Argumentation LogicsLecture 3:

Abstract argumentationpreferred semantics

Henry PrakkenChongqing

May 29, 2010

Contents Review of stable semantics

Definitions A problem

Preferred semantics Labelling-based Extension-based

Abstract argumentation: general remarks on semantics

Status of arguments: abstract semantics (Dung 1995)

INPUT: an abstract argumentation theory AAT = Args,Defeat

OUTPUT: An assignment of the status ‘in’ or ‘out’ to all members of Args So: semantics specifies conditions for

labeling the ‘argument graph’. Should capture reinstatement:A B C

Possible labeling conditions Every argument is either ‘in’ or ‘out’.

1. An argument is ‘in’ iff all arguments defeating it are ‘out’.

2. An argument is ‘out’ iff it is defeated by an argument that is ‘in’.

Works fine with:

But not with:

A B C

A B

Two solutions Change conditions so that always a unique status

assignment results

Use multiple status assignments:

and

A B C

A BA B

A B C

A B

A problem(?) with grounded semantics

We have: We want(?):

A B

C

D

A B

C

D

Multiple labellings

A B

C

D

A B

C

D

Stable status assignments (Below is AAT = Args,Defeat implicit) A stable status assignment is a partition of Args

into sets In and Out such that:1. An argument is in In iff all arguments defeating it are

in Out.2. An argument is in Out iff it is defeated by an

argument that is in In.

A is justified if A is In in all s.a. A is overruled if A is Out in all s.a. A is defensible if A is In in some but not all s.a.

Stable extensions Dung (1995):

S is conflict-free if no member of S defeats a member of S

S is a stable extension if it is conflict-free and defeats all arguments outside it

Now: S is a stable argument extension if (In,Out) is a

stable status assignment and S = In.

Proposition 4.3.4: S is a stable argument extension iff S is a stable extension

Stable status assignments:a problem

A stable status assignment is a partition of Args into sets In and Out such that:1. An argument is in In iff all arguments

defeating it are in Out.2. An argument is in Out iff it is defeated by

an argument that is in In.

A B

C

Stable status assignments:a problem

A stable status assignment is a partition of Args into sets In and Out such that:1. An argument is in In iff all arguments

defeating it are in Out.2. An argument is in Out iff it is defeated by

an argument that is in In.

A B

C

Stable status assignments:a problem

A stable status assignment is a partition of Args into sets In and Out such that:1. An argument is in In iff all arguments

defeating it are in Out.2. An argument is in Out iff it is defeated by

an argument that is in In.

A B

C

Stable status assignments:a problem

A stable status assignment is a partition of Args into sets In and Out such that:1. An argument is in In iff all arguments

defeating it are in Out.2. An argument is in Out iff it is defeated by

an argument that is in In.

A B

C

Stable status assignments:a problem

A stable status assignment is a partition of Args into sets In and Out such that:1. An argument is in In iff all arguments

defeating it are in Out.2. An argument is in Out iff it is defeated by

an argument that is in In.

A B

C

Status assignments A status assignment assigns to zero or more members of

Args either the status In or Out (but not both) such that:1. An argument is in In iff all arguments defeating it are in Out.2. An argument is in Out iff it is defeated by an argument that is in

In.

Let Undecided = Args / (In Out): A status assignment is stable if Undecided = .

In is a stable argument extension A status assignment is preferred if Undecided is -minimal.

In is a preferred argument extension A status assignment is grounded if Undecided is -maximal.

In is the grounded argument extension

A B

C D E

1. An argument is In iff all arguments defeating it are Out.2. An argument is Out iff it is defeated by an argument that is In.

Grounded s.a. minimise node colouring Preferred s.a maximise node colouring

Preferred extensions Dung (1995):

S is conflict-free if no member of S defeats a member of S S is admissible if it is conflict-free and all its members are

acceptable wrt S S is a preferred extension if it is -maximally admissible

Recall: S is a preferred (grounded) argument extension if (In,Out)

is a preferred (grounded) status assignment and S = In.

Proposition 4.3.13(16): S is a preferred (grounded) argument extension iff S is a preferred (grounded) extension

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Admissible?

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Admissible?

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Admissible?

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Admissible?

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Preferred? S is preferred if it is maximally admissible

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Preferred? S is preferred if it is maximally admissible

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Preferred? S is preferred if it is maximally admissible

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Grounded? S is groundeded if it is the smallest set s.t. A S iff S defends A

A B

C D E

S defends A if all defeaters of A are defeated by a member of S

S is admissible if it is conflict-free and defends all its members

Grounded? S is groundeded if it is the smallest set s.t. A S iff S defends A

A B

C D E

1. An argument is In if all arguments defeating it are Out.2. An argument is Out if it is defeated by an argument that is In.

F

Properties Every admissible set is included in a preferred

extension The grounded extension is unique Every stable extension is preferred (but not v.v.) There exists at least one preferred extension The grounded extension is a subset of all preferred

and stable extensions Every AAT without infinite defeat paths has a unique

extension (which is the same in all semantics) Every AAT without odd defeat cycles has a stable

extension ...

Self-defeating arguments again

Recall: A is justified if A is In in all s/p/g.s.a. A is overruled if A is Out in all s/p/g.s.a. A is defensible if A is In in some but not in

all s/p/g.s.a.

In grounded and preferred semantics self-defeating arguments are not always overruled

They can make that there are no stable extensions