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

2. 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. 3 Definitions & Problematic  1234 121314232434 123234134124 1234  F M(F) 123 134 124 234 Closure system Meet-irreducible elements Implicational base

4. 4  F M(F) 123 134 124 23 4 2434 234 4  1  123 12131423 123134124 1234 Closure system Meet-irreducible elements Implicational base

5. 5  F M(F) 123 124 14 23 4  1  1 23 4 121314232434 123234134124 1234 13  2 Closure system Meet-irreducible elements Implicational base

6. 6  F M(F) 123 124 14 23 4  1  1 23 4 121314232434 123234134124 1234 13  2 Closure system Meet-irreducible elements Implicational base 34  2

7. 7  F M(F) 123 124 14 23 4  1  1 23 4 121314232434 123234134124 1234 13  2 Closure system Meet-irreducible elements Minimal Implicational base 34  2

8. 8  F M(F) 123 124 14 23 4  1  1 23 4 121314232434 123234134124 1234 13  2 ? ? Polynomial

9. 9  M(F) 123 124 134 234 Polynomial F  1234 121314232434 123234134124 1234 Input size : n Output size : 0 Process 2 n operations

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

11. 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. 12  1 2 34 1314232434 123 234 134124 1234  1 2 13 1234 24 12 123124 4 134 3 13 1234 14 3  1312  1234 4  1234

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

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

15. 15 Modify  J without modify   We look for   - equivalent closure systems Example of application :  F M(F)  {a  b} M(F’) JJ  F’ Interesting if |F’| ≤ |F| |M(F’) | ≤ |M(F)|

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

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

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

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

20. 20 Characterization : cover relation in C  (F) (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  Result a  b may be added without modification of  , and F’ covers F in C  (F) iff

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

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

23. 23 124  1234 3  123 124  1234 3  123 123 1 2 1214 1234 123 2 1 1224 1234 4  14 4  24 Family of   - equivalent closure systems is not a closure system F F’ 3  123 4  1234 123 1 2 12 1234 F’’ F’’ is not   - equivalent to F

24. 24 Characterization : cover relation in C  (F) (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  Result a  b may be added without modification of  , and F’ covers F in C  (F) iff Conditions on implications

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

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

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

28. 28 we can add an implication a  b 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

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