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Re: new version of semantic layering document

From: pat hayes <phayes@ai.uwf.edu>
Date: Mon, 23 Sep 2002 20:06:56 -0500
Message-Id: <p05111b47b9b5562713b6@[65.217.30.172]>
To: "Peter F. Patel-Schneider" <pfps@research.bell-labs.com>
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 14:37:31 -0500
>
>>  >On a quick read through of part of the new document I found the following
>>  >issues.  I'm assuming that rdfs:subClassOf, rdfs:subPropertyOf,
>>  >rdfs:domain, and rdfs:range are strengthened to iff.
>>
>>  Yes for the subs, as the document says; no for domain and range. See
>>  my message to Jeremy.
>
>If it is ``No'' for domain and range, then there are quite a few triples in
>the table near the end of section 3 that are not valid.

I think most of them are, but you may be right. I will check them all 
one at a time. This will take a while though. In the meantime, 
(everyone) treat this table as provisional.

>
>>  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.

>
>[...]
>
>>  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?

>[...]
>
>>  >- 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>}.

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.

>
>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). 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.

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?

Pat

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Received on Monday, 23 September 2002 21:06:49 GMT

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