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L3: ABSTRACT ARGUMENTATION(more advanced concepts)

Dov M. Gabbay Massimiliano Giacomin

COURSE: INTRODUCTION TO FORMAL ARGUMENTATION

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• Abstract argumentation

- main argumentation semantics

- dialectical proof theories

- principles and properties

- extensions to the model and semantics

Plan of the lecture

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A growing set of properties (1)

• Semantics properties in Dung’s AF have been proposed (explicitly and/or implicitly) in several papers with several different aims, e.g.

• In defining a semantics, referring (at least implicitly) to properties is unavoidable, e.g.

- grounded extension: minimal complete- preferred extension: maximal admissible- semistable extensions: admissible maximizing the range

• Identifying a general schema for argumentation semantics

- [Baroni & Giacomin 2005]: SCC-recursiveness to capture most semantics in a general schema, exhibiting an incremental fashion

- [Liao et al. 2011]: the SCC-recursive schema is exploited topartially reuse extensions in case of modifications of the AF

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A growing set of properties (2)

• In [Baroni & Giacomin 2005], a formal notion of skepticism hasbeen introduced to compare argumentation semantics

• In [Baroni & Giacomin 2007], a number of principles has beenproposed to evaluate argumentation semantics

• In [Baroni et al. 201]: two properties have been introduced:- decomposability in order to analyze the semantics

behavior in terms of composition of local behaviors- transparency in order to investigate replacements of

partial argumentation frameworks

Explicit considerations of properties

HERE JUST SOME PROPERTIES

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DIRECTIONALITY

INTUITION

Given a set which does not receive attacks from outside, the evaluationof its arguments (through extensions) should not be affected by theexternal arguments

Definition

" AF, " U “unattacked set” of AF,

{(E Ç U) | E ÎℰS(AF)} = ℰS(AF )U

• The intersections of the extensions with an unattacked part of the AF are the same whatever is the remaining part of the AF and coincide with the extensions of the restricted AF

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Directionality: example with preferred semantics

da b

da b

da b

da b g

da b g

da b g

da b g

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Semistable semantics is not directional (1)

a b g

Example

ONLY ONE SEMISTABLE EXTENSION FOR THE UNATTACKED SET

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Semistable semantics is not directional (2)

a b g

d

Example

BUT ALSO THIS ONE IF d IS INTRODUCED!

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SCC-RECURSIVENESS

l The intersection of each global extension with each SCC satisfies a (relatively complex) constraint involving a general scheme and a semantics specific function

l Two main features:- incremental computation of extensions following SCCs- recursiveness

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Example with preferred semantics (1)

S1

S3

S5

S6

S7

S4

S2

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Example with preferred semantics (2)

S1

S3

S5

S6

S7

S4

S2

There is exactly one choice for E Ç S1

There is exactlyone choice for E Ç S2

(the empty set)

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Example with preferred semantics (3)

S1

S3

S5

S6

S7

S4

S2

RecursiveCall

RecursiveCall

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Example with preferred semantics (4)

S1

S3

S5

S6

S7

S4

S2

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Example with preferred semantics (5)

S5

Focus on the recursive call on S5*

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Example with preferred semantics (6)

S1

S3

S5

S6

S7

S4

S2

RecursiveCall

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Example with preferred semantics (7)

S1

S3

S5

S6

S7

S4

S2

S7

OR

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Directionality and SCC-Recursiveness

• Two properties slightly related:- DIRECTIONALITY:

The intersections of the extensions with an unattacked part (a set of SCCs) coincide with the extensions of the restricted AF

- SCC-RECURSIVENESS:The intersection of each global extension with each SCCsatisfies a (relatively complex) constraint. This in particularapplies to initial SCCs…

• However, neither of them entails the other:- IDEAL SEMANTICS is DIRECTIONAL BUT NOT SCC-RECURSIVE- STABLE SEMANTICS is NOT DIRECTIONAL BUT SCC-RECURSIVE

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Ideal semantics: directional/not SCC-recursive (1)

b

a

gWhateverelse

“Local” Ideal extension = Æ

DIRECTIONALITY entails that the global ideal extensiondoes not include alfa, beta, gamma!

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Ideal semantics: directional/not SCC-recursive (2)

b

a

g d e

Ideal extension = Æ

b

a

g d e

Ideal extension = {e}

Local informationfrom the first parts is the same

The second parts are the same …

but the outcome is different !

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Stable semantics: SCC-recursive / not directional (1)

a b g

“FIRST” STRONGLY CONNECTED COMPONENT:

the base function givestwo “local stable extensions”

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Stable semantics: SCC-recursive / not directional (2)

a b gPROPAGATING

the firstchoice

a b gPROPAGATING

the secondchoice

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Stable semantics: SCC-recursive / not directional (3)

a b g

a b gPROPAGATING

the secondchoice

BASE FUNCTIONRETURNS THE EMPTY VALUE!

ONE LOCAL EXTENSION “GETS LOST”, THERE IS ONE STABLE EXTENSION

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Stable semantics: SCC-recursive / not directional (4)

a b g

UNATTACKED SET: TWO STABLE EXTENSIONS

a b

a bThe intersection withthe global stable extensionsincludes this only!

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On the difference between directionality and SCC-Recursiveness (1)

a

e

a

e

a

e

Whateverelse

DIRECTIONALITY

• Refers to DIFFERENT AFs• INDIFFERENCE TO CHANGE:

what we compute locally ispreserved at the global level

• BUT does not embed a notionof incremental construction:

- does not ensure thatinformation embedded inthe local extensions issufficient to proceed!

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On the difference between directionality and SCC-Recursiveness (2)

SCC-Recursiveness

• Refers to ONE AF• EMBEDS a notion of incremental computation:

local extensions embed all necessary info• BUT does not require uniform outcomes on

equal restrictions of distinct frameworks:- it is true that we can start with some

semantics-specific local computations- but there is no guarantee that the local

extensions are preserved at the global level

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On the difference between directionality and SCC-Recursiveness (2)

SCC-Recursiveness

• Refers to ONE AF• EMBEDS a notion of incremental computation:

local extensions embed all necessary info• BUT does not require uniform outcomes on

equal restrictions of distinct frameworks:- it is true that we can start with some

semantics-specific local computations- but there is no guarantee that the local

extensions are preserved at the global level

• IF a semantics is SCC-recursive and its base function is UNIVERSALLY DEFINED then the semantics is directional

Intuitively: by SCC-Recursiveness we know that the choices for theunattacked set correspond to the extensions of the restriction, andall of them can be “extended” to a global extension

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Evaluation of existing semantics (partial)

Grounded,Preferred,CF2

Ideal Stable Semistable

DIRECTIONALITY Yes Yes No NoSCC_RECURSIVENESS Yes No Yes No

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Properties at work: dynamic computation

• Exogenous addition/deletion of arguments and attacks

leading from the original AF to a modified AF*

• The division-based method: rather than computing the extensions of

the AF* from scratch, partial reuse the extensions of the AF:

- identify the unaffected and affected part of the AF

- the unaffected part is a set of SCCs (unattacked set) of AF:

project the extensions of the AF to the unaffected part,

obtaining the “local extensions”. (DIRECTIONALITY)

- in AF*, the unaffected part is a set of SCCs that precede (or are

incomparable with) the SCCs in the affected part:

extend the “local extensions” to the affected part,

obtaining the “global” extensions of AF*. (SCC-RECURSIVENESS)

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An example with preferred semantics (1)

a b

d

g e

z

OLD AF: PREFERRED EXTENSIONS

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An example with preferred semantics (2)

a b

d

g e

z

OLD AF: PREFERRED EXTENSIONS

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An example with preferred semantics (3)

a b

z

NEW AF

d

g e

h

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An example with preferred semantics (4)

a b

z

NEW AF

d

g e

hUnaffected

partAffected

part

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An example with preferred semantics (5)

a b

z

PROJECTION: LE1

d

g e

Unaffectedpart

Affectedpart

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An example with preferred semantics (6)

a b

z

PROJECTION: LE2

Unaffectedpart

d

g e

Affectedpart

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An example with preferred semantics (7)

a b

z

EXTENSION of LE1: FIRST CHOICE

d

g e

hUnaffected

partAffected

part

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An example with preferred semantics (8)

a b

z

EXTENSION of LE1: SECOND CHOICE

d

g e

hUnaffected

partAffected

part

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An example with preferred semantics (9)

a b

z

EXTENSION of LE2: UNIQUE CHOICE

d

g e

hUnaffected

partAffected

part

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Ordering semantics wrt skepticism

A two steps definition

A basic skepticism relation between sets of extensionsE

E1 E2E E2 is ‘at least as committed as’ E1

A skepticism relation between semantics induced by

S1 S2 [S2 is at least as committed as S1]

iff " AF ℰS1(AF) ℰS2(AF)

S E

S

E

[Baroni & Giacomin, 2009]

(a labelling-based counterpart can be defined in a similar way)

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Skepticism comparison: skeptical viewpoint on justification

• Comparison between two extensions E1 and E2:

• Comparison between two sets of extensions:

E1Í E2

" E2 ÎE2 $ E1 ÎE1 : E1Í E2:E2E1

E1’

E1’’E2’

E2’’’

E1 can includeunrelated extensions

• Comparison between two semantics:

�E\+

S1 �S\+ S2 i↵ 8AF 8E2 2 ES2(AF )9E1 2 ES1(AF ) : E1 ✓ E2

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Skepticism comparison: credulous viewpoint on justification

• Comparison between two extensions E1 and E2:

• Comparison between two sets of extensions:

E1Í E2

" E1 ÎE1 $ E2 ÎE2: E1Í E2:E2E1

E1’

E1’’E2’

E2’’’

E2 can includeunrelated extensions

• Comparison between two semantics:

�E[+

S1 �S[+ S2 i↵ 8AF 8E1 2 ES1(AF )9E2 2 ES2(AF ) : E1 ✓ E2

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Hasse diagrams

GR

ID

SST=ST

PR

GR

ID

PR

SST=ST

�S[+

�S\+

(skeptical perspective) (credulous perspective)

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• Abstract argumentation

- main argumentation semantics

- dialectical proof theories

- principles and properties

- extensions to the model and semantics

> alternative semantics

> gradual/ranking-based semantics

> numbers assigned to arguments/attacks

> extending the model: attacks to attacks, support

Plan of the lecture

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CF2 semantics: motivation

Preferred/stable/semistable semantics and cycles

ba

ba

ba

a

g

b

a

g

b

A different treatment for even and odd-length cycles.Is it just a matter of symmetry and elegance?

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Preferred/Semistable Semantics and cycles

ba d1 d2

VS

ℰPR(AF) =

{{a, d1}, {a, d2},

{b, d2} }

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Preferred/Semistable Semantics and cycles

b

a

g d1 d2

VS

ba d1 d2

VS

ℰPR(AF) =

{{a, d1}, {a, d2},

{b, d2} }

ℰPR(AF) = {{d2}}

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Preferred/Semistable Semantics and cycles

b

a

g d1 d2

d1 d2ga

b

d

VS

ba d1 d2

VS

ℰPR(AF) =

{{a, d1}, {a, d2},

{b, d2} }

ℰPR(AF) = {{d2}}

ℰPR(AF) =

{{a, g, d2},

{b, d, d1}, {b, d, d2} }

NOTE: grounded semantics yields the empty set in all cases

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An instantition (1)

Rob says Jones unrel.

Jones unreliable

Smith says Rob unrel.

Rob unreliable

Jones says Smith unrel.

Smith unreliable

Smith says it’s raining

It’sraining

Bob says it’s not raining

It’s notraining

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An instantiation(2)

Rob says Fred unrel.

Fred unreliable

Smith says Rob unrel.

Rob unreliable

Jones says Smith unrel.

Smith unreliable

Smith says it’s raining

It’sraining

Bob says it’s not raining

Fred says Jones unrel.

Jones unreliable

It’s notraining

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Preferred Semantics and Floating Arguments again…

b

a

g d

VS

b

a

g d f

[ two preferred extensions]

[empty set is the unique preferred extension]

NB: same behavior for semistable semantics, stable semantics clashes, grounded and ideal semantics yields the empty set in both cases

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Other semantics in the previous example

++ Grounded semantics yields the empty set in all cases(homogeneous treatment of odd and even length cycles)

-- Stable semantics clashes in the odd-length cycle cases

-- Semistable semantics yields the same extensions as preferredsemantics (non homogeneous treatment)

-- Ideal semantics yields the empty set in the even-length cyclecases, a unique justified argument in the odd-length ones(non homogeneous treatment)

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CF2 semantics: the basic idea

b

a

g

b

a

g

b

a

g

b

a

g

Maximal conflict-free sets (NOT ADMISSIBLE!!!)

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CF2 semantics: the definition

EÎ ℰCF2(AF) iff:

- E Î MCF(AF) if |SCCSAF| = 1

- " S Î SCCSAF(EÇS) Î ℰCF2(AF UP_AF(S,E)) otherwise

S

UP_AF(S,E)

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b

a

g

b

a

g

b

a

g

b

a

g

Maximal conflict-free sets

CF2 semantics and odd-length cycles (1)

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b

ag f1 f2

Yields several extensionsÞ all arguments not justified

in both cases

f1 f2gab

d

{g,f2}, {a,f1}, {a,f2}, {b,f1}, {b,f2}

{a,g,f2}, {b,d,f1}, {b,d,f2}

CF2 semantics and odd-length cycles (2)

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Floating arguments with a three-length cycle

b

ag d f

Extensions: {g,f}, {a,f}, {b,f}

b

ag d f

b

ag d f

b

ag d f

Defeat status

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A possible semantics team

GROUNDED

PREFERRED

STABLE SEMISTABLE

IDEAL

STAGE

PRUDENTROBUST

CF2

SUSTAINABLE

RESOLUTION

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• Abstract argumentation

- main argumentation semantics

- dialectical proof theories

- principles and properties

- extensions to the model and semantics

> alternative semantics

> gradual/ranking-based semantics

> numbers assigned to arguments/attacks

> extending the model: attacks to attacks, support

Plan of the lecture

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Epistemic vs Practical reasoning

EPISTEMIC REASONING: reasoning about what to believe

gb

b: it will be dry in London (there will be sunshine according to BBC)

g: it will be wet in London (there will be rain according to CNN)

skeptical acceptance appropriate (both arguments undecided)

PRACTICAL REASONING: reasoning about what to do

b: give aspirin (to prevent blood clotting)

g: give chlopidrogel (to prevent blood clotting)

credulous acceptance more appropriate (choose one argument!)

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Value-based argumentation frameworks (1)

ba c

VAF = <A, ®, V, val, P>

V1 V2 Audience a: total ordering >a

between values

• Arguments promote social values• An attack a1® a2 succeeds only if it is not the case that

val(a2) >a val(a1)• A specific audience expresses a total preference order between values

[Bench-Capon, 2003]

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Value-based argumentation frameworks (2)

ba c

VAF = <A, ®, V, val, P>

V1 V2 Audience a: total orderingsbetween values

[Bench-Capon, 2003]

EXAMPLE (Modgil 2010)

a: give aspirin (to prevent blood clotting)b: give chlopidrogel (to prevent blood clotting)c: chlopidrogel is prohibitively expensiveV1: improving the (specific) patient’s healthV2: cost (save money for all patients)

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Value-based argumentation frameworks (3)

ba c

V1 V2

Audience V1 (patient’s health)>V2 (cost)

ba c

V1 V2

Audience V2 (cost)>V1 (patient’s health)

Aspirin and Chlopidrogelequally preferable(choose a preferred extension)

(Less expensive) Aspirin is the unique option

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Modelling attacks to attacks (1)

Representing preferences between arguments at the argument level⇒ possibility of arguing about (and reasoning with) preferences

ABA: BBC forecast sunshine ⇒ dry in LondonB: CNN forecast rain ⇒ wet in LondonC: I feel BBC as more trustworthy than CNN

C

C expresses the preference of A wrt B ⇒ it must attack the attack from B to A

NB: C does not attack B, in fact - if A was not the case, B should be accepted (and C is still believed)- the same if an independent argument D attacks A

(e.g. BBC did not really forecast sunshine)

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Modelling attacks to attacks (2)

ABA: BBC forecast sunshine ⇒ dry in LondonB: CNN forecast rain ⇒ wet in LondonC: I feel BBC as more trustworthy than CNNC’: statistically CNN more accurate than BBC

C C’ [C and C’ express contradictory conclusions⇒ they attack each other]

…EXAMPLE CONTINUED…

Approaches to model attacks to attacks

• EAF [Modgil, 2009]• AFRA [Baroni et al., 2011]• Meta-argumentation [Boella et al. 2009, Gabbay et al. 2009]

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A B

V2>V1 V1>V2

A1: V1>V2

(V1) (V1)

From VAFs to AFs with attacks to attacks (1)

A2: V2>V1

BACK TO THE EXAMPLE

C(V2)

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A B

V2>V1 V1>V2

A1: V1>V2

(V1) (V1)

A2: V2>V1

FIRST EXTENSION (AUDIENCE)

C(V2)

From VAFs to AFs with attacks to attacks (2)

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A B

V2>V1 V1>V2

A1: V1>V2

(V1) (V1)

A2: V2>V1

SECOND EXTENSION (AUDIENCE)

C(V2)

From VAFs to AFs with attacks to attacks (3)

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Other extensions to Dung’s model

• Preference Argumentation Framework

• Modelling collective attack (e.g. conjunctive attack)

• Adding a support relation besides attack

• Modelling gradual attacks

• Integrating AF with

- probability

- possibility

- fuzziness

• …