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The Rule Interchange Format (RIF) Basic Logic Dialect (BLD)
(RIF-BLD) is a format for
interchanging logical rules over the Web. Rules that are exchanged
using RIF may refer to external data sources and may be based on
data models that are represented using a language different from
RIF. The Resource Description Framework RDF (RDF-Concepts) is a Web-based language for the
representation and exchange of data; RDF Schema
(RDFS) (RDF-Schema) and the
OWL Web Ontology Language
(OWL-Reference) are
Web-based languages for representing and exchanging ontologies
(i.e., data models). This document specifies how combinations of
RIF BLD documents and RDF data and RDFS and OWL ontologies are
interpreted; i.e., it specifies how RIF interoperates with RDF/OWL.RDF,
RDFS, and OWL.
The RIF working group plans to develop further dialects besides BLD, most notably a dialect based on Production Rules (RIF-PRD); these dialects are not necessarily extensions of BLD. Future versions of this document may address compatibility of these dialects with RDF and OWL. In the remainder, RIF is understood to refer to RIF BLD (RIF-BLD).
RDF data and RDFS and OWL ontologies are represented using
RDF graphs. Several syntaxes have been proposed for the
exchange of RDF graphs, the normative syntax being RDF/XML
(RDF-Syntax). RIF does not
provide a format for exchanging RDF graphs, since this would be a duplication. Instead,graphs; it is assumed that RDF
graphs are exchanged using RDF/XML, or any other syntax that can be
used for representing or exchanging RDF graphs.
A typical scenario for the use of RIF with RDF/OWL is the
exchange of rules that use RDF data or anand/or RDFS or OWL ontology:ontologies:
an interchange partner A has a rules language that is
RDF/OWL-aware, i.e., it supports the use of RDF data, it uses an
RDFS or OWL ontology, or it extends RDF(S)/OWL. A sends its
rules using RIF, possibly with references to the appropriate RDF
graph(s), to partner B. B receives the rules and
retrieves the referenced RDF graph(s) (published as, e.g., RDF/XML
(RDF-SYNTAX)). The rules are
translated to the internal rules language of B and are
processed, together with the RDF graphs, using the RDF/OWL-aware
rule engine of B. The use case Vocabulary Mapping for Data Integration (RIF-UCR) is an example of the
interchange of RIF rules that use RDF data and RDFS ontologies.
A specialization of this scenario is the publication of RIF rules that refer to RDF graphs; publication is a special kind of interchange: one to many, rather than one-to-one. When a rule publisher A publishes its rules on the Web, it is hoped that there are several consumers that retrieve the RIF rules and RDF graphs from the Web, translate the RIF rules to their own rules language, and process them together with the RDF graphs in their own rules engine. The use case Publishing Rules for Interlinked Metadata (RIF-UCR) illustrates the publication scenario.
Another specialization of the exchange scenario is the Interchange of Rule Extensions to OWL (RIF-UCR). The intention of the rule publisher in this scenario is to extend an OWL ontology with rules: interchange partner A has a rules language that extends OWL. A splits its ontology+rules description into a separate OWL ontology and a RIF document, publishes the OWL ontology, and sends (or publishes) the RIF document, which includes a reference to the OWL ontology. A consumer of the rules retrieves the OWL ontology and translates the ontology and document into a combined ontology+rules description in its own rule extension of OWL.
A RIF document that refers to (imports) RDF graphs and/or
RDFS/OWL ontologies, or any use of a RIF document with RDF graphs,
is viewed as a combination of a document and a number of graphs and
ontologies. This document specifies how, in such a combination, the
document and the graphs and ontologies interoperate in a technical
sense, i.e., the conditions under which the combination is
satisfiable (i.e., consistent), as well as the entailments (i.e.,
logical consequences) of the combination. The interaction between
RIF and RDF/OWL is realized by connecting the model theory of RIF
(RIF-BLD) with the model
theories of RDF (RDF-Semantics) and OWL (OWL-Semantics), respectively.
The notation of certain symbols,symbols in RIF, particularly IRIs and plain
literals, in RIFis slightly different from the notation in RDF/OWL. This difference isThese
differences are illustrated in the Section Symbols in RIF Versus RDF/OWL.
The RDF semantics specification (RDF-Semantics) defines four notions of entailment for RDF graphs. The OWL semantics specification (OWL-Semantics) defines two notions of entailment for OWL ontologies, namely OWL Lite/DL and OWL Full. This document specifies the interaction between RIF and RDF/OWL for all six notions. The Section RDF Compatibility is concerned with the combination of RIF and RDF/RDFS. The combination of RIF and OWL is addressed in the Section OWL Compatibility. The semantics of the interaction between RIF and OWL DL is close in spirit to (SWRL).
RIF provides a mechanism for referring to (importing) RDF graphs
and a means for specifying the context of this import, which
corresponds to the intended entailment regime. The Section Importing RDF Graphsand OWL in
RIF specifies how such import statements are used for
representing RIF-RDF and RIF-OWL combinations.
The Appendix: Embeddings (Informative) describes how reasoning with combinations of RIF rules with RDF and a subset of OWL DL can be reduced to reasoning with RIF documents, which can be seen as a guide to describing how a RIF processor could be turned into an RDF/OWL-aware RIF processor. This reduction can be seen as a guide for interchange partners that do not have RDF-aware rule systems, but want to be able to process RIF rules that refer to RDF graphs. In terms of the aforementioned scenario: if the interchange partner B does not have an RDF/OWL-aware rule system, but B can process RIF rules, then the appendix explains how B's rule system could be used for processing RIF-RDF.
Throughout this document the following conventions are used when writing RIF and RDF statements in examples and definitions.
Where RDF/OWL has four kinds of constants: URI references (i.e., IRIs), plain literals without language tags, plain literals with language tags and typed literals (i.e., Unicode sequences with datatype IRIs) (RDF-Concepts), RIF has one kind of constants: Unicode sequences with symbol space IRIs (DTB).
Symbol spaces can be seen as groups of constants. Every datatype is a symbol space, but there are symbol spaces that are not datatypes. For example, the symbol space rif:iri groups all IRIs. The correspondence between constant symbols in RDF graphs and RIF documents is explained in Table 1.
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 (RIF-DTB), used throughout this
document, corresponds with the syntax for IRIs and plain literals
in (Turtle ).), a commonly used
syntax for RDF.
The correspondence between constant symbols in RDF graphs and RIF documents is explained in Table 1 . Table 1. Correspondence between RDF andRIF symbols.does not have a notion corresponding to RDF
Symbol Example RIF Symbol Example IRI <http://www.w3.org/2007/rif> IRI <http://www.w3.org/2007/rif> Plain literal without language tag "literal string" String "literal string" Plain literal with language tag "literal string"@en String plus language tag in symbol space rif:text "literal string@en"^^rif:text Constant "1"^^xs:integer Symbol in symbol space "1"^^xs:integer RIF does not have a notion corresponding to RDF blank nodes .blank nodes. RIF local symbols, written _symbolname, have
some commonality with blank nodes; like the blank node label, the
name of a local symbol is not exposed outside of the document.
However, in contrast to blank nodes, which are essentially
existentially quantified variables, RIF local symbols are
constant symbols. In many applications and deployment
scenarios, this difference may be inconsequential. However the
results will differ when an RDF graph is used in a non-assertional
context, such as in a query pattern.
Finally, variables in the bodies of RIF rules or in query patterns may be existentially quantified, and are thus similar to blank nodes; however, RIF BLD does not allow existentially quantified variables to occur in rule heads.
This section specifies how a RIF document interacts with a set of RDF graphs in a RIF-RDF 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,?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 somesomething, denoted here by a blank node thatnode,
which has the name "John", and a RIF document that
contains the rules
Forall?x,?x ?y ( ?x[rdf:type ->ex:nameBearer]ex:named] :- ?x[ex:hasName -> ?y] ) Forall?x,?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:nameBearerex:named and
http://a has a property http://p with value
y.
From this combination the following RIF condition formula can be derived:
Exists ?z ( And( ?z[rdf:type ->ex:nameBearer]ex:named] <http://a>[<http://p> -> ?z] ))
as can the following RDF triples:
_:y rdf:typeex:nameBearerex:named . <http://a> <http://p> "John" .
However, there is no RIF constant symbol t such that
t[rdf:type -> ex:nameBearer]ex:named] can be derived, because there
is no constant that represents the name-bearer.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 RIF-RDF combinations are formally defined. Finally, definitions related to datatypes and typed literals are reviewed.
An RDF vocabulary V consists of the following sets of names:
The syntax of the names in these sets is defined inSee RDF Concepts
and Abstract Syntax (RDF-Concepts ). Besides) for precise definitions of these concepts.
Besides the sets of 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 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 RIF-RDF combination consists of a RIF document and zero or more RDF graphs. Formally:
Definition. A RIF-RDF combination is a pair < R,S>, where R is a RIF document and S is a set of generalized RDF graphs of a vocabulary V. ☐
When clear from the context, RIF-RDF combinations are referred to simply as combinations.
Even though RDF allows the use of arbitrary datatype IRIs in typed literals, not all such datatype IRIs are recognized in the semantics. In fact, simple entailment does not recognize any datatype and RDF and RDFS entailment recognize only the datatype rdf:XMLLiteral. To facilitate discussing datatypes, and specifically datatypes supported in specific contexts (required for D-entailment), the notion of datatype maps (RDF-Semantics) is used.
A datatype map is a partial mapping from IRIs to datatypes.
RDFS, specifically D-entailment, allows the use of arbitrary
datatype maps, as long as therdf:XMLLiteral datatypeis includedin the domain
of the map. RIF BLD additionallyrequires the followinga number of additional datatypes to be
included: xs:string , xs:decimal , xs:time , xs:date , xs:dateTime , and rif:text ;included; these datatypesare the RIF-required datatypes .(RIF-DTB).
When checking consistency of a conforming datatype map iscombination <
R,S> or entailment of a datatype map that recognizes at least the RIF-required datatypes. Editor's Note:graph S or RIF
formula φ by a combination < R,S>, the listset of
required data typesconsidered
datatypes is to be replaced with a link. Definition. Let T bethe smallest set of considereddatatypes (cf. Section 5 of ( RIF-BLD )), i.e., Tthat
includes at leastall data typesRIF-required datatypes, all datatypes used in
R, all datatypes used in the combination under considerationdocuments imported into R, and all datatypes required for RIF-BLD ( RIF-DTB ).used in φ (when
considering entailment of φ).
Definition. Let T be a set of datatypes. A
datatype
map D is aconforming datatype mapwith
T if it satisfies the following conditions:
Editor's Note:Note that it follows from the terminology "considered datatype" might change ifdefinition that every datatype
used in the terminologyRIF document in the combination or the entailed RIF
formula (when considering entailment questions) is changedincluded in BLD.any
datatype map conforming to the notionset of well-typed literal loosely correspond withconsidered datatypes. There
may be datatypes used in an RDF graph in the notion of legal symbolcombination that this
not included in RIF:such a datatype map.
Definition. Given a conformingdatatype map D, a typed literal
(s, d) is a well-typed literal
if
The semantics of RIF documents and RDF graphs are defined in terms of model theories. The semantics ofRIF-RDF combinations is defined through a
combination of the RIF and RDF model theories, using a notion of
common models. These models are then used to define
satisfiability and entailment in the usual way. Combined entailment
extends both entailment in RIF and entailment in RDF.
The RDF Semantics document (RDF-Semantics) defines four normative kinds of interpretations, as well as corresponding notions of satisfiability and entailment:
Those four types of interpretations are reflected in the definitions of satisfaction and entailment in this section.
This section defines the notion of common-rif-rdf-interpretation, which is an interpretation of a RIF-RDF combination. This common-rif-rdf-interpretation is the basis for the definitions of satisfaction and entailment in the following sections.
The correspondence between RIF semantic
structures (interpretations) and RDF
interpretations is defined through a number of conditions that
ensure the correspondence in the interpretation of names (i.e.,
IRIs and literals) and formulas, i.e., the correspondence between
RDF triples of the form s p o and RIF frames of the form
s'[p' -> o'], where s', p', and
o' are RIF symbols corresponding to the RDF symbols
s, p, and o, respectively (cf. Table 1the
Section Symbols in RIF Versus RDF/OWL).
The notions of RDF interpretation and RIF semantic structure (interpretation) are briefly reviewed below.
As defined in (RDF-Semantics), a simple interpretation of a vocabulary V is a tuple I=< IR, IP, IEXT, IS, IL, LV >, where
Rdf-, rdfs-, and D-interpretations are simple interpretations that satisfy certain conditions:
As defined in (RIF-BLD), a semantic structure is a tuple of the form I = <TV, DTS, D, Dind, Dfunc, IC, IV, IF, Iframe, ISF, Isub, Iisa, I=, Iexternal, Itruth>. The specification of RIF-RDF compatibility is only concerned with DTS, D, IC, IV, Iframe, Isub, Iisa, and Itruth. The other mappings that are parts of a semantic structure are not used in the definition of combinations.
Recall that Const is the set of constant symbols and Var is the set of variable symbols in RIF.
For the purpose of the interpretation of imported documents, RIF BLD defines the notion of semantic multi-structures, which are nonempty sets {I1, ..., In} of semantic structures that are identical in all respects with the exception of the interpretation of local constants.
Given a semantic multi-structure I={I1, ..., In}, we use the symbol I to denote both the multi-structure and the common part of the individual structures I1, ..., In.
Definition. A common-rif-rdf-interpretation is a pair (I, I), where I is a semantic multi-structure and I is an RDF interpretation of a vocabulary V, such that the following conditions hold:
Editor's Note: Make sure the concept of "considered datatype" is consistent with the terminology defined in BLD.Condition 1 ensures that the combination of resources and
properties corresponds exactly to the RIF domain; note that if I is
an rdf-, rdfs-, or D-interpretation, IP is a subset of IR, and thus
IR=Dind. Condition 2 ensures that the set
of RDF properties at least includes all elements that are used as
properties in frames in the RIF domain. Condition 3 ensures that
all concrete values in Dind are included
in LV (by definition, the value spaces of all considered datatypes
are included in Dind). Condition 4 ensures
that RDF triples are interpreted in the same way as frame formulas.
Condition 5 ensures that IRIs are interpreted in the same way.
Condition 6 ensures that typed literals are interpreted in the same
way. Note that no correspondences are defined for the mapping of
names in RDF that are not symbols of RIF, e.g., ill-typed literals
and RDF URI references that are not absolute IRIs. Condition 7
ensures that typing in RDF and typing in RIF correspond, i.e.,
a rdf:type b is true iff a # b is true. Finally,
condition 8 ensures that whenever a RIF subclass statement holds,
the corresponding RDF subclass statement holds as well, i.e., a
rdfs:subClassOf b is true if a ## b is true.
One consequence of
conditions 5 and 6 is that IRIs of the form http://iri and
typed literals of the form "http://iri"^^rif:iri that
occur in an RDF graph are treated the same in RIF-RDF combinations,
even if the RIF Document is empty. Similar 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 tripletriples
<http://a> <http://p> "http://b"^^rif:iri . <http://a> <http://p> "abc" .
This combination allows the derivation of, among other things,
the following triple:triples:
<http://a> <http://p> <http://b> . <http://a> <http://p> "abc"^^xs:string .
as well as the following frame formula:formulas:
<http://a>[<http://p> -> <http://b>] <http://a>[<http://p> -> "abc"]
The notion of satisfiability refers to the conditions under which a common-rif-rdf-interpretation (I, I) is a model of a combination < R, S>. The notion of satisfiability is defined for all four entailment regimes of RDF (simple, RDF, RDFS, and D). The definitions are all analogous. Intuitively, a common-rif-rdf-interpretation (I, I) satisfies a combination < R, S> if I is a model of R and I satisfies S. Formally:
Definition. A common-rif-rdf-interpretation (I, I)
satisfies a
RIF-RDF combination
C=< R, S > if I is a model
of R and I satisfies
every RDF graph S in S; in this case
(I, I) is called a simple-model, or
model, of C, and C is satisfiable.
(I, I) satisfies a generalized RDF graph
S if I satisfies S. (I, I) satisfies an existentially closed RIF-BLDa
condition formula φ if
TValI(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 rule
Forall ("a"="b")
does not have a model, since the symbols "a" and "b" are mapped to the (distinct) character strings "a" and "b", respectively, in every semantic structure.
Rdf-, rdfs-, and D-satisfiability are defined through additional restrictions on I:
Definition. A model (I, I) of a combination C is an rdf-model of C if I is an rdf-interpretation; in this case C is rdf-satisfiable. ☐
Definition. A model (I, I) of a combination C is an rdfs-model of C if I is an rdfs-interpretation; in this case C is rdfs-satisfiable. ☐
Definition. Given a conforming datatype map D, a modelLet (I, I) be a model of a combination C and let D
be a datatype map conforming with the set of datatypes in I.
(I, I) is a D-model of C if I is a D-interpretation;
in this case C is D-satisfiable. ☐
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,Let C be a RIF-RDF combination C D-entailscombination, let S
be a generalized RDF
graph, let φ be a condition
formula, and let D be a datatype map conforming with the set
of considered
datatypes. C D-entails S if every
D-model of C satisfies S. Likewise,
C D-entails an existentially closed RIF-BLD condition formulaφ 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). ☐
The syntax for exchanging OWL ontologies is based on RDF graphs. Therefore, RIF-OWL-combinations are combinations of RIF documents and sets of RDF graphs, analogous to RIF-RDF combinations. This section specifies how RIF documents and OWL ontologies interoperate in such combinations.
OWL (OWL-Reference) specifies three increasingly expressive species, namely Lite, DL, and Full. OWL Lite is a syntactic subset of OWL DL, but the semantics is the same (OWL-Semantics). Since every OWL Lite ontology is an OWL DL ontology, the Lite species is not considered separately in this document.
Syntactically speaking, OWL DL is a subset of OWL Full, but the semantics of the DL and Full species are different (OWL-Semantics). While OWL DL has an abstract syntax with a direct model-theoretic semantics, the semantics of OWL Full is an extension of the semantics of RDFS, and is defined on the RDF syntax of OWL. Consequently, the OWL Full semantics does not extend the OWL DL semantics; however, all derivations sanctioned by the OWL DL semantics are sanctioned by the OWL Full semantics.
Finally, the OWL DL RDF syntax, which is based on the OWL abstract syntax, does not extend the RDF syntax, but rather restricts it: every OWL DL ontology is an RDF graph, but not every RDF graph is an OWL DL ontology. OWL Full and RDF have the same syntax: every RDF graph is an OWL Full ontology and vice versa. This syntactical difference is reflected in the definition of RIF-OWL compatibility: combinations of RIF with OWL DL are based on the OWL abstract syntax, whereas combinations with OWL Full are based on the RDF syntax.
Since the OWL Full syntax is the same as the RDF syntax and the OWL
Full semantics is an extension of the RDF semantics, the definition
of RIF-OWL Full compatibility is a straightforward extension of
RIF-RDF compatibility. Defining RIF-OWL DL compatibility in the
same way would entail losing certain semantic properties of OWL DL.
One of the main reasons for this is the difference in the way
classes and properties are interpreted in OWL Full and OWL DL. In
the Full species, classes and properties are both interpreted as
objects in the domain of interpretation, which are then associated
with subsets of and binary relations over the domain of
interpretation using rdf:type and the extension function
IEXT, as in RDF. In the DL species, classes and properties are
directly interpreted as subsets of andof, respectively binary relations
over the domain. The latterThis is a key property of Description Logic
semantics thatand enables the use of Description Logic reasoning
techniques for processing OWL DL descriptions. Defining RIF-OWL DL
compatibility as an extension of RIF-RDF compatibility would define
a correspondence between OWL DL statements and RIF frame formulas.
Since RIF frame formulas are interpreted using an extension
function, the same way as in RDF, defining the correspondence
between them and OWL DL statements would change the semantics of
OWL statements, even if the RIF document iswere empty.
A RIF-OWL combination that is faithful to the OWL DL semantics requires interpreting classes and properties as sets and binary relations, respectively, suggesting that correspondence could be defined with unary and binary predicates. It is, however, also desirable that there be uniform syntax for the RIF component of both OWL DL and RDF/OWL Full combinations, because one may not know at time of writing the rules which type of inference will be used. Consider, for example, an RDF graph S with the following statement
a rdf:type C .
and a RIF document with the rule
Forall ?x (?x[rdf:type -> D] :- ?x[rdf:type -> C])
The combination of the two, according to the specification of RDF Compatibility, allows deriving
a rdf:type D .
Now, the RDF graph S is also an OWL DL ontology. Therefore, one would expect the triple to be derived by RIF-OWL DL combinations as well.
To ensure that the RIF-OWL DL combination is faithful to the OWL DL
semantics and to enable using the same, or similar, rules with both
OWL DL and RDF/OWL Full, the interpretation of frame formulas
s[p -> o] in the RIF-OWL DL combinations is slightly
different from their interpretation in RIF BLD and syntactical
restrictions are imposed on the use of variables, function terms,
and frame formulas.
Note that the abstract syntax form of OWL DL allows so-called
punning (this is not allowed in the RDF syntax), i.e., the
same IRI may be used in an individual position, a property
position, and a class position; the interpretation of the IRI
depends on its context. Since combinations of RIF and OWL DL are
based on the abstract syntax of OWL DL, punning may also be used in
these combinations.
In this document, we are using OWL to refer to OWL 1.as specified
in (OWL-Semantics). 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 OWL2OWL 2 the support for punning may be
beneficial, and that there might be particular problemsbeneficial. In using section 3.2.2.3, sinceaddition the semantics of annotation properties
might be different than in OWL 1.OWL, so there might be particular
problems if these properties are considered in combinations, as in
the Section Annotation
properties.
Since RDF graphs and OWL Full ontologies cannot be distinguished, the syntax of RIF-OWL Full combinations is the same as the syntax of RIF-RDF combinations.
The syntax of OWL ontologies in RIF-OWL DL combinations is
specified by the abstract syntax of OWL DL. Certain restrictions
are imposed on the syntax of the RIF rules in combinations with OWL
DL. Specifically, the only terms allowed in class and property
positions in frame formulas are constant symbols. Definition.A conditionDL-frame
formula φis a DL-condition if for everyframe formula a[b1 -> c] in φc1
... bn -> cn] such that n≥1
and for every bi, with 1≤i≤n,
it holds that bi is a constant and if
bi = rdf:type, then
ci is a constant.
Definition. A condition formula φ is a DL-condition if every frame formula in φ is a DL-frame formula. ☐
Definition. A RIF-BLD
document formula R is a RIF-BLD DL-document formula
if forevery frame formula a[b -> c]in every rule ofR it holds that b is a constant and if b = rdf:type , then cis a constant.DL-frame formula.
☐
Definition. A RIF-OWL-DL-combination
is a pair < R,O>, where R is a RIF-BLD DL-document formula and
O is a set of OWL
DL ontologies in abstract syntax form of aan
OWL vocabulary V. ☐
When clear from the context, RIF-OWL-DL-combinations are referred to simply as combinations.
In the literature, several restrictions on the use of variables
in combinations of rules and Description Logics have been
identified (Motik05, Rosati06) for the purpose of decidableeffective
reasoning. TheseThis section specifies such safeness restrictions are specifiedfor
RIF-OWL-DL combinations.
Given a set of OWL DL ontologies in abstract syntax form
O, a variable ?x in a RIF rule Q
thenH :- ifB is DL-safe if it occurs in
an atomic formula in ifB that is not of the form s[P ->
o] or s[rdf:type -> A], where P or
A, respectively, occurs in one of the ontologies in
O. A RIF rule Q thenH :- ifB is
DL-safe, given O if every variable that occurs
in thenH :- ifB is DL-safe. A RIF rule Q
thenH :- ifB is weakly DL-safe, given
O if every variable that occurs in thenH is DL-safe and every variable in if that is not DL-safe occurs only in atomic formulas in if that are of the form s[P -> o] or s[rdf:type -> A] , where P or A , respectively, occurs in one of the ontologies in O .DL-safe.
Definition. A RIF-OWL-DL-combination < R,O> is DL-safe if every rule in R is DL-safe, given O. A RIF-OWL-DL-combination < R,O> is weakly DL-safe if every rule in R is weakly DL-safe, given O. ☐
Editor'sFeature At Risk #1: Safeness
Note: This feature is "at risk" and may be removed from this specification based on feedback. Please send feedback to public-rif-comments@w3.org.
The above definition of DL-safeness is intended to identify a
fragment of RIF-OWL DL combinations for which implementation is
easier than full RIF-OWL DL. This definition should be considered
AT RISK and may changebecome stricter based on implementation
experience.
The semantics of RIF-OWL Full combinations is a straightforward extension of the Semantics of RIF-RDF Combinations.
The semantics of RIF-OWL-DL-combinations cannot straightforwardly extend the semantics of RIF RDF combinations, because OWL DL does not extend the RDF semantics. In order to keep the syntax of the rules uniform between RIF-OWL-Full- and RIF-OWL-DL-combinations, the semantics of RIF frame formulas is slightly altered in RIF-OWL-DL-combinations.
A D-interpretation I is an OWL Full interpretation if it interprets the OWL vocabulary and it satisfies the conditions in the sections 5.2 and 5.3 in (OWL Semantics).
The semantics of RIF-OWL Full combinations is a straightforward extension of the semantics of RIF-RDF combinations. It is based on the same notion of common-interpretations, but defines additional notions of satisfiability and entailment.
Definition. Given a conforming datatype map D, a common-rif-rdf-interpretationLet (I, I) be a common-rif-rdf-interpretation that is an OWL-Full-modela model of a RIF-RDF combination C=<
R, S > if I isand let D be a modeldatatype map conforming with the set
of Rdatatypes in I. (I, I) is an
owl-full-model
of C if I is an
OWL Full interpretation , and I satisfies every RDF graph S in S ;with respect to D; in this case C is
OWL-Full-satisfiable
.with respect to D. ☐
Definition. Given a conforming datatype map D,Let C be a RIF-RDF combination C OWL-Full-entailscombination, let S
be a generalized RDF
graph, let φ be a condition
formula, and let D be a datatype map conforming with the set
of considered
datatypes. C owl-full-entails
S with respect to D if every owl-full-model of C satisfies S. Likewise, C
owl-full-entails an existentially closed RIF-BLD condition formulaφ with respect to D 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. Analogous to a
common-rif-rdf-interpretation, there is the notion of
common-rif-dl-interpretations,common-rif-owl-dl-interpretations, which are pairs of RIF and OWL
DL interpretations, and which define a number of conditions that
relate these interpretations to each other. In contrast to RIF-RDF
combinations, the conditions below define a correspondence between
the interpretation of OWL DL classes and properties and RIF unary
and binary predicates.
The modification of the semantics of RIF frame formulas is achieved by modifying the mapping function for frame formulas (Iframe), and leaving the RIF BLD semantics (RIF-BLD) otherwise unchanged.
Namely, frame formulas of the form s[rdf:type -> o] are interpreted as membership of s in the set denoted by o and frame formulas of the form s[p -> o], where p is not rdf:type, as membership of the pair (s, o) in the binary relation denoted by p.
Definition. A dl-semantic
structure is a tuple I =
<TV, DTS, D,
Dind, Dfunc,
IC, IV,
IF, Iframe',
ISF, Isub,
Iisa, I=,
Iexternal,
Itruth>, where
Iframe' is a mapping from
Dind to total functions of the form
SetOfFiniteFrame'BagsSetOfFiniteBags(D × D) →
D, such that for each pair (a, b) in
SetOfFiniteFrame'BagsSetOfFiniteBags(D × D) holds
that if a≠IC(rdf:type),
then b in Dind; all other
elements of the structure are defined as in RIF semantic
structures.
A dl-semantic
multi-structure is a nonempty set of dl-semantic
structures {I1, ...,
In} that are identical in all respects
except that the mappings I C1C,
..., I CnC might differ on the
constants in Const that belong to the rif:local
symbol space. ☐
Given a dl-semantic multi-structure I={I1, ..., In}, we use the symbol I to denote both the multi-structure and the common part of the individual structures I1, ..., In.
We define I(o[a1->v1 ... ak->vk]) = Iframe(I(o))({<I(a1),I(v1)>, ..., <I(an),I(vn)>}). The truth valuation function TValI is then defined as in RIF BLD.
Definition. A dl-semantic multi-structure I is a model of a RIF-BLD DL-document formula R if TValI(R)=t. ☐
As defined in (OWL-Semantics), an abstract OWL interpretation with respect to a datatype map D, with vocabulary V is a tuple I=< R, EC, ER, L, S, LV >, where
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 conformingdatatype map D, a common-rif-dl-interpretationcommon-rif-owl-dl-interpretation
with respect to D is a pair (I, I), where
I is a dl-semantic multi-structure and I is an
abstract OWL interpretation with respect to D of a vocabulary
V, such that the following conditions hold
Condition 12 ensures that the relevant parts of the domains of
interpretation are the same. Condition 23 ensures that the
interpretation (extension) of an OWL DL class u
corresponds to the interpretation of frames of the form
?x[rdf:type -> <u>]. Condition 34 ensures that the
interpretation (extension) of an OWL DL object or datatype property
u corresponds to tothe interpretation of frames of the form
?x[<u> -> ?y]. Condition 45 ensures that
typed literals of the form (s, d) in OWL DL are
interpreted in the same way as constants of the form
"s"^^d in RIF. Finally, condition 56 ensures that
individual identifiers in the OWL ontologies and the RIF documents
are interpreted in the same way.
Using the definition of common-rif-dl-interpretation,common-rif-owl-dl-interpretation,
satisfaction, models, and entailment are defined in the usual
way:
Definition. Given a conforming datatype map D,A common-rif-dl-interpretationcommon-rif-owl-dl-interpretation (I, I)
with respect to a datatype map D is an owl-dl-model of a RIF-OWL-DL-combination
C=< R, O > if I is a model of R and I
satisfies every OWL
DL ontology in abstract syntax form O in O; in
this case C is owl-dl-satisfiable .with
respect to D. (I, I) is an owl-dl-model
of an OWL
DL ontology in abstract syntax form O if I
satisfies O. (I, I) is an
owl-dl-model of an existentially closed RIF-BLD condition φa DL-condition formula φ if
TValI(φ)=t. ☐
Definition. Given a conforming datatype map D, a RIF-OWL-DL-combinationLet C OWL-DL-entailsbe a RIF-OWL-DL-combination, let
O be an OWL
DL ontology in abstract syntax form, let φ be a DL-condition formula, and let D be
a datatype map conforming with the set of considered datatypes. C
owl-dl-entails
O with respect to D if every
common-rif-owl-dl-interpretation with respect to D that is an
owl-dl-model of C is an
owl-dl-model of O.
Likewise, C owl-dl-entails an existentially closed DL-condition formulaφ with respect to D if
every common-rif-owl-dl-interpretation with respect to D that is an
owl-dl-model of C is an
owl-dl-model of φ.
☐
Recall that in an abstract OWL interpretation I the sets O, which is used for interpreting individuals,O and LV,
which isare used for interpretinginterpreting, respectively literals (data
values), are disjoint and that EC maps class identifiers to subsets
of O and datatype identifiers to subsets of LV. The disjointness
entails that data values cannot be members of a class and
individuals cannot be members of a datatype.
In RIF, variable quantification ranges over
Dind. So, the same variable may be
assigned to an abstract individual or a concrete data value.
Additionally, RIF constants (e.g., IRIs) denoting individuals can
be written in place of a data value, such as the value of a
data-valued property or in datatype membership statements;
similarly for constants denoting data values. Such statements
cannot be satisfied in any common-rif-dl-interpretation,common-rif-owl-dl-interpretation, due to
the constraints on the EC and ER functions. The following example
illustrates several such statements.
Consider the datatype xs:string and a RIF-OWL DL combination consisting of the set containing only the OWL DL ontology
ex:myiri rdf:type ex:A .
and a RIF document containing the following fact
ex:myiri[rdf:type -> xs:string]
This combination is not owl-dl-satisfiable, because ex:myiri is an individual identifier and S maps individual identifiers to elements in O, which is disjoint from the elements in the datatype xs:string.
Consider a RIF-OWL DL combination consisting of the set containing only the OWL DL ontology
ex:hasChild rdf:type owl:ObjectProperty .
and a RIF document containing the following fact
ex:myiri[ex:hasChild -> "John"]
This combination is not owl-dl-satisfiable, because ex:hasChild is an object property, and values of object properties may not be concrete data values.
Consider a RIF-OWL DL combination consisting of the OWL DL ontology
SubClassof(ex:A ex:B)
and a RIF document containing the following rule
Forall ?x (?x[rdf:type -> ex:A])
This combination is not owl-dl-satisfiable, because the rule requires every element,
including every concrete data value, to be a member of the class
ex:A. However, the mapping EC in any abstract OWL
interpretation requires every member of ex:A to be an
element of O, and concrete data values may notcannot be members of O.
Note that the above definition of RIF-OWL DL compatibility does
not consider ontology and annotation properties, in contrast to the
definition of compatibility of RIF with OWL Full, where there is no
clear distinction between annotation and ontology properties and
other kinds of properties. Therefore, it is not possible to
"access" or use the values of these properties in the RIF rules.
This limitation is overcome in the following definition. It is
envisioned that the user will choose whether annotation and
ontology properties are to be considered. It is currently expectedmight be the case that
OWL 2, the successor of OWL
currently under development, will not define athe semantics for
annotation and ontology properties;properties in the same way as OWL;
therefore, the below definition cannotmay not be extended to the case ofextensible towards OWL
2.
Definition. Given a conformingdatatype map D, a common-rif-dl-interpretationcommon-rif-owl-dl-interpretation (I, I) is a
common-dl-annotation-interpretation
with respect to D if the following condition holds
6. ER(p) = set of all pairs (k,
l) in O × O such that
Itruth(Iframe'(IC(<p> ))())({(k,
l ))}) ) = t (true), for every IRI p in
V. ☐
Condition 6, which strengthens condition 3, ensures that the
interpretation of all properties (also annotation and ontology
properties) in the OWL DL ontologies corresponds with their
interpretation in the RIF rules.
Definition. GivenA conformingcommon-dl-annotation-interpretation with respect to a
datatype map D, a common-DL-annotation-interpretationD (I, I) is an owl-dl-annotation-model
of a RIF-OWL-DL-combination C=< R, O > if
I is a model of
R and I
satisfies every OWL
DL ontology in abstract syntax form O in O; in
this case C is owl-dl-annotation-satisfiable.
☐
Definition. Given a conforming datatype map D, a RIF-RDF combinationLet C OWL-DL-annotation-entailsbe a RIF-OWL-DL-combination, let
O be an OWL
DL ontology in abstract syntax form, let φ be a DL-condition formula, and let D be
a datatype map conforming with the set of considered datatypes. C
owl-dl-annotation-entails
O with respect to D if every
common-rif-owl-dl-interpretation with respect to D that is an
owl-dl-annotation-model of C is an owl-dl-model of O.
Likewise, C owl-dl-annotation-entails an existentially closed RIF-BLD condition formulaφ with respect
to D if every common-rif-owl-dl-interpretation with respect to D
that is an owl-dl-annotation-model of C is an owl-dl-model of φ.
☐
The difference between the two kinds of OWL DL entailment can be illustrated using an example. Consider the following OWL DL ontology in abstract syntax form
Ontology (ex:myOntology Annotation(dc:title "Example ontology"))
which defines an ontology with a single annotation (title). Consider also a document consisting of the following rule:
Forall ?x ?y ( ?x[ex:hasTitle -> ?y] :- ?x[dc:title -> ?y])
which says that whenever something has a dc:title, it has the same ex:hasTitle.
The combination of the ontology and the document owl-dl-annotation-entails the RIF condition formula ex:myOntology[ex:hasTitle -> "Example ontology"]; the combination does not owl-dl-entail the formula.
In the previous sections, RIF-RDF Combinations and RIF-OWL combinations were defined in an abstract way, as pairs of documents and sets of RDF graphs/OWL ontologies. In addition, different semantics were specified based on the various RDF and OWL entailment regimes. RIF provides a mechanism for explicitly referring to (importing) RDF graphs from documents and specify the intended profile (entailment regime) through the use of Import statements.
This section specifies how RIF documents with such import statements are interpreted.
A RIF
document contains a number of Import statements. Unary
Import statements are used for importing RIF documents,
and the interpretation of these statements is defined in (RIF-BLD). This section defines the
interpretation of two-ary Import statements:
Import(t1 p1) ... Import(tn pn)
Here, ti is an IRI constant of the form <absolute-IRI>, where absolute-IRI is the location of an RDF graph to be imported, and pi is an IRI constant denoting the profile to be used.
The profile determines which notions of model, satisfiability and entailment must be used. For example, if a RIF document R imports an RDF graph S with the profile RDFS, the notions of rdfs-model, rdfs-satisfiability, and rdfs-entailment should be used with the combination <R, {S}>.
Profiles are assumed to be ordered. In case several graphs are imported in a document, and these imports specify different profile, the highest of these profiles is used. For example, if a RIF document R imports an RDF graph S1 with the profile RDF and an RDF graph S2 with the profile OWL Full, the notions of owl-full-model, owl-full-satisfiability, and owl-full-entailment must be used with the combination <R, {S1, S2}>.
Finally, if a RIF document R imports an RDF graph S with the profile OWL DL, R must be a RIF-BLD DL-document formula, S must be the translation to RDF of an OWL DL ontology in abstract syntax form O, and the notions of owl-dl-model, owl-dl-satisfiability, and owl-dl-entailment must be used with the combination <R, {O}>.
RIF defines a specific profile for each of the notions of
satisfiability and entailment of combinations, as well as twoone
generic profiles for RDF and OWL, respectively.profile. The use of a specific profile specifies how a
combination should be interpreted and a receiver should reject a combination withinterpreted. If a specific profile itcannot handle.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 his or her 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/rif-import-profile#Simple | simple-model | satisfiability | simple-entailment |
rdf | http://www.w3.org/2007/rif-import-profile#RDF | rdf-model | rdf-satisfiability | rdf-entailment |
rdfs | http://www.w3.org/2007/rif-import-profile#RDFS | rdfs-model | rdfs-satisfiability | rdfs-entailment |
D | http://www.w3.org/2007/rif-import-profile#D | d-model | d-satisfiability | d-entailment |
OWL DL | http://www.w3.org/2007/rif-import-profile#OWL-DL | owl-dl-model | owl-dl-satisfiability | owl-dl-entailment |
OWL DL annotation | http://www.w3.org/2007/rif-import-profile#OWL-DL-annotation | owl-dl-annotation-model | owl-dl-annotation-satisfiability | owl-dl-annotation-entailment |
OWL Full | http://www.w3.org/2007/rif-import-profile#OWL-Full | owl-full-model | owl-full-satisfiability | owl-full-entailment |
Profiles that are defined for combinations of DL-documentsDL-document
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 following table listsuse of the generic
profiles in RIF along with the IRI of the profile. Note that the use of a genericprofile 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/rif-import-profile#Generic |
Let R be a RIF document such that
Import(<u1> <p1>) ... Import(<un> <pn>)
are the two-ary import statements in R and all imported documents and let Profile be the set of profiles corresponding to the IRIs p1,...,pn.
If Profile contains only specific profiles, then:
If Profile contains a generic profile, then the
combination
C=<R,{S1,....,Sn}>,
where S1,....,Sn are RDF graphs
accessible from the locations
u1,...,unu1,...,un and C 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 Sainte-Marie, 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 RIF-OWL task force have contributed to the OWL Compatibility section in this document: Mike Dean, Peter F. Patel-Schneider, and Ulrike Sattler.
The regular attendees at meetings of the Rule Interchange Format
(RIF) Working Group at the time of the publication were: Adrian
Paschke (REWERSE), Axel Polleres (DERI), Chris Welty (IBM),
Christian de Sainte Marie (ILOG), Dave Reynolds (HP), Gary Hallmark
(ORACLE), Harold Boley (NRC), Hassan Aït-Kaci (ILOG), Igor Mozetic
(JFI), John Hall (OMG), Jos de Bruijn (FUB), Leora Morgenstern
(IBM), Michael Kifer (Stony Brook), Mike Dean (BBN), Sandro Hawke
(W3C/MIT), and Stella Mitchell (IBM).
RIF-RDF combinations can be embedded into RIF documents in a
fairly straightforward way, thereby demonstrating how a
RIF-compliant translator without native support for RDF can process
RIF-RDF combinations. RIF-OWL combinations cannot be embedded in
RIF, in the general case. However, there is a subset of RIF-OWLOWL DL, the
so-called DLP subset (DLP), for
which RIF-OWL DL combinations that can be embedded.
This appendix illustrates embeddings into RIF BLD of simple, RDF, and RDFS entailment for RIF-RDF combinations and OWL DL entailment for RIF-OWL DL combinations, restricted to the DLP subset of OWL DL.The embeddings are defined using the embedding function tr,
which maps symbols, triples, RDF graphs, and OWL DL ontologies in
abstract syntax form to RIF symbols, statements, and documents,
respectively.
Besides the namespace prefix isprefixes defined in the Overview, the
following namespace prefix is used in this appendix: pred
refers to the RIF namespace for built-in predicates
http://www.w3.org/2007/rif-builtin-predicate# (RIF-DTB).
7.1 Embedding RIF-RDF CombinationsTo facilitate the embeddingdefinition of RIF-RDF combinations 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. 7.1.1 Embedding Symbols Given a combination C=< R , S >,the function tr maps RDF symbolsembeddings we define the notion
of a vocabulary V andmerge of RIF formulas.
Definition. Let
R={R1,...,Rn} be a set
of blank nodes B to RIF symbols, as defined in following table. Mapping RDF symbols to RIF. RDF Symbol RIF Symbol Mapping IRI idocument,
group, and rule formulas, such that there are no prefix or base
directives or relative IRIs in V U Constant with symbol space rif:iri tr( i ) = <i> Blank node xR and
directive11, ...,
directivenm are all the import
directives occurring in B Variable symbol ? x tr( x ) = ? x Plain literal without a language tag xxxdocument formulas in V PL Constant withR. The
datatype xs:string tr( "xxx" ) = "xxx" Plain literal with a language tagmerge of R,
denoted merge ( xxxR), is defined as
Document(directive11 ...
directivenm
Group(R*1 ...
R*n)), lang )where R*i is
obtained from Ri in V PL Constant withthe datatype rif:text tr( "xxx"@lang ) = "xxx@lang"^^rif:text Well-typed literal ( s , u ) in V TL Constant withfollowing way:
Note that the mapping function trrequirement that no prefix or based directives or
relative IRIs are included in any of the formulas to be merged is
extendednot a real limitation, since compact IRIs can be rewritten to
embed triplesabsolutes 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 statements. Finally, twoBLD that is at risk. However, equality is
not a crucial feature for the embeddings; removing equality from
embedded combinations is fairly straightforward.
RIF-RDF combinations are embedded through embeddings of graphs
and tr Q embedaxiomatization of simple, RDF, and RDFS entailment.
The embedding is not defined for combinations that include
infinite RDF graphs as RIF documentsand conditions, respectively. The following section shows how these embeddings can be usedfor reasoningcombinations that include RDF graphs
with combinations. We define two mappings forRDF graphs, one (tr R ) in which variablesURI references that are Skolemized, i.e., replaced with constant symbols, and one (tr Q ) in which variablesnot absolute IRIs (see the
End note on RDF URI
references) or plain literals that are existentially quantified.not in the function sk takes aslexical space
of the xs:string
datatype (XML-Schema2). In
addition, for the embedding of RDFS entailment, each datatype must
have an argumentassociated 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 formula φ with variablescombination C=< R,S>, the function tr
maps RDF symbols of a vocabulary V and returnsa formula φ', whichset of blank nodes
B to RIF symbols, as defined in following table.
In the table, the mapping tr' is obtained from R by replacing every variable symbol ? xan injective function that maps
typed literals to new constants in R with <new-iri> ,the rif:local symbol
space, where new-iria 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 globally unique IRI.constant from a typed
literal.
RDF Symbol | RIF |
Mapping |
---|---|---|
|
Constant with symbol space rif:iri | tr( |
Blank node _:x in B | Variable symbol ?x | tr( |
Plain literal without a language tag xxx in VPL | Constant with the datatype xs:string | tr( |
Plain literal with a language tag "xxx"@lang in VPL | Constant with the datatype rif:text | tr("xxx"@lang) = "xxx@lang"^^rif:text |
Well-typed literal "s"^^u in VTL | Constant with the symbol space u | tr("s"^^u) = "s"^^u |
Non-well-typed literal "s"^^u in VTL | Local constant s-u' 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 (trR) graphs are embedded as RIF documents and variables are skolemized, i.e., replaced with new constant symbols. In the other (trQ) 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 <new-iri>, where new-iri 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 trR(S) | trR(S) = sk(Document (Group (Forall
(tr( |
Graph S | Condition (query) trQ(S) | trQ(S) = Exists
tr( |
Even thoughThe semantics of the RDF vocabulary does not need to be
axiomatized for simple entailment,entailment. Nonetheless, the connection
between RIF class membership and subclass statements and the RDF
type and subclass statements needs to be axiomatized.axiomatization. We define:
Rsimple | = | 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 simple-entailment of
combinations can be reduced to checking entailment of RIF
conditions by using the embeddings of RDF graphs defined in the previous section. Theorem A RIF-RDF combination < R ,{ S1 ,..., Sn }> is satisfiable iff there is a semantic multi-structure I that is a model of R , R simple , tr R ( S1 ), ..., and tr R ( Sn )). Proof. The theorem follows immediately from the following theorem and the observation that a combination (respectively, document) is satisfiable (respectively, has a model) if it does not entail the condition formula "a"="b" .above.
Theorem A RIF-RDF combination
C=<R,{ S1S1,..., SnSn}>
simple-entails a
generalized RDF
graph T if and only if (merge({R union,
trR( S1 ) union ... unionS1), ...,
trR( Sn ))Sn)}) entails
trQ(T). C simple-entails an existentially closed RIF-BLDa condition
formula φ if and only if (merge({R union,
Rsimple union, trR( S1 ) union ... unionS1),
..., trR( Sn ))Sn}) entails
φ.
Editor'sNote:Formulationoftheentailmenttheoremistobeupdatedwithanotionofmergeofrulesets.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({Runion, Rsimpleunion, trR(S1)union...unionS1), ..., trR(Sn))Sn)}) with R'.
(=>) Assume R' does not entail φ. This means there is some semantic multi-structure I that is a model of R', but not of φ. Consider the pairinterpretation(I, I), where I is the interpretation defined as follows:Clearly, (I, I) is a common-rif-rdf-interpretation: conditions 1-6 in the definition are satisfied by construction of I and conditions 7 and 8 are satisfied by condition 4 and by the fact that I is a model of Rsimple.
- IR=Dind,
- IP is the set of all k in Dind such that there exist some a, b in Dind and Itruth(Iframe(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 Dind, such that Itruth(Iframe(k)(a,b))=t,
- IS(i) = IC(<i>) for every absolute IRI i in VU, and
- IL((s, d)) = IC(tr("s"^^d)) for every typed literal (s, d) in VTL.
Consider a graphSiSi in{S1,...,Sn}{S1,...,Sn}. Letx1,...,xmx1,..., xm be the blank nodes inSiSi and letu1,...,umu1,..., um be the new IRIs that were obtained from the variables?x1,..., ?xm?x1,..., ?xm through the skolemization in trR(SiSi), i.e., ui=sk(?xi). Now, let A be a mapping from blank nodes to elements in Dind such that A(xjxj)=IC(ujuj) for every blank nodexjxj inSiSi. From the fact that I is a model of trR(SiSi) and by construction of I it follows that [I+A] satisfiesSi,Si (see Section 1.5 of (RDF-Semantics))), and so I satisfiesSiSi.
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 trQ(T)and), which means there is a semantic multi-structure I that is a model of R', but not of trQ(T). The common-rif-rdf-interpretation (I, I) is obtained in the same way as above, and so clearly satisfies C.
We proceed by contradiction. Assume I satisfies T. This means there is some mapping A from the blank nodesx1,...,xmx1,...,xm in T to objects in Dind such that [I+A] satisfies T.
Consider now the semantic multi-structure I*, which is the same as I, with the exception of the mapping I*V on the variables?x1,...,?xm?x1,...,?xm, which is defined as follows: I*V(?xj?xj)=A(xj)xj) for each blank nodexjxj in S. By construction of I and since [I+A] satisfies T we can conclude that I* is a model of And(tr(t1t1)... tr(tmtm)), and so I is a model of trQ(T), violating the assumption that it is not. Therefore, (I, I) does not satisfy T and C does not entailφ.T.
(<=) Assume C does not entail φ. This means there is some common-rif-rdf-interpretation (I, I) that satisfies C such that I is not a model of φ.
Consider the semantic multi-structureI$I', which is exactly the sameI$I, except for the mappingI$I'C on new IRIs that were introduced in skolemization. The mapping of these new IRIs is defined as follows:
For each graphSiSi in{S1,...,Sn}{S1,...,Sn}, letx1,...,xmx1,..., xm be the blank nodes inSiSi and letu1,...,umu1,..., um be the new IRIs that were obtained from the variables?x1,..., ?xm?x1,..., ?xm through the skolemization in trR(SiSi). Now, since I satisfiesSiSi, there must be a mapping A from blank nodes to elements in Dind such that [I+A] satisfiesSiSi. We defineI$I'C(ujuj)=A(xjxj) for every blank nodexjxj inSiSi.
By assumption,I$I' is a model of R (recall thatI$I' differs from I only on the newIRIs).IRIs, which are not in R). Clearly,I$I' is also a model of Rsimple, by conditions 7, 8, and 4 in the definition of common-rif-rdf-interpretation.
From the fact that I satisfiesSiSi and by construction ofI$I' it follows thatI$I' is a model of trR(SiSi). So,I$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$I' is not the model of φ.
Therefore, R' does not entail φ.
Assume now that C does not entail Tand, which means there is a common-rif-owl-dl-interpretation (I, I) that satisfies C, but I does not satisfy T. We obtainI$I' from I in the same way as above, and so clearly satisfies R'. It can be shown analogous to the (=>) direction that ifI$I' is a model of trQ(T), then there is a blank node mapping A such that [I+A] satisfies T, and thus I satisfies S, violating the assumption that it does not. Therefore,I$I' is not a model of trQ(T) and thus R' does not entail trQ(T).7.1.4EmbeddingRDFEntailmentWeaxiomatizethesemanticsoftheRDFvocabularyusingthefollowingRIFrules.Weassumethatex:illxmlisnotusedinanydocument.RRDF=Rsimpleunion((Forall(tr(spo.)))foreveryRDFaxiomatictriplespo.)union((Forall(ex:illxml(tr("s"^^rdf:XMLLiteral))))foreverynon-well-typedliteraloftheform(s,rdf:XMLLiteral)inVTL)union(Forall?x(?x[rdf:type->rdf:Property]:-Exists?y?z(?y[?x->?z])),Forall?x(?x[rdf:type->rdf:XMLLiteral]:-pred:isXMLLiteral(?x)),Forall?x("a"="b":-And(?x[rdf:type->rdf:XMLLiteral]ex:illxml(?x))))Here,inconsistenciesmayoccurifnon-well-typedXMLliterals,axiomatizedusingtheex:illxmlpredicate,areintheclassextensionofrdf:XMLLiteral.Ifthissituationoccurs,"a"="b"isderived,whichisaninconsistencyinRIF.☐
Theorem A RIF-RDF combination
<R,{ S1,...,SnS1,...,Sn}> is
rdf-satisfiablesatisfiable iff there
is a semantic multi-structure I that is a
model of merge({R, R RDFsimple,
trR( S1S1), ...,
andtrR( Sn )).Sn)}).
Proof. The theorem follows immediately from thefollowingprevious theorem and the observation that a combination (respectively, RIF document) isrdf-satisfiablesatisfiable (respectively, has a model) if and only if it does not entail the condition formula "a"="b".TheoremARIF-RDFcombinationC=<R,{S1,...,Sn}>rdf-entailsageneralizedRDFgraphTiff(R☐
We abbreviate ( R union R RDF union tr R ( S1 ) union ... union tr R ( Sn )) with R'. The proof is then obtained fromaxiomatize the proofsemantics of correspondence for simple entailment inthe previous section withRDF vocabulary using the
following modifications: (*) in the (=>) directionRIF rules. We additionally need to showassume that I is an rdf-interpretation and (**) inthe (<=) direction we need to slightly extend the definition of I$ to account forpredicate symbol
ex:illxml and show that I$is a model of R RDF . (*)not used in any RIF document.
To show that I is an rdf-interpretation,finitely embed RDF entailment, we need to show that I satisfiesconsider a subset
of the RDF axiomatic triples andtriples. Given a combination C, the
RDF semantic conditions . Satisfactioncontext of C includes C and all graphs/formulas considered
for entailment checking. The axiomatic triples follows immediately from the inclusionset of tr( t ) in RRDF forfinite-axiomatic triples is
the smallest set such that:
Let T be the definitionset of satisfactionconsidered datatypes. We assume that each datatype in RIF BLD, I C ( " xxx "^^rdf:XMLLiteral ) isT has
an associated capitalized short name Datatype
(e.g., the XML valueshort name of xxx,xs:string is String) and
a guard pred:isDatatype, which
can be used to test whether a particular object is clearly in LV, by definitiona value of I.the
final partdatatype; see (RIF-DTB) for
definitions of condition 2 is satisfied byguards for the second rule inRIF-required datatypes.
RRDF | = | merge ({Rsimple} union ({Forall (tr(s p o . Forall ?x (?x[rdf:type -> rdf:Property] :- Exists ?y ?z (?y[?x -> ?z])), } union {Forall ?x ("a"="b" :- And(ex:illxml(?x) Or(pred:isDT1(?x) ... pred:isDTn(?x)))), where DT1,...,DTn are the capitalized short names of the datatypes in |
Here, inconsistencies may occur if non-well-typed XML literalliterals,
axiomatized using the ex:illxml predicate, are in the
class extension of rdf:XMLLiteral , then I would not be a model of this rule).. If this establishes the fact that Isituation
occurs, "a"="b" is an rdf-interpretation. (**) Recall that, by assumption, ex:illxmlderived, which is not usedan inconsistency in
RIF.
Theorem A RIF-RDF combination
C=<R . Therefore, changing satisfaction of atomic formulas concerning ex:illxml does not affect satisfaction of R . We assume that k is,{S1,...,Sn}>
rdf-entails a unique element, i.e., no other constant is mapped to k . We define I$ C ( k ) as follows: For every non-well-typed literal of the formgeneralized RDF graph
T iff merge({RRDF, R,
trR(S , rdf:XMLLiteral ) such that I$ C (tr( s^^rdf:XMLLiteral ))= l we define I truth1), ...,
trR( I$ FSn)}) entails
trQ( k )( l ))=T ; I$ truth ( I$ F ( k )(m))= f for any other object m in D ind . Consider). C rdf-entails a condition
formula φ iff merge({RRDF . Satisfaction of, R simple was established in the proof in the previous section. Satisfaction of the facts corresponding,
trR(S1), ...,
trR(Sn)}) entails
φ.
Proof. In the proof we abbreviate merge({RRDF, R, trR(S1), ..., trR(Sn)}) with R'.
The proof is then obtained from the proof of correspondence for simple entailment in the previous section with the following modifications: (*) in the (=>) direction we additionally need to extend I to ensure it satisfies the RDF axiomatic triples and show that I is an rdf-interpretation and (**) inI$followsimmediatelyfromthe (<=) direction we need to slightly extend the definition ofcommon-rif-rdf-interpretationI' to account forex:illxml
and show that I' is a model of RRDF.
(*) For any positive integer j such that rdf:_j does not occur in thefactcontext of C, I and I are extended such that IS(rdf:_j)=IC(rdf:_j)=IC(rdf:_m) (see the definition of finite-axiomatic triples above for the definition of m). Clearly, this does not affect satisfaction of R' or non-satisfaction of φ, respectively trQ(T). To show that I is an rdf-interpretation,andthuswe need to show that I satisfiesallthe RDF axiomatictriples.Sincerdf:XMLLiteralisarequireddatatype,thesetofnon-well-typedXMLliteralsisthesameastriples and thesetofill-typedXMLliterals.RDF semantic conditions.
Satisfaction of theex:illxmlfactsinRRDFthenaxiomatic triples follows immediately from thedefinitionofI$.Satisfactioninclusion ofthefirst,second,andthirdruletr(t) in RRDFfollowstraightforwardlyfromthefor every RDFsemanticconditions1,2,and3.Thisestablishesfinite-axiomatic triple t, the fact thatI$I is a model of RRDF.7.1.5EmbeddingRDFSEntailmentWeaxiomatizethesemantics, and construction of I, and theRDF(S)vocabularyusingextension of I to satisfy thefollowingRIFrules.LetTbeinfinite axiomatic triples. Consider thesetofconsidereddatatypes(cf.Section5of(RIF-BLD)),i.e.,Tincludesatleastalldatatypesusedthree RDF semantic conditions:
1 x is in thecombinationunderconsiderationIP if andalldatatypesrequiredforRIF-BLD(RIF-DTB).Editor'sNote:Theterminology"considereddatatype"mightchangeonly iftheterminology<x, I(rdf:Property
)> ischangedinBLD.By(RIF-DTB),eachdatatypeIEXT(I(rdf:type
))2 If "
xxx"^^rdf:XMLLiteral
is inThasanassociatedlabelDATATYPE(e.g.,V and xxx is a well-typed XML literal string, then
(a) IL(
"
xxx"^^rdf:XMLLiteral
) is thelabelXML value ofxs:stringisStringxxx;
(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 andaguardpred:isDATATYPE,whichcanbeusedtotestwhetheraparticularobjectxxx isavaluean ill-typed 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
datatype.Editor'sNote:Verifythatthesethingsaredefinedfirst rule intheDTBdocumentbeforepublication.RRDFS=RRDFunion((Foralltr(spo.))foreveryRDFSaxiomatictriplespoin I and construction of I; specifically the second bullet.
Consider a well-typed XML literal"
xxx"^^rdf:XMLLiteral
.)union(Forall?x(?x[rdf:type->rdfs:Resource]),Forall?u?v?x?y(?u[rdf:type->?y]:-And(?x[rdfs:domain->?y]?u[?x->?v])),Forall?u?v?x?y(?v[rdf:type->?y]:-And(?x[rdfs:range->?y]?u[?x->?v])),Forall?x(?x[rdfs:subPropertyOf->?x]:-?x[rdf:type->rdf:Property]),Forall?x?y?z(?x[rdfs:subPropertyOf->?z]:-And(?x[rdfs:subPropertyOf->?y]?y[rdfs:subPropertyOf->?z])),Forall?x?y?z1?z2(?z1[?y->?z2]:-And(?x[rdfs:subPropertyOf->?y]?z1[?x->?z2])),Forall?x(?x[rdfs:subClassOf->rdfs:Resource]:-?x[rdf:type->rdfs:Class]),Forall?x?y?z(?z[rdf:type->?y]:-And(?x[rdfs:subClassOf->?y]?z[rdf:type->?x])),Forall?x(?x[rdfs:subClassOf->?x]:-?x[rdf:type->rdfs:Class]),Forall?x?y?z(?x[rdfs:subClassOf->?z]:-And(?x[rdfs:subClassOf->?y]?y[rdfs:subClassOf->?z])),Forall?x(?x[rdfs:subPropertyOf->rdfs:member]:-?x[rdf:type->rdfs:ContainerMembershipProperty]),Forall?x(?x[rdfs:subClassOf->rdfs:Literal]:-?x[rdf:type->rdfs:Datatype]),Forall?x("a"="b":-And(?x[rdf:type->rdfs:Literal]ex:illxml(?x))))union(Forall ?x(?x[rdf:type->rdfs:Literal] :-Or(pred:isDT1(?x)...pred:isDTn(?x))),whereDT1,...,DTnareBy thelabelsdefinition ofthedatatypessatisfaction inT)TheoremARIF-RDFcombination<R,{S1,...,Sn}>RIF BLD, IC("
xxx"^^rdf:XMLLiteral
) isrdfs-satisfiableifthe XML value of xxx (condition 2a), andonlyifthereisasemanticmulti-structureIthatclearly in LV (condition 2b), by definition of I. Condition 2c isamodelsatisfied by satisfaction of the second rule in R,RRDFS,trRDF in I.
Satisfaction of 3a follows from satisfaction of the fourth rule in R(S1),...,RDF andtrR(Sn)).Proof.thetheoremdefinition of LV. (3b) followsimmediatelyfrom satisfaction of thefollowingtheoremandtheobservationthatacombination(respectively,document)isrdfs-satisfiable(respectively,hasamodel)ifitdoesnotentailtheconditionformula"a"="b".TheoremARIF-RDFcombinationC=<R,{S1,...,Sn}>rdfs-entailsageneralizedRDFgraphTifandonlyif(RRDFSunionRuniontrR(S1)union...uniontrR(Sn))entailstrQ(T).Crdfs-entailsanexistentiallyclosedRIF-BLDconditionformulaφifandonlyif(RRDFSunionRuniontrR(S1)union...uniontrR(Sn))entailsφ.Proof.third rule intheproofweabbreviate(RunionRRDFSuniontrR(S1)union...uniontrR(Sn))withR'.TheproofisthenobtainedfromtheproofofcorrespondenceforRDFentailment(3b) intheprevioussectionwiththefollowingmodifications:(*)I (if there were a non-well-typed XML literal in the(=>)directionweneedtoslightlyamendthedefinitionclass extension ofrdf:XMLLiteral
, then Itoaccountforrdfs:Literalandshowwould not be a model of this rule). This establishes the fact that I is anrdfs-interpretationandrdf-interpretation.
(**) Recall that, by assumption, ex:illxml is not used inthe(<=)directionR. Therefore, changing satisfaction of atomic formulas concerning ex:illxml does not affect satisfaction of R. Weneedtoshowassume thatI$I'C(ex:illxml)=k is amodelofRRDFSunique element, i.e., no other constant is mapped to k.
(*)WeamendthedefinitionofIbychangingthedefinitionofLVtothefollowing:LV=(uniondefine I'F(k) as follows: For every non-well-typed literal of thevaluespacesofallconsidereddatatypes)union(setofallkinDindform (s, rdf:XMLLiteral) such that I'C(tr(s^^rdf:XMLLiteral))=l we define Itruth(Iframe(ICI'F(rdf:type))(k,IC(rdfs:Literal)))=)(l))=t).Clearly,thischangedoesnoteffect; I'truth(I'F(k)(m))=f for each other object m in Dind.Consider RRDF. Satisfaction of Rsimple was established in the
RDFsemanticconditions1and2.Toseethatcondition3isstillsatisfied,considersomeilltypedXMLliteralt.Then,ex:illxml(tr(t))issatisfiedproof inI.Iftr(t)[rdf:type->rdfs:Literal]werethe previous section. Satisfaction of the facts corresponding tobesatisfiedaswell,then,bythesecondlastruleRDF axiomatic triples in I' follows immediately from the definition ofRRDFS,"a"="b"wouldbesatisfied,whichcannotbethecase.Therefore,tr(t)[rdf:type->rdfs:Literal]isnotsatisfiedcommon-rif-rdf-interpretation andthusIL(t)isnotinICEXT(rdfs:Literal).And,sinceIL(t)isnotinthevaluespaceofanyconsidereddatatype,itisnotinLV.Toshowfact that I is anrdfs-interpretation,weneedtoshowthatIsatisfiestheRDFSaxiomatictriplesrdf-interpretation, andthethus satisfies all RDFsemanticconditions.axiomatic triples.
Satisfaction of theaxiomatictriplesex:illxml facts in RRDF follows immediately from theinclusiondefinition oftr(t)inRRDFSforeveryRDFSaxiomatictriplet,I'. Satisfaction of the first, second, and third rule in RRDF follow straightforwardly from the RDF semantic conditions 1, 2, and 3. This establishes the fact that I' is a model of RRDF. ☐
Theorem A RIF-RDF combination
<R,{S1,...,Sn}> is
rdf-satisfiable iff
there is a semantic multi-structure I that is a
model of merge({R RDFS, RRDF,
trR(S1), ...,
trR(Sn)}).
Proof. The theorem follows immediately from the previous theorem andconstructionofI.ConsidertheRDFSsemanticconditions:1xobservation that a combination (respectively, RIF document) isinICEXT(y)rdf-satisfiable (respectively, has a model) if and only if<x,y>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 finite-axiomatic
triples is in IEXT(I( rdf:type )) IC = ICEXT(I( rdfs:Class )) IR = ICEXT(I( rdfs:Resource )) LV = ICEXT(I( rdfs:Literal )) 2 If <x,y>the smallest set such that:
The definitionset of considered datatypes T is defined as before.
RRDFS | |
((Forall tr(s p o .))
) union |
Theorem A RIF-RDF combination
C=<R,{S1,...,Sn}>
rdfs-entails a generalized RDF graph
T if and only if merge({R, RRDFS,
it must consequently be the case that I truthtrR( I frameS1), ...,
trR( I CSn)}) entails
trQ( rdf:type ))( k , IT). C rdfs-entails a condition
formula φ if and only if merge({R,
RRDFS, trR( rdfs:Literal )))= tS1), ...,
trR(Sn)}) entails
φ.
Proof. In the proof we abbreviate merge({R,andthuskRRDFS, trR(S1), ..., trR(Sn)}) with R'.
The proof isinICEXT(I(rdfs:Literal)).So,Isatisfiescondition1.Satisfactionofconditions2through10inIfollowsimmediatelythen obtained fromsatisfactioninIthe proof of correspondence for RDF entailment in the2ndthroughprevious section with the12thrulefollowing modifications: (*) in the (=>) direction we need to slightly amend the definition ofRRDFS.ThisestablishesthefactI to account for rdfs:Literal and show that I is anrdfs-interpretation.rdfs-interpretation and (**)Considerin the (<=) direction we need to show that I' is a model of RRDFS.
SatisfactionofRRDFwasestablishedinthe[[SWC#proof-rdf-entailment|proof]in(*) We amend theprevioussection.Satisfactiondefinition of I by changing thefactscorrespondingdefinition of LV to theRDFSaxiomatictriplesinI$followsimmediatelyfromthedefinitionfollowing:Clearly, this change does not effect satisfaction of the
- LV=(union of
common-rif-rdf-interpretationandthefactvalue spaces of all considered datatypes) union (set of all k in Dind such that Iisanrdfs-interpretation,andthussatisfiesallRDFSaxiomatictriples.truth(Iframe(IC(rdf:type))(k,IC(rdfs:Literal)))=t).1stthroughthe12th,second,RDF axiomatic triples andthirdruleinRRDFSfollowstraightforwardlyfromtheRDFSsemantic conditions 1through10.Satisfactionofthe13thrulefollowsfromthefactthat,givenanill-typedand 2. To see that condition 3 is still satisfied, consider some non-well-typed XML literal t. Then, ex:illxml(tr(t)) is satisfied in I. If tr(t)[rdf:type -> rdfs:Literal] were to be satisfied as well, then, by the second last rule in the definition of RRDFS, "a"="b" would be satisfied, which cannot be the case. Therefore, tr(t)[rdf:type -> rdfs:Literal] is not satisfied and thus IL(t) is not inLV(byRDFsemanticcondition3),ICEXT(rdfs:Literal)=LV,and). And, since IL(t) is not in thefactvalue space of any considered datatype, it is not in LV. To show thattheex:illxmlpredicateI isonlytrueonill-typedXMLliterals.Finally,an rdfs-interpretation, we need to show that I satisfies the RDFS axiomatic triples and the RDF semantic conditions.
Satisfaction of thelastruleaxiomatic triples follows immediately from the inclusion of tr(t) in RRDFSfollowsfromfor every RDFS finite-axiomatic triple t, the fact thatICEXT(rdfs:Literal)=LV,thedefinitionofLVasI is asupersetmodel oftheunionRRDFS, construction of I, and thevaluespacesextension ofalldatatypes,andI in thedefinitionproof of thepred:isDpredicates.ThisestablishesRDF entailment embedding. Consider thefactthatI$isamodelofRRDFS.7.2EmbeddingRIF-OWLDLCombinationsItsemantic conditions:
1 (a) x is knownthatexpressiveDescriptionLogiclanguagessuchasOWLDLcannotbestraightforwardlyembeddedintotypicalruleslanguagessuchasRIFBLD.inthissectionwethereforeconsiderasubsetofOWLDLICEXT(y) if and only if <x,y> is inRIF-OWLDLcombinations.WedefineOWLDLP,whichIEXT(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 inspiredbyso-calledDescriptionLogicprograms(DLP),in IEXT(I(rdfs:domain
)) anddefinehowreasoningwithRIF-OWLDLPcombinationscanbereducedtoreasoningwithRIF.7.2.1IdentifyingOWLDLPOWLDLPrestrictstheOWLDLabstractsyntax(OWL-Semantics),removingdisjunction<u,v> is in IEXT(x) then u is in ICEXT(y)3 If <x,y> is in IEXT(I( rdfs:range
)) andextensionalquantificationfromconsequentsofimplications<u,v> is in IEXT(x) then v is in ICEXT(y)4 IEXT(I( rdfs:subPropertyOf
)) is transitive andremovingnegationreflexive on IP5 If <x,y> is in IEXT(I( rdfs:subPropertyOf
)) then x andequality.Thesemanticsy are in IP and IEXT(x) is a subset ofOWLDLPIEXT(y)6 If x is thesameasOWLDL.Definition.AnOWLDLontologyinabstractsyntaxformIC then <x, I(rdfs:Resource
)> isanOWLDLPontologyin IEXT(I(rdfs:subClassOf
))7 If itrespectsthegrammarbelow.☐Thebasicsyntax<x,y> is in IEXT(I(rdfs:subClassOf
)) then x and y are in IC and ICEXT(x) is a subset ofontologiesICEXT(y)8 IEXT(I( rdfs:subClassOf
)) is transitive andidentifiersreflexive 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
sameasforOWLDL.ontology::='Ontology('[ontologyID]{directive}')'directive::='Annotation('ontologyPropertyIDontologyID')'|'Annotation('annotationPropertyIDURIreference')'|'Annotation('annotationPropertyIDdataLiteral')'|'Annotation('annotationPropertyIDindividual')'|axiom|factdatatypeID::=URIreferenceclassID::=URIreferenceindividualID::=URIreferenceontologyID::=URIreferencedatavaluedPropertyID::=URIreferenceindividualvaluedPropertyID::=URIreferenceannotationPropertyID::=URIreferenceontologyPropertyID::=URIreferencedataLiteral::=typedLiteral|plainLiteraltypedLiteral::=lexicalForm^^URIreferenceplainLiteral::=lexicalForm|lexicalForm@languageTaglexicalForm::=asfirst rule inRDF,aunicodestringthe definition of RRDFS it must be the case that every element k innormalformClanguageTag::=asDind is inRDF,anXMLlanguagetagFactsarethesameasforOWLDL,exceptICEXT(I(rdfs:Resource)). Since IR=Dind, it follows thatequalityandinequality(SameIndividualandDifferentIndividual),aswellasindividualswithoutanidentifierarenotallowed.fact::=individualindividual::='Individual('individualID{annotation}{'type('type')'}{value}')'IR = ICEXT(I(rdfs:Resource
)), establishing 1c. Clearly, every object in ICEXT(I(rdfs:Literal
)) is in LV, by definition. Consider any value::='value('individualvaluedPropertyIDindividualID')'|'value('individualvaluedPropertyIDindividual')'|'value('datavaluedPropertyIDdataLiteral')'type::=RdescriptionThemainrestrictionsposedk in LV. ByOWLDLPontheOWLDLsyntaxareondescriptionsandaxioms.Specifically,weneedtodistinguishbetweendescriptionswhichareallowedontheright-handside(Rdescription)andthoseallowedondefinition, either k is in theleft-handside(Ldescription)value space ofsubclassstatements.WestartwithdescriptionsthatmaybeallowedonbothsidesdataRange::=datatypeID|'rdfs:Literal'description::=classID|restriction|'intersectionOf('{description}')'restriction::='restriction('datavaluedPropertyIDdataRestrictionComponent{dataRestrictionComponent}')'|'restriction('individualvaluedPropertyIDindividualRestrictionComponent{individualRestrictionComponent}')'dataRestrictionComponent::='value('dataLiteral')'individualRestrictionComponent::='value('individualID')'Wethenproceedsome considered datatype or Itruth(Iframe(IC(rdf:type))(k,IC(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 Itruth(IF(IC(pred:isD))(k))=t. By the last rule in RRDFS, it must consequently be the case that Itruth(Iframe(IC(rdf:type))(k,IC(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 RRDFS. This establishes the fact that I is an rdfs-interpretation.
(**) Consider RRDFS. Satisfaction of RRDF was established in the proof in the previous section. Satisfaction of the facts corresponding to the RDFS axiomatic triples in I' follows immediately from the definition of common-rif-rdf-interpretation and the fact that I is an rdfs-interpretation, and thus satisfies all RDFS axiomatic triples.
Satisfaction of the 1st through the 12th rule in RRDFS follow straightforwardly from the RDFS semantic conditions 1 through 10. Satisfaction of the 13th rule follows from the fact that, given an ill-typed XML literal t, IL(t) is not in LV (by RDF semantic condition 3), ICEXT(rdfs:Literal)=LV, and the fact that the ex:illxml predicate is only true on ill-typed XML literals. Finally, satisfaction of the last rule in RRDFS 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 RRDFS. ☐
Theorem A RIF-RDF combination <R,{S1,...,Sn}> is rdfs-satisfiable if and only if there is a semantic multi-structure I that is a model of merge({R, RRDFS, trR(S1), ..., trR(Sn)}).
Proof. The theorem follows immediately from the previous theorem and the observation that a combination (respectively, RIF document) is rdfs-satisfiable (respectively, has a model) if and only if it does not entail the condition formula "a"="b". ☐
It is known that expressive Description Logic languages such as OWL DL cannot be straightforwardly embedded into typical rules languages such as RIF BLD (RIF-BLD), because of features such as disjunction and negation.
In this section we consider a subset of OWL DL in RIF-OWL DL combinations. We define OWL DLP, which is inspired by so-called Description Logic Programs (DLP), and define how reasoning with RIF-OWL DLP combinations can be reduced to reasoning with RIF.
The embedding of RIF-OWL DL combinations is not defined for combinations that include infinite OWL ontologies and for combinations that include ontologies with RDF URI references that are not absolute IRIs or plain literals that are not in the lexical space of the xs:string datatype. In addition, each datatype used in the combination must have associated positive and negative guard predicates (DTB).
OWL DLP restricts the OWL DL abstract syntax (OWL-Semantics), disallowing disjunction and extensional quantification in consequents of implications, as well as negation and equality. The semantics of OWL DLP is that of OWL DL.
The syntax is defined through an EBNF grammar, which is derived from the grammar of the OWL abstract syntax (OWL-Semantics). Any OWL DL ontology in abstract syntax form that conforms to this grammar is an OWL DLP ontology.
The basic syntax of ontologies and identifiers is the same as for OWL DL.
ontology ::= 'Ontology(' [ ontologyID ] { directive } ')' directive ::= 'Annotation(' ontologyPropertyID ontologyID ')' | 'Annotation(' annotationPropertyID absolute-IRI ')' | 'Annotation(' annotationPropertyID dataLiteral ')' | 'Annotation(' annotationPropertyID individual ')' | axiom | fact
datatypeID ::= absolute-IRI classID ::= absolute-IRI individualID ::= absolute-IRI ontologyID ::= absolute-IRI datavaluedPropertyID ::= absolute-IRI individualvaluedPropertyID ::= absolute-IRI annotationPropertyID ::= absolute-IRI ontologyPropertyID ::= absolute-IRI
dataLiteral ::= typedLiteral | plainLiteral typedLiteral ::= lexicalForm^^absolute-IRI plainLiteral ::= lexicalForm | lexicalForm@languageTag lexicalForm ::= as in RDF, a unicode string in normal form C languageTag ::= as in RDF, an XML language tag
Facts are the same as for OWL DL, except that equality and
inequality (SameIndividual and
DifferentIndividual), as well as individuals without an
identifier, are not allowed.
fact ::= individual individual ::= 'Individual(' individualID { annotation } { 'type(' type ')' } { value } ')' value ::= 'value(' individualvaluedPropertyID individualID ')' | 'value(' individualvaluedPropertyID individual ')' | 'value(' datavaluedPropertyID dataLiteral ')'
type ::= Rdescription
The main restrictions posed by OWL DLP on the OWL DL syntax are on descriptions and axioms. Specifically, OWL DLP distinguishes between descriptions that are allowed on the right-hand side (Rdescription) and those allowed on the left-hand side (Ldescription) of subclass statements.
We start with descriptions that may be allowed on both sides
dataRange ::= datatypeID | 'rdfs:Literal'
description ::= classID | restriction | 'intersectionOf(' { description } ')'
restriction ::= 'restriction(' datavaluedPropertyID dataRestrictionComponent { dataRestrictionComponent } ')' | 'restriction(' individualvaluedPropertyID individualRestrictionComponent { individualRestrictionComponent } ')' dataRestrictionComponent ::= 'value(' dataLiteral ')' individualRestrictionComponent ::= 'value(' individualID ')'
We then proceed with the individual sides
Ldescription ::= description | Lrestriction | 'unionOf(' { Ldescription } ')' | 'intersectionOf(' { Ldescription } ')' | 'oneOf(' { individualID } ')'
Lrestriction ::= 'restriction(' datavaluedPropertyID LdataRestrictionComponent { LdataRestrictionComponent } ')' | 'restriction(' individualvaluedPropertyID LindividualRestrictionComponent { LindividualRestrictionComponent } ')' LdataRestrictionComponent ::= 'someValuesFrom(' dataRange ')' | 'value(' dataLiteral ')' LindividualRestrictionComponent ::= 'someValuesFrom(' Ldescription ')' | 'value(' individualID ')'
Rdescription ::= description | Rrestriction | 'intersectionOf(' { Rdescription } ')'
Rrestriction ::= 'restriction(' datavaluedPropertyID RdataRestrictionComponent { RdataRestrictionComponent } ')' | 'restriction(' individualvaluedPropertyID RindividualRestrictionComponent { RindividualRestrictionComponent } ')' RdataRestrictionComponent ::= 'allValuesFrom(' dataRange ')' | 'value(' dataLiteral ')' RindividualRestrictionComponent ::= 'allValuesFrom(' Rdescription ')' | 'value(' individualID ')'
Finally, we turn to axioms. We start with class axioms.
axiom ::= 'Class(' classID ['Deprecated'] 'complete' { annotation } { description } ')' axiom ::= 'Class(' classID ['Deprecated'] 'partial' { annotation } { Rdescription } ')'
axiom ::= 'DisjointClasses(' Ldescription Ldescription { Ldescription } ')' | 'EquivalentClasses(' description { description } ')' | 'SubClassOf(' Ldescription Rdescription ')'
axiom ::= 'Datatype(' datatypeID ['Deprecated'] { annotation } )'
Property axioms in OWL DLP restrict those in OWL DL by disallowing functional and inverse functional properties, because these involve equality.
axiom ::= 'DatatypeProperty(' datavaluedPropertyID ['Deprecated'] { annotation } { 'super(' datavaluedPropertyID ')'} { 'domain(' description ')' } { 'range(' dataRange ')' } ')' | 'ObjectProperty(' individualvaluedPropertyID ['Deprecated'] { annotation } { 'super(' individualvaluedPropertyID ')' } [ 'inverseOf(' individualvaluedPropertyID ')' ] [ 'Symmetric' ] [ 'Transitive' ] { 'domain(' description ')' } { 'range(' description ')' } ')' | 'AnnotationProperty(' annotationPropertyID { annotation } ')' | 'OntologyProperty(' ontologyPropertyID { annotation } ')'
axiom ::= 'EquivalentProperties(' datavaluedPropertyID datavaluedPropertyID { datavaluedPropertyID } ')' | 'SubPropertyOf(' datavaluedPropertyID datavaluedPropertyID ')' | 'EquivalentProperties(' individualvaluedPropertyID individualvaluedPropertyID { individualvaluedPropertyID } ')' | 'SubPropertyOf(' individualvaluedPropertyID individualvaluedPropertyID ')'
Definition. An OWL DL ontology in abstract syntax form is an OWL DLP ontology if it conforms with the grammar above. ☐
Recall that the semantics of frame formulas in DL-document formulas is different from the semantics of frame formulas in RIF documents. Nonetheless, DL-document formulas can be embedded into RIF documents, by translating frame formulas to predicate formulas. The mapping tr is the identity mapping on all RIF formulas, with the exception of frame formulas, as defined in the following table.
In the table, the mapping tr' is an injective function that maps constants to new constants, i.e., constants that are not used in the original document or its vicinity (e.g., entailed or entailing formula). It "generates" a new constant from an existing one.
RIF Construct | Mapping |
---|---|
Term t | tr(t)=t |
Atomic formula φ that is not a frame formula | tr(φ)=φ |
a[b1->c1 ... bn->cn], with n≥2 | tr(a[b1->c1 ... bn->cn])=And( tr(a[b1->c1]) ... tr(a[bn->cn])) |
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(φ1 ... φn) | tr(And(φ1 ... φn))=And(tr(φ1) ... tr(φn)) |
Or(φ1 ... φn) | tr(Or(φ1 ... φn))=Or(tr(φ1) ... tr(φn)) |
φ1 :- φ2 | tr(φ1 :- φ2)=tr(φ1) :- tr(φ2) |
Forall ?V1 ... ?Vn(φ) | tr(Forall ?V1 ... ?Vn(φ))=Forall ?V1 ... ?Vn(tr(φ)) |
Group(φ1 ... φn) | tr(Group(φ1 ... φn))=Group(tr(φ1) ... tr(φn)) |
Document(directive1 ... directiven Γ) | tr(Document(directive1 ... directiven Γ))=Document(directive1 ... directiven tr(Γ)) |
For the purpose of making statements about this embedding, we define a notion of entailment for DL-document formulas.
Definition. A RIF-BLD DL-document formula R dl-entails a DL-condition φ if for every dl-semantic multi-structure I that is a model of R it holds that TValI(φ)=t. ☐
The following lemma establishes faithfulness with respect to entailment of the embedding.
RIF-BLD DL-document formula Lemma A RIF-BLD DL-document formula R dl-entails a DL-condition φ if and only if tr(R) entails tr(φ).
Proof. We prove both directions by contradiction: if the entailment does not hold on the one side, we show that it also does not hold on the other.
(=>) Assume tr(R) does not entail tr(φ). This means there is some semantic multi-structure I = <TV, DTS, D, Dind, Dfunc, IC, IV, IF, Iframe, ISF, Isub, Iisa, I=, Iexternal, Itruth> that is a model of tr(R), but not of tr(φ).
Consider the dl-semantic multi-structure I* = <TV, DTS, D, Dind, Dfunc, I*C, IV, IF, I*frame', ISF, Isub, Iisa, I=, Iexternal, Itruth>, with I*C and I*frame' defined as follows: Let t be an element in D such that Itruth(t)=t and let f in D be such that Itruth(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')=IC(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 Dind, if Itruth(IF(IC(c'))(k))=t, I*frame'(k)((IC(rdf:type), IC(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 Dind × Dind, if Itruth(IF(IC(b'))(k,l))=t, I*frame'(k)((IC(b),l))=t,
- if I*frame'(k)((b1,...,bn))=t and I*frame'(k)((c1,...,cm))=t for any two finite bags (b1,...,bn) and (c1,...,cm), then I*frame'(k)((b1,...,bn,c1,...,cm))=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 DL-condition, I* is a model of a[b -> c] iff I is a model of tr(a[b -> c]). This argument straightforwardly extends to the case of frames with multiple bis and cis, since in RIF semantic structures the following condition is required to hold: TValI(a[b1->c1 ... bn->cn]) = t if and only if TValI(a[b1->c1]) = ... = TValI(a[bn->cn]) = t (RIF-BLD).
Consider the case b=rdf:type. Then,
I* is a model of a[b -> c] iff Itruth(I*frame'(I(a))(IC(rdf:type),IC(c)))=t.
From the definition of I* we obtain
Itruth(I*frame'(I(a))(IC(rdf:type),IC(c)))=t iff I*frame'(I(a))(IC(rdf:type),IC(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))(IC(rdf:type),IC(c))=t iff Itruth(IF(IC(tr'(c)))(I(a)))=t.
Finally, since tr(a[b -> c])=tr'(c)(a), we obtain
Itruth(IF(IC(tr'(c)))(I(a)))=t iff I is a model of tr(a[b -> c]).
From this chain of equivalences follows that I* is a model of a[b -> c] iff I is a model of tr(a[b -> c]).
The argument for the case b≠rdf:type is analogous, thereby obtaining (+).
(<=) Assume R does not dl-entail φ. This means there is some dl-semantic multi-structure I = <TV, DTS, D, Dind, Dfunc, IC, IV, IF, Iframe', ISF, Isub, Iisa, I=, Iexternal, Itruth> that is a model of R, but not of φ. Let B be the set of constant symbols occurring in the frame formulas of the forms a[rdf:type -> b] and a[b -> c] in R or φ.
Consider the semantic multi-structure I* = <TV, DTS, D, Dind, Dfunc, I*C, IV, I*F, I*frame, ISF, Isub, Iisa, I=, Iexternal, Itruth>. Let t and f in D be such that Itruth(t)=t and Itruth(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)=IC(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 Dind, if Itruth(Iframe'(k)((IC(rdf:type), IC(c)))=t, I*F(I*C(tr'(c)))(k)=t; I*F(I*C(tr'(c)))(k')=f for any other k' in Dind,
- for every b in B, given a pair (k, l) in Dind × Dind, if Itruth(Iframe(k)((IC(b),l)))=t, I*F(tr'(b))(k,l)=t;I*F(tr'(b))(k',l')=f for any other pair (k', l') in Dind × Dind, and
- I*F(c')=IF(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 | trN(
Ontology( [ ontologyID ]
directive1
...
directiven )
) |
Ontology( trN(directive1) ... trN(directiven) ) |
|
2 | trN(Annotation( ... )) | ||
3 | trN(
Individual( individualID
annotation1
...
annotationn
type1
...
typem
value1
...
valuek )
) |
trN(Individual( individualID type1 )) ... trN(Individual( individualID typem )) trN(Individual( individualID value1 )) ... trN(Individual( individualID valuek )) |
|
4 | trN(
Individual( individualID
type(intersectionOf(
description1
...
descriptionn
))
) |
trN(Individual( individualID type(description1) )) ... trN(Individual( individualID type(descriptionn) )) |
|
5 | trN(
Individual( individualID type(X))) |
Individual( individualID type(X)) |
X is a classID or value restriction |
6 | trN(
Individual( individualID type(restriction(propertyID allValuesFrom(X))))) |
trN(
SubClassOf( oneOf(individualID) restriction(propertyID allValuesFrom(X))) ) |
|
7 | trN(
Individual( individualID value(propertyID b))) |
Individual( individualID value(propertyID b)) |
b is an individualID or dataLiteral |
8 | trN(
Individual( individualID1 value(propertyID Individual( individualID2 ... )))) |
trN(
Individual( individualID1 value(propertyID individualID2) ) trN(Individual( individualID2 ... )) |
|
9 | trN(
Class( classID [Deprecated]
complete
annotation1
...
annotationn
description1
...
descriptionm )
) |
trN(
EquivalentClasses(classID
intersectionOf(description1
...
descriptionm )
) |
|
10 | trN(
Class( classID [Deprecated]
partial
annotation1
...
annotationn
description1
...
descriptionm )
) |
trN(
SubClassOf(classID
intersectionOf(description1
...
descriptionm )
) |
|
11 | trN(
DisjointClasses(
description1
...
descriptionm )
) |
trN(SubClassOf(intersectionOf(description1
description2) owl:Nothing))
... trN(SubClassOf(intersectionOf(description1 descriptionm) owl:Nothing)) ... trN(SubClassOf(intersectionOf(descriptionm-1 descriptionm) owl:Nothing)) |
|
12 | trN(
EquivalentClasses(
description1
...
descriptionm )
) |
trN(SubClassOf(description1 description2)) trN(SubClassOf(description2 description1)) ... trN(SubClassOf(descriptionm-1 descriptionm)) trN(SubClassOf(descriptionm descriptionm-1)) |
|
13 | trN(
SubClassOf(description X)) |
SubClassOf(description X) |
X is a description that does not contain intersectionOf |
14 | trN(
SubClassOf(description
...intersectionOf(
description1
...
descriptionn
)...)
) |
trN(SubClassOf(description ...description1...)) ... trN(SubClassOf(description ...descriptionn...)) |
|
15 | trN(Datatype( ... )) | ||
16 | trN(
DatatypeProperty( propertyID [ Deprecated ]
annotation1
...
annotationn
super(superproperty1)
...
super(superpropertym)
domain(domaindescription1)
...
domain(domaindescriptionj)
range(rangedescription1)
...
range(rangedescriptionk) )
) |
SubPropertyOf(propertyID superproperty1)
...
SubPropertyOf(propertyID superpropertym)
trN(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescription1)) ... trN(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescriptionj)) trN(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescription1))) ... trN(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescriptionk))) |
|
17 | trN(
ObjectProperty( propertyID [ Deprecated ]
annotation1
...
annotationn
super(superproperty1)
...
super(superpropertym)
[ inverseOf( inversePropertyID ) ]
[ Symmetric ]
[ Transitive ]
domain(domaindescription1)
...
domain(domaindescriptionl)
range(rangedescription1)
...
range(rangedescriptionk) )
) |
SubPropertyOf(propertyID superproperty1)
...
SubPropertyOf(propertyID superpropertym)
trN(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescription1)) ... trN(SubClassof(restriction(propertyID someValuesFrom(owl:Thing)) domaindescriptionl)) trN(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescription1))) ... trN(SubClassof(owl:Thing restriction(propertyID allValuesFrom(rangedescriptionk))) ObjectProperty( propertyID [ inverseOf( inversePropertyID ) ] ) ObjectProperty( propertyID [ Symmetric ] ) ObjectProperty( propertyID [ Transitive ] ) |
|
18 | trN(
EquivalentProperties(
property1
...
propertym )
) |
trN(SubPropertyOf(property1 property2)) trN(SubPropertyOf(property2 property1)) ... trN(SubPropertyOf(propertym-1 propertym)) trN(SubPropertyOf(propertym propertym-1)) |
The result of the normalization is a set of individual sides Ldescription ::= description | Lrestriction | 'unionOf(' { Ldescription } ')' | 'intersectionOf(' { Ldescription } ')' | 'oneOf(' { individualID } ')' Lrestriction ::= 'restriction(' datavaluedPropertyID LdataRestrictionComponent { LdataRestrictionComponent } ')' | 'restriction(' individualvaluedPropertyID LindividualRestrictionComponent { LindividualRestrictionComponent } ')' LdataRestrictionComponent ::= 'someValuesFrom(' dataRange ')' | 'value(' dataLiteral ')' LindividualRestrictionComponent ::= 'someValuesFrom(' description ')' | 'value(' individualID ')' Rdescription ::= description | Rrestriction | 'intersectionOf(' { Rdescription } ')' Rrestriction ::= 'restriction(' datavaluedPropertyID RdataRestrictionComponent { RdataRestrictionComponent } ')' | 'restriction(' individualvaluedPropertyID RindividualRestrictionComponent { RindividualRestrictionComponent } ')' RdataRestrictionComponent ::= 'allValuesFrom(' dataRange ')' | 'value(' dataLiteral ')' RindividualRestrictionComponent ::= 'allValuesFrom(' description ')' | 'value(' individualID ')' Finally,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,{O1,...,On}>, where O1,...,On are OWL DLP ontologies that do not import ontologies, C owl-dl-entails φ iff C'=<R,{trN(O1),...,trN(On)}> owl-dl-entails φ.
Proof. Weturnprove both directions by contradiction: if the entailment does not hold on the one side, we show that it also does not hold on the other.
(=>) Assume C' does not owl-dl-entail φ. This means there is a common-rif-owl-dl-interpretation (I, I) that is a model of C', but I is not a model of φ.
Consider the pair (I, I*), where I* is obtained from I by suitably extending EC and ER toaxioms.Westartwithclassaxioms.axiom::='Class('classID['Deprecated']'complete'{annotation}{description}')'axiom::='Class('classID['Deprecated']'partial'{annotation}{Rdescription}')'axiom::='DisjointClasses('LdescriptionLdescription{Ldescription}')'|'EquivalentClasses('description{description}')'|'SubClassOf('LdescriptionRdescription')'axiom::='Datatype('datatypeID['Deprecated']{satisfy the annotation})'PropertyaxiomsinOWLDLPrestrictthoseproperties. Clearly, (I, I*) is a common-rif-owl-dl-interpretation, since the extension realized inOWLDLI* does not affect any of the conditions on common-rif-owl-dl-interpretations. Bydisallowingfunctionalthe interpretation of axioms andinversefunctionalproperties,becausetheseinvolveequality.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('datavaluedPropertyIDdatavaluedPropertyID{datavaluedPropertyID}')'|'SubPropertyOf('datavaluedPropertyIDdatavaluedPropertyID')'|'EquivalentProperties('individualvaluedPropertyIDindividualvaluedPropertyID{individualvaluedPropertyID}')'|'SubPropertyOf('individualvaluedPropertyIDindividualvaluedPropertyID')'7.2.2EmbeddingRIFDL-documentsintoRIFBLDRecallfacts and the EC extension table in sections 3.3 and 3.2 in (OWL-Semantics) it is easy to verify that, for any directive d in I, if I satisfies trN(d), I* satisfies d. Therefore, I* satisfies O1,..., and On, and thus (I, I*) satisfies C. Since I is not a model of φ, C does not owl-dl-entail φ.
(<=) Assume C does not owl-dl-entail φ. This means there is a common-rif-owl-dl-interpretation (I, I) thatthesemanticsis a model offrameformulasinDL-documentsC, but I isdifferentfromnot a model of φ. It is easy to verify, by thesemanticsinterpretation offrameformulasinRIFBLD.Frameformulasaxioms and facts and the EC extension table inDL-documentsareembeddedaspredicatessections 3.3 and 3.2 inRIFBLD.Themapping(OWL-Semantics), that I satisfies trN(O1),..., and trN(On). So, (I, I) is a model of C', and thus C' does not owl-dl-entail φ. ☐
We now proceed with the identity mapping on allembedding of normalized OWL DLP
ontologies into RIF formulas, withDL-document formulas. The exceptionembedding is an
extension of frame formulas,the embedding function tr. The embeddings of IRIs and
literals is as defined in the following table. Mapping RIF DL-documents to RIF documents. RIF Construct Mapping Term x tr( x )= x Atomic formula xSection Embedding Symbols.
In the following, let T be the set of considered datatypes union
the set of datatypes used in any ontology 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
framevalue of the datatype (cf. (RIF-DTB)).
# | Normalized OWL | RIF RIF-BLD DL-document formula | Condition on translation |
---|---|---|---|
1 | trO(
Ontology(
directive1
...
directiven
)
) |
trO(directive1)
...trO(directiven) |
|
2 | trO(
Individual( individualID type(A) )) |
tr( |
A is a |
3 | trO(
Individual( individualID type(restriction(propertyID value(b))) )) |
tr( |
|
4 | trO(
Individual( individualID value(propertyID b) )) |
tr(individualID)[tr(propertyID) -> tr(b)] |
|
5 | trO(
SubPropertyOf(property1 property2)
) |
Forall ?x ?y (?x[tr(property2) -> ?y] :- ?x[tr(property1) -> ?y]) |
|
6 | trO(
ObjectProperty(propertyID)) |
||
7 | trO(
ObjectProperty(property1
inverseOf(property2) )
) |
Forall ?x ?y (?y[tr(property2) -> ?x] :- ?x[tr(property1) -> ?y]) Forall ?x ?y (?y[tr(property1) -> ?x] :- ?x[tr(property2) -> ?y]) |
|
8 | trO(
ObjectProperty(propertyID Symmetric )) |
Forall ?x ?y (?y[tr(propertyID) -> ?x] :- ?x[tr(propertyID) -> ?y]) |
|
9 | trO(
ObjectProperty(propertyID Transitive )) |
Forall ?x ?y ?z (?x[tr(propertyID) -> ?z] :- And( ?x[tr(propertyID) -> ?y] ?y[tr(propertyID) -> ?z])) |
|
10 | trO(
SubClassOf(description1 description2)
) |
trO(description1, |
|
11 | trO(A |
?x[rdf:type -> tr(A |
A |
12 |
trO(description1, |
Forall ?x (trO(X, ?x) :- trO(description1, ?x ) |
X is a |
13 |
trO(description1 |
|
X is a classID, datatypeID or value restriction |
14 | trO( |
|
|
15 | tr |
|
|
16 | trO(oneOf(value1 ...
|
Or( ?x =
trO(value1 |
|
17 | tr |
Exists ?y(And(?x[tr(propertyID)
|
|
18 | trO(restriction(propertyID value(valueID)), ?x) |
|
Besides the embedding in the previous table, we also need an axiomatization of some of the aspects of the OWL DL semantics, e.g., separation between individual and datatype domains. This axiomatization is defined relative to an OWL vocabulary V and a datatype map D, which includes all datatypes in T. In the table, for a given datatype d, L2V(d) is the lexical-to-value mapping of d.
ROWL-DL(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=trN( Class( classID [Deprecated] partial annotationO).
The following lemma establishes faithfulness of the embedding.
Normalized Combination
Embedding Lemma Given a datatype map D conforming with
T, a RIF-OWL-DL-combination
C=<R,{O1 ... annotation,...,On description}>,
where O1 ... description m ) ) tr,...,On are normalized
OWL DLP ontologies
with vocabulary V, owl-dl-entails a DL-condition φ with respect to D iff
merge({R, ROWL-DL( SubClassOf(classID intersectionOf(description 1 ... description m ) )V),
tr NO( DisjointClasses( descriptionO1 ... description m ) )), ...,
trO(On)}) dl-entails φ.
Proof. We prove both directions by contradiction: if the entailment does not hold on one side, we show that it also does not hold on the other.
In the proof we abbreviate merge({R, ROWL-DL(SubClassOf(intersectionOf(description1description2)owl:Nothing))...V), trNO(SubClassOf(intersectionOf(descriptionO1descriptionm)owl:Nothing))...), ..., trNO(SubClassOf(intersectionOf(descriptionm-1descriptionm)owl:Nothing))trOn)} with R'.
(=>) Assume R' does not dl-entail φ. This means there is a dl-semantic multi-structure I = <TV, DTS, D, Dind, Dfunc, IC, IV, IF, Iframe', ISF, Isub, Iisa, I=, Iexternal, Itruth> that is a model of R', but not of φ.
Consider the pair (EquivalentClasses(description1...descriptionm)I*,I), where I* = <TV, DTS, D*, D*ind union (union of the value spaces of all datatypes in the range of D), Dfunc, IC, IV, IF, Iframe', ISF, Isub, Iisa, I=, Iexternal, Itruth> is such thatand I = <R, EC, ER, L, S, LV> is a tuple defined as follows:
- D*ind=Dind 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 VL and seven sets of IRIs, VC, VD, VI, VDP, VIP, VAP, and VO, which are the sets of class, datatype, individual, data-valued property, individual-valued property, annotation property, and ontology identifiers. According to its definition, an abstract OWL interpretation with respect to a datatype map D must fulfill the following conditions, where L(d) denotes the lexical space, V(d) denotes the value space and L2V(d) denotes to lexical-to-value mapping of a datatype d:
- 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')
trN(SubClassOf(description1description2= the value space of D(d'), if D(d') is defined,- EC(c)
trN= set of all objects k such that Itruth(SubClassOf(description2description1))...trNIframe'(SubClassOf(descriptionm-1descriptionm))trNIC(SubClassOf(descriptionmdescriptionm-1))trNrdf:type))(k,IC(SubClassOf(descriptionX))SubClassOf(descriptionX)Xisadescription<c>))) = t, for every class identifier or datatype identifier c≠rdfs:Literal in V thatdoesis notcontainintersectionOftrN(SubClassOf(description...intersectionOf(description1...descriptionn)...)in the domain of D,- ER(p)
trN= set of all pairs (SubClassOf(description...description1...)k, l)...trNsuch that Itruth(SubClassOf(description...descriptionn...))trNIframe'(Datatype(...))trNIC(DatatypeProperty(propertyID[Deprecated]annotation1...annotationnsuper(superproperty1)...super(superpropertym)domain(domaindescription1<p>))( k, l ))) = t (true), for every data valued and individual valued property identifier p in V;- L((s, d)) = IC("s"^^d)
...domain(domaindescriptionjfor every well-typed literal (s, d)range(rangedescription1in V;- S(i)
...range(rangedescriptionk= IC(<i>) for every IRI i in V.Condition 1
- 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 : VC → 2O
- EC : VD → 2LV
- ER : VDP → 2O×LV
- ER : VIP → 2O×O
- ER : VAP ∪ { rdf:type } → 2R×R
- ER : VOP → 2R×R
- L : TL → LV, where TL is the set of typed literals in VL
- S : VI ∪ VC ∪ VD ∪ VDP ∪ VIP ∪ VAP ∪ VO ∪ { owl:Ontology, owl:DeprecatedClass, owl:DeprecatedProperty } → R
- S(VI) ⊆ O
- EC(owl:Thing)
SubPropertyOf(propertyIDsuperproperty1= O ⊆ R, where O is nonempty and disjoint from LV- EC(owl:Nothing)
...SubPropertyOf(propertyIDsuperpropertym= { }- EC(rdfs:Literal)
trN(SubClassof(restriction(propertyIDsomeValuesFrom(owl:Thing))domaindescription= 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
))...trN(SubClassof(restriction(propertyIDsomeValuesFrom(owl:Thing))domaindescriptionj))trNis met because D is a nonempty set. Clearly LV is a subset of R and contains the value spaces for each datatype in D, which include the sets of Unicode strings and pairs of Unicode strings and language tags, since the xs:string and rif:text datatypes are included in D, by the fact that D is conforming and the two datatypes are RIF-required; therefore, condition 2 is met.
When referring to rules in the remainder we mean rules in ROWL-DL(SubClassof(owl:Thingrestriction(propertyIDallValuesFrom(rangedescription1))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)...trNit holds that k in EC(owl:Thing). But if k in EC(c), then, by definition, Itruth(SubClassof(owl:Thingrestriction(propertyIDallValuesFrom(rangedescriptionk)))trNIframe'(ObjectProperty(propertyID[Deprecated]annotation1...annotationnsuper(superproperty1)...super(superpropertym)[inverseOf(inversePropertyID)][Symmetric][Transitive]domain(domaindescription1)...domain(domaindescriptionl)range(rangedescription1)...range(rangedescriptionIC(rdf:type))(k)))SubPropertyOf(propertyIDsuperproperty1)...SubPropertyOf(propertyIDsuperpropertym)trN,IC(SubClassof(restriction(propertyIDsomeValuesFrom(owl:Thing))domaindescription1))...trN<C>))) = t. But then, by rule (iii), it must be the case that Itruth(SubClassof(restriction(propertyIDsomeValuesFrom(owl:Thing))domaindescriptionl))trNIframe'(SubClassof(owl:Thingrestriction(propertyIDallValuesFrom(rangedescription1)))...trNIC(SubClassof(owl:Thingrestriction(propertyIDallValuesFrom(rangedescriptionrdf:type))(k)))ObjectProperty(propertyID[inverseOf(inversePropertyID)])ObjectProperty(propertyID[Symmetric])ObjectProperty(propertyID[Transitive])trN,IC(EquivalentProperties(property1...propertym))trN<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 Itruth(SubPropertyOf(property1property2))trNIframe'(SubPropertyOf(property2property1))...trNIC(SubPropertyOf(propertym-1propertym))trNrdf:type))(k,IC(SubPropertyOf(propertympropertym-1))<Diri>))) = t, but also Itruth(IF(IC(isNotDT))(k)) = t (since theresultvalue space of thenormalizationdatatype is asetsubset ofindividualpropertyvalue,individualtyping,subclass,subproperty,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 andpropertyinverse,symmetry6 can be shown similarly, exploiting rules (v), (vi), (vii), andtransitivestatements.7.2.3.2EmbeddingWenowproceedwith(xv).
ER maps annotation and ontology properties to theembeddingofnormalizedOWLDLPontologiesintoaRIFDL-document.empty set, so conditions 7 and 8 are trivially satisfied.
IC maps well-typed literals "s"^^d to objects in theembeddingextendsvalue space of d. Since L is defined in terms of IC and since theembeddingfunctiontr.value spaces of all datatypes are included in LV, condition 9 is satisfied.
Condition 10 is clearly satisfied by theembeddingsdefinition ofIRIsS andliteralssince 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 isasdefinedno element in theSectionEmbeddingSymbols.LetTbevalue 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 thesetdefinition ofconsidereddatatypes(cf.Section5IC; observe that for every typed literal "v"^^d' must hold that d' is in the domain of(RIF-BLD)),D, since D includes all datatypes under consideration.
Assume there exists an ill-typed literal "v"^^d' in V, i.e.,Tincludesatleastalldatatypesusedv is not in thecombinationunderconsiderationandalldatatypesrequiredforRIF-BLD(RIF-DTBlexical space D(d').Editor'sNote:Theterminology"considereddatatype"mightchangeifSince I satisfies rule (xiv), "a"="b" must be satisfied, which is a contradiction. So, there is no ill-typed literal and thus condition 18 is satisfied.
This establishes theterminologyfact that I ischangedan abstract OWL interpretation.
Consider now any ontology O in {O1,...,On}. To establish that I satisfies O, we need to establish five conditions (cf. Section 3.4 inBLD.By(RIF-DTB),OWL-Semantics)):Conditions 1
- each
datatypeURI reference inThasanassociatedlabelDATATYPE(e.g.,thelabelofxs:stringisString)andpositiveandnegativeguardspred:isDATATYPEandpred:isNotDATATYPE,whichcanbeO usedtotestwhetheras aparticularobjectis(resp.,class ID (datatype ID, individual ID, data-valued property ID, individual-valued property ID, annotation property ID, annotation ID, ontology ID) belongs to VC (VD, VI, VDP, VIP, VAP, VO, respectively);- each literal in O belongs to VL;
- I satisfies each directive in O, except for Ontology Annotations;
- there is
not)avaluesome o ∈ R with <o,S(owl:Ontology)> ∈ ER(rdf:type) such that for each Ontology Annotation of thedatatype.Editor'sNote:Verifyform Annotation(p v), <o,S(v)> ∈ ER(p) and thatthesethingsaredefinedintheDTBdocumentbeforepublication.EmbeddingOWLDLP.NormalizedOWLRIFDL-documenttrif O(directive1...directivehas name n, then S(n)tr= o(; and- I satisfies each ontology mentioned in an owl:imports annotation directive of O.
)...trand 2 are satisfied by the fact O(is an ontology of vocabulary V.
Conditions 4 and 5 are trivially satisfied, because normalized OWL DLP ontologies do not contain annotations and do not have names.
Consider any directiven)trd in O; d has one of the following forms (cf. the right 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 Itruth(
- class membership statement of the form Individual (
Individual(individualID type(A) ))tr(individualID)[rdf:type->tr(A)], where A is aclassIDtrO(Individual(individualIDtype(restriction(propertyIDvalue(b)))))tr(individualID)[tr(propertyID)->tr(b)]trO(Individual(individualIDvalue(propertyIDb)))tr(individualID)[tr(propertyID)->tr(b)]trO(SubPropertyOf(property1property2))Forall ?x ?y(?x[tr(property2)-> ?y] :- ?x[tr(property1)-> ?y])trO(ObjectProperty(propertyID))trO(ObjectProperty(property1inverseOf(property2)))Forall ?x ?y(?y[tr(class ID,- membership of value restriction,
- property
2)-> ?x] :- ?x[tr(value statement,- subproperty statement,
- inverse property
1)-> ?y])Forall ?x ?y(?y[tr(statement,- symmetric property
1)-> ?x] :- ?x[tr(statement,- transitive property
2)-> ?y])trO(ObjectProperty(propertyIDSymmetric))Forall ?x ?y(?y[tr(propertyID)-> ?x] :- ?x[tr(propertyIDstatement, or- subclass statement SubClassOf(X Y)
-> ?y])trO.ObjectProperty(propertyIDTransitive))Forall ?x ?y ?z(?x[tr(propertyID)-> ?z] :-And( ?x[tr(propertyID)-> ?y] ?y[tr(propertyID)-> ?z]))trOIframe'(SubClassOf(description1description2))trOIC(description1,description2,?x)trOrdf:type))(IC(description1,X,?x)Forall ?x<individualID>),IC(trO<A>))) = t. Consequently, IC(X,?x<individualID>):-trOis in EC(<A>). Since, in addition, S(<individualID>)=IC(description1,?x<individualID>), we have that S(<individualID>)Xis in EC(<A>), and thus d is satisfied in I. Similar for statements of the forms 2 and 3.
Consider aclassID,datatypeIDorvaluerestrictiontrOsubproperty statement SubPropertyOf(p q) and a pair (description1,restriction(property1allValuesFrom(...restriction(propertynallValuesFrom(X))...)),?xk, l)Forall ?x ?y1... ?ynin ER(<p>). Then, by construction of I, Itruth(trOIframe'(XIC(<p>))( k,?yn):-And(trOl ))) = t. But, by tr(d), it must be the case that also Itruth(description1Iframe'(IC(<q>))( k,?x)?x[tr(property1l ))) = t. But then, (k,l)-> ?y1] ?y1[tr(property2must 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)-> ?y2]... ?yn-1[tr(propertynand consider any k in EC(X), where EC is as in the EC Extension Table in (OWL-Semantics). We show, by induction, that I* satisfies trO(X)-> ?yn]))when ?x is assigned to k.
If X is a classID,datatypeIDorthen satisfaction of tr(X) follows by an analogous argument as that for directives of form 1. Similar for value restrictions. If X is a some-value restrictiontrO(of type Z on a property p,?xthen there must be some object l such that (k,l)?x[rdf:type->tr(Ain ER(p)]Asuch that l isaclassIDordatatypeIDtrOin EC(Z). By induction we have satisfaction of tr(Z) for some variable assignment. Then, by definition of I, we have Itruth(intersectionOf(description1...descriptionnIframe'(IC(<p>))( k,?x)And(l )) = t (true), thereby establishing satisfaction of trO(description1,?xX)...in I*. This extends straightforwardly to union, intersection, and one-of descriptions.
By satisfaction of trO(d), we have that trO(descriptionn,?x)Y)tris necessarily satisfied for ?x assigned to k. By an argument analogous to the argument above, we obtain that k is in EC(Y).
This establishes satisfaction of d in I.
We obtain that every directive is satisfied in I, thereby obtaining satisfaction of condition 2. Therefore, O, and thus every ontology in C, is satisfied in I. Clearly, I* satisfies R and not φ, so (I*, I) satisfies R and not φ. We conclude that C does not entail φ.
(<=) Assume C does not owl-dl-entail φ. This means there is a common-rif-owl-dl-interpretation (unionOf(description1...descriptionnI,?x)Or(I) that is an owl-dl-model of C, but I is not a model of φ. To show that R' does not entail φ, we show that I is a model of R'.
R is satisfied in I by assumption. Satisfaction of trO(description1,?x)...trO(descriptionn,?x)i)trcan be shown analogously to establishment of satisfaction in I of Oi in the (=>) direction. We now establish satisfaction of the rules in ROWL-DL(oneOf(value1...valuen,?xV).
(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)Or( ?x=trO((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 value1)... ?x=trO(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 valuen))trO(restriction(propertyIDsomeValuesFrom(description)),?x)Exists ?y(And(?x[tr(propertyID)-> ?y]trO(description,?y)))trO(restriction(propertyIDvalue(valueID)),space of a datatype. (xii) follows from condition 15; i.e., there is no assignment for the variable ?x)?x[tr(propertyID)->tr(valueID)]Besidesthat is both a member of theembeddingvalue space of the datatype and is in its class extension and thus theprevioustable,wealsoneedanaxiomatizationantecedent ofsomethe rule will never be satisfied and rule is always satisfied. (xiii) follows from condition 12 and the fact that LV is a superset of theaspectsunion of all value spaces. (xiv) follows from theOWLDLsemantics,e.g.,separationbetweenindividualanddatatypedomains.Thisaxiomatizationfact that there isdefinedrelativetoanno ill-typed literal, since such a literal would either violate condition 16 or condition 18 on abstract OWLvocabularyV.interpretations.
This establishes satisfaction of ROWL-DL(V)=(Forall ?x("a"="b" :- ?x[rdf:type->owl:Nothing]),Forall ?x("a"="b" :-And(?x[rdf:type->rdfs:Literal] ?x[rdf:type->owl:Thing])),(Forall ?x(?x[rdf:type->owl:Thing] :- ?x[rdf:type->C]))foreveryclassnameCV), and thus R', inV,(Forall ?x(?x[rdf:type->rdfs:Literal] :- ?x[rdf:type->D]))foreveryI. Therefore, R' does not entail φ. ☐
The following theorems establish faithfulness of the full embedding of RIF-OWL DLP combinations into RIF.
Theorem Given a datatype namemap D in V, ( Forall ?x ?y (?x[rdf:type -> owl:Thing] :- ?x[rdf:type -> P]) ) for every property name P in V, ( Forall ?x ?y (?y[rdf:type -> owl:Thing] :- ?x[rdf:type -> P]) ) for every individual-valued property name P in V, ( Forall ?x ?y (?y[rdf:type -> rdfs:Literal] :- ?x[rdf:type -> P]) ) for every data-valued property name P in V,conforming with T, a
RIF-OWL-DL-combination
C=<R,{O1,...,On}>,
where O1,...,On are OWL DLP ontologies with
vocabulary V that do not import other ontologies, owl-dl-entails a DL-condition
formula φ with respect to D iff tr(merge({R,
ROWL-DL( ForallV),
trO(trN( tr( i ) [rdf:type -> owl:Thing]) ) for every individual name i in V,O1)), ...,
trO(trN( ForallOn))})) entails
tr(φ).
Proof. By the Normalization Lemma,
C=<R,{O1,...,On}> owl-dl-entails φ iff <R,{trN(tr(s^^u)[rdf:type->u']))foreverywell-typedliterals^^uanddatatypeidentifieru'inVsuchthatL2VuO1),...,trN(s)isinOn)}> owl-dl-entails φ.
Then, by thevaluespaceofu'Normalized Combination Embedding Lemma,
<R,{trN(ForallO1),...,trN(On)}> owl-dl-entails φ iff merge({R, ROWL-DL(tr(t)[rdf:type->u']))foreveryplainliteraltanddatatypeidentifieru'inVsuchthattisin), trO(trN(O1)), ..., trO(trN(On))}) dl-entails φ.
Finally, by thevaluespaceofu'RIF-BLD DL-document formula Lemma,
(Forall ?x([rdf:type->rdfs:Literal] :- ?x[rdf:type->Diri]))foreveryconsidereddatatypewithidentifierDirimerge({R, ROWL-DL(Forall ?x("a"="b" :-And(?x[rdf:type->Diri]isNotD(?x)])))foreveryconsidereddatatypewithidentifierDiriandlabelD,V), trO(trN(Forall("a"="b" :-tr(t)[rdf:type->rdfs:Literal]))foreveryill-typedliteraltO1)), ..., trO(trN(On))}) dl-entails φ iff tr(merge({R, ROWL-DL(V), trO(trN(O1)), ..., trO(trN(On))})) entails tr(φ).
This chain of equivalences establishes the theorem. ☐
Theorem Given a datatype map D conforming with T, a
RIF-OWL-DL-combination
<R,{ O1O1,..., OnOn}>,
where O1O1,..., OnOn are OWL DLP ontologies with
vocabulary V,V that do not import other ontologies, is
owl-dl-satisfiable
with respect to D iff there is a semantic multi-structure I that is a
model of tr(tr(merge({R union,
ROWL-DL (V) union(V),
trO(trN( O1 )) union ... unionO1)), ...,
trO(trN( On ))).On))}).
Proof. The theorem follows immediately from thefollowingprevious theorem and the observation that a combination (respectively, document) is owl-dl-satisfiable (respectively, has a model) if and only if it does not entail the condition formula "a"="b".TheoremAnOWL-DL-satisfiableRIF-OWL-DL-combinationC=<R,{O1,...,On}>,whereO1,...,OnareOWLDLPontologieswithvocabularyV,OWL-DL-entailsanexistentiallyclosedRIF-BLDconditionformulaφifftr(RunionROWL-DL(V)uniontrO(trN(O1))union...uniontrO(trN(On)))entailsφ.Proof.Theproofofthetheoremconsistsofthreesteps.Wefirstshow(*)GivenanOWLDLPontologyO,OandtrN(O)thesamemodels.IntheremainderweassumethatallOWLDLPontologiesarenormalized.Wethenshow(**)C=<R,{O1,...,On}>OWL-DL-entailsφifandonlyifforeverymodelofRunionROWL-DL(V)uniontrO(trN(O1))union...uniontrO(trN(On))holdsthatTValI(φ)=t.Finally,weshow(***)tr(RunionROWL-DL(V)uniontrO(trN(O1))union...uniontrO(trN(On)))entailsφifandonlyifforeverymodelofRunionROWL-DL(V)uniontrO(trN(O1))union...uniontrO(trN(On))holdsthatTValI(φ)=t.8☐
RDF URI References: There are certain RDF URI references that are not absolute IRIs (e.g., those containing spaces). It is possible to use such RDF URI references in RDF graphs that are combined with RIF rules. However, such URI references cannot be represented in RIF rules and their use in RDF is discouraged.
Generalized RDF graphs: Standard RDF graphs, as defined in (RDF-Concepts), do not allow the use of literals in subject and predicate positions and blank nodes in predicate positions. The RDF Core working group has listed two issues questioning the restrictions that literals may not occur in subject and blank nodes may not occur in predicate positions in triples. Anticipating lifting of these restrictions in a possible future version of RDF, we use the more liberal notion of generalized RDF graph. We note that the definitions of interpretations, models, and entailment in the RDF semantics document (RDF-Semantics) also apply to such generalized RDF graphs.
We note that every standard RDF graph is a generalized RDF graph. Therefore, our definition of combinations applies to standard RDF graphs as well.
We note also that the notion of generalized RDF graphs is more liberal than the notion of RDF graphs used by SPARQL; generalized RDF graphs additionally allow blank nodes and literals in predicate positions.