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problems with RDF when building an axiomatization or model theory for DAML-ONT

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Mon, 04 Dec 2000 10:54:04 -0500
To: www-rdf-logic@w3.org
Message-Id: <20001204105404S.pfps@research.bell-labs.com>
I was looking over the combined axiomatization of RDF, RDF Schema, and
DAML-ONT and noticed a major number with equality.  

The axiomatization provided does not supply an axiomatization for
equality.  One might think that it would be easy to provide a simple
axiomatization, namely one that requires that equal objects have the same
properties.  Equality and inequality could then be explicitly specified for
object that cannot be deduced to be unequal by means of their properties.

This simple version of inequality does not work because RDF
includes multi-sets (a.k.a. bags).  Well then, perhaps we can include a
simple version of set theory, something like two multi-sets are equal
whenever they contain the same elements.  

This extension does not work because RDF multi-sets can include themselves.
(At least there is no prohibition of this, and RDFS includes set-like
constructs that ``include'' themselves, for example the class ``CLASS''.)
To get around this one might use something like hypersets, but now we have
to determine which non-standard version of sets to use.

These problems do not surface in RDF and RDFS because they do not include
any operations on multi-sets nor do they include any constructs that need
to know the cardinality of multi-sets.  However, DAML-ONT has lots of
places where the cardinality of a set (or collection) must be determined.
Therefore an axiomatization of DAML-ONT or a model-theory for DAML-ONT that
includes constructs from RDF has to get this right.

So my question is, I guess, whether anyone has any solution that gives
either a good axiomatization of RDF and RDFS or a good model theory for RDF
and RDFS.  Without such, any attempt to extend RDF or RDFS will be
seriously flawed.

Peter Patel-Schneider 
Received on Monday, 4 December 2000 10:56:01 GMT

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