1. Panel Discussion on Granular Computing at RSCTC2004 J. T. Yao University of Regina Email: jtyao@cs.uregina.ca Web: http://www2.cs.uregina.ca/~jtyao

2. What is Granular Computing? “There are three basic concepts that underline human cognition: granulation, organization and causation. Informally, granulation involves decomposition of whole into parts; Organization involves integration of parts into whole; Causation involves association of causes with effects. Granulation of an object A leads to a collection of granules of A, with a granule being a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality” (Zadeh 1997)

3. What is Granular Computing An umbrella term to cover any theories, methodologies, techniques, and tools that make use of granules in problem solving. A subset of the universe is called a granule in granular computing. Basic ingredients of granular computing are subsets, classes, and clusters of a universe.

4. Granule Computing and Data Mining A concept is understood as a unit of thoughts that consists of two parts, the intension and extension of the concept. The intension of a concept consists of all properties or attributes that are valid for all those objects to which the concept applies. The extension of a concept is the set of objects or entities which are instances of the concept. A rule can be expressed in the form, φ=>ψ where φ and ψ are intensions of two concepts. Rules are interpreted using extensions of the two concepts.

5. How do Rough Sets Contribute to Granular Computing? Zadeh define Granular Computing in BISC/SIG on GrC as “a superset of the theory of fuzzy information granulation, rough set theory and interval computations, and is a subset of granular mathematics”.

6. Zadeh’s Fuzzy GrC Model Granules are constructed and defined based on the concept of generalized constraints. Relationships between granules are represented in terms of fuzzy graphs or fuzzy if then rules. A granule is defined by a fuzzy set G = {X | X isr R}

7. Pawlak’s Rough Set Model Granulation: Universe => granules Some granules can only be approximately described. Rough sets can deal with approximation of information granulation. In the case one cannot describe X using E

8. Information Tables U: a finite nonempty set of objects. A t : a finite nonempty set of attributes. L: a language defined using attributes in A t. V a : a nonempty set of values for a ∊ A t I a : U → V a is an information function.

9. Concept Formation Atomic formula: a=v (a ∊ A t, v ∊ V a ) If φ, ψ are formulas, so is φ ∧ ψ If a formula is a conjunction of atomic formulas we call it a conjunctor. Meaning of a formula: m(φ)={x ∊ U | x ⊨ φ} x ⊨ a=v iff I a (x)=v A definable concept is a pair (φ, m(φ)) φ is the intension of the concept m(φ) is the extension of the concept

10. Classification Problems Assume that each object is associated with a unique class label. Objects are divided into disjoint classes which form a partition of the universe. The set of attributes is expressed as A t = F ∪ {class}, where F is the set of attributes used to describe the objects. To find classification rules of the form, φ ⇒ class = c i, where φ is a formula over F and c i is a class label.

11. Solution to Classification Problems The partition solution to a consistent classification problem is a conjunctively definable partition π such that π ≼ π class. The covering solution to a consistent classification problem is a conjunctively definable covering  such that  ≼ π class.

12. A construction algorithm Construct the family of basic concept with respect to atomic formulas: BC(U) = (a=v, m (a = v)) | a  F, v  V a } Set the unused basic concepts to the set of basic concepts: UBC(U) = BC(U). Set the granule network to GN = ({U},  ), which is a graph consists of only one node and no arc. While the set of smallest granules in GN is not a covering solution of the classification problem do the following: Compute the fitness of each unused basic concept. Select the basic concept C=(a=v, m(a=v)) with maximum value of fitness. Set UBC(U) = UBC(U) - {C}. Modify the granule network GN by adding new nodes which are the intersection of m(a=v) and the original nodes of GN; connect the new nodes by arcs labelled by a = v.

13. References Pawlak, Z., Granularity of knowledge, indiscernibility and rough sets, IEEE International Conference on Fuzzy Systems, 106-110, 1998. Yao, J.T., Yao, Y.Y. A granular computing approach to machine learning, FSKD'02, 732-736, 2002. Yao, Y.Y. Granular computing: basic issues and possible solutions, JCIS (I), 86-189, 2000. Zadeh, L.A. Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets and Systems, 19, 111-127, 1997.