Enforcement in Argumentation is a kind of Update an axiomatic - - PowerPoint PPT Presentation

Enforcement is a kind of Update

Enforcement in Argumentation is a kind of Update

an axiomatic approach of enforcement

P . Bisquert, C. Cayrol, F . Dupin de Saint-Cyr, MC. Lagasquie

IRIT, Toulouse University, France

July 2013

Sintelnet workshop: Believing, planning, acting, revising

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 1 / 30

Enforcement is a kind of Update

A lawyer during a trial

Lawyer

blabla . . . my client . . . innocent . . . blabla . . . guilty

Opposite party audience

a b c d

A lawyer makes her final address to an audience ; O (opposite party) knows the corresponding argumentation system (AS). O can compute the accepted arguments in AS.

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 2 / 30

Enforcement is a kind of Update

Lawyer’s example

Lawyer Opposite party

e attacks c

audience

a b c d

O wants to force the audience to accept specific arguments. She has to make a change to the public system :

◮ by adding an argument

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 3 / 30

Enforcement is a kind of Update

Lawyer’s example

Lawyer Opposite party

e attacks c

audience

a b c d e

O wants to force the audience to accept specific arguments. She has to make a change to the public system :

◮ by adding an argument

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 3 / 30

Enforcement is a kind of Update

Lawyer’s example

Lawyer Opposite party

• bjection

against c

audience

a b c d

O wants to force the audience to accept specific arguments. She has to make a change to the public system :

◮ by adding an argument ◮ or by doing an objection about an argument (to remove it)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 3 / 30

Enforcement is a kind of Update

Lawyer’s example

Lawyer Opposite party

• bjection

against c

audience

a b d

O wants to force the audience to accept specific arguments. She has to make a change to the public system :

◮ by adding an argument ◮ or by doing an objection about an argument (to remove it)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 3 / 30

Enforcement is a kind of Update

Agent

a b c d e ∧¬on(e) a(d) ∧ on(b)

• bjection against c

Target

a b c d

Agent :

◮ has a private argumentation system (her knowledge) ◮ has a goal w.r.t. the target ◮ agent should respect some constraints

⇒ notion of executable operation

Target = public argumentation system (state of the dialog)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 4 / 30

Enforcement is a kind of Update

Summary

1

Framework

2

Towards Generalized Enforcement

3

Generalized Update Postulates

4

Concluding Remarks

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 5 / 30

Enforcement is a kind of Update

Summary

1

Framework Abstract argumentation Goals Change in Argumentation

2

Towards Generalized Enforcement

3

Generalized Update Postulates

4

Concluding Remarks

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 6 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

• Mr. X is not guilty of the murder of Mrs. X

Universe

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

• Mr. X is not guilty of the murder of Mrs. X

1

• Mr. X is guilty of the murder of Mrs. X

Universe

1

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

• Mr. X is not guilty of the murder of Mrs. X

1

• Mr. X is guilty of the murder of Mrs. X

2

• Mr. X’s business associate has sworn that he

met him at the time of the murder.

Universe

1 2

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

• Mr. X is not guilty of the murder of Mrs. X

1

• Mr. X is guilty of the murder of Mrs. X

2

• Mr. X’s business associate has sworn that he

met him at the time of the murder.

3

• Mr. X associate’s testimony is suspicious due to

their close working business relationship

Universe

3 1 2

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

• Mr. X is not guilty of the murder of Mrs. X

1

• Mr. X is guilty of the murder of Mrs. X

2

• Mr. X’s business associate has sworn that he

met him at the time of the murder.

3

• Mr. X associate’s testimony is suspicious due to

their close working business relationship

4

• Mr. X loves his wife. A man who loves his wife

cannot be her killer.

Universe

4 3 1 2

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

• Mr. X is not guilty of the murder of Mrs. X

1

• Mr. X is guilty of the murder of Mrs. X

2

• Mr. X’s business associate has sworn that he

met him at the time of the murder.

3

• Mr. X associate’s testimony is suspicious due to

their close working business relationship

4

• Mr. X loves his wife. A man who loves his wife

cannot be her killer.

5

• Mr. X has a reputation for being promiscuous.

Universe

5 4 3 1 2

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

• Mr. X is not guilty of the murder of Mrs. X

1

• Mr. X is guilty of the murder of Mrs. X

2

• Mr. X’s business associate has sworn that he

met him at the time of the murder.

3

• Mr. X associate’s testimony is suspicious due to

their close working business relationship

4

• Mr. X loves his wife. A man who loves his wife

cannot be her killer.

5

• Mr. X has a reputation for being promiscuous.

6

• Mr. X had no interest to kill Mrs. X, since he was

not the beneficiary of her life insurance

Universe

5 6 4 3 1 2

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

• Mr. X is not guilty of the murder of Mrs. X

1

• Mr. X is guilty of the murder of Mrs. X

2

• Mr. X’s business associate has sworn that he

met him at the time of the murder.

3

• Mr. X associate’s testimony is suspicious due to

their close working business relationship

4

• Mr. X loves his wife. A man who loves his wife

cannot be her killer.

5

• Mr. X has a reputation for being promiscuous.

6

• Mr. X had no interest to kill Mrs. X, since he was

not the beneficiary of her life insurance

7

• Mr. X is known to be venal and his “love” for a

very rich woman could be only lure of profit.

Universe

5 7 6 4 3 1 2

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Abstract argumentation

A universe Arg, Rel = all arguments and their interactions.

Definition (Argumentation graph (AS))

An argumentation graph G is a pair (A, R) A ⊆ Arg arguments (finite) R ⊆ Rel ∩ (A × A) Γ = all argumentation graphs w.r.t. the universe.

2 7 4 1 2 4 1

G1 G2

Universe

5 7 6 4 3 1 2

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 7 / 30

Enforcement is a kind of Update

Argumentation graph example

Agent O knows all the arguments of the universe. But O is not sure about the content of the jury’s system. She hesitates between two graphs :

2 7 4 1 2 4 1

G1 G2

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 8 / 30

Enforcement is a kind of Update

Acceptability of an argument

“semantics” : way to compute particular sets of arguments (extensions)

◮ a set of arguments Y defends an argument x, denoted

x ∈ F(Y) iff ∀y s.t. (y, x) ∈ R, ∃z ∈ Y s.t. (z, y) ∈ R.

◮ a grounded extension : conflict free set of arguments = the

least (wrt inclusion) fix point for F

◮ a preferred extension : conflict free set of arguments

defending all its elements and maximal (wrt inclusion)

◮ a stable extension S : conflict free set of arguments

attacking each arguments outside of it.

x is accepted if it belongs to the extension(s) (x ∈ E)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 9 / 30

Enforcement is a kind of Update

Goals

Egrd E′

grd a b c d a b d

Change

G G′

Several types of goals (based on the typology presented in [Bisquert et al., 2013b]) :

◮ Argument goals : x ∈ E′, x /

∈ E′

◮ Extension goals : E = E′, E ⊆ E′, E ⊂ E′, E′ ⊆ E, E′ ⊂ E,

E′ = ∅, E′ = ∅...

How can an agent achieve her goal ? What operation can she use ?

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 10 / 30

Enforcement is a kind of Update

Change in argumentation

([Cayrol et al., 2010]) : four elementary change operations.

◮ adding/removing an argument with related attacks, ◮ adding/removing an attack.

Here, purpose of enforcement :

◮ an agent may act on a target argumentation system ◮ agent has a goal ◮ agent should respect some constraints

⇒ notion of executable operation.

Simplifications for the presentation of examples :

◮ focus on addition and removal of arguments ◮ for attacks addition/removal (see Zhiqiang Zhuang)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 11 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• p, x, att allowed for k iff x ∈ Ak and att ⊆ RAk
• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• p, x, att allowed for k iff x ∈ Ak and att ⊆ RAk
• p, x, att executable by k on G iff :
• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• p, x, att allowed for k iff x ∈ Ak and att ⊆ RAk
• p, x, att executable by k on G iff :

◮ op = ⊕ : x ∈ A and ∀(u, v) ∈ att, (u ∈ A or v ∈ A)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• p, x, att allowed for k iff x ∈ Ak and att ⊆ RAk
• p, x, att executable by k on G iff :

◮ op = ⊕ : x ∈ A and ∀(u, v) ∈ att, (u ∈ A or v ∈ A) ◮ op = ⊖ : x ∈ A.

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• p, x, att allowed for k iff x ∈ Ak and att ⊆ RAk
• p, x, att executable by k on G iff :

◮ op = ⊕ : x ∈ A and ∀(u, v) ∈ att, (u ∈ A or v ∈ A) ◮ op = ⊖ : x ∈ A.

• = op, x, att executable by k on G provides

a new system G′ = o(G) = A′, RA′ :

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• p, x, att allowed for k iff x ∈ Ak and att ⊆ RAk
• p, x, att executable by k on G iff :

◮ op = ⊕ : x ∈ A and ∀(u, v) ∈ att, (u ∈ A or v ∈ A) ◮ op = ⊖ : x ∈ A.

• = op, x, att executable by k on G provides

a new system G′ = o(G) = A′, RA′ :

◮ op = ⊕ : G′ = A ∪ {x}, RA ∪ {att}

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• p, x, att allowed for k iff x ∈ Ak and att ⊆ RAk
• p, x, att executable by k on G iff :

◮ op = ⊕ : x ∈ A and ∀(u, v) ∈ att, (u ∈ A or v ∈ A) ◮ op = ⊖ : x ∈ A.

• = op, x, att executable by k on G provides

a new system G′ = o(G) = A′, RA′ :

◮ op = ⊕ : G′ = A ∪ {x}, RA ∪ {att} ◮ op = ⊖ :

G′ = A \ {x}, RA \ {(u, v) ∈ RA|u = x or v = x}

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operation

k = agent, Gk = Ak, RAk her AS, G = A, RA any AS. elementary operation o = op, x, att

• p ∈ {⊕, ⊖},

x ∈ Arg, att ⊆ Rel and

◮ op = ⊕ : ∀(u, v) ∈ att, (u = v) and (u = x or v = x) ◮ op = ⊖ : att = ∅

• p, x, att allowed for k iff x ∈ Ak and att ⊆ RAk
• p, x, att executable by k on G iff :

◮ op = ⊕ : x ∈ A and ∀(u, v) ∈ att, (u ∈ A or v ∈ A) ◮ op = ⊖ : x ∈ A.

• = op, x, att executable by k on G provides

a new system G′ = o(G) = A′, RA′ :

◮ op = ⊕ : G′ = A ∪ {x}, RA ∪ {att} ◮ op = ⊖ :

G′ = A \ {x}, RA \ {(u, v) ∈ RA|u = x or v = x}

program = sequence of executable operations

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 12 / 30

Enforcement is a kind of Update

Executable operations : example

Given the universe :

5 6 3 2 7 4 1

⊕, 2, {(2, 1)}, ⊖, 3, ∅, ⊖, 4, ∅ and ⊕, 5, {(5, 4)} are elementary operations With GA :

5 6 3 4 1

⊖, 3, ∅, ⊖, 4, ∅ and ⊕, 5, {(5, 4)} are allowed for Agent A (arguments she knows). On the target G :

4 1

⊖, 4, ∅ and ⊕, 5, {(5, 4)} are executable by A on G.

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 13 / 30

Enforcement is a kind of Update

Summary

1

Framework

2

Towards Generalized Enforcement Belief Change Theory Generalized Enforcement Operator

3

Generalized Update Postulates

4

Concluding Remarks

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 14 / 30

Enforcement is a kind of Update

Belief Change Theory

initial belief about the state

• f the world

ϕ piece of info describing a belief evolution new belief about the state

• f the world

ϕ′ = ϕ ⋆ α Revision by α change of the belief about the state of the world

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 15 / 30

Enforcement is a kind of Update

Belief Change Theory

initial belief about the state

• f the world

ϕ piece of info describing a state evolution new belief about the state

• f the world

ϕ′ = ϕ ⋄ α Update by α change of the state of the world

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 15 / 30

Enforcement is a kind of Update

Belief Change Theory : Example

There is either an apple or a banana in the box but not both. I learn that there was no apple in the box. initial belief about the world a ⊕ b = (a ∧ ¬b) ∨ (¬a ∧ b) new belief about the world (a ⊕ b) ⋄ ¬a Revision by ¬a (¬a ∧ b)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 16 / 30

Enforcement is a kind of Update

Belief Change Theory : Example

There is either an apple or a banana in the box but not both. I learn that the apple stealer has visited the box initial state of the world a ⊕ b = (a ∧ ¬b) ∨ (¬a ∧ b) new state of the world (a ⊕ b) ⋄ ¬a Update by ¬a (¬a ∧ ¬b) ∨ (¬a ∧ b)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 16 / 30

Enforcement is a kind of Update

“Semantics” in Belief Change

Revision/Update operator : formula + formula → formula Representation theorem : set of axioms related to a preorder on worlds.

◮ revision : [ϕ ⋆ α] = {ω′ ∈ [α], “closer” from ϕ }

= {ω′ ∈ [α], ∀ω′′ ∈ [α], ω′ ϕ ω′′}

◮ update :

[ϕ ⋄ α] = ∪ω∈[ϕ]{ω′ ∈ [α], “closer” from ω} ∪ω∈[ϕ]{ω′ ∈ [α], ∀ω′′ ∈ [α], ω′ ω ω′′}

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 17 / 30

Enforcement is a kind of Update

Parallel enforcement/update

graphs worlds formula characterizing a set of graphs formula characterizing a set of worlds. Enforcement Update

• Init. knowledge :

set of AS set of worlds Input : goal new info Constraints : set of none (every transitions update is achievable)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 18 / 30

Enforcement is a kind of Update

Generalized Enforcement

Requirement : Generalized Enforcement is based on

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 19 / 30

Enforcement is a kind of Update

Generalized Enforcement

Requirement : Generalized Enforcement is based on

◮ a propositional language L able to describe any AS and its

accepted arguments,

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 19 / 30

Enforcement is a kind of Update

Generalized Enforcement

Requirement : Generalized Enforcement is based on

◮ a propositional language L able to describe any AS and its

accepted arguments,

◮ a characteristic function f associated with L , such that

∀G ∈ Γ, [f(G)] = {G}.

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 19 / 30

Enforcement is a kind of Update

Generalized Enforcement

Requirement : Generalized Enforcement is based on

◮ a propositional language L able to describe any AS and its

accepted arguments,

◮ a characteristic function f associated with L , such that

∀G ∈ Γ, [f(G)] = {G}.

restriction on authorized changes : let T ⊆ Γ × Γ a set of allowed transitions

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 19 / 30

Enforcement is a kind of Update

Generalized Enforcement

Requirement : Generalized Enforcement is based on

◮ a propositional language L able to describe any AS and its

accepted arguments,

◮ a characteristic function f associated with L , such that

∀G ∈ Γ, [f(G)] = {G}.

restriction on authorized changes : let T ⊆ Γ × Γ a set of allowed transitions examples :

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 19 / 30

Enforcement is a kind of Update

Generalized Enforcement

Requirement : Generalized Enforcement is based on

◮ a propositional language L able to describe any AS and its

accepted arguments,

◮ a characteristic function f associated with L , such that

∀G ∈ Γ, [f(G)] = {G}.

restriction on authorized changes : let T ⊆ Γ × Γ a set of allowed transitions examples :

◮ T k

e = {(G, G′) ∈ Γ × Γ, ∃o s.t. o is an elementary

• peration executable by Agent k on G s.t. o(G) = G′}.
• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 19 / 30

Enforcement is a kind of Update

Generalized Enforcement

Requirement : Generalized Enforcement is based on

◮ a propositional language L able to describe any AS and its

accepted arguments,

◮ a characteristic function f associated with L , such that

∀G ∈ Γ, [f(G)] = {G}.

restriction on authorized changes : let T ⊆ Γ × Γ a set of allowed transitions examples :

◮ T k

e = {(G, G′) ∈ Γ × Γ, ∃o s.t. o is an elementary

• peration executable by Agent k on G s.t. o(G) = G′}.

◮ T k

p = {(G, G′) ∈ Γ × Γ, ∃p s.t. p is a program executable

by agent k on G s.t. p(G) = G′}

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 19 / 30

Enforcement is a kind of Update

Generalized Enforcement

Requirement : Generalized Enforcement is based on

◮ a propositional language L able to describe any AS and its

accepted arguments,

◮ a characteristic function f associated with L , such that

∀G ∈ Γ, [f(G)] = {G}.

restriction on authorized changes : let T ⊆ Γ × Γ a set of allowed transitions examples :

◮ T k

e = {(G, G′) ∈ Γ × Γ, ∃o s.t. o is an elementary

• peration executable by Agent k on G s.t. o(G) = G′}.

◮ T k

p = {(G, G′) ∈ Γ × Γ, ∃p s.t. p is a program executable

by agent k on G s.t. p(G) = G′}

◮ Baumann’s normal expansion : TB = {(G, G′) ∈ Γ × Γ, with

G = (A, RA) and G′ = (A′, RA′) s.t. A A′}.

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 19 / 30

Enforcement is a kind of Update

Generalized Enforcement : definition

Given a set of authorized transitions T ⊆ Γ × Γ L × , L L ϕ information about a target AS α goal ϕ♦Tα AS in which α holds, that can be obtained by a change in T ( ) General Enforcement

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 20 / 30

Enforcement is a kind of Update

Example

Formula known by Agent O about the Jury’s AS : ϕ = on(0) ∧ on(1) ∧ on(2) ∧ on(4)∧ ¬on(3) ∧ ¬on(5) ∧ ¬on(6) ∧ (on(7) ∨ ¬on(7)). Two possible graphs :

2 7 4 1 2 4 1

Agent O wants to enforce acceptation of 1 when 2 and 4 are present (w.r.t. the grounded semantics) with a program executable

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 21 / 30

Enforcement is a kind of Update

Example

Formula known by Agent O about the Jury’s AS : ϕ = on(0) ∧ on(1) ∧ on(2) ∧ on(4)∧ ¬on(3) ∧ ¬on(5) ∧ ¬on(6) ∧ (on(7) ∨ ¬on(7)). Two possible graphs :

5 3 2 7 4 1 5 3 2 4 1

Agent O wants to enforce acceptation of 1 when 2 and 4 are present (w.r.t. the grounded semantics) with a program executable ϕ♦T O

p (a(1) ∧ on(2) ∧ on(4) ) |

= on(3) ∧ on(5) ∧ a(1).

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 21 / 30

Enforcement is a kind of Update

Summary

1

Framework

2

Towards Generalized Enforcement

3

Generalized Update Postulates Update Postulates Postulates for Generalized Enforcement

4

Concluding Remarks

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 22 / 30

Enforcement is a kind of Update

Update Postulates

U1 : ϕ ⋄ α | = α U2 : ϕ | = α ⇒ [ϕ ⋄ α] = [ϕ] U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ U4 : [ϕ] = [ψ] and [α] = [β] ⇒ [ϕ ⋄ α] = [ψ ⋄ β] U5 : (ϕ ⋄ α) ∧ β | = ϕ ⋄ (α ∧ β) U8 : [(ϕ ∨ ψ) ⋄ α] = [(ϕ ⋄ α) ∨ (ψ ⋄ α)] U9 : if card([ϕ]) = 1 then [(ϕ ⋄ α) ∧ β] = ∅ ⇒ ϕ ⋄ (α ∧ β) |

= (ϕ ⋄ α) ∧ β

Theorem ([Katsuno and Mendelzon, 1991])

∃ an operator ⋄ : L × L → L satisfying U1, U2, U3, U4, U5, U8, U9 iff ∃ a faithful assignment : ω → a complete pre-order ω s.t. [ϕ ⋄ α] =

• ω∈[ϕ]

{ω′ ∈ [α] s.t. ∀ω′′ ∈ [α], ω′ ω ω′′} (“faithful”= ∀ω, ω′ ∈ Ω, ω ≺ω ω′)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 23 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ⋄ α | = α

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ⋄ α] = [ϕ] (optional : inertia)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ (every transition is possible)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ ⇒ E3 : [ϕ♦Tα] = ∅ iff (ϕ, α) | = T

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ ⇒ E3 : [ϕ♦Tα] = ∅ iff (ϕ, α) | = T ✓ U4 : [ϕ] = [ψ] and [α] = [β] ⇒ [ϕ ⋄ α] = [ψ ⋄ β]

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ ⇒ E3 : [ϕ♦Tα] = ∅ iff (ϕ, α) | = T ✓ U4 : [ϕ] = [ψ] and [α] = [β] ⇒ [ϕ ♦T α] = [ψ ♦T β] ✗ U5 : (ϕ ⋄ α) ∧ β | = ϕ ⋄ (α ∧ β) (not compatible with enforcement failure)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ ⇒ E3 : [ϕ♦Tα] = ∅ iff (ϕ, α) | = T ✓ U4 : [ϕ] = [ψ] and [α] = [β] ⇒ [ϕ ♦T α] = [ψ ♦T β] ✗ U5 : (ϕ ⋄ α) ∧ β | = ϕ ⋄ (α ∧ β) ⇒ E5 : if card([ϕ]) = 1 then (ϕ♦Tα) ∧ β | = ϕ♦T(α ∧ β)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ ⇒ E3 : [ϕ♦Tα] = ∅ iff (ϕ, α) | = T ✓ U4 : [ϕ] = [ψ] and [α] = [β] ⇒ [ϕ ♦T α] = [ψ ♦T β] ✗ U5 : (ϕ ⋄ α) ∧ β | = ϕ ⋄ (α ∧ β) ⇒ E5 : if card([ϕ]) = 1 then (ϕ♦Tα) ∧ β | = ϕ♦T(α ∧ β) ✗ U8 : [(ϕ ∨ ψ) ⋄ α] = [(ϕ ⋄ α) ∨ (ψ ⋄ α)] (not compatible with enforcement failure

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ ⇒ E3 : [ϕ♦Tα] = ∅ iff (ϕ, α) | = T ✓ U4 : [ϕ] = [ψ] and [α] = [β] ⇒ [ϕ ♦T α] = [ψ ♦T β] ✗ U5 : (ϕ ⋄ α) ∧ β | = ϕ ⋄ (α ∧ β) ⇒ E5 : if card([ϕ]) = 1 then (ϕ♦Tα) ∧ β | = ϕ♦T(α ∧ β) ✗ U8 : [(ϕ ∨ ψ) ⋄ α] = [(ϕ ⋄ α) ∨ (ψ ⋄ α)] ⇒ E8 if ([ϕ] = ∅ and [ϕ♦Tα] = ∅) or ([ψ] = ∅ and [ψ♦Tα] = ∅) then [(ϕ ∨ ψ)♦Tα] = ∅ else [(ϕ ∨ ψ)♦Tα] = [(ϕ♦Tα) ∨ (ψ♦Tα)]

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ ⇒ E3 : [ϕ♦Tα] = ∅ iff (ϕ, α) | = T ✓ U4 : [ϕ] = [ψ] and [α] = [β] ⇒ [ϕ ♦T α] = [ψ ♦T β] ✗ U5 : (ϕ ⋄ α) ∧ β | = ϕ ⋄ (α ∧ β) ⇒ E5 : if card([ϕ]) = 1 then (ϕ♦Tα) ∧ β | = ϕ♦T(α ∧ β) ✗ U8 : [(ϕ ∨ ψ) ⋄ α] = [(ϕ ⋄ α) ∨ (ψ ⋄ α)] ⇒ E8 if ([ϕ] = ∅ and [ϕ♦Tα] = ∅) or ([ψ] = ∅ and [ψ♦Tα] = ∅) then [(ϕ ∨ ψ)♦Tα] = ∅ else [(ϕ ∨ ψ)♦Tα] = [(ϕ♦Tα) ∨ (ψ♦Tα)] ✓ U9 : if card([ϕ]) = 1 then [(ϕ ⋄ α) ∧ β] = ∅ ⇒ ϕ ⋄ (α ∧ β) | = (ϕ ⋄ α) ∧ β

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Generalized Enforcement Postulates

✓ U1 : ϕ ♦T α | = α ∼ U2 : ϕ | = α ⇒ [ϕ ♦T α] = [ϕ] (optional : inertia) ✗ U3 : [ϕ] = ∅ and [α] = ∅ ⇒ [ϕ ⋄ α] = ∅ ⇒ E3 : [ϕ♦Tα] = ∅ iff (ϕ, α) | = T ✓ U4 : [ϕ] = [ψ] and [α] = [β] ⇒ [ϕ ♦T α] = [ψ ♦T β] ✗ U5 : (ϕ ⋄ α) ∧ β | = ϕ ⋄ (α ∧ β) ⇒ E5 : if card([ϕ]) = 1 then (ϕ♦Tα) ∧ β | = ϕ♦T(α ∧ β) ✗ U8 : [(ϕ ∨ ψ) ⋄ α] = [(ϕ ⋄ α) ∨ (ψ ⋄ α)] ⇒ E8 if ([ϕ] = ∅ and [ϕ♦Tα] = ∅) or ([ψ] = ∅ and [ψ♦Tα] = ∅) then [(ϕ ∨ ψ)♦Tα] = ∅ else [(ϕ ∨ ψ)♦Tα] = [(ϕ♦Tα) ∨ (ψ♦Tα)] ✓ U9 : if card([ϕ]) = 1 then [(ϕ ♦T α) ∧ β] = ∅ ⇒ ϕ ♦T (α ∧ β) | = (ϕ ♦T α) ∧ β

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 24 / 30

Enforcement is a kind of Update

Representation Theorem

Proposition (minimality)

U1 is implied by E3, E5 and E8. E3, U4, E5, E8, U9 constitute a minimal set.

Theorem ([Bisquert et al., 2013a])

∃ an operator ♦T : L × L → L satisfying E3, U4, E5, E8, U9 iff ∃ an assignment respecting T : G → a complete pre-order G s.t. [f(G) ♦T α] = { G1 ∈ [α] s.t. (G, G1) ∈ T and ∀G2 ∈ [α] s.t.(G, G2) ∈ T, G1 G G2 } [ϕ ♦T α] = ∅ if ∃G ∈ [ϕ] s.t. [f(G)♦Tα] = ∅

• G∈[ϕ][f(G) ♦T α]
• therwise

(“respecting T”= if (G, G1) ∈ T and (G, G2) ∈ T then G1 ≺G G2)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 25 / 30

Enforcement is a kind of Update

Representation Theorem (continued)

Enforcement postulates are more general than KM postulates If T = Γ × Γ then ♦T satisfies U2, E3, U4, E5, E8, U9 iff ⋄ satisfies U1, U2, U3, U4, U5, U8 and U9 Baumann’s enforcement by normal expansion = particular enforcement operator ♦T : L × L → L such that T = TB.

Baumann’s language : initial system = {ϕ ∈ Lon, card([ϕ]) = 1} enforced formulas = only conjunctions of positive literals of La where La and Lon propositional languages based only on a(x) and

• n on(x).
• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 26 / 30

Enforcement is a kind of Update

Summary

1

Framework

2

Towards Generalized Enforcement

3

Generalized Update Postulates

4

Concluding Remarks

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 27 / 30

Enforcement is a kind of Update

Conclusion

What did we do ? An extension of classical enforcement : ability to remove an argument generalize what can be enforced (any goal expressed in propositional logic) possibility to restrict the authorized changes :

◮ authorized changes = set of possible transitions

⇒ restriction to specific changes (e.g. additions or elementary

• perations)

⇒ restriction to arguments that are allowed to be added/removed.

claim : “enforcement is a kind of update”

◮ axiomatic approach ◮ kind of update more general than classical update (transition

constraints)

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 28 / 30

Enforcement is a kind of Update

Conclusion

What are we going to do ? Examples are in a simplified logical language

◮ focus only on changes about arguments ◮ fixed attack relation.

Results hold on any given propositional logic (with an f). ⇒ Choose a logic in which attacks are encoded. Our postulates concern changes in any kind of graphs ⇒ Find postulates more specific for argumentation dynamics :

◮ take into account the particularities of argumentation ◮ introduce semantics notions in the postulates

⇒ Counterpart of enforcement for revision instead of update.

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 29 / 30

Enforcement is a kind of Update

References I

Bisquert, P ., Cayrol, C., Dupin de Saint Cyr Bannay, F., and Lagasquie-Schiex, M.-C. (2013a). Axiomatic Approach of Enforcement in Argumentation : Enforcement in Argumentation is a kind of Update. Rapport de recherche RR- -2013-24- -FR, IRIT, Université Paul Sabatier, Toulouse. Bisquert, P ., Cayrol, C., Dupin de Saint Cyr Bannay, F., and Lagasquie-Schiex, M.-C. (2013b). Characterizing change in abstract argumentation systems. In Simari, G. and Fermé, E., editors, Trends in Belief Revision and Argumentation Dynamics, pages 1–30. College Publications, http ://www.collegepublications.co.uk/. Cayrol, C., Dupin de Saint Cyr, F., and Lagasquie-Schiex, M.-C. (2010). Change in abstract argumentation frameworks : Adding an argument. Journal of Artificial Intelligence Research, 38 :49–84. Katsuno, H. and Mendelzon, A. O. (1991). On the difference between updating a knowledge base and revising it. In Proc. of KR, pages 387–394.

• F. Dupin de Saint-Cyr

Sintelnet workshop: Believing, planning, acting, revising July 2013 30 / 30