Granular Computing: A new problem Solving Paradigm
Tsau Young (T.Y.) Lin
Department of Computer Science
San Jose State University
San Jose, CA 95192
Outline
1. SummaryRough Computing
Equivalence Relation
Neighborhood Concept
Binary Relation
2. Details
Rough Computing
A partition is a set of (1) disjoint subsets,
(2) a cover
Class A
Class B
f, g, h i, j, k
Class Cl, m, n
Rough Computing
• X Y (equivalence)
if and only if
• both belong to the same class
Rough Computing
An Equivalence Relation
Class A
Class B
f g h i j k
Class Cl m n
Equivalence Relation
• X X (Reflexive)
• X Y implies Y X (Symmetric)
• X Y, Y Z implies X Z (Transitive)
Neighborhood Concept
• i j k Class B
i j k Class B In spite of a technical error,
• the idea was, and still is, fascinating
Introduction
• An aggressive model (ACWSP) was proposed by Lin the same year (1989) that keeps the same spirit and corrects the error
• Lost some Strength
Introduction
• Lost Interests until
• A practical way of construction ACWSP was introduced 2000
Brewer and Nash Requirements
• A set of impenetrable Chinese Great Walls
• No corporate data that are in conflict can be stored in the same side of Walls
Brewer and Nash -Theory
• Corporate data are decomposed into
Conflict of Interest Classes(CIR-classes)
• Walls are built around the CIR-classes
• Corporate data is called an object (tradition)
BN -Theory
All objects
Class A
Class B
f, g, h i, j, k
Class Cl, m, n
Is CIR Transitive?
• US (conflict) Russia
• UK Russia
• UK ? US
Is CIR Reflexive?
• US (conflict) US ?
• Is CIR self conflicting?
Is CIR Symmetric?
• US (conflict) USSR
implies
• USSR (conflict) US ?
• YES
BN -Theory• Can they be partitioned?
BN -Theory
CUS, Russia
UK?
France, German
CIR-classes
• CIR classes do overlap (Conflict of Interests)
US, UK, Iraq, . . .
USSR
CIR & IAR
• Complement of CIR: an equivalence relation
Iraq, . . .US, UK, . . .
German, . . .
ACWSP
• CIR: Anti-reflexive, symmetric, anti-transitive
CIR-class IJAR-classes
IJAR-classes
ACWSP
• CIR: Anti-reflexive, symmetric, anti-transitive
CIR-class IJAR-classes
IJAR-classes
ACWSP
• CIR: Anti-reflexive, symmetric, anti-transitive
CIR-class IJAR-classes
IJAR-classes
Trojan Horses
Direct Information flow(DIF)
Professor
Grader
StudentsCIF
DIF Trojan horse(DIF)
ACWSP
• CIR (with three conditions) only allows information sharing within one IJAR-class
• An IJAR-class is an equivalence class; so there is no danger the information will spill to outside.
• No Trojan horses could occur
SCWSP
• Simple CWSP (SVWSP)
No DIF: x y (direct information flow)(x, y) CIR
ACWSP
• Strong CWSP(ACSWP)
No CIF: x . . . y ((composite) information flow)
(x, y) CIR
ACWSP
• Theorem If CIR is anti-reflexive, symmetric and anti-transitive, then
• Simple CWSP Strong CWSP
ACWSP
CIF =a sequence of DIFs
CIF: X=X0 X1 . . . Xn=Y
YCIRX
• To derive a contradiction
ACWSP
• X=X0 X1 implies
X1 [X] CIRX = CIRX1
. . .
Y=Xn [X] CIRX = CIRXn = CIRY
Y CIRX Contradiction