RE: RDF-Based Semantics and n-ary dataranges

Hi Peter,

I see a lot of "no"s to all of my statements, but I am not quite certain
that I understand your arguments.

Do you claim that a set such as

  S := { 1, <2,3>, <4,5,6> }

is allowed to be a subset of the data domain? I mean, where "<,>" is not
just some random way to write down composite data values (such as the "/" in
terms like "2/3" for expressing rationals), but it is really meant to be the
n-tuple operator appearing in statements such as 

  <x,y> in DR

?

Or, more generally, do you say that all n-ary data ranges (or better, their
"value spaces") must be subsets of the data domain?

Let me say that I would not be particularly happy with this sort of
"ontological mixup" (what about sets like "{ 1, {2}, {3,<{4,<5,6>}},7>},
{1,2}x{3,4,5} }"? they should then consequently be allowed as subsets of the
data domain as well!), but telling my reasons would probably lead to far. 

However, what is more important, I then do not correctly understand the
Direct Semantics:

  Direct Semantics, 2.2.2: "Data Ranges"
  <http://www.w3.org/2007/OWL/wiki/Semantics#Data_Ranges>

[[
    All datatypes in OWL 2 are unary, so each datatype DT is interpreted as
a unary 
    relation over ?_D — that is, a set (DT)^DT subset ?_D. 
    Data ranges, however, can be n-ary, as this allows implementations 
    to extend OWL 2 with built-in operations such as comparisons or
arithmetic. 
--> An n-ary data range DR is interpreted as an n-ary relation (DR)^DT over
?_D.
                                                      ^^^^^^^^         ^^^^
]]

Until now, I have understood the word "over" in the context of the word
"relation" to mean

  (DR)^DT subset (?_D)^n

But you seem to suggest that "over" means

  (DR)^DT subset ?_D

without the exponent "n", meaning that dataranges, independent on their
arity, are always subsets of the data domain?

Since I am an official reviewer of the Direct Semantics, I feel obliged to
ask for clarification of the Direct Semantics in this point. Whatever the
actual meaning will be in the end, the RDF-Based Semantics will then need to
be aligned with the Direct Semantics.

Michael

>-----Original Message-----
>From: Peter F. Patel-Schneider [mailto:pfps@research.bell-labs.com]
>Sent: Wednesday, April 01, 2009 11:10 PM
>To: Michael Schneider
>Cc: public-owl-wg@w3.org
>Subject: Re: RDF-Based Semantics and n-ary dataranges
>
>From: "Michael Schneider" <schneid@fzi.de>
>Subject: RE: RDF-Based Semantics and n-ary dataranges
>Date: Wed, 1 Apr 2009 21:44:54 +0200
>
>>>-----Original Message-----
>>>From: public-owl-wg-request@w3.org [mailto:public-owl-wg-
>request@w3.org]
>>>On Behalf Of Ian Horrocks
>>>Sent: Wednesday, April 01, 2009 8:47 PM
>>>To: W3C OWL Working Group
>>>Subject: RDF-Based Semantics and n-ary dataranges
>>>
>>>We didn't manage to conclude this discussion.
>>>
>>>Summary of (my understanding of) the discussion so far:
>
>[...]
>
>>>* the structure of n-ary restrictions is defined in SS&FS, but
>>>(hopefully) only the unary case can occur in conforming ontologies
>>>(as above)
>>>* Michael believes that as a result the RDF-Based semantics is broken
>>
>> Yes, it is _syntactically_ broken. It essentially contains an
>expression of
>> the form
>>
>>   "<x1,...,xn> in S"
>>
>> where "S" is defined to denote a subset of the object domain.
>
>I still don't understand why this can be considered to be syntactically
>or semantically or even pragmatically broken.
>
>It is entirely possible to have an OWL 2 Full interpretation
>	I = <IR, IP, IEXT, IS, IL, LV>
>where LV and thus IR contains not only things like the integers but
>also things like pairs, triples, quads, quints, ... over the integers
>(or over reals, or over complex numbers, or even over elements in
>IR-LV).
>
>However, even if LV only contains "standard" data values there is
>nothing wrong with asking whether LV contains a tuple.  This is a
>perfectly well-formed question even in this case, it is just that the
>answer is then always false.  (Which is, of course, the expected and
>desirable answer.)
>
>> If something like this would be written in the Direct Semantics, you
>would
>> certainly be horrified.
>
>Why?  Again, the answer would just be false.
>
>> And so you should be for the RDF-Based Semantics as
>> well.
>
>I'm certainly not horrified, and I don't see why anyone would be
>horrified.
>
>> Because this has nothing to do with the distinction between the Direct
>> Semantics and the RDF-Based Semantics. It only has to do with what can
>be
>> written syntactically in the set theory that underlies both our
>semantics.
>
>There is nothing in even set theory that requires that the atomic set
>elements not have some internal structure.
>
>> (There are other problems as well, but I think this is the simplest
>one to
>> acknowledge.)
>
>I don't see this problem, nor can I think of any other problems.
>
>> The problem is: Interpretation function under the semantics of RDF are
>> restricted to interpret names by individuals (instances of the domain
>IR).
>> In addition (in RDFS), there are two functions that allow me to
>/indirectly/
>> talk about subsets of the domain IR (the class extension function
>> "ICEXT()"), and subsets of the product IRxIR (the property extension
>> function "IEXT()"). But there is not yet such a function (or a
>collection of
>> functions) that allow me to talk about subsets of the products IR^n
>for
>> arbitrary n.
>
>I don't follow this reasoning at all.  Certainly there is nothing (so
>far) that requires tuples to be present in IR, but there is also nothing
>(so far) that forbids tuples from being present in IR.
>
>> So the underlying logic may allow me to write statements as above, at
>least
>> for an "S" representing a set of n-ary tuples. The problem is that I
>do not
>> reach this functionality of the underlying logic from within the
>current
>> framework of the RDFS semantics. So I need to extend this framework.
>This is
>> what I suggest to do (before April 15th...).
>
>Again, I don't think that any change is required.  As far as I can see,
>the RDF-Based Semantics is currently entirely coherent.
>
>>>* Peter doesn't agree.
>
>Yep.
>
>>>Comments?
>>>
>>>Ian
>>>
>
>> Cheers,
>> Michael
>
>peter

--
Dipl.-Inform. Michael Schneider
Research Scientist, Dept. Information Process Engineering (IPE)
Tel  : +49-721-9654-726
Fax  : +49-721-9654-727
Email: michael.schneider@fzi.de
WWW  : http://www.fzi.de/michael.schneider
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