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Comment on Last Call Working Draft of RDF Semantics document concerning treating classes and properties as objects

From: Jeff Z Pan <pan@cs.man.ac.uk>
Date: Fri, 14 Feb 2003 21:15:22 -0800
Message-ID: <004601c2d4b1$44223e70$6401a8c0@percival>
To: <www-rdf-comments@w3.org>

Recent research (http://dl-web.man.ac.uk/rdfsfa/paper.htm) has shown that
RDF Schema (RDFS) has a non-standard metamodeling architecture, which makes
some elements in the model have dual roles in the RDFS specification.  As a
result, this can be confusing and difficult to understand and,  more
importantly, the specification of its semantics requires a non-standard
model theory - RDF MT. RDF MT justifies dual roles by "treating classes and
properties as objects", which seems to be ok within RDFS. However, this
leads to semantic problems when we extend RDFS with more expressive FOL
constructs, and extend RDF MT to the so called ``RDF+ MT'' to give meaning
to this extended language:

*Problem 1 (Too few entailments) [1]: In "RDF+ MT", closure rules are used
to represent semantic conditions to facilitate entailment. However, if the
expected class and/or property  objects are not in the universe, then the
related closure rules are not expressible, which hence leads to too few
entailments. E.g., does "John type (A and B and C)" entail "John type (B
and A)"? No, because the class object "B and A" might not exist in every
interpretation which satisfies " John type (A and B and C)" .

In order to fix the problem, one can/should also introduce, besides the
closure rules, comprehension axioms to add all the possible missing objects
into the universe, e.g. the "B and A" object in the above example. In order
to support FOL constructs in the extended language, a large and complex set
of comprehension axioms (and closure rules) must be added  to capture the
meaning and characteristic of its additional constructs. It is very
difficult to get them right, and it is more difficult to prove that they are
correct (adding enough but not too many objects into the universe).

*Problem 2 (Contradiction classes)[1,2]: Since class and property objects
are distinguished from their extensions, one can then, in the extended
language, define a class eg:C as an instance of itself, and add a
cardinality constraint ``=0'' on the rdf:type property that pointing to
itself. It is impossible for one to determine the membership of this class.
If an object is an instance of this class, then  it isn't, because instances
should have no rdf:type property pointing to itself. But if it isn't then it
is. This is a contradiction class.

According to the comprehension axioms, we must add all possible class
objects into the domain, and the above contradiction class is one of them.
In this way, all the interpretations will have such contradiction classes,
and thus have ill-defined class memberships.

*Problem 3 (Size of the Universe)[3,4]: treating classes and properties as
objects also triggers a problem if we set constraints on the size of the
universe. E.g. is it possible to have an interpretation s.t. Thing <=
{John}, "John type Person", "John type not Car"? It is possible in FOL, but
not in "RDF+ MT". In FOL, the interpretation of John, I(John), is the only
object in the universe; the interpretation of Person, I(Person), is a set
with only one element I(John); the interpretation of Car, I(Car), is an
empty set. In "RDF+ MT", since classes are also objects, John, Person and
Car should all be interpreted as the only one object in the universe.
However,  since I(John) is in ICEXT(I(Person)), but not in ICEXT(I(Car)),
I(Person) and I(Car) should be different. Thus there should be at least two
objects in the universe. In other words, the required interpretation is
impossible in ``RDF+ MT''.

This problem shows that the interpretation of RDF MT has different features
than the interpretation of standard FOL model theoretic semantics. This
raises the question as to whether it is possible to layer FOL languages on
top of both the syntax and semantics of RDFS.

More details about the above three problems can be found in [4].

As a consequence of these problems, when DAML+OIL is layering on top
of RDFS, it uses the syntax of RDFS only, but defines its own semantics [6]
 for the ontological primitives of RDFS. Similarly, when the Web Ontology
Language (OWL) is layering on top of RDFS, (slightly but necessarily)
different semantics are defined for OWL DL and OWL Full [7].

Some people believe that Lbase [8] can provide a framework for specifying
the semantics of all of the Web ontology languages in a uniform and coherent
way. However, that might not be so straight forward. To make the model
theory of Lbase as the model theory of all the semantic web languages so
as to make it possible to use a *single* inference mechanism to work on
different languages [8], Lbase must at least be expressive enough to allow
 mapping from RDF constructs into its constructs. But according to the
 translation rules from RDF graph to Lbase in [9], e.g., rdfs:subClassOf is
directly mapped to logical implication in Lbase, but must be explicitly
or axiomatised as, e.g.rdfs:subClassOf(class1,class2). That is, the
 rdfs:subClassOf from RDF and the logical implication from FOL are not
 to the same thing in Lbase, which means there are no (or far less than
 semantic interoperability between RDFS and FOL in the Lbase framework.[3]
So the problem of layering FOL on top of RDFS still exists.

An alternative approach is RDFS(FA) (http://dl-web.man.ac.uk/rdfsfa/) [5,4],
which as a sub-language of RDFS has a relatively standard model theoretic
semantics, such that FOLs, e.g. DAML+OIL and OWL DL, can be built on top of
both the syntax and semantics of RDFS(FA).

Jeff Z. Pan  ( http://DL-Web.man.ac.uk/ )
Computer Science Dept., The University of Manchester

[1] http://www-db.research.bell-labs.com/user/pfps/papers/dl-2002.ps
[2] http://www-db.research.bell-labs.com/user/pfps/papers/layering.ps
[6] http://www.daml.org/2001/03/model-theoretic-semantics.html
[7] http://www-db.research.bell-labs.com/user/pfps/owl/semantics/
[8] http://www.w3.org/2002/06/lbase/20030116Snapshot.html
[9] http://www.w3.org/TR/rdf-mt/#Lbase
Received on Friday, 14 February 2003 16:11:53 UTC

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