rst09 panel - Rough Sets Theory
rst09 panel - Rough Sets Theory
rst09 panel - Rough Sets Theory
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Semantics of <strong>Rough</strong> <strong>Sets</strong><br />
From theory to applications (through<br />
semantics understanding).<br />
Motivated by Dubois, D. and Prade, P.<br />
The three semantics of fuzzy set,<br />
Fuzzy <strong>Sets</strong> and Systems, 90, 141-150,<br />
1997.
<strong>Rough</strong> sets and concepts<br />
<strong>Rough</strong> sets for modeling concepts, or<br />
more precisely the extensions of<br />
concepts.
Concepts<br />
A concept, in the classical view, is<br />
defined by a pair of intension and<br />
extension.<br />
The extension: instances to which the<br />
concept applies.<br />
The intension: a set of singly necessary<br />
and jointly sufficient conditions that<br />
describe the instances of the concept.
Concepts: idealization<br />
Concepts are assumed to have welldefined<br />
boundaries and their<br />
extensions can be precisely defined by<br />
sets of objects.
Concepts: reality<br />
Concepts in many practical situations<br />
cannot be precisely defined and their<br />
extensions may not be given exactly by<br />
sets of objects.
Concepts: generalization<br />
Concepts with grey/gradual boundaries.<br />
Partially known concepts.<br />
Undefinable concepts.<br />
Approximate concepts.<br />
… … … …<br />
* Last three situations were discussed by Marek, V.W.<br />
and Truszczynski, M. Contributions to the theory of<br />
rough sets, Fundamenta Informaticae, 39(4) , 389 -<br />
409, 1999.
Concepts with grey/gradual boundaries<br />
<br />
There is not a well-defined boundary that<br />
differentiates the instances from the noninstances<br />
of the concept. For some objects,<br />
the concept only applies partially instead of<br />
fully.<br />
Fuzzy sets.<br />
* Zadeh, L.A. Fuzzy sets, Information and Control, 8, 338–353,<br />
1965.<br />
* Dubois, D. and Prade, P. Fuzzy rough sets and rough fuzzy<br />
sets, International Journal of General Systems, 17, 191-209.
Partially known concepts<br />
An object may actually be either an<br />
instance or not an instance of a<br />
concept.<br />
Due to a lack of information and<br />
knowledge, one can only express the<br />
state of instance and non-instance for<br />
some objects, instead of all objects.
Partially known concepts<br />
<br />
One has a partially known concept defined by a<br />
lower bound and upper bound of its extension.<br />
Interval sets<br />
<br />
Y.Y. Yao, Interval-set algebra for qualitative knowledge<br />
representation, Proceedings of the Fifth International<br />
Conference on Computing and Information, 370-374, 1993.
Undefinable concepts<br />
In general, the intension of a concept<br />
may be defined by using a logical<br />
language, such as the decision logic<br />
language in rough set theory and<br />
knowledge representation languages of<br />
description logic.
Undefinable concepts<br />
It may happen that some objects<br />
cannot be differentiated due to the use<br />
of a fixed and limited set of attributes<br />
for their description. The language<br />
may not be able to define certain sets<br />
of objects that are the extensions of<br />
some concepts. That is, we have<br />
some undefinable concepts with<br />
respect to a particular language..
Undefinable concepts<br />
<br />
We have to approximate undefinable<br />
concepts by definable concepts.<br />
<strong>Rough</strong> sets.<br />
<br />
Pawlak, Z. <strong>Rough</strong> sets, International Journal of<br />
Computer and Information Sciences, 11, 341-356,<br />
1982.
Approximate concepts<br />
One knows the exact extension of a<br />
concept and the language can<br />
precisely define the concept.<br />
The description may be too complex to<br />
be of any practical value; it may be<br />
difficult to understand and manipulate.
Approximate concepts<br />
<br />
This may require us to approximate the<br />
concept by some concepts with simple<br />
descriptions.<br />
<strong>Rough</strong> sets.<br />
* Marek, V.W. and Truszczynski, M. Contributions to<br />
the theory of rough sets, Fundamenta Informaticae,<br />
39(4) , 389 - 409, 1999.
Thank You!<br />
http://www.cs.uregina.ca/~yyao/rough_set.html