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Re: Problems with dark triples approach

From: Pat Hayes <phayes@ai.uwf.edu>
Date: Wed, 17 Apr 2002 18:52:33 -0500
Message-Id: <p05101516b8e3b70fc0a1@[65.217.30.94]>
To: "Jeremy Carroll" <jjc@hplb.hpl.hp.com>
Cc: www-webont-wg@w3.org
>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.

As you say below, dark triples isn't a magic wand. In fact *in 
itself* it doesn't actually *solve* anything. BUt it does at least 
open the possibility of constructing a solution: it removes what is 
otherwise a rigid and impassable barrier to any possible solution.

>
>These are:
>
>1: OWL syntax cannot be extended using RDF/RDFS/OWL mechanisms.
>
>2: We still need a theory of classes.

These are other issues, but lets talk about them by all means.

>
>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.

OK, but....

>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

...why do you call this 'extend' ? Seems to me you were just *using* 
DAML+OIL, not extending it. DAML allows you to define new classes and 
properties freely, that isn't doing any extending.

>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.

Right, any assertional language would.

>
>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.

?? Why do you call these 'semantic'? They are just part of the 
language. Of course they have a semantics, but so does the rest of it.

>
>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.

The semantics of OWL syntax would need to be defined this way, of 
course, since OWL is more expressive than RDF. But once that is done 
(in the OWL MT, presumably) you can use OWL constructs without any 
further need to alter the language.

>
>Building on top of dark triples seems to be a commitment to not permitting
>this extensibility.

No, on the contrary, it is designed to make such extensibility easier 
to achieve without getting clashes between different interpretations 
at different levels.

>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.

Set theories are like that. Get used to it: nobody has come up with a 
definitive proof of the consistency of ZF, and since Cohen's work 
probably nobody ever will.

>
>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.

I see it as a problem with the basic assumption that a web language 
should be a kind of set theory. If we regard it simply as a 
restricted subset of first-order logic, all of these problems just 
evaporate. They are artifacts of taking the 'class' way of talking 
too seriously.

>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.

More like 60, actually, if you go back to Frege.

>Given
>that we have the prior work to guide us, we may be able to look for a factor
>of 10 speed up.

What this optimism doesn't take into account is that the search for a 
fully adequate set theory was basically abandoned by most 
philosophers of mathematics about 25 years ago. (For more on this 
topic than anyone probably wants to know, subscribe to the FOM 
mailing list.)

>
>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:

No, wait. There are 2 separate issues here: the need for a clean 
account of layering, and the idea of basing the semantics on FOL 
rather than set theory. The dark triples are needed for the former, 
not the latter.

>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.

Of course you can take a subPropertyOf an OWL property, why should 
you not be able to? I don't see the relevance here.

>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.

I don't quite see why. The web is open in the sense that anyone can 
say anything on it, but that also means that they can close their own 
quantifier scopes. If I say all people are mammals, that doesn't give 
you a licence to insert "..and have green heads" into my universal 
quantifier.

Pat

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Received on Wednesday, 17 April 2002 19:52:40 GMT

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