RIF RDF and OWL Compatibility

- This version:
- http://www.w3.org/2005/rules/wg/draft/ED-rif-rdf-owl-20080219/
- Latest editor's draft:
- http://www.w3.org/2005/rules/wg/draft/rif-rdf-owl/

- Editors:
- Jos de Bruijn, Free University of Bozen/Bolzano

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

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

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

The Rule Interchange Format (RIF) Working Group seeks public feedback on these Working Drafts. Please send your comments to public-rif-comments@w3.org (public archive). If possible, please offer specific changes to the text that would address your concern.

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

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

## Contents |

The Rule Interchange Format (RIF), specifically the Basic Logic Dialect (BLD) [ RIF-BLD ], defines a means for the interchange of (logical) rules over the Web. Rules which are exchanged using RIF may refer to external data sources and may be based on certain data models which are represented using a language different from RIF. The Resource Description Framework RDF [ RDF-Concepts ] is a Web-based language for the representation and exchange of data; RDF Schema (RDFS) [ RDF-Schema ] and the Web Ontology Language OWL [ OWL-Reference ] are Web-based languages for the representation and exchange of ontologies (i.e., data models). This document specifies how combinations of RIF BLD Rule sets and RDF data and RDFS and OWL ontologies must be interpreted.

The RIF working group plans to develop further dialects besides BLD, most notably a dialect based on Production Rules [ RIF-PRD ]; these dialects are not necessarily extensions of BLD. Future versions of this document will address compatibility of these dialects with RDF and OWL as well. In the remainder of the document, when mentioning RIF, we mean RIF BLD [ RIF-BLD ].

RDF data and RDFS and OWL ontologies are represented using
*RDF graphs*. Several syntaxes have been proposed for the
exchange of RDF graphs, the normative syntax being RDF/XML [
RDF-Syntax ], which is an
XML-based format. RIF does not provide a format for exchanging RDF;
instead, it is assumed that RDF graphs are exchanged using RDF/XML,
or any other syntax that can be used for representing or exchanging
RDF graphs.

This document does not, as yet, define whether or how RDF documents/graphs should be referred to from RIF rule sets. The specification of combinations in this document does not depend on (the existence of) this mechanism: it applies in case an RIF rule sets explicity points to (one or more) RDF documents, but also in case the references to the RDF document(s) are not interchanged using RIF, but using some other (out of bounds) mechanism.

A typical scenario for the use of RIF with RDF/OWL includes the
exchange of rules which either use RDF data or an RDFS or OWL
ontology. In terms of rule interchange the scenario is the
following: interchange partner *A* has a rules language which
is RDF/OWL-aware, i.e., it allows to use RDF data, it uses an RDFS
or OWL ontology, or it extends RDF(S)/OWL. *A* sends its rules
(using RIF), possibly with a reference to the appropriate RDF
graph(s), to partner *B*. *B* can now translate the RIF
rules into its own rules language, retrieve the RDF graph(s) (which
is published most likely using RDF/XML), and process the rules in
its own rule engine, which is also RDF/OWL-aware. The use case
Vocabulary Mapping for Data Integration [ RIF-UCR ] is an example of the
interchange of RIF rules which use RDF graphs.

A specialization of this use case is the publication of RIF
rules which refer to RDF graphs (notice that publication is a
specific kind of interchange). In such a scenario, a rule publisher
*A* publishes its rules on the Web. There may be several
consumers who retrieve the RIF rules and RDF graphs from the Web,
and translate the RIF rules to their own rules languages. The use
case
Publishing Rules for Interlinked Metadata [ RIF-UCR ] is an example of the
publication of RIF rules related to RDF graphs.

Another specialization of this use case is the
interchange of rule extensions to OWL [ RIF-UCR ]. The intention of the rule publisher in this
scenario is to extend an OWL ontology with rules: interchange
partner *A* has a rules language that extends OWL. *A*
splits its ontology+rules description into a separate OWL ontology
and an RIF rule set, publishes the OWL ontology, and sends (or
publishes) the RIF rule set, which includes a reference to the OWL
ontology. The consumers of rules retrieves the OWL ontology, and
translates the ontology and the rule set into a combined
ontology+rules description in its own rules language which also
extends OWL.

An RIF rule set which refers to RDF graphs, or any use of an RIF
rule set with RDF graphs, is viewed as a combination of an RIF rule
set and a number of RDF graphs. This document specifies how, in
such a combination, the rule set interacts with the RDF graphs.
With "interaction" we mean the conditions under which the
combination is satisfiable, as well as the entailments defined for
the combination. The interaction between RIF and RDF and OWL is
realized by connecting the model theory of RIF (specified in [
RIF-BLD ]) with the model
theories of RDF (specified in [ RDF-Semantics ]) and OWL (specified in [ OWL-Semantics ]),
respectively.

The RDF semantics specification [ RDF-Semantics ] defines 4 notions of entailment for RDF graphs. At this stage it has not yet been decided which of these notions are of interest in RIF. Therefore, we specify the interaction between RIF and RDF for all 4 notions.

**EDITOR'S NOTE:** Currently, this document only defines how
combinations of RIF rule sets and RDF/OWL should be interpreted; it
does not suggest how references to RDF graphs are specified in RIF,
nor does it specify which of the RDF entailment regimes (simple,
RDF, RDFS, or D) should be used. Possible ways to refer to RDF
graphs and RDFS/OWL ontologies include annotations in RIF rule sets
and extensions of the syntax of RIF. Note that no agreement has yet
been reached on this issue, and that especially the issue of the
specification of entailment regimes is controversial (see
http://lists.w3.org/Archives/Public/public-rif-wg/2007Jul/0030.html
and the ensuing thread). See the Annotations
page for a proposal for extending RIF with annotations.

The #Appendix: Embedding RDF Combinations describes how
reasoning with combinations of RIF rules with RDF can be reduced to
reasoning with RIF rule sets, which can be seen as a guide to
describing how an RIF processor could be turned into an RDF-aware
RIF processor. This reduction can be seen as a guide for
interchange partners which do not have RDF-aware rule systems, but
still want to be able to process RIF rules which refer to RDF
graphs. In terms of the scenario above: if the interchange partner
*B* does not have an RDF-aware rule system, but *B* can
process RIF rules, then the appendix explains how the rule system
could be used for processing combinations.

**EDITOR'S NOTE:** The future status of the appendix with the
embedding is
uncertain. The appendix is not about interchange, but rather
about possible implementation, so it can be argued that it should
not be included in this document. On the other hand, many think the
appendix is useful. If we decide not to include it in this
document, we might consider publishing it as a separate note (not
recommendation-track document).

When speaking about RDF compatibility in RIF, we speak about RIF-RDF combinations, which are combinations of RIF rule sets and sets of RDF graphs. This section specifies how, in such a combination, the rule set and the graphs interact. In other words, how rules can "access" data in the RDF graphs and how additional conclusions which may be drawn from the RIF rules are reflected in the RDF graphs.

There is a correspondence between constant symbols in RIF rule sets and names in RDF graphs. The following table explains the correspondences of symbols.

RDF Symbol | Example | RIF Symbol | Example |
---|---|---|---|

Absolute IRI | <http://www.w3.org/2007/rif> |
Absolute IRI | "http://www.w3.org/2007/rif"^^rif:iri |

Plain literal without a language tag | "literal string" |
String in the symbol space xsd:string |
"literal string"^^xsd:string |

Plain literal with a language tag | "literal string"@en |
String plus language tag in the symbol space
rif:text |
"literal string@en"^^rif:text |

Literal with a datatype | "1"^^xsd:integer |
Symbol in a symbol space | "1"^^xsd:integer |

There is, furthermore, 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 which contains the triples

john brotherOf jack . jack parentOf mary .

saying that `john` is a brother of `jack` and
`jack` is a parent of `mary`, and an RIF rule set
which contains the rule

Forall ?x, ?y, ?z (?x["uncleOf"^^rif:iri -> ?z] :- And( ?x["brotherOf"^^rif:iri -> ?y] ?y["parentOf"^^rif:iri -> ?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 we can
derive the RIF frame formula `"john"^^rif:iri["uncleOf"^^rif:iri
-> "mary"^^rif:iri]`, as well as the RDF triple `john
uncleOf marry`.

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 an RIF rule using a variable in a rule.

Typed literals in RDF may be *ill-typed*, which means that
the literal string is not part of the lexical space of the datatype
under consideration. Examples of such ill-typed literals are
`"abc"^^xsd:integer`, `"2"^^xsd:boolean`, and
`"<non-valid-XML"^^rdf:XMLLiteral`. Rules which include
ill-typed symbols are not well-formed RIF rules, so there are no
RIF symbols which correspond to ill-typed literals. However,
variables may quantify over such literals. The following example
illustrates the interaction between RDF and RIF in the face of
ill-typed literals and blank nodes.

Consider a combination of an RDF graph which contains the triple

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

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

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

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

From this combination we can derive the RIF condition formulas

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

as well as the RDF triples

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

However, `"http://a"^^rif:iri["http://p"^^rif:iri ->
"a"^^xsd:integer]` cannot be derived, because it is not a
well-formed RIF formula, due to the fact that `"a"` is not
an integer; it is not in the lexical space of the datatype
`xsd:integer`.

This remainder of this section formally defines combinations of RIF
rules with RDF graphs and the semantics of such combinations.
Combinations are pairs of RIF rule sets and sets 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 definitions related to datatypes and ill-typed literals are reviewed. Finally, RIF-RDF combinations are formally defined.

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

- absolute
IRIs
*V*, (correspond to the Concepts and Abstract Syntax term "_{U}`RDF URI references`"; see the [ End note on RDF URI references ])

- plain
literals
*V*(i.e., character strings with an optional language tag), and_{PL}

- typed
literals
*V*(i.e., pairs of character strings and datatype IRIs)._{TL}

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

**DEFINITION:**Given an RDF vocabulary*V*, a*generalized RDF graph*of*V*is a set of*generalized RDF triples*`s p o .`, where`s`,`p`and`o`are names in*V*or blank nodes.

(See the [ End note on generalized RDF graphs ])

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

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

RDFS, specifically D-entailment, allows the use of arbitrary
datatype maps, as long as the `rdf:XMLLiteral` datatype is
considered. RIF BLD additionally requires the following datatypes
to be considered: `xsd:long`, `xsd:string`,
`xsd:integer`, `xsd:decimal`, `xsd:time`,
`xsd:dateTime`, and `rif:text`; we call these
datatypes the *RIF-required
datatypes*. We define the notion of a conforming datatype
map as a datatype map which recognizes at least the RIF-required
datatypes.

**DEFINITION:**A datatype map D is a*conforming datatype map*if it satisfies the following conditions- 1. No RIF-supported symbol space which is not an RIF-required
datatype (these are
`rif:local`and`rif:iri`in RIF BLD) is in the domain of D. - 2. The IRIs identifying all RIF-required datatypes are in the domain of D.
- 3. D maps IRIs identifying XML schema datatypes to the
respective data
types [ XML-SCHEMA2 ],
`rdf:XMLLiteral`to the rdf:XMLLiteral datatype [ RDF-Concepts ], and`rif:text`to the rif:text primitive datatype [ RIF-BLD ].

We now define the notions of well- and ill-typed literals, which loosely correspond to the notions of well-formed and ill-formed symbols in RIF.

**DEFINITION:**Given a conforming datatype map D, a typed literal (`s`,`d`) is a*well-typed literal*if- 1.
`d`is in the domain of D and`s`is in the lexical space of D(`d`), - 2.
`d`is the IRI of a symbol space supported by RIF BLD and`s`is in the lexical space of the symbol space, or - 3.
`d`is not in the domain of D and does not identify a symbol space supported by RIF.

Otherwise (`s`, `d`) is an *ill-typed literal*.

We now formally define combinations.

**DEFINITION:**An*RIF-RDF combination*is a pair <*R*,**S**>, where*R*is a*Rule set*and**S**is a set of generalized RDF graphs of a vocabulary*V*.

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

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

The RDF Semantics document [ RDF-Semantics ] defines 4 (normative) kinds of interpretations, as well as corresponding notions of satisfiability and entailment:

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

This distinction is reflected in the definitions of satisfaction and entailment in this section.

We define the notion of *common interpretation*, which is
an interpretation of an RIF-RDF combination. This common
interpretation is the basis for the definitions satisfaction and
entailment in the following sections.

The correspondence between RIF semantic structures and RDF
interpretations is defined through a number of conditions which
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.

We first review the notions of RDF interpretations and RIF semantic structures, after which we define common interpretations.

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

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

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

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

As defined in [ RIF-BLD ], a
semantic structure is a tuple of the form * I* =
<

is the set of datatypes, which have associated datatype identifiers,**DTS**is a non-empty set (the domain),**D****I**_{C}is a mapping from`Const`to,**D****I**_{V}is a mapping from`Var`to, and**D****I**_{frame}is (in BLD) a mapping fromto truth-valued functions of the form**D**×**D**→**D**.**TV**

**DEFINITION:**A*common interpretation*is a pair (, I), where**I**= <**I**,**TV**,**DTS**,**D****I**_{C},**I**_{V},**I**_{F},**I**_{frame},**I**_{SF},**I**_{sub},**I**_{isa},**I**_{=},**I**_{Truth}> is an RIF semantic structure and I=<IR, IP, IEXT, IS, IL, LV> is an RDF interpretation of a vocabulary*V*, such that the following conditions hold- 1. IR is a subset of
;**D** - 2. IP is a superset of the set of all
`k`insuch that there exist**D**`a`,`b`inand**D****I**_{frame}(`k`)(`a`,`b`)=t (i.e. the truth value of**I**_{frame}(`k`)(`a`,`b`) is*true*); - 3. (IR union IP) =
;**D** - 4. LV is a subset of IR and a superset of (
intersection (union of all value spaces**D**));**DS** - 5. IEXT(k) = the set of all pairs (
`a`,`b`), with`a`,`b`in, such that**D****I**_{slot}(`k`)(`a`,`b`)=**t**, for every`k`in;**D** - 6. IS(
`i`) =**I**_{C}(`"i"^^rif:iri`) for every absolute IRI`i`in*V*;_{U} - 7. IL((
`s`,`d`)) =**I**_{C}(`"s"^^d`) for every well-typed literal (`s`,`d`) in*V*;_{TL} - 8. IEXT(IS(
`rdf:type`)) is equal to the set of all pairs`<a,b>`in×**D**such that**D****I**_{isa}(< a,b >)=t; and - 9. IEXT(IS(
`rdfs:subClassOf`)) is a superset of the set of all pairs`<a,b>`in×**D**such that**D****I**_{sub}(< a,b >)=t.

Condition 1 ensures that all resources in an RDF interpretation
correspond to elements in the RIF domain. Condition 2 ensures that
the set of properties at least includes all elements which are used
as properties in the RIF domain. Condition 3 ensures that the
combination of resources and properties corresponds exactly to the
RIF domain; note that if I is an rdf-, rdfs-, or D-interpretation,
IP is a subset of IR, and thus IR=* D*. Condition 4
ensures that all concrete values in

One consequence of conditions 6 and 7 is that IRIs of the form
`http://iri` and typed literals of the form
`"http://iri"^^rif:iri` which occur in an RDF
graph are treated the same in RIF-RDF combinations, even if the RIF
component is empty. For example, consider an RIF-RDF combination
with an empty rule set and an RDF graph which contains the
triple

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

This combination allows to derive, among other things, the following triples:

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

as well as the following frame formula:

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

We now define the notion of satisfiability for common
interpretations, i.e., the conditions under which a common
interpretation (* I*, I) is a model of a combination
<

**DEFINITION:**A common interpretation (, I)**I***simple-satisfies*an RIF-RDF combination C=<*R*,**S**> ifsatisfies**I***R*and I satisfies every RDF graph*S*in**S**; in this case (, I) is called a**I***simple model*, or*model*, of C, and C is*satisfiable*. (, I) satisfies a generalized RDF graph**I***S*if I satisfies*S*. (, I) satisfies a closed RIF condition formula φ if**I****I**_{truth}(φ)=*t*.

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

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

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

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

**DEFINITION:**A model (, I) of a combination C**I***rdf-satisfies*C if I is an rdf-interpretation; in this case (, I) is called an**I***rdf-model*of C, and C is*rdf-satisfiable*.**DEFINITION:**A model (, I) of a combination C**I***rdfs-satisfies*C if I is an rdfs-interpretation; in this case (, I) is called an**I***rdfs-model*of C, and C is*rdfs-satisfiable*.**DEFINITION:**Given a conforming datatype map D, a model (, I) of a combination C**I***D-satisfies*C if I is a D-interpretation; in this case (, I) is called a**I***D-model*of C, and C is*D-satisfiable*.

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

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

The other notions of entailment are defined analogously:

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

Two kinds of combinations of RIF rules with OWL ontologies are considered. The combination of RIF rules with the Full species of OWL is a straightforward extension of RIF-RDF compatibility (see the definition below), in which RDF triples correspond to RIF frame formulas. The combination of RIF rules with the DL and Lite species of OWL is slightly different; OWL classes and properties correspond to RIF unary and binary predicates, respectively. The discrepancy between the two kinds of combinations is overcome by interpreting frame formulas as unary and binary predicates and imposing certain restrictions on the use of variables in the rules.

OWL [ OWL-Reference ] specifies three increasingly expressive species, namely Lite, DL, and Full.

OWL Lite is a syntactic subset of OWL DL, but the semantics is the same [ OWL-Semantics ]. Since every OWL Lite ontology is an OWL DL ontology, the Lite species is not explicitly considered in the remainder.

Syntactically speaking, OWL DL is a subset of OWL Full. The semantics of the DL and Full species are different, though [ OWL-Semantics ]. While OWL DL has an abstract syntax with a direct model-theoretic semantics, the semantics of OWL Full is an extension of the semantics of RDFS, and is defined on the RDF syntax of OWL. Consequently, the OWL Full semantics does not extend the OWL DL semantics; however, every OWL DL entailment is an OWL Full entailment.

Finally, the OWL DL RDF 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.

Note that the abstract syntax form of OWL DL allows so-called
*punning* (this is not allowed in the RDF syntax), i.e., the
same IRI may be used in an individual position, a property
position, and a class position; the interpretation of the IRI
depends on its context. Since combinations of RIF and OWL DL are
based on the abstract syntax of OWL DL, punning may also be used in
these combinations. This paves the way towards combination with
OWL 1.1, which is envisioned to
allow punning in all its syntaxes.

Since RDF graphs and OWL Full ontologies cannot be distinguished, we use the notion of RIF-RDF combinations for the syntax of combinations of RIF rule sets with OWL Full ontologies.

For the combination of RIF rule sets with OWL DL ontologies we define the notion of RIF-OWL DL combinations based on the abstract syntax of OWL DL. We need to furthermore impose certain restrictions on the syntax of the rules. Specifically, we can only allow constant symbols in class and property positions.

**DEFINITION:**An RIF rule set*R*is a*DL Rule set*if for every frame formula`a [ b -> c ]`in every rule of*R*it holds that`b`is a constant and if`b`=`rdf:type`, then`c`is a constant.

**DEFINITION:**An*RIF-OWL DL combination*is a pair <*R*,**O**>, where*R*is a*DL Rule set*and**O**is a set of OWL DL ontologies in abstract syntax form of a vocabulary*V*.

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

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

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

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

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

**DEFINITION:**Given a conforming datatype map D, an RIF-RDF combination C*OWL Full entails*a generalized RDF graph*S*if every OWL Full model of C satisfies*S*. Likewise, C*OWL Full entails*a closed RIF condition formula φ if every OWL Full model of C satisfies φ.

The semantics of RIF-OWL DL combinations is similar in spirit to the semantics of RIF-RDF combinations. We define a notion of common interpretations, which are pairs of RIF and OWL DL interpretations, and define a number of conditions which relate these interpretations. In contrast to RIF-RDF combinations, the conditions below define a correspondence between the interpretation of OWL DL classes and properties and RIF unary and binary predicates.

It is now the case that elementary class and property statements
in OWL DL of the forms `A` and `P` correspond to the
unary and binary predicates expressions in RIF of the forms
`A(?x)` and `P(?x,?y)`, whereas elementary statements
in OWL Full, which are triples, correspond to frame formulas in
RIF, e.g., a class membership statement `x rdf:type A`
corresponds to `x[rdf:type -> A]`. Therefore, rules which
essentially express the same thing will look quite different,
depending on whether they are used in OWL DL and OWL Full
ontologies. For example, in an RIF-OWL DL combination, the uncle
rule looks something like:

hasUncle(?x,?y) :- And(hasParent(?x,?y) hasBrother(?y,?z))

whereas, in an RIF-OWL Full combination, the rule will look something like:

?x[hasUncle -> ?y] :- And(?x[hasParent -> ?y] ?y[hasBrother -> ?z])

To overcome this problem we define a slightly modified semantics
for RIF rules to enable the use of the latter kind of rules in
RIF-OWL DL combinations. The modified semantics essentially
corresponds to a rewriting of atomic formulas of the form
`x[rdf:type -> y]` to `y(x)` and `x[p ->
y]` to `p(x, y)`.

We define a new truth valuation function for RIF formulas, which is the same as the truth valuation function defined in RIF-BLD, with the exception of frame formulas.

**DEFINITION: Given an RIF semantic structure***I***= <***TV***,***DTS***,***D***,***I*_{C},*I*_{V},*I*_{F},*I*_{frame},*I*_{SF},*I*_{sub},*I*_{isa},*I*_{=},*I*_{Truth}>, the truth valuation function*I*_{T-DL}is obtained by modifying the truth valuation of frame formulas in*I*_{Truth}in the following way:*I*_{T-DL}(`t [ rdf:type -> A ]`) =*I*_{R}(`A`)(`t`) and*I*_{T-DL}(`t`) =_{1}[ P -> t_{2}]*I*_{R}(`P`)(`t`)._{1}, t_{2}

**DEFINITION: We say that***I*Q*DL satisfies*a ruleQ*then*:-*if*, where**is a quantification prefix for all the variables in the rule, if***I*^{*}_{T-DL}(then) ≥*I*^{*}_{T-DL}(if) for every*I*^{*}that agrees with*I***everywhere except possibly on some variables mentioned in**Q**.***I***is a**R*DL model*of a rule set**if it DL satisfies every rule in the set.**

**DEFINITION:**Given a conforming datatype map D, a*common DL interpretation*is a pair (, I), where**I**= <**I**,**TV**,**DTS**,**D****I**_{C},**I**_{V},**I**_{F},**I**_{frame},**I**_{SF},**I**_{sub},**I**_{isa},**I**_{=},**I**_{Truth}> is an RIF semantic structure and I=<R, EC, ER, L, S, LV> is an abstract OWL interpretation with respect to D of a vocabulary*V*, such that the following conditions hold- 1. R=
;**D** - 2. LV is a subset of R and contains the value spaces of all data types in D;
- 3. EC(
`u`) = set of all objects`k`in R such that**I**_{R}(`"u"^^rif:iri`)(`k`) =**t**(true), for every IRI`u`in*V*; - 4. ER(
`u`) = set of all tuples (`k`,`l`) such that**I**_{R}(`"u"^^rif:iri`)(`k`,`l`) =**t**(true), for every data valued and individual valued property identifier`u`in*V*; - 5. L((
`s`,`d`)) =**I**_{C}(`"s"^^d`) for every well-typed literal (`s`,`d`) in*V*; - 6. S(
`i`) =**I**_{C}(`"i"^^rif:iri`) for every IRI`i`in*V*.

Condition 1 ensures that the domains of interpretation are the
same. Condition 2 ensures that the set of literal values includes
the value spaces of all considered datatypes. Condition 3 ensures
that the interpretation (extension) of an OWL DL class `u`
corresponds to the interpretation of the unary predicate with the
same name in RIF. Condition 4 ensures that the interpretation
(extension) of an OWL DL object or datatype property `u`
corresponds to the interpretation of the binary predicates with the
same name in RIF. 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 rule sets are interpreted in the same
way.

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

**DEFINITION:**Given a conforming datatype map D, a common DL interpretation (, I)**I***OWL DL satisfies*an RIF-OWL DL combination C=<*R*,**O**> ifDL satisfies**I***R*and I satisfies every OWL DL ontology in abstract syntax form*O*in**O**; in this case (, I) is called an**I***OWL DL model*of C, and C is*OWL DL satisfiable*. (, I) satisfies an OWL DL ontology in abstract syntax form**I***O*if I satisfies*O*. (, I) satisfies a closed RIF condition formula φ if**I****I**_{T-DL}(φ)=**t**.

**DEFINITION:**Given a conforming datatype map D, an RIF-OWL DL combination C*OWL DL entails*an OWL DL ontology in abstract syntax form*O*if every OWL DL model of C satisfies*S*. Likewise, C*OWL DL entails*a closed RIF condition formula φ if every OWL DL model of C satisfies φ.

Note that the above definition of RIF-OWL DL compatibility does not consider ontology and annotation properties, in contrast to the definition of compatibility of RIF with OWL Full, where there is no clear distinction between annotation and ontology properties and other kinds of properties. Therefore, it is not possible to "access" or use the values of these properties in the RIF rules. This limitation is overcome in the following definition. It is envisioned that the user will choose whether annotation and ontology properties are to be considered. It is noted that it is envisioned that OWL 1.1 will not define a semantics for annotation and ontology properties; therefore, the below definition cannot be extended to the case of OWL 1.1.

**DEFINITION:**Given a conforming datatype map D, a common DL interpretation (, I) is a**I***common DL annotation interpretation*if the following condition holds- 7. ER(
`u`) = set of all tuples (`k`,`l`) such that**I**_{R}(`"u"^^rif:iri`)(`k`,`l`) =**t**(true), for every IRI`u`in*V*.

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

**DEFINITION:**Given a conforming datatype map D, a common DL annotation interpretation (, I)**I***OWL DL annotation satisfies*an RIF-OWL DL combination C=<*R*,**O**> ifsatisfies**I***R*and I satisfies every OWL DL ontology in abstract syntax form*O*in**O**; in this case (, I) is called an**I***OWL DL annotation model*of C, and C is*OWL DL annotation satisfiable*.

**DEFINITION:**Given a conforming datatype map D, an RIF-RDF combination C*OWL DL annotation entails*an OWL DL ontology and abstract syntax form*O*if every OWL DL annotation model of C satisfies*O*. Likewise, C*OWL DL annotation entails*a closed RIF condition formula φ if every OWL DL annotation model of C satisfies φ.

We illustrate the difference between the two kinds of OWL DL entailment 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 rule set which consists 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 rule set OWL DL
annotation entails the RIF condition formula
`ex:myOntology[ex:hasTitle -> "Example
ontology"^^xsd:string]`; the combination does not OWL DL entail
the formula.

**[OWL-Semantics]***OWL Web Ontology Language Semantics and Abstract Syntax*, P. F. Patel-Schneider, P. Hayes, I. Horrocks, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-semantics-20040210/. Latest version available at http://www.w3.org/TR/owl-semantics/.- [RDF-CONCEPTS]
*Resource Description Framework (RDF): Concepts and Abstract Syntax*, G. Klyne, J. Carroll (Editors), W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/. Latest version available at http://www.w3.org/TR/rdf-concepts/.- [RDF-SEMANTICS]
*RDF Semantics*, P. Hayes, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-mt-20040210/. Latest version available at http://www.w3.org/TR/rdf-mt/.- [RIF-BLD]
*RIF Basic Logic Dialect*, H. Boley, M. Kifer (Editors), W3C Editor's Draft, http://www.w3.org/2005/rules/wg/wiki/BLD. Accessed on 2008-02-13T17:00 UTC.- [XML-SCHEMA2]
*XML Schema Part 2: Datatypes*, W3C Recommendation, World Wide Web Consortium, 2 May 2001. This version is http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/. Latest version available at http://www.w3.org/TR/xmlschema-2/.

- [RDF-Schema]
*RDF Vocabulary Description Language 1.0: RDF Schema*, D. Brickley, R.V. Guha, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-schema-20040210/. Latest version available at http://www.w3.org/TR/rdf-schema/.- [RDF-SYNTAX]
*RDF/XML Syntax Specification (Revised)*, D. Beckett, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-syntax-grammar-20040210/. Latest version available at http://www.w3.org/TR/rdf-syntax-grammar/.- [RFC-3066]
*RFC 3066 - Tags for the Identification of Languages*, H. Alvestrand, IETF, January 2001. This document is http://www.isi.edu/in-notes/rfc3066.txt.- [RIF-PRD]
*RIF Production Rule dialect*, C. de Sainte Marie (Editors), Editor's Draft. Latest version available at http://www.w3.org/2005/rules/wg/wiki/PRdialect.- [RIF-UCR]
*RIF Use Cases and Requirements*, A. Ginsberg, D. Hirtle, F. !McCabe, P.-L. Patranjan (Editors), W3C Working Draft, 10 July 2006, http://www.w3.org/TR/2006/WD-rif-ucr-20060710/. Latest version available at http://www.w3.org/TR/rif-ucr.- [OWL-Reference]
*OWL Web Ontology Language Reference*, M. Dean, G. Schreiber, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-ref-20040210/. Latest version available at http://www.w3.org/TR/owl-ref/.

RIF-RDF combinations can be embedded into RIF Rule sets in a fairly straightforward way, thereby demonstrating how an RIF-compliant translator without native support for RDF can process RIF-RDF combinations.

For the embedding we use the concrete syntax of RIF and the N-Triples syntax for RDF.

Throughout this section the function tr is defined, which maps symbols, triples, and RDF graphs to RIF symbols, statements, and rule sets.

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.

RDF Symbol | RIF Symbol | Mapping |
---|---|---|

IRI i in V_{U} |
Constant with symbol space rif:iri |
tr(i) = "i"^^rif:iri |

Blank node x<em> in
<em>B<em></td> |
Variable symbols ?<em>x |
tr(x) = ?x |

Plain literal without a language tag xxx in
V_{PL} |
Constant with the datatype xsd:string |
tr("xxx") = "xxx"^^xsd:string |

Plain literal with a language tag (xxx,lang)
in V_{PL} |
Constant with the datatype rif:text |
tr("xxx"@lang) = "xxx@lang"^^rif:text |

Well-typed literal (s,u) in
V_{TL} |
Constant with the symbol space u |
tr("s"^^u) = "s"^^u |

Ill-typed literal (s,u) in
V_{TL} |
Constant s^^u' with symbol space rif:local
which is not used in C |
tr("s"^^u) = "s^^u'"^^rif:local |

The embedding is not defined for combinations which include RDF graphs with RDF URI references which are not absolute IRIs.

The mapping function tr is extended to embed triples as RIF
statements. Finally, two embedding functions, tr_{R} and
tr_{Q} embed RDF graphs as RIF rule sets and conditions,
respectively. The following section shows how these embeddings can
be used for reasoning with combinations.

We define two mappings for RDF graphs, one (tr_{R}) in
which variables are Skolemized, i.e. replaced with constant
symbols, and one (tr_{Q}) in which variables are
existentially quantified.

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

</table>

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

**Theorem** A combination C=<R,{S1,...,Sn}>
simple-entails a generalized RDF graph S iff (R union
tr_{R}(S1) union ... union tr_{R}(Sn)) entails
tr_{Q}(S). C simple-entails an RIF condition φ iff (R union
tr_{R}(S1) union ... union tr_{R}(Sn)) entails
φ.

The embeddings of RDF and RDFS entailment require a number of built-in predicate symbols to be available to appropriately deal with literals.

**EDITORS NOTE:** It is not yet clear which built-in
predicates will be available in RIF. Therefore, the built-ins
mentioned in this section may change. Furthermore, built-ins may be
axiomatized if they are not provided by the language.

Given a vocabulary *V*,

- the unary predicate wellxml
_{V}/1 is interpreted as the set of XML values, - the unary predicate illxml
_{V}/1 is interpreted as the set of objects corresponding to ill-typed XML literals in*V*, and_{TL} - the unary predicate illD
_{V}/1 is interpreted as the set of objects corresponding to ill-typed literals in*V*, and_{TL} - the unary predicate lit/1 is interpreted as the union of the value spaces of all data types.

We axiomatize the semantics of the RDF vocabulary using the following RIF rules and conditions.

The compact URIs used in the RIF rules in this section and the next are short for the complete URIs with the rif:iri datatype, e.g. rdf:type is short for "http://www.w3.org/1999/02/22-rdf-syntax-ns#type"^^rif:iri

RDF Construct | RIF Construct | Mapping |
---|---|---|

Triple s p o . |
Property frame tr(s)[tr(p)
-> tr(o)] |
tr(s p o .) = tr(s)[tr(p)
-> tr(o)] |

Graph S | Rule set tr_{R}(S) |
tr_{R}(S) = the set of all sk(Forall tr(s p o
.)) such that s p o . is a triple in S |

Graph S | Condition (query) tr_{Q}(S) |
tr_{Q}(S) = Exists tr(x1<tt>),
..., </tt>tr(xn<tt>)
And(</tt>tr(t1<tt>) ...
</tt>tr(tm<tt>))</tt>, where
x1, ..., xn are the blank nodes occurring in S
and t1, ..., tm are the triples in S |

R^{RDF} |
= | (Forall tr(s p o .)) for every <a
href="http://www.w3.org/TR/rdf-mt/#RDF_axiomatic_triples">RDF
axiomatic triple</a> s p o .) union( Forall ?x "1"^^xsd:integer="2"^^xsd:integer :-
And(?x[rdf:type -> rdf:XMLLiteral] illxml(?x))) |

**Theorem** A combination <R,{S1,...,Sn}> is
rdf-satisfiable iff (*R ^{RDF}* union R union
tr

**Theorem** A combination C=<R,{S1,...,Sn}> rdf-entails
a generalized RDF graph T iff (*R ^{RDF}* union R union
tr

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

R^{RDFS} |
= | R union^{RDF}( Forall ?x "1"^^xsd:integer="2"^^xsd:integer :-
And(?x[rdf:type -> rdfs:Literal] illxml(?x))) |

**Theorem** A combination <R_{1},{S1,...,Sn}> is
rdfs-satisfiable iff (*R ^{RDFS}* union R

**Theorem** A combination <R,{S1,...,Sn}> rdfs-entails
generalized RDF graph T iff (*R ^{RDFS}* union R union
tr

We axiomatize the semantics of the data types using the following RIF rules and conditions.

R^{D} |
= | R union^{RDFS}( Forall ?x "1"^^xsd:integer="2"^^xsd:integer :-
And(?x[rdf:type -> rdfs:Literal] illD(?x))`)) |

**Theorem** A combination <R,{S1,...,Sn}>, where R does
not contain the equality symbol, is D-satisfiable iff
(*R ^{D}* union R union tr

**EDITOR'S NOTE:** Since this condition is very complex we
might consider discarding this theorem, and suggest the above set
of rules (*R ^{D}*) as an approximation of the
semantics.

**Theorem** A D-satisfiable combination <R,{S1,...,Sn}>,
where R does not contain the equality symbol, D-entails a
generalized RDF graphs T iff (*R ^{D}* union R union
tr

**EDITOR'S NOTE:** The restriction to equality-free rule sets
is necessary because D-interpretations impose stronger conditions
on the interpretation of typed literals in case different datatype
URIs are equal than RIF does.

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

**Generalized RDF
graphs**: Standard RDF
graphs, as defined in [ RDF-Concepts ], do not allow the use of literals in subject
and predicate positions and blank nodes in predicate positions. The
RDF Core working
group has listed two issues
questioning the restrictions that literals
may not occur in subject and blank
nodes may not occur in predicate positions in triples.
Anticipating lifting of these restrictions in a possible future
version of RDF, we use the more liberal notion of
*generalized* RDF graph. We note that the definitions of
interpretations, models, and entailment in the RDF semantics
document [ RDF-Semantics
] also apply to such generalized RDF graphs.

We note that every standard RDF graph is a generalized RDF graph. Therefore, our definition of combinations applies to standard RDF graphs as well. We note also that the notion of generalized RDF graphs is more liberal than the notion of RDF graphs used by SPARQL; generalized RDF graphs additionally allow blank nodes and literals in predicate positions.