About the family of Closure Systems preserving non unit implications in the Guigues-Duquenne Base

1. About the family of Closure Systems preserving non unit implications in the Guigues-Duquenne Base Alain Gély & Lhouari Nourine LIMOS – Clermont-Ferrand - France ICFCA’06 - Dresden

2. Definitions & Problematic Incremental Approach Implications in the Guigues-Duquenne base Results about the family of closure systems preserving non unit implications in the Guigues-Duquenne Base. Conclusions & Perspectives

3. Definitions & Problematic 1234 123 134 124 234 12 13 14 23 24 34 1 2 3 4  F Closure system 123 134 M(F)  124 234 Meet-irreducible elements Implicational base

4. 1234 123 134 124 234 12 13 14 23 24 34 1 2 3 4  F Closure system 123 4 1 134 M(F)  124 23 Meet-irreducible elements Implicational base

5. 1234 123 134 124 234 12 13 14 23 24 34 4 1 2 3  F Closure system 123 4 1 124 13  2 M(F)  14 23 Meet-irreducible elements Implicational base

6. 1234 123 134 124 234 12 13 14 23 24 34 4 1 2 3  F Closure system 123 4 1 124 13  2 M(F)  14 34 2 23 Meet-irreducible elements Implicational base

7. 1234 123 134 124 234 12 13 14 23 24 34 4 1 2 3  F Closure system 123 4 1 124 13  2 M(F)  14 34 2 23 Meet-irreducible elements Minimal Implicational base

8. 1234 123 134 124 234 12 13 14 23 24 34 4 1 2 3  Polynomial Polynomial Polynomial Polynomial ? ? F 123 4 1 124 13  2 M(F)  14 23

9. 1234 123 134 124 234 12 13 14 23 24 34 1 2 3 4  F Polynomial Polynomial Process 2n operations 123 124 M(F)  134 234 Output size : 0 Input size : n

10. ? ? A question arise : What happen if the set of implications is modified… For the implicational base For the meet-irreducible elements Incremental Approach 123 4 1 124 13  2 M(F)  14 +/- x  y 23

11. What about Implications in the Guigues-Duquenne base To study changes in an implicationnal base, We need to choose a canonical minimal base : The Guigues-Duquenne base

12. 1234 123 124 134 13 14 24 12 4 3 1234 1234 123 134 124 234 13 12 13 14 23 24 34 1 2 3 4 1 2   4  1234 3  13 12  1234

13. Size of premise > 1 Premise is singleton  = J   Unit implications Non unit implications Canonical minimum base(Guigues-Duquenne base) [Guigues & Duquenne 86] 1234 Let F be a closure system  = { P P | P a pseudo-closed set of F } is a minimum implicational base for F. 13 1 2  3  13 12  1234 4  1234

14. Unit Implications • Implications in J are • Easy to compute from M(F) • In polynomial number relative to M(F) Non Unit Implications Implications in may be • Not easy to compute from M(F) • In exponential number relative to M(F)

15. Interesting if J  Modify Jwithout modify We look for  - equivalent closure systems Example of application : • |F’| ≤ |F| • |M(F’) | ≤ |M(F)| F’ F  M(F’) M(F)  {a  b}

16. Modification of   Add an implication a b : 123 J 2  23 {a b} shall not be a Guigues-Duquenne Base 123 13 23 12 13 23 12 1 2 3 1 2 3 Add Unit Implications  12  123 J 2  3

17. 1. Closure of P may change Premise is not anymore a pseudo-closed set because… Three cases of problem P  P  2. It is not anymore a quasi-closed set 3. It remain a quasi-closed set, but not minimal

18. Result Keep conclusion Remains a quasi-closed set Remains a minimal quasi-closed set Characterization :  -equivalence addinga  b a  b may be added without modification of  iff For all P  P  (i) if a P then b P (ii) if a P then b P (iii) if a j , j P, then (jb) ≠P

19. 123 123 12 12 1 123 cover relation in C(F)  = {} J = {3  123, 2  12}  = {} J = {3  123, 2  12, 112}  = {} J = {3  123, 2  123, 1123}

20. Result Characterization : cover relation in C(F) a  b may be added without modification of  , andF’ covers FinC(F)iff (i’)For all P  P , P≠ a , if a P then b P (ii)For all P  P if a P then b P (iii’)For all P  P  , if a  P then (ab) ≠P

21. 3 1 1 2 3 2 123 12 123 13 23 1 2 3 3 2 3 1 123 123 12 123 12 123 13 23 3 23 3 13 1 2 1 2 3 1 3 2 123 12 123 3 123 1 2

22. 1234 3  13 4  24 13 24 12 1234 1 2 Family of J - equivalent closure systems is a closure system [Nation & Pogel 97] 1234 1234 3  13 3  13 123 4  24 124 4  24 13 13 24 12 123 24 12 124 124 1234 123 1234 2 1 2 1

23. 1234 1234 123 123 14 12 24 12 2 1 1 2 1234 F’’ is not  - equivalent to F 123 3 123 12 4 1234 2 1 F’’ Family of  - equivalent closure systems is not a closure system 124  1234 124  1234 3 123 3 123 4 14 4 24 F F’

24. Result Conditions on implications Characterization : cover relation in C(F) a  b may be added without modification of  , andF’ covers FinC(F)iff (i’)For all P  P , P≠ a , if a P then b P (ii)For all P  P if a P then b P (iii’)For all P  P  , if a  P then (ab) ≠P

25. 123 (i’) et (ii)  Isomorphism between A and A* A 13 23 12 A* 1 2 3 (iii’)  A  (A* B) F F Detection : can I add the implication a  b ? (using only M(F) ) Athe closure of a, B the closure of b in F example 3  1 A*immediate predecessor ofA in F A family of sets F such that a F and b F A* family of closed sets F such that A* F, a  F and b F A  (A* B) B = A = A* =

26. 23 A* 3 Reduction from F to F’ A family of sets F such that a F and b F example 3  1 123 A 13 12 1 2

27. A A* Reduction from F to F’ Evolution of meet-irreducible elements

28. Sufficient and necessary conditions in polynomial time Transformation of the data in polynomial time New data is smaller than the original one Closure system is smaller than the original one Interesting method to reduce data Conclusion we can add an implicationa  b

29. Perspectives • Links between implications in  and J • What happen with other bases ? • Structural Properties of C(F) • Efficient algorithms to add an implication