Re: Large OWL structures

From: pat hayes <phayes@ai.uwf.edu>
Subject: large OWL structures (was: Re: new version of semantic layering document)
Date: Mon, 23 Sep 2002 21:49:00 -0500

[...]

> >You have not nearly shown that this can be completed to a Large OWL
> >interpretation.
> 
> You are a very hard man to please, Peter.

Not in the slightest.  You are proposing a complex, recursive,
self-referential construction as the model theory for Large OWL.  I'm only
asking that you show that this construction is well behaved.   And, so far,
I'm only asking for the most basic notion of well-behavedness.  If this
construction is to be used as a basis for part of the semantic web, there
are many more characteristics that it should have.

> >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 think it is {a}, but I confess to not being entirely certain until 
> I have checked the details.

These details need to be checked.

> >>  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.
> 
> Right, I realize that. That is why it will take me a day or so. I 
> need to consider things like
> 
> _:x owl:onProperty owl:onProperty .
> 
> for example.
> 
> >This would
> >mean that Large OWL would not be trivial.

[Trivial in the sense of having no interpretations.]

> I never said it was trivial.  RDFS isn't trivial. What I am certain 
> of, and what seems kind of obvious to me, is that these structures 
> always exist. Look, how could they possibly not? Almost every 
> mathematical structure ever thought of can be constructed, perhaps in 
> simulacrum, in ZF-AC. The only significant exceptions are Topos 
> constructs, and those only because they make exorbitant claims to 
> being both extraordinarily infinite and completely self-describing. 
> There is nothing in OWL remotely like this; and even the Topos can be 
> fitted into ZF+AC if one is willing to admit that certain large 
> categories can be thought of as sets. The entire OWL/RDFS universe 
> fits inside the iterated power set of IR, for goodness sake. 

Sure the universe does, but you haven't shown that the universe can be
augmented with rdf:type and rdfs:subClassOf extensions that meet the
conditions imposed by the RDFS and OWL semantic constraints.

> None of 
> this is very hard mathematics, just tedious.

Some of it might be hard.  Take, for example, a very cut down and slightly
modified version of OWL where all there is is a restriction that requires
exactly three subclasses, a restriction that requires exactly zero
subclasses, a principle that different restrictions have to have
different denotations, and a principle that there be domain elements whose
class extension is any finite subset of the domain of discourse.

Can an interpretation be found for this?  It is not trivial, and certainly
can't use ZF-AC as a magic wand.  One problem is that if the zero-subclass
restriction is used as the empty set, then the three-subclass restriction
can interfere with its own class extension.  Without a domain element for
the three-subclass restriction, no domain elements have three subclasses,
so the three-subclass restriction's class extension should be the empty
set.  However, then there are two classes whose class extension is the
empty set, so the domain elements corresponding to singleton sets get three
subclasses, violating the semantic condition for the three-subclass
restriction.

> >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?
> 
> No. I will sketch you a satisfying interpretation after dinner. In 
> fact there is one in an earlier message, where the set/property a is 
> equinumerous with its own class of subclasses.

You have not shown that this is a satisfying interpretation.

> >  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.
> 
> Well, tell me what you think is wrong with it. It seems obvious 
> enough to me. We start with some sets, and some mappings over them. 
> We then need to be certain the the closure of those sets under those 
> mappings exists. How could it not? 

First of all, not any such closure works.  For example, the extensional
closure, where a class is identified with its class extension, doesn't work
because it violates the functionality of IRP.  Fixing this takes you out of
a direct appeal to set theory already.  Then you might not be able to build
a subClassOf extension that meets the semantic conditions from the OWL
restrictions on subClassOf.

> When I asked you a few weeks ago 
> why these issues didnt crop up in your MT for abstract OWL, you said 
> they just didnt arise, which is in a sense true, but in that very 
> same sense they do not arise here either. 

In my model theory the problems don't exist because classes and
restrictions are not in the domain of discourse.  My domain of discourse is
just a flat set, everything else is built up from that foundation. 

> They don't arise because we 
> no longer feel the need to constantly prove that sets exist. We know 
> that it makes sense to talk about the sets defined in terms of other 
> sets in 'reasonable' ways. We know that ZF+AC can establish the 
> existence of almost any set that anyone will ever want. It can 
> certainly handle restrictions defined over the positive integers and 
> functions on subsets of a fixed set, which is all that OWL needs.

OK, ZF+AC gets you part of the way.  I've never disputed that.  Now you
need to show how to build a Large OWL interpretation on this basis.

[...]

> >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.
> 
> Actually, I think that is *all* that I need to do. If there is one, 
> then your concerns about large OWL/RDFS being uninterpretable are 
> clearly unwarranted.

I have other concerns as well, as I have indicated.

> >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.
> 
> I think your insistence on seeing examples rather than accepting a 
> general argument is quixotic and wrong-headed, rather like someone 
> who doesnt believe that equations have solutions until he is shown a 
> few. But I will do my best to convince you.

Well, I wouldn't need an example, per se, but you haven't provided anything
even close to an argument that the structures that you are creating can be
completed in a way that satisfies all the semantic conditions from Large OWL.

> >  Further, the
> >mere fact that I am asking is a bad sign.
> 
> I tend to agree, but maybe not about what it is sign of.

................

> Pat

peter

Received on Tuesday, 24 September 2002 08:37:17 UTC