Granular Computing: A new problem Solving Paradigm

Tsau Young (T.Y.) Lin

Department of Computer Science

San Jose State University

San Jose, CA 95192

tylin@cs.sjsu.edu

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