1. 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. 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. 3 Outline 1. Introduction

4. 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. 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. 6 Granular computing IEEE GrC-conference http://www.cs.sjsu.edu/~grc/.

7. 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. 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. 9 Granular computing Granulation seems to be a natural problem-solving methodology deeply rooted in human thinking.

10. 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. 11 Partition theory Mathematicians have idealized the granulation into Partition (at least back to Euclid)

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

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

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

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

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

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

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

19. 19 Outline 2. New Theory - Granular Computing 2

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

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

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

23. 23 Outline 3. A Formal Theory

24. 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. 25 (Single Level) Granulation A granulation is a map: p  V  B(p)  2 U where B(p) could be an empty set.

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

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

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

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

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

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

32. 32 Development 4. Incremental Development 2

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

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

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

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

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

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

39. 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. 40 New territories Granulation (not Partition) B 0 = [0] 4  {5, 9}, B 1 = [1] 4 ={..., 1, 5, 9,...}, B 2 = [2] 4  {7}, B 3 = [3] 4  {6}.

41. 41 New territories Granulation (not Partition) B 0  B 1 = {5, 9}, B 2  B 3 = {6,7}, Could we define a quotient set ?

42. 42 New territories If {B 0, B 1, B 2, B 3 } is a quotient set, then B 0 and B 1 are distinct elements, so B 0  B 1 (= {5, 9}) should be empty {B 0, B 1, B 2, B 3 } is NOT a set

43. 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. 44 New formal theory New view of the universe

45. 45 Granulated/clustered space Let V be a set of object with granulation B: V  B(p)  2 V 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. 46 Classical Paradigm What do we have? 1. (Divide) Partitioning 2. Quotient Set (Knowledge level) 3. Integration (of subtasks and quotient task)

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

48. 48 Integration-Approximations Some Comments on approximations

49. 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. 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)

51. 51 Upper/Closure approximations Cl (X)=  i C i (X) (Sierpenski-topology) Where C i (X)= C(…(C(X))…) (transfinite steps) Cl (X) is closed.

52. 52 New View Divide (and Conquer) Partition of set  (generalize) ? Partition of B-space (topological partition)

53. 53 New View:B-space The pair (V, B) is the universe, namely an object is a pair (p, B(p)) where B: V  2 V ;  p  B(p) is a granulation

54. 54 Derived Partitions The inverse images of B is a partition (an equivalence relation) C ={C p | C p =B –1 (B p ) p  V}

55. 55 Derived Partitions C p is called the center class of B p A member of C p is called a center.

56. 56 Derived Partitions The center class C p consists of all the points that have the same granule Center class C p = {q | B q = B p }

57. 57 C-quotient set The set of center classes C p is a quotient set Iran, Iraq..US, UK,... Russia, Korea

58. 58 New Problem Solving Paradigm (Divide and) Conquer Quotient set  Topological Quotient space

59. 59 Neighborhood of center class C (in the case B is not reflexive) B-granule/neighborhoodC-classes

60. 60 Neighborhood of center class B-granule C-classes

61. 61 Topological partition C p -classes B-granule/neighborhood

62. 62 New Problem Solving Paradigm (Divide and) Conquer Quotient set  Topological Quotient space

63. 63 Topological partition C p -classes B-granule/neighborhood

64. 64 Topological partition C p -classes B-granule/neighborhood

65. 65 Topological partition C p -classes B-granule/neighborhood

66. 66 Topological Table (2-column) 2-columns Binary relation for Column I US  CXCX West CXCX C Y (  B X ) UK  CXCX West CXCX C Z (  B X ) Iran  CYCY M-east CYCY C X (  B Y ) Iraq  CYCY M-east CYCY C Z (  B Y ) Russia  CzCz East CZCZ C X (  B z ) Korea  CzCz East CZCZ C y (  B z )

67. 67 Future Direction Topological Reduct Topological Table processing

68. 68 Application 1: CWSP In UK, a financial service company may consulted by competing companies. Therefore it is vital to have a lawfully enforceable security policy. 3

69. 69 Background Brewer and Nash (BN) proposed Chinese Wall Security Policy Model (CWSP) 1989 for this purpose

70. 70 Policy: Simple CWSP (SCWSP) "Simple Security", BN asserted that "people (agents) are only allowed access to information which is not held to conflict with any other information that they (agents) already possess."

71. 71 A little Fomral Simple CWSP(SCWSP): No single agent can read data X and Y that are in CONFLICT

72. 72 Formal SCWSP SCWSP says that a system is secure, if “(X, Y)  CIR  X NDIF Y “ CIR=Conflict of Interests Binary Relation NDIF=No direct information flow

73. 73 Formal Simple CWSP SCWSP says that a system is secure, if “(X, Y)  CIR  X NDIF Y “ “(X, Y)  CIR  X DIF Y “ CIR=Conflict of Interests Binary Relation

74. 74 More Analysis SCWSP requires no single agent can read X and Y, but do not exclude the possibility a sequence of agents may read them Is it secure?

75. 75 Aggressive CWSP (ACWSP) The Intuitive Wall Model implicitly requires: No sequence of agents can read X and Y: A 0 reads X=X 0 and X 1, A 1 reads X 1 and X 1,... A n reads X n =Y

76. 76 Composite Information flow Composite Information flow(CIF) is a sequence of DIFs, denoted by  such that X=X 0  X 1 ...  X n =Y And we write X CIF Y NCIF: No CIF

77. 77 Composition Information Flow Aggressive CWSP says that a system is secure, if “(X, Y)  CIR  X NCIF Y “ “(X, Y)  CIR  X CIF Y “

78. 78 The Problem Simple CWSP  ? Aggressive CWSP This is a malicious Trojan Horse problem

79. 79 Need ACWSP Theorem Theorem If CIR is anti-reflexive, symmetric and anti-transitive, then Simple CWSP  Aggressive CWSP

80. 80 C and CIR classes CIR: Anti-reflexive, symmetric, anti-transitive CIR-class C p -classes

81. 81 Application 2 Association mining by Granular/Bitmap computing

82. 82 Fundamental Theorem Theorem 1: All isomorphic relations have isomorphic patterns

83. 83 Illustrations:Table K v1v1  TWENTYMARNY) v2v2  TENMARSJ) v3v3  TENFEBNY) v4v4  TENFEBLA) v5v5  TWENTYMARSJ) v6v6  TWENTYMARSJ) v7v7  TWENTYAPRSJ) v8v8  THIRTYJANLA) v9v9  THIRTYJANLA)

84. 84 Illustrations: Table K’ v1v1  203rdNew York) v2v2  103rdSan Jose) v3v3  102ndNew York) v4v4  102ndLos Angels) v5v5  203rdSan Jose) v6v6  203rdSan Jose) v7v7  204thSan Jose) v8v8  301stLos Angels) v9v9  301stLos Angels)

85. 85 Illustrations: Patterns in K v1v1  TWENTYMAR NY) v2v2  TEN MARSJ) v3v3  TENFEBNY) v4v4  TENFEBLA) v5v5  TWENTYMARSJ) v6v6  TWENTYMARSJ) v7v7  TWENTYAPR SJ) v8v8  THIRTYJAN LA) v9v9  THIRTYJAN LA)

86. 86 Isomorphic 2-Associations K Count K’ (TWENTY, MAR) 3(20, 3rd) (MAR, SJ)3(3rd, San Jose) (TWENTY, SJ)3(20, San Jose)

87. 87 Canonical Model Bitmaps in Granular Forms Patterns in Granular Forms

88. 88 Table K’ v1v1  203rd v2v2  103rd v3v3  102nd v4v4  102nd v5v5  203rd v6v6  203rd v7v7  204th v8v8  301st v9v9  301st

89. 89 Illustration: K  GDM K GDM v1v1  203rd{v 1 v 5 v 6 v 7 }{v 1 v 2 v 5 v 6 } v2v2  103rd{v 2 v 3 v 4 }{v 1 v 2 v 5 v 6 } v3v3  102nd{v 2 v 3 v 4 }{v 3 v 4 } v4v4  102nd{v 2 v 3 v 4 }{v 3 v 4 } v5v5  203rd{v 1 v 5 v 6 v 7 }{v 1 v 2 v 5 v 6 } v6v6  203rd{v 1 v 5 v 6 v 7 }{v 1 v 2 v 5 v 6 } v7v7  204th{v 1 v 5 v 6 v 7 }{v 7 } v8v8  301st{v 8 v 9 } v9v9  301st{v 8 v 9 }

90. 90 Illustration: K  GDM K GDM v1v1  203rd(100011100)(110011000) v2v2  103rd(011100000)(110011000) v3v3  102nd(011100000)(001100000) v4v4  102nd(011100000)(001100000) v5v5  203rd(100011100)(110011000) v6v6  203rd(100011100)(110011000) v7v7  204th(100011100)(110011000) v8v8  301st(000000011) v9v9  301st(000000011)

91. 91 GranularData Model (of K’ ) NAMEElementary Granules 10(011100000)={v 2 v 3 v 4 } 20(100011100) ={v 1 v 5 v 6 v 7 } 30(000000011)={v 8 v 9 } 1st(000000011)={v 8 v 9 } 2nd(001100000)={v 3 v 4 } 3rd(110011000)={v 1 v 2 v 5 v 6 } 4th(110011000)={v 7 }

92. 92 Associations in Granular Forms KCardinality of Granules (20, 3rd) |{v 1 v 5 v 6 v 7 }  {v 1 v 2 v 5 v 6 }|= |{v 1 v 5 v 6 }|=3 (10, 2 nd ) |{v 2 v 3 v 4 }  {v 3 v 4 }|= |{v 3 v 4 }|=2 (30, 1 st ) |{v 8 v 9 }  {v 8 v 9 }|= |{v 8 v 9 }|=2

93. 93 Associations in Granular Forms KCardinality of Granules (20, 3rd) |{v 1 v 5 v 6 v 7 }  {v 1 v 2 v 5 v 6 }|= |{v 1 v 5 v 6 }|=3 (3rd, SJ) |{v 1 v 2 v 5 v 6 }  {v 2 v 5 v 6 v 7 }|= |{v 2 v 5 v 6 }|=3 (20, SJ) |{v 1 v 5 v 6 v 7 }  {v 2 v 5 v 6 v 7 }|= |{v 5 v 6 v 7 }|= 3

94. 94 Fundamental Theorems 1. All isomorphic relations are isomorphic to the canonical model (GDM) 2. A granule of GDM is a high frequency pattern if it has high support.

95. 95 Relation Lattice Theorems 1. The granules of GDM generate a lattice of granules with join =  and meet= . This lattice is called Relational Lattice by Tony Lee (1983) 2. All elements of lattice can be written as join of prime (join-irreducible elements) (Birkoff & MacLane, 1977, Chapter 11)

96. 96 Find Association by Linear Inequalities Theorem. Let P 1, P 2,  are primes (join-irreducible) in the Canonical Model. then G= x 1 * P 1  x 2 * P 2   is a High Frequency Pattern, If |G|= x 1 * |P 1 | +x 2 * |P 2 | +   th, ( x j is binary number)

97. 97 Join-irreducible elements 10  1 st {v 2 v 3 v 4 }  {v 8 v 9 }=  20  1 st {v 1 v 5 v 6 v 7 }  {v 8 v 9 }=  30  1 s {v 8 v 9 }  {v 8 v 9 }= {v 8 v 9 } 10  2 nd {v 2 v 3 v 4 }  {v 3 v 4 }= {v 3 v 4 } 20  2 nd {v 1 v 5 v 6 v 7 }  {v 3 v 4 }=  30  2nd{v 8 v 9 }  {v 3 v 4 }=  10  3 rd {v 2 v 3 v 4 }  {v 1 v 2 v 5 v 6 }= {v 2 } 20  3 rd {v 1 v 5 v 6 v 7 }  {v 1 v 2 v 5 v 6 }= {v 1 v 5 v 6 } 30  3 rd {v 8 v 9 }  {v 1 v 2 v 5 v 6 }=  10  4th{v 2 v 3 v 4 }  {v 7 }=  20  4th{v 1 v 5 v 6 v 7 }  {v 7 }= {v 7 } 30  4th{v 8 v 9 }  {v 7 }= 

98. 98 AM by Linear Inequalities |x 1 *{v 1 v 5 v 6 }=(20, 3rd) +x 2 *{v 2 } =(10, 3rd) +x 3 *{v 3 v 4 }=(10, 2nd) +x 4 *{v 7 } =(20, 4th) +x 5 *{v 8 v 9 } =(30, 1 st )| = x 1 *3+x 2 *1+x 3 *2+x 4 *1+ x 5 *2

99. 99 AM by Linear Inequalities |x 1 *{v 1 v 5 v 6 }+x 2 *{v 2 }+x 3 *{v 3 v 4 }+x 4 *{v 7 }+x 5 *{v 8 v 9 }| = x 1 *3+x 2 *1+x 3 *2+x 4 *1+ x 5 *2 1. x 1 =1 2. x 2 =1, x 3 =1, or x 2 =1, x 5 =1 3. x 3 =1, x 4 =1 or x 3 =1, x 5 =1 4. x 4 =1, x 5 =1

100. 100 AM by Linear Inequalities |x 1 *{v 1 v 5 v 6 }+x 2 *{v 2 }+x 3 *{v 3 v 4 }+x 4 *{v 7 }+x 5 *{v 8 v 9 }| = x 1 *3+x 2 *1+x 3 *2+x 4 *1+ x 5 *2 1. x 1 =1 |1*{v 1 v 5 v 6 } | = 1*3=3 (20, 3rd) |{v 1 v 5 v 6 v 7 }  {v 1 v 2 v 5 v 6 }|= |{v 1 v 5 v 6 }|=3

101. 101 AM by Linear Inequalities |x 1 *{v 1 v 5 v 6 }+x 2 *{v 2 }+x 3 *{v 3 v 4 }+x 4 *{v 7 }+x 5 *{v 8 v 9 }| = x 1 *3+x 2 *1+x 3 *2+x 4 *1+ x 5 *2 x 2 =1, x 3 =1, or x 2 =1, x 5 =1 |x 2 *{v 2 }+x 3 *{v 3 v 4 }| =(10  20, 3 rd ) |x 2 *{v 2 }+x 5 *{v 8 v 9 }| =( 10, 2nd)  (10, 3 rd ) x 3 =1, x 4 =1 or x 3 =1, x 5 =1 x 4 =1, x 5 =1

102. 102 AM by Linear Inequalities |x 1 *{v 1 v 5 v 6 }+x 2 *{v 2 }+x 3 *{v 3 v 4 }+x 4 *{v 7 }+x 5 *{v 8 v 9 }| = x 1 *3+x 2 *1+x 3 *2+x 4 *1+ x 5 *2 x 3 =1, x 4 =1 or x 3 =1, x 5 =1 | x 3 *{v 3 v 4 }+x 4 *{v 7 }| =(10, 2nd  3 rd ) | x 3 *{v 3 v 4 }+x 5 *{v 8 v 9 }| =( 10, 2nd)  (30, 1st) x 4 =1, x 5 =1

103. 103 AM by Linear Inequalities |x 1 *{v 1 v 5 v 6 }+x 2 *{v 2 }+x 3 *{v 3 v 4 }+x 4 *{v 7 }+x 5 *{v 8 v 9 }| = x 1 *3+x 2 *1+x 3 *2+x 4 *1+ x 5 *2 x 4 =1, x 5 =1 | x 3 *{v 3 v 4 }+x 5 *{v 8 v 9 }| =( 20, 4st)  (30, 1st)