[Top] [All Lists]

Re: [ontolog-forum] Question: more on N-ary?

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Chris Menzel <cmenzel@xxxxxxxx>
Date: Thu, 14 Jul 2005 18:43:09 -0500
Message-id: <20050714234309.GV903@xxxxxxxx>
On Thu, Jul 14, 2005 at 03:42:00PM -0700, Duane Nickull wrote:
> Can anyone provide a clear concise distinction between a symmetric and 
> coreflexive relationships?  The gist seems to be that both express a 
> relationship wherefor all x and y in X it holds that if xRy then x = y. 
> It appears that one states "holds xRy and yRx" while the other equates 
> equality "x=y"    (01)

Symmetry and coreflexivity are properties of binary relations.  A binary
relation R is symmetric just in case, whenever it holds of the pair
(a,b) it also holds of (b,a); formally:    (02)

(<=> (Symmetric R)
     (forall (?x ?y)
             (=> (R a b)
                 (R b a))))    (03)

Note this is conditional -- a symmetric relation R needn't hold between
all, or even any, pairs (a,b); but whenever it *does* hold of (a,b), it
must hold of (b,a) as well.  An example of a symmetric relation is
"brother-of".    (04)

A relation is coreflexive just in case it holds of (a,b) only if a is
identical to b:    (05)

(<=> (Coreflexive R)
     (forall (?x ?y)
             (=> (R a b)
                 (= a b))))    (06)

Coreflexive relations are pretty rare in formal ontology, as they are,
as Duane notes, simply a restricted form of identity, and there aren't
many (any?) intuitive relations that play that role.  It's easy enough
to cook one up, of course:    (07)

(<=> (CabbageIdentical ?x ?y)
     (and (Cabbage ?x)
          (Cabbage ?y)
          (= ?x ?y)))    (08)

Thus, a and b are CabbageIdentical just in case they are the same head
of cabbage, and that is obviously coreflexive.  But it's hard to see how
such a relation could be useful for anything.  The broader property of
reflexivity is much more common and much more useful:    (09)

(<=> (Reflexive R)
     (forall (?x) (R ?x ?x)))    (010)

> Also, in SUMO, InverseRelation is expressed as:
> (=>
>   (inverse ?REL1 ?REL2)
>   (forall (?INST1 ?INST2)
>      (<=>
>         (holds ?REL1 ?INST1 ?INST2)
>         (holds ?REL2 ?INST2 ?INST1))))
> The distinctions between the 3 seem to be very subtle. InverseRelation 
> and AsymmetricRelation are opposite? I find it a bit confusing.    (011)

The inverse R' of a relation R holds of the pair (b,a) if and only if R
holds of (a,b).  Thus, if (as in set theory) we simply identify a binary
relation with the set of ordered pairs it holds of, the inverse of a
relation R is the relation that results from "flipping" each pair in R
so that the first element of the pair becomes the second and vice versa.    (012)

Unlike symmetry and coreflexivity, which, as noted, are *properties* of
binary relations, inverse is itself a *binary relation* between two
binary relations.    (013)

Chris Menzel    (014)

Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/ 
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (015)

<Prev in Thread] Current Thread [Next in Thread>