W3C


RIF RDF and OWL Compatibility

W3C Editor's Draft 18 May09 June 2008

This version:
http://www.w3.org/2005/rules/wg/draft/ED-rif-rdf-owl-20080518/http://www.w3.org/2005/rules/wg/draft/ED-rif-rdf-owl-20080609/
Latest editor's draft:
http://www.w3.org/2005/rules/wg/draft/rif-rdf-owl/
Previous version:
http://www.w3.org/TR/2008/WD-rif-rdf-owl-20080415/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

Copyright © 2008 W3C® (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.

Abstract

Rules interchanged using the Rule Interchange Format RIF may depend on or be used in combination with RDF data and RDF Schema or OWL data models (ontologies). This document, developed by the Rule Interchange Format (RIF) Working Group, specifies the interoperation between RIF and the data and ontology languages RDF, RDFS,RDF Schema, and OWL.

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May Be Superseded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/.

This document is being published as one of a set of 56 documents:

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

RIF Use Cases and RequirementsPlease Comment By 2008-05-252008-06-13

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

No Endorsement

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

Patents

This document was produced by a group operating under the 5 February 2004 W3C Patent Policy. W3C maintains a public list of any patent disclosures made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent. An individual who has actual knowledge of a patent which the individual believes contains Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.

Contents

1 Overview of RDF and OWL Compatibility

The Rule Interchange Format (RIF), specifically the(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 Rulesetsdocuments and RDF data and RDFS and OWL ontologies are interpreted; i.e., it specifies how RIF interoperates with RDF/OWL.

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

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

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

A specialization of this scenario is the publication of RIF rules that refer to RDF graphs: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 ruleset,document, publishes the OWL ontology, and sends (or publishes) the RIF ruleset,document, which includes a reference to the OWL ontology. A consumer of the rules retrieves the OWL ontology and translates the ontology and rulesetdocument into a combined ontology+rules description in its own rule extension of OWL.


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


Throughout this documentThe following conventions are used when writing RIFnotation of certain symbols, particularly IRIs and RDF statementsplain literals, in examples and definitions.RIF constantsis slightly different from the notation in RDF/OWL. This difference is illustrated in the symbol space rif:iri , i.e., constants that are absolute IRIs, are abbreviated. Specifically, constants ofSection Symbols in RIF Versus RDF/OWL.

The form " absolute-IRI "^^rif:iri are written as compact IRIsRDF semantics specification ( CURIE ), i.e., as prefix : localname , where prefix is understood to refer to an IRI namespace-IRI , and prefix : localname standsRDF-Semantics) defines four notions of entailment for RDF graphs. The IRIOWL semantics specification ( absolute-IRIOWL-Semantics) obtained by concatenating namespace-IRIdefines two notions of entailment for OWL ontologies, namely OWL Lite/DL and localname . RDF triples are written usingOWL Full. This document specifies the Turtle syntax ( Turtle ):interaction between RIF and RDF/OWL for all six notions. The purposesSection RDF Compatibility is concerned with the combination of this document, triples are written as s p o , where s, p, o are IRIs delimited with ' < 'RIF and ' > ', compact IRIs prefix : localname , or typed literals "literal" ^^ datatype-IRI . The following namespace prefixes are used throughout this document: ex refers to the example namespace http://example.org/example# , xsd refers to the XML schema namespace http://www.w3.org/2001/XMLSchema# , rdf refers toRDF/RDFS. 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# ,combination of RIF and OWL refers tois addressed in the Section OWL namespace http://www.w3.org/2002/07/owl# , and rif refers to the RIF namespace http://www.w3.org/2007/rif#Compatibility. The RDF semantics specification ( RDF-Semantics ) defines four notions of entailment for RDF graphs. The OWLsemantics specification ( OWL-Semantics ) defines two notionsof entailment for OWL ontologies, namely OWL Lite/DL and OWL Full. This document specifiesthe interaction between RIF and RDF/OWL for all six notions.OWL DL is close in spirit to (SWRL).

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

The Appendix: Embeddings (Informative) describes how reasoning with combinations of RIF rules with RDF and a subset of OWL DL can be reduced to reasoning with RIF rulesets,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.

2 RDF CompatibilityThroughout this section specifies how adocument the following conventions are used when writing RIF ruleset interacts with a set ofand RDF graphs in a RIF-RDF combination. In other words, how rules can "access" datastatements in the RDF graphsexamples and how additional conclusions that may be drawn from thedefinitions.

2 Symbols in RIF Versus RDF/OWL (Informative)

Where RDF/OWL has four kinds of constants: URI references (i.e., IRIs), plain literalliterals without language tag "literal string" String in the symbol space xsd:string "literal string"^^xsd:stringtags, plain literalliterals with language tag "literal string"@en String plus language tag in the symbol space rif:text "literal string@en"^^rif:text Literaltags and typed literals (i.e., Unicode sequences with datatype "1"^^xsd:integer Symbol inIRIs) (RDF-Concepts), RIF has one kind of constants: Unicode sequences with symbol space "1"^^xsd:integer There is, furthermore, a correspondence between statements in RDF graphs and certain kindsIRIs (DTB).

Symbol spaces can be seen as groups of formulas in RIF. Namely, thereconstants. Every datatype is a correspondence between RDF triples ofsymbol space, but there are symbol spaces that are not datatypes. For example, the form s p osymbol space rif:iri groups all IRIs. The shortcut syntax for IRIs and RIF frame formulasstrings (RIF-DTB), used throughout this document, corresponds with the syntax for IRIs and plain literals in (Turtle).

saying that ex:john is a brother of ex:jackThe correspondence between constant symbols in RDF graphs and ex:jackRIF documents is a parent of ex:mary ,explained in Table 1.

Table 1. Correspondence between RDF and RIF symbols.
RDF Symbol Example RIF Symbol Example
IRI <http://www.w3.org/2007/rif> IRI <http://www.w3.org/2007/rif>
Plain literal without language tag "literal string" String "literal string"
Plain literal with language tag "literal string"@en String plus language tag in symbol space rif:text "literal string@en"^^rif:text
Constant "1"^^xs:integer Symbol in symbol space "1"^^xs:integer

RIF does not have a notion corresponding to RDF blank nodes. RIF ruleset that contains the rule Forall ?x, ?y, ?z (?x[ex:uncleOf -> ?z] :- And(?x[ex:brotherOf -> ?y] ?y[ex:parentOf -> ?z])) which says that wheneverlocal symbols, written _symbolname, have some x is a brothercommonality with blank nodes; like the blank node label, the name of some y and y isa parentlocal symbol is not exposed outside of some z , then x is an uncle of z . From this combination the RIF frame formula :john[:uncleOf -> :mary] , as well asthe RDF triple :john :uncleOf :mary , can be derived. Note that blank nodes cannot be referenced directly from RIF rules, sincedocument. However, in contrast to blank nodesnodes, which are essentially existentially quantified variables, RIF local to a specific RDF graph.symbols are constant symbols. Finally, variables in the bodies of RIF rules do, however, range over objects denoted by blank nodes. So, it is possiblemay be existentially quantified, and are thus similar to "access" an object denoted by ablank node from anodes; however, RIF BLD does not allow existentially quantified variables to occur in rule usingheads.

3 RDF Compatibility

This section specifies how a variableRIF document interacts with a set of RDF graphs in a rule. Typed literalsRIF-RDF combination. In other words, how rules can "access" data in the RDF graphs and how additional conclusions that may be ill-typed , which means thatdrawn from the literal stringRIF rules are reflected in the RDF graphs.

There is not parta correspondence between statements in RDF graphs and certain kinds of the lexical spaceformulas in RIF. Namely, there is a correspondence between RDF triples of the datatype under consideration. Examplesform s p o and RIF frame formulas of such ill-typed literals are "abc"^^xsd:integerthe form s'[p' -> o'], where s', "2"^^xsd:booleanp', and "<non-valid-XML"^^rdf:XMLLiteral . Ill-typed literalso' are not expectedRIF symbols corresponding to be used very often. However, asthe RDF recommendation ( RDF-Concepts ) allows creating RDF graphs with ill-typed literals, their occurrence cannot be completely ruled out. Rules that include ill-typed symbols are not legal RIF rules, so there are no RIFsymbols that correspond to ill-typed literals. As withs, p, and o, respectively. This means that whenever a triple s p o is satisfied, the corresponding RIF frame formula s'[p' -> o'] is satisfied, and vice versa.

Consider, for example, a combination of an RDF graph that contains the triples

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

saying that ex:john is a brother of ex:jack and ex:jack is a parent of ex:mary, and a RIF document that contains the rule

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

which says that whenever some x is a brother of some y and y is a parent of some z, then x is an uncle of z. From this combination the RIF frame formula :john[:uncleOf -> :mary], as well as the RDF triple :john :uncleOf :mary, can be derived.

Note that blank nodes,nodes cannot be referenced directly from RIF rules, since blank nodes are local to a specific RDF graph. Variables doin RIF rules do, however, range over objects denoted by such literals.blank nodes. So, it is possible to "access" an object denoted by a blank node from a RIF rule using a variable in a rule.

The following example illustrates the interaction between RDF and RIF in the face of ill-typed literals andblank nodes.

Consider a combination of an RDF graph that contains the triple

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

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

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

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

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

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

as can the following RDF triples:

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

However, "http://a"^^rif:iri["http://p"^^rif:irithere is no RIF constant symbol t such that t[rdf:type -> "a"^^xsd:integer] cannotex:nameBearer] can be derived, because itthere is not a well-formed RIF formula, due to the factno constant that "a" is not an integer; it is not in the lexical space ofrepresents the datatype xsd:integer .name-bearer.


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

2.13.1 Syntax of RIF-RDF Combinations

This section first reviews the definitions of RDF vocabularies and RDF graphs, after which definitions related to datatypes and ill-typedtyped literals are reviewed. Finally, RIF-RDF combinations are formally defined.

2.1.13.1.1 RDF Vocabularies and Graphs

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

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

DEFINITION: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:  ☐

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))

2.1.23.1.2 Datatypes and Typed Literals

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

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

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

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

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

  1. No RIF-supported symbol space that is not a datatype (these are rif:local and rif:iri in RIF BLD) is in the domain of D.The IRIs identifying all RIF-requireddatatypes in T are in the domain of D.
  2. D maps IRIs identifying XML schema datatypes to the respective data types ( XML-SCHEMA2 ), rdf:XMLLiteraleach IRI in its domain to the rdf:XMLLiteralcorresponding datatype ( RDF-Concepts ), and rif:text toin T.   ☐

Editor's Note: The rif:text primitive datatype ( RIF-BLD ).terminology "considered datatype" might change if the notionsterminology is changed in BLD.

The notion of well- and ill-typed literalswell-typed literal loosely correspond towith the notionsnotion of legal and illegal symbolssymbol in RIF:

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

  1. d is in the domain of D and s is in the lexical space of D(d) or
  2. d is the IRI of a symbol space supportedrequired by RIF BLD and s is in the lexical space of the symbol space. Otherwise ( s , d ) is an ill-typed literal . Editor's Note: The definition of "ill-typed literal" should be revisited. Namely, not all literals that are not well-typed are ill-typed. Perhaps the notion of ill-typed literal is not necessary at all. 2.1.3  ☐

3.1.3 RIF-RDF Combinations

A RIF-RDF combination consists of a RIF rulesetdocument and zero or more RDF graphs. Formally:

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

2.23.2 Semantics of RIF-RDF Combinations

The semantics of RIF rulesetsdocuments and RDF graphs are defined in terms of model theories. The semantics of RIF-RDF combinations is defined through a combination of the RIF and RDF model theories, using a notion of common models. These models are then used to define satisfiability and entailment in the usual way. Combined entailment extends both entailment in RIF and entailment in RDF.

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

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

2.2.13.2.1 Interpretations

This section defines the notion of common-rif-rdf-interpretation, which is an interpretation of a RIF-RDF combination. This common-rif-rdf-interpretation is the basis for the definitions of satisfaction and entailment in the following sections.

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

2.2.1.13.2.1.1 RDF and RIF Interpretations

The notions of RDF interpretation and RIF semantic structure (interpretation) are briefly reviewed below.

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

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

As defined in (RIF-BLD), a semantic structure is a tuple of the form I = <TV, DTS, D, Dind, Dfunc, IC, IV, IF, Iframe, ISF, Isub, Iisa, I=, Iexternal, Itruth>. The specification of RIF-RDF compatibility is only concerned with DTS, D, IC, IV, Iframe, Isub, Iisa, and Itruth. 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.

2.2.1.2 Common RIF-RDF Interpretations DEFINITION: A common-rif-rdf-interpretation is a pair ( I , I), where I = < TV , DTS , D , D ind , D func , I CFor the purpose of the interpretation of imported documents, RIF BLD defines the notion of semantic multi-structures, which are nonempty sets {I V1, ..., I F ,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 frame ,={I SF1, ..., I sub ,n}, we use the symbol I isa ,to denote both the multi-structure and the common part of the individual structures I =1, ..., In.

3.2.1.2 Common RIF-RDF Interpretations

Definition. A common-rif-rdf-interpretation is a pair (I external, I), where I truth >is a RIFsemantic structuremulti-structure and I=<IR, IP, IEXT, IS, IL, LV>I is an RDF interpretation of a vocabulary V, such that the following conditions hold:

  1. (IR union IP) = Dind;
  2. IP is a superset of the set of all k in Dind such that there exist some a, b in Dind and Itruth(Iframe(k)(a,b))=t;
  3. LV is a superset of ( D ind intersection(union of the value spaces of all datatypes in DTS ));considered datatypes);
  4. IEXT(k) = the set of all pairs (a, b), with a, b, and k in Dind, such that Itruth(Iframe(k)(a,b))=t;
  5. IS(i) = IC( "i"^^rif:iri<i>) for every absolute IRI i in VU;
  6. IL((s, d)) = IC("s"^^d) for every well-typed literal (s, d) in VTL;
  7. IEXT(IS(rdf:type)) is equal to the set of all pairs <a,b>(a, b) in Dind × Dind such that Itruth(Iisa (< a,b >))=(a,b))=t; and
  8. IEXT(IS(rdfs:subClassOf)) is a superset of the set of all pairs <a,b>(a, b) in Dind × Dind such that Itruth(Isub (< a,b >))=(a,b))=t.   ☐

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

Condition 1 ensures that the combination of resources and properties corresponds exactly to the RIF domain; note that if I is an rdf-, rdfs-, or D-interpretation, IP is a subset of IR, and thus IR=Dind. 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 Dind are included in LV.LV (by definition, the value spaces of all considered datatypes are included in Dind). 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 RulesetDocument is empty. For example, consider the combination of an empty rulesetdocument and an RDF graph that contains the triple

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

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

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

as well as the following frame formula:

 "http://a"^^rif:iri ["http://p"^^rif:iri<http://a>[<http://p> ->  "http://b"^^rif:iri] 2.2.2<http://b>]

3.2.2 Satisfaction and Models

The notion of satisfiability refers to the conditions under which a common-rif-rdf-interpretation (I, I) is a model of a combination < R, S>. The notion of satisfiability is defined for all four entailment regimes of RDF (simple, RDF, RDFS, and D). The definitions are all analogous. Intuitively, a common-rif-rdf-interpretation (I, I) satisfies a combination < R, S> if I is a model of R and I satisfies S. Formally:

DEFINITION: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 aan existentially closed RIF-BLD condition formula φ if TValI(φ)=t.   ☐

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

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

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

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

DEFINITION: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:  ☐

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:  ☐

Definition. Given a conforming datatype map D, a model (I, I) of a combination C is a D-model of C if I is a D-interpretation; in this case C is D-satisfiable. 2.2.3  ☐

3.2.3 Entailment

Using the notions of models defined above, entailment is defined in the usual way, i.e., through inclusion of sets of models.

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


The other notions of entailment are defined analogously:

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

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

Definition. A combination C rdfs-entails S (resp., φ) if every rdfs-model of C satisfies S (resp., φ). 3  ☐

4 OWL Compatibility

The syntax for exchanging OWL ontologies is based on RDF graphs. Therefore, RIF-OWL-combinations are combinations of RIF rulesetsdocuments and sets of RDF graphs, analogous to RIF-RDF combinations. This section specifies how RIF rulesetsdocuments and OWL ontologies interoperate in such combinations.

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

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

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


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

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

a rdf:type C .

This statement says that the set denoted by C contains at least one element that is denoted by a . The corresponding RIF frame formula is a[rdf:type -> C] The terms a , rdf:type , and C are all interpreted as elements in the individual domain, and the pair of elements denoted by a and C is in the extension of the element denoted by rdf:type . This semantic discrepancy has practical implications in terms of entailments. Consider, for example, an OWL DL ontology with two class membership statements a rdf:type C . D rdf:type owl:Class . and a RIF ruleset Forall ?x ?y ?x=?y which says that every element is the same as every other element (note that such statements can also be written in OWL using owl:Thing and owl:hasValue ). From the naïve combination of the two one can derive C=D , and indeed a rdf:type D . This derivation is not sanctioned by the OWL DL semantics, because even if every element is the same as every other element, the class D might be interpreted as the empty set. A RIF-OWL combination that is faithful to the OWL DL semantics requires interpreting classes and properties as sets and binary relations, respectively, suggesting that correspondence could be defined with unary and binary predicates. It is, however, also desirable that there be uniform syntax for the RIF component of both OWL DL and RDF/OWL Full combinations, because one may not know at time of writing the rules which type of inference will be used. Consider, for example, an RDF graph S with the following statement a rdf:type C . and a RIF ruleset withand a RIF document with the rule

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

The combination of the two, according to the specification of RDF Compatibility, allows deriving

a rdf:type D .

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


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


Note that the abstract syntax form of OWL DL allows so-called punning (this is not allowed in the RDF syntax), i.e., the same IRI may be used in an individual position, a property position, and a class position; the interpretation of the IRI depends on its context. Since combinations of RIF and OWL DL are based on the abstract syntax of OWL DL, punning may also be used in these combinations.

In this paves the way towards combination withdocument, we are using OWL to refer to OWL 1. While OWL 2 , whichis envisioned to allowstill in development it is unclear how RIF will interoperate with it. At the time of writing, we believe that with OWL2 the support for punning may be beneficial, and that there might be particular problems in all its syntaxes. Editor's Note:using section 3.2.2.3, since the semantics of RIF-OWL DL combinations is similarannotation properties might be different than in spirit to the Semantic Web Rule Language proposal. However, a reference to SWRL from the above text does not seem appropriate. 3.1OWL 1.

4.1 Syntax of RIF-OWL Combinations

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

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

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

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

Definition. A RIF-OWL-DL-combination is a pair < R,O>, where R is a DL-RulesetDL-Document and O is a set of OWL DL ontologies in abstract syntax form of a vocabulary V.   ☐

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

3.1.14.1.1 Safeness Restrictions

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

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

Editor's Note: It is not strictly necessary to disallow disjunctions in the definition, but it would make the definition a lot more complex. It would require defining the disjunctive normal form of a condition formula and defining safeness with respect to each disjunct. Given that the safeness restriction is meant for implementation purposes, and that converting rules to disjunctive normal form is extremely expensive, it is probably a reasonable restriction to disallow disjunction. DEFINITION:Definition. A RIF-OWL-DL-combination < R,O> is DL-safe if every rule in R is DL-safe, given O. A RIF-OWL-DL-combination < R,O> is weakly DL-safe if every rule in R is weakly DL-safe, given O.   ☐

Editor's Note: Do we want additional safeness restrictions to ensure that variables do not crossThe abstract-concrete domain boundary? 3.2above definition of DL-safeness is intended to identify a fragment of RIF-OWL DL combinations for which implementation is easier than full RIF-OWL DL. This definition should be considered AT RISK and may change based on implementation experience.

4.2 Semantics of RIF-OWL Combinations

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

The semantics of RIF-OWL-DL-combinations cannot straightforwardly 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.

3.2.14.2.1 OWL Full

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

Definition. Given a conforming datatype map D, a RIF-RDF combination C OWL-Full-entails a generalized RDF graph S if every OWL-Full-model of C satisfies S. Likewise, C OWL-Full-entails aan existentially closed RIF-BLD condition formula φ if every OWL-Full-model of C satisfies φ. 3.2.2  ☐

4.2.2 OWL DL

The semantics of RIF-OWL-DL-combinations is similar in spirit to the semantics of RIF-RDF combinations. Analogous to a common-rif-rdf-interpretation, there is the notion of common-rif-dl-interpretations, which are pairs of RIF and OWL DL interpretations, and which define a number of conditions that relate these interpretations to each other. In contrast to RIF-RDF combinations, the conditions below define a correspondence between the interpretation of OWL DL classes and properties and RIF unary and binary predicates.
3.2.2.14.2.2.1 Modified Semantics for RIF Frame Formulas

The modification of the semantics of RIF frame formulas is achieved by modifying the mapping function for frame formulas (Iframe), 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:Definition. A RIFDL-semantic structure is a tuple I = <TV, DTS, D, Dind, Dfunc, IC, IV, IF, Iframe', ISF, Isub, Iisa, I=, Iexternal, Itruth>, where Iframe' is a mapping from Dind to total functions of the form SetOfFiniteFrame'Bags(D × D) → D, such that for each pair (a, b) in SetOfFiniteFrame'Bags(D × D) holds that if a≠IC(rdf:type), then b in Dind; all other elements of the structure are defined as in RIF semantic structures.

We define I ( o[a 1 ->v 1 ...A k ->v k ] ) = I frame ( I ( o ))({< I (DL-semantic multi-structure is a 1 ),nonempty set of DL-semantic structures {I ( v1 )>,, ..., < I ( a n ),I ( vn )>}). The truth valuation} that are identical in all respects except that the mappings IC1, ..., ICn might differ on the constants in Const that belong to the rif:local symbol space.   ☐

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

We define I(o[a1->v1 ... ak->vk]) = Iframe(I(o))({<I(a1),I(v1)>, ..., <I(an),I(vn)>}). The truth valuation function TValI is then defined as in RIF BLD.

DEFINITION:Definition. A RIFDL-semantic structuremulti-structure I is a model of a DL-RulesetDL-Document R if TValI(R)=t. 3.2.2.2  ☐

4.2.2.2 Semantics of RIF-OWL DL Combinations

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

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:Definition. Given a conforming datatype map D, a common-rif-dl-interpretation is a pair (I, I), where I = < TV , DTS , D , I C , I V , I F , I frame' , I SF , I sub , I isa , I = , I Truth >is a RIFDL-semantic structuremulti-structure and I=<R, EC, ER, L, S, LV>I is an abstract OWL interpretation with respect to D of a vocabulary V, such that the following conditions hold

  1. (O union LV)=Dind;
  2. EC( uc) = set of all objects k in O such that Itruth(Iframe'(IC(rdf:type))(k,IC( "u"^^rif:iri<c>))) = t, for every class identifier uc in V;
  3. ER( up) = set of all pairs (k, l) in O × (O union LV) such that Itruth(Iframe'(IC( "u"^^rif:iri<p>))( k, l ))) = t (true), for every data valued and individual valued property identifier up in V;
  4. L((s, d)) = IC("s"^^d) for every well-typed literal (s, d) in V;
  5. S(i) = IC( "i"^^rif:iri<i>) for every individual identifier i in V.   ☐

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


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

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

Definition. Given a conforming datatype map D, a RIF-OWL-DL-combination C OWL-DL-entails an OWL DL ontology in abstract syntax form O if every OWL-DL-model of C is an OWL-DL-model of O. Likewise, C OWL-DL-entails aan existentially closed RIFDL-condition formula φ if every OWL-DL-model of C is an OWL-DL-model of φ.   ☐


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

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

Consider the datatype xsd:stringxs: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 rulesetdocument containing the following fact

ex:myiri[rdf:type ->  xsd:string]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 xsd:stringxs: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 rulesetdocument containing the following fact

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

 ex:A rdfs:subClassOf ex:BSubClassof(ex:A ex:B)

and a RIF rulesetdocument containing the following rule

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

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

3.2.2.34.2.2.3 Annotation properties

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 is currently expected that OWL 2 will not define a semantics for annotation and ontology properties; therefore, the below definition cannot be extended to the case of OWL 2.

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

  6. ER( up) = set of all pairs (k, l) in O × O such that Itruth(Iframe'(IC( "u"^^rif:iri<p>))( k, l) ) = t (true), for every IRI up in V.   ☐


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

DEFINITION:Definition. Given a conforming datatype map D, a common-DL-annotation-interpretation (I, I) is an OWL-DL-annotation-model of a RIF-OWL-DL-combination C=< R, O > if I is a DL-modelmodel 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:  ☐

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

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

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

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

Forall  ?x,?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 rulesetdocument OWL-DL-annotation-entails the RIF condition formula ex:myOntology[ex:hasTitle -> "Example ontology"^^xsd:string]ontology"]; the combination does not OWL-DL-entail the formula.

45 Importing RDF Graphs in RIF

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

This section specifies how RIF rulesetsdocuments with such import statements should beare interpreted.


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

 Import(t 1 p 1 )Import(t1 p1)
  ...
 Import(t n p n )Import(tn pn)

Here, t iti is an IRI constant "of the form <absolute-IRI "^^rif:iri>, where absolute-IRI is the location of an RDF graph to be importedimported, and p ipi is an IRI constant denoting the profile to be used.

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

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

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

4.15.1 Profiles of Imports

RIF defines a specific profile for each of the notions of satisfiability and entailment of combinations, as well as two generic profiles for RDF and OWL, respectively. The use of a specific profile specifies how a combination should be interpreted and a receiver should reject a combination with a profile it cannot handle. The use of a generic profile implies that a receiver may interpret the combination to the best of his ability.

4.1.1 Specific ProfilesThe use of profiles is not restricted to the profiles specified in this document. Any profile that is used with RIF must specify an IRI that identifies it and notions of model, satisfiability, and entailment for combinations.

5.1.1 Specific Profiles

The following table lists the specific profiles,profiles defined by RIF, the IRIs of these profiles, and the notions of model, satisfiability, and entailment the shouldthat must be used with the profile.

Specific profiles in RIF.
Profile IRI of the Profile Model Satisfiability Entailment
simple http://www.w3.org/2008/rif-import-profile#Simplehttp://www.w3.org/2007/rif-import-profile#Simple simple-model satisfiability simple-entailment
rdf http://www.w3.org/2008/rif-import-profile#RDFhttp://www.w3.org/2007/rif-import-profile#RDF rdf-model rdf-satisfiability rdf-entailment
rdfs http://www.w3.org/2008/rif-import-profile#RDFShttp://www.w3.org/2007/rif-import-profile#RDFS rdfs-model rdfs-satisfiability rdfs-entailment
D http://www.w3.org/2008/rif-import-profile#Dhttp://www.w3.org/2007/rif-import-profile#D d-model d-satisfiability d-entailment
OWL DL http://www.w3.org/2008/rif-import-profile#OWL-DLhttp://www.w3.org/2007/rif-import-profile#OWL-DL OWL-DL-model OWL-DL-satisfiability OWL-DL-entailment
OWL DL annotation http://www.w3.org/2008/rif-import-profile#OWL-DL-annotationhttp://www.w3.org/2007/rif-import-profile#OWL-DL-annotation OWL-DL-annotation-model OWL-DL-annotation-satisfiability OWL-DL-annotation-entailment
OWL Full http://www.w3.org/2008/rif-import-profile#OWL-Fullhttp://www.w3.org/2007/rif-import-profile#OWL-Full OWL-Full-model OWL-Full-satisfiability OWL-Full-entailment

Profiles that are defined for combinations of DL-documents and OWL ontologies in abstract syntax form are called DL profiles. Of the mentioned profiles, the profiles OWL DL and OWL DL annotation are DL profiles.

The profiles are ordered as follows, where '<' reads "is lower than":

simple < rdf < rdfs < D < OWL Full

OWL DL < OWL DL annotation < OWL Full

4.1.25.1.2 Generic Profiles

The following table lists the generic profiles in RIF along with the IRI of the profile. Note that the use of a generic profile does not imply the use of a specific notion of model, satisfiability, and entailment.

Generic profiles in RIF.
Profile IRI of the Profile
RDF http://www.w3.org/2008/rif-import-profile#RDF 4.2Generic http://www.w3.org/2007/rif-import-profile#Generic

5.2 Interpretation of Profiles

Let R be a RIF ruleset withdocument such that

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

are the two-ary import statements Import("u 1 "^^rif:iri "p 1 "^^rif:iri) ... Import("u n "^^rif:iri "p n "^^rif:iri)in R and all imported documents and let Profile be the set of profiles corresponding to the IRIs p 1 ,...,p np1,...,pn.

If Profile contains only specific profiles, then:

If Profile contains a generic profile, then the combination C=<R,{S1,....,Sn}>, where S1,....,Sn are RDF graphs accessible from the locations u 1 ,...,u nu1,...,un and C may be interpreted according to the highest among the specific profiles in Profile or any higher profile. 5.

6 References

5.16.1 Normative References

[OWL-Semantics](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](RDF-Concepts)
Resource Description Framework (RDF): Concepts and Abstract Syntax, G. Klyne, J. Carroll (Editors), W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/. Latest version available at http://www.w3.org/TR/rdf-concepts/.
[RDF-Semantics](RDF-Semantics)
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-BLD)
RIF Basic Logic Dialect Harold Boley, Michael Kifer, eds. W3C Editor's Draft, 18 May09 June 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-bld-20080518/http://www.w3.org/2005/rules/wg/draft/ED-rif-bld-20080609/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-bld/.
[XML-Schema2](RIF-DTB)
RIF Data Types and Built-Ins Axel Polleres, Harold Boley, Michael Kifer, eds. W3C Editor's Draft, 09 June 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-dtb-20080609/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-dtb/.
(XML-Schema2)
XML Schema Part 2: Datatypes, W3C Recommendation, 2 May 2001. This version is http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/. Latest version available at http://www.w3.org/TR/xmlschema-2/.

5.26.2 Informational References

[CURIE](CURIE)
CURIE Syntax 1.0, M. Birbeck and S. McCarron, W3C Working Draft, 26 November 2007, http://www.w3.org/TR/2007/WD-curie-20071126/. Latest version available at http://www.w3.org/TR/curie/.
[DLP](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](Motik05)
Query Answering for OWL-DL with rules, B. Motik, U. Sattler, R. Studer, Journal of Web Semantics 3(1): 41-60, 2005.
[RDF-Schema](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-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)
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-PRD)
RIF Production Rule Dialect , C.Christian de Sainte Marie, Editor,eds. W3C Editor's Draft.Draft, 09 June 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-prd-20080609/. Latest version available at http://www.w3.org/2005/rules/wg/wiki/PRdialecthttp://www.w3.org/2005/rules/wg/draft/rif-prd/.
[RIF-UCR](RIF-UCR)
RIF Use Cases and Requirements, A. Ginsberg, D. Hirtle, F. McCabe, P.-L. Patranjan, Editors, W3C Working Draft, 10 July 2006, http://www.w3.org/TR/2006/WD-rif-ucr-20060710/. Latest version available at http://www.w3.org/TR/rif-ucr/.
[OWL-Reference] OWL Web Ontology Language Reference , M. Dean, G. Schreiber, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-ref-20040210/ . Latest version available at http://www.w3.org/TR/owl-ref/ . [Rosati06](Rosati06)
DL+log: Tight Integration of Description Logics and Disjunctive Datalog, R. Rosati, Proceedings of the 10th International Conference on Principles of Knowledge Representation and Reasoning, pp. 68-78, 2005.
[Turtle](SWRL)
SWRL: A Semantic Web Rule Language Combining OWL and RuleML, I. Horrocks, P. F. Patel-Schneider, H. Boley, S. Tabet, B., and M. Dean, W3C Member Submission, 21 May 2004, http://www.w3.org/Submission/2004/SUBM-SWRL-20040521/. Latest version available at http://www.w3.org/Submission/SWRL/.
(Turtle)
Turtle - Terse RDF Triple Language, D. Beckett and T. Berners-Lee, W3C Team Submission, 14 January 2008, http://www.w3.org/TeamSubmission/2008/SUBM-turtle-20080114/. Latest version available at http://www.w3.org/TeamSubmission/turtle/.

67 Appendix: Embeddings (Informative)

RIF-RDF combinations can be embedded into RIF RulesetsDocuments 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 RIF-OWL DL combinations that can be embedded.

ThroughoutThis sectionappendix illustrates embeddings into RIF BLD of simple, RDF, and RDFS entailment for RIF-RDF combinations and OWL DL entailment for RIF-OWL DL combinations, restricted to the DLP subset of OWL DL.

The embeddings are defined using the embedding function tr is defined,tr, which maps symbols, triples, RDF graphs, and OWL DL ontologies in abstract syntax form to RIF symbols, statements, and rulesets. 6.1documents, respectively.


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

7.1 Embedding RIF-RDF Combinations

The embedding of RIF-RDF combinations 6.1.1is 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.

7.1.1 Embedding Symbols

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.

Mapping RDF symbols to RIF.
RDF Symbol RIF Symbol Mapping
IRI i in VU Constant with symbol space rif:iri tr(i) = "i"^^rif:iri<i>
Blank node x in B Variable symbolssymbol ?x tr(x) = ?x
Plain literal without a language tag xxx in VPL Constant with the datatype xsd:stringxs:string tr("xxx") = "xxx"^^xsd:string"xxx"
Plain literal with a language tag (xxx,lang) in VPL Constant with the datatype rif:text tr("xxx"@lang) = "xxx@lang"^^rif:text
Well-typed literal (s,u) in VTL Constant with the symbol space u tr("s"^^u) = "s"^^u
Ill-typedNon-well-typed literal (s,u) in VTL Local constant s^^u' with symbol space rif:localthat is not used in C tr("s"^^u) = "s^^u'"^^rif:local

The embedding is not defined for combinations that include RDF graphs with RDF URI references that are not absolute IRIs. 6.1.27.1.2 Embedding Triples and Graphs

The mapping function tr is extended to embed triples as RIF statements. Finally, two embedding functions, trR and trQ embed RDF graphs as RIF rulesetsdocuments and conditions, respectively. The following section shows how these embeddings can be used for reasoning with combinations.

We define two mappings for RDF graphs, one (trR) in which variables are Skolemized, i.e., replaced with constant symbols, and one (trQ) in which variables are existentially quantified.

The function sk takes as an argument a formula Rφ with variables,variables and returns a formula R',φ', which is obtained from R by replacing every variable symbol ?x in R with "new-iri"^^rif:iri<new-iri>, where new-iri is a new globally unique IRI.

RDF Construct RIF Construct Mapping
Triple s p o . Frame formula tr(s)[tr(p) -> tr(o)] tr(s p o .) = tr(s)[tr(p) -> tr(o)]
Graph S RulesetDocument trR (S)(S) trR (S)(S) = the set of all sk(Forallsk(Document (Group (Forall (tr( s p o .t1)) ... Forall (tr(t1)) ))), where s p o . is a triplet1, ..., tm are the triples in S
Graph S Condition (query) trQ (S)(S) trQ (S)(S) = Exists tr(x1) , ...,... tr(xn) And((And(tr(t1) ... tr(tm )), where x1))), where x1, ..., xn are the blank nodes occurring in S and t1, ..., tm are the triples in S

6.1.37.1.3 Embedding Simple Entailment

Even though the semantics of the RDF vocabulary does not need to be axiomatized for simple entailment, the connection between RIF class membership and subclass statements and the RDF type and subclass statements needs to be axiomatized.

Rsimple = Document(Group(Forall ?x ?y (?x[rdf:type -> ?y] :- ?x # ?y)
   Forall ?x ?y (?x # ?y :- ?x[rdf:type -> ?y])

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

The following theorem shows how checking simple-entailment of combinations can be reduced to checking entailment of RIF conditions by using the embeddings of RDF graphs ofdefined in the previous section.

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

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

Theorem A RIF-RDF combination C=<R,{S1,...,Sn}>C=<R,{S1,...,Sn}> simple-entails a generalized RDF graph S iff (RT if and only if (R union trR (S1)(S1) union ... union trR (Sn))(Sn)) entails trQ (S).(T). C simple-entails a RIFan existentially closed RIF-BLD condition formula φ iff (Rif and only if (R union Rsimple union trR (S1)(S1) union ... union trR (Sn))(Sn)) entails φ.

6.1.4 Built-ins required The embeddingsEditor's Note: Formulation of RDF and RDFSthe entailment require a number of built-in predicate symbolstheorem is to be available to appropriately dealupdated with literals. Editor's Note:a notion of merge of rule sets.

Proof. We prove both directions by contradiction: if the entailment does not hold on one side, we show that it isalso does not yet clear which built-in predicates will be available in RIF. Therefore,hold on the built-ins mentionedother.
In this section may change. Furthermore, built-ins may be axiomatized if they are not provided by the language. Given a vocabulary V , the unary predicate wellxml V /1 is interpreted asthe set of XML valuesproof we abbreviate ( RDF-Concepts ), the unary predicate illxml V /1R union Rsimple union trR(S1) union ... union trR(Sn)) with R'.

(=>) Assume R' does not entail φ. This means there is interpreted as the setsome semantic multi-structure I that is a model of objects corresponding to ill-typed XML literals in V TL , andR', but not of φ. Consider the unary predicate illD V /1pair interpretation (I, I), where I is interpreteddefined as follows: Clearly, ( Forall ?x ?x[rdf:type -> rdf:Property] :- Exists ?y,?z (?y[?x -> ?z])I, Forall ?x ?x[rdf:type -> rdf:XMLLiteral] :- wellxml(?x) , Forall ?x "1"^^xsd:integer="2"^^xsd:integer :- And(?x[rdf:type -> rdf:XMLLiteral] illxml(?x)) ) TheoremI) is a combination <R,{S1,...,Sn}>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 rdf-satisfiable iff ( R RDF union R union tra model of R (S1) union ... unionsimple.

Consider a graph Si in {S1,...,Sn}. Let x1,..., xm be the blank nodes in Si and let u1,..., um be the new IRIs that were obtained from the variables ?x1,..., ?xm through the skolemization in trR (Sn)) has a model. Theorem(Si). Now, let A combination C=<R,{S1,...,Sn}> rdf-entailsbe a generalized RDF graph T iffmapping from blank nodes to elements in Dind such that A(xj)=IC( R RDF union R unionuj) for every blank node xj in Si. From the fact that I is a model of trR (S1) union ... union tr(Si) and by construction of I it follows that [I+A] satisfies Si, and so I satisfies Si.

We have that I is a model of R (Sn)) entails, 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 trQ (T). C simple-entails a RIF condition φ iff( R RDF union R union tr R (S1) union ... union tr R (Sn)) entails φ. 6.1.6 Embedding RDFS Entailment We axiomatize the semanticsT) and I is a model of R', but not of trQ(T). The RDF(S) vocabulary usingcommon-rif-rdf-interpretation (I, I) is obtained in the following RIF rulessame way as above, and conditions. R RDFS = R RDF union ( Forall tr( s p oso clearly satisfies C.
We proceed by contradiction. Assume I satisfies T. )) for every RDFS axiomatic triple s p oThis means there is some mapping A from the blank nodes x1,...,xm in T to objects in Dind such that [I+A] satisfies T.
) union ( Forall ?x ?x[rdf:type -> rdfs:Resource] , Forall ?u,?v,?x,?y ?u[rdf:type -> ?y] :- And(?x[rdfs:domain -> ?y] ?u[?x -> ?v])Consider now the semantic multi-structure I*, Forall ?u,?v,?x,?y ?v[rdf:type -> ?y] :- And(?x[rdfs:range -> ?y] ?u[?x -> ?v])which is the same as I, Forall ?x ?x[rdfs:subPropertyOf -> ?x] :- ?x[rdf:type -> rdf:Property]with the exception of the mapping I*V on the variables ?x1,...,?xm, Forall ?x,?y,?z ?x[rdfs:subPropertyOf -> ?z] :-which is defined as follows: I*V(?xj)=A(xj) for each blank node xj in S. By construction of I and (?x[rdfs:subPropertyOf -> ?y] ?y[rdfs:subPropertyOf -> ?z])since [I+A] satisfies T we can conclude that I* is a model of And(tr(t1)... tr(tm)), 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]so I is a model of trQ(T), violating the assumption that it is not. Therefore, (I, Forall ?x,?y,?z ?z[rdf:type -> ?y] :-I) does not satisfy T and (?x[rdfs:subClassOf -> ?y] ?z[rdf:type -> ?x])C does not entail φ.

(<=) Assume C does not entail φ. This means there is some common-rif-rdf-interpretation (I, Forall ?x ?x[rdfs:subClassOf -> ?x] :- ?x[rdf:type -> rdfs:Class]I) that satisfies C such that I is not a model of φ.
Consider the semantic multi-structure I$, Forall ?x,?y,?z ?x[rdfs:subClassOf -> ?z] :- And (?x[rdfs:subClassOf -> ?y] ?y[rdfs:subClassOf -> ?z])which is exactly the same I$, Forall ?x ?x[rdfs:subPropertyOf -> rdfs:member] :- ?x[rdf:type -> rdfs:ContainerMembershipProperty] , Forall ?x ?x[rdfs:subClassOf -> rdfs:Literal] :- ?x[rdf:type -> rdfs:Datatype]except for the mapping I$C on new IRIs that were introduced in skolemization. The mapping of these new IRIs is defined as follows:
For each graph Si in {S1,...,Sn}, Forall ?x ?x[rdf:type -> rdfs:Literal] :- lit(?x)let x1,..., xm be the blank nodes in Si and let u1,..., um be the new IRIs that were obtained from the variables ?x1,..., ?xm through the skolemization in trR(Si). Now, since I satisfies Si, Forall ?x "1"^^xsd:integer="2"^^xsd:integer :- And(?x[rdf:type -> rdfs:Literal] illxml(?x)) ) Theoremthere must be a combination <R 1 ,{S1,...,Sn}> is rdfs-satisfiable iffmapping A from blank nodes to elements in Dind such that [I+A] satisfies Si. We define I$C(uj)=A(xj) for every blank node xj in Si.

By assumption, I$ is a model of R RDFS union R 1 union tr(recall that I$ differs from I only on the new IRIs). Clearly, I$ is also a model of R (S1) union ... unionsimple, by conditions 7, 8, and 4 in the definition of common-rif-rdf-interpretation.
From the fact that I satisfies Si and by construction of I$ it follows that I$ is a model of trR (Sn)) has a model. Theorem(Si). So, I$ is a combination <R,{S1,...,Sn}> rdfs-entailsmodel of R'. Since I is not a generalized RDF graphmodel 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 iffand ( R RDFS union R union tr R (S1) union ... union tr R (Sn)) entails tr Q (T).I, I) satisfies C, but I does not satisfy T. We obtain I$ from I in the same way as above, and so clearly satisfies R'. It can be shown analogous to the (=>) direction that if I$ is a model of trQ(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 trQ(T) and thus R' does not entail trQ(T).

7.1.4 Embedding RDF Entailment

We axiomatize the semantics of the RDF vocabulary using the following RIF rules. We assume that ex:illxml is not used in any document.

RRDF = Rsimple union

((Forall (tr(s p o .))) for every RDF axiomatic triple s p o .) union
((Forall (ex:illxml(tr("s"^^rdf:XMLLiteral)))) for every non-well-typed literal of the form (s, rdf:XMLLiteral) in VTL) union
(

   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 <R,{S1,...,Sn}> is rdf-satisfiable iff there is a semantic multi-structure I that is a model of R, RRDF, trR(S1), ..., and trR(Sn)).

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

Theorem A RIF-RDF combination C=<R,{S1,...,Sn}> rdf-entails a generalized RDF graph T iff (RRDF union R union trR(S1) union ... union trR(Sn)) entails trQ(T). C rdf-entails an existentially closed RIF-BLD condition formula φ iff (RRDF union R union trR(S1) union ... union trR(Sn)) entails φ.

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

(*) 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 RRDF for every RDF axiomatic triple t, the fact that I is a model of RRDF, and construction of I. Consider the three RDF semantic conditions:
1 x is in IP if and only if <x, I(rdf:Property)> is in IEXT(I(rdf:type))
2 If "xxx"^^rdf:XMLLiteral is in V and xxx is a well-typed XML literal string, then

IL("xxx"^^rdf:XMLLiteral) is the XML value of xxx;
IL("xxx"^^rdf:XMLLiteral) is in LV;
IEXT(I(rdf:type)) contains <IL("xxx"^^rdf:XMLLiteral), I(rdf:XMLLiteral)>
3 If "xxx"^^rdf:XMLLiteral is in V and xxx is an ill-typed XML literal string, then

IL("xxx"^^rdf:XMLLiteral) is not in LV;

IEXT(I(rdf:type)) does not contain <IL("xxx"^^rdf:XMLLiteral), I(rdf:XMLLiteral)>.

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

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

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

7.1.5 Embedding RDFS Entailment

We axiomatize the semantics of the RDF(S) vocabulary using the following RIF rules.

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

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

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

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

RRDFS = RRDF union

((Forall tr(s p o .)) for every RDFS axiomatic triple s p o .) union
(

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

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

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

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

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

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

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

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

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

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

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

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

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

) union

(Forall ?x (?x[rdf:type -> rdfs:Literal] :- Or(pred:isDT1(?x) ... pred:isDTn(?x))), where DT1,...,DTn are the labels of the datatypes in T)


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

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

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

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

(*) We amend the definition of I by changing the definition of LV to the following: Clearly, this change does not effect satisfaction of the RDF semantic conditions 1 and 2. To see that condition 3 is still satisfied, consider some ill typed XML literal t. Then, ex:illxml(tr(t)) is satisfied in I. If tr(t)[rdf:type -> rdfs:Literal] were to be satisfied as well, then, by the second last rule in the definition of RRDFS, "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 RRDFS for every RDFS axiomatic triple t, the fact that I is a model of RRDFS, and construction of I. Consider the RDFS semantic conditions:
1 x is in ICEXT(y) if and only if <x,y> is in IEXT(I(rdf:type))

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

The first and second part of condition 1 are simply definitions of ICEXT and IC, respectively. Since I satisfies the first rule in the definition of RRDFS it must be the case that every element k in Dind is in ICEXT(I(rdfs:Resource)). Since IR=Dind, it follows that IR = ICEXT(I(rdfs:Resource)). Clearly, every object in ICEXT(I(rdfs:Literal)) is in LV, by definition. Consider any value k in LV. By definition, either k is in the value space of some considered datatype or Itruth(Iframe(IC rdfs-entails a RIF condition φ iff( R RDFS union R union tr R (S1) union ... union tr R (Sn)) entails φ. 6.1.7 Embedding D-Entailment We axiomatizerdf:type))(k,IC(rdfs:Literal)))=t. In the semantics oflatter case, clearly k is in ICEXT(I(rdfs:Literal)). In the data types usingformer case, k is in the following RIF rulesvalue space of some datatype with some label D, and conditions. Rthus Itruth(IF(IC(pred:isD =))(k))=t. By the last rule in RRDFS union ( Forall u[rdf:type -> rdfs:Datatype] | for every IRI u in, it must consequently be the domain of D) unioncase that Itruth( Forall "s"^^u[rdf:type -> "u"^^rif:iri] | for every well-typed literalIframe( sIC(rdf:type))(k, u ) in V TL ) unionIC( Forall ?x, ?y dt(?x,?y) :- And(?x[rdf:type -> ?y] ?y[rdf:type -> rdfs:Datatype])rdfs:Literal)))=t, Forall ?x "1"^^xsd:integer="2"^^xsd:integer :- And(?x[rdf:type -> rdfs:Literal] illD(?x)) ) Theorem A combination <R,{S1,...,Sn}>, whereand thus k is in ICEXT(I(rdfs:Literal)). So, I satisfies condition 1.

Satisfaction of conditions 2 through 10 in I follows immediately from satisfaction in I of the 2nd through the 12th rule in the definition of R does not containRDFS. This establishes the equality symbol,fact that I is D-satisfiable iff ( R D union R union tran rdfs-interpretation.

(**) Consider R (S1) union ... union trRDFS. Satisfaction of R (Sn))RDF was established in the [[SWC#proof-rdf-entailment|proof] in the previous section. Satisfaction of the facts corresponding to the RDFS axiomatic triples in I$ follows immediately from the definition of common-rif-rdf-interpretation and the fact that I is satisfiablean rdfs-interpretation, and does not entail Exists ?x And(dt(?x,u) dt(?x,u')) for any two URIs uthus satisfies all RDFS axiomatic triples.
Satisfaction of the 1st through the 12th, second, and u'third rule in RRDFS follow straightforwardly from the domainRDFS semantic conditions 1 through 10. Satisfaction of D such thatthe value spaces of D( u ) and D( u' ) are disjoint, and does not entail Exists ?x dt(s^^u,"u'"^^rif:iri) for any ( s13th rule follows from the fact that, given an ill-typed XML literal t, uIL(t) is not in V TLLV (by RDF semantic condition 3), ICEXT(rdfs:Literal)=LV, and u' inthe domain of D suchfact that sthe ex:illxml predicate is notonly true on ill-typed XML literals. Finally, satisfaction of the last rule in RRDFS follows from the lexical space of D( u' ). Editor's Note: Since this condition is very complex we might consider discarding this theorem, and suggestfact that ICEXT(rdfs:Literal)=LV, the above setdefinition of rules ( R D )LV as an approximation of the semantics. Theorema D-satisfiable combination <R,{S1,...,Sn}>, where R does not containsuperset of the equality symbol, D-entails a generalized RDF graph T iff ( R D union R union tr R (S1) union ...union tr R (Sn)) entails tr Q (T). C D-entails a RIF condition φ iff ( Rof the value spaces of all datatypes, and the definition of the pred:isD union R union tr R (S1) union ... union tr R (Sn)) entails φ. Editor's Note:predicates. This establishes the restriction to equality-free rulesetsfact that I$ is necessary because, in case different datatype URIs are equal, D-interpretations impose stronger conditions on the interpretationa model of typed literals than RIF does. 6.2RRDFS.

7.2 Embedding RIF-OWL DL Combinations

It is known that expressive Description Logic languages such as OWL DL cannot be straightforwardly embedded into typical rules languages such as RIF BLD.

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

6.2.17.2.1 Identifying OWL DLP

Our definition ofOWL DLP removesrestricts the OWL DL abstract syntax (OWL-Semantics), removing disjunction and extensional quantification from consequents of implications and removesremoving negation and equality. We introduceThe semantics of OWL DLP through its abstract syntax, whichis a subset ofthe abstract syntax ofsame as OWL DL.

The semantics ofDefinition. An OWL DL ontology in abstract syntax form is an OWL DLP isontology if it respects the same as OWL DL.grammar below.   ☐

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

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

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

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


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

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

type ::= Rdescription

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

We start with descriptions that may be allowed on both sides

dataRange ::= datatypeID | 'rdfs:Literal'

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

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

We then proceed with the individual sides

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



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



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

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

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

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

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



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

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

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

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

6.2.27.2.2 Embedding RIF DL-rulesetsDL-documents into RIF BLD

Recall that the semantics of frame formulas in DL-rulesetsDL-documents is different from the semantics of frame formulas in RIF BLD.

Frame formulas in DL-rulesetsDL-documents are embedded as predicates in RIF BLD. The mapping tr is the identity mapping on all RIF formulas, with the exception of frame formulas.formulas, as defined in the following table.

Mapping RIF DL-rulesetsDL-documents to RIF rulesets.documents.
RIF Construct Mapping
Term x tr(x)=x
Atomic formula x that is not a frame formula tr(x)=x
a[b -> c], where a,c are terms and b ≠ rdf:type is a constant tr(a[b -> c])=b'(a,c), where b' is a constant symbol obtained from b that does not occur in the original rulesetdocument or the ontologies
a[rdf:type -> c], where a is a term and c is a constant tr(a[rdf:type -> c])=c'(a), where c' is a constant symbol obtained from c that does not occur in the original rulesetdocument or the ontologies
Exists ?V1 ... ?Vn(φ) tr(Exists ?V1 ... ?Vn(φ))=Exists ?V1 ... ?Vn(tr(φ))
And(φ1 ... φn) tr(And(φ1 ... φn))=And(tr(φ1) ... tr(φn))
Or(φ1 ... φn) tr(Or(φ1 ... φn))=Or(tr(φ1) ... tr(φn))
φ1 :- φ2 tr(φ1 :- φ2)=tr(φ1) :- tr(φ2)
Forall ?V1 ... ?Vn(φ) tr(Forall ?V1 ... ?Vn(φ))=Forall ?V1 ... ?Vn(tr(φ))
Ruleset(φDocument(φ1 ... φn) tr( Ruleset(φDocument(φ1 ... φn))= Ruleset(tr(φDocument(tr(φ1) ... tr(φn))

6.2.37.2.3 Embedding OWL DLP into RIF BLD

The embedding of OWL DLP into RIF BLD has two stages: normalization and embedding.
6.2.3.17.2.3.1 Normalization

Normalization splits the OWL axioms so that the mapping of the individual axioms results in rules. Additionally, it simplifies the abstract syntax and removes annotations.

Editor's Note: Embedding OWL-DL-annotation semantics would require maintaining the annotation properties.Normalizing OWL DLP.
Complex OWL Normalized OWL
trN( 
Ontology( [ ontologyID ] 
 directive1
 ... 
 directiven )
)		 trN(directive1) 
...
trN(directiven)
trN(Annotation( ... ))
trN( 
Individual( individualID 
  annotation1 
  ...
  annotationn 
  type1
  ...
  typem
  value1
  ...
  valuek )
)

trN(Individual( individualID type1 ))

  ...

trN(Individual( individualID typem ))

Individual( individualID value1 )
  ...
Individual( individualID valuek )
trN( 
Individual( individualID 
  type(intersectionOf(
   description1
   ...
   descriptionn
  ))
)

trN(Individual( individualID type(description1) ))

  ...

trN(Individual( individualID type(descriptionn) ))
trN( 
Individual( individualID type(X))
)		 
Individual( individualID type(X))
X is a classID or value restriction
trN( 
Individual( individualID 
 type(restriction(propertyID 
 allValuesFrom(X))))
)		 trN( 
SubClassOf( oneOf(individualID) 
  restriction(propertyID allValuesFrom(X)))

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

trN(

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

trN(

SubClassOf(classID  
 intersectionOf(description1
  ...
  descriptionm )
)
trN( 
DisjointClasses(
  description1
  ...
  descriptionm )
)		 trN(SubClassOf(intersectionOf(description1 description2) owl:Nothing)) 
 ...

trN(SubClassOf(intersectionOf(description1 descriptionm) owl:Nothing))

 ...

trN(SubClassOf(intersectionOf(descriptionm-1 descriptionm) owl:Nothing))
trN( 
EquivalentClasses(
  description1
  ...
  descriptionm )
)

trN(SubClassOf(description1 description2)) trN(SubClassOf(description2 description1))

  ...

trN(SubClassOf(descriptionm-1 descriptionm)) trN(SubClassOf(descriptionm descriptionm-1))
trN( 
SubClassOf(description X)
)		 
SubClassOf(description X)
X is a description that does not contain intersectionOf
trN( 
SubClassOf(description 
  ...intersectionOf(
    description1 
    ...
    descriptionn
  )...)
)

trN(SubClassOf(description ...description1...))

...

trN(SubClassOf(description ...descriptionn...))
trN(Datatype( ... ))
trN( 
DatatypeProperty( propertyID [ Deprecated ]
  annotation1 
  ...
  annotationn 
  super(superproperty1)
  ...
  super(superpropertym)
  domain(domaindescription1)
  ...
  domain(domaindescriptionj)
  range(rangedescription1)
  ...
  range(rangedescriptionk) )
)		 
SubPropertyOf(propertyID superproperty1)
  ... 
SubPropertyOf(propertyID superpropertym)

trN(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescription1))

  ...

trN(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescriptionj))

trN(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescription1)))

  ...

trN(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescriptionk)))
trN( 
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)

trN(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescription1))

  ...

trN(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescriptionl))

trN(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescription1)))

  ...

trN(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescriptionk)))

ObjectProperty( propertyID
  [ inverseOf( inversePropertyID ) ] )
ObjectProperty( propertyID [ Symmetric ] )
ObjectProperty( propertyID [ Transitive ] )
trN( 
EquivalentProperties(
  property1
  ...
  propertym )
)

trN(SubPropertyOf(property1 property2)) trN(SubPropertyOf(property2 property1))

  ...

trN(SubPropertyOf(propertym-1 propertym)) trN(SubPropertyOf(propertym propertym-1))

The result of the normalization is a set of individual property value, individual typing, subclass, subproperty, and property inverse, symmetry and transitive statements.
6.2.3.27.2.3.2 Embedding

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

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

Editor's Note: This embedding assumes that for a given datatype identifier D , there is unary built-in predicate is D , calledThe "positive guard" for D , whichterminology "considered datatype" might change if the terminology is always interpreted aschanged in BLD.

By (RIF-DTB), each datatype in T has an associated label DATATYPE (e.g., the value spacelabel of thexs:string is String) and positive and negative guards pred:isDATATYPE denoted by Dand there is a built-in isNot D , called the "negative guard" for Dpred:isNotDATATYPE, which can be used to test whether a particular object is always interpreted as the complement of the(resp., is not) a value spaceof the datatype denoted by D .data type.

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

Embedding OWL DLP.
Normalized OWL RIF DL-rulesetDL-document
trO( 
directive1 
 ... 
directiven
)		 trO(directive1) 
...
trO(directiven)
trO( 
Individual( individualID type(A) )
)

tr(individualID)[rdf:type -> tr(A)]

 A is a classID
trO( 
Individual( individualID 
  type(restriction(propertyID value(b))) )
)

tr(individualID)[tr(propertyID) -> tr(b)]
trO( 
Individual( individualID 
  value(propertyID b) )
)

tr(individualID)[tr(propertyID) -> tr(b)]
trO( 
SubPropertyOf(property1 property2)
)

Forall ?x, ?yForall ?x ?y (?x[tr(property2) -> ?y] :- ?x[tr(property1) -> ?y])
trO( 
ObjectProperty(propertyID)
)
trO( 
ObjectProperty(property1
  inverseOf(property2) )
)

Forall ?x, ?yForall ?x ?y (?y[tr(property2) -> ?x] :- ?x[tr(property1) -> ?y])

Forall ?x, ?yForall ?x ?y (?y[tr(property1) -> ?x] :- ?x[tr(property2) -> ?y])
trO( 
ObjectProperty(propertyID
  Symmetric )
)

Forall ?x, ?yForall ?x ?y (?y[tr(propertyID) -> ?x] :- ?x[tr(propertyID) -> ?y])
trO( 
ObjectProperty(propertyID
  Transitive )
)

Forall ?x, ?y, ?zForall ?x ?y ?z (?x[tr(propertyID) -> ?z] :- And( ?x[tr(propertyID) -> ?y] ?y[tr(propertyID) -> ?z]))
trO( 
SubClassOf(description1 description2)
)

trO(description1,description2,?x)
trO(description1,X,?x)

Forall ?x (trO(X, ?x ) :- tr O ( description 1 , ?x ) X is a classID or value restriction tr O ( description 1 , D , ?x ) Forall ?x ( tr O ( owl:Nothing , ?x ) :- And( isNot D (?x) tr O ( description 1 , ?x ) ) D is a datatypeID and isNot D is the "negative guard" for D tr O ( description 1 , restriction(property 1 allValuesFrom(...restriction(property n allValuesFrom( X )) ...)) , ?x ) Forall ?x, ?y 1 , ..., ?y n ( tr O ( X , ?y n?x) :- And(trO(description1, ?x )

 ?x[ tr( property 1 ) -> ?y 1 ] ?y 1 [ tr( property 2 ) -> ?y 2 ] ... ?y n-1 [ tr( property n ) -> ?y n ]))X is a classIDclassID, datatypeID or value restriction
trO(description1,restriction(property1 allValuesFrom(...restriction(propertyn allValuesFrom( DX)) ...)),?x)

Forall ?x, ?yForall ?x ?y1 , ..., ?y... ?yn (trO( owl:NothingX, ?yn) :- And( trO(description1, ?x)?x[tr(property1) -> ?y1] ?y1[tr(property2) -> ?y2] ... ?yn-1[tr(propertyn) -> ?yn ] isNot D (?y n ))) D]))

 X is a classID, datatypeID or value restriction
trO(A,?x)

?x[rdf:type -> tr(A)]

 A is a classID tr O ( D , ?x ) is D (?x) D is aor datatypeID
and is D is the "guard" for the datatypetrO(intersectionOf(description1 ... descriptionn, ?x)

And(trO(description1, ?x) ... trO(descriptionn, ?x))
trO(unionOf(description1 ... descriptionn, ?x)

Or(trO(description1, ?x) ... trO(descriptionn, ?x))
trO(oneOf(value1 ... valuen, ?x)

Or( ?x = trO(value1) ... ?x = trO(valuen))
trO(restriction(propertyID someValuesFrom(description)), ?x)

Exists ?y(And(?x[tr(propertyID) -> ?y] trO(description, ?y) ))
trO(restriction(propertyID value(valueID)), ?x)

?x[tr(propertyID) -> tr(valueID) ]

tr OBesides 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.

ROWL-DL(V) = ( owl:ThingForall ?x ("a"="b" :- ?x[rdf:type -> owl:Nothing]),

?xForall ?x ("a"="b" :- And(?x[rdf:type -> rdfs:Literal] ?x[rdf:type -> owl:Thing])),
(Forall ?x (?x[rdf:type -> owl:Thing] :- ?x[rdf:type -> C])) ?x = ?x tr Ofor every class name C in V,
(Forall ?x (?x[rdf:type -> rdfs:Literal] :- ?x[rdf:type -> D])) for every datatype name D in V,
(Forall ?x ?y (?x[rdf:type -> owl:Thing] :- ?x[rdf:type -> P])) for every property name P in V,
(Forall ?x ?y (?y[rdf:type -> owl:Thing] :- ?x[rdf:type -> P])) for every individual-valued property name P in V,
(Forall ?x ?y (?y[rdf:type -> rdfs:Literal] :- ?x[rdf:type -> P])) for every data-valued property name P in V,
(Forall (tr(i)[rdf:type -> owl:Thing])) for every individual name i in V,
(Forall (tr(s^^u)[rdf:type -> u'])) for every well-typed literal s^^u and datatype identifier u' in V such that L2Vu(s) is in the value space of u',
(Forall (tr(t)[rdf:type -> u'])) for every plain literal t and datatype identifier u' in V such that t is in the value space of u',
( owl:NothingForall ?x ([rdf:type -> rdfs:Literal] :- ?x[rdf:type -> Diri])) for every considered datatype with identifier Diri,
?x(Forall ?x ("a"="b" :- And(?x[rdf:type -> Diri] isNotD(?x)]))) "1"^^xsd:integer="2"^^xsd:integer 6.2.4 Reasoningfor every considered datatype with RIF-OWL DLP Combinationsidentifier Diri and label D,
(Forall ("a"="b" :- tr(t)[rdf:type -> rdfs:Literal])) for every ill-typed literal t,

Theorem A RIF-OWL-DL-combination <R ,{O 1 ,...,O n,{O1,...,On}>, where O 1 ,...,O nO1,...,On are OWL DLP ontologies,ontologies with vocabulary V, is OWL-DL-satisfiable iff there is a semantic multi-structure I that is a model of tr(R union ROWL-DL(V) union trO(trN (O 1(O1)) union ... union trO(trN (O n )))(On))).

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

Theorem An OWL-DL-satisfiable RIF-OWL-DL-combination C=<R ,{O 1 ,...,O n }>,,{O1,...,On}>, where O 1 ,...,O nO1,...,On are OWL DLP ontologies,ontologies with vocabulary V, OWL-DL-entails aan existentially closed RIFRIF-BLD condition formula φ iff tr(R union ROWL-DL(V) union trO(trN (O 1(O1)) union ... union trO(trN (O n(On))) entails φ.

7Proof. The proof of the theorem consists of three steps. We first show
(*) Given an OWL DLP ontology O, O and trN(O) the same models. In the remainder we assume that all OWL DLP ontologies are normalized.
We then show
(**) C=<R,{O1,...,On}> OWL-DL-entails φ if and only if for every model of R union ROWL-DL(V) union trO(trN(O1)) union ... union trO(trN(On)) holds that TValI(φ)=t.
Finally, we show
(***) tr(R union ROWL-DL(V) union trO(trN(O1)) union ... union trO(trN(On))) entails φ if and only if for every model of R union ROWL-DL(V) union trO(trN(O1)) union ... union trO(trN(On)) holds that TValI(φ)=t.

8 End Notes

RDF URI References: There are certain RDF URI references that are not absolute IRIs (e.g., those containing spaces). It is possible to use such RDF URI references in RDF graphs that are combined with RIF rules. However, such URI references cannot be represented in RIF rules and their use in RDF is discouraged.

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.