RE: RDF-Based Semantics ready for WG-internal Review

Hi Uli!

Uli Sattler wrote:

>> 2) Nary Datatypes:
>> Recently, I had some discussion with Ivan on the semantics for the n-
>> ary stub. The problem is that simply following the way as it is
>> defined in the Direct Semantics is technically not possible for the
>> RDF-Based Semantics, and it would be a non-trivial task to extend
>> the basic framework of the RDF Semantics to support this (though
>> possible in principle).
>
>out of curiosity: what causes this difficulty?

Let's compare the situation with the Direct Semantics: For example, when I write (in Turtle, omitting declaration triples)

      :w rdf:type [ 
        owl:someValuesFrom :D
        owl:onProperties ( :dp1 ... :dpn )
      ]

Under the Direct Semantics, this means that for the denotation of ":w", named ":w^I", there exist values v1,...,vn in the data domain DELTA_D, such that

     :w^I :dpi^I vi , for 1 <= i <= n

and

      <v1,...,vn> in :D^I

In order to make the latter expression meaningful, D^I must be a subset of DELTA_D^n, i.e. the n-ary set product of the data domain DELTA_D.
 
In the RDF(S) semantics, there is no direct support for talking about subsets of the n-ary (data-) domain (for a given n). Actually, under the RDFS semantics, names always denote individuals, and classes are individuals that maintain a link to a subset of the domain (being called the "class extensions" of these class individuals), and properties are linked to a set of pairs of individuals (being called the "property extensions" of the property individuals). One refers to the actual set and tuple set by applying functions "ICEXT(x)" and "IEXT(x)" to the individual x, respectively. What's actually missing is a notion of an "n-ary extension" of an individual, to make an individual into an "n-ary range". This can, of course, be introduced, though it would be a new idea in the RDF world (but so be it).

But there are some difficulties with the definition of such n-ary extensions. When I want to claim that an individual d may have an n-ary extension INEXT_n(d) consisting of tuples <x1,...,xn> in LV^n (LV is the data domain in RDF), then, AFAICT, I have to extend the basic model of a so called "Simple Interpretation", which underlies every RDF-based language so far, and this would *really* be new. Actually, it has not been necessary to extend Simple Interpretations for RDFS, when classes were introduced, because it was possible to define the class extensions of class individuals in terms of what was already available. But I don't see how this would work for n-aries.

On the other hand, adding datatype maps to Simple Interpretations in the RDF Semantics (for "D-entailment" = RDFS + datatypes) can also be regarded as an extension to the basic notion of a "Simple Interpretation". So, yes, I can do this and add an "n-ary extension context" to each OWL 2 Full interpretation: every individual may have an n-ary extension for a given n, and possibly even for more than one such n (if there are more, they would be mutually independent of each other). In most cases, these n-ary extensions are "semantically invisible" and would not need to be taken into account by implementations, unless one uses them by using the n-ary stub. In this case, it would be up to the reasoner implementation how to interpret a given n-ary datarange name, and, in particular, the reasoner would have to know the "n" of the n-ary restriction (just as in the case of the Direct Semantics). 

I think this might work, but I will further have to check. Anyway, it can hardly become more broken than it is at the moment. :-]

Michael

--
Dipl.-Inform. Michael Schneider
Research Scientist, Dept. Information Process Engineering (IPE)
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Email: michael.schneider@fzi.de
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