rst09 panel - Rough Sets Theory

Semantics of Rough Sets
From theory to applications (through
semantics understanding).
Motivated by Dubois, D. and Prade, P.
The three semantics of fuzzy set,
Fuzzy Sets and Systems, 90, 141-150,
1997.

Rough sets and concepts
Rough sets for modeling concepts, or
more precisely the extensions of
concepts.

Concepts
A concept, in the classical view, is
defined by a pair of intension and
extension.
The extension: instances to which the
concept applies.
The intension: a set of singly necessary
and jointly sufficient conditions that
describe the instances of the concept.

Concepts: idealization
Concepts are assumed to have welldefined
boundaries and their
extensions can be precisely defined by
sets of objects.

Concepts: reality
Concepts in many practical situations
cannot be precisely defined and their
extensions may not be given exactly by
sets of objects.

Concepts: generalization
Concepts with grey/gradual boundaries.
Partially known concepts.
Undefinable concepts.
Approximate concepts.
… … … …
* Last three situations were discussed by Marek, V.W.
and Truszczynski, M. Contributions to the theory of
rough sets, Fundamenta Informaticae, 39(4) , 389 -
409, 1999.

Concepts with grey/gradual boundaries
There is not a well-defined boundary that
differentiates the instances from the noninstances
of the concept. For some objects,
the concept only applies partially instead of
fully.
Fuzzy sets.
* Zadeh, L.A. Fuzzy sets, Information and Control, 8, 338–353,
1965.
* Dubois, D. and Prade, P. Fuzzy rough sets and rough fuzzy
sets, International Journal of General Systems, 17, 191-209.

Partially known concepts
An object may actually be either an
instance or not an instance of a
concept.
Due to a lack of information and
knowledge, one can only express the
state of instance and non-instance for
some objects, instead of all objects.

Partially known concepts
One has a partially known concept defined by a
lower bound and upper bound of its extension.
Interval sets
Y.Y. Yao, Interval-set algebra for qualitative knowledge
representation, Proceedings of the Fifth International
Conference on Computing and Information, 370-374, 1993.

Undefinable concepts
In general, the intension of a concept
may be defined by using a logical
language, such as the decision logic
language in rough set theory and
knowledge representation languages of
description logic.

Undefinable concepts
It may happen that some objects
cannot be differentiated due to the use
of a fixed and limited set of attributes
for their description. The language
may not be able to define certain sets
of objects that are the extensions of
some concepts. That is, we have
some undefinable concepts with
respect to a particular language..

Undefinable concepts
We have to approximate undefinable
concepts by definable concepts.
Rough sets.
Pawlak, Z. Rough sets, International Journal of
Computer and Information Sciences, 11, 341-356,
1982.

Approximate concepts
One knows the exact extension of a
concept and the language can
precisely define the concept.
The description may be too complex to
be of any practical value; it may be
difficult to understand and manipulate.

Approximate concepts
This may require us to approximate the
concept by some concepts with simple
descriptions.
Rough sets.
* Marek, V.W. and Truszczynski, M. Contributions to
the theory of rough sets, Fundamenta Informaticae,
39(4) , 389 - 409, 1999.

Thank You!
http://www.cs.uregina.ca/~yyao/rough_set.html