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Re: new version of semantic layering document

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Mon, 23 Sep 2002 21:48:31 -0400 (EDT)
Message-Id: <20020923.214831.52156628.pfps@research.bell-labs.com>
To: phayes@ai.uwf.edu
Cc: www-webont-wg@w3.org

From: pat hayes <phayes@ai.uwf.edu>
Subject: Re: new version of semantic layering document
Date: Mon, 23 Sep 2002 20:06:56 -0500

> >From: pat hayes <phayes@ai.uwf.edu>
> >Subject: Re: new version of semantic layering document
> >Date: Mon, 23 Sep 2002 14:37:31 -0500


> >>  BTW, strengthening domain and range to IFF seems to me to be a really
> >>  bad idea, in general: it completely destroys the idea of conjunctive
> >>  range/domain assertions. We put the current semantics into RDFS in
> >>  order to conform to the DAML conjunctive semantics, so why do you
> >>  want to blow this out of the water in OWL?
> >
> >I don't understand why you are saying this?  How does making domain and
> >range IFF destroy the conjunctive semantics?
> Because if superclasses of ranges are also ranges, then there is no 
> information to be gained by asserting more about a range. All 
> properties range over the universe. The whole point of allowing 
> conjunctive semantics on ranges is so that one can accumulate 
> information that allows one to pin down a range more precisely.


If I say

	ex:bar rdfs:subClassOf ex:baa .
	ex:baz rdfs:subClassOf ex:bar .
	ex:foo rdfs:range ex:bar .

Then no matter whether ex:foo has an IFF reading or an ONLY IF reading
any object of an ex:foo triple must belong to both ex:bar and ex:baa

	ex:bar rdfs:subClassOf ex:baa .
	ex:baz rdfs:subClassOf ex:bar .
	ex:foo rdfs:range ex:bar .
	ex:a ex:foo ex:b .

RDFS entails

	ex:b rdf:type ex:bar .
	ex:b rdf:type ex:baa .

The only difference is that under an IFF reading graph A RDFS entails

	ex:foo rdfs:range ex:baa .

If I say instead

	ex:bar rdfs:subClassOf ex:baa .
	ex:baz rdfs:subClassOf ex:bar .
	ex:foo rdfs:range ex:bar .
	ex:foo rdfs:range ex:baz .

then more information has definitely been gained, as C RDFS entails

		ex:foo rdfs:range ex:baz .

which was not an RDFS entailment of A.


	ex:bar rdfs:subClassOf ex:baa .
	ex:baz rdfs:subClassOf ex:bar .
	ex:foo rdfs:range ex:bar .
	ex:foo rdfs:range ex:baz .
	ex:a ex:foo ex:b .


	ex:b rdf:type ex:baz .

which is not an RDFS entailment of  B.


> >>  OK, its easy to make that change if you prefer. It seems odd to say
> >>  that an empty class is disjoint with itself, is all.
> >
> >Why is this odd?
> Maybe it isnt. OK, you win on this one. That means then
> aaa owl:disjointWith owl:Nothing .
> is always true for any aaa , right?

Yes, as it should be.


> >>  >- There are many restrictions in Large OWL that may be problematic because
> >>  >   their presence may affect their extension.  For example, what is the
> >>  >   class extension of
> >>  >       _:x owl:onProperty rdfs:subClassOf .
> >>  >       _:x owl:maxCardinality 57 .
> >>
> >>  Its the class of all RDFS classes which have no more than 57
> >>  subclasses, right? That might be empty, for all I know. (Oh no, wait,
> >>  all the empty classes are in it.) But Im sure it *exists*. In fact,
> >>  you could toss in things like transfinite cardinalities (as long as
> >>  they are describable) and I would still be sure the restriction
> >>  classes would exist.
> >
> >>  >   Can it be shown that there are *no* problems in determining the class
> >>  >   extensions of all the restrictions that mention the RDF and RDFS
> >>  >   structural properties and that thus may interfere with their own class
> >>  >   extension?
> >>
> >>  Im not sure what you mean by 'interfere with'. One can certainly
> >>  create inconsistencies by asserting that, say, rdfs:subClassOf is the
> >  > same as some odd restriction. But just defining new restriction
> >  > classes in terms of the RDFS/OWL vocabulary doesn't in itself
> >  > interfere with that vocabulary in any sense I can think of.
> >
> >In Large OWL the class of all classes with no more than n subclasses
> >exists in all interpretations, for n any non-negative integer.
> Right. Some of them might be empty, of course. And one needs to be 
> careful about 'all classes'; the universe is required to be closed 
> under formation of restrictions, but it might not contain all the 
> sets that a set-theorist might want to believe in.
> >Some of
> >these classes may be subclasses of some other of them.  Can you show that
> >there is not some interaction between all these classes, and other such
> >Large OWL restrictions, that makes it impossible to have a consistent view
> >of all such OWL classes?
> Im not sure what you mean by interaction. Of course there is no 
> guarantee that any particular large-OWL graph will be consistent. But 
> some of them will be, for sure. For example,
> _:x owl:onProperty _:x .
> _:x owl:allValuesFrom _:x .
> _:x rdf:type owl:transitiveProperty .
> _:x owl:disjointWith owl:Nothing .
> is large-OWL consistent: it can be satisfied by an interpretation 
> whose 'core' (ie apart from all the RDFS machinery) is this:
> IR= IC= {a,b}+IOR, IOR = {<a,{a}>,<a,{ }>}, IRP(x) = a
> IP={a}
> ICEXT(a)={a}, ICEXT(b)={ } (just to provide an empty class) and 
> ICEXT(<x,y>)=y in IOR
> IEXT(a)={<a, a>}
> IEXT(I(owl:allValuesFrom))={<a, a>, <b, b>}
> and similarly for the others, and
> IEXT(I(owl:minCardinality)) = {<0, a>, <1, b>, <2, b>,....}
> and also similarly for the others, and the obvious extensions for the 
> unions and intersections and so on. The universe needs to be 
> populated by a lot more entities for the OWL and RDFS vocabulary to 
> denote, but they don't need to be in any of the class or property 
> extensions so they won't affect the closure conditions. For example, 
> I(rdf:subClassOf) has got to be something, but all that really 
> matters about it is that IEXT(I(rdfs:subClassOf)) = {<a, a>, <b, a>}.

You have not nearly shown that this can be completed to a Large OWL
interpretation.  What is the class extension of the required domain element
that consists of precisely those domain elements that have at most 57
superclasses, for example?

> I tell you what, If I define a complete satisfying interpretation for 
> this plus the entire RDFS/OWL vocabulary, in full detail, will that 
> satisfy you? It will take me a day or so.

That would be a start.  However, you need an interpretation for *all* the
OWL restrictions that are in every Large OWL interpretation.  This would
mean that Large OWL would not be trivial.  

However, even that would not be enough.  For example, suppose that I say

	ex:Person owl:subClassOf _:x .
	_:x owl:onProperty ex:parent .
	_:x owl:allValuesFrom ex:Person .

which would be a natural thing to say about people.  Is this a
contradiction in Large OWL?  If it is, then Large OWL is not very useful,
though it might be non-trivial.  You would have to show that such natural
OWL graphs are not contradictions.

> >The kind of problem that you could get into is if there was some break
> >point where ``less than n subclasses'' was empty but ``less than n+1
> >subclasses'' was not, except that the existence of  ``less than n
> >subclasses'' messed up this reasoning.
> The basic point is that all these constructions fit within standard 
> ZF+AC set theory, so as long as the basic domains are all sets - 
> which they are by decree - then certainly the required semantic 
> *structures* exist which satisfy all the closure rules, given all the 
> basic interpretation machinery  (the IEXT, IRP, ICEXT mappings and so 
> on). 

I don't buy this.  

> Whether or not that basic machinery can be so arranged as to 
> satisfy all the triples of a particular graph is of course a 
> different question.

Agreed, but that is not what I am trying to get at.

> It might be worth rewriting the MT so as to make this 
> structure/interpretation distinction more clear. That is the standard 
> way to do it, as you know, but I thought it was too formal for the 
> RDF MT. Now we have gotten this complicated, maybe that needs to be 
> re-thought. The idea would be to have semantic mappings corresponding 
> to all the class constructors as part of the structure, and to 
> require the universe to be closed under them in an appropriate way. 
> Call that an OWL structure. Then *independently* say that an OWL 
> structure satisfies a graph when the graph comes out to be true when 
> you interpret the OWL vocabulary in the appropriate way. This treats 
> all the OWL vocabulary as a logical vocabulary, in effect. Right now 
> we have these two aspects - closure of the universe and satisfaction 
> of the graph - kind of mixed together. If they were separated then it 
> would be clearer, I think, that the structures always exist. But 
> since you and maybe Ian are the only people who need to be convinced 
> of this, I wonder if its really worth all the trouble....:-) 
> Actually, a serious point: once the MT gets this complicated, I 
> wonder if an Lbase-style way of giving the semantics might not be 
> preferable: its easier to understand, easier to check and just as 
> formally precise. And it doesnt require us to treat the entire 
> namespace as logical vocabulary. If SW languages ever get more 
> complicated than OWL, I wouldnt want to be in the MT business any 
> more.
> >
> >[...]
> >
> >>  >PS:  I would much have preferred to try to get a version of the previous
> >>  >      document that didn't have any known bugs before making such drastic
> >>  >      changes.
> >>
> >>  Well, the previous approach seemed to be hopeless, and this is really
> >>  only a simplified version phrased somewhat differently. The only
> >>  major change was realizing that we could impose simplified
> >>  restriction closure conditions on large OWL, and let it support all
> >>  the intuitive inferences. (That is what took the weekend.)
> >
> >But this is precisely where I cannot see a way of showing that the model
> >theory is non-trivial.
> Well, Im not sure what would convince you. One can construct little 
> interpretations, like the above, to show that it can be done in some 
> cases; and we all agree that it can't be done in all cases. So where 
> is your boundary?

Well showing that there was at least one Large OWL interpretation would be
a good start.  I am definitely not convinced that there is even one.
Showing that a good selection of interesting Large OWL graphs are
non-contradictory would go almost all the way to convincing me.

It is not as if I am asking for anything special, after all.  Further, the
mere fact that I am asking is a bad sign.

> Pat

Received on Monday, 23 September 2002 21:48:40 UTC

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