(R) or tr(φ) such that b'=tr'(b) for some constant b, and every pair (k, l) in D_ind × D_ind, if I_truth(I_F(I_C(b'))(k,l))=t, I*_frame'(k)((I_C(b),l))=t,
• if I*_frame'(k)((b₁,...,b_n))=t and I*_frame'(k)((c₁,...,c_m))=t for any two finite bags (b₁,...,b_n) and (c₁,...,c_m), then I*_frame'(k)((b₁,...,b_n,c₁,...,c_m))=t, and
• I*_frame'(b)=f for any other bag b.

Observe that tr(R) and tr(φ) do not include frame formulas.
To show that I* is a model of R and not of φ, we only need to show that (+) for any frame formula a[b -> c] that is a DL-condition, I* is a model of a[b -> c] iff I is a model of tr(a[b -> c]). This argument straightforwardly extends to the case of frames with multiple b_is and c_is, since in RIF semantic structures the following condition is required to hold: TVal_I(a[b₁->c₁ ... b_n->c_n]) = t if and only if TVal_I(a[b₁->c₁]) = ... = TVal_I(a[b_n->c_n]) = t [RIF-BLD].

Consider the case b=rdf:type. Then,
I* is a model of a[b -> c] iff I_truth(I*_frame'(I(a))(I_C(rdf:type),I_C(c)))=t.
From the definition of I* we obtain
I_truth(I*_frame'(I(a))(I_C(rdf:type),I_C(c)))=t iff I*_frame'(I(a))(I_C(rdf:type),I_C(c))=t.
By definition of the embedding, we know that tr'(c) is used as unary predicate symbol in tr(R) or tr(φ). From the definition of I* we obtain
I*_frame'(I(a))(I_C(rdf:type),I_C(c))=t iff I_truth(I_F(I_C(tr'(c)))(I(a)))=t.
Finally, since tr(a[b -> c])=tr'(c)(a), we obtain
I_truth(I_F(I_C(tr'(c)))(I(a)))=t iff I is a model of tr(a[b -> c]).
From this chain of equivalences follows that I* is a model of a[b -> c] iff I is a model of tr(a[b -> c]).

The argument for the case b≠rdf:type is analogous, thereby obtaining (+).

(<=) Assume R does not dl-entail φ. This means there is some dl-semantic multi-structure 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> that is a model of R, but not of φ. Let B be the set of constant symbols occurring in the frame formulas of the forms a[rdf:type -> b] and a[b -> c] in R or φ.
Consider the semantic multi-structure 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>. Let t and f in D be such that I_truth(t)=t and I_truth(f)=f. We define I*_C, I*_frame, and I*_F as follows:

• I*_C(tr'(b))=I*_C(b) for any b in B ; I*_C(c)=I_C(c) for any c not in B,
• I*_frame(b)=f for any finite bag b of D, and
• I*_F is defined as follows:
  • for every c in B, given an object k in D_ind, if I_truth(I_frame'(k)((I_C(rdf:type), I_C(c)))=t, I*_F(I*_C(tr'(c)))(k)=t; I*_F(I*_C(tr'(c)))(k')=f for any other k' in D_ind,
  • for every b in B, given a pair (k, l) in D_ind × D_ind, if I_truth(I_frame(k)((I_C(b),l)))=t, I*_F(tr'(b))(k,l)=t;I*_F(tr'(b))(k',l')=f for any other pair (k', l') in D_ind × D_ind, and
  • I*_F(c')=I_F(c') for every other constant c'.

Observe that R and φ do not include predicate formulas involving derived constant symbols tr'(b) or tr'(c). The remainder of the proof is analogous to the (=>) direction.   ☐

Ontology( [ ontologyID ] 
 directive1
 ... 
 directiven )

Ontology( 

...

)

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

Individual( individualID 
 value(propertyID b))

Individual( individualID 
 value(propertyID b))

Individual( individualID1 
 value(propertyID 
 Individual( individualID2 ... )))

Individual( individualID1 
 value(propertyID individualID2)

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 )

  ...

Proof. We prove both directions by contradiction: if the entailment does not hold on the one side, we show that it also does not hold on the other.

(=>) Assume C' does not owl-dl-entail φ. This means there is a common-rif-owl-dl-interpretation (I, I) that is a model of C', but I is not a model of φ.
Consider the pair (I, I*), where I* is obtained from I by suitably extending EC and ER to satisfy the annotation properties. Clearly, (I, I*) is a common-rif-owl-dl-interpretation, since the extension realized in I* does not affect any of the conditions on common-rif-owl-dl-interpretations. By the interpretation of axioms and facts and the EC extension table in sections 3.3 and 3.2 in [OWL-Semantics] it is easy to verify that, for any directive d in I, if I satisfies tr_N(d), I* satisfies d. Therefore, I* satisfies O₁,..., and O_n, and thus (I, I*) satisfies C. Since I is not a model of φ, C does not owl-dl-entail φ.

(<=) Assume C does not owl-dl-entail φ. This means there is a common-rif-owl-dl-interpretation (I, I) that is a model of C, but I is not a model of φ. It is easy to verify, by the interpretation of axioms and facts and the EC extension table in sections 3.3 and 3.2 in [OWL-Semantics], that I satisfies tr_N(O₁),..., and tr_N(O_n). So, (I, I) is a model of C', and thus C' does not owl-dl-entail φ.   ☐

We now proceed with the embedding of normalized OWL DLP ontologies into RIF DL-document formulas. The embedding is an extension of the embedding function tr. The embeddings of IRIs and literals is as defined in the Section Embedding Symbols.

In the following, let T be the union of the set of considered datatypes and the set of datatypes used in the ontologies under consideration. As with the RDFS embedding, we assume that each datatype in T has an associated capitalized short name Datatype and a positive guard pred:isDatatype. In addition, we assume each datatype has a negative guard pred:isNotDatatype, which can be used to test whether a particular object is not a value of the datatype (cf. [RIF-DTB]).

Ontology(
 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 and a datatype map D, which includes all datatypes in T. In the table, for a given datatype d, L2V(d) is the lexical-to-value mapping of d.

We call an OWL DLP ontology O normalized if it is the same as its normalization, i.e., O=tr_N(O).

The following lemma establishes faithfulness of the embedding.

Normalized Combination Embedding Lemma Given a datatype map D conforming with T, a RIF-OWL-DL-combination C=<R,{O₁,...,O_n}>, where O₁,...,O_n are normalized OWL DLP ontologies with vocabulary V, owl-dl-entails a DL-condition φ with respect to D iff merge({R, R^OWL-DL(V), tr_O(O₁), ..., tr_O(O_n)}) dl-entails φ.

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 merge({R, R^OWL-DL(V), tr_O(O₁), ..., tr_O(O_n)} with R'.

(=>) Assume R' does not dl-entail φ. This means there is a dl-semantic multi-structure 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> that is a model of R', but not of φ.
Consider the pair (I*,I), where I* = <TV, DTS, D*, D*_ind, D_func, I_C, I_V, I_F, I_frame', I_SF, I_sub, I_isa, I₌, I_external, I_truth> is such that

• D*_ind=D_ind union (union of the value spaces of all datatypes in the range of D) and
• D*=D union D*_ind

and I = <R, EC, ER, L, S, LV> is a tuple defined as follows:

• R=D*,
• LV=(union of the value spaces of all datatypes in the range of D),
• O=EC(owl:Thing),
• EC(rdfs:Literal)=LV,
• EC(d') = the value space of D(d'), if D(d') is defined,
• EC(c) = set of all objects k such that I_truth(I_frame'(I_C(rdf:type))(k,I_C(<c>))) = t, for every class identifier or datatype identifier c≠rdfs:Literal in V that is not in the domain of D,
• ER(p) = set of all pairs (k, l) 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;
• L((s, d)) = I_C("s"^^d) for every well-typed literal (s, d) in V;
• S(i) = I_C(<i>) for every IRI i in V.

Recall that an OWL vocabulary V consists of a set of literals V_L and seven sets of IRIs, V_C, V_D, V_I, V_DP, V_IP, V_AP, and V_O, which are the sets of class, datatype, individual, data-valued property, individual-valued property, annotation property, and ontology identifiers. According to its definition, an abstract OWL interpretation with respect to a datatype map D must fulfill the following conditions, where L(d) denotes the lexical space, V(d) denotes the value space and L2V(d) denotes to lexical-to-value mapping of a datatype d:

1. R is a nonempty set,
2. LV is a subset of R that contains the set of Unicode strings, the set of pairs of Unicode strings and language tags, and the value spaces of all datatypes in D,
3. EC : V_C → 2^O
4. EC : V_D → 2^LV
5. ER : V_DP → 2^O×LV
6. ER : V_IP → 2^O×O
7. ER : V_AP ∪ { rdf:type } → 2^R×R
8. ER : V_OP → 2^R×R
9. L  : TL → LV, where TL is the set of typed literals in V_L
10. S  : V_I ∪ V_C ∪ V_D ∪ V_DP ∪ V_IP ∪ V_AP ∪ V_O ∪ { owl:Ontology, owl:DeprecatedClass, owl:DeprecatedProperty } → R
11. S(V_I) ⊆ O
12. EC(owl:Thing) = O ⊆ R, where O is nonempty and disjoint from LV
13. EC(owl:Nothing) = { }
14. EC(rdfs:Literal) = LV
15. If D(d') = d then EC(d') = V(d)
16. If D(d') = d then L("v"^^d') ∈ V(d)
17. If D(d') = d and v ∈ L(d) then L("v"^^d') = L2V(d)(v)
18. If D(d') = d and v ∉ L(d) then L("v"^^d') ∈ R - LV

Condition 1 is met because D is a nonempty set. Clearly LV is a subset of R and contains the value spaces for each datatype in D, which include the sets of Unicode strings and pairs of Unicode strings and language tags, since the xs:string and rif:text datatypes are included in D, by the fact that D is conforming and the two datatypes are RIF-required; therefore, condition 2 is met.
When referring to rules in the remainder we mean rules in R^OWL-DL(V), unless otherwise specified.
To establish satisfaction of condition 3, observe that, by definition, O=EC(owl:Thing). So, for a given class name C we only need to establish that for any k in EC(c) it holds that k in EC(owl:Thing). But if k in EC(c), then, by definition, I_truth(I_frame'(I_C(rdf:type))(k,I_C(<C>))) = t. But then, by rule (iii), it must be the case that I_truth(I_frame'(I_C(rdf:type))(k,I_C(<owl:Thing>))) = t, and thus k in EC(owl:Thing).
Consider a datatype identifier Diri and associated short name DT and an object k not in LV such that k in EC(Diri). This means that I_truth(I_frame'(I_C(rdf:type))(k,I_C(<Diri>))) = t, but also I_truth(I_F(I_C(isNotDT))(k)) = t (since the value space of the datatype is a subset of LV). But then "a"="b" must be satisfied in I*, by rule (xii), which is clearly a contradiction. This establishes satisfaction of condition 4.
Satisfaction of conditions 5 and 6 can be shown similarly, exploiting rules (v), (vi), (vii), and (xv).
ER maps annotation and ontology properties to the empty set, so conditions 7 and 8 are trivially satisfied.
I_C maps well-typed literals "s"^^d to objects in the value space of d. Since L is defined in terms of I_C and since the value spaces of all datatypes are included in LV, condition 9 is satisfied.
Condition 10 is clearly satisfied by the definition of S and since R=D*.
Satisfaction of condition 11 follows straightforwardly from rule (viii) and the definition of O.
EC(owl:Thing) = O subset R, by definition. Then, by rule (xiii), there is no element in the value space of any datatype that is in O. Consequently, O is disjoint from LV. This establishes satisfaction of condition 12.
Satisfaction of condition 13 follows straightforwardly from rule (i); satisfaction of conditions 14 and 15 is immediate by definition of I.
Conditions 16 and 17 are satisfied by definition of L and the definition of I_C; observe that for every typed literal "v"^^d' must hold that d' is in the domain of D, since D includes all datatypes under consideration.
Assume there exists an ill-typed literal "v"^^d' in V, i.e., v is not in the lexical space D(d'). Since I satisfies rule (xiv), "a"="b" must be satisfied, which is a contradiction. So, there is no ill-typed literal and thus condition 18 is satisfied.
This establishes the fact that I is an abstract OWL interpretation.

Consider now any ontology O in {O₁,...,O_n}. To establish that I satisfies O, we need to establish five conditions (cf. Section 3.4 in [OWL-Semantics]):

1. each URI reference in O used as a class ID (datatype ID, individual ID, data-valued property ID, individual-valued property ID, annotation property ID, annotation ID, ontology ID) belongs to V_C (V_D, V_I, V_DP, V_IP, V_AP, V_O, respectively);
2. each literal in O belongs to V_L;
3. I satisfies each directive in O, except for Ontology Annotations;
4. there is some o ∈ R with <o,S(owl:Ontology)> ∈ ER(rdf:type) such that for each Ontology Annotation of the form Annotation(p v), <o,S(v)> ∈ ER(p) and that if O has name n, then S(n) = o; and
5. I satisfies each ontology mentioned in an owl:imports annotation directive of O.

Conditions 1 and 2 are satisfied by the fact O is an ontology of vocabulary V.
Conditions 4 and 5 are trivially satisfied, because normalized OWL DLP ontologies do not contain annotations and do not have names.
Consider any directive d in O; d has one of the following forms (cf. the second column of Table Normalizing OWL DLP):

1. class membership statement of the form Individual ( individualID type(A) ), where A is a class ID,
2. membership of value restriction,
3. property value statement,
4. subproperty statement,
5. inverse property statement,
6. symmetric property statement,
7. transitive property statement, or
8. subclass statement SubClassOf(X Y).

If d is of form 1, then we have that tr(d)=<individualID>[rdf:type -> <A>] is satisfied in I*, and thus I_truth(I_frame'(I_C(rdf:type))(I_C(<individualID>),I_C(<A>))) = t. Consequently, I_C(<individualID>) is in EC(<A>). Since, in addition, S(<individualID>)=I_C(<individualID>), we have that S(<individualID>) is in EC(<A>), and thus d is satisfied in I. Similar for statements of the forms 2 and 3.
Consider a subproperty statement SubPropertyOf(p q) and a pair (k, l) in ER(<p>). Then, by construction of I, I_truth(I_frame'(I_C(<p>))( k, l ))) = t. But, by tr(d), it must be the case that also I_truth(I_frame'(I_C(<q>))( k, l ))) = t. But then, (k,l) must be in ER(<q>), by construction of I. So, I satisfies d. Similar for statements of the forms 5, 6, and 7.

Consider the case that d is a subclass statement SubClassOf(X Y) and consider any k in EC(X), where EC is as in the EC Extension Table in [OWL-Semantics]. We show, by induction, that I* satisfies tr_O(X) when ?x is assigned to k.
If X is a classID, then satisfaction of tr(X) follows by an analogous argument as that for directives of form 1. Similar for value restrictions. If X is a some-value restriction of type Z on a property p, then there must be some object l such that (k,l) in ER(p) such that l is in EC(Z). By induction we have satisfaction of tr(Z) for some variable assignment. Then, by definition of I, we have I_truth(I_frame'(I_C(<p>))( k, l )) = t (true), thereby establishing satisfaction of tr_O(X) in I*. This extends straightforwardly to union, intersection, and one-of descriptions.
By satisfaction of tr_O(d), we have that tr_O(Y) is necessarily satisfied for ?x assigned to k. By an argument analogous to the argument above, we obtain that k is in EC(Y).
This establishes satisfaction of d in I.

We obtain that every directive is satisfied in I, thereby obtaining satisfaction of condition 2. Therefore, O, and thus every ontology in C, is satisfied in I. Clearly, I* satisfies R and not φ, so (I*, I) satisfies R and not φ. We conclude that C does not entail φ.

(<=) Assume C does not owl-dl-entail φ. This means there is a common-rif-owl-dl-interpretation (I, I) that is an owl-dl-model of C, but I is not a model of φ. To show that R' does not entail φ, we show that I is a model of R'.
R is satisfied in I by assumption. Satisfaction of tr_O(O_i) can be shown analogously to establishment of satisfaction in I of O_i in the (=>) direction. We now establish satisfaction of the rules in R^OWL-DL(V).
(i) follows immediately from the fact that EC(owl:Nothing)={}. (ii) follows from conditions 14 and 12 on abstract OWL interpretations. (iii) follows from the fact that EC maps class names to subsets of O=EC(owl:Thing) (conditions 3, 12 on abstract OWL interpretations). (iv) follows from condition 14 on abstract OWL interpretations and the fact that LV is a superset of the value spaces of all datatypes (by conditions 15 and 4 on abstract OWL interpretations). (v) follows from conditions 12, 5, and 6. (vi) and (vii) follow from conditions 12, 14, 5, and 6. (viii) follows from condition 11. (ix) follows from conditions 16 and 15. (x) follows from the fact that LV includes all plain literals (condition 2) and condition 17. (xi) follows from conditions 15, 14, and the fact that LV is a superset of the value space of a datatype. (xii) follows from condition 15; i.e., there is no assignment for the variable ?x that is both a member of the value space of the datatype and is in its class extension and thus the antecedent of the rule will never be satisfied and rule is always satisfied. (xiii) follows from condition 12 and the fact that LV is a superset of the union of all value spaces. (xiv) follows from the fact that there is no ill-typed literal, since such a literal would either violate condition 16 or condition 18 on abstract OWL interpretations.

This establishes satisfaction of R^OWL-DL(V), and thus R', in I. Therefore, R' does not entail φ.   ☐

The following theorems establish faithfulness of the full embedding of RIF-OWL DLP combinations into RIF.

Theorem Given a datatype map D conforming with T, a RIF-OWL-DL-combination C=<R,{O₁,...,O_n}>, where O₁,...,O_n are OWL DLP ontologies with vocabulary V that do not import other ontologies, owl-dl-entails a DL-condition formula φ with respect to D iff tr(merge({R, R^OWL-DL(V), tr_O(tr_N(O₁)), ..., tr_O(tr_N(O_n))})) entails tr(φ).

Proof. By the Normalization Lemma,
C=<R,{O₁,...,O_n}> owl-dl-entails φ iff <R,{tr_N(O₁),...,tr_N(O_n)}> owl-dl-entails φ.
Then, by the Normalized Combination Embedding Lemma,
<R,{tr_N(O₁),...,tr_N(O_n)}> owl-dl-entails φ iff merge({R, R^OWL-DL(V), tr_O(tr_N(O₁)), ..., tr_O(tr_N(O_n))}) dl-entails φ.
Finally, by the RIF-BLD DL-document formula Lemma,
merge({R, R^OWL-DL(V), tr_O(tr_N(O₁)), ..., tr_O(tr_N(O_n))}) dl-entails φ iff tr(merge({R, R^OWL-DL(V), tr_O(tr_N(O₁)), ..., tr_O(tr_N(O_n))})) entails tr(φ).
This chain of equivalences establishes the theorem.   ☐

Theorem Given a datatype map D conforming with T, a RIF-OWL-DL-combination <R,{O₁,...,O_n}>, where O₁,...,O_n are OWL DLP ontologies with vocabulary V that do not import other ontologies, is owl-dl-satisfiable with respect to D iff there is a semantic multi-structure I that is a model of tr(merge({R, R^OWL-DL(V), tr_O(tr_N(O₁)), ..., tr_O(tr_N(O_n))}).

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

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.

OWL 2: 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 OWL 2 the support for punning may