ISSUE 131 (OWL R Unification): Different semantics on syntactic fragment

Dear WG!

I was able to find an example, which demonstrates that the semantics of OWL R DL and OWL R Full differ on the syntactic fragment of a unified OWL R language (see below). The example shows that entailments exist w.r.t. the semantics of OWL R DL (actually, the OWL 2 DL semantics), for which there is no equivalent sequence of rule applications in OWL R Full. This means that the semantics of OWL R ontologies, which match the syntactic fragment of OWL R, cannot be defined in parallel by both the DL semantics *and* the ruleset.

An alternative approach would be to define the semantics of OWL R on the syntactic fragment by the OWL 2 DL semantics only. However, I would expect that this would lead to significant acceptance problems for implementers, who want to base the implementation of their OWL R reasoner on the OWL R ruleset (and for customers/users of such reasoners, either). 

An implementer, who plans to create a reasoner based on the ruleset, certainly wants to be ensured that his reasoner implements *all* of OWL R, not only a subset, i.e. is complete w.r.t. the OWL R semantics. It would at least be very confusing and hard to accept that it is not sufficient to just implement the "OWL R rules", in order to create a complete OWL R reasoner. But with my counter example, it now turns out that completeness will not be achievable for such a rulebased reasoner.

Even worse, having such a concrete counter example makes it easy to create a proper *superset* of the current ruleset, which will still be sound w.r.t. the OWL R semantics. This means that the implementer cannot even claim that his reasoner is at least the "best possible" triple-rule based implementation of OWL R.

The only thing which the implementer can claim is that his reasoner is a "valid" OWL R reasoner, i.e. is sound w.r.t. the OWL R semantics. But this will also be the case for any reasoner, which is based on an arbitrary *subset* of the OWL R rules, and even for trivial reasoners, which do not produce any inferences at all. So, OWL R soundness alone isn't a very interesting property of an OWL R reasoner in practice. 

>From this p.o.v., I would expect that, after a unification, implementers will not regard the OWL R ruleset to be relevant as a base for the implementation of OWL R reasoners. IMO, the OWL R rules will then be no more than just an arbitrary example for how one MAY create a valid OWL R reasoner. Without completeness guarantees, this ruleset won't be better than any other ruleset which provides formal OWL R soundness. Compared with the ruleset's current status in OWL R Full, where it specifies the semantics of the language, this would be a significant demotion.

   * * *

The example below is not simply a bug in the current ruleset, which could easily be fixed. Instead, it shows two general problems: 

 (1) The ruleset would need to be able to create fresh bNodes on the right hand side of derivations. To my understanding, it was a design goal to avoid this. At least, none of the current rules creates such new bNodes on the consequent side.

 (2) Since the semantics of the current OWL R DL language is the DL semantics, it allows for very flexible reasoning with complex class expressions, as long as an ontology conforms to OWL R DL's syntactic restrictions. The current triple ruleset isn't able to compete with this amount of flexibility. And don't believe that this can be overcome. 

Here is the example. We start from the RDF graph

  G_L := {
    ex:C rdf:type owl:Class .
    ex:D rdf:type owl:Class .
    ex:C rdfs:subClassOf ex:D .
  }

This is a valid OWL R DL ontology in RDF graph form, since only named
classes are used on the LHS and RHS of the subclass axiom (see sec. 4.2.3 
of [1]).

The second RDF graph is

  G_R := {
    ex:C rdf:type owl:Class .
    ex:D rdf:type owl:Class .
    _:X owl:intersectionOf ( ex:C ex:D ) .
    _:X rdfs:subClassOf ex:D .
  }
 
Again, this is a valid OWL R DL ontology, because this graph RDF-maps to the
following expression in Functional Syntax (omitting declarations):

  SubClassOf( IntersectionOf(ex:C ex:D) ex:D )

According to sec. 4.2.3 of [1], intersections of named classes may occur on
the LHS of subclass axioms.

The semantic meaning of these two graphs is given in OWL R DL by applying
the OWL 2 DL semantics. According to the OWL 2 DL semantics, the entailment

  G_L |= G_R

is true, because from 

  C subset D

follows

  (C and D) subset D

However, there is no sequence of applications of OWL R Full rules, which
starts in G_L, and which produces G_R (or a super graph of G_R) as a result.
Actually, there is no single triple rule in OWL R Full that would allow to
produce the bNode '_:X', which occurs in G_R but not in G_L. 

Regards,
Michael

[1] <http://www.w3.org/2007/OWL/wiki/Profiles#Class_Expressions_3>

--
Dipl.-Inform. Michael Schneider
FZI Forschungszentrum Informatik Karlsruhe
Abtl. Information Process Engineering (IPE)
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Email: Michael.Schneider@fzi.de
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