Belief Change Tools for Negotiation in
Multi-Agent Systems
Sébastien Konieczny
CNRS - CRIL, Lens, France
[email protected]
1 / 20
Belief change for solving conflicts
Ally, Brian and Charles have to decide what they will do this night. Brian and
Ally want to go to the restaurant and to the cinema. Charles does not want to
go out this night and so he does not want to go nor to the restaurant nor to the
cinema.
Ally
Brian
Charles
/ /
2 / 20
Belief change for solving conflicts
Ally, Brian and Charles have to decide what they will do this night. Brian and
Ally want to go to the restaurant and to the cinema. Charles does not want to
go out this night and so he does not want to go nor to the restaurant nor to the
cinema.
Ally
Brian
Ideal setting
• Aggregation
Merging
Charles
/ /
Real setting
• Negotiation
Conciliation
2 / 20
Merging
• Contradictory pieces of information (beliefs, goals, . . .) coming from
different sources
3 / 20
Merging
• Contradictory pieces of information (beliefs, goals, . . .) coming from
different sources
Propositional Logic
3 / 20
Merging
• Contradictory pieces of information (beliefs, goals, . . .) coming from
different sources
Propositional Logic
no priority (same reliability, hierarchical importance, ...)
3 / 20
Merging
• Contradictory pieces of information (beliefs, goals, . . .) coming from
different sources
Propositional Logic
no priority (same reliability, hierarchical importance, ...)
ϕ1
a, b → c
ϕ2
a, b
ϕ3
¬a
4(ϕ1 t ϕ2 t ϕ3 ) =
3 / 20
Merging
• Contradictory pieces of information (beliefs, goals, . . .) coming from
different sources
Propositional Logic
no priority (same reliability, hierarchical importance, ...)
ϕ1
a, b → c
ϕ2
a, b
ϕ3
¬a
4(ϕ1 t ϕ2 t ϕ3 ) = b → c, b
3 / 20
Merging
• Contradictory pieces of information (beliefs, goals, . . .) coming from
different sources
Propositional Logic
no priority (same reliability, hierarchical importance, ...)
ϕ1
a, b → c
ϕ2
a, b
ϕ3
¬a
4(ϕ1 t ϕ2 t ϕ3 ) = b → c, b, a
3 / 20
Merging
• Applications :
Distributed information systems
I
I
Databases
Multi-agent systems
• Propositional bases can encode different types of information :
knowledge
beliefs
goals
rules / laws
...
4 / 20
Definitions
• A base ϕ is a finite set of propositional formulae.
• A profile E is a multi-set of bases : E = {ϕ1 , . . . , ϕn }.
•
V
E denotes the conjunction of the bases of E.
V
• A profile E is consistent if and only if E is consistent.
V
We will note mod(E) instead of mod( E).
5 / 20
Merging
E = {ϕ1 , . . . , ϕn }
6 / 20
Merging
Profile
E = {ϕ1 , . . . , ϕn }
6 / 20
Merging
Profile
E = {ϕ1 , . . . , ϕn }
µ
6 / 20
Merging
Profile
E = {ϕ1 , . . . , ϕn }
µ
Integrity Constraints
6 / 20
Merging
Profile
E = {ϕ1 , . . . , ϕn }
−→ 4µ (E)
µ
Integrity Constraints
6 / 20
Merging
Profile
E = {ϕ1 , . . . , ϕn }
−→ 4µ (E)
µ
Integrity Constraints
Merged base
6 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
7 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
(IC0) 4µ (E) ` µ
7 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
(IC0) 4µ (E) ` µ
(IC1) If µ is consistent, then 4µ (E) is consistent
7 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
(IC0) 4µ (E) ` µ
(IC1) If µ is consistent, then 4µ (E) is consistent
V
V
(IC2) If E is consistent with µ, then 4µ (E) = E ∧ µ
7 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
(IC0) 4µ (E) ` µ
(IC1) If µ is consistent, then 4µ (E) is consistent
V
V
(IC2) If E is consistent with µ, then 4µ (E) = E ∧ µ
(IC3) If E1 ↔ E2 and µ1 ↔ µ2 , then 4µ1 (E1 ) ↔ 4µ2 (E2 )
7 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
(IC0) 4µ (E) ` µ
(IC1) If µ is consistent, then 4µ (E) is consistent
V
V
(IC2) If E is consistent with µ, then 4µ (E) = E ∧ µ
(IC3) If E1 ↔ E2 and µ1 ↔ µ2 , then 4µ1 (E1 ) ↔ 4µ2 (E2 )
(IC4) If ϕ ` µ and ϕ0 ` µ, then 4µ (ϕ t ϕ0 ) ∧ ϕ 0 ⊥ ⇒ 4µ (ϕ t ϕ0 ) ∧ ϕ0 0 ⊥
7 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
(IC0) 4µ (E) ` µ
(IC1) If µ is consistent, then 4µ (E) is consistent
V
V
(IC2) If E is consistent with µ, then 4µ (E) = E ∧ µ
(IC3) If E1 ↔ E2 and µ1 ↔ µ2 , then 4µ1 (E1 ) ↔ 4µ2 (E2 )
(IC4) If ϕ ` µ and ϕ0 ` µ, then 4µ (ϕ t ϕ0 ) ∧ ϕ 0 ⊥ ⇒ 4µ (ϕ t ϕ0 ) ∧ ϕ0 0 ⊥
(IC5) 4µ (E1 ) ∧ 4µ (E2 ) ` 4µ (E1 t E2 )
7 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
(IC0) 4µ (E) ` µ
(IC1) If µ is consistent, then 4µ (E) is consistent
V
V
(IC2) If E is consistent with µ, then 4µ (E) = E ∧ µ
(IC3) If E1 ↔ E2 and µ1 ↔ µ2 , then 4µ1 (E1 ) ↔ 4µ2 (E2 )
(IC4) If ϕ ` µ and ϕ0 ` µ, then 4µ (ϕ t ϕ0 ) ∧ ϕ 0 ⊥ ⇒ 4µ (ϕ t ϕ0 ) ∧ ϕ0 0 ⊥
(IC5) 4µ (E1 ) ∧ 4µ (E2 ) ` 4µ (E1 t E2 )
(IC6) If 4µ (E1 ) ∧ 4µ (E2 ) is consistent, then 4µ (E1 t E2 ) ` 4µ (E1 ) ∧ 4µ (E2 )
7 / 20
Logical Characterization
4 is an Integrity Constraint merging operator (IC merging operator) if and only
if it satisfies the following properties :
(IC0) 4µ (E) ` µ
(IC1) If µ is consistent, then 4µ (E) is consistent
V
V
(IC2) If E is consistent with µ, then 4µ (E) = E ∧ µ
(IC3) If E1 ↔ E2 and µ1 ↔ µ2 , then 4µ1 (E1 ) ↔ 4µ2 (E2 )
(IC4) If ϕ ` µ and ϕ0 ` µ, then 4µ (ϕ t ϕ0 ) ∧ ϕ 0 ⊥ ⇒ 4µ (ϕ t ϕ0 ) ∧ ϕ0 0 ⊥
(IC5) 4µ (E1 ) ∧ 4µ (E2 ) ` 4µ (E1 t E2 )
(IC6) If 4µ (E1 ) ∧ 4µ (E2 ) is consistent, then 4µ (E1 t E2 ) ` 4µ (E1 ) ∧ 4µ (E2 )
(IC7) 4µ1 (E) ∧ µ2 ` 4µ1 ∧µ2 (E)
(IC8) If 4µ1 (E) ∧ µ2 is consistent, then 4µ1 ∧µ2 (E) ` 4µ1 (E)
7 / 20
Majority vs Arbitration
Ally, Brian and Charles have to decide what they will do this night. Brian and
Ally want to go to the restaurant and to the cinema. Charles does not want to
go out this night and so he does not want to go nor to the restaurant nor to the
cinema.
Ally
Brian
Charles
/ /
8 / 20
Majority vs Arbitration
Ally, Brian and Charles have to decide what they will do this night. Brian and
Ally want to go to the restaurant and to the cinema. Charles does not want to
go out this night and so he does not want to go nor to the restaurant nor to the
cinema.
Ally
Majority
Ally
Brian
Charles
Brian
Charles
/ /
restaurant and cinema
++
++
––
8 / 20
Majority vs Arbitration
Ally, Brian and Charles have to decide what they will do this night. Brian and
Ally want to go to the restaurant and to the cinema. Charles does not want to
go out this night and so he does not want to go nor to the restaurant nor to the
cinema.
Ally
Majority
Ally
Brian
Charles
Brian
restaurant and cinema
++
++
––
Charles
Arbitration
Ally
Brian
Charles
/ /
restaurant xor cinema
+
+
+
8 / 20
Majority - Arbitration
(Maj)
∃n 4µ (E1 t E2 n ) ` 4µ (E2 )
. An IC merging operator is a majority operator if it satisfies (Maj).
9 / 20
Majority - Arbitration
(Maj)
∃n 4µ (E1 t E2 n ) ` 4µ (E2 )
. An IC merging operator is a majority operator if it satisfies (Maj).
(Arb)
4µ1 (ϕ1 ) ↔ 4µ2 (ϕ2 )
4µ1 ↔¬µ2 (ϕ1 t ϕ2 ) ↔ (µ1 ↔ ¬µ2 )
µ1 0 µ2
µ2 0 µ1




⇒ 4µ1 ∨µ2 (ϕ1 t ϕ2 ) ↔ 4µ1 (ϕ1 )



. An IC merging operator is an arbitration operator if it satifies (Arb).
9 / 20
Model-Based Merging
Idea : Select the interpretations that are the most plausible for a given profile.
10 / 20
Model-Based Merging
Idea : Select the interpretations that are the most plausible for a given profile.
ω ≤dEx ω 0 iff dx (ω, E) ≤ dx (ω 0 , E)
10 / 20
Model-Based Merging
Idea : Select the interpretations that are the most plausible for a given profile.
ω ≤dEx ω 0 iff dx (ω, E) ≤ dx (ω 0 , E)
dx can be computed using : • a distance between interpretations d
• an aggregation function f
10 / 20
Model-Based Merging
Idea : Select the interpretations that are the most plausible for a given profile.
ω ≤dEx ω 0 iff dx (ω, E) ≤ dx (ω 0 , E)
dx can be computed using : • a distance between interpretations d
• an aggregation function f
• Distance between interpretations
d(ω, ω 0 ) = d(ω 0 , ω)
d(ω, ω 0 ) = 0 iff ω = ω 0
10 / 20
Model-Based Merging
Idea : Select the interpretations that are the most plausible for a given profile.
ω ≤dEx ω 0 iff dx (ω, E) ≤ dx (ω 0 , E)
dx can be computed using : • a distance between interpretations d
• an aggregation function f
• Distance between interpretations
d(ω, ω 0 ) = d(ω 0 , ω)
d(ω, ω 0 ) = 0 iff ω = ω 0
• Distance between an interpretation and a base
d(ω, ϕ) = minω0 |=ϕ d(ω, ω 0 )
10 / 20
Model-Based Merging
Idea : Select the interpretations that are the most plausible for a given profile.
ω ≤dEx ω 0 iff dx (ω, E) ≤ dx (ω 0 , E)
dx can be computed using : • a distance between interpretations d
• an aggregation function f
• Distance between interpretations
d(ω, ω 0 ) = d(ω 0 , ω)
d(ω, ω 0 ) = 0 iff ω = ω 0
• Distance between an interpretation and a base
d(ω, ϕ) = minω0 |=ϕ d(ω, ω 0 )
• Distance between an interpretation and a profile
dd,f (ω, E) = f (d(ω, ϕ1 ), . . . d(ω, ϕn ))
10 / 20
Model-Based Merging
• Examples of aggregation function :
max, leximax, Σ, Σn , leximin, . . .
11 / 20
Model-Based Merging
• Examples of aggregation function :
max, leximax, Σ, Σn , leximin, . . .
• Let d be any distance between interpretations.
4d,max operators satisfy (IC0-IC5), (IC7), (IC8) and (Arb).
4d,GMIN operators are IC merging operators.
4d,GMAX operators are arbitration operators.
n
4d,Σ and 4d,Σ operators are majority operators.
11 / 20
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 (x1 , . . . , x, . . . , xn ) ≤ f (x1 , . . . , y , . . . , xn )
(monotony)
• f (x1 , . . . , xn ) = 0 if and only if x1 = . . . = xn = 0
(minimality)
• f (x) = x
(identity)
12 / 20
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 (x1 , . . . , x, . . . , xn ) ≤ f (x1 , . . . , y , . . . , xn )
(monotony)
• f (x1 , . . . , xn ) = 0 if and only if x1 = . . . = xn = 0
(minimality)
• f (x) = x
(identity)
Theorem Let d be a distance between interpretation and f be an aggregation
function, then the operateur 4d,f satisfies properties (IC0), (IC1), (IC2), (IC7)
et (IC8).
12 / 20
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 (x1 , . . . , x, . . . , xn ) ≤ f (x1 , . . . , y , . . . , xn )
(monotony)
• f (x1 , . . . , xn ) = 0 if and only if x1 = . . . = xn = 0
(minimality)
• f (x) = x
(identity)
Theorem Let d be a distance between interpretation and f be an aggregation
function, then the operateur 4d,f satisfies properties (IC0), (IC1), (IC2), (IC7)
et (IC8).
Theorem The operateur 4d,f satisfies properties (IC0-IC8) if and only if f
satisfies :
• For any permutation σ, f (x1 , . . . , xn ) = f (σ(x1 , . . . , xn ))
(symmetry)
• If f (x1 , . . . , xn ) ≤ f (y1 , . . . , yn ), then f (x1 , . . . , xn , z) ≤ f (y1 , . . . , yn , z)
(composition)
• If f (x1 , . . . , xn , z) ≤ f (y1 , . . . , yn , z), then f (x1 , . . . , xn ) ≤ f (y1 , . . . , yn )
(decomposition)
12 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
ϕ1
3
ϕ2
3
ϕ3
0
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
ddH ,Max
ddH ,Σ
ddH ,Σ2
ddH ,GMax
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
ϕ1
3
ϕ2
3
ϕ3
0
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
ddH ,Max
3
ddH ,Σ
ddH ,Σ2
ddH ,GMax
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
ϕ1
3
ϕ2
3
ϕ3
0
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
ddH ,Max
3
ddH ,Σ
8
ddH ,Σ2
ddH ,GMax
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
ϕ1
3
ϕ2
3
ϕ3
0
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
ddH ,Max
3
ddH ,Σ
8
ddH ,Σ2
22
ddH ,GMax
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
ϕ1
3
ϕ2
3
ϕ3
0
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
ddH ,Max
3
ddH ,Σ
8
ddH ,Σ2
22
ddH ,GMax
(3,3,2,0)
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
(0, 0, 0, 1)
(0, 0, 1, 0)
(0, 0, 1, 1)
(0, 1, 0, 0)
(0, 1, 0, 1)
(0, 1, 1, 1)
(1, 0, 0, 0)
(1, 0, 0, 1)
(1, 0, 1, 1)
(1, 1, 0, 1)
(1, 1, 1, 1)
ϕ1
3
3
2
2
2
2
1
2
2
1
1
0
ϕ2
3
3
2
2
2
2
1
2
2
1
1
0
ϕ3
0
1
1
2
1
2
3
1
2
3
3
4
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
3
1
2
1
2
1
2
3
2
2
1
ddH ,Max
3
3
2
2
2
2
3
2
3
2
3
4
ddH ,Σ
8
10
6
8
6
8
6
7
9
7
7
5
ddH ,Σ2
22
28
10
16
10
16
12
13
21
15
15
17
ddH ,GMax
(3,3,2,0)
(3,3,3,1)
(2,2,1,1)
(2,2,2,2)
(2,2,1,1)
(2,2,2,2)
(3,1,1,1)
(2,2,2,1)
(3,2,2,2)
(3,2,1,1)
(3,2,1,1)
(4,1,0,0)
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
(0, 0, 0, 1)
(0, 0, 1, 0)
(0, 0, 1, 1)
(0, 1, 0, 0)
(0, 1, 0, 1)
(0, 1, 1, 1)
(1, 0, 0, 0)
(1, 0, 0, 1)
(1, 0, 1, 1)
(1, 1, 0, 1)
(1, 1, 1, 1)
ϕ1
3
3
2
2
2
2
1
2
2
1
1
0
ϕ2
3
3
2
2
2
2
1
2
2
1
1
0
ϕ3
0
1
1
2
1
2
3
1
2
3
3
4
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
3
1
2
1
2
1
2
3
2
2
1
ddH ,Max
3
3
2
2
2
2
3
2
3
2
3
4
ddH ,Σ
8
10
6
8
6
8
6
7
9
7
7
5
ddH ,Σ2
22
28
10
16
10
16
12
13
21
15
15
17
ddH ,GMax
(3,3,2,0)
(3,3,3,1)
(2,2,1,1)
(2,2,2,2)
(2,2,1,1)
(2,2,2,2)
(3,1,1,1)
(2,2,2,1)
(3,2,2,2)
(3,2,1,1)
(3,2,1,1)
(4,1,0,0)
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
(0, 0, 0, 1)
(0, 0, 1, 0)
(0, 0, 1, 1)
(0, 1, 0, 0)
(0, 1, 0, 1)
(0, 1, 1, 1)
(1, 0, 0, 0)
(1, 0, 0, 1)
(1, 0, 1, 1)
(1, 1, 0, 1)
(1, 1, 1, 1)
ϕ1
3
3
2
2
2
2
1
2
2
1
1
0
ϕ2
3
3
2
2
2
2
1
2
2
1
1
0
ϕ3
0
1
1
2
1
2
3
1
2
3
3
4
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
3
1
2
1
2
1
2
3
2
2
1
ddH ,Max
3
3
2
2
2
2
3
2
3
2
3
4
ddH ,Σ
8
10
6
8
6
8
6
7
9
7
7
5
ddH ,Σ2
22
28
10
16
10
16
12
13
21
15
15
17
ddH ,GMax
(3,3,2,0)
(3,3,3,1)
(2,2,1,1)
(2,2,2,2)
(2,2,1,1)
(2,2,2,2)
(3,1,1,1)
(2,2,2,1)
(3,2,2,2)
(3,2,1,1)
(3,2,1,1)
(4,1,0,0)
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
(0, 0, 0, 1)
(0, 0, 1, 0)
(0, 0, 1, 1)
(0, 1, 0, 0)
(0, 1, 0, 1)
(0, 1, 1, 1)
(1, 0, 0, 0)
(1, 0, 0, 1)
(1, 0, 1, 1)
(1, 1, 0, 1)
(1, 1, 1, 1)
ϕ1
3
3
2
2
2
2
1
2
2
1
1
0
ϕ2
3
3
2
2
2
2
1
2
2
1
1
0
ϕ3
0
1
1
2
1
2
3
1
2
3
3
4
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
3
1
2
1
2
1
2
3
2
2
1
ddH ,Max
3
3
2
2
2
2
3
2
3
2
3
4
ddH ,Σ
8
10
6
8
6
8
6
7
9
7
7
5
ddH ,Σ2
22
28
10
16
10
16
12
13
21
15
15
17
ddH ,GMax
(3,3,2,0)
(3,3,3,1)
(2,2,1,1)
(2,2,2,2)
(2,2,1,1)
(2,2,2,2)
(3,1,1,1)
(2,2,2,1)
(3,2,2,2)
(3,2,1,1)
(3,2,1,1)
(4,1,0,0)
13 / 20
Example
µ = ((S ∧ T ) ∨ (S ∧ P) ∨ (T ∧ P)) → I
ϕ1 = ϕ2 = S ∧ T ∧ P
ϕ3 = ¬S ∧ ¬T ∧ ¬P ∧ ¬I
ϕ4 = T ∧ P ∧ ¬I
(0, 0, 0, 0)
(0, 0, 0, 1)
(0, 0, 1, 0)
(0, 0, 1, 1)
(0, 1, 0, 0)
(0, 1, 0, 1)
(0, 1, 1, 1)
(1, 0, 0, 0)
(1, 0, 0, 1)
(1, 0, 1, 1)
(1, 1, 0, 1)
(1, 1, 1, 1)
ϕ1
3
3
2
2
2
2
1
2
2
1
1
0
ϕ2
3
3
2
2
2
2
1
2
2
1
1
0
ϕ3
0
1
1
2
1
2
3
1
2
3
3
4
mod(ϕ1 ) = {(1, 1, 1, 1), (1, 1, 1, 0)}
mod(ϕ3 ) = {(0, 0, 0, 0)}
mod(ϕ4 ) = {(1, 1, 1, 0), (0, 1, 1, 0)}
ϕ4
2
3
1
2
1
2
1
2
3
2
2
1
ddH ,Max
3
3
2
2
2
2
3
2
3
2
3
4
ddH ,Σ
8
10
6
8
6
8
6
7
9
7
7
5
ddH ,Σ2
22
28
10
16
10
16
12
13
21
15
15
17
ddH ,GMax
(3,3,2,0)
(3,3,3,1)
(2,2,1,1)
(2,2,2,2)
(2,2,1,1)
(2,2,2,2)
(3,1,1,1)
(2,2,2,1)
(3,2,2,2)
(3,2,1,1)
(3,2,1,1)
(4,1,0,0)
13 / 20
Merging in other frameworks
• Merging of weighted formulae
Benferhat-Dubois-Kaci-Prade [2000,2002,2003]
Meyer [2001]
• First order logic
Gorogiannis-Hunter [2008]
• Logic programs
Delgrande-Schaub-Tompits-Woltran [2009]
Hué-Papini-Würbel [2009]
• Constraints Networks
Condotta-Kaci-Marquis-Schwind [2009]
• Argumentation systems
Dung : arguments + attack relation [AAAI’05, AIJ-07]
I
I
Partial argumentation frameworks (PAF)
Edition distances
Weighted Argumentation Frameworks [IJCAI’15]
Merging Extensions [KR’16]
14 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
Revision
(ϕ01 ∗ ϕ∆0 , . . . , ϕ0n ∗ ϕ∆0 )
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
Revision
(ϕ01 ∗ ϕ∆0 , . . . , ϕ0n ∗ ϕ∆0 )
(ϕ11 , . . . , ϕ1n )
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
Revision
(ϕ01 ∗ ϕ∆0 , . . . , ϕ0n ∗ ϕ∆0 )
(ϕ11 , . . . , ϕ1n )
ϕ∆1
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
Revision
(ϕ01 ∗ ϕ∆0 , . . . , ϕ0n ∗ ϕ∆0 )
(ϕ11 , . . . , ϕ1n )
ϕ∆1
(ϕk1 , . . . , ϕkn )
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
Revision
(ϕ01 ∗ ϕ∆0 , . . . , ϕ0n ∗ ϕ∆0 )
(ϕ11 , . . . , ϕ1n )
ϕ∆1
(ϕk1 , . . . , ϕkn )
ϕ∆k
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
Revision
(ϕ01 ∗ ϕ∆0 , . . . , ϕ0n ∗ ϕ∆0 )
(ϕ11 , . . . , ϕ1n )
ϕ∆1
(ϕk1 , . . . , ϕkn )
Merging
ϕ∆k
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
Revision
(ϕ01 ∗ ϕ∆0 , . . . , ϕ0n ∗ ϕ∆0 )
(ϕ11 , . . . , ϕ1n )
Conciliation
ϕ∆1
(ϕk1 , . . . , ϕkn )
ϕ∆k
15 / 20
Iterated Merging - Conciliation
• Iterated Merging Operators
(ϕ01 , . . . , ϕ0n )
Merging
ϕ∆0
Revision
(ϕ01 ∗ ϕ∆0 , . . . , ϕ0n ∗ ϕ∆0 )
(ϕ11 , . . . , ϕ1n )
Conciliation
ϕ∆1
(ϕk1 , . . . , ϕkn )
ϕ∆k
• Merging
• Conciliation
(ϕ1 , . . . , ϕn ) −→ ϕ∆
(ϕ1 , . . . , ϕn ) −→ (ϕ∗1 , . . . , ϕ∗n )
15 / 20
Negotiation - Conciliation
Let E = (ϕ1 , . . . , ϕn ) be a profile of belief/goal bases.
Two questions :
• What are the beliefs/goals of the group of agents ?
Merging (vote, social choice, MCDM, . . .)
• Can the agents find a consensual position ?
Conciliation (negotiation, bargaining, . . .)
16 / 20
A Game between Sources
• Negotiation :
• Some sources have to concede to solve the conflicts
17 / 20
A Game between Sources
• Negotiation :
• Some sources have to concede to solve the conflicts
• The idea :
• Each source gives her base
• Contest between the bases :
The weakest ones loose
The loosers have to concede (logical weakening)
• Ends when a compromise is reached
17 / 20
A Game between Sources
• Negotiation :
• Some sources have to concede to solve the conflicts
• The idea :
• Each source gives her base
• Contest between the bases :
The weakest ones loose
The loosers have to concede (logical weakening)
• Ends when a compromise is reached
Definition A Belief Game Model is a pair N = hg, Hi where g is a choice
function and H is a weakening function.
The solution to a belief profile E for a Belief Game Model N = hg, Hi, noted
N (E), is the belief profile E N , defined as :
• E0 = E
• E i+1 = Hg(E i ) (E i )
• E N is the first E i that is consistent
17 / 20
A Game between Sources
• Negotiation :
• Some sources have to concede to solve the conflicts
• The idea :
• Each source gives her base
• Contest between the bases :
The weakest ones loose
The loosers have to concede (logical weakening)
• Ends when a compromise is reached
Definition A Belief Game Model is a pair N = hg, Hi where g is a choice
function and H is a weakening function.
The solution to a belief profile E for a Belief Game Model N = hg, Hiunder the
integrity constraints µ, noted N µ (E), is the belief profile E N µ , defined as :
• E0 = E
• E i+1 = Hg(E i ) (E i )
• E N µ is the first E i that is consistent with µ
17 / 20
A Game between Sources
• Negotiation :
• Some sources have to concede to solve the conflicts
• The idea :
• Each source gives her base
• Contest between the bases :
The weakest ones loose
The loosers have to concede (logical weakening)
• Ends when a compromise is reached
Definition A Belief Game Model is a pair N = hg, Hi where g is a choice
function and H is a weakening function.
The solution to a belief profile E for a Belief Game Model N = hg, Hiunder the
integrity constraints µ, noted N µ (E), is the belief profile E N µ , defined as :
• E0 = E
• E i+1 = Hg(E i ) (E i )
• E N µ is the first E i that is consistent with µ
17 / 20
Belief Game Model
A choice function is a function g : E → E such that :
• g(E) ⊆ E
• If
V
E 6≡ >, then ∃ϕ ∈ g(E) s.t. ϕ 6≡ >
• If E ↔ E 0 , then g(E) ↔ g(E 0 )
A weakening function is a function H : K → K such that :
• ϕ ` H(ϕ)
• If ϕ = H(ϕ), then ϕ ↔ >
• If ϕ ↔ ϕ0 , then H(ϕ) ↔ H(ϕ0 )
18 / 20
Example : Database Class [Revesz, 1994]
• g = dDΣ , H = δ
ϕ1 = {100, 001, 101}
ϕ2 = {010, 001}
ϕ3 = {111}
mod(ϕ1 ∧ ϕ2 ∧ ϕ3 ) = ∅
19 / 20
Example : Database Class [Revesz, 1994]
• g = dDΣ , H = δ
ϕ1 = {100, 001, 101}
ϕ2 = {010, 001}
ϕ3 = {111}
mod(ϕ1 ∧ ϕ2 ∧ ϕ3 ) = ∅
ϕ1
ϕ1
ϕ2
ϕ3
0
1
ϕ2
0
1
ϕ3
1
1
Σ
1
1
2
g
•
19 / 20
Example : Database Class [Revesz, 1994]
• g = dDΣ , H = δ
ϕ1 = {100, 001, 101}
ϕ2 = {010, 001}
ϕ3 = {111}
ϕ3 = {111, 011, 101, 110}
mod(ϕ1 ∧ ϕ2 ∧ ϕ3 ) = ∅
ϕ1
ϕ1
ϕ2
ϕ3
0
1
ϕ2
0
1
ϕ3
1
1
Σ
1
1
2
g
•
19 / 20
Example : Database Class [Revesz, 1994]
• g = dDΣ , H = δ
ϕ1 = {100, 001, 101}
ϕ2 = {010, 001}
ϕ3 = {111}
ϕ3 = {111, 011, 101, 110}
mod(ϕ1 ∧ ϕ2 ∧ ϕ3 ) = ∅
ϕ1
ϕ1
ϕ2
ϕ3
0
0
ϕ2
0
1
ϕ3
0
1
Σ
0
1
1
g
•
•
19 / 20
Example : Database Class [Revesz, 1994]
• g = dDΣ , H = δ
ϕ1 = {100, 001, 101}
ϕ2 = {010, 001}
ϕ2 = {010, 001, 110, 000, 011, 101}
ϕ3 = {111}
ϕ3 = {111, 011, 101, 110}
ϕ3 = {111, 011, 101, 110, 001, 010, 100}
mod(ϕ1 ∧ ϕ2 ∧ ϕ3 ) = ∅
ϕ1
ϕ1
ϕ2
ϕ3
0
0
ϕ2
0
1
ϕ3
0
1
Σ
0
1
1
g
•
•
19 / 20
Example : Database Class [Revesz, 1994]
• g = dDΣ , H = δ
ϕ1 = {100, 001, 101}
ϕ2 = {010, 001}
ϕ2 = {010, 001, 110, 000, 011, 101}
ϕ3 = {111}
ϕ3 = {111, 011, 101, 110}
ϕ3 = {111, 011, 101, 110, 001, 010, 100}
mod(ϕ1 ∧ ϕ2 ∧ ϕ3 ) = {001, 101}
ϕ1
ϕ1
ϕ2
ϕ3
0
0
ϕ2
0
1
ϕ3
0
1
Σ
0
1
1
g
•
•
19 / 20
Merging - Conciliation
• Merging
Postulates - Representation Theorems [JLC’02] [AIJ’04]
Manipulability issues [JAIR’07]
Truth-tracking issues [ECAI’10]
Egalitarism issues [KR’14]
Judgement Aggregation [AAMAS’14, AAMAS’15]
• Conciliation
Iterated Belief Merging [JLC’07]
Belief Negotiation Games [JANCL’04]
Confluence operators [AMAI’13]
Belief Revision Games [AAAI’15, IJCAI’16]
20 / 20