Granular Computing: A New Problem Solving Paradigm

1. Granular Computing: A New Problem Solving Paradigm Tsau Young (T.Y.) Lin tylin@cs.sjsu.edu dr.tylin@sbcglobal.net Computer Science Department, San Jose State University, San Jose, CA 95192, and Berkeley Initiative in Soft Computing, UC-Berkeley, Berkeley, CA 94720

2. Outline 1. Introduction 2. Intuitive View of Granular Computing 3. A Formal Theory 4. Incremental Development 4.1. Classical Problem Solving Paradigm 4.2. New View of the Universe 4.3. New Problem Solving Paradigm 2

3. Outline 1. Introduction

4. Granular computing The term granular computing is first used by this speaker in 1996-97 to label a subset of Zadeh’s granular mathematics as his research topic in BISC. (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.)

5. Granular computing Since, then, it has grown into an active research area: • Books, sessions, workshops • IEEE task force(send e-mails to join the task force, please include • Full name, affiliation, and E-mail

6. Granular computing IEEE GrC-conference http://www.cs.sjsu.edu/~grc/.

7. Granular computing Historical Notes 1. Zadeh (1979) Fuzzy sets and granularity 2. Pawlak, Tony Lee (1982):Partition Theory(RS) 3. Lin 1988/9: Neighborhood Systems(NS) and Chinese Wall (a set of binary relations. A non-reflexive. . .) 4. Stefanowski 1989 (Fuzzified partition) 5. Qing Liu &Lin 1990 (Neighborhood system)

8. Granular computing Historical Notes 6. Lin (1992):Topological and Fuzzy Rough Sets 7. Lin & Liu (1993): Operator View of RS and NS 8. Lin & Hadjimichael (1996): Non-classificatory hierarchy

9. Granular computing Granulation seems to be a natural problem-solving methodology deeply rooted in human thinking.

10. Granular computing Human body has been granulated into head, neck, and etc. (there are overlapping areas) The notion is intrinsically fuzzy, vague, and imprecise.

11. Partition theory Mathematicians have idealized the granulation into • Partition (at least back to Euclid)

12. Partition theory Mathematicians have developed it into a fundamental problem solving methodology in mathematics.

13. Partition theory • Rough Set community has applied the idea into Computer Science with reasonable results.

14. Partition theory But Partition requires • Absolutely no overlapping among granules (equivalence classes)

15. More General Theory • Partitions is too restrictive for real world applications.

16. More General Theory • Even in natural science, classification does permit a small degree of overlapping;

17. More General Theory • There are beings that are both proper subjects of zoology and botany.

18. More General Theory • A more general theory is needed

19. Outline 2. New Theory - Granular Computing 2

20. Zadeh’s Intuitive notion Granulation involves partitioning a class of objects(points) into granules,

21. Zadeh’s Intuitive notion with a granule being a clump of objects (points) which are drawn together by indistinguishability, similarity or functionality.

22. Formalization • We will present a formal theory, we believe, that has captures quite an essence of Zadeh’s idea (but not full)

23. Outline • 3. A Formal Theory

24. (Single Level) Granulation Consider two universes(classical sets): 1. V is a universe of objects 2. U is a data/information space 3. To each object p  V, we associate at most one granule  U; The granule is a classical/fuzzy subset.

25. (Single Level) Granulation A granulation is a map: p  V  B(p)  2U where B(p) could be an empty set.

26. (Single Universe) Granulation A (single universe) granulation is a map: p  V  B(p)  2V (U=V) where B(p) is a granule/neighborhood of objects.

27. (Single Level) Granulation Intuitively B(p) is the collection of objects that are drawn towards p

28. Granulation - Binary Relation The collection B={(p, x) | x B(p)  p  V} VU is a binary relation

29. (Single Level) Granulation If B is an equivalence relation the collection {B(p)} is a partition

30. More General Case If we consider a set of Bj of binary relations(drawn by various “forces”, such as indistinguishability, similarity or functionality) then

31. More General Case we have the association p NS(p)={Nj(p)|Nj(p)={x | (p, x) Bj} j runs through an index}. is called multiple level granulation and form a neighborhood system (pre-topological space).

32. Development 4. Incremental Development 2

33. Classical Paradigm What do we have? 1. (Divide) Partitioning 2. Quotient Set (Knowledge level) 3. Integration (of subtasks and quotient task)

34. What do we have? Classical Paradigm 1. Partition of a classical set (Divide) Absolutely no overlapping among granules

35. Some Mathematics A partition Granule B i, j, k f, g, h Granule C GranuleA l, m, n

36. Some Mathematics Partition  Equivalence relation • X  Y (Equivalence Relation) if and only if • both belong to the same class/granule

37. Equivalence Relation Generalized Identity • X  X (Reflexive) • X  Y implies Y X (Symmetric) • X  Y, Y Z implies X  Z (Transitive)

38. Example Partition [0]4 = {. . . , 0, 4, 8, . . .}, [1]4 = {. . . , 1, 5, 9, . . .}, [2]4 = {. . . , 2, 6, 10, . . .}, [3]4 = {. . . , 3, 7, 11, . . .}.

39. Quotient set { [0]4 , [1]4 , [2]4, [3]4 } [0]4+[1]4 =[1]4 [4]4+[5]4 =[9]4 [1]4 = [9]4

40. New territories Granulation (not Partition) B0 = [0]4{5, 9}, B1 = [1]4 ={. . . , 1, 5, 9, . . .}, B2 = [2]4{7}, B3 = [3]4{6}.

41. New territories Granulation (not Partition) B0 B1= {5, 9}, B2 B3 = {6,7}, Could we define a quotient set ?

42. New territories If {B0, B1, B2, B3} is a quotient set, then B0 andB1 are distinct elements, so B0  B1 (= {5, 9}) should be empty {B0, B1, B2, B3 } is NOT a set

43. New Paradigm • In general, classical scheme is unavailable for general granulation • We will show that: classical scheme can be extended to single level granulation

44. New formal theory • New view of the universe

45. Granulated/clustered space Let V be a set of object with granulation B: V  B(p)  2V V=(V, B) is a granulated/clustered space, called B-space (a pre-topological space). V is approximation space (called A-space) if B is a partition.

46. Classical Paradigm What do we have? 1. (Divide) Partitioning 2. Quotient Set (Knowledge level) 3. Integration (of subtasks and quotient task)

47. What are in the new paradigm? 1. Partition of B-space (Divide) 2. Quotient B-space (Knowledge) 3. Integration-Approximation (and extension)

48. Integration-Approximations Some Comments on approximations

49. Lower/Interior approximations B(p), p  V, be a granule L(X)=  {B(p) | B(p)  X} (Pawlak) I(X)= {p | B(p)  X} (Lin-topology)

50. Upper/Closure approximations Let B(p), p  V, be an elementary granule U(X)= {B(p) | B(p)  X = } (Pawlak) C(X)= {p | B(p)  X = } (Lin-topology)