Download - Argumentation Logics Lecture 3: Abstract argumentation semantics (3) Henry Prakken Chongqing May 28, 2010

Argumentation LogicsLecture 3:

Abstract argumentationsemantics (3)

Henry PrakkenChongqing

May 28, 2010

Contents

Review of grounded, stable and preferred semantics Labelling-based

Stable and preferred semantics Extension-based

Correspondence between labelling-based and extension-based semantics

Concluding remarks on semantics of abstract argumentation.

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’.

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’.

Produces unique labelling with:

But produces two labellings with:

A B C

A BA 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 assigns to all members

of Args either the status In or Out (but not both) such that :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.

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 status assignments:a problem

A stable status assignment assigns to all members of Args either the status In or Out (but not both) such that: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.

A B

C

Stable status assignments:a problem

A stable status assignment assigns to all members of Args either the status In or Out (but not both) such that: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.

A B

C

D

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

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 labelling Preferred s.a maximise node labelling

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 labelling Preferred s.a maximise node labelling

Correspondence between labelling-based and extension-

based semantics ofabstract argumentation

Bart Verheij (1996)Hadassah Jakobovits (1999)

Martin Caminada (2006)

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

Grounded extensions again

Dung (1995): Construct a sequence such that:

S0: the empty set Si+1: Si + all arguments in Args that are defended by Si

The endpoint of the sequence is the grounded extension

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

grounded status assignment and S = In.

Proposition 2.3.16: S is a grounded argument extension iff S is a grounded extension

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

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

stable status assignment and S = In.

Proposition 2.3.4: S is a stable argument extension iff S is a stable extension

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 argument extension if (In,Out) is a

preferred status assignment and S = In.

Proposition 2.3.13: S is a preferred argument extension iff S is a preferred 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



Preferred?S is preferred if it is maximally admissible

A B

C D E



Grounded?

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 defeat cycles of odd length has a

stable extension ...

Self-defeating arguments again

Recall (for preferred and stable semantics): A is justified if A is In in all s/p.s.a. A is overruled if A is Out in all s/p.s.a. A is defensible if A is In in some but not in

all s/p.s.a.

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

They can make that there are no stable extensions

A B

Which semantics is the “right” one?

Alternative semantics may each have their use in different contexts E.g. in criminal procedure the burden of

proof is on the prosecution, so grounded semantics with justified arguments is suitable.

Or in decision making a choice must be made between alternative ways to achieve one’s goals, so preferred semantics with defensible arguments is suitable.

Download - Argumentation Logics Lecture 3: Abstract argumentation semantics (3) Henry Prakken Chongqing May 28, 2010

Top Related