1. 1 Granular and Rough Computing: Incremental Development Tsau Young (T.Y.) Lin tylin@cs.sjsu.edu Computer Science Department, San Jose State University, San Jose, CA 95192, and Berkeley Initiative in Soft Computing, UC-Berkeley, Berkeley, CA 94720

2. 2 Introduction 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.)

3. 3 Granular computing IEEE GrC-conference http://www.cs.sjsu.edu/~grc/.

4. 4 Historical Notes 1. Zadeh (1979) Fuzzy Sets and Information granularity(about Dempster- Shaffer Theory(DST))

5. 5 Notes Dempster-Shaffer Theory(DST) Note: In general, basic probability assignment (bpa)  classical probability(cp) but...

6. 6 Notes Dempster-Shaffer Theory(DST) 2. Note, but if the given focal elements are mutually disjoints, then bpa=cp. In this case 2.1. Bel = inner probability(lower bound) 2.2. Pl=Outer probability (upper bound) (This is NOT general cases—Common errors)

7. 7 Historical Notes 2. Pawlak (1982 Dec) 3. Tony Lee (1983 Jan) Study of relations via partitions

8. 8 Historical Notes Pawlak: Rough Sets, Information systems, Approximations Lee: Algebraic Theory of Relational Databases

9. 9 Historical Notes 4a T. Y. Lin 1988-89: Neighborhood Systems(NS) (  a set of general binary relations) 4b T. Y. Lin (1989) Chinese Wall Security Model (A study of non-reflexive, symmetric, non-transitive binary relation) 5. Stefanowski ( 1989) about Fuzzified Partitions

10. 10 Historical Notes 6. Lin, Qing Liu & James Huang (1990): Neighborhood system &RS) 7. Lin (1992):Topological and Fuzzy Rough Sets Lin said, Intuitively, closure is smallest closed set—incorrect. This is true for topological spaces, not for NS-space

11. 11 Historical Notes 8. Lin & Liu (1993): Operator View of RS and NS Important Results: 8.1. Axiomatic View of RS 8.2. clopen space defines a partition:

12. 12 Historical Notes 8.3. Simple Proof of item 2 Consider connected components(CC) of clopen space. From general topology theory, 8.3.1. CC is closed. 8.3.2. CCs intersect each other only on boundary. Since CC is also open, and open set has no boundary. So CC is disjoint and hence is a partition.

13. 13 Historical Notes 9. Lin & Hadjimichael (1996): Non-classificatory hierarchy This paper builds a tree for nested binary relations; This result is obvious for equivalence relations but is very hard for general binary relations

14. 14 Granular computing Granulation seems to be a natural problem-solving methodology deeply rooted in human thinking.

15. 15 Granular computing Human body has been granulated into head, neck, and etc. (there are overlapping areas) The notion is intrinsically fuzzy, vague, and imprecise.

16. 16 Partition theory Mathematicians have idealized the granulation into Partition (at least as far back to Euclid)

17. 17 Partition theory Mathematicians have developed it into a fundamental problem solving methodology in mathematics.

18. 18 Partition theory Rough Set community has applied the idea into Computer Science with reasonable results, called R ough S et T heory (RST)

19. 19 Key Views in RST 1. Granulation=Partition: E 2. (V, E): Approximation Space 3. Representations (Information Systems/Tables) 4. Table Processing( Knowledge Processing ) 4.1. Reducts and Core 4.2. Value Reducts(Data Mining)

20. 20 Key Views in RST 5. Granular/Rough Logic Theory.

21. 21 1. Granulation=Partition A Partition of V: Class A Class B f, g, h i, j, k Class C l, m, n

22. 22 RS Approximations Upper approximation Lower approximation

23. 23 Lower/Interior Approximations L(X)=  {B(p) | B(p)  X} (Pawlak) I(X)= {p | B(p)  X} (Lin – topology based) L = I in RS theory

24. 24 Upper/Closure Approximations U(X)=  {B(p) | B(p)  X   } (Pawlak) C(X)= {p | B(p)  X   } (Lin - topology based ) (A Closure,C(X), may not be closed;error in Lin 1992) U=C if B is a partition

25. 25 Research Issues In general case, the coverings((  {B(p)) are not unique, so Pawlak’s notion of Upper/Lower approximations need “MAX” or “MIN” U(X)= MIN (  {B(p) | B(p)  X   } ) L(X)= MAX (  {B(p) | B(p)  X}) Please do some research (see measure theory)

26. 26 Research Issues From topological space point of view, Pawlak Style approximations should be replaced by Lin’s style in Topological View.

27. 27 Other Closure approximations Cl (X)=  i C i (X) (Sierpenski defined) where C i (X)= C(…C(C(X))…) (transfinite steps) Cl (X) is closed. This is the closure in classical topology.

28. 28 Other Closure approximations May consider finite steps Cls: Cl k (X)=  i=1 k C i (X) Chinese Wall Security Policy Model (Consider all Cl k, k =1, 2,...)

29. 29 Representation Theory Given a partition(named COLOR) Class A Class B f, g, h i, j, k Class C l, m, n

30. 30 Representation Theory Review RS approaches: V: the universe, V, of entities {f,g,h,I,j,k,l,m} and is partitioned into A, B, C equivalence classes

31. 31 Representation Theory (information table) All classes A, B, C are named Yellow, Red, Blue. First tuple means (see next page) “f “ is a member of “A,” so its COLOR is “Yellow” Here COLOR is the name of the given Partition (equivalence relation)

32. 32 Summarized in Table Format f AYellow g A h A i BRed j B k B l CBlue m C n C

33. 33 One columns Information Table f Yellow The left hand side is a g Yellowtable in which the h YellowUniverse V of entities i Red is represented by one j Red column, called COLOR. k Red Each row represent l Bluethe color of one entity m Blue n

34. 34 Representation Theory for granulation Lat few pages explain how RS has handled Knowledge representations We will Extend the idea to binary relations

35. 35 Representation Theory Given a granulation(has overlapping) Neighborhood A Neighborhood B f, g, h i, j, k, l Neighborhood C m, n

36. 36 Neighborhood of x, N(x) x has N(x)Is in N(x)Name of N(x) fAA U gAA U hAA U iBA, B V jBB V kBB V lCB,C W m CCW n CCW

37. 37 The Center set of N(x) x has N(x) fA f, g, h have the neighborhood A gA Each of f, g, h is the center of A hA The set of all centers is called iB the center set of A and is jB denoted by C(A) kB So i,j,k form the center set C(B) lC l, m, n form the center set C(C) m C n C

38. 38 Topological Information Table I x Name of N(x) The left hand side f U is a table in which g U the Universe V of entities h U is represented by one i V column, called ?????. j V Each row represent one k V entity Syntactically is the l W same as information table; mW However U, V, W are nWRelated

39. 39 Binary Relation B1 (approach one) Name The left hand side is a table that U U Represents the binary relation of the U V interactions among attribute values V V (See Lin 1998) V U (U, V) is in B1, if U  V   V W W W W V (W, V) is in B1, if W  V  

40. 40 Topological Information Table II xCenter set Name of C(-) The left hand side fC(A) U’ is a table in which the first gC(A) U’ row means: an entity “f” is in hC(A) U’ a unique center set “C(A)” and iC(B) V’ “C(A)” has unique name “U’ ” jC(B) V’ kC(B) V’ lC(C) W’ mC(C)W’ nC(C)W’

41. 41 Topological Information Table II x Name of C(-) The left hand side is a table in which f U’ the Universe V of entities is represented g U’ by one column h U’ Each row represents an entity and i V’ the name of its center set j V’ k l W’ m n

42. 42 Binary Relation B2 (approach Two) Name The left hand side is a table that U’ represents the binary relation of the U’ V’ interactions among attribute values V’ V Lin 2004 (IEEE-CIS news letter) V’ U (U’, V’) is in B2, if A  C(B)   V’ W(A is neighborhood of member in U’) W’ WV’ is name of C(B) W’ V (V’, U’) is in B2, if B  C(A)  

43. 43 B1 and B2 1.B1 is symmetric, and B1 does not describe the interactions among U, V, W completely. 2.B2 may not be symmetric,moreover

44. 44 B1 and B2 If the given granulation under consideration is symmetric then B2 does describe the interactions among U’, V’, W’ completely. (In fact B2 is the “quotient” of the granulation)