# SWC/OWL-Compatibility

This page contains a proposal for the specification of RIF-OWL compatibility, taking into account both the DL and Full species of OWL, for consideration by the RIF working group. The choices of possible ways to define combinations and considerations which led to the proposal on the page are documented here.

As is the case for the specification of RIF-RDF compatibility, the page specifies the semantics of combinations of OWL ontologies and RIF rule sets. The specification is agnostic to the way such combinations are represented. See also the Overview of RDF and OWL compatibility.

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

## Contents

## OWL Species

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 in the remainder.

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

Finally, the OWL DL RDF syntax does not extend the RDF syntax, but rather restricts it: every OWL DL ontology is an RDF graph, but not every RDF graph is an OWL DL ontology. OWL Full and RDF have the same syntax: every RDF graph is an OWL Full ontology, and vice versa.

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

## Syntax

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

For the combination of RIF rule sets with OWL DL ontologies we define the notion of RIF-OWL DL combinations based on the abstract syntax of OWL DL. We need to furthermore impose certain restrictions on the syntax of the rules.

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

An *RIF-OWL DL combination* is a pair < *R*,**O**>, where *R* is a *DL Rule set* and **O** is a set of OWL DL ontologies in abstract syntax form of a vocabulary *V*. When clear from the context, RIF-OWL DL combinations are referred to simply as *combinations*.

## Semantics

### OWL Full

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.

Given a conforming datatype map D, a common interpretation (* I*, I)

*OWL Full satisfies*an RIF-RDF combination C=<

*R*,

**S**> if

*satisfies*

**I***R*, I is an OWL Full interpretation, and I satisfies every RDF graph

*S*in

**S**; in this case (

*, I) is called an*

**I***OWL Full model*of C, and C is

*OWL Full satisfiable*.

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

### OWL DL

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

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

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

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

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

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

#### Modified Semantics for RIF Frame Formulas

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

Given an RIF semantic structure * I* = <

*,*

**D**

**I**_{C},

**I**_{V},

**I**_{F},

**I**_{R},

**I**_{slot},

**I**_{SF},

**I**_{SR},

**I**_{sub},

**I**_{isa}>, the truth valuation function

**I**_{T-DL}is obtained by modifying the truth valuation of frame formulas in

**I**_{Truth}in the following way:

**I**_{T-DL}(

`t [ rdf:type -> A ]`) =

**I**_{R}(

`A`)(

`t`) and

**I**_{T-DL}(

`t`) =

_{1}[ P -> t_{2}]

**I**_{R}(

`P`)(

`t`).

_{1}, t_{2}We say that **I***DL satisfies* a rule **Q** *then* :- *if*, where **Q** is a quantification prefix for all the variables in the rule, if **I**^{*}_{T-DL}(then) ≥ **I**^{*}_{T-DL}(if) for every **I**^{*} that agrees with * I* everywhere except possibly on some variables mentioned in

**Q**.

*is a*

**I***DL model*of a rule set

**R**if it DL satisfies every rule in the set.

#### Semantics of RIF-OWL DL Combinations

Given a conforming datatype map D, a *common DL interpretation* is a pair (* I*, I), where

*= <*

**I***,*

**D**

**I**_{C},

**I**_{V},

**I**_{F},

**I**_{R},

**I**_{slot},

**I**_{SF},

**I**_{SR},

**I**_{sub},

**I**_{isa}> is an RIF semantic structure and I=<R, EC, ER, L, S, LV> is an abstract OWL interpretation with respect to D of a vocabulary

*V*, such that the following conditions hold:

- 1. R=
;**D** - 2. LV is a subset of R and contains the value spaces of all data types in D;
- 3. EC(
`u`) = set of all objects`k`in R such that**I**_{R}(`"u"^^rif:iri`)(`k`) =**t**(true), for every IRI`u`in*V*; - 4. ER(
`u`) = set of all tuples (`k`,`l`) such that**I**_{R}(`"u"^^rif:iri`)(`k`,`l`) =**t**(true), for every data valued and individual valued property identifier`u`in*V*; - 5. L((
`s`,`d`)) =**I**_{C}(`"s"^^d`) for every well-typed literal (`s`,`d`) in*V*; - 6. S(
`i`) =**I**_{C}(`"i"^^rif:iri`) for every IRI`i`in*V*.

Condition 1 ensures that the domains of interpretation are the same. Condition 2 ensures that the set of literal values includes the value spaces of all considered datatypes. Condition 3 ensures that the interpretation (extension) of an OWL DL class `u` corresponds to the interpretation of the unary predicate with the same name in RIF. Condition 4 ensures that the interpretation (extension) of an OWL DL object or datatype property `u` corresponds to the interpretation of the binary predicates with the same name in RIF. Condition 5 ensures that typed literals of the form (`s`, `d`) in OWL DL are interpreted in the same way as constants of the form `"s"^^d` in RIF. Finally, condition 6 ensures that individual identifiers in the OWL ontologies and the RIF rule sets are interpreted in the same way.

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

Given a conforming datatype map D, a common DL interpretation (* I*, I)

*OWL DL satisfies*an RIF-OWL DL combination C=<

*R*,

**O**> if

*DL satisfies*

**I***R*and I satisfies every OWL DL ontology in abstract syntax form

*O*in

**O**; in this case (

*, I) is called an*

**I***OWL DL model*of C, and C is

*OWL DL satisfiable*. (

*, I) satisfies an OWL DL ontology in abstract syntax form*

**I***O*if I satisfies

*O*. (

*, I) satisfies a closed RIF condition formula φ if*

**I**

**I**_{T-DL}(φ)=

*t*.

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

##### Annotation properties

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

Given a conforming datatype map D, a common DL interpretation (* I*, I) is a

*common DL annotation interpretation*if the following condition holds:

- 7. ER(
`u`) = set of all tuples (`k`,`l`) such that**I**_{R}(`"u"^^rif:iri`)(`k`,`l`) =**t**(true), for every IRI`u`in*V*.

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

Given a conforming datatype map D, a common DL annotation interpretation (* I*, I)

*OWL DL annotation satisfies*an RIF-OWL DL combination C=<

*R*,

**O**> if

*satisfies*

**I***R*and I satisfies every OWL DL ontology in abstract syntax form

*O*in

**O**; in this case (

*, I) is called an*

**I***OWL DL annotation model*of C, and C is

*OWL DL annotation satisfiable*.

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