Specification From Examples

1. Specification From Examples Julia Johnson Dept. of Math & Computer Science Laurentian University Sudbury, Ontario Canada

2. Problem To describe system characteristics by providing examples of systems that exhibit those characteristics.

3. Outline • Problem Statement • Criticism of Existing Solutions • Suggested Solution 3.1 Rough Sets 3.2 Strength of a Rule 3.3 Rough Mereology 3.4 RM System Specification • Conclusions

4. ag … ag1 ag2 agm … … … ag21 ag20 ag11 ag1n ag12 agm1 agmp

5. M L B

6. µB(B3,B1) = B≥ .25 B1 B3 µL(L3,L2) = L≥ .4 L2 L3

7. µM(C5,C1) = C1 C5 M≥ .14

8. Rough Mereology Mereology≡ Theory of “Part of” relation, Lesniewski Rough Mereology – Theory of Relation “Part of toadegree”, Polkowski & Skowron Applications of Rough Mereology Control – Skowron & Polkowski 1994 Warsaw Politecnica Building – Poitr 1998-99 Polish Academy of Science Scheduling – Johnson 1998-99 University of Regina/University of Waterloo µ (x,y) is read “the degree in which x is a part of y” -the rough inclusion function

9. For each construction of objects from sub-objects, we form a vector, B L M Where if M1 = O(B1L1) and M2 = O(B2L2) Then B = µB(B1,B2) L = µL(L1,L2) M = µM(M1,M2) M2 is constructed from B2 and L2

10. The vector means If µB(B1,B2) >B (B1 is part of B2 to degree at least B) And µL(L1,L2) >L (L1 is part of L2 to degree at least L) Then µM(M1,M2) >M (M1 is part of M2 to degree at least M)

11. µB(B3,B1) = B≥ .25 Rough Mereology B1 B3 µL(L2,L2) = L≥ 1 L2 L2

12. µM(C4,C1) = C1 C4 M≥ .28

13. Some Properties of µ • (A) µ(x,y) Є [0,1] • (B) µ(x,x) = 1 • If µ(x,y) = 1 then µ(z,y) > µ(z,x) for each object z • A null object is any object n which satisfies • (D) µ(n,y) = 1 for every object y

14. L M f B B1 B2 B3. . .Bn L1 L2 L3. . .Ln M1 M2 M3. . .Mn We wish to learn functions f from a set of vectors.

15. Back to the Problem at Hand To describe system characteristics by providing examples of systems that exhibit those characteristics. To determine system cost by providing examples of systems whose design, maintenance and overall costs are known.

16. Maintenance requirements Maintk and Maintq similar to degree at least Maint, possibly k=q • Cost1 and Cost2 , respectively, of the two systems O(Design1, Maint1) and O(Design2, Maint2) similar by at least Cost. • Specs Designi and Designj similar to degree at least Design , i,j not necessarily distinct Suppose we know the following:

17. Design Cost f Maint Design1Design2Design3 . . .Designn Maint1Maint2Maint3 . . .Maintn Cost1Cost2Cost3 . . . Costn We wish to learn function f from a set of vectors.

18. Table 1: Criteria Inferred from Application Data

19. (ResponseTime, slow) and (Throughput, low) (Acceptable, no), (ResponseTime, fast) and (Memory, medium) (Acceptable, yes), (Throughput, high) and (Memory, large) (Acceptable, no). Uncertain (or possible) rules are: (ResponseTime, fast) and (Throughput, high) (Acceptable, yes), (ResponseTime, fast) and (Throughput, high) (Acceptable, no).

20. Table 3: Rough Inclusion for Table 1

21. Table 2: Criteria Dictated by the Customer

22. Table 4: Rough Inclusion for Table 2

23. Table 5: Maintenance Criteria

24. Table 6: Rough Inclusion for Table 5

25. Table 7: Partial List of Arguments for µ

26. Table 8: Partial List of Arguments for 

27. RM – ‘acceptable degree’ ? µ (X, Y) threshold vectors composition of objects agents message passing What? How?

28. Simplicity user – satisfaction learnability ease of use comprehensibility user - friendliness µ (X, Y) threshold vectors composition of objects agents message passing What? How?

29. Summary & Conclusions Our objective is to describe system characteristics such as user friendliness by providing examples of systems that exhibit such characteristics. The computer recognizes a pattern and generates rules for what a user friendly system, for example, would be. This is possible because computers are able to provide imprecise solutions to problems. We have demonstrated the feasibility of applying rough sets/rough mereology to the problem of systems requirements systems.