Re: querying DAML+OIL syntax

From: Peter F. Patel-Schneider ([email protected])
Date: 12/04/01 

[Pat may have already seen my replies to Dan, but I thought that I would
put down more of my thinking behind this issue.   peter]

From: Pat Hayes <[email protected]>
Subject: Re: querying DAML+OIL syntax
Date: Mon, 3 Dec 2001 23:08:50 -0600

> >From: Pat Hayes <[email protected]>
> >Subject: Re: querying DAML+OIL syntax
> >Date: Thu, 29 Nov 2001 21:56:39 -0600
> >
[...]
> 
> >  > If we don't have any relationships in the model theory, how will we
> >>  entail ANYTHING?
> >
> >There are lots of entailments, it is just that you end up with too few,
> >because the syntax triples pollute the interpretations.
> 
> Right. But we ought to be able to just get round this by defining a 
> semantic extension to RDF, ie a notion of DAML+OIL-entailment in RDF 
> which treats the syntactic encoding appropriately. It won't look much 
> like RDF entailment, to be sure, but I think it could be described in 
> terms of operations on RDF triples; not all of them RDF-valid, of 
> course, but one cannot expect miracles. The relevant notion of 
> 'closure' would add all the permutations of syntactically ordered but 
> semantically unordered things, for example. In this way DAML+OIL 
> would be an extension to RDFS in the same kind of way that RDFS is an 
> extension to RDF. This would imply that any DAML+OIL-savvy RDF engine 
> would have to be able to recognize the triples that were to be 
> treated specially, of course; but that is hardly news, right?
> 
> I may still be missing something vital in your argument: if so, please say so.

In my initial argument I did not think of the approach of, roughly,
axiomatizing DAML+OIL syntax in the semantics.  When Dan first brought it
up I was rather taken aback, and had serious esthetic problems with it.
Even so, I looked at the techical details of doing this, and was proceeding
along those lines, until .....

To get everything to work you have to 

1/ give all the DAML+OIL constructors and relationships, such as unionOf
   and complementOf, a syntax in the form of triples, which has already
   been done;

2/ require that all interpretations include analogues all syntax, that is
   lists, unionOf constructs for all lists of daml classes, complementOf
   for all daml classes, etc., etc.; and

3/ give semantic conditions for the resulting relationships in the model
   theory, which turn out to be just the recasting of the existing semantic
   conditions.

OK so far?  No!  We have ended up with semantic paradoxes!

Let's try to construct an interepretation that contains a complementOf
loop.  If an object is in the CEXT of the node in the loop it isn't, but if
it isn't then it is.  Ooops.

In fact, we are in really bad shape, because all our interpretations have
these loops.  Why?  Well we just said that we had analogues for all syntax
in all interpretations, and this includes 
	A complementOf A .
If this is just syntax, then it is unsatisfiable (assuming a non-empty
domain requirement).  However the elevation of syntax to semantics in
DAML+OIL results in very similar problems to those it also produces in
more-standard logics. 

> Pat

peter

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