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Information Fusion and Social Choice S ebastien Konieczny CNRS - - PowerPoint PPT Presentation

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Information Fusion and Social Choice S ebastien Konieczny CNRS - CRIL, Lens, France konieczny@cril.fr COST-ADT Doctoral School on computational Social Choice 1 / 43 Merging Contradictory pieces of information (beliefs, goals, . . . )

  1. Majority - Arbitration n ) ⊢ △ µ ( E 2 ) ∃ n △ µ ( E 1 ⊔ E 2 (Maj) ⊲ An IC merging operator is a majority operator if it satisfies ( Maj ) .  △ µ 1 ( ϕ 1 ) ↔ △ µ 2 ( ϕ 2 )   △ µ 1 ↔¬ µ 2 ( ϕ 1 ⊔ ϕ 2 ) ↔ ( µ 1 ↔ ¬ µ 2 )  (Arb) ⇒ △ µ 1 ∨ µ 2 ( ϕ 1 ⊔ ϕ 2 ) ↔ △ µ 1 ( ϕ 1 ) µ 1 � µ 2   µ 2 � µ 1  ⊲ An IC merging operator is an arbitration operator if it satifies ( Arb ) . 9 / 43

  2. Syncretic Assignment A syncretic assignment is a function mapping each profile E to a total pre-order ≤ E over interpretations such that : = E and ω ′ | = E , then ω ≃ E ω ′ 1) If ω | = E and ω ′ �| = E , then ω < E ω ′ 2) If ω | 3) If E 1 ≡ E 2 , then ≤ E 1 = ≤ E 2 = ϕ 1 ∃ ω ′ | = ϕ 2 ω ′ ≤ ϕ 1 ⊔ ϕ 2 ω 4) ∀ ω | 5) If ω ≤ E 1 ω ′ and ω ≤ E 2 ω ′ , then ω ≤ E 1 ⊔ E 2 ω ′ 6) If ω < E 1 ω ′ and ω ≤ E 2 ω ′ , then ω < E 1 ⊔ E 2 ω ′ 10 / 43

  3. Syncretic Assignment A syncretic assignment is a function mapping each profile E to a total pre-order ≤ E over interpretations such that : = E and ω ′ | = E , then ω ≃ E ω ′ 1) If ω | = E and ω ′ �| = E , then ω < E ω ′ 2) If ω | 3) If E 1 ≡ E 2 , then ≤ E 1 = ≤ E 2 = ϕ 1 ∃ ω ′ | = ϕ 2 ω ′ ≤ ϕ 1 ⊔ ϕ 2 ω 4) ∀ ω | 5) If ω ≤ E 1 ω ′ and ω ≤ E 2 ω ′ , then ω ≤ E 1 ⊔ E 2 ω ′ 6) If ω < E 1 ω ′ and ω ≤ E 2 ω ′ , then ω < E 1 ⊔ E 2 ω ′ A majority syncretic assignment is a syncretic assignment which satisfies : 7) If ω < E 2 ω ′ , then ∃ n ω < E 1 ⊔ E 2 n ω ′ 10 / 43

  4. Syncretic Assignment A syncretic assignment is a function mapping each profile E to a total pre-order ≤ E over interpretations such that : = E and ω ′ | = E , then ω ≃ E ω ′ 1) If ω | = E and ω ′ �| = E , then ω < E ω ′ 2) If ω | 3) If E 1 ≡ E 2 , then ≤ E 1 = ≤ E 2 = ϕ 1 ∃ ω ′ | = ϕ 2 ω ′ ≤ ϕ 1 ⊔ ϕ 2 ω 4) ∀ ω | 5) If ω ≤ E 1 ω ′ and ω ≤ E 2 ω ′ , then ω ≤ E 1 ⊔ E 2 ω ′ 6) If ω < E 1 ω ′ and ω ≤ E 2 ω ′ , then ω < E 1 ⊔ E 2 ω ′ A majority syncretic assignment is a syncretic assignment which satisfies : 7) If ω < E 2 ω ′ , then ∃ n ω < E 1 ⊔ E 2 n ω ′ A fair syncretic assignment is a syncretic assignment which satisfies : ω < ϕ 1 ω ′   ω < ϕ 2 ω ′′  ⇒ ω < ϕ 1 ⊔ ϕ 2 ω ′ 8) ω ′ ≃ ϕ 1 ⊔ ϕ 2 ω ′′ 10 / 43

  5. Arbitration ϕ 1 ϕ 2 11 / 43

  6. Arbitration Y s ϕ 1 ϕ 2 11 / 43

  7. Arbitration D s Y s ϕ 1 ϕ 2 11 / 43

  8. Arbitration R D s s Y s ϕ 1 ϕ 2 11 / 43

  9. Arbitration R D s s Y R s s ϕ 1 ϕ 2 11 / 43

  10. Arbitration R D s D s Y R s s s ϕ 1 ϕ 2 11 / 43

  11. Arbitration R Y D s s D s Y R s s s ϕ 1 ϕ 2 11 / 43

  12. Arbitration R Y D s s D s Y R s s s ϕ 1 ϕ 2 Y R s s ϕ 1 ⊔ ϕ 2 11 / 43

  13. Arbitration R Y D s s D s Y R s s s ϕ 1 ϕ 2 Y R D s s s ϕ 1 ⊔ ϕ 2 11 / 43

  14. Representation Theorem Theorem An operator is an IC merging operator if and only if there exists a syncretic assignment that maps each profile E to a total pre-order ≤ E such that mod ( △ µ ( E ))) = min ( mod ( µ ) , ≤ E ) . 12 / 43

  15. Representation Theorem Theorem An operator is an IC merging operator (respectively IC majority merging operator or IC arbitration operator) if and only if there exists a syncretic assignment (respectively majority syncretic assignment or fair syncretic assignment) that maps each profile E to a total pre-order ≤ E such that mod ( △ µ ( E ))) = min ( mod ( µ ) , ≤ E ) . 12 / 43

  16. Model-Based Merging Idea : Select the interpretations that are the most plausible for a given profile. 13 / 43

  17. Model-Based Merging Idea : Select the interpretations that are the most plausible for a given profile. E ω ′ iff d x ( ω, E ) ≤ d x ( ω ′ , E ) ω ≤ d x 13 / 43

  18. Model-Based Merging Idea : Select the interpretations that are the most plausible for a given profile. E ω ′ iff d x ( ω, E ) ≤ d x ( ω ′ , E ) ω ≤ d x d x can be computed using : • a distance between interpretations d • an aggregation function f 13 / 43

  19. Model-Based Merging Idea : Select the interpretations that are the most plausible for a given profile. E ω ′ iff d x ( ω, E ) ≤ d x ( ω ′ , E ) ω ≤ d x d x can be computed using : • a distance between interpretations d • an aggregation function f • Distance between interpretations d ( ω, ω ′ ) = d ( ω ′ , ω ) d ( ω, ω ′ ) = 0 iff ω = ω ′ 13 / 43

  20. Model-Based Merging Idea : Select the interpretations that are the most plausible for a given profile. E ω ′ iff d x ( ω, E ) ≤ d x ( ω ′ , E ) ω ≤ d x d x can be computed using : • a distance between interpretations d • an aggregation function f • Distance between interpretations d ( ω, ω ′ ) = d ( ω ′ , ω ) d ( ω, ω ′ ) = 0 iff ω = ω ′ • Distance between an interpretation and a base = ϕ d ( ω, ω ′ ) d ( ω, ϕ ) = min ω ′ | 13 / 43

  21. Model-Based Merging Idea : Select the interpretations that are the most plausible for a given profile. E ω ′ iff d x ( ω, E ) ≤ d x ( ω ′ , E ) ω ≤ d x d x can be computed using : • a distance between interpretations d • an aggregation function f • Distance between interpretations d ( ω, ω ′ ) = d ( ω ′ , ω ) d ( ω, ω ′ ) = 0 iff ω = ω ′ • Distance between an interpretation and a base = ϕ d ( ω, ω ′ ) d ( ω, ϕ ) = min ω ′ | • Distance between an interpretation and a profile d d , f ( ω, E ) = f ( d ( ω, ϕ 1 ) , . . . d ( ω, ϕ n )) 13 / 43

  22. Model-Based Merging • Examples of aggregation function : max, leximax , Σ , Σ n , leximin , . . . 14 / 43

  23. Model-Based Merging • Examples of aggregation function : max, leximax , Σ , Σ n , leximin , . . . • Let d be any distance between interpretations. △ d , max operators satisfy (IC0-IC5), (IC7), (IC8) and (Arb). △ d , G MIN operators are IC merging operators. △ d , G MAX operators are arbitration operators. △ d , Σ and △ d , Σ n operators are majority operators. 14 / 43

  24. Model-Based Merging An aggregation function f is a function that associates a positive number to any tuple of positive numbers such that : • If x ≤ y , then f ( x 1 , . . . , x , . . . , x n ) ≤ f ( x 1 , . . . , y , . . . , x n ) (monotony) • f ( x 1 , . . . , x n ) = 0 if and only if x 1 = . . . = x n = 0 (minimality) • f ( x ) = x (identity) 15 / 43

  25. Model-Based Merging An aggregation function f is a function that associates a positive number to any tuple of positive numbers such that : • If x ≤ y , then f ( x 1 , . . . , x , . . . , x n ) ≤ f ( x 1 , . . . , y , . . . , x n ) (monotony) • f ( x 1 , . . . , x n ) = 0 if and only if x 1 = . . . = x n = 0 (minimality) • f ( x ) = x (identity) Theorem Let d be a distance between interpretation and f be an aggregation function, then the operateur △ d , f satisfies properties (IC0), (IC1), (IC2), (IC7) et (IC8). 15 / 43

  26. Model-Based Merging An aggregation function f is a function that associates a positive number to any tuple of positive numbers such that : • If x ≤ y , then f ( x 1 , . . . , x , . . . , x n ) ≤ f ( x 1 , . . . , y , . . . , x n ) (monotony) • f ( x 1 , . . . , x n ) = 0 if and only if x 1 = . . . = x n = 0 (minimality) • f ( x ) = x (identity) Theorem Let d be a distance between interpretation and f be an aggregation function, then the operateur △ d , f satisfies properties (IC0), (IC1), (IC2), (IC7) et (IC8). Theorem The operateur △ d , f satisfies properties (IC0-IC8) if and only if f satisfies : • For any permutation σ , f ( x 1 , . . . , x n ) = f ( σ ( x 1 , . . . , x n )) (symmetry) • If f ( x 1 , . . . , x n ) ≤ f ( y 1 , . . . , y n ) , then f ( x 1 , . . . , x n , z ) ≤ f ( y 1 , . . . , y n , z ) (composition) • If f ( x 1 , . . . , x n , z ) ≤ f ( y 1 , . . . , y n , z ) , then f ( x 1 , . . . , x n ) ≤ f ( y 1 , . . . , y n ) (decomposition) 15 / 43

  27. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I ϕ 4 = T ∧ P ∧ ¬ I 16 / 43

  28. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } 16 / 43

  29. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 16 / 43

  30. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 16 / 43

  31. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 8 16 / 43

  32. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 8 22 16 / 43

  33. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 8 22 (3,3,2,0) 16 / 43

  34. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 8 22 (3,3,2,0) ( 0 , 0 , 0 , 1 ) 3 3 1 3 3 10 28 (3,3,3,1) ( 0 , 0 , 1 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 0 , 1 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 0 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 1 , 0 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 1 , 1 ) 1 1 3 1 3 6 12 (3,1,1,1) ( 1 , 0 , 0 , 0 ) 2 2 1 2 2 7 13 (2,2,2,1) ( 1 , 0 , 0 , 1 ) 2 2 2 3 3 9 21 (3,2,2,2) ( 1 , 0 , 1 , 1 ) 1 1 3 2 2 7 15 (3,2,1,1) ( 1 , 1 , 0 , 1 ) 1 1 3 2 3 7 15 (3,2,1,1) ( 1 , 1 , 1 , 1 ) 0 0 4 1 4 5 17 (4,1,0,0) 16 / 43

  35. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 8 22 (3,3,2,0) ( 0 , 0 , 0 , 1 ) 3 3 1 3 3 10 28 (3,3,3,1) ( 0 , 0 , 1 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 0 , 1 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 0 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 1 , 0 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 1 , 1 ) 1 1 3 1 3 6 12 (3,1,1,1) ( 1 , 0 , 0 , 0 ) 2 2 1 2 2 7 13 (2,2,2,1) ( 1 , 0 , 0 , 1 ) 2 2 2 3 3 9 21 (3,2,2,2) ( 1 , 0 , 1 , 1 ) 1 1 3 2 2 7 15 (3,2,1,1) ( 1 , 1 , 0 , 1 ) 1 1 3 2 3 7 15 (3,2,1,1) ( 1 , 1 , 1 , 1 ) 0 0 4 1 4 5 17 (4,1,0,0) 16 / 43

  36. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 8 22 (3,3,2,0) ( 0 , 0 , 0 , 1 ) 3 3 1 3 3 10 28 (3,3,3,1) ( 0 , 0 , 1 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 0 , 1 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 0 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 1 , 0 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 1 , 1 ) 1 1 3 1 3 6 12 (3,1,1,1) ( 1 , 0 , 0 , 0 ) 2 2 1 2 2 7 13 (2,2,2,1) ( 1 , 0 , 0 , 1 ) 2 2 2 3 3 9 21 (3,2,2,2) ( 1 , 0 , 1 , 1 ) 1 1 3 2 2 7 15 (3,2,1,1) ( 1 , 1 , 0 , 1 ) 1 1 3 2 3 7 15 (3,2,1,1) ( 1 , 1 , 1 , 1 ) 0 0 4 1 4 5 17 (4,1,0,0) 16 / 43

  37. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 8 22 (3,3,2,0) ( 0 , 0 , 0 , 1 ) 3 3 1 3 3 10 28 (3,3,3,1) ( 0 , 0 , 1 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 0 , 1 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 0 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 1 , 0 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 1 , 1 ) 1 1 3 1 3 6 12 (3,1,1,1) ( 1 , 0 , 0 , 0 ) 2 2 1 2 2 7 13 (2,2,2,1) ( 1 , 0 , 0 , 1 ) 2 2 2 3 3 9 21 (3,2,2,2) ( 1 , 0 , 1 , 1 ) 1 1 3 2 2 7 15 (3,2,1,1) ( 1 , 1 , 0 , 1 ) 1 1 3 2 3 7 15 (3,2,1,1) ( 1 , 1 , 1 , 1 ) 0 0 4 1 4 5 17 (4,1,0,0) 16 / 43

  38. Example µ = (( S ∧ T ) ∨ ( S ∧ P ) ∨ ( T ∧ P )) → I ϕ 1 = ϕ 2 = S ∧ T ∧ P mod ( ϕ 1 ) = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) } ϕ 3 = ¬ S ∧ ¬ T ∧ ¬ P ∧ ¬ I mod ( ϕ 3 ) = { ( 0 , 0 , 0 , 0 ) } ϕ 4 = T ∧ P ∧ ¬ I mod ( ϕ 4 ) = { ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 0 ) } ϕ 1 ϕ 2 ϕ 3 ϕ 4 d d H , Max d d H , Σ d d H , Σ 2 d d H , GMax ( 0 , 0 , 0 , 0 ) 3 3 0 2 3 8 22 (3,3,2,0) ( 0 , 0 , 0 , 1 ) 3 3 1 3 3 10 28 (3,3,3,1) ( 0 , 0 , 1 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 0 , 1 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 0 , 0 ) 2 2 1 1 2 6 10 (2,2,1,1) ( 0 , 1 , 0 , 1 ) 2 2 2 2 2 8 16 (2,2,2,2) ( 0 , 1 , 1 , 1 ) 1 1 3 1 3 6 12 (3,1,1,1) ( 1 , 0 , 0 , 0 ) 2 2 1 2 2 7 13 (2,2,2,1) ( 1 , 0 , 0 , 1 ) 2 2 2 3 3 9 21 (3,2,2,2) ( 1 , 0 , 1 , 1 ) 1 1 3 2 2 7 15 (3,2,1,1) ( 1 , 1 , 0 , 1 ) 1 1 3 2 3 7 15 (3,2,1,1) ( 1 , 1 , 1 , 1 ) 0 0 4 1 4 5 17 (4,1,0,0) 16 / 43

  39. Formula-Based Merging [BKM91,BKMS92] Idea : Select some formulae from the union of the bases of the profile MAXCONS ( E , µ ) = { M ⊆ � E ∪ µ s.t. • M � ⊥ • µ ⊆ M • ∀ M ⊂ M ′ ⊆ � E ∪ µ M ′ ⊢ ⊥} 17 / 43

  40. Formula-Based Merging [BKM91,BKMS92] Idea : Select some formulae from the union of the bases of the profile MAXCONS ( E , µ ) = { M ⊆ � E ∪ µ s.t. • M � ⊥ • µ ⊆ M • ∀ M ⊂ M ′ ⊆ � E ∪ µ M ′ ⊢ ⊥} △ C 1 µ ( E ) = MAXCONS ( E , µ ) 17 / 43

  41. Formula-Based Merging [BKM91,BKMS92] Idea : Select some formulae from the union of the bases of the profile MAXCONS ( E , µ ) = { M ⊆ � E ∪ µ s.t. • M � ⊥ • µ ⊆ M • ∀ M ⊂ M ′ ⊆ � E ∪ µ M ′ ⊢ ⊥} △ C 1 µ ( E ) = MAXCONS ( E , µ ) △ C 3 µ ( E ) = { M : M ∈ MAXCONS ( E , ⊤ ) and M ∧ µ consistent } 17 / 43

  42. Formula-Based Merging [BKM91,BKMS92] Idea : Select some formulae from the union of the bases of the profile MAXCONS ( E , µ ) = { M ⊆ � E ∪ µ s.t. • M � ⊥ • µ ⊆ M • ∀ M ⊂ M ′ ⊆ � E ∪ µ M ′ ⊢ ⊥} △ C 1 µ ( E ) = MAXCONS ( E , µ ) △ C 3 µ ( E ) = { M : M ∈ MAXCONS ( E , ⊤ ) and M ∧ µ consistent } △ C 4 µ ( E ) = MAXCONS card ( E , µ ) 17 / 43

  43. Formula-Based Merging [BKM91,BKMS92] Idea : Select some formulae from the union of the bases of the profile MAXCONS ( E , µ ) = { M ⊆ � E ∪ µ s.t. • M � ⊥ • µ ⊆ M • ∀ M ⊂ M ′ ⊆ � E ∪ µ M ′ ⊢ ⊥} △ C 1 µ ( E ) = MAXCONS ( E , µ ) △ C 3 µ ( E ) = { M : M ∈ MAXCONS ( E , ⊤ ) and M ∧ µ consistent } △ C 4 µ ( E ) = MAXCONS card ( E , µ ) △ C 5 µ ( E ) = { M ∧ µ : M ∈ MAXCONS ( E , ⊤ ) and M ∧ µ consistent } if this set is nonempty and µ otherwise. 17 / 43

  44. Formula-Based Merging [BKM91,BKMS92] Idea : Select some formulae from the union of the bases of the profile MAXCONS ( E , µ ) = { M ⊆ � E ∪ µ s.t. • M � ⊥ • µ ⊆ M • ∀ M ⊂ M ′ ⊆ � E ∪ µ M ′ ⊢ ⊥} △ C 1 µ ( E ) = MAXCONS ( E , µ ) △ C 3 µ ( E ) = { M : M ∈ MAXCONS ( E , ⊤ ) and M ∧ µ consistent } △ C 4 µ ( E ) = MAXCONS card ( E , µ ) △ C 5 µ ( E ) = { M ∧ µ : M ∈ MAXCONS ( E , ⊤ ) and M ∧ µ consistent } if this set is nonempty and µ otherwise. IC0 IC1 IC2 IC3 IC4 IC5 IC6 IC7 IC8 MI Maj △ C 1 √ √ √ √ √ √ √ △ C 3 √ √ √ √ √ △ C 4 √ √ √ √ √ √ △ C 5 √ √ √ √ √ √ √ √ 17 / 43

  45. Formula-Based Merging : Selection Functions Idea : Use a selection function to choose only the best maxcons. 18 / 43

  46. Formula-Based Merging : Selection Functions Idea : Use a selection function to choose only the best maxcons. • Partial-meet versus full-meet revision operators 18 / 43

  47. Formula-Based Merging : Selection Functions Idea : Use a selection function to choose only the best maxcons. • Partial-meet versus full-meet revision operators • Take into account the distribution of the information among the sources 18 / 43

  48. Formula-Based Merging : Selection Functions Idea : Use a selection function to choose only the best maxcons. • Partial-meet versus full-meet revision operators • Take into account the distribution of the information among the sources Example : Consider a profile E and a maxcons M : • dist ∩ ( M , ϕ ) = | ϕ ∩ M | 18 / 43

  49. Formula-Based Merging : Selection Functions Idea : Use a selection function to choose only the best maxcons. • Partial-meet versus full-meet revision operators • Take into account the distribution of the information among the sources Example : Consider a profile E and a maxcons M : • dist ∩ ( M , ϕ ) = | ϕ ∩ M | • dist ∩ , Σ ( M , E ) = � ϕ ∈ E dist ∩ ( M , ϕ ) 18 / 43

  50. Formula-Based Merging : Selection Functions Idea : Use a selection function to choose only the best maxcons. • Partial-meet versus full-meet revision operators • Take into account the distribution of the information among the sources Example : Consider a profile E and a maxcons M : • dist ∩ ( M , ϕ ) = | ϕ ∩ M | • dist ∩ , Σ ( M , E ) = � ϕ ∈ E dist ∩ ( M , ϕ ) IC0 IC1 IC2 IC3 IC4 IC5 IC6 IC7 IC8 MI Maj △ C 1 √ √ √ √ √ √ √ △ d √ √ √ √ √ √ √ △ S , Σ √ √ √ √ √ √ √ △ ∩ , Σ √ √ √ √ √ √ √ √ 18 / 43

  51. Example ϕ 1 ϕ 2 ϕ 3 a , b → c ¬ a a , b 19 / 43

  52. Example ϕ 1 ϕ 2 ϕ 3 a , b → c ¬ a a , b △ C 1 ⊤ ( E ) = MAXCONS ( E , ⊤ ) = {{ a , b → c , b } , 19 / 43

  53. Example ϕ 1 ϕ 2 ϕ 3 a , b → c ¬ a a , b △ C 1 ⊤ ( E ) = MAXCONS ( E , ⊤ ) = {{ a , b → c , b } , {¬ a , b → c , b }}} 19 / 43

  54. Example ϕ 1 ϕ 2 ϕ 3 a , b → c ¬ a a , b △ C 1 ⊤ ( E ) = MAXCONS ( E , ⊤ ) = {{ a , b → c , b } , {¬ a , b → c , b }}} 2 1 ϕ 1 19 / 43

  55. Example ϕ 1 ϕ 2 ϕ 3 a , b → c ¬ a a , b △ C 1 ⊤ ( E ) = MAXCONS ( E , ⊤ ) = {{ a , b → c , b } , {¬ a , b → c , b }}} 2 1 ϕ 1 ϕ 2 2 1 19 / 43

  56. Example ϕ 1 ϕ 2 ϕ 3 a , b → c ¬ a a , b △ C 1 ⊤ ( E ) = MAXCONS ( E , ⊤ ) = {{ a , b → c , b } , {¬ a , b → c , b }}} 2 1 ϕ 1 ϕ 2 2 1 ϕ 3 0 1 19 / 43

  57. Example ϕ 1 ϕ 2 ϕ 3 a , b → c ¬ a a , b △ C 1 ⊤ ( E ) = MAXCONS ( E , ⊤ ) = {{ a , b → c , b } , {¬ a , b → c , b }}} 2 1 ϕ 1 ϕ 2 2 1 ϕ 3 0 1 Σ 4 3 19 / 43

  58. Example ϕ 1 ϕ 2 ϕ 3 a , b → c ¬ a a , b △ C 1 ⊤ ( E ) = MAXCONS ( E , ⊤ ) = {{ a , b → c , b } , {¬ a , b → c , b }}} 2 1 ϕ 1 ϕ 2 2 1 ϕ 3 0 1 Σ 4 3 19 / 43

  59. Merging • Formula-based Merging ⊲ Selection of maximal consistent subsets of formulas in the union of bases. 20 / 43

  60. Merging • Formula-based Merging ⊲ Selection of maximal consistent subsets of formulas in the union of bases. – Distribution of information – Bad logical properties + Inconsistent bases 20 / 43

  61. Merging • Formula-based Merging • Model-based Merging ⊲ Selection of maximal consistent ⊲ Selection of preferred models for subsets of formulas in the union the bases. of bases. – Distribution of information – Bad logical properties + Inconsistent bases 20 / 43

  62. Merging • Formula-based Merging • Model-based Merging ⊲ Selection of maximal consistent ⊲ Selection of preferred models for subsets of formulas in the union the bases. of bases. – Distribution of information + Distribution of information – Bad logical properties + Good logical properties + Inconsistent bases – Inconsistent bases 20 / 43

  63. Merging • Formula-based Merging • Model-based Merging ⊲ Selection of maximal consistent ⊲ Selection of preferred models for subsets of formulas in the union the bases. of bases. – Distribution of information + Distribution of information – Bad logical properties + Good logical properties + Inconsistent bases – Inconsistent bases • DA 2 Operators 20 / 43

  64. Merging • Formula-based Merging • Model-based Merging ⊲ Selection of maximal consistent ⊲ Selection of preferred models for subsets of formulas in the union the bases. of bases. – Distribution of information + Distribution of information – Bad logical properties + Good logical properties + Inconsistent bases – Inconsistent bases • DA 2 Operators + Distribution of information + Good logical properties + Inconsistent bases 20 / 43

  65. DA 2 Operators Let d be a distance between interpretations and f and g be two aggregation functions. The DA 2 merging operator △ d , f , g ( E ) is defined by : µ For each ϕ i = { α i , 1 , . . . , α i , n i } d ( ω, α i , 1 ) , . . . , d ( ω, α i , n i ) 21 / 43

  66. DA 2 Operators Let d be a distance between interpretations and f and g be two aggregation functions. The DA 2 merging operator △ d , f , g ( E ) is defined by : µ For each ϕ i = { α i , 1 , . . . , α i , n i } d ( ω, ϕ i ) = f ( d ( ω, α i , 1 ) , . . . , d ( ω, α i , n i )) 21 / 43

  67. DA 2 Operators Let d be a distance between interpretations and f and g be two aggregation functions. The DA 2 merging operator △ d , f , g ( E ) is defined by : µ For each ϕ i = { α i , 1 , . . . , α i , n i } d ( ω, ϕ i ) = f ( d ( ω, α i , 1 ) , . . . , d ( ω, α i , n i )) Let E = { ϕ 1 , . . . , ϕ n } d ( ω, E ) = g ( d ( ω, ϕ 1 ) , . . . , d ( ω, ϕ m )) 21 / 43

  68. DA 2 Operators Let d be a distance between interpretations and f and g be two aggregation functions. The DA 2 merging operator △ d , f , g ( E ) is defined by : µ For each ϕ i = { α i , 1 , . . . , α i , n i } d ( ω, ϕ i ) = f ( d ( ω, α i , 1 ) , . . . , d ( ω, α i , n i )) Let E = { ϕ 1 , . . . , ϕ n } d ( ω, E ) = g ( d ( ω, ϕ 1 ) , . . . , d ( ω, ϕ m )) mod ( △ d , f , g ( E ))) = { ω ∈ mod ( µ ) | d ( ω, E ) is minimal } µ 21 / 43

  69. Example ϕ 1 ϕ 2 ϕ 3 ϕ 4 a , b , c , a ∧ ¬ b a , b ¬ a , ¬ b a , a → b 22 / 43

  70. Example ϕ 1 ϕ 2 ϕ 3 ϕ 4 a , b , c , a ∧ ¬ b a , b ¬ a , ¬ b a , a → b = c MAXCONS = c MAXCONS card 22 / 43

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