1 Granular Computing: Formal Theory & Applications Tsau Young (‘T. Y.’) Lin GrC Society and Computer Science Department, San Jose State University San Jose, CA 95192, USA ;

2 Outline 1.Introduction 2.GrC on the web 3.Formal Theory 4. Applications 5. Conclusions Scope of GrC

3 A Bit History Zadeh’s GrM granular mathematics T.Y. Lin  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, )

4 GrC on the Web W eb Page is a linearly ordered Text. 5 th GrC Model

5 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.

6 Granular Structure(GrS) Wall Street InformationSecurity FinanceIndustry 2-tuples are generalized equivalence classes of size 2

7 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. that was based on the ancient intent of 2. Wall Street is a shorthand for US finance industry. Its E-security has applied security policy that was based on the ancient intent of Chinese wall. is 3. Wall Street represents an abstract concept of finance industry. Its information security policy is Chinese wall.

8 Granular Structure(GrS) 4-tuples are generalized equivalence classes securitypolicyChinawall WallStreetFinanceIndustry

9 GrC model for the Web GrC model for the Web U = a set of keywords GrS= a collection of 1-ary relation: frequent keywords 2-ary relation: freq keyword pairs...

10 Geometric View tuples  Simplexes GrS  Simplicial Complex an amazing fact!

11 It is derived from Data Mining: Apriori principle = geometry: Closed condition Is it God’s will?

12 Concept Analysis Simplex, as an ordered keyword set, represents a Concept in the web

13 Concept: 1-simplex Concept: 1-simplex Wall    Street Wall Street is a simplex represents the concept of financial industry

14 Concept: 1-simplex Concept: 1-simplex Finance    Industry Finance Industry (Stemming)

15 Concept: 3-simplex Concept: 3-simplex Wall China Security Policy

16 Concept Analysis The Concepts on the web forms a simplicial complex So we can use geometry to analyze the knowledge structure of the web

17 Knowledge Structure Simplicial Complex Knowledge Structure Simplicial Complex a b c d x z y w h f g e Open tetrahedron 1 Open tetrahedron 2

18 Knowledge Structure Indexing the Concepts by indexing the concepts... we are building Knowledge Based Search Engine

19 Google and etc index only the keyword, which is 0-dimensional subcomplex of the simplicial complex!

20 The output will be clustered by primitive concepts T. Y. LIN  – Tung Yen Lin –Tsau Young Lin...

21 Formal Theory

22 Formal Theory Use Category Theory to formalize the universe of discourse

23 Model in Category Theory Model in Category Theory GrC Model (U, β) U = a set of objects U i i=1, 2, … in abstract category β=a set of relation objects

24 Non Commutative Granules: Category of Sets (5th GrC): the Web Functions (6th GrC) Turing machines(7th GrC)

25 Other Applications

26 Other Applications 2. Information Flow Security Solve 30 years outstanding Problem; IEEE SMC rd GrC Model

27 Other Applications 3. In the category of Turing machines Expressing DNA sequences by finite automata 7 th GrC Model

28 Other Applications 3. In the category of Turing machines Identify authorship by expressing the stops words by finite automata 7 th GrC Model

29 Other Applications 4. In the category of Functions Patterns in numerical sequences (1999) 6 th GrC Model

30 Conclusions

31 Scope of GrC Scope of GrC Ltofi Zadeh: “ TFIG... 1.mathematical in nature 1.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.

32 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, The World of Mathematics (ed J Newman) , 1956

33 Scope of GrC Mathematically o incorrect o un-substantiated opinions are not considered (verbally add...

34 Scope of GrC Rough set /computing (RS) has been a guide for GrC, but 2. GrC  beyond RS

35 Scope of GrC The time for “BS” theory has gone Please Read the Fallacies in GrC2008

36 Thanks !

37 Thanks !

38 Thanks !

39 Key Components (skip) 1.GrC Model (U, β): 2.Two Operations: (skip) Granulation Integration (Important in DB) (IBM Almaden Project)

40 Key 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

41 Key Components 4. Four Structures Granular structure/variable (Zadeh) Quotient Structure (QS - Zhang) Knowledge Structure (KS - Pawlak) Linguistic Structure/variable(Zadeh) (From TY Lin’s home page  granular computing conference 2009  GrC Information Center  Click here for a formal theory in First paragraph.) Click here for a formal theory

42 Two Important Structures Quotient Structure (QS) Each granule  a point Interactions are axiomatized

43 Knowledge Structure Each point  a concept Concept interactions  QS Concepts are attribute values in Rough Set Theory (RS)