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Problems with dark triples approach

From: Jeremy Carroll <jjc@hplb.hpl.hp.com>
Date: Wed, 17 Apr 2002 14:14:16 +0100
To: <www-webont-wg@w3.org>
Message-ID: <JAEBJCLMIFLKLOJGMELDOELDCDAA.jjc@hplb.hpl.hp.com>

As my idea of the problem that we are trying to solve is clarifying
(hopefully not too incorrectly) I am beginning to see problems with the dark
triples solution to that problem.

These are:

1: OWL syntax cannot be extended using RDF/RDFS/OWL mechanisms.

2: We still need a theory of classes.


In order:

Extensibility
=============

The first ontology I wrote using DAML started by taking a subclass
daml:Class, and subproperties of daml:Property etc, and a subclass of
daml:Ontology.

The motive was that the ontology I was creating was to represent the
conceptualization of e-mail in Microsoft's Outlook product and I wished to
extend the DAML+OIL mechanisms to include OLE mappings for the properties.

From the point of view of computer science this seems a logical and natural
thing to do; and a functionality that I hope that OWL can support.

From the point of view of the semantic web I see extensibility in all
possible directions as being a fundamental design obligation.

This relied on *semantic* mechanisms such as subPropertyOf and subClassOf.

If the semantics of OWL is defined directly on top of the graph syntax then
this does not work. Rather, to have this work, we would need the conditions
on an OWL interpretation to be essentially semantic constraints, like the
additional constraints on RDFS interpretations in Pat's RDF Model Theory.

Building on top of dark triples seems to be a commitment to not permitting
this extensibility. If this is correct then I am unhappy with the proposal
to use dark triples to address the semantic layering problems.

Theory of Classes
=================

(Summary dark triples ain't no magic wand).

As we are all aware, a naive set theory is problematic.
As Peter has shown, the theory of classes in DAML+OIL can be thought of as
problematic if one presupposes that the ability to write a description of a
class is sufficient to ensure its existence.
It is at least plausible, that even without qualified cardinality
constraints, a new version of Peter's paradox can be found.

In axiomatic set theory, the class of syntactic expressions that actually
correspond to sets that exist is restricted in some way. As a footnote I
include a version from von Neumann-Bernays-Gödel set theory [1]. In a
semantic web based theory such restrictions are very hard to state because
they need to be robust against the open world assumption. As an example, in
DAML+OIL we can have an innocuous looking qualified cardinality constraint:


   foo:r, rdf:type, owl:Restriction .
    foo:r, daml:onProperty, foo:bar .
    foo:r, daml:maxCardinalityQ, "0" .
    foo:r, daml:hasClassQ, :_3 .
      :_3, daml:oneOf, :_4 .
        :_4, daml:first, foo:singleton .
	:_4, daml:rest, daml:nil .


when combined with another innocuous document

rdf:type daml:samePropertyAs foo:bar .
foo:singleton daml:sameInstanceAs foo:r .

we have the Patel-Schneider paradox.

I don't see this as a problem relating to RDF, but relating to the radical
open world assumption that, for me, characterises the semantic web.

At this stage my assumption is that an adequate theory of classes for the
semantic web will be a major research undertaking, on a par with creating an
adequate set theory. I believe that the latter took about 30 years. Given
that we have the prior work to guide us, we may be able to look for a factor
of 10 speed up.

There appeared to be agreement at the f2f, that a first order theory (aka my
solipsistic stuff [2]):
- does clarify OWL semantics without contradicting our set theoretic
intuitions
- is the theory used by DAML+OIL
- does not contain an adequate theory of classes capturing our set theoretic
intuitions

Personally I would feel happier with that solution than paying either of the
prices that my analsysis suggests for a dark triple based theory of classes:
viz:
either:
- a significant delay to the WG product in order for the SEM focus area to
undertake a research project
or:
- the inability to meaningful take a subPropertyOf the properties used in
constructing an owl ontology.


Jeremy

[1] Elliot Mendelson, Introduction to Mathematical Logic, 2nd edition, 1979,
p 178.

[Within a first-order theory NBG ... proposed by von Neumann and ... R.
Robinson, Bernays and Gödel]

PROPOSITION 4.4 Let phi(X1,....Xn,Y1,...Ym) be a well-formed formula the
variables of which occur among X1,...Xn,Y1,....Ym and in which only set
variables are quantified (i.e. phi can be abbreviated in such a way that
only set variables are quantified). We call such a well-formed formula
*predicative*. Then,

turnstile (EZ)(x1)...(xn)(<x1,...,xn> memberOf Z iff
phi(x1,...,xn,Y1,...,Ym))


Jeremy: I am trying to draw attention to the syntactic constraint to do with
quantified variables in the wff; I see this constraint as alien in nature to
how I perceive the open world of the semantic web.

[2]

http://lists.w3.org/Archives/Public/www-webont-wg/2002Mar/0179.html
Received on Wednesday, 17 April 2002 09:16:34 GMT

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