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Re: [ontolog-forum] Ontology of Rough Sets

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Sat, 15 Jan 2011 12:42:41 -0600
Message-id: <46322F1A-67C6-46A9-AEA2-441F3452E41C@xxxxxxxx>
On Jan 15, 2011, at 5:11 AM, Tara Athan wrote:
Christopher Menzel wrote:
On Jan 14, 2011, at 4:16 PM, Rich Cooper wrote:



Agreed, rough set theory is a different view of sets, not probabilistic, not fuzzy,

Rough sets differ formally from fuzzy sets in some important ways, but they are certainly kissin' cousins.  Both were designed to deal with vagueness and imperfect knowledge, and the idea of the boundary region of a rough set is meant to do the same sort of work that fuzzy membership was designed to do.

I disagree,

With what?  Certainly not with my claim that rough set theory was designed to deal with vagueness and uncertainty; Pawlak explicitly cites that as his motivation for rough set theory. Perhaps you are reading too much into my claim that the idea of the boundary region of a rough set is meant to do the same sort of work that fuzzy membership was designed to do.  That is not to say that there is no difference between them.  It is to say only that (a) both theories are designed to deal with imperfect knowledge and (b) the central mechanisms for doing so are, respectively, fuzzy membership in Zadeh's theory and boundary regions in Pawlak's. I find this claim complete uncontroversial.

and I think a number of practitioners of these methods would also. See
DUBOIS, D. and H. PRADE. 1990. ROUGH FUZZY SETS AND FUZZY ROUGH SETS*. International Journal of General Systems, 17(2), pp.191 - 209.

The methods for handling rough sets may look superficially like a specific case of a discontinuous fuzzy membership function, but rough and fuzzy sets were designed to deal with different kinds of uncertainty. In order to perform classification under both kinds of uncertainty, these authors developed rough fuzzy sets or fuzzy rough sets.

Their abstract says it well

"The notion of a rough set introduced by Pawlak has often been compared to that of a fuzzy set, sometimes with a view to prove that one is more general, or, more useful than the other.

Well, I never said anything about usefulness, but it was Pawlak himself who proved that rough set theory is a generalization of fuzzy set theory. To prove that it was, of course, was not his motivation for the theory. But given what Pawlak proved, it follows trivially that rough set theory (suitably restricted) can in principle be applied anywhere that fuzzy set theory can be applied.  That is not to say that it should be.  As Pawlak notes, the intuitions underlying rough set theory are rather different than those underlying fuzzy set theory and, consequently, despite the fact that rough set theory generalizes fuzzy set theory, there might well be modeling situations in which fuzzy set theory would be the more appropriate choice.

In this paper we argue that both notions aim to different purposes. Seen this way, it is more natural to try to combine the two models of uncertainty (vagueness and coarseness) rather than to have them compete on the same problems.

This appears to be the authors' central thesis.  I don't know why you seem to think it is any way inconsistent with what I said.  The idea that the two theories are competitors was no part of my claim.

According to Google Scholar, this paper has been cited 734 times, so I'd say quite a few folks found this view worthwhile.

Which, of course, might be relevant if, again, the paper were inconsistent with anything I'd claimed.

Chris Menzel

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