# Re: RDF-Based Semantics and n-ary dataranges

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Fri, 03 Apr 2009 07:18:09 -0400 (EDT)
Message-Id: <20090403.071809.193487976.pfps@research.bell-labs.com>

```From: "Michael Schneider" <schneid@fzi.de>
Subject: RE: RDF-Based Semantics and n-ary dataranges
Date: Fri, 3 Apr 2009 12:31:30 +0200

> 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
>
> ?

Absolutely.  This is a perfectly fine data domain in RDF and thus in the OWL
2 RDF-Based semantics and, further, there is no reason not to have parts
of the RDF-Based semantics that look inside these elements of the
domain.  In fact, the entire idea behind datatypes is precisely that those
elements of the data domain that matter do have some internal structure.

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

The interpretation of every datatype and data range (unary or n-ary, it
doesn't matter) would be a subset 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.

It is not that such sets *should* be allowed to be subsets of the data
domain, it is that they *currently* are allowed as subsets of the data
domain.  There is nothing in the RDF semantics that prohibits elements
of the data domain from being *anything* - they could be integers, they
could be pairs of integers, they could be infinite strings, they could
even be actual people.

> 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?

This is about the direct semantics, which may (and indeed does) have a
different basis, so the question is not very germane here.

> 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.

The two semantics do have to be aligned, true, but that doesn't mean
that the two semantics have to look completely the same.  They already
look quite different in many areas but nonetheless end up being in close
alignment.

> Michael

peter

>>-----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
>>
>>> 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.
>>