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)
Click here for a formal theory-this article will appear in Springer’s Encyclopedia of
Complexity and System Science.
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
