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Granular Computing:Granular Computing:Formal Theory & Applications

Tsau Young (‘T. Y.’) LinGrC Society

andComputer Science Department,

San Jose State UniversitySan Jose, CA 95192, USA

tylin@cs.sjsu.edu ; prof.tylin@gmail.com

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OutlineOutline1. Introduction

2. GrC on the web

3. Formal Theory

4. Applications

5. Conclusions• Scope of GrC

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A Bit HistoryA Bit HistoryZadeh’s GrM granular mathematics

T.Y. Lin 1996-97

GrC Granular Computing

(Zadeh, L.A. (1998) Some reflections on soft computing, granular computing and their roles in the conception, design and utilization of information/intelligent systems, Soft Computing, 2, 23-25.)

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GrC GrC on theon the Web WebWeb Page is

a linearly ordered Text.

5th GrC Model

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1. Wall Street is a symbol for American

financial industry. Most of the computer systems for those financial institute have employed information flow security policy.

2. Wall Street is a shorthand for US financial industry. Its E-security has applied security policy that was based on the ancient intent of Chinese wall.

3. Wall Street represents an abstract concept of financial industry. Its information security policy is Chinese wall.

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Granular Structure(GrS)Granular Structure(GrS)

Wall StreetInformation Security

Finance Industry

2-tuples are generalized equivalence classes of size 2

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1. Wall Street is a symbol for American

finance industry. Most of the computer systems for those financial institute have employed information flow security policy.

2. Wall Street is a shorthand for US finance industry. Its E-security has applied security policy that was based on the ancient intent that was based on the ancient intent ofof Chinese wall.

3. Wall Street represents an abstract concept of finance industry. Its information security policy isis Chinese wall.

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Granular Structure(GrS)Granular Structure(GrS)

4-tuples are generalized equivalence classes

security policy China wall

Wall Street Finance Industry

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GrC GrC model for the Web model for the Web

U = a set of keywords

GrS= a collection of • 1-ary relation: frequent keywords

• 2-ary relation: freq keyword pairs

. . .

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Geometric ViewGeometric View

• tuples Simplexes

• GrS Simplicial Complex

• an amazing fact!

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It is derived from

Data Mining: Apriori principle =geometry: Closed condition

Is it God’s will?

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Concept AnalysisConcept Analysis

• Simplex, as an ordered keyword set, represents

• a Concept in the web

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Concept: 1-simplexConcept: 1-simplex

Wall Street

Wall Street is a simplex

represents the concept of financial industry

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Concept: 1-simplexConcept: 1-simplex

Finance Industry

Finance Industry (Stemming)

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Concept: 3-simplexConcept: 3-simplex

Wall

China

Security

Policy

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Concept AnalysisConcept Analysis

• The Concepts on the web forms a simplicial complex

• So we can use geometry to analyze the knowledge structure of the web

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Knowledge Structure Knowledge Structure Simplicial Complex Simplicial Complex

a

b

c

d

x

z

y

w

h

f g

e

Open tetrahedron 1

Open tetrahedron 2

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Knowledge Structure Knowledge Structure Indexing the ConceptsIndexing the Concepts

• by indexing the concepts . . . we are building

• Knowledge Based Search Engine

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• Google and etc

index only the keyword, which is 0-dimensional subcomplex of the simplicial complex!

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• The output will be clustered

by primitive concepts

• T. Y. LIN – Tung Yen Lin– Tsau Young Lin . . .

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Formal Theory

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Formal TheoryFormal Theory• Use Category Theory

• to formalize the universe of discourse

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Model in Category TheoryModel in Category Theory

GrC Model (U, β)U = a set of objects Ui i=1, 2,

… in abstract category

β=a set of relation objects

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Non Commutative Granules:Non Commutative Granules:

Category of

• Sets (5th GrC): the Web

• Functions (6th GrC)

• Turing machines(7th GrC)

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Other Applications

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Other ApplicationsOther Applications2. Information Flow Security

• Solve 30 years outstanding Problem; IEEE SMC 2009

• 3rd GrC Model

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Other ApplicationsOther Applications3. In the category of Turing machines

• Expressing DNA sequences by finite automata

• 7th GrC Model

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Other ApplicationsOther Applications3. In the category of Turing machines

• Identify authorship by expressing the stops words by finite automata

• 7th GrC Model

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Other ApplicationsOther Applications4. In the category of Functions

• Patterns in numerical sequences (1999)

• 6th GrC Model

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Conclusions

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Scope of GrCScope of GrC

Ltofi Zadeh: “TFIG...1.mathematical in

nature1.Zadeh, L.A. (1997) ‘Toward a theory of fuzzy information granulation and its centrality in

human reasoning and fuzzy logic’, Fuzzy Sets and Systems, Vol. 90, pp.111–127.

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John von Neumann (1941):

”organisms ... made up of parts” (granulation)

Axiomatic Method 1. von Neumann J(1941): The General and Logical Theory of Automata in: Cerebral

Mechanisms in Behavior, pp. 1-41, Wiley, 1941. The World of Mathematics (ed J Newman) 2070-2098, 1956

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Scope of GrCScope of GrCMathematically

o incorrect

o un-substantiated opinions are not considered(verbally add. . .

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Scope of GrCScope of GrCRough set /computing (RS) has been

a guide for GrC, but

2. GrC beyond RS

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Scope of GrCScope of GrCThe time for “BS” theory has

gone

Please Read the Fallacies in

GrC2008

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Thanks !

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Thanks !

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Thanks !

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Key Components (skip)Key Components (skip)

1. GrC Model (U, β):

2. Two Operations: (skip)

• Granulation

• Integration (Important in DB) (IBM Almaden Project)

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Key ComponentsKey Components

3. Three Semantic Views on β• Knowledge Engineering (This talk)• Uncertainty Theory Zadeh and Lin’s initial idea • How-to-solve/compute-it

Polya 1945

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Key ComponentsKey Components4. Four Structures• Granular structure/variable (Zadeh)• Quotient Structure (QS - Zhang) • Knowledge Structure (KS - Pawlak) • Linguistic Structure/variable(Zadeh)http://xanadu.cs.sjsu.edu/~grc/grcinfo_center/1Linabs_william.pdf(From TY Lin’s home page granular computing conference 2009 GrC Information

Center Click here for a formal theory in First paragraph.)

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Two Important StructuresTwo Important Structures

Quotient Structure (QS)

• Each granule a point

• Interactions are axiomatized

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Knowledge StructureKnowledge Structure

• Each point a concept

• Concept interactions QS

• Concepts are attribute values in Rough Set Theory (RS)