RIF RDF and OWL Compatibility

W3C Editor's Draft 18 May17 July 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-20080717/

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-20080709/ (color-coded diff)

Editors:

Jos de Bruijn, Free University of Bozen/Bolzano

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

Abstract

Status of this Document

May Be Superseded

This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

Set of Documents

This document is being published as one of a set of 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 Data Types and Built-InsProduction Rule Dialect
6. RIF Use CasesDatatypes and RequirementsBuilt-Ins 1.0

Please Comment By 2008-05-252008-07-21

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

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

Patents

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The Rule Interchange Format (RIF), 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.RDF, RDFS, and OWL.

The RIF working group plans to develop further dialects besides BLD, most notably a dialect based on Production Rules ([RIF-PRD );]; these dialects are not necessarily extensions of BLD. Future versions of this document may address compatibility of these dialects with RDF and OWL 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,graphs; it is assumed that RDF graphs are exchanged using RDF/XML, or any other syntax that can be used for representing or exchanging RDF graphs.

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

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

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

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

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

• All RIF rulesstatements are reflected inwritten using the RDF graphs. There is a correspondence between constant symbols inRIF rulesetspresentation syntax [RIF-BLD]. Where possible, this document uses the shortcut syntax for IRIs and namesstrings as defined in [RIF-DTB].
• RDF graphs.triples are written using the Turtle syntax [Turtle]: for the purposes of this document, triples are written as s p o, where s, p, o are blank nodes _:x, IRIs delimited with '<' and '>', compact IRIs prefix:localname, plain literals without language tags "literal", plain literals with language tags "literal"@lang, or typed literals "literal"^^datatype-IRI.
• The following table explainsnamespace prefixes are used throughout this document: ex refers to the correspondencesexample namespace http://example.org/example#, xs refers to the XML schema namespace http://www.w3.org/2001/XMLSchema#, rdf refers to the RDF namespace http://www.w3.org/1999/02/22-rdf-syntax-ns#, rdfs refers to the RDFS namespace http://www.w3.org/2000/01/rdf-schema#, owl refers to the OWL namespace http://www.w3.org/2002/07/owl#, and rif refers to the RIF namespace http://www.w3.org/2007/rif#.

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

Symbol spaces can be seen as groups of constants. Every datatype is a symbol space, but there are symbol spaces that are not datatypes. For example, the symbol space rif:iri groups all IRIs. The correspondence between constant symbols in RDF graphs and RIF documents is explained in Table 1.

The shortcut syntax for IRIs and certain kinds of formulas in RIF. Namely, there is a correspondence between RDF triples ofstrings [RIF-DTB], used throughout this document, corresponds with the form s p osyntax for IRIs and RIF frame formulasplain literals in [Turtle], a commonly used syntax for RDF.

• IRIs, i.e., constants of the form s'[p' -> o']"absolute-IRI"^^rif:iri may be written as <absolute-IRI> or as compact IRIs [CURIE], i.e., as prefix:localname, where s' , p'prefix is understood to refer to an IRI namespace-IRI, and o' areprefix:localname stands for the IRI (absolute-IRI) obtained by concatenating namespace-IRI and localname.
• Strings, i.e., constants of the form "my string"^^xs:string may be written as "my string".

RIF symbolsdoes not have a notion corresponding to theRDF blank nodes. RIF local symbols s, p , and owritten _symbolname, respectively. This means that wheneverhave some commonality with blank nodes; like the blank node label, the name of a triple s p olocal symbol is satisfied,not exposed outside of the correspondingdocument. However, in contrast to blank nodes, which are essentially existentially quantified variables, RIF frame formula s'[p' -> o'] is satisfied,local symbols are constant symbols. In many applications and vice versa. Consider, for example, a combination ofdeployment scenarios, this difference may be inconsequential. However the results will differ when an RDF graph that contains the triples ex:john ex:brotherOf ex:jack . ex:jack ex:parentOf ex:mary . saying that ex:johnis used in a brother of ex:jack and ex:jack isnon-assertional context, such as in a parentquery pattern.

Finally, variables in the bodies of ex:mary ,RIF rules or in query patterns may be existentially quantified, and aare thus similar to blank nodes; however, RIF ruleset that contains the rule Forall ?x, ?y, ?z (?x[ex:uncleOf ->BLD does not allow existentially quantified variables to occur in rule heads.

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

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

_:x ex:hasName "John" . 

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

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

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

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

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

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

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

The syntax of the names in these sets is definedIn RDF Concepts and Abstract Syntax ( RDF-Concepts ). Besides these names,addition, there is an infinite set of blank nodes, which is disjoint from the sets of literalsnames. See RDF Concepts and IRIs. DEFINITION:Abstract Syntax [RDF-Concepts] for precise definitions of these concepts.

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

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 )) 2.1.2 Datatypes)

Definition. A RIF-RDF combination is a pair < R,S>, where R is a RIF document and S is a set of generalized RDF graphs of a vocabulary V.   ☐

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

Even though RDF allows the use of arbitrary datatype IRIs in typed literals, not all such datatype IRIs are recognized in the semantics. In fact, simple entailment does not recognize any datatype and RDF and RDFS entailment recognize only the datatype rdf:XMLLiteral. 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 therdf:XMLLiteral datatypeis includedin the domain of the map. RIF BLD additionallyrequires the followinga number of additional datatypes to be included: xsd:string , xsd:decimal , xsd:time , xsd:date , xsd:dateTime , and rif:text ;included; these datatypesare the RIF-required datatypes .[RIF-DTB].

When checking consistency of a conforming datatype map iscombination < R,S> or entailment of a datatype map that recognizes at leastgraph S or RIF formula φ by a combination < R,S>, the set of considered datatypes is the union of the set of RIF-required datatypes. DEFINITION:datatypesand the sets of datatypes used in R, the documents imported into R, and φ (when considering entailment of φ).

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

1. No RIF-supported symbol space that is notEvery IRI identifying a datatype (these are rif:local and rif:iridatatypes in RIF BLD)T is in the domain of D.
2. The IRIs identifying all RIF-required datatypes are in the domain of D.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.   ☐

Note that it follows from the rif:text primitivedefinition that every datatype ( RIF-BLD ).used in the notions of well- and ill-typed literals loosely correspond toRIF document in the notionscombination or the entailed RIF formula (when considering entailment questions) is included in any datatype map conforming to the set of legal and illegal symbolsconsidered datatypes. There may be datatypes used in an RDF graph in the combination that are not included in RIF: DEFINITION:such a datatype map.

Definition. Given a conformingdatatype 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 RIF-RDF Combinations A RIF-RDF combination consists of a RIF ruleset and zero or more RDF graphs. Formally: DEFINITION: A RIF-RDF combination is a pair < R , S >, where R is a RIF ruleset and S is a set of generalized RDF graphs of a vocabulary V . When clear from the context, RIF-RDF combinations are referred to simply as combinations . 2.2  ☐

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

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

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

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

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

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

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

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

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

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

• DTS is the set of datatypes, which have associated datatype identifiers,
• D is a set (the domain),
• D_ind is a non-empty subset of D,
• D_func is a non-empty subset of D,
• I_C is a mapping from individual identifiersconstants to D such that constants in Constindividual position are mapped to D_ind and from function symbolsconstants in Constfunction positions are mapped to D_func,
• I_V is a mapping from Var to D_ind,
• I_frame is a mapping from D_ind to functions of the form SetOfFiniteBags(D_ind × D_ind) → D,
• I_sub is a mapping from D_ind × D_ind to D,
• I_isa is a mapping from D_ind × D_ind to D, and
• I_truth is a mapping from D to TV.

2.2.1.2 Common RIF-RDF Interpretations DEFINITION: A common-rif-rdf-interpretation is a pair ( IFor the purpose of the interpretation of imported documents, RIF BLD defines the notion of semantic multi-structures, I), wherewhich are nonempty sets {I _{= < TV , DTS , D , D ind , D func , I C1}, ..., I _{V ,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 F ,={I _frame1, ..., I _{SF ,n}}, we use the symbol I sub ,to denote both the multi-structure and the common part of the individual structures I _isa1, ..., I _{= ,n}.

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

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

One consequence of conditions 5 and 6 is that IRIs of the form http://iri and typed literals of the form "http://iri"^^rif:iri that occur in an RDF graph are treated the same in RIF-RDF combinations, even if the RIF Rulesetdocument is empty. Similarly for plain literals without language tags of the form "mystring" and typed literals of the form "mystring"^^xs:string. For example, consider the combination of an empty rulesetdocument and an RDF graph that contains the tripletriples

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

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

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

as well as the following frame formula: "http://a"^^rif:iri ["http://p"^^rif:iriformulas:

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

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 a closed RIF-BLDcondition formula φ if TVal_I(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)fact

"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: Given a conforming datatype map D, a model  ☐

Definition. Let (I, I) be a model of a combination C and let D be a datatype map conforming with the set of datatypes in I. (I, I) is a D-model of C if I is a D-interpretation; in this case C is D-satisfiable. 2.2.3  ☐

DEFINITION: Given a conforming datatype map D,Definition. Let C be a RIF-RDF combination C D-entailscombination, let S be a generalized RDF graph S if every D-model of C satisfies S . Likewise, C D-entails, let φ be a closed RIF-BLDcondition φ if every D-model of C satisfiesformula, and let D be a datatype map conforming with the set of considered datatypes. C D-entails S if every D-model of C satisfies S. Likewise, C D-entails φ if every D-model of C satisfies φ.   ☐

The other notions of entailment are defined analogously:

DEFINITION: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  ☐

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 latterThis is a key property of Description Logic semantics thatand enables the use of Description Logic reasoning techniques for processing OWL DL descriptions. Defining RIF-OWL DL compatibility as an extension of RIF-RDF compatibility would define a correspondence between OWL DL statements and RIF frame formulas. Since RIF frame formulas are interpreted using an extension function, the same way as in RDF, defining the correspondence between them and OWL DL statements would change the semantics of OWL statements, even if the RIF ruleset isdocument were empty.

Consider, for example, an OWL DL ontology with a class membership statementA rdf:type C . This statement says that the set denoted by C contains at least one elementRIF-OWL combination that is denoted by a . The corresponding RIF frame formula is a[rdf:type -> C]faithful to the terms a , rdf:type ,OWL DL semantics requires interpreting classes and C are all interpretedproperties as elements in the individual domain,sets and the pair of elements denoted bybinary relations, respectively, suggesting that a correspondence could be defined with unary and C is inbinary predicates. It is, however, also desirable that there be uniform syntax for the extensionRIF component of both OWL DL and RDF/OWL Full combinations, because one may not know at time of writing the element denoted by rdf:type . This semantic discrepancy has practical implications in termsrules which type of entailments.inference will be used. Consider, for example, an OWL DL ontologyRDF graph S with two class membership statementsthe following statement

a rdf:type C .

D rdf:type owl:Class .and a RIF rulesetdocument with the rule

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

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

a rdf:type D .

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

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

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

In this paves the way towards combination withdocument, we are using OWL 2 ,to refer to OWL as specified in [OWL-Semantics], as opposed to OWL 2, which is envisioned to allow punningcurrently in all its syntaxes. Editor's Note:the semanticsprocess of RIF-OWL DL combinations is similar in spirit tobeing defined by the Semantic Web Rule Language proposal. However, a reference to SWRL fromOWL working group. (See the above text does not seem appropriate. 3.1End note on OWL 2)

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

The syntax of OWL ontologies in RIF-OWL DL combinations is specified by the abstract syntax of OWL DL. Certain restrictions are imposed on the syntax of the RIF rules in combinations with OWL DL. Specifically, the only terms allowed in class and property positions in frame formulas are constant symbols. DEFINITION:A RIF-BLD condition φDL-frame formula is a RIF DL-condition if for everyframe formula a[b₁ -> c] in φ it holds thatc₁ ... b _{is a constant andn} -> c_n] such that n≥1 and for every b_i, with 1≤i≤n, it holds that b_i is a constant and if b_i = rdf:type, then c_i is a constant.

DEFINITION:Definition. A RIF-BLD ruleset Rcondition formula φ is a DL-RulesetDL-condition if forevery frame formula a[b -> c]in every rule ofφ is a DL-frame formula.   ☐

Definition. A RIF-BLD document formula R it holds that bis a constant andRIF-BLD DL-document formula if b = rdf:type , then cevery frame formula in R is a constant. DEFINITION:DL-frame formula.   ☐

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

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

In the literature, several restrictions on the use of variables in combinations of rules and Description Logics have been identified ([Motik05, Rosati06 )] for the purpose of decidableeffective reasoning. TheseThis section specifies such safeness restrictions are specifiedfor RIF-OWL-DL combinations.

Given a set of OWL DL ontologies in abstract syntax form O, a variable ?x in a RIF rule Q thenH :- ifB is DL-safe if it occurs in an atomic formula in ifB 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 thenH :- ifB is DL-safe, given O if every variable that occurs in thenH :- ifB is DL-safe. A RIF rule Q thenH :- ifB 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, itH is probably a reasonable restriction to disallow disjunction. DEFINITION:DL-safe.

Definition. A RIF-OWL-DL-combination < R,O> is DL-safe if every rule in R is DL-safe, given O. A RIF-OWL-DL-combination < R,O> is weakly DL-safe if every rule in R is weakly DL-safe, given O. Editor's Note: Do we want additional  ☐

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

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

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

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

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

Definition. Let C be a RIF-RDF combination C OWL-Full-entailscombination, let S be a generalized RDF graph, let φ be a condition formula, and let D be a datatype map conforming with the set of considered datatypes. C owl-full-entails S with respect to D if every owl-full-model of C satisfies S. Likewise, C owl-full-entails a closed RIF-BLD conditionφ with respect to D if every owl-full-model of C satisfies φ. 3.2.2  ☐

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

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

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

A dl-semantic multi-structure is a nonempty set of dl-semantic structures {I₁, ..., I_n} that are identical in all respects except that the mappings I_1C, ..., I_nC might differ on the constants in Const that belong to the rif:local symbol space.   ☐

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

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

DEFINITION:Definition. A RIFdl-semantic structuremulti-structure I is a model of a DL-RulesetRIF-BLD DL-document formula R if TVal_I(R)=t. 3.2.2.2  ☐

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

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

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

DEFINITION:Definition. Given a conformingdatatype map D, a common-rif-dl-interpretationcommon-rif-owl-dl-interpretation with respect to D 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. D is conforming with the datatypes in I;
2. (O union LV)=D_ind;
3. EC( uc) = set of all objects k in Osuch that I_truth(I_frame'(k)({(I_C(rdf:type ))( k ,),I_C( "u"^^rif:iri )))<c>))})) = t, for every class or datatype identifier uc in V;
4. ER( up) = set of all pairs (k, l) in O × (O union LV)such that I_truth(I_frame'(k)({(I_C( "u"^^rif:iri ))( k ,<p>), l ))))})) = t (true), for every data valued and individual valued property identifier up in V;
5. L((s, d)) = I_C("s"^^d) for every well-typed literal (s, d) in V;
6. S(i) = I_C( "i"^^rif:iri<i>) for every individual identifier i in V.   ☐

Condition 12 ensures that the relevant parts of the domains of interpretation are the same. Condition 23 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 34 ensures that the interpretation (extension) of an OWL DL object or datatype property u corresponds to tothe interpretation of frames of the form ?x["u"^^rif:iri?x[<u> -> ?y]. Condition 45 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 56 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,common-rif-owl-dl-interpretation, satisfaction, models, and entailment are defined in the usual way:

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

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

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

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

Consider the datatype 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 notcannot be members of O.

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

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

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

Condition 6,7, which strengthens condition 3, ensures that the interpretation of all properties (also annotation and ontology properties) in the OWL DL ontologies corresponds with their interpretation in the RIF rules.

DEFINITION: GivenDefinition. A common-dl-annotation-interpretation with respect to a conformingdatatype map D, a common-DL-annotation-interpretationD (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: Given a conforming datatype map D, a RIF-RDF combination  ☐

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

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

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

which defines an ontology with a single annotation (title). Consider also a 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.

In the previous sectionssections, RIF-RDF Combinations and RIF-OWL combinations were defined in an abstract way, as pairs consisting of rulesetsa RIF document and setsa set of RDF graphs/OWL ontologies. In addition, different semantics were specified based on the various RDF and OWL entailment regimes. RIF provides a mechanism for explicitly referring to (importing) RDF graphs from rulesetsdocuments and specifyspecifying 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 shouldmust be used with the combination <R, {S}>.

Profiles are ordered as defined later in casethis section. If several graphs are imported in a ruleset,document, and these imports specify different profile,profiles, 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-RulesetRIF-BLD DL-document formula, S must be the translation to RDF of an OWL DL ontology in abstract syntax form O, and the notions of owl-dl-model, owl-dl-satisfiability, and owl-dl-entailment shouldmust be used with the combination <R, {O}>.

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 n<p1>,...,<pn>.

If Profile contains only specific profiles, then:

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

• If Profile contains only the OWLDL profiles and OWL DL annotation profiles, thenthe combination C=< R ,{ ORDF graphs accessible from the locations u₁ ,...., O,...,u_n }>, whereare OWL DL ontologies in graph form such that O₁,....,O_n are the OWL DLsame ontologies in abstract syntax form corresponding to the OWL DL ontologies in graph form accessible fromform, then the locations ucombination C=<R,{O₁ ,...,u,....,O_n and C should}> must be interpreted according to the highest among the profiles in Profile.

• Otherwise, the combination C=<R,{S₁,....,S_n}>, where S₁,....,S_n are RDF graphs accessible from the locations u₁,...,u_n and C should, must be interpreted according to the lowesthighest profile that is higher than or equal to all profilesin Profile.

If Profile contains a generic profile, then the combination C=<R,{S₁,....,S_n}>, where S₁,....,S_n are RDF graphs accessible from the locations u₁,...,u_n and C, may be interpreted according to the highest among the specific profiles in Profile or any higher profile. 5 References 5.1 Normative References [OWL-Semantics] OWL Web Ontology Language Semantics and Abstract Syntax , P. F. Patel-Schneider, P. Hayes, I. Horrocks, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-semantics-20040210/ . Latest version available at http://www.w3.org/TR/owl-semantics/ . [RDF-Concepts] Resource Description Framework (RDF): Concepts and Abstract Syntax , G. Klyne, J. Carroll (Editors), W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/ . Latest version available at http://www.w3.org/TR/rdf-concepts/.

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

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

[OWL-Semantics]

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

[RDF-Concepts]

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

[RDF-Semantics]

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

[RIF-BLD]

RIF Basic Logic Dialect Harold Boley, Michael Kifer, eds. W3C Editor's Draft, 18 May17 July 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-bld-20080518/http://www.w3.org/2005/rules/wg/draft/ED-rif-bld-20080717/. Latest version available at http://www.w3.org/2005/rules/wg/draft/rif-bld/.

[XML-Schema2] XML Schema Part 2:[RIF-DTB]

RIF Datatypes ,and Built-Ins 1.0 Axel Polleres, Harold Boley, Michael Kifer, eds. W3C Recommendation, 2 May 2001. This version is http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/Editor's Draft, 17 July 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-dtb-20080717/. Latest version available at http://www.w3.org/TR/xmlschema-2/http://www.w3.org/2005/rules/wg/draft/rif-dtb/.

[CURIE]

CURIE Syntax 1.0, M. Birbeck andBirbeck, S. McCarron, W3C Working Draft, 26 November 2007,6 May 2008, http://www.w3.org/TR/2007/WD-curie-20071126/. Latest version available at http://www.w3.org/TR/curie/.

[DLP]

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

[Motik05]

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

[OWL-Reference]

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

[RDF-Schema]

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

[RDF-Syntax]

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

[RFC-3066]

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

[RIF-PRD]

RIF Production Rule Dialect , C.Christian de Sainte Marie, Editor,eds. W3C Editor's Draft.Draft, 17 July 2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-prd-20080717/. Latest version available at http://www.w3.org/2005/rules/wg/wiki/PRdialecthttp://www.w3.org/2005/rules/wg/draft/rif-prd/.

[RIF-UCR]

RIF Use Cases and Requirements , A. Ginsberg, D.Adrian Paschke, David Hirtle, F. McCabe, P.-L.Allen Ginsberg, Paula-Lavinia Patranjan, Editors,Frank McCabe, eds. W3C WorkingEditor's Draft, 1017 July 2006, http://www.w3.org/TR/2006/WD-rif-ucr-20060710/2008, http://www.w3.org/2005/rules/wg/draft/ED-rif-ucr-20080717/. 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/http://www.w3.org/2005/rules/wg/draft/rif-ucr/.

[Rosati06]

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

[SWRL]

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

[Turtle]

Turtle - Terse RDF Triple Language, D. Beckett andBeckett, 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/.

6[XML-Schema2]

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

RIF-RDF combinations can be embedded into RIF 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 OWL DL, the so-called DLP subset [DLP], for which RIF-OWL DL combinations that can be embedded.

Throughout this sectionThe 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. 6.1 Embedding RIF-RDF Combinations 6.1.1 Embedding Symbols Given a combination C=< Rdocuments, respectively.

Besides the namespace prefixes defined in the Overview, S >,the function tr maps RDF symbolsfollowing namespace prefix is used in this appendix: pred refers to the RIF namespace for built-in predicates http://www.w3.org/2007/rif-builtin-predicate# [RIF-DTB].

To facilitate the definition of the embeddings we define the notion of a vocabulary V andmerge of RIF formulas.

Definition. Let R={R₁,...,R_n} be 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 idocument, group, and rule formulas, such that there are no prefix or base directives or relative IRIs in V U Constant with symbol space rif:iri tr( i ) = "i"^^rif:iri Blank node xR and directive₁₁, ..., directive_nm are all the import directives occurring in B Variable symbols ? x tr( x ) = ? x Plain literal without a language tag xxxdocument formulas in V PL Constant withR. The datatype xsd:string tr( "xxx" ) = "xxx"^^xsd:string Plain literal with a language tag ( xxxmerge of R, lang ) in V PL Constant with the datatype rif:text tr( "xxx"@lang ) = "xxx@lang"^^rif:text Well-typed literaldenoted merge ( sR), is defined as Document(directive₁₁ ... directive_nm Group(R*₁ ... R*_n)), u )where R*_i is obtained from R_i in V TL Constant withthe symbol space u tr( "s"^^u ) = "s"^^u Ill-typed literal ( sfollowing way:

• if R_i is a document formula of the form Document(directive_i1 ... directive_im Γ), uthen R*_i=Γ;
• if R_i is a group formula, then R*_i=R_i;
• and if R_i is a RIF-BLD rule, then R*_i=Group(R_i) in V TL Constant s^^u' with symbol space rif:local.   ☐

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