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

2 Outline 1.A Bit History 2.Scope of GrC 3. GrC on the Web(2/16;tokenizer; Index; TFIDF) 4. Formal GrC Theory 5. Conclusions-Applications

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 John von Neumann (1941): ” organisms... made up of parts ” (granulation) Method: Axiomatic 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

5 Lotfi Zadeh : Partitioning... into granules. A granule is a clump of bjects, which are drawn together by... functionality

6 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. Neumann: A xiomatic (math)..

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

8 Scope of GrC Rough set /computing (RS) is GrC has served as guides, but focus is beyond RS

9 Scope of GrC Please Read the Fallacies in GrC2008

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

11 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.

12 Granule: 2-ary Relation Wall Street InformationSecurity FinanceIndustry 3 generalized equivalence classes (of size 2)

13 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.

14 Granule: 4-nary Relation securitypolicyChinawall One Generalized Equivalence Class of size 4

15 GrC on the Web GrC on the Web The Universe is: U = the set of keywords in the web pages (use TFIDF or... to find them).

16 2. Granular Structure β Granules are tuples in U  U ... organized into subsets (=relations) of U  U ...

17 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

18 Formal GrC Theory 1.GrC Model (U, β): GrC on the Web 2.Two Operations: (skip) Granulation and Integration 3. Three Semantic Views on β Knowledge Engineering (considering) Uncertainty Theory How-to-solve/compute-it Four Structures

19 Formal GrC Theory 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

20 Formal GrC Theory Quotient Structure (QS) Each granule  a point Interactions are axiomatized

21 Formal GrC Theory 3. QS=KS: each point  a concept Concept interactions from QS In RS, concepts are labeled by attribute values; No interactions

22 Formal GrC Theory Linguistic Structure: granule  words from the precisitated natural language.

23 QS= KS in the Web QS= KS in the Web QS: a tuple (a simplex) is a point in an ordered simplicial complex KS: Each simplex represents a concept in the web defined by the ordered keyword set (tuple)

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

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

26 4-nary Relation securitypolicyChinawall Represent the Concept: Chinese Wall Security Policy

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

28 Knowledge Structure of the Web is a Simplicial Complex of Concepts

29 QS= KS in the Web QS= KS in the Web a b c d x z y w h f g e Open tetrahedron 1 Open tetrahedron 2

30 Applications By indexing the concepts in simplicial complex, we are building 1. Knowledge Based Search Engine

31 The output will be clustered by primitive concepts... Next Generation Search Engine -

32 TYLIN has over millions items It will be group into Tung Yen Lin Tsau Young Lin

33 The output will be clustered by primitive concepts... TYLIN will be group into Tung Yen Lin Tsau Young Lin And many others

34 Other Applications 2. Information Flow Security 3 rd GrC model Solve 30 years outstanding Problem; IEEE SMC 2009

35 Applications 3. Approximation Theory in the category of Turing machines 7 th GrC Model 3a. Expressing DNA sequences by finite automata 2009

36 3b. Identify authorships of books: Harry Potter c. Intrusion Detection System 2005 (authorships of programs)

37 4. Approximation Theory in the category of Functions 6 th GrC Model Patterns in numerical sequences (1999)

38 Thanks !