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LANG: inverse/onOf interaction

From: Ian Horrocks <horrocks@cs.man.ac.uk>
Date: Thu, 13 Jun 2002 13:33:25 +0100 (BST)
To: www-webont-wg@w3c.org
Message-ID: <Pine.LNX.4.44.0206131326520.13525-100000@galahad.cs.man.ac.uk>

I promised to explain in a bit more detail why inverse properties and
oneOf interact adversely (at least as far as reasoning support is
concerned).

The combination of these features is very powerful, and it is easy to
see how they can be used to build ontologies for which reasoning will
be very difficult. One example is the use of a so called "spy-point"
to limit the maximum size of the domain of discourse. This is done,
e.g., as follows:

1. Add an axiom asserting that every element in the domain of
discourse is related to the individual "Spy" via some property, i.e.,
add the axiom Thing subClassOf (hasValue P Spy).

2. Add an axiom asserting that Spy is related to at most n distinct
individuals via the inverse of the P property, i.e., Spy type
(maxCardinality (inverse P) n).

It may be counter-intuitive, but the ability to fix the size of the
domain of discourse to some arbitrary size makes reasoning much
harder.  From a practical point of view, this can be seen in many
"strange" additional inferences: e.g., for ANY property R, the class
defined by a restriction (minCardinality R n+1) becomes incoherent
(because there can only be n distinct objects in the domain of
discourse). From a theoretical point of view, the worst case
complexity of basic inference tasks (e.g., class consistency) jumps
from Exptime (for DAML+OIL without inverse or without oneOf), to
NExpTime for full DAML+OIL. If fact, inference is already NExpTime
hard for the basic ALC description logic (booleans plus hasClass and
toClass) with the addition of inverse and oneOf.

In fact, the problem boils down to the loss of the tree model
property. The language without oneOf has the (transitive) tree model
property, that is, any class that has a model has a (transitive) tree
model. This means that decision procedures can restrict their
attention to this kind of model without sacrificing
completeness/corectness. With oneOf but no inverse, things are a bit
more complex, but a similar restriced model property can be
demonstrated.

You can find some more information, including pointers to various
other relevant papers, in [1] (sorry for the self citation).

I hope this helps - no doubt you will let me know if you have any
queries.

Ian

[1] http://www.cs.man.ac.uk/~horrocks/Publications/download/2001/ijcai01.pdf
Received on Thursday, 13 June 2002 08:32:46 GMT

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