Granular Computing:

1. Granular Computing: Formal Theory & Applications Tsau Young (‘T. Y.’) Lin Computer Science Department, San Jose State University San Jose, CA 95192, USA tylin@cs.sjsu.edu ; prof.tylin@gmail.com

2. Outline • A Bit History • 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’sGrM 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.)

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, 1941. The World of Mathematics (ed J Newman) 2070-2098, 1956

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

6. 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: Axiomatic (math). .

7. Scope of GrC Mathematically • incorrect • 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 2. . . . beyond RS

9. Scope of GrC Please Read the Fallacies in GrC2008

10. GrC on the Web • Web Page is a linearly orderedText. • 5th GrC Model

11. 1. Wall Street is a symbol for American financial industry. Most of the computer systems for those financial institute have employed informationflow 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 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. 2. Wall Street is a shorthand for US finance industry. Its E-security has applied security policythat was based on the ancient intent ofChinese wall. 3. Wall Street represents an abstract concept of finance industry. Its information security policy is Chinese wall.

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

15. 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 GrC Model (U, β) U =a set of objectsUi i=1, 2, … in abstract category β=a set of relation objects

18. Formal GrC Theory • GrC Model (U, β): GrC on the Web • 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) 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.)

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: 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 Represent the Concept: Chinese Wall Security Policy

27. Concept: 3-simplex Policy Security China Wall

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

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

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 • 3rd GrC model • Solve 30 years outstanding Problem; • IEEE SMC 2009

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

36. 3b. Identify authorships of books: Harry Potter 2008 3c. Intrusion Detection System 2005 (authorships of programs)

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

38. Thanks !