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The Rule Interchange Format (RIF) Basic Logic Dialect (BLD) [RIFBLD] 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 [RDFConcepts] is a Webbased language for the representation and exchange of data; RDF Schema (RDFS) [RDFSchema] and the OWL Web Ontology Language [OWLReference] are Webbased languages for representing and exchanging ontologies (i.e., data models). This document specifies how combinations of RIF BLD documents and RDF data and RDFS and OWL ontologies are interpreted; i.e., it specifies how RIF interoperates with RDF, RDFS, and OWL.
The RIF working group plans to develop further dialects besides BLD, most notably a dialect based on Production Rules [RIFPRD]; these dialects are not necessarily extensions of BLD. Future versions of this document may address compatibility of these dialects with RDF and OWL. In the remainder, RIF is understood to refer to RIF BLD [RIFBLD].
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 [RDFSyntax]. RIF does not provide a format for exchanging RDF graphs; it is assumed that RDF graphs are exchanged using RDF/XML, or any other syntax that can be used for representing or exchanging RDF graphs.
A typical scenario for the use of RIF with RDF/OWL is the exchange of rules that use RDF data and/or RDFS or OWL ontologies: an interchange partner A has a rules language that is RDF/OWLaware, 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 [RDFSyntax]). The rules are translated to the internal rules language of B and are processed, together with the RDF graphs, using the RDF/OWLaware rule engine of B. The use case Vocabulary Mapping for Data Integration [RIFUCR] is an example of the interchange of RIF rules that use RDF data and RDFS ontologies.
A specialization of this scenario is the publication of RIF rules that refer to RDF graphs; publication is a special kind of interchange: one to many, rather than onetoone. 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 [RIFUCR] illustrates the publication scenario.
Another specialization of the exchange scenario is the Interchange of Rule Extensions to OWL [RIFUCR]. The intention of the rule publisher in this scenario is to extend an OWL ontology with rules: interchange partner A has a rules language that extends OWL. A splits its ontology+rules description into a separate OWL ontology and a RIF document, publishes the OWL ontology, and sends (or publishes) the RIF document, which includes a reference to the OWL ontology. A consumer of the rules retrieves the OWL ontology and translates the ontology and document into a combined ontology+rules description in its own rule extension of OWL.
A RIF document that refers to (imports) RDF graphs and/or
RDFS/OWL ontologies, or any use of a RIF document with RDF graphs,
is viewed as a combination of a document and a number of graphs and
ontologies. This document specifies how, in such a combination, the
document and the graphs and ontologies interoperate in a technical
sense, i.e., the conditions under which the combination is
satisfiable (i.e., consistent), as well as the entailments (i.e.,
logical consequences) of the combination. The interaction between
RIF and RDF/OWL is realized by connecting the model theory of RIF
[RIFBLD] with the model
theories of RDF [RDFSemantics] and OWL [OWLSemantics], respectively.
The notation of certain symbols in RIF, particularly IRIs and plain
literals, is slightly different from the notation in RDF/OWL. These
differences are illustrated in the Section Symbols in RIF Versus RDF/OWL.
The RDF semantics specification [RDFSemantics] defines four notions of entailment for RDF graphs. The OWL semantics specification [OWLSemantics] defines two notions of entailment for OWL ontologies, namely OWL Lite/DL and OWL Full. This document specifies the interaction between RIF and RDF/OWL for all six notions. The Section RDF Compatibility is concerned with the combination of RIF and RDF/RDFS. The combination of RIF and OWL is addressed in the Section OWL Compatibility. The semantics of the interaction between RIF and OWL DL is close in spirit to [SWRL].
RIF provides a mechanism for referring to (importing) RDF graphs and a means for specifying the context of this import, which corresponds to the intended entailment regime. The Section Importing RDF and OWL in RIF specifies how such import statements are used for representing RIFRDF and RIFOWL combinations.
The Appendix: Embeddings (Informative) describes how reasoning with combinations of RIF rules with RDF and a subset of OWL DL can be reduced to reasoning with RIF documents, which can be seen as a guide to describing how a RIF processor could be turned into an RDF/OWLaware RIF processor. This reduction can be seen as a guide for interchange partners that do not have RDFaware 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/OWLaware rule system, but B can process RIF rules, then the appendix explains how B's rule system could be used for processing RIFRDF.
Throughout this document the following conventions are used when writing RIF and RDF statements in examples and definitions.
Where RDF/OWL has four kinds of constants: URI references (i.e., IRIs), plain literals without language tags, plain literals with language tags and typed literals (i.e., Unicode sequences with datatype IRIs) [RDFConcepts], RIF has one kind of constants: Unicode sequences with symbol space IRIs [RIFDTB].
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.
RDF Symbol  Example  RIF Symbol  Example 

IRI  <http://www.w3.org/2007/rif>  Constant in the rif:iri symbol space  "http://www.w3.org/2007/rif"^^rif:iri 
Plain literal without language tag  "literal string"  Constant in the xs:string symbol space  "literal string"^^xs:string 
Plain literal with language tag  "literal string"@en  String plus language tag in symbol space rif:text  "literal string@en"^^rif:text 
Typed literal  "1"^^xs:integer  Constant with symbol space  "1"^^xs:integer 
The shortcut syntax for IRIs and strings [RIFDTB], used throughout this document, corresponds with the syntax for IRIs and plain literals in [Turtle], a commonly used syntax for RDF.
RIF does not have a notion corresponding to RDF blank nodes. RIF local symbols, written _symbolname, have some commonality with blank nodes; like the blank node label, the name of a local symbol is not exposed outside of the document. However, in contrast to blank nodes, which are essentially existentially quantified variables, RIF local symbols are constant symbols. In many applications and deployment scenarios, this difference may be inconsequential. However the results will differ when an RDF graph is used in a nonassertional context, such as in a query pattern.
Finally, variables in the bodies of RIF rules or in query patterns may be existentially quantified, and are thus similar to blank nodes; however, RIF BLD does not allow existentially quantified variables to occur in rule heads.
This section specifies how a RIF document interacts with a set of RDF graphs in a RIFRDF combination. In other words, how rules can "access" data in the RDF graphs and how additional conclusions that may be drawn from the RIF rules are reflected in the RDF graphs.
There is a correspondence between statements in RDF graphs and certain kinds of formulas in RIF. Namely, there is a correspondence between RDF triples of the form s p o and RIF frame formulas of the form s'[p' > o'], where s', p', and o' are RIF symbols corresponding to the RDF symbols s, 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 cannot be referenced directly from RIF rules, since blank nodes are local to a specific RDF graph. Variables in RIF rules do, however, range over objects denoted by 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 blank nodes.
Consider a combination of an RDF graph that contains the triple
_:x ex:hasName "John" .
saying that there is something, denoted here by a blank node, which has the name "John", and a RIF document that contains the rules
Forall ?x ?y ( ?x[rdf:type > ex:named] : ?x[ex:hasName > ?y] ) Forall ?x ?y ( <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:named and http://a has a property http://p with value y.
From this combination the following RIF condition formulas can be derived:
Exists ?z ( And( ?z[rdf:type > ex:named] <http://a>[<http://p> > ?z] )) <http://a>[<http://p> > "John"]
as can the following RDF triples:
_:y rdf:type ex:named . <http://a> <http://p> "John" .
However, there is no RIF constant symbol t such that t[rdf:type > ex:named] can be derived, because there is no constant that represents the named individual.
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 document 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.
This section first reviews the definitions of RDF vocabularies and RDF graphs, after which RIFRDF combinations are formally defined. The section concludes with a review of definitions related to datatypes and typed literals.
An RDF vocabulary V consists of the following sets of names:
In addition, there is an infinite set of blank nodes, which is disjoint from the sets of names. See RDF Concepts and Abstract Syntax [RDFConcepts] for precise definitions of these concepts.
Definition. Given an RDF vocabulary V, a generalized RDF triple of V is a statement of the form s p o, where s, p and o are names in V or blank nodes. ☐
Definition. Given an RDF vocabulary V, a generalized RDF graph is a set of generalized RDF triples of V. ☐
(See the End note on generalized RDF graphs)
A RIFRDF combination consists of a RIF document and zero or more RDF graphs. Formally:
Definition. A RIFRDF 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, RIFRDF combinations are referred to simply as combinations.
Even though RDF allows the use of arbitrary datatype IRIs in typed literals, not all such datatype IRIs are recognized in the semantics. In fact, simple entailment does not recognize any datatype and RDF and RDFS entailment recognize only the datatype rdf:XMLLiteral. To facilitate discussing datatypes, and specifically datatypes supported in specific contexts (required for Dentailment), the notion of datatype maps [RDFSemantics] is used.
A datatype map is a partial mapping from IRIs to datatypes.
RDFS, specifically Dentailment, allows the use of arbitrary datatype maps, as long as rdf:XMLLiteral is in the domain of the map. RIF BLD requires a number of additional datatypes to be included; these are the RIFrequired datatypes [RIFDTB].
When checking consistency of a combination < R,S> or entailment of a graph S or RIF formula φ by a combination < R,S>, the set of considered datatypes is the union of the set of RIFrequired datatypesand the sets of datatypes used in R, the documents imported into R, and φ (when considering entailment of φ).
Definition. Let T be a set of datatypes. A datatype map D is conforming with T if it satisfies the following conditions:
Note that it follows from the definition that every datatype used in the RIF document in the combination or the entailed RIF formula (when considering entailment questions) is included in any datatype map conforming to the set of considered datatypes. There may be datatypes used in an RDF graph in the combination that are not included in such a datatype map.
Definition. Given a datatype map D, a typed literal (s, d) is a welltyped literal if
The semantics of RIFRDF 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 [RDFSemantics] 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.
This section defines the notion of commonrifrdfinterpretation, which is an interpretation of a RIFRDF combination. This commonrifrdfinterpretation 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. the Section Symbols in RIF Versus RDF/OWL).
The notions of RDF interpretation and RIF semantic structure (interpretation) are briefly reviewed below.
As defined in [RDFSemantics], a simple interpretation of a vocabulary V is a tuple I=< IR, IP, IEXT, IS, IL, LV >, where
Rdf, rdfs, and Dinterpretations are simple interpretations that satisfy certain conditions:
As defined in [RIFBLD], 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 RIFRDF 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.
For the purpose of the interpretation of imported documents, RIF BLD defines the notion of semantic multistructures, which are nonempty sets {I_{1}, ..., I_{n}} of semantic structures that are identical in all respects with the exception of the interpretation of local constants.
Given a semantic multistructure I={I_{1}, ..., I_{n}}, we use the symbol I to denote both the multistructure and the common part of the individual structures I_{1}, ..., I_{n}.
Definition. A commonrifrdfinterpretation is a pair (I, I), where I is a semantic multistructure and I is an RDF interpretation of a vocabulary V, such that the following conditions hold:
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 Dinterpretation, IP is a subset of IR, and thus IR=D_{ind}. Condition 2 ensures that the set of RDF properties at least includes all elements that are used as properties in frames in the RIF domain. Condition 3 ensures that all concrete values in D_{ind} are included in LV (by definition, the value spaces of all considered datatypes are included in D_{ind}). Condition 4 ensures that RDF triples are interpreted in the same way as frame formulas. Condition 5 ensures that IRIs are interpreted in the same way. Condition 6 ensures that typed literals are interpreted in the same way. Note that no correspondences are defined for the mapping of names in RDF that are not symbols of RIF, e.g., illtyped 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 RIFRDF combinations, even if the RIF document is empty. Similarly for plain literals without language tags of the form "mystring" and typed literals of the form "mystring"^^xs:string. For example, consider the combination of an empty document and an RDF graph that contains the triples
<http://a> <http://p> "http://b"^^rif:iri . <http://a> <http://p> "abc" .
This combination allows the derivation of, among other things, the following triples:
<http://a> <http://p> <http://b> . <http://a> <http://p> "abc"^^xs:string .
as well as the following frame formulas:
<http://a>[<http://p> > <http://b>] <http://a>[<http://p> > "abc"]
The notion of satisfiability refers to the conditions under which a commonrifrdfinterpretation (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 commonrifrdfinterpretation (I, I) satisfies a combination < R, S> if I is a model of R and I satisfies S. Formally:
Definition. A commonrifrdfinterpretation (I, I) satisfies a RIFRDF 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 simplemodel, or model, of C, and C is satisfiable. (I, I) satisfies a generalized RDF graph S if I satisfies S. (I, I) satisfies a condition formula φ if TVal_{I}(φ)=t. ☐
Notice that not every combination is satisfiable. In fact, not every RIF document has a model. For example, the document consisting of the fact
"a"="b"
does not have a model, since the symbols "a" and "b" are mapped to the (distinct) character strings "a" and "b", respectively, in every semantic structure.
Rdf, rdfs, and Dsatisfiability are defined through additional restrictions on I:
Definition. A model (I, I) of a combination C is an rdfmodel of C if I is an rdfinterpretation; in this case C is rdfsatisfiable. ☐
Definition. A model (I, I) of a combination C is an rdfsmodel of C if I is an rdfsinterpretation; in this case C is rdfssatisfiable. ☐
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 Dmodel of C if I is a Dinterpretation; in this case C is Dsatisfiable. ☐
Using the notions of models defined above, entailment is defined in the usual way, i.e., through inclusion of sets of models.
Definition. Let C be a RIFRDF combination, let S be a generalized RDF graph, let φ be a condition formula, and let D be a datatype map conforming with the set of considered datatypes. C Dentails S if every Dmodel of C satisfies S. Likewise, C Dentails φ if every Dmodel of C satisfies φ. ☐
The other notions of entailment are defined analogously:
Definition. A combination C simpleentails S (resp., φ) if every simple model of C satisfies S (resp., φ). ☐
Definition. A combination C rdfentails S (resp., φ) if every rdfmodel of C satisfies S (resp., φ). ☐
Definition. A combination C rdfsentails S (resp., φ) if every rdfsmodel of C satisfies S (resp., φ). ☐
The syntax for exchanging OWL ontologies is based on RDF graphs. Therefore, a RIFOWLcombination consists of a RIF document and a set of RDF graphs, analogous to a RIFRDF combination. This section specifies how RIF documents and OWL ontologies interoperate in such combinations.
OWL [OWLReference] 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 [OWLSemantics]. 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 [OWLSemantics]. While OWL DL has an abstract syntax with a direct modeltheoretic 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 RIFOWL 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 RIFOWL Full compatibility is a straightforward extension of
RIFRDF compatibility. Defining RIFOWL DL compatibility in the
same way would entail losing certain semantic properties of OWL DL.
One of the main reasons for this is the difference in the way
classes and properties are interpreted in OWL Full and OWL DL. In
the Full species, classes and properties are both interpreted as
objects in the domain of interpretation, which are then associated
with subsets of and binary relations over the domain of
interpretation using rdf:type and the extension function
IEXT, as in RDF. In the DL species, classes and properties are
directly interpreted as subsets of and binary relations over the
domain. This is a key property of Description Logic semantics and
enables the use of Description Logic reasoning techniques for
processing OWL DL descriptions. Defining RIFOWL DL compatibility
as an extension of RIFRDF compatibility would define a
correspondence between OWL DL statements and RIF frame formulas.
Since RIF frame formulas are interpreted using an extension
function, the same way as in RDF, defining the correspondence
between them and OWL DL statements would change the semantics of
OWL statements, even if the RIF document were empty.
A RIFOWL combination that is faithful to the OWL DL semantics requires interpreting classes and properties as sets and binary relations, respectively, suggesting that a correspondence could be defined with unary and binary predicates. It is, however, also desirable that there be uniform syntax for the RIF component of both OWL DL and RDF/OWL Full combinations, because one may not know at time of writing the rules which type of inference will be used. Consider, for example, an RDF graph S with the following statement
a rdf:type C .
and a RIF document with the rule
Forall ?x (?x[rdf:type > D] : ?x[rdf:type > C])
The combination of the two, according to the specification of RDF Compatibility, allows deriving
a rdf:type D .
Now, the RDF graph S is also an OWL DL ontology. Therefore, one would expect the triple to be derived by RIFOWL DL combinations as well.
To ensure that the RIFOWL 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 RIFOWL DL combinations is slightly
different from their interpretation in RIF BLD and syntactical
restrictions are imposed on the use of variables and function terms
in frame formulas.
Note that the abstract syntax form of OWL DL allows socalled
punning (this is not allowed in the RDF syntax), i.e., the
same IRI may be used in an individual position, a property
position, and a class position; the interpretation of the IRI
depends on its context. Since combinations of RIF and OWL DL are
based on the abstract syntax of OWL DL, punning may also be used in
these combinations.
In this document, we are using OWL to refer to OWL as specified in [OWLSemantics], as opposed to OWL 2, which is currently in the process of being defined by the OWL working group. (See the End note on OWL 2)
Since RDF graphs and OWL Full ontologies cannot be distinguished, the syntax of RIFOWL Full combinations is the same as the syntax of RIFRDF combinations.
The syntax of OWL ontologies in RIFOWL DL combinations is specified by the abstract syntax of OWL DL. Certain restrictions are imposed on the syntax of the RIF rules in combinations with OWL DL. Specifically, the only terms allowed in class and property positions in frame formulas are constant symbols. A DLframe formula is a frame formula a[b_{1} > c_{1} ... b_{n} > c_{n}] such that n≥1 and for every b_{i}, with 1≤i≤n, it holds that b_{i} is a constant and if b_{i} = rdf:type, then c_{i} is a constant.
Definition. A condition formula φ is a DLcondition if every frame formula in φ is a DLframe formula. ☐
Definition. A RIFBLD document formula R is a RIFBLD DLdocument formula if every frame formula in R is a DLframe formula. ☐
Definition. A RIFOWLDLcombination is a pair < R,O>, where R is a RIFBLD DLdocument formula and O is a set of OWL DL ontologies in abstract syntax form of an OWL vocabulary V. ☐
When clear from the context, RIFOWLDLcombinations are referred to simply as combinations.
In the literature, several restrictions on the use of variables in combinations of rules and Description Logics have been identified [Motik05, Rosati06] for the purpose of effective reasoning. This section specifies such safeness restrictions for RIFOWLDL combinations.
Given a set of OWL DL ontologies in abstract syntax form O, a variable ?x in a RIF rule Q H : B is DLsafe if it occurs in an atomic formula in B that is not of the form s[P > o] or s[rdf:type > A], where P or A occurs in one of the ontologies in O. A RIF rule Q H : B is DLsafe, given O if every variable that occurs in H : B is DLsafe. A RIF rule Q H : B is weakly DLsafe, given O if every variable that occurs in H is DLsafe.
Definition. A RIFOWLDLcombination < R,O> is DLsafe if every rule in R is DLsafe, given O. A RIFOWLDLcombination < R,O> is weakly DLsafe if every rule in R is weakly DLsafe, given O. ☐
Feature At Risk #1: Safeness
Note: This feature is "at risk" and may be removed from this specification based on feedback. Please send feedback to publicrifcomments@w3.org.
The above definition of DLsafeness is intended to identify a fragment of RIFOWL DL combinations for which implementation is easier than full RIFOWL DL. This definition should be considered AT RISK and may become stricter based on implementation experience.
The semantics of RIFOWL Full combinations is a straightforward extension of the Semantics of RIFRDF Combinations.
The semantics of RIFOWLDLcombinations 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 RIFOWLFull and RIFOWLDLcombinations, the semantics of RIF frame formulas is slightly altered in RIFOWLDLcombinations.
A Dinterpretation 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 RIFOWL Full combinations is a straightforward extension of the semantics of RIFRDF combinations. It is based on the same notion of commoninterpretations, but defines additional notions of satisfiability and entailment.
Definition. Let (I, I) be a commonrifrdfinterpretation that is a model of a RIFRDF combination C=< R, S > and let D be a datatype map conforming with the set of datatypes in I. (I, I) is an owlfullmodel of C if I is an OWL Full interpretation with respect to D; in this case C is OWLFullsatisfiable with respect to D. ☐
Definition. Let C be a RIFRDF combination, let S be a generalized RDF graph, let φ be a condition formula, and let D be a datatype map conforming with the set of considered datatypes. C owlfullentails S with respect to D if every owlfullmodel of C satisfies S. Likewise, C owlfullentails φ with respect to D if every owlfullmodel of C satisfies φ. ☐
The semantics of RIFOWLDLcombinations is similar in spirit to the semantics of RIFRDF combinations. Analogous to a commonrifrdfinterpretation, there is the notion of commonrifowldlinterpretations, 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 RIFRDF combinations, the conditions below define a correspondence between the interpretation of OWL DL classes and properties and RIF unary and binary predicates.
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 [RIFBLD] otherwise unchanged.
Namely, frame formulas of the form s[rdf:type > o] are interpreted as membership of s in the set denoted by o and frame formulas of the form s[p > o], where p is not rdf:type, as membership of the pair (s, o) in the binary relation denoted by p.
Definition. A dlsemantic
structure is a tuple I =
<TV, DTS, D,
D_{ind}, D_{func},
I_{C}, I_{V},
I_{F}, I_{frame'},
I_{SF}, I_{sub},
I_{isa}, I_{=},
I_{external},
I_{truth}>, where
I_{frame'} is a mapping from
D_{ind} to total functions of the form
SetOfFiniteBags(D × D) →
D, such that for each pair (a, b) in
SetOfFiniteBags(D × D) it
holds that if
a≠I_{C}(rdf:type), then
b in D_{ind}; all other elements of
the structure are defined as in RIF semantic
structures.
A dlsemantic multistructure is a nonempty set of dlsemantic structures {I_{1}, ..., I_{n}} that are identical in all respects except that the mappings I_{1}_{C}, ..., I_{n}_{C} might differ on the constants in Const that belong to the rif:local symbol space. ☐
Given a dlsemantic multistructure I={I_{1}, ..., I_{n}}, we use the symbol I to denote both the multistructure and the common part of the individual structures I_{1}, ..., I_{n}.
We define I(o[a_{1}>v_{1} ... a_{k}>v_{k}]) = I_{frame'}(I(o))({<I(a_{1}),I(v_{1})>, ..., <I(a_{n}),I(v_{n})>}). The truth valuation function TVal_{I} is then defined as in RIF BLD.
Definition. A dlsemantic multistructure I is a model of a RIFBLD DLdocument formula R if TVal_{I}(R)=t. ☐
As defined in [OWLSemantics], 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. Given a datatype map D, a commonrifowldlinterpretation with respect to D is a pair (I, I), where I is a dlsemantic multistructure and I is an abstract OWL interpretation with respect to D of a vocabulary V, such that the following conditions hold
Condition 2 ensures that the relevant parts of the domains of interpretation are the same. Condition 3 ensures that the interpretation (extension) of an OWL DL class u corresponds to the interpretation of frames of the form ?x[rdf:type > <u>]. Condition 4 ensures that the interpretation (extension) of an OWL DL object or datatype property u corresponds to the interpretation of frames of the form ?x[<u> > ?y]. Condition 5 ensures that typed literals of the form (s, d) in OWL DL are interpreted in the same way as constants of the form "s"^^d in RIF. Finally, condition 6 ensures that individual identifiers in the OWL ontologies and the RIF documents are interpreted in the same way.
Using the definition of commonrifowldlinterpretation,
satisfaction, models, and entailment are defined in the usual
way:
Definition. A commonrifowldlinterpretation (I, I) with respect to a datatype map D is an owldlmodel of a RIFOWLDLcombination 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 owldlsatisfiable with respect to D. (I, I) is an owldlmodel of an OWL DL ontology in abstract syntax form O if I satisfies O. (I, I) is an owldlmodel of a DLcondition formula φ if TVal_{I}(φ)=t. ☐
Definition. Let C be a RIFOWLDLcombination, let O be an OWL DL ontology in abstract syntax form, let φ be a DLcondition formula, and let D be a datatype map conforming with the set of considered datatypes. C owldlentails O with respect to D if every commonrifowldlinterpretation with respect to D that is an owldlmodel of C is an owldlmodel of O. Likewise, C owldlentails φ with respect to D if every commonrifowldlinterpretation with respect to D that is an owldlmodel of C is an owldlmodel of φ. ☐
Recall that in an abstract OWL interpretation I the sets O and LV,
which are used for interpreting, respectively, literals (data
values), are disjoint and that EC maps class identifiers to subsets
of O and datatype identifiers to subsets of LV. The disjointness
entails that data values cannot be members of a class and
individuals cannot be members of a datatype.
In RIF, variable quantification ranges over D_{ind}. So, the same variable may be assigned to an abstract individual or a concrete data value. Additionally, RIF constants (e.g., IRIs) denoting individuals can be written in place of a data value, such as the value of a datavalued property or in datatype membership statements; similarly for constants denoting data values. Such statements cannot be satisfied in any commonrifowldlinterpretation, due to the constraints on the EC and ER functions. The following example illustrates several such statements.
Consider the datatype xs:string and a RIFOWL DL combination consisting of the set containing only the OWL DL ontology
ex:myiri rdf:type ex:A .
and a RIF document containing the following fact
ex:myiri[rdf:type > xs:string]
This combination is not owldlsatisfiable, because ex:myiri is an individual identifier and S maps individual identifiers to elements in O, which is disjoint from the elements in the datatype xs:string.
Consider a RIFOWL DL combination consisting of the set containing only the OWL DL ontology
ex:hasChild rdf:type owl:ObjectProperty .
and a RIF document containing the following fact
ex:myiri[ex:hasChild > "John"]
This combination is not owldlsatisfiable, because ex:hasChild is an object property, and values of object properties may not be concrete data values.
Consider a RIFOWL DL combination consisting of the OWL DL ontology
SubClassof(ex:A ex:B)
and a RIF document containing the following rule
Forall ?x (?x[rdf:type > ex:A])
This combination is not owldlsatisfiable, because the rule requires every element, including every concrete data value, to be a member of the class ex:A. However, the mapping EC in any abstract OWL interpretation requires every member of ex:A to be an element of O, and concrete data values cannot be members of O.
Note that the above definition of RIFOWL DL compatibility does not consider ontology and annotation properties, in contrast to the definition of compatibility of RIF with OWL Full, where there is no clear distinction between annotation and ontology properties and other kinds of properties. Therefore, it is not possible to "access" or use the values of these properties in the RIF rules. This limitation is overcome in the following definition. It is envisioned that the user will choose whether annotation and ontology properties are to be considered. It might be the case that OWL 2, the successor of OWL currently under development, will not define the semantics for annotation and ontology properties in the same way as OWL; therefore, the below definition may not extend to cover OWL 2.
Definition. Given a datatype map D, a commonrifowldlinterpretation (I, I) is a commondlannotationinterpretation with respect to D if the following condition holds
7. ER(p) = set of all pairs (k, l) in O × O such that I_{truth}(I_{frame'}(k)({(I_{C}(<p>), l)}) ) = t (true), for every IRI p in V. ☐
Condition 7, which strengthens condition 3, ensures that the
interpretation of all properties (also annotation and ontology
properties) in the OWL DL ontologies corresponds with their
interpretation in the RIF rules.
Definition. A commondlannotationinterpretation with respect to a datatype map D (I, I) is an owldlannotationmodel of a RIFOWLDLcombination 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 owldlannotationsatisfiable. ☐
Definition. Let C be a RIFOWLDLcombination, let O be an OWL DL ontology in abstract syntax form, let φ be a DLcondition formula, and let D be a datatype map conforming with the set of considered datatypes. C owldlannotationentails O with respect to D if every commonrifowldlinterpretation with respect to D that is an owldlannotationmodel of C is an owldlmodel of O. Likewise, C owldlannotationentails φ with respect to D if every commonrifowldlinterpretation with respect to D that is an owldlannotationmodel of C is an owldlmodel of φ. ☐
The difference between the two kinds of OWL DL entailment can be illustrated using an example. Consider the following OWL DL ontology in abstract syntax form
Ontology (ex:myOntology Annotation(dc:title "Example ontology"))
which defines an ontology with a single annotation (title). Consider also a document consisting of the following rule:
Forall ?x ?y ( ?x[ex:hasTitle > ?y] : ?x[dc:title > ?y])
which says that whenever something has a dc:title, it has the same ex:hasTitle.
The combination of the ontology and the document owldlannotationentails the RIF condition formula ex:myOntology[ex:hasTitle > "Example ontology"]; the combination does not owldlentail the formula.
In the previous sections, RIFRDF Combinations and RIFOWL combinations were defined in an abstract way, as pairs consisting of a RIF document and a set of RDF graphs/OWL ontologies. In addition, different semantics were specified based on the various RDF and OWL entailment regimes. RIF provides a mechanism for explicitly referring to (importing) RDF graphs from documents and specifying the intended profile (entailment regime) through the use of Import statements.
This section specifies how RIF documents with such import statements are interpreted.
A RIF
document contains a number of Import statements. Unary
Import statements are used for importing RIF documents,
and the interpretation of these statements is defined in [RIFBLD]. This section defines the
interpretation of twoary Import statements:
Import(t1 p1) ... Import(tn pn)
Here, ti is an IRI constant of the form <absoluteIRI>, where absoluteIRI is the location of an RDF graph to be imported, and pi is an IRI constant denoting the profile to be used.
The profile determines which notions of model, satisfiability and entailment must be used. For example, if a RIF document R imports an RDF graph S with the profile RDFS, the notions of rdfsmodel, rdfssatisfiability, and rdfsentailment must be used with the combination <R, {S}>.
Profiles are ordered as defined later in this section. If several graphs are imported in a document, and these imports specify different profiles, the highest of these profiles is used. For example, if a RIF document R imports an RDF graph S_{1} with the profile RDF and an RDF graph S_{2} with the profile OWL Full, the notions of owlfullmodel, owlfullsatisfiability, and owlfullentailment must be used with the combination <R, {S_{1}, S_{2}}>.
Finally, if a RIF document R imports an RDF graph S with the profile OWL DL, R must be a RIFBLD DLdocument formula, S must be the translation to RDF of an OWL DL ontology in abstract syntax form O, and the notions of owldlmodel, owldlsatisfiability, and owldlentailment must be used with the combination <R, {O}>.
RIF defines a specific profile for each of the notions of satisfiability and entailment of combinations, as well as one generic profile. The use of a specific profile specifies how a combination should be interpreted. If a specific profile cannot be handled by a receiver, the combination should be rejected. The use of a generic profile implies that a receiver may interpret the combination to the best of its ability.
The use of profiles is not restricted to the profiles specified in this document. Any specific profile that is used with RIF must specify an IRI that identifies it and associated notions of model, satisfiability, and entailment for combinations.
The following table lists the specific profiles defined by RIF, the IRIs of these profiles, and the notions of model, satisfiability, and entailment that must be used with the profile.
Profile  IRI of the Profile  Model  Satisfiability  Entailment 

simple  <http://www.w3.org/2007/rifimportprofile#Simple>  simplemodel  satisfiability  simpleentailment 
rdf  <http://www.w3.org/2007/rifimportprofile#RDF>  rdfmodel  rdfsatisfiability  rdfentailment 
rdfs  <http://www.w3.org/2007/rifimportprofile#RDFS>  rdfsmodel  rdfssatisfiability  rdfsentailment 
D  <http://www.w3.org/2007/rifimportprofile#D>  dmodel  dsatisfiability  dentailment 
OWL DL  <http://www.w3.org/2007/rifimportprofile#OWLDL>  owldlmodel  owldlsatisfiability  owldlentailment 
OWL DL annotation  <http://www.w3.org/2007/rifimportprofile#OWLDLannotation>  owldlannotationmodel  owldlannotationsatisfiability  owldlannotationentailment 
OWL Full  <http://www.w3.org/2007/rifimportprofile#OWLFull>  owlfullmodel  owlfullsatisfiability  owlfullentailment 
Profiles that are defined for combinations of DLdocument formulas 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
RIF specifies one generic profile. The use of the generic profile does not imply the use of a specific notion of model, satisfiability, and entailment.
Profile  IRI of the Profile 

Generic  <http://www.w3.org/2007/rifimportprofile#Generic> 
Let R be a RIF document such that
Import(<u1> <p1>) ... Import(<un> <pn>)
are the twoary import statements in R and all imported documents and let Profile be the set of profiles corresponding to the IRIs <p1>,...,<pn>.
If Profile contains only specific profiles, then:
If Profile contains a generic profile, then the combination C=<R,{S_{1},....,S_{n}}>, where S_{1},....,S_{n} are RDF graphs accessible from the locations u_{1},...,u_{n}, may be interpreted according to the highest among the specific profiles in Profile.
This document is the product of the Rules Interchange Format (RIF) Working Group (see below), the members of which deserve recognition for their time and commitment to RIF. Special thanks to Mike Dean, Michael Kifer, Stella Mitchell, Axel Polleres, and Dave Reynolds, for their thorough reviews and insightful discussions; the working group chairs, Chris Welty and Christian de SainteMarie, for their selfless and inspirational leadership through the long and difficult trials leading to this draft; and W3C team contact Sandro Hawke, a constant source of ideas, help, and feedback.
The following members of the joint RIFOWL task force have contributed to the OWL Compatibility section in this document: Mike Dean, Peter F. PatelSchneider, 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 AïtKaci (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).
RIFRDF combinations can be embedded into RIF documents in a fairly straightforward way, thereby demonstrating how a RIFcompliant translator without native support for RDF can process RIFRDF combinations. RIFOWL combinations cannot be embedded in RIF, in the general case. However, there is a subset of OWL DL, the socalled DLP subset [DLP], for which RIFOWL DL combinations that can be embedded.
The embeddings are defined using the embedding function tr, which maps symbols, triples, RDF graphs, and OWL DL ontologies in abstract syntax form to RIF symbols, statements, and documents, respectively.
Besides the namespace prefixes defined in the Overview, the
following namespace prefix is used in this appendix: pred
refers to the RIF namespace for builtin predicates
http://www.w3.org/2007/rifbuiltinpredicate# [RIFDTB].
To facilitate the definition of the embeddings we define the notion
of a merge of RIF formulas.
Definition. Let R={R_{1},...,R_{n}} be a set of document, group, and rule formulas, such that there are no prefix or base directives or relative IRIs in R and directive_{11}, ..., directive_{nm} are all the import directives occurring in document formulas in R. The merge of R, denoted merge (R), is defined as Document(directive_{11} ... directive_{nm} Group(R*_{1} ... R*_{n})), where R*_{i} is obtained from R_{i} in the following way:
Note that the requirement that no prefix or based directives or relative IRIs are included in any of the formulas to be merged is not a real limitation, since compact IRIs can be rewritten to absolutes IRIs, as can relative IRIs by exploiting a base directive or the location of the document.
Editor's Note: We note here is that the embeddings in this appendix use equality, which is a feature of RIF BLD that is at risk. However, equality is not a crucial feature for the embeddings; removing equality from embedded combinations is fairly straightforward.
RIFRDF combinations are embedded through embeddings of graphs and axiomatization of simple, RDF, and RDFS entailment.
The embedding is not defined for combinations that include infinite RDF graphs and for combinations that include RDF graphs with RDF URI references that are not absolute IRIs (see the End note on RDF URI references) or plain literals that are not in the lexical space of the xs:string datatype [XMLSchema2]. In addition, for the embedding of RDFS entailment, each datatype must have an associated guard predicate.
In the remainder of this section we first define the embedding of symbols, triples, and graphs, after which we define the axiomatization of simple, RDF, and RDFS entailment of combinations and, finally, demonstrate faithfulness of the embedding.
Given a combination C=< R,S>, the function tr maps RDF symbols of a vocabulary V and a set of blank nodes B to RIF symbols, as defined in following table.
In the table, the mapping tr' is an injective function that maps typed literals to new constants in the rif:local symbol space, where a new constant is a constant that is not used in the documents or its vicinity (e.g., entailed formula or entailing combination). It "generates" a new constant from a typed literal.
RDF Symbol  RIF Symbol  Mapping 

IRI i in V_{U}  Constant with symbol space rif:iri  tr(i) = <i> 
Blank node _:x in B  Variable symbol ?x  tr(_:x) = ?x 
Plain literal without a language tag xxx in V_{PL}  Constant with the datatype xs:string  tr("xxx") = "xxx" 
Plain literal with a language tag "xxx"@lang in V_{PL}  Constant with the datatype rif:text  tr("xxx"@lang) = "xxx@lang"^^rif:text 
Welltyped literal "s"^^u in V_{TL}  Constant with the symbol space u  tr("s"^^u) = "s"^^u 
Nonwelltyped literal "s"^^u in V_{TL}  Local constant su' that is not used in C and is obtained from "s"^^u  tr("s"^^u) = tr'("s"^^u) 
This section extends the mapping function tr to triples as RIF statements and defines two embedding functions for RDF graphs. In the one embedding (tr_{R}) graphs are embedded as RIF documents and variables are skolemized, i.e., replaced with new constant symbols. In the other (tr_{Q}) graphs are embedded as condition formulas and variables are existentially quantified. The following sections show how these embeddings can be used for reasoning with combinations.
For skolemization we assume a function sk that takes as an argument a formula φ with variables and returns a formula φ', which is obtained from an RIF document R by, for every variable symbol ?x, replacing ?x with <newiri>, where newiri is a new globally unique IRI, i.e., it does not occur in the graph or its vicinity (e.g., entailing combination or entailed graph/formula).
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  Document tr_{R}(S)  tr_{R}(S) = sk(Document (Group (tr(t_{1}) ... tr(t_{1})) )), where t_{1}, ..., t_{m} are the triples in S 
Graph S  Condition (query) tr_{Q}(S)  tr_{Q}(S) = Exists tr(x_{1}) ... tr(x_{n}) (And(tr(t_{1}) ... tr(t_{m}))), where x_{1}, ..., x_{n} are the blank nodes occurring in S and t_{1}, ..., t_{m} are the triples in S 
The semantics of the RDF vocabulary does not need to be axiomatized for simple entailment. Nonetheless, the connection between RIF class membership and subclass statements and the RDF type and subclass statements needs axiomatization. We define:
R^{simple}  =  Document(Group(Forall ?x ?y (?x[rdf:type
> ?y] : ?x # ?y)Forall ?x ?y (?x # ?y : ?x[rdf:type > ?y]) 
The following theorem shows how checking simpleentailment of combinations can be reduced to checking entailment of RIF conditions by using the embeddings of RDF graphs defined above.
Theorem A RIFRDF combination C=<R,{S_{1},...,S_{n}}> simpleentails a generalized RDF graph T if and only if merge({R, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}) entails tr_{Q}(T). C simpleentails a condition formula φ if and only if merge({R, R^{simple}, tr_{R}(S_{1}), ..., tr_{R}(S_{n}}) entails φ.
Proof. We prove both directions by contradiction: if the entailment does not hold on one side, we show that it also does not hold on the other. We first consider condition formulas (the second part of the theorem), after which we consider graphs (the first part of the theorem).
In the proof we abbreviate merge({R, R^{simple}, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}) with R'.
(=>) Assume R' does not entail φ. This means there is some semantic multistructure I that is a model of R', but not of φ. Consider the pair (I, I), where I is the interpretation defined as follows:Clearly, (I, I) is a commonrifrdfinterpretation: conditions 16 in the definition are satisfied by construction of I and conditions 7 and 8 are satisfied by condition 4 and by the fact that I is a model of R^{simple}.
 IR=D_{ind},
 IP is the set of all k in D_{ind} such that there exist some a, b in D_{ind} and I_{truth}(I_{frame}(k)(a,b))=t,
 LV=(union of the value spaces of all considered datatypes),
 IEXT(k) = the set of all pairs (a, b), with a, b, and k in D_{ind}, such that I_{truth}(I_{frame}(k)(a,b))=t,
 IS(i) = I_{C}(<i>) for every absolute IRI i in V_{U}, and
 IL((s, d)) = I_{C}(tr("s"^^d)) for every typed literal (s, d) in V_{TL}.
Consider a graph S_{i} in {S_{1},...,S_{n}}. Let x_{1},..., x_{m} be the blank nodes in S_{i} and let u_{1},..., u_{m} be the new IRIs that were obtained from the variables ?x_{1},..., ?x_{m} through the skolemization in tr_{R}(S_{i}), i.e., u_{i}=sk(?x_{i}). Now, let A be a mapping from blank nodes to elements in D_{ind} such that A(x_{j})=I_{C}(u_{j}) for every blank node x_{j} in S_{i}. From the fact that I is a model of tr_{R}(S_{i}) and by construction of I it follows that [I+A] satisfies S_{i} (see Section 1.5 of [RDFSemantics])), and so I satisfies S_{i}.
We have that I is a model of R, by assumption. So, (I, I) satisfies C. Again, by assumption, I is not a model of φ. Therefore, C does not entail φ.
Assume now that R' does not entail tr_{Q}(T), which means there is a semantic multistructure I that is a model of R', but not of tr_{Q}(T). The commonrifrdfinterpretation (I, I) is obtained in the same way as above, and so clearly satisfies C.
We proceed by contradiction. Assume I satisfies T. This means there is some mapping A from the blank nodes x_{1},...,x_{m} in T to objects in D_{ind} such that [I+A] satisfies T.
Consider now the semantic multistructure I*, which is the same as I, with the exception of the mapping I*_{V} on the variables ?x_{1},...,?x_{m}, which is defined as follows: I*_{V}(?x_{j})=A(x_{j}) for each blank node x_{j} in S. By construction of I and since [I+A] satisfies T we can conclude that I* is a model of And(tr(t_{1})... tr(t_{m})), and so I is a model of tr_{Q}(T), violating the assumption that it is not. Therefore, (I, I) does not satisfy T and C does not entail T.
(<=) Assume C does not entail φ. This means there is some commonrifrdfinterpretation (I, I) that satisfies C such that I is not a model of φ.
Consider the semantic multistructure I', which is exactly the same I, except for the mapping I'_{C} on new IRIs that were introduced in skolemization. The mapping of these new IRIs is defined as follows:
For each graph S_{i} in {S_{1},...,S_{n}}, let x_{1},..., x_{m} be the blank nodes in S_{i} and let u_{1},..., u_{m} be the new IRIs that were obtained from the variables ?x_{1},..., ?x_{m} through the skolemization in tr_{R}(S_{i}). Now, since I satisfies S_{i}, there must be a mapping A from blank nodes to elements in D_{ind} such that [I+A] satisfies S_{i}. We define I'_{C}(u_{j})=A(x_{j}) for every blank node x_{j} in S_{i}.
By assumption, I' is a model of R (recall that I' differs from I only on the new IRIs, which are not in R). Clearly, I' is also a model of R^{simple}, by conditions 7, 8, and 4 in the definition of commonrifrdfinterpretation.
From the fact that I satisfies S_{i} and by construction of I' it follows that I' is a model of tr_{R}(S_{i}). So, I' is a model of R'. Since I is not a model of φ and φ does not contain any of the new IRIs, I' is not the model of φ.
Therefore, R' does not entail φ.
Assume now that C does not entail T, which means there is a commonrifowldlinterpretation (I, I) that satisfies C, but I does not satisfy T. We obtain I' from I in the same way as above, and so clearly satisfies R'. It can be shown analogous to the (=>) direction that if I' is a model of tr_{Q}(T), then there is a blank node mapping A such that [I+A] satisfies T, and thus I satisfies S, violating the assumption that it does not. Therefore, I' is not a model of tr_{Q}(T) and thus R' does not entail tr_{Q}(T). ☐
Theorem A RIFRDF combination <R,{S_{1},...,S_{n}}> is satisfiable iff there is a semantic multistructure I that is a model of merge({R, R^{simple}, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}).
Proof. The theorem follows immediately from the previous theorem and the observation that a combination (respectively, RIF document) is satisfiable (respectively, has a model) if and only if it does not entail the condition formula "a"="b". ☐
We axiomatize the semantics of the RDF vocabulary using the following RIF rules. We assume that the predicate symbol ex:illxml is not used in any RIF document.
To finitely embed RDF entailment, we need to consider a subset of the RDF axiomatic triples. Given a combination C, the context of C includes C and all graphs/formulas considered for entailment checking. The set of RDF finiteaxiomatic triples is the smallest set such that:
Let T be the set of considered datatypes. We assume that each datatype in T has an associated capitalized short name Datatype (e.g., the short name of xs:string is String) and a guard pred:isDatatype, which can be used to test whether a particular object is a value of the datatype; see [RIFDTB] for definitions of guards for the RIFrequired datatypes.
R^{RDF}  =  merge ({R^{simple}} union ({tr(s p o .)} for every RDF finiteaxiomatic triple
s p o .) union Forall ?x (?x[rdf:type > rdf:Property] : Exists ?y ?z (?y[?x > ?z])), } union {Forall ?x ("a"="b" : And(ex:illxml(?x) Or(pred:isDT1(?x) ... pred:isDTn(?x)))), where DT1,...,DTn are the capitalized short names of the datatypes in T}) 
Here, inconsistencies may occur if nonwelltyped 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 RIFRDF combination C=<R,{S_{1},...,S_{n}}> rdfentails a generalized RDF graph T iff merge({R^{RDF}, R, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}) entails tr_{Q}(T). C rdfentails a condition formula φ iff merge({R^{RDF}, R, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}) entails φ.
Proof. In the proof we abbreviate merge({R^{RDF}, R, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}) with R'.
The proof is then obtained from the proof of correspondence for simple entailment in the previous section with the following modifications: (*) in the (=>) direction we additionally need to extend I to ensure it satisfies the RDF axiomatic triples and show that I is an rdfinterpretation and (**) in the (<=) direction we need to slightly extend the definition of I' to account forex:illxml
and show that I' is a model of R^{RDF}.
(*) For any positive integer j such that rdf:_j does not occur in the context of C, I and I are extended such that IS(rdf:_j)=I_{C}(rdf:_j)=I_{C}(rdf:_m) (see the definition of finiteaxiomatic triples above for the definition of m). Clearly, this does not affect satisfaction of R' or nonsatisfaction of φ, respectively tr_{Q}(T). To show that I is an rdfinterpretation, we need to show that I satisfies the RDF axiomatic triples and the RDF semantic conditions.
Satisfaction of the axiomatic triples follows immediately from the inclusion of tr(t) in R^{RDF} for every RDF finiteaxiomatic triple t, the fact that I is a model of R^{RDF}, and construction of I, and the extension of I to satisfy the infinite axiomatic triples. Consider the three RDF semantic conditions:
1 x is in IP if and only if <x, I( rdf:Property
)> is in IEXT(I(rdf:type
))2 If "
xxx"^^rdf:XMLLiteral
is in V and xxx is a welltyped XML literal string, then
(a) IL(
"
xxx"^^rdf:XMLLiteral
) is the XML value of xxx;
(b) IL("
xxx"^^rdf:XMLLiteral
) is in LV;
(c) IEXT(I(rdf:type
)) contains <IL("
xxx"^^rdf:XMLLiteral
), I(rdf:XMLLiteral
)>3 If "
xxx"^^rdf:XMLLiteral
is in V and xxx is an illtyped XML literal string, then
(a) IL(
(b) IEXT(I("
xxx"^^rdf:XMLLiteral
) is not in LV;rdf:type
)) does not contain <IL("
xxx"^^rdf:XMLLiteral
), I(rdf:XMLLiteral
)>.Satisfaction of condition 1 follows from satisfaction of the first rule in R^{RDF} in I and construction of I; specifically the second bullet.
Consider a welltyped XML literal"
xxx"^^rdf:XMLLiteral
. By the definition of satisfaction in RIF BLD, I_{C}("
xxx"^^rdf:XMLLiteral
) is the XML value of xxx (condition 2a), and is clearly in LV (condition 2b), by definition of I. Condition 2c is satisfied by satisfaction of the second rule in R^{RDF} in I.
Satisfaction of 3a follows from satisfaction of the fourth rule in R^{RDF} and the definition of LV. (3b) follows from satisfaction of the third rule in R^{RDF} (3b) in I (if there were a nonwelltyped XML literal in the class extension ofrdf:XMLLiteral
, then I would not be a model of this rule). This establishes the fact that I is an rdfinterpretation.
(**) Recall that, by assumption, ex:illxml is not used in R. Therefore, changing satisfaction of atomic formulas concerning ex:illxml does not affect satisfaction of R. We assume that I'_{C}(ex:illxml)=k is a unique element, i.e., no other constant is mapped to k.
We define I'_{F}(k) as follows: For every nonwelltyped literal of the form (s, rdf:XMLLiteral) such that I'_{C}(tr(s^^rdf:XMLLiteral))=l we define I_{truth}(I'_{F}(k)(l))=t; I'_{truth}(I'_{F}(k)(m))=f for each other object m in D_{ind}.Consider R^{RDF}. Satisfaction of R^{simple} was established in the proof in the previous section. Satisfaction of the facts corresponding to the RDF axiomatic triples in I' follows immediately from the definition of commonrifrdfinterpretation and the fact that I is an rdfinterpretation, and thus satisfies all RDF axiomatic triples.
Satisfaction of the ex:illxml facts in R^{RDF} follows immediately from the definition of I'. Satisfaction of the first, second, and third rule in R^{RDF} follow straightforwardly from the RDF semantic conditions 1, 2, and 3. This establishes the fact that I' is a model of R^{RDF}. ☐
Theorem A RIFRDF combination <R,{S_{1},...,S_{n}}> is rdfsatisfiable iff there is a semantic multistructure I that is a model of merge({R, R^{RDF}, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}).
Proof. The theorem follows immediately from the previous theorem and the observation that a combination (respectively, RIF document) is rdfsatisfiable (respectively, has a model) if and only if it does not entail the condition formula "a"="b". ☐
We axiomatize the semantics of the RDF(S) vocabulary using the following RIF rules.
Similar to the case for RDF, the set of RDFS finiteaxiomatic triples is the smallest set such that:
The set of considered datatypes T is defined as before.
R^{RDFS}  =  merge({R^{RDF}} union (tr(s p o .) for every RDFS finiteaxiomatic triple
s p o .) union Forall ?x (?x[rdf:type > rdfs:Resource]), ) union 
Theorem A RIFRDF combination C=<R,{S_{1},...,S_{n}}> rdfsentails a generalized RDF graph T if and only if merge({R, R^{RDFS}, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}) entails tr_{Q}(T). C rdfsentails a condition formula φ if and only if merge({R, R^{RDFS}, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}) entails φ.
Proof. In the proof we abbreviate merge({R, R^{RDFS}, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}) with R'.
The proof is then obtained from the proof of correspondence for RDF entailment in the previous section with the following modifications: (*) in the (=>) direction we need to slightly amend the definition of I to account for rdfs:Literal and show that I is an rdfsinterpretation and (**) in the (<=) direction we need to show that I' is a model of R^{RDFS}.
(*) We amend the definition of I by changing the definition of LV to the following:Clearly, this change does not effect satisfaction of the RDF axiomatic triples and the semantic conditions 1 and 2. To see that condition 3 is still satisfied, consider some nonwelltyped XML literal t. Then, ex:illxml(tr(t)) is satisfied in I. If tr(t)[rdf:type > rdfs:Literal] were to be satisfied as well, then, by the second last rule in the definition of R^{RDFS}, "a"="b" would be satisfied, which cannot be the case. Therefore, tr(t)[rdf:type > rdfs:Literal] is not satisfied and thus IL(t) is not in ICEXT(rdfs:Literal). And, since IL(t) is not in the value space of any considered datatype, it is not in LV. To show that I is an rdfsinterpretation, we need to show that I satisfies the RDFS axiomatic triples and the RDF semantic conditions.
 LV=(union of the value spaces of all considered datatypes) union (set of all k in D_{ind} such that I_{truth}(I_{frame}(I_{C}(rdf:type))(k,I_{C}(rdfs:Literal)))=t).
Satisfaction of the axiomatic triples follows immediately from the inclusion of tr(t) in R^{RDFS} for every RDFS finiteaxiomatic triple t, the fact that I is a model of R^{RDFS}, construction of I, and the extension of I in the proof of the RDF entailment embedding. Consider the RDFS semantic conditions:
1 (a) x is in ICEXT(y) if and only if <x,y> is in IEXT(I( rdf:type
))
(b) IC = ICEXT(I(rdfs:Class
))
(c) IR = ICEXT(I(rdfs:Resource
))
(d) LV = ICEXT(I(rdfs:Literal
))2 If <x,y> is in IEXT(I( rdfs:domain
)) and <u,v> is in IEXT(x) then u is in ICEXT(y)3 If <x,y> is in IEXT(I( rdfs:range
)) and <u,v> is in IEXT(x) then v is in ICEXT(y)4 IEXT(I( rdfs:subPropertyOf
)) is transitive and reflexive on IP5 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 IC9 If x is in ICEXT(I( rdfs:ContainerMembershipProperty
)) then:
< x, I(rdfs:member
)> is in IEXT(I(rdfs:subPropertyOf
))10 If x is in ICEXT(I( rdfs:Datatype
)) then <x, I(rdfs:Literal
)> is in IEXT(I(rdfs:subClassOf
))Conditions 1a and 1b are simply definitions of ICEXT and IC, respectively. Since I satisfies the first rule in the definition of R^{RDFS} it must be the case that every element k in D_{ind} is in ICEXT(I(rdfs:Resource)). Since IR=D_{ind}, it follows that IR = ICEXT(I(
rdfs:Resource
)), establishing 1c. Clearly, every object in ICEXT(I(rdfs:Literal
)) is in LV, by definition. Consider any value k in LV. By definition, either k is in the value space of some considered datatype or I_{truth}(I_{frame}(I_{C}(rdf:type))(k,I_{C}(rdfs:Literal)))=t. In the latter case, clearly k is in ICEXT(I(rdfs:Literal
)). In the former case, k is in the value space of some datatype with some label D, and thus I_{truth}(I_{F}(I_{C}(pred:isD))(k))=t. By the last rule in R^{RDFS}, it must consequently be the case that I_{truth}(I_{frame}(I_{C}(rdf:type))(k,I_{C}(rdfs:Literal)))=t, and thus k is in ICEXT(I(rdfs:Literal
)). This establishes satisfaction of condition 1d in I.Satisfaction in I of conditions 2 through 10 follows immediately from satisfaction in I of the 2nd through the 12th rule in the definition of R^{RDFS}. This establishes the fact that I is an rdfsinterpretation.
(**) Consider R^{RDFS}. Satisfaction of R^{RDF} was established in the proof in the previous section. Satisfaction of the facts corresponding to the RDFS axiomatic triples in I' follows immediately from the definition of commonrifrdfinterpretation and the fact that I is an rdfsinterpretation, and thus satisfies all RDFS axiomatic triples.
Satisfaction of the 1st through the 12th rule in R^{RDFS} follow straightforwardly from the RDFS semantic conditions 1 through 10. Satisfaction of the 13th rule follows from the fact that, given an illtyped XML literal t, IL(t) is not in LV (by RDF semantic condition 3), ICEXT(rdfs:Literal)=LV, and the fact that the ex:illxml predicate is only true on illtyped XML literals. Finally, satisfaction of the last rule in R^{RDFS} follows from the fact that ICEXT(rdfs:Literal)=LV, the definition of LV as a superset of the union of the value spaces of all datatypes, and the definition of the pred:isD predicates. This establishes the fact that I' is a model of R^{RDFS}. ☐
Theorem A RIFRDF combination <R,{S_{1},...,S_{n}}> is rdfssatisfiable if and only if there is a semantic multistructure I that is a model of merge({R, R^{RDFS}, tr_{R}(S_{1}), ..., tr_{R}(S_{n})}).
Proof. The theorem follows immediately from the previous theorem and the observation that a combination (respectively, RIF document) is rdfssatisfiable (respectively, has a model) if and only if it does not entail the condition formula "a"="b". ☐
It is known that expressive Description Logic languages such as OWL DL cannot be straightforwardly embedded into typical rules languages such as RIF BLD [RIFBLD], because of features such as disjunction and negation.
In this section we consider a subset of OWL DL in RIFOWL DL combinations. We define OWL DLP, which is inspired by socalled Description Logic Programs [DLP], and define how reasoning with RIFOWL DLP combinations can be reduced to reasoning with RIF.
The embedding of RIFOWL DL combinations is not defined for combinations that include infinite OWL ontologies and for combinations that include ontologies with RDF URI references that are not absolute IRIs or plain literals that are not in the lexical space of the xs:string datatype. In addition, each datatype used in the combination must have associated positive and negative guard predicates [DTB].
OWL DLP restricts the OWL DL abstract syntax [OWLSemantics], disallowing disjunction and extensional quantification in consequents of implications, as well as negation and equality. The semantics of OWL DLP is that of OWL DL.
The syntax is defined through an EBNF grammar, which is derived from the grammar of the OWL abstract syntax [OWLSemantics]. Any OWL DL ontology in abstract syntax form that conforms to this grammar is an OWL DLP ontology.
The basic syntax of ontologies and identifiers is the same as for OWL DL.
ontology ::= 'Ontology(' [ ontologyID ] { directive } ')' directive ::= 'Annotation(' ontologyPropertyID ontologyID ')'  'Annotation(' annotationPropertyID absoluteIRI ')'  'Annotation(' annotationPropertyID dataLiteral ')'  'Annotation(' annotationPropertyID individual ')'  axiom  fact
datatypeID ::= absoluteIRI classID ::= absoluteIRI individualID ::= absoluteIRI ontologyID ::= absoluteIRI datavaluedPropertyID ::= absoluteIRI individualvaluedPropertyID ::= absoluteIRI annotationPropertyID ::= absoluteIRI ontologyPropertyID ::= absoluteIRI
dataLiteral ::= typedLiteral  plainLiteral typedLiteral ::= lexicalForm^^absoluteIRI plainLiteral ::= lexicalForm  lexicalForm@languageTag lexicalForm ::= as in RDF, a unicode string in normal form C languageTag ::= as in RDF, an XML language tag
Facts are the same as for OWL DL, except that equality and
inequality (SameIndividual and
DifferentIndividual), as well as individuals without an
identifier, are not allowed.
fact ::= individual individual ::= 'Individual(' individualID { annotation } { 'type(' type ')' } { value } ')' value ::= 'value(' individualvaluedPropertyID individualID ')'  'value(' individualvaluedPropertyID individual ')'  'value(' datavaluedPropertyID dataLiteral ')'
type ::= Rdescription
The main restrictions posed by OWL DLP on the OWL DL syntax are on descriptions and axioms. Specifically, OWL DLP distinguishes between descriptions that are allowed on the righthand side (Rdescription) and those allowed on the lefthand side (Ldescription) of subclass statements.
We start with descriptions that may be allowed on both sides
dataRange ::= datatypeID  'rdfs:Literal'
description ::= classID  restriction  'intersectionOf(' { description } ')'
restriction ::= 'restriction(' datavaluedPropertyID dataRestrictionComponent { dataRestrictionComponent } ')'  'restriction(' individualvaluedPropertyID individualRestrictionComponent { individualRestrictionComponent } ')' dataRestrictionComponent ::= 'value(' dataLiteral ')' individualRestrictionComponent ::= 'value(' individualID ')'
We then proceed with the individual sides
Ldescription ::= description  Lrestriction  'unionOf(' { Ldescription } ')'  'intersectionOf(' { Ldescription } ')'  'oneOf(' { individualID } ')'
Lrestriction ::= 'restriction(' datavaluedPropertyID LdataRestrictionComponent { LdataRestrictionComponent } ')'  'restriction(' individualvaluedPropertyID LindividualRestrictionComponent { LindividualRestrictionComponent } ')' LdataRestrictionComponent ::= 'someValuesFrom(' dataRange ')'  'value(' dataLiteral ')' LindividualRestrictionComponent ::= 'someValuesFrom(' Ldescription ')'  'value(' individualID ')'
Rdescription ::= description  Rrestriction  'intersectionOf(' { Rdescription } ')'
Rrestriction ::= 'restriction(' datavaluedPropertyID RdataRestrictionComponent { RdataRestrictionComponent } ')'  'restriction(' individualvaluedPropertyID RindividualRestrictionComponent { RindividualRestrictionComponent } ')' RdataRestrictionComponent ::= 'allValuesFrom(' dataRange ')'  'value(' dataLiteral ')' RindividualRestrictionComponent ::= 'allValuesFrom(' Rdescription ')'  'value(' individualID ')'
Finally, we turn to axioms. We start with class axioms.
axiom ::= 'Class(' classID ['Deprecated'] 'complete' { annotation } { description } ')' axiom ::= 'Class(' classID ['Deprecated'] 'partial' { annotation } { Rdescription } ')'
axiom ::= 'DisjointClasses(' Ldescription Ldescription { Ldescription } ')'  'EquivalentClasses(' description { description } ')'  'SubClassOf(' Ldescription Rdescription ')'
axiom ::= 'Datatype(' datatypeID ['Deprecated'] { annotation } )'
Property axioms in OWL DLP restrict those in OWL DL by disallowing functional and inverse functional properties, because these involve equality.
axiom ::= 'DatatypeProperty(' datavaluedPropertyID ['Deprecated'] { annotation } { 'super(' datavaluedPropertyID ')'} { 'domain(' description ')' } { 'range(' dataRange ')' } ')'  'ObjectProperty(' individualvaluedPropertyID ['Deprecated'] { annotation } { 'super(' individualvaluedPropertyID ')' } [ 'inverseOf(' individualvaluedPropertyID ')' ] [ 'Symmetric' ] [ 'Transitive' ] { 'domain(' description ')' } { 'range(' description ')' } ')'  'AnnotationProperty(' annotationPropertyID { annotation } ')'  'OntologyProperty(' ontologyPropertyID { annotation } ')'
axiom ::= 'EquivalentProperties(' datavaluedPropertyID datavaluedPropertyID { datavaluedPropertyID } ')'  'SubPropertyOf(' datavaluedPropertyID datavaluedPropertyID ')'  'EquivalentProperties(' individualvaluedPropertyID individualvaluedPropertyID { individualvaluedPropertyID } ')'  'SubPropertyOf(' individualvaluedPropertyID individualvaluedPropertyID ')'
Definition. An OWL DL ontology in abstract syntax form is an OWL DLP ontology if it conforms with the grammar above. ☐
Recall that the semantics of frame formulas in DLdocument formulas is different from the semantics of frame formulas in RIF documents. Nonetheless, DLdocument formulas can be embedded into RIF documents, by translating frame formulas to predicate formulas. The mapping tr is the identity mapping on all RIF formulas, with the exception of frame formulas, as defined in the following table.
In the table, the mapping tr' is an injective function that maps constants to new constants, i.e., constants that are not used in the original document or its vicinity (e.g., entailed or entailing formula). It "generates" a new constant from an existing one.
RIF Construct  Mapping 

Term t  tr(t)=t 
Atomic formula φ that is not a frame formula  tr(φ)=φ 
a[b_{1}>c_{1} ... b_{n}>c_{n}], with n≥2  tr(a[b_{1}>c_{1} ... b_{n}>c_{n}])=And( tr(a[b_{1}>c_{1}]) ... tr(a[b_{n}>c_{n}])) 
a[b > c], where a and c are terms and b ≠ rdf:type is a constant  tr(a[b > c])=tr'(b)(a,c) 
a[rdf:type > c], where a is a term and c is a constant  tr(a[rdf:type > c])=tr'(c)(a) 
Exists ?V1 ... ?Vn(φ)  tr(Exists ?V1 ... ?Vn(φ))=Exists ?V1 ... ?Vn(tr(φ)) 
And(φ_{1} ... φ_{n})  tr(And(φ_{1} ... φ_{n}))=And(tr(φ_{1}) ... tr(φ_{n})) 
Or(φ_{1} ... φ_{n})  tr(Or(φ_{1} ... φ_{n}))=Or(tr(φ_{1}) ... tr(φ_{n})) 
φ_{1} : φ_{2}  tr(φ_{1} : φ_{2})=tr(φ_{1}) : tr(φ_{2}) 
Forall ?V1 ... ?Vn(φ)  tr(Forall ?V1 ... ?Vn(φ))=Forall ?V1 ... ?Vn(tr(φ)) 
Group(φ_{1} ... φ_{n})  tr(Group(φ_{1} ... φ_{n}))=Group(tr(φ_{1}) ... tr(φ_{n})) 
Document(directive_{1} ... directive_{n} Γ)  tr(Document(directive_{1} ... directive_{n} Γ))=Document(directive_{1} ... directive_{n} tr(Γ)) 
For the purpose of making statements about this embedding, we define a notion of entailment for DLdocument formulas.
Definition. A RIFBLD DLdocument formula R dlentails a DLcondition φ if for every dlsemantic multistructure I that is a model of R it holds that TVal_{I}(φ)=t. ☐
The following lemma establishes faithfulness with respect to entailment of the embedding.
RIFBLD DLdocument formula Lemma A RIFBLD DLdocument formula R dlentails a DLcondition φ if and only if tr(R) entails tr(φ).
Proof. We prove both directions by contradiction: if the entailment does not hold on the one side, we show that it also does not hold on the other.
(=>) Assume tr(R) does not entail tr(φ). This means there is some semantic multistructure I = <TV, DTS, D, D_{ind}, D_{func}, I_{C}, I_{V}, I_{F}, I_{frame}, I_{SF}, I_{sub}, I_{isa}, I_{=}, I_{external}, I_{truth}> that is a model of tr(R), but not of tr(φ).
Consider the dlsemantic multistructure I* = <TV, DTS, D, D_{ind}, D_{func}, I*_{C}, I_{V}, I_{F}, I*_{frame'}, I_{SF}, I_{sub}, I_{isa}, I_{=}, I_{external}, I_{truth}>, with I*_{C} and I*_{frame'} defined as follows: Let t be an element in D such that I_{truth}(t)=t and let f in D be such that I_{truth}(f)=f.Observe that tr(R) and tr(φ) do not include frame formulas.
 for every constant c' used as unary or binary predicate symbol in tr(R) or tr(φ) such that c'=tr'(c) for some constant c, I*_{C}( c')=I_{C}(c); I*_{C}(c*)=I*_{C}(c*) for every other constant c*;
 for every constant c' used as unary predicate symbol in tr(R) or tr(φ) such that c'=tr'(c) for some constant c, and every object k in D_{ind}, if I_{truth}(I_{F}(I_{C}(c'))(k))=t, I*_{frame'}(k)((I_{C}(rdf:type), I_{C}(c))=t,
 for every constant b' used as binary predicate symbol in tr(R) or tr(φ) such that b'=tr'(b) for some constant b, and every pair (k, l) in D_{ind} × D_{ind}, if I_{truth}(I_{F}(I_{C}(b'))(k,l))=t, I*_{frame'}(k)((I_{C}(b),l))=t,
 if I*_{frame'}(k)((b_{1},...,b_{n}))=t and I*_{frame'}(k)((c_{1},...,c_{m}))=t for any two finite bags (b_{1},...,b_{n}) and (c_{1},...,c_{m}), then I*_{frame'}(k)((b_{1},...,b_{n},c_{1},...,c_{m}))=t, and
 I*_{frame'}(b)=f for any other bag b.
To show that I* is a model of R and not of φ, we only need to show that (+) for any frame formula a[b > c] that is a DLcondition, I* is a model of a[b > c] iff I is a model of tr(a[b > c]). This argument straightforwardly extends to the case of frames with multiple b_{i}s and c_{i}s, since in RIF semantic structures the following condition is required to hold: TVal_{I}(a[b_{1}>c_{1} ... b_{n}>c_{n}]) = t if and only if TVal_{I}(a[b_{1}>c_{1}]) = ... = TVal_{I}(a[b_{n}>c_{n}]) = t [RIFBLD].
Consider the case b=rdf:type. Then,
I* is a model of a[b > c] iff I_{truth}(I*_{frame'}(I(a))(I_{C}(rdf:type),I_{C}(c)))=t.
From the definition of I* we obtain
I_{truth}(I*_{frame'}(I(a))(I_{C}(rdf:type),I_{C}(c)))=t iff I*_{frame'}(I(a))(I_{C}(rdf:type),I_{C}(c))=t.
By definition of the embedding, we know that tr'(c) is used as unary predicate symbol in tr(R) or tr(φ). From the definition of I* we obtain
I*_{frame'}(I(a))(I_{C}(rdf:type),I_{C}(c))=t iff I_{truth}(I_{F}(I_{C}(tr'(c)))(I(a)))=t.
Finally, since tr(a[b > c])=tr'(c)(a), we obtain
I_{truth}(I_{F}(I_{C}(tr'(c)))(I(a)))=t iff I is a model of tr(a[b > c]).
From this chain of equivalences follows that I* is a model of a[b > c] iff I is a model of tr(a[b > c]).
The argument for the case b≠rdf:type is analogous, thereby obtaining (+).
(<=) Assume R does not dlentail φ. This means there is some dlsemantic multistructure I = <TV, DTS, D, D_{ind}, D_{func}, I_{C}, I_{V}, I_{F}, I_{frame'}, I_{SF}, I_{sub}, I_{isa}, I_{=}, I_{external}, I_{truth}> that is a model of R, but not of φ. Let B be the set of constant symbols occurring in the frame formulas of the forms a[rdf:type > b] and a[b > c] in R or φ.
Consider the semantic multistructure I* = <TV, DTS, D, D_{ind}, D_{func}, I*_{C}, I_{V}, I*_{F}, I*_{frame}, I_{SF}, I_{sub}, I_{isa}, I_{=}, I_{external}, I_{truth}>. Let t and f in D be such that I_{truth}(t)=t and I_{truth}(f)=f. We define I*_{C}, I*_{frame}, and I*_{F} as follows:Observe that R and φ do not include predicate formulas involving derived constant symbols tr'(b) or tr'(c). The remainder of the proof is analogous to the (=>) direction. ☐
 I*_{C}(tr'(b))=I*_{C}(b) for any b in B ; I*_{C}(c)=I_{C}(c) for any c not in B,
 I*_{frame}(b)=f for any finite bag b of D, and
 I*_{F} is defined as follows:
 for every c in B, given an object k in D_{ind}, if I_{truth}(I_{frame'}(k)((I_{C}(rdf:type), I_{C}(c)))=t, I*_{F}(I*_{C}(tr'(c)))(k)=t; I*_{F}(I*_{C}(tr'(c)))(k')=f for any other k' in D_{ind},
 for every b in B, given a pair (k, l) in D_{ind} × D_{ind}, if I_{truth}(I_{frame}(k)((I_{C}(b),l)))=t, I*_{F}(tr'(b))(k,l)=t;I*_{F}(tr'(b))(k',l')=f for any other pair (k', l') in D_{ind} × D_{ind}, and
 I*_{F}(c')=I_{F}(c') for every other constant c'.
The embedding of OWL DLP into RIF BLD has two stages: normalization and embedding.
Normalization splits the OWL axioms so that the later mapping to RIF of the individual axioms results in rules. Additionally, it simplifies the axioms and removes annotations.
#  Complex OWL  Normalized OWL  Condition on translation 

1  tr_{N}(
Ontology( [ ontologyID ]
directive_{1}
...
directive_{n} )
) 
Ontology( tr_{N}(directive_{1}) ... tr_{N}(directive_{n}) ) 

2  tr_{N}(Annotation( ... ))  
3  tr_{N}(
Individual( individualID
annotation_{1}
...
annotation_{n}
type_{1}
...
type_{m}
value_{1}
...
value_{k} )
) 
tr_{N}(Individual( individualID type_{1} )) ... tr_{N}(Individual( individualID type_{m} )) tr_{N}(Individual( individualID value_{1} )) ... tr_{N}(Individual( individualID value_{k} )) 

4  tr_{N}(
Individual( individualID
type(intersectionOf(
description_{1}
...
description_{n}
))
) 
tr_{N}(Individual( individualID type(description_{1}) )) ... tr_{N}(Individual( individualID type(description_{n}) )) 

5  tr_{N}(
Individual( individualID type(X))) 
Individual( individualID type(X)) 
X is a classID or value restriction 
6  tr_{N}(
Individual( individualID type(restriction(propertyID allValuesFrom(X))))) 
tr_{N}(
SubClassOf( oneOf(individualID) restriction(propertyID allValuesFrom(X))) ) 

7  tr_{N}(
Individual( individualID value(propertyID b))) 
Individual( individualID value(propertyID b)) 
b is an individualID or dataLiteral 
8  tr_{N}(
Individual( individualID1 value(propertyID Individual( individualID2 ... )))) 
tr_{N}(
Individual( individualID1 value(propertyID individualID2) ) tr_{N}(Individual( individualID2 ... )) 

9  tr_{N}(
Class( classID [Deprecated]
complete
annotation_{1}
...
annotation_{n}
description_{1}
...
description_{m} )
) 
tr_{N}( EquivalentClasses(classID
intersectionOf(description_{1}
...
description_{m} )
) 

10  tr_{N}(
Class( classID [Deprecated]
partial
annotation_{1}
...
annotation_{n}
description_{1}
...
description_{m} )
) 
tr_{N}( SubClassOf(classID
intersectionOf(description_{1}
...
description_{m} )
) 

11  tr_{N}(
DisjointClasses(
description_{1}
...
description_{m} )
) 
tr_{N}(SubClassOf(intersectionOf(description_{1}
description_{2}) owl:Nothing))
... tr_{N}(SubClassOf(intersectionOf(description_{1} description_{m}) owl:Nothing)) ... tr_{N}(SubClassOf(intersectionOf(description_{m1} description_{m}) owl:Nothing)) 

12  tr_{N}(
EquivalentClasses(
description_{1}
...
description_{m} )
) 
tr_{N}(SubClassOf(description_{1} description_{2})) tr_{N}(SubClassOf(description_{2} description_{1})) ... tr_{N}(SubClassOf(description_{m1} description_{m})) tr_{N}(SubClassOf(description_{m} description_{m1})) 

13  tr_{N}(
SubClassOf(description X)) 
SubClassOf(description X) 
X is a description that does not contain intersectionOf 
14  tr_{N}(
SubClassOf(description
...intersectionOf(
description_{1}
...
description_{n}
)...)
) 
tr_{N}(SubClassOf(description ...description_{1}...)) ... tr_{N}(SubClassOf(description ...description_{n}...)) 

15  tr_{N}(Datatype( ... ))  
16  tr_{N}(
DatatypeProperty( propertyID [ Deprecated ]
annotation_{1}
...
annotation_{n}
super(superproperty_{1})
...
super(superproperty_{m})
domain(domaindescription_{1})
...
domain(domaindescription_{j})
range(rangedescription_{1})
...
range(rangedescription_{k}) )
) 
SubPropertyOf(propertyID superproperty_{1})
...
SubPropertyOf(propertyID superproperty_{m})
tr_{N}(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescription_{1})) ... tr_{N}(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescription_{j})) tr_{N}(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescription_{1}))) ... tr_{N}(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescription_{k}))) 

17  tr_{N}(
ObjectProperty( propertyID [ Deprecated ]
annotation_{1}
...
annotation_{n}
super(superproperty_{1})
...
super(superproperty_{m})
[ inverseOf( inversePropertyID ) ]
[ Symmetric ]
[ Transitive ]
domain(domaindescription_{1})
...
domain(domaindescription_{l})
range(rangedescription_{1})
...
range(rangedescription_{k}) )
) 
SubPropertyOf(propertyID superproperty_{1})
...
SubPropertyOf(propertyID superproperty_{m})
tr_{N}(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescription_{1})) ... tr_{N}(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescription_{l})) tr_{N}(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescription_{1}))) ... tr_{N}(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescription_{k}))) ObjectProperty( propertyID [ inverseOf( inversePropertyID ) ] ) ObjectProperty( propertyID [ Symmetric ] ) ObjectProperty( propertyID [ Transitive ] ) 

18  tr_{N}(
EquivalentProperties(
property_{1}
...
property_{m} )
) 
tr_{N}(SubPropertyOf(property_{1} property_{2})) tr_{N}(SubPropertyOf(property_{2} property_{1})) ... tr_{N}(SubPropertyOf(property_{m1} property_{m})) tr_{N}(SubPropertyOf(property_{m} property_{m1})) 
The result of the normalization is a set of individual property value, individual typing, subclass, subproperty, and property inverse, symmetry and transitive statements.
The following lemma establishes the fact that, for the purpose of entailment, the ontologies in a combination may be replaced by their normalization.
Normalization Lemma Given a combination C=<R,{O_{1},...,O_{n}}>, where O_{1},...,O_{n} are OWL DLP ontologies that do not import ontologies, C owldlentails φ iff C'=<R,{tr_{N}(O_{1}),...,tr_{N}(O_{n})}> owldlentails φ.
Proof. We prove both directions by contradiction: if the entailment does not hold on the one side, we show that it also does not hold on the other.
(=>) Assume C' does not owldlentail φ. This means there is a commonrifowldlinterpretation (I, I) that is a model of C', but I is not a model of φ.
Consider the pair (I, I*), where I* is obtained from I by suitably extending EC and ER to satisfy the annotation properties. Clearly, (I, I*) is a commonrifowldlinterpretation, since the extension realized in I* does not affect any of the conditions on commonrifowldlinterpretations. By the interpretation of axioms and facts and the EC extension table in sections 3.3 and 3.2 in [OWLSemantics] it is easy to verify that, for any directive d in I, if I satisfies tr_{N}(d), I* satisfies d. Therefore, I* satisfies O_{1},..., and O_{n}, and thus (I, I*) satisfies C. Since I is not a model of φ, C does not owldlentail φ.
(<=) Assume C does not owldlentail φ. This means there is a commonrifowldlinterpretation (I, I) that is a model of C, but I is not a model of φ. It is easy to verify, by the interpretation of axioms and facts and the EC extension table in sections 3.3 and 3.2 in [OWLSemantics], that I satisfies tr_{N}(O_{1}),..., and tr_{N}(O_{n}). So, (I, I) is a model of C', and thus C' does not owldlentail φ. ☐
We now proceed with the embedding of normalized OWL DLP ontologies into RIF DLdocument formulas. The embedding is an extension of the embedding function tr. The embeddings of IRIs and literals is as defined in the Section Embedding Symbols.
In the following, let T be the union of the set of considered datatypes and the set of datatypes used in the ontologies under consideration. As with the RDFS embedding, we assume that each datatype in T has an associated capitalized short name Datatype and a positive guard pred:isDatatype. In addition, we assume each datatype has a negative guard pred:isNotDatatype, which can be used to test whether a particular object is not a value of the datatype (cf. [RIFDTB]).
#  Normalized OWL  RIF RIFBLD DLdocument formula  Condition on translation 

1  tr_{O}(
Ontology(
directive_{1}
...
directive_{n}
)
) 
tr_{O}(directive_{1})
...tr_{O}(directive_{n}) 

2  tr_{O}(
Individual( individualID type(A) )) 
tr(individualID)[rdf:type > tr(A)] 
A is a classID 
3  tr_{O}(
Individual( individualID type(restriction(propertyID value(b))) )) 
tr(individualID)[tr(propertyID) > tr(b)] 

4  tr_{O}(
Individual( individualID value(propertyID b) )) 
tr(individualID)[tr(propertyID) > tr(b)] 

5  tr_{O}(
SubPropertyOf(property_{1} property_{2})
) 
Forall ?x ?y (?x[tr(property_{2}) > ?y] : ?x[tr(property_{1}) > ?y]) 

6  tr_{O}(
ObjectProperty(propertyID)) 

7  tr_{O}(
ObjectProperty(property_{1}
inverseOf(property_{2}) )
) 
Forall ?x ?y (?y[tr(property_{2}) > ?x] : ?x[tr(property_{1}) > ?y]) Forall ?x ?y (?y[tr(property_{1}) > ?x] : ?x[tr(property_{2}) > ?y]) 

8  tr_{O}(
ObjectProperty(propertyID Symmetric )) 
Forall ?x ?y (?y[tr(propertyID) > ?x] : ?x[tr(propertyID) > ?y]) 

9  tr_{O}(
ObjectProperty(propertyID Transitive )) 
Forall ?x ?y ?z (?x[tr(propertyID) > ?z] : And( ?x[tr(propertyID) > ?y] ?y[tr(propertyID) > ?z])) 

10  tr_{O}(
SubClassOf(description_{1} description_{2})
) 
tr_{O}(description_{1},description_{2},?x) 

11  tr_{O}(A,?x) 
?x[rdf:type > tr(A)] 
A is a classID or datatypeID; x is a variable name 
12  tr_{O}(description_{1},X,?x) 
Forall ?x (tr_{O}(X, ?x) : tr_{O}(description_{1}, ?x ) 
X is a classID, datatypeID or value restriction 
13  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}) : And( tr_{O}(description_{1}, ?x)?x[tr(property_{1}) > ?y_{1}] ?y_{1}[tr(property_{2}) > ?y_{2}] ... ?y_{n1}[tr(property_{n}) > ?y_{n}])) 
X is a classID, datatypeID or value restriction 
14  tr_{O}(intersectionOf(description_{1} ... description_{n}), ?x) 
And(tr_{O}(description_{1}, ?x) ... tr_{O}(description_{n}, ?x')) 
x is a variable name 
15  tr_{O}(unionOf(description_{1} ... description_{n}), ?x) 
Or(tr_{O}(description_{1}, ?x) ... tr_{O}(description_{n}, ?x)) 
x is a variable name 
16  tr_{O}(oneOf(value_{1} ... value_{n}), ?x) 
Or( ?x = tr(value_{1}) ... ?x = tr(value_{n})) 
x is a variable name 
17  tr_{O}(restriction(propertyID someValuesFrom(description)), ?x) 
Exists ?y(And(?x[tr(propertyID) > ?y] tr_{O}(description, ?y) )) 
x is a variable name 
18  tr_{O}(restriction(propertyID value(valueID)), ?x) 
?x[tr(propertyID) > tr(valueID) ] 
x is a variable name 
Besides the embedding in the previous table, we also need an axiomatization of some of the aspects of the OWL DL semantics, e.g., separation between individual and datatype domains. This axiomatization is defined relative to an OWL vocabulary V and a datatype map D, which includes all datatypes in T. In the table, for a given datatype d, L2V(d) is the lexicaltovalue mapping of d.
R^{OWLDL}(V)  =  merge({(i) (Forall ?x
("a"="b" : ?x[rdf:type > owl:Nothing]), (ii) Forall ?x ("a"="b" : And(?x[rdf:type >
rdfs:Literal] ?x[rdf:type > owl:Thing])), 
We call an OWL DLP ontology O normalized if it is the same as its normalization, i.e., O=tr_{N}(O).
The following lemma establishes faithfulness of the embedding.
Normalized Combination Embedding Lemma Given a datatype map D conforming with T, a RIFOWLDLcombination C=<R,{O_{1},...,O_{n}}>, where O_{1},...,O_{n} are normalized OWL DLP ontologies with vocabulary V, owldlentails a DLcondition φ with respect to D iff merge({R, R^{OWLDL}(V), tr_{O}(O_{1}), ..., tr_{O}(O_{n})}) dlentails φ.
Proof. We prove both directions by contradiction: if the entailment does not hold on one side, we show that it also does not hold on the other.
In the proof we abbreviate merge({R, R^{OWLDL}(V), tr_{O}(O_{1}), ..., tr_{O}(O_{n})} with R'.
(=>) Assume R' does not dlentail φ. This means there is a dlsemantic multistructure I = <TV, DTS, D, D_{ind}, D_{func}, I_{C}, I_{V}, I_{F}, I_{frame'}, I_{SF}, I_{sub}, I_{isa}, I_{=}, I_{external}, I_{truth}> that is a model of R', but not of φ.
Consider the pair (I*,I), where I* = <TV, DTS, D*, D*_{ind}, D_{func}, I_{C}, I_{V}, I_{F}, I_{frame'}, I_{SF}, I_{sub}, I_{isa}, I_{=}, I_{external}, I_{truth}> is such thatand I = <R, EC, ER, L, S, LV> is a tuple defined as follows:
 D*_{ind}=D_{ind} union (union of the value spaces of all datatypes in the range of D) and
 D*=D union D*_{ind}
Recall that an OWL vocabulary V consists of a set of literals V_{L} and seven sets of IRIs, V_{C}, V_{D}, V_{I}, V_{DP}, V_{IP}, V_{AP}, and V_{O}, which are the sets of class, datatype, individual, datavalued property, individualvalued property, annotation property, and ontology identifiers. According to its definition, an abstract OWL interpretation with respect to a datatype map D must fulfill the following conditions, where L(d) denotes the lexical space, V(d) denotes the value space and L2V(d) denotes to lexicaltovalue mapping of a datatype d:
 R=D*,
 LV=(union of the value spaces of all datatypes in the range of D),
 O=EC(owl:Thing),
 EC(rdfs:Literal)=LV,
 EC(d') = the value space of D(d'), if D(d') is defined,
 EC(c) = set of all objects k such that I_{truth}(I_{frame'}(I_{C}(rdf:type))(k,I_{C}(<c>))) = t, for every class identifier or datatype identifier c≠rdfs:Literal in V that is not in the domain of D,
 ER(p) = set of all pairs (k, l) such that I_{truth}(I_{frame'}(I_{C}(<p>))( k, l ))) = t (true), for every data valued and individual valued property identifier p in V;
 L((s, d)) = I_{C}("s"^^d) for every welltyped literal (s, d) in V;
 S(i) = I_{C}(<i>) for every IRI i in V.
Condition 1 is met because D is a nonempty set. Clearly LV is a subset of R and contains the value spaces for each datatype in D, which include the sets of Unicode strings and pairs of Unicode strings and language tags, since the xs:string and rif:text datatypes are included in D, by the fact that D is conforming and the two datatypes are RIFrequired; therefore, condition 2 is met.
 R is a nonempty set,
 LV is a subset of R that contains the set of Unicode strings, the set of pairs of Unicode strings and language tags, and the value spaces of all datatypes in D,
 EC : V_{C} → 2^{O}
 EC : V_{D} → 2^{LV}
 ER : V_{DP} → 2^{O×LV}
 ER : V_{IP} → 2^{O×O}
 ER : V_{AP} ∪ { rdf:type } → 2^{R×R}
 ER : V_{OP} → 2^{R×R}
 L : TL → LV, where TL is the set of typed literals in V_{L}
 S : V_{I} ∪ V_{C} ∪ V_{D} ∪ V_{DP} ∪ V_{IP} ∪ V_{AP} ∪ V_{O} ∪ { owl:Ontology, owl:DeprecatedClass, owl:DeprecatedProperty } → R
 S(V_{I}) ⊆ O
 EC(owl:Thing) = O ⊆ R, where O is nonempty and disjoint from LV
 EC(owl:Nothing) = { }
 EC(rdfs:Literal) = LV
 If D(d') = d then EC(d') = V(d)
 If D(d') = d then L("v"^^d') ∈ V(d)
 If D(d') = d and v ∈ L(d) then L("v"^^d') = L2V(d)(v)
 If D(d') = d and v ∉ L(d) then L("v"^^d') ∈ R  LV
When referring to rules in the remainder we mean rules in R^{OWLDL}(V), unless otherwise specified.
To establish satisfaction of condition 3, observe that, by definition, O=EC(owl:Thing). So, for a given class name C we only need to establish that for any k in EC(c) it holds that k in EC(owl:Thing). But if k in EC(c), then, by definition, I_{truth}(I_{frame'}(I_{C}(rdf:type))(k,I_{C}(<C>))) = t. But then, by rule (iii), it must be the case that I_{truth}(I_{frame'}(I_{C}(rdf:type))(k,I_{C}(<owl:Thing>))) = t, and thus k in EC(owl:Thing).
Consider a datatype identifier Diri and associated short name DT and an object k not in LV such that k in EC(Diri). This means that I_{truth}(I_{frame'}(I_{C}(rdf:type))(k,I_{C}(<Diri>))) = t, but also I_{truth}(I_{F}(I_{C}(isNotDT))(k)) = t (since the value space of the datatype is a subset of LV). But then "a"="b" must be satisfied in I*, by rule (xii), which is clearly a contradiction. This establishes satisfaction of condition 4.
Satisfaction of conditions 5 and 6 can be shown similarly, exploiting rules (v), (vi), (vii), and (xv).
ER maps annotation and ontology properties to the empty set, so conditions 7 and 8 are trivially satisfied.
I_{C} maps welltyped literals "s"^^d to objects in the value space of d. Since L is defined in terms of I_{C} and since the value spaces of all datatypes are included in LV, condition 9 is satisfied.
Condition 10 is clearly satisfied by the definition of S and since R=D*.
Satisfaction of condition 11 follows straightforwardly from rule (viii) and the definition of O.
EC(owl:Thing) = O subset R, by definition. Then, by rule (xiii), there is no element in the value space of any datatype that is in O. Consequently, O is disjoint from LV. This establishes satisfaction of condition 12.
Satisfaction of condition 13 follows straightforwardly from rule (i); satisfaction of conditions 14 and 15 is immediate by definition of I.
Conditions 16 and 17 are satisfied by definition of L and the definition of I_{C}; observe that for every typed literal "v"^^d' must hold that d' is in the domain of D, since D includes all datatypes under consideration.
Assume there exists an illtyped literal "v"^^d' in V, i.e., v is not in the lexical space D(d'). Since I satisfies rule (xiv), "a"="b" must be satisfied, which is a contradiction. So, there is no illtyped literal and thus condition 18 is satisfied.
This establishes the fact that I is an abstract OWL interpretation.
Consider now any ontology O in {O_{1},...,O_{n}}. To establish that I satisfies O, we need to establish five conditions (cf. Section 3.4 in [OWLSemantics]):Conditions 1 and 2 are satisfied by the fact O is an ontology of vocabulary V.
 each URI reference in O used as a class ID (datatype ID, individual ID, datavalued property ID, individualvalued property ID, annotation property ID, annotation ID, ontology ID) belongs to V_{C} (V_{D}, V_{I}, V_{DP}, V_{IP}, V_{AP}, V_{O}, respectively);
 each literal in O belongs to V_{L};
 I satisfies each directive in O, except for Ontology Annotations;
 there is some o ∈ R with <o,S(owl:Ontology)> ∈ ER(rdf:type) such that for each Ontology Annotation of the form Annotation(p v), <o,S(v)> ∈ ER(p) and that if O has name n, then S(n) = o; and
 I satisfies each ontology mentioned in an owl:imports annotation directive of O.
Conditions 4 and 5 are trivially satisfied, because normalized OWL DLP ontologies do not contain annotations and do not have names.
Consider any directive d in O; d has one of the following forms (cf. the second column of Table Normalizing OWL DLP):If d is of form 1, then we have that tr(d)=<individualID>[rdf:type > <A>] is satisfied in I*, and thus I_{truth}(I_{frame'}(I_{C}(rdf:type))(I_{C}(<individualID>),I_{C}(<A>))) = t. Consequently, I_{C}(<individualID>) is in EC(<A>). Since, in addition, S(<individualID>)=I_{C}(<individualID>), we have that S(<individualID>) is in EC(<A>), and thus d is satisfied in I. Similar for statements of the forms 2 and 3.
 class membership statement of the form Individual ( individualID type(A) ), where A is a class ID,
 membership of value restriction,
 property value statement,
 subproperty statement,
 inverse property statement,
 symmetric property statement,
 transitive property statement, or
 subclass statement SubClassOf(X Y).
Consider a subproperty statement SubPropertyOf(p q) and a pair (k, l) in ER(<p>). Then, by construction of I, I_{truth}(I_{frame'}(I_{C}(<p>))( k, l ))) = t. But, by tr(d), it must be the case that also I_{truth}(I_{frame'}(I_{C}(<q>))( k, l ))) = t. But then, (k,l) must be in ER(<q>), by construction of I. So, I satisfies d. Similar for statements of the forms 5, 6, and 7.
Consider the case that d is a subclass statement SubClassOf(X Y) and consider any k in EC(X), where EC is as in the EC Extension Table in [OWLSemantics]. We show, by induction, that I* satisfies tr_{O}(X) when ?x is assigned to k.
If X is a classID, then satisfaction of tr(X) follows by an analogous argument as that for directives of form 1. Similar for value restrictions. If X is a somevalue restriction of type Z on a property p, then there must be some object l such that (k,l) in ER(p) such that l is in EC(Z). By induction we have satisfaction of tr(Z) for some variable assignment. Then, by definition of I, we have I_{truth}(I_{frame'}(I_{C}(<p>))( k, l )) = t (true), thereby establishing satisfaction of tr_{O}(X) in I*. This extends straightforwardly to union, intersection, and oneof descriptions.
By satisfaction of tr_{O}(d), we have that tr_{O}(Y) is necessarily satisfied for ?x assigned to k. By an argument analogous to the argument above, we obtain that k is in EC(Y).
This establishes satisfaction of d in I.
We obtain that every directive is satisfied in I, thereby obtaining satisfaction of condition 2. Therefore, O, and thus every ontology in C, is satisfied in I. Clearly, I* satisfies R and not φ, so (I*, I) satisfies R and not φ. We conclude that C does not entail φ.
(<=) Assume C does not owldlentail φ. This means there is a commonrifowldlinterpretation (I, I) that is an owldlmodel of C, but I is not a model of φ. To show that R' does not entail φ, we show that I is a model of R'.
R is satisfied in I by assumption. Satisfaction of tr_{O}(O_{i}) can be shown analogously to establishment of satisfaction in I of O_{i} in the (=>) direction. We now establish satisfaction of the rules in R^{OWLDL}(V).
(i) follows immediately from the fact that EC(owl:Nothing)={}. (ii) follows from conditions 14 and 12 on abstract OWL interpretations. (iii) follows from the fact that EC maps class names to subsets of O=EC(owl:Thing) (conditions 3, 12 on abstract OWL interpretations). (iv) follows from condition 14 on abstract OWL interpretations and the fact that LV is a superset of the value spaces of all datatypes (by conditions 15 and 4 on abstract OWL interpretations). (v) follows from conditions 12, 5, and 6. (vi) and (vii) follow from conditions 12, 14, 5, and 6. (viii) follows from condition 11. (ix) follows from conditions 16 and 15. (x) follows from the fact that LV includes all plain literals (condition 2) and condition 17. (xi) follows from conditions 15, 14, and the fact that LV is a superset of the value space of a datatype. (xii) follows from condition 15; i.e., there is no assignment for the variable ?x that is both a member of the value space of the datatype and is in its class extension and thus the antecedent of the rule will never be satisfied and rule is always satisfied. (xiii) follows from condition 12 and the fact that LV is a superset of the union of all value spaces. (xiv) follows from the fact that there is no illtyped literal, since such a literal would either violate condition 16 or condition 18 on abstract OWL interpretations.
This establishes satisfaction of R^{OWLDL}(V), and thus R', in I. Therefore, R' does not entail φ. ☐
The following theorems establish faithfulness of the full embedding of RIFOWL DLP combinations into RIF.
Theorem Given a datatype map D conforming with T, a RIFOWLDLcombination C=<R,{O_{1},...,O_{n}}>, where O_{1},...,O_{n} are OWL DLP ontologies with vocabulary V that do not import other ontologies, owldlentails a DLcondition formula φ with respect to D iff tr(merge({R, R^{OWLDL}(V), tr_{O}(tr_{N}(O_{1})), ..., tr_{O}(tr_{N}(O_{n}))})) entails tr(φ).
Proof. By the Normalization Lemma,
C=<R,{O_{1},...,O_{n}}> owldlentails φ iff <R,{tr_{N}(O_{1}),...,tr_{N}(O_{n})}> owldlentails φ.
Then, by the Normalized Combination Embedding Lemma,
<R,{tr_{N}(O_{1}),...,tr_{N}(O_{n})}> owldlentails φ iff merge({R, R^{OWLDL}(V), tr_{O}(tr_{N}(O_{1})), ..., tr_{O}(tr_{N}(O_{n}))}) dlentails φ.
Finally, by the RIFBLD DLdocument formula Lemma,
merge({R, R^{OWLDL}(V), tr_{O}(tr_{N}(O_{1})), ..., tr_{O}(tr_{N}(O_{n}))}) dlentails φ iff tr(merge({R, R^{OWLDL}(V), tr_{O}(tr_{N}(O_{1})), ..., tr_{O}(tr_{N}(O_{n}))})) entails tr(φ).
This chain of equivalences establishes the theorem. ☐
Theorem Given a datatype map D conforming with T, a RIFOWLDLcombination <R,{O_{1},...,O_{n}}>, where O_{1},...,O_{n} are OWL DLP ontologies with vocabulary V that do not import other ontologies, is owldlsatisfiable with respect to D iff there is a semantic multistructure I that is a model of tr(merge({R, R^{OWLDL}(V), tr_{O}(tr_{N}(O_{1})), ..., tr_{O}(tr_{N}(O_{n}))}).
Proof. The theorem follows immediately from the previous theorem and the observation that a combination (respectively, document) is owldlsatisfiable (respectively, has a model) if and only if it does not entail the condition formula "a"="b". ☐
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 [RDFConcepts], 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 [RDFSemantics] also apply to such generalized RDF graphs.
We note that every standard RDF graph is a generalized RDF graph. Therefore, our definition of combinations applies to standard RDF graphs as well. We note also that the notion of generalized RDF graphs is more liberal than the notion of RDF graphs used by SPARQL; generalized RDF graphs additionally allow blank nodes and literals in predicate positions.
OWL 2: While OWL 2 is still in development it is unclear how RIF will interoperate with it. At the time of writing, we believe that with OWL 2 the support for punning may
be beneficial. In addition the semantics of annotation properties might be different than in OWL, so there might be particular problems if these properties are considered in combinations, as in the Section Annotation properties.