Is subclassing "strict order" or is it reflexive? RE: [sc34wg3] New SAM PSIs

[Bernard Vatant]

Does Murray need a little help from the maths'teacher? :))

Well, what is the type of class-subclass relation?

-- An "order" relation, noted <=3D herafter, with the three following
properties ...

1. Transitivity: IF (a <=3D b) and (b <=3D c) THEN (a <=3D c)
2. Reflexivity: (For all x) (x <=3D x)
3. Antisymmetry: IF (a <=3D b) and (b <=3D a) THEN (a =3D b)

-- Or a "strict order" relation, noted < hereafter, with the following
properties ...=20

1. Transitivity: IF (a < b) and (b < c) THEN (a < c)
2. (For all x) (x < x) is false=20
3. IF (a < b), THEN (b < a) is false

Note that a "strict order" relation is *not* an "order" relation - speak
about consistency in maths vocabulary :(

I understand, Murray, you consider class-subclass relation as a "strict
order", and do not understand how it can be considered an "order".

In fact, to any "strict order" relation can be associated an "order"
relation, and vice-versa, by turning reflexivity on and off. It happens
that "strict order" is more difficult to manage since its definition
needs negation, so "order" is used most of the time. It's the case for
set inclusion for instance. That's what OWL does for class-subclass.
Antisymmetry seems weird to begin with, but it is the best way to check
for loops, and collapse them if necessary to a single point.=20

In fact what happens in checking order integrity is generally not the
obvious situation:

(a <=3D b) and (b <=3D a)
=20
But loops like:

a0 <=3D a1  ;  a1 <=3D a2 ; ... ; an <=3D a0

In such a case:
- Either the ai have been previously declared as distinct, and it's a
reported error.
- Or all ai are declared equal, by consequence of antissymetry +
transitivity (demonstration is easy ...)

That's what "checking for loops" means, for OWL class-subclass like for
any order relation: same issue e.g. with "whole-part" and same way to
deal with it.

Does that help ?

Bernard

_________________________________

Bernard Vatant
Senior Consultant - Knowledge Engineering
www.mondeca.com
_____________________________________

| -----Original Message-----
| From: sc34wg3-admin@isotopicmaps.org [mailto:sc34wg3-
| admin@isotopicmaps.org] On Behalf Of Murray Altheim
| Sent: jeudi 13 f=E9vrier 2003 18:52
| To: sc34wg3@isotopicmaps.org
|=20
| Rath, Holger (empolis KL) wrote:
| > Murray Altheim [mailto:m.altheim@open.ac.uk] wrote:
| >
| >>Lars Marius Garshol wrote:
| >>
| >>>* Murray Altheim
| >>>|
| >>>| First, I don't understand what a "subtype loop" is logically,
| >>>| and I can't quite think what it could be, in FOL terms. Steve?
| >>>
| >>>A subtype loop looks as follows in LTM syntax:
| >>>
| >>>  super-sub(a : superclass, b : subclass)
| >>>  super-sub(b : superclass, a : subclass)
| >>>
| >>>(Assuming, of course, that 'super-sub', 'superclass', and
'subclass'
| >>>have been assigned the correct PSIs.)
| >>
| >>Okay, so this is something that is syntactically possible but
| >>logically impossible. I'm not sure why it's proposed, unless
| >>there's some domain when something's superclass can also be a
| >>subclass of it. I can't think of any examples.
| >
| > DAML+OIL defines the equality between classes that way.
|=20
| Regardless of what DAML+OIL does, what does it *mean*? I'm
| confused. It's nonsense, AFAIK:
|=20
|       A is superclass of subclass B
|       B is superclass of subclass A
|=20
| That doesn't make logical sense. (I must be missing something
| here, as defining equality of two classes based on them being
| both superclasses and subclasses of each other seems to ignore
| the very definitions of superclass and subclass.)
|=20
| Murray
|=20
| ......................................................................
| Murray Altheim                  <http://kmi.open.ac.uk/people/murray/>
| Knowledge Media Institute
| The Open University, Milton Keynes, Bucks, MK7 6AA, UK
|=20
|      "In Las Vegas Mr Gates also demonstrated a prototype
|       fridge magnet which can be programmed to receive traffic
|       reports, sports results and advertisements from local
|       restaurants using the same FM signal as the wristwatch."
|                                   -- The Guardian, 10 Jan 2003.
|=20
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