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### What is granular computing? (updated 1/17/2009)

Informally, Granular computing (GrC) is a computing theory based on granulation (=granular structure)

The intuition is: element of the universe is data, some subsets (granules) of data represent  basic units of KNOWN knowledge, LACK of knowledge (Uncertainty) or a subproblem (in Divide and Conquer)

#### I. Some Examples/Applications from classical Database Theory (and Data Mining) Click for more details

A. Simplest granulation is a partition (Rough Set Theory is based on partition)

A partition P of a classical set U is a collection of subsets that are mutually disjoint and their union is U. Each subset is called an equivalence class.

Observe that a partition defines and is defined by an equivalence relation R.

B. Relational Table (relation instance) has a very natural granular structure;

(Pawlak Theorem). n-column relational table  « (U, E1, E2, . . .),  where each E1, E2, . . . are equivalence relations induced by attributes A1, A2, . . ..

C. Generalize the equivalence relation to binary relation:

Relational Tables with a binary relation on each attribute domain; the binary  relation carries additional information or knowledge,

(Extended Pawlak Theorem)  Such a relation  « (U, B1, B2,     ),  where B1, B2, are binary relations that represents certain semantics

David Hsiao, Seymour Ginsburg and Richard Hull (equivalence relations and partial orderings)

D. Generalize the binary relation to pre-topological spaces

Relational Tables with a neighborhood system (a pre-topological space) on each attribute domain, that carries carries additional information or knowledge,,

(Extended Pawlak Theorem)  Such a relation  « (U, N1, N2,     ),  where N1, N2, are neighborhood systems(pre-topological spaces) that carries some semantic information or knowledge.

Wesely Chu and T. Y. Lin

E Generalize the binary relation to n-nary relations

Relational Tables with a n-ary Relation on each attribute domain  that carries additional information or knowledge,

(Extended Pawlak Theorem)  Such a relation  « (U, G1, G2,     ),  where G1, G2, are general  n-ary relations (different n for different G) that represents certain semantics in each attribute domain

Note:  the granular structure {G1, G2, . .} is mathematically the same as a relational database  and the relational structure of  the First Order Logic. (all are sets of relation instances)

However, their respective semantics are very different.  Due to this syntactic similarity, many database technologies are readily available to GrC   .

II. Two Major Applications