1. Granular Computing: A new problem Solving Paradigm Tsau Young (T.Y.) Lin Department of Computer Science San Jose State University San Jose, CA 95192 tylin@cs.sjsu.edu

2. Outline 1. Summary Rough Computing Equivalence Relation Neighborhood Concept Binary Relation 2. Details

3. Rough Computing A partition is a set of (1) disjoint subsets, (2) a cover Class A Class B f, g, h i, j, k Class C l, m, n

4. Rough Computing X  Y (equivalence) if and only if both belong to the same class

5. Rough Computing An Equivalence Relation Class A Class B f  g  h i  j  k Class C l  m  n

6. Equivalence Relation X  X (Reflexive) X  Y implies Y  X (Symmetric) X  Y, Y  Z implies X  Z (Transitive)

7. Neighborhood Concept i  j  k Class B In spite of a technical error, the idea was, and still is, fascinating

8. Introduction An aggressive model (ACWSP) was proposed by Lin the same year (1989) that keeps the same spirit and corrects the error Lost some Strength

9. Introduction Lost Interests until A practical way of construction ACWSP was introduced 2000

10. Brewer and Nash Requirements A set of impenetrable Chinese Great Walls No corporate data that are in conflict can be stored in the same side of Walls

11. Brewer and Nash -Theory Corporate data are decomposed into Conflict of Interest Classes(CIR-classes) Walls are built around the CIR-classes Corporate data is called an object (tradition)

12. BN -Theory All objects Class A Class B f, g, h i, j, k Class C l, m, n

13. Is CIR Transitive? US  (conflict) Russia UK  Russia UK  ? US

14. Is CIR Reflexive? US  (conflict) US ? Is CIR self conflicting?

15. Is CIR Symmetric? US  (conflict) USSR implies USSR  (conflict) US ? YES

16. BN -Theory Can they be partitioned? BN -Theory C US, Russia UK? France, German

17. CIR-classes CIR classes do overlap (Conflict of Interests) US, UK, Iraq,... USSR

18. CIR & IAR Complement of CIR: an equivalence relation Iraq,...US, UK,... German,...

19. ACWSP CIR: Anti-reflexive, symmetric, anti-transitive CIR-class IJAR-classes

20. ACWSP CIR: Anti-reflexive, symmetric, anti-transitive CIR-class IJAR-classes

21. ACWSP CIR: Anti-reflexive, symmetric, anti-transitive CIR-class IJAR-classes

22. Trojan Horses Direct Information flow(DIF) Professor Grader StudentsCIF DIFTrojan horse( DIF )

24. ACWSP CIR (with three conditions) only allows information sharing within one IJAR-class An IJAR-class is an equivalence class; so there is no danger the information will spill to outside. No Trojan horses could occur

25. SCWSP Simple CWSP (SVWSP) No DIF: x  y (direct information flow)  (x, y)  CIR

26. ACWSP Strong CWSP(ACSWP) No CIF: x ...  y ((composite) information flow)  (x, y)  CIR

27. ACWSP Theorem If CIR is anti-reflexive, symmetric and anti-transitive, then Simple CWSP  Strong CWSP

28. ACWSP CIF =a sequence of DIFs CIF: X=X 0  X 1 ...  X n =Y Y  CIR X To derive a contradiction

29. ACWSP X=X 0  X 1 implies X 1  [X] CIR X = CIR X1... Y=X n  [X] CIR X = CIR Xn = CIR Y Y  CIR X Contradiction