Change in abstract bipolar argumentation systems

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Change in abstract bipolar argumentation systems

Claudette Cayrol, Marie-Christine Lagasquie-Schiex

To cite this version:

Claudette Cayrol, Marie-Christine Lagasquie-Schiex. Change in abstract bipolar argumentation

sys-tems. [Research Report] IRIT-2015-02, IRIT - Institut de recherche en informatique de Toulouse.

2015. �hal-02884082�

Claudette Cayrol,

M-Christine Lagasquie-S hiex

IRIT, Université PaulSabatier,

118 route de Narbonne,31062 Toulouse, Fran e { ayrol,lagasq}irit.fr

Te h. Report IRIT RR- -2015- -02- -FR

An argumentationsystem anundergo hanges(additionorremovalof arguments, ad-dition or removal of intera tions), parti ularly in multiagent systems. In this paper, we areinterestedin the hange on erningabstra t bipolarargumentationsystems,i.e. argu-mentationsystemsusingtwokindsofintera tion: atta ksand supports. Weproposesome hara terizationsof this hange thatuseand extendpreviousresultsdenedin the aseof Dungabstra t argumentationsystems.

1 Introdu tion 1 2 Abstra t bipolar argumentation system 2 2.1 Abstra targumentation system . . . 2 2.2 Abstra tbipolar argumentation system. . . 3

3 Dynami s inargumentation systems 5

4 Denition of a hange operation taking into a ount support 7 5 Chara terizing the addition of an argument and a support 9 5.1 Case ofan added supportedargument . . . 10 5.2 Case ofan added supporting argument . . . 10

6 Con lusion and future works 11

Themainfeatureofargumentation istheabilitytodealwith in ompleteand/or ontradi tory information, espe iallyfor reasoning

[Dung,

1995; Amgoud andCayrol, 2002℄.

Moreover, argu-mentation an be usedto formalize dialoguesbetween severalagentsbymodeling theex hange of arguments in, e.g., negotiation between agents

[

Amgoud et al., 2000 ℄

. An argumentation system(ASfor short) onsistsof a olle tion ofarguments intera ting withea hother through a relation ree ting oni ts between them, alled atta k. The issue of argumentation is then to determine a eptable sets of arguments (i.e., sets able to defend themselves olle tively while avoiding internal atta ks), alled extensions, and thus to rea h a oherent on lusion. Another formof analysisof anAS isthestudy of the parti ular statusof ea hargument based on its membership to theextensions. Formal frameworks have greatly eased themodeling and study ofAS. Inparti ular, theframework of

[Dung,1995℄

allows for abstra ting from the  on- rete meaning of thearguments and relies only onbinary intera tions thatmay exist between them. Thisapproa h enables the userto fo usonother aspe tsof argumentation, in luding its dynami side. Indeed, inthe ourse of a dis ussion or due to the a quisition of new pie es of information,anAS anundergo hanges su hasthe additionofa newargument ortheremoval of an argument onsidered asillegal. Thisis parti ularly important for dialogs ina multiagent system sin e it is unrealisti to onsider that the argumentation system ree ting the dialog an be stati ally dened. Thus, it is interesting to study these hanges, to hara terize them bygivingproperties des ribinga hange operation andtoprovide onditions underwhi h these propertieshold. Thishasbeendoneinseveralpapers,espe ially

[Bisquert etal.,

2013℄,

forDung AS withonlyatta ks.

In thispaper,we areinterested intheextension ofthis work tobipolar AS(BASfor short), i.e. AS augmentedwitha se ondkind of intera tion, thesupportrelation. This relation repre-sents a positive intera tion between arguments and has been rst introdu ed by

[

Kara apilidis and Papadias, 2001; Verheij,

2003℄. In

[Cayrol

and Lagasquie-S hiex, 2005℄,

the support rela-tion is left general so that the resulting bipolar framework keeps a high level of abstra tion. However thereis nosingle interpretationof the support, and anumber ofresear hers proposed spe ializedvariantsofthesupportrelation: dedu tivesupport

[Boella etal., 2012℄, ne essary sup-port [

NouiouaandRis h,2010;NouiouaandRis h,2011 ℄

,evidentialsupport [

OrenandNorman, 2008; Oren et al., 2010

℄

. Ea h spe ialization an be asso iated with an appropriate modelling using appropriate omplex atta ks. These proposalshave been developed quite independently, based on dierent intuitions and with dierent formalizations.

[Cayrol

and Lagasquie-S hiex, 2013

℄

presentsa omparative studyinordertorestatethese proposalsina ommon setting,the bipolar argumentation framework. The idea is to keep the original arguments, to add omplex atta ks dened bythe ombination oftheoriginal atta ks and thesupports,and to modifythe lassi al notions of a eptability. An important ontribution of

[

Cayrol and Lagasquie-S hiex, 2013℄

istohighlight akindofdualitybetweenthededu tive andthene essaryinterpretationsof support, whi hresults ina dualityinthemodelling by omplex atta ks. Handling supportis a growing on ern:

[

Polbergand Oren,2014 ℄

givesa translation between ne essarysupports and evidential supports;

[Prakken, 2014℄

proposes ajusti ation of thene essarysupportusing the notionofsubarguments;

[Nouioua,2013℄

studiesanextensionofthene essarysupport;

[Gabbay , 2013

℄

givesalogi alstudy ofbipolarsystems; [

Cohenetal.,2014 ℄

proposesageneralframework for taking into a ount re ursive atta ks and supports. However, there is no work on erning thestudyofthe dynami sofabipolarASwhileitisanessentialissueformodellingthea tions of theparti ipants toa multiagent system:

Journalist

J

1

(Argument

a

):

I

isimportant, we must publish it;

Journalist

J

2

(Argument

b

):

I

is about a person

X

, it is forbidden to publish without the agreement of the on erned person and

X

disagrees withthe publi ation;

Journalist

J

1

(Argument

c

):

X

isa publi person (she is the PrimeMinister); in this ase, heragreement is notmandatory;

Journalist

J

2

(Argument

d

): However, I have heard about X'sresignation;

Journalist

J

3

(Argument

e

): Inow understand whyCNNhas announ ed yesterday the post-ponement of the Coun il of Ministers;

Journalist

J

4

(Argument

f

): However, yesterday was April Fools' Day; so CNN news an-noun ed yesterday are not reliable.

Thisexample illustrates a typi al situationbetween agents that ex hangeargumentsin order to take a de ision (here, publish or not publish information

I

). In this dialog, one an see arguments (here, informal arguments orresponding to pie es of dialog), atta ks (for instan e Argument

b

atta ks Argument

a

), supports (between Argument

d

and Argument

e

); and the dynami s of argumentation is illustrated by the dynami s of the dialog: at ea h step of the dialog, the global argumentation system evolves (here, by the addition of an argument and an intera tion).

In this paper, we dene the update of BAS and hara terize it in a spe ial ase: a BAS redu ed to an AS that is hanged by the introdu tion of a new argument that intera ts with another argument using supports. Su han update isrealized usinga ombinationof theworks of both domains(bipolar argumentation and dynami s ofargumentation).

Some ba kground is given in Se tion 2 for AS and BAS, and in Se tion 3 for hange op-erations. Then Se tion 4 proposes a hange operation on erning a BAS. Chara terizations of thisnew hange operationarepresented inSe tion 5. Finally,Se tion 6 on ludesand suggests perspe tivesof our work. The proofsof our results aregiven inAppendix A.

2 Abstra t bipolar argumentation system

Thebipolarargumentation framework extendsDung's argumentation framework.

2.1 Abstra t argumentation system

Dung's abstra t framework onsists of a set of arguments and only one type of intera tion between these arguments, these intera tions representing atta ks.

Def. 1 (Dung AS) ADungargumentation system(AS,forshort) isapair

hA, Ri

where

A

is a niteandnon-emptyset of argumentsand

R

isabinaryrelationover

A

(a subsetof

A

× A

), alled the atta krelation.

An argumentation system an be represented by a dire tedgraph denoted by

G

, alledthe intera tion graph, in whi h nodes represent arguments and edges are dened by the atta k relation:

∀a, b ∈ A

,

aRb

is representedby

a

6→ b

.

Def. 2 (Admissibility) GivenAS

= hA, Ri

and

S

⊆ A

,

• S

is oni t-free in AS if and only if (i for short) there are no arguments

a, b

∈ S

, su h that (s.t. for short)

aRb

.

• a ∈ A

is a eptable in AS with respe t to(wrt for short)

S

i

∀b ∈ A

s.t.

bRa

,

∃c ∈ S

s.t.

cRb

.

F

denotes the hara teristi fun tion of AS dened by

∀S ⊆ A

,

F(S) = {x

s.t.

x

is a eptable in ASwrt

S}

.

• S

isadmissible inAS i

S

is oni t-free andea h argument in

S

isa eptable inAS wrt

S

.

Standardsemanti s introdu edbyDung(preferred, stable,grounded)enableto hara terize admissible sets of arguments that satisfy a form of optimality (see

[Baroni et al.,

2011℄ for a surveyof semanti s inabstra targumentation systems).

Def. 3 (Extensions) Given AS

= hA, Ri

and

S

⊆ A

,

• S

isa preferredextension of AS i it isa maximal (wrt

⊆

) admissible set in AS.

• S

isastableextension ofASiit is oni t-free andforea h

a

6∈ S

,there is

b

∈ S

s.t.

bRa

.

• S

isthegroundedextension of AS iit isthe least xpoint of

F

.

Ex. 2 Let AS be dened by

A

= {a, b, c, d, e}

and

R

= {(a, b)

,

(b, a)

,

(b, c)

,

(c, d)

,

(d, e)

,

(e, c)}

andrepresented bythe followinggraph. There are twopreferred extensions(

{a}

and

{b, d}

),one stable extension (

{b, d}

)and the grounded extension

= ∅

.

a

b

c

d

e

/ / / / / /

Thestatusofanargumentisdetermined byitsmembershiptotheextensionsofthesele ted semanti s: e.g.,anargumentisskepti allya epted(resp.  redulously)ifitbelongstoallthe extensions(resp. atleastto oneextension)andreje ted ifitdoesnotbelongtoanyextension.

Some interesting propertieshave been identied: Prop. 1

[Dung, 1995℄

1. There is at least one preferred extension, always a unique grounded extension, while there maybe zero, one or several stable extensions.

2. Ea h admissible set is in luded in a preferred extension.

3. Ea h stable extension isa preferred extension, the onverse isfalse. 4. The grounded extensionis in luded in ea h preferred extension.

5. Ea h argument whi h is not atta ked belongs to the grounded extension (hen e to ea h pre-ferred andto ea h stable extension).

6. If

R

isnite,thenthegroundedextension anbe omputedbyiterativelyapplyingthefun tion

F

fromthe empty set.

7. If

A

isnon empty, then a stable extension isalways non empty. Prop. 2

[Dunne

andBen h-Capon,2001; Dunne andBen h-Capon, 2002℄

1. If

G

ontains no y le, then

hA, Ri

has a unique preferred extension, whi h is also the grounded extensionand the unique stable extension.

2. If

{}

is theunique preferred extension of

hA, Ri

, then

G

ontains an odd-length y le. 3. If

hA, Ri

has no stable extension,then

G

ontains an odd-length y le.

4. If

G

ontains no odd-length y le, then preferred andstable extensions oin ide. 5. If

G

ontains no even-length y le, then

hA, Ri

has a unique preferred extension.

2.2 Abstra t bipolar argumentation system Theabstra tbipolarargumentationframeworkpresentedin

[Cayrol

andLagasquie-S hiex, 2010℄ extendsDung'sframeworkinordertotake into a ount bothnegativeintera tionsexpressedby the atta k relation and positive intera tions expressed by a support relation (see

[

Amgoud et al.,

2008℄

Def. 4 (BAS) A bipolar argumentation system (BAS, for short) is a tuple

hA, R

att

, R

sup

i

where

A

is a nite and non-empty set of arguments,

R

att

is a binary relation over

A

alled the atta krelation and

R

sup

is a binaryrelation over

A

alled the supportrelation.

A BAS an still be represented bya dire ted graph

G

b

alled the bipolar intera tion graph, withtwo kindsof edges. Let

a

and

b

∈ A

,

aR

att

b

(resp.

aR

sup

b

) means that

a

atta ks

b

(resp.

a

supports

b

) anditis representedby

a

6→ b

(resp. by

a

→ b

).

Among the dierent variantsdened for interpreting a supportbetween arguments,

[Boella et al., 2012

℄

proposed the notion of dedu tive support. This notion is intended to enfor e the following onstraint: If

bR

sup

c

then the a eptan e of

b

implies the a eptan e of

c

, and as a onsequen e the non-a eptan e of

c

implies the non-a eptan e of

b

. The support used in Example1 anbe onsideredasadedu tiveone(If

X

hasresignedthentheCoun ilofMinisters mustbepostponed):

Ex.1 ( ont'd)The bipolar argumentation system orresponding to the editorial board an be represented by:

f

/

e

d

/

c

/

b

/

a

Then, in order to ompute semanti s of a BAS, one of the main proposals is to translate the BAS into an AS expressing the new atta ks due to the presen e of supports (this kind of attening is studied for instan e in

[Gabbay ,

2013℄).

In the ase of dedu tive support, two kinds of atta k an be added. The rst one, alled mediated atta k, orresponds to the ase when

bR

sup

c

and

aR

att

c

: the a eptan e of

a

implies the non-a eptan e of

c

and sothe non-a eptan e of

b

:

Def. 5 (Mediated atta k)

[Boella etal.,

2012℄

Let BAS

= hA, R

att

, R

sup

i

. There is a medi-atedatta kfrom

a

to

b

i thereisa sequen e

a

1

R

sup

. . . R

sup

a

n−1

, and

a

n

R

att

a

n−1

,

n

≥ 3

,with

a

1

= b

,

a

n

= a

.

M

R

sup

R

att

denotesthe set of mediated atta ks generated by

R

sup

on

R

att

.

Moreover,the dedu tiveinterpretationofsupportjustiestheintrodu tionofanotheratta k ( alled supported atta k in

[Cayrol and Lagasquie-S hiex, 2010℄): if

aR

sup

c

and

cR

att

b

, the a eptan eof

a

impliesthea eptan e of

c

and the a eptan eof

c

impliesthenon-a eptan e of

b

;so, thea eptan eof

a

impliesthe non-a eptan e of

b

.

Def. 6 (Supported atta k)

[Cayrol

and Lagasquie-S hiex, 2010℄

Let BAS

= hA, R

att

, R

sup

i

. Thereisa supportedatta kfrom

a

to

b

ithereisasequen e

a

1

R

sup

. . . R

sup

a

n−1

R

att

a

n

,

n

≥ 3

, with

a

1

= a

,

a

n

= b

.

S

Rsup

R

att

denotesthe set of supported atta ks generated by

R

sup

on

R

att

. So,thededu tiveinterpretationofsupportprodu esnewkindsofatta k,from

a

to

b

,inthe following ases:

Supportedatta ks: Mediatedatta ks:

a

...

c

/

b

b

...

c

a

/ By iterating the onstru tion, d-atta ks an bedened: Def. 7 (d-atta ks)

[

Cayrol andLagasquie-S hiex, 2013 ℄

Let BAS

= hA, R

att

, R

sup

i

with

R

sup being a set of dedu tivesupports. There existsa d-atta k from

a

to

b

i

•

either

aR

att

b

, or

aS

R

sup

R

att

b

, or

aM

R

sup

R

att

b

(Basi ase),

•

orthere existsanargument

c

s.t. there isasequen e ofsupports from

a

to

c

and

c

d-atta ks

•

or there exists an argument

c

s.t.

a

d-atta ks

c

andthere is a sequen e of supports from

b

to

c

(Case 2).

D

Rsup

_R

att

denoted the set of d-atta ks generated by

R

sup on

R

att .

hA, D

Rsup

R

att

i

is alled the de-du tive asso iatedDung ASof BASand denoted by AS

BAS .

Ex.1 ( ont'd) The dedu tive asso iated Dung AS an be represented by (a mediated atta k appears from

f

to

d

):

f

/

e

d

/

c

/

b

/

a

/

Then, in this system, using for instan e the preferred semanti s, one an on lude to the a eptability of

a

(so the information

I

will be published).

Note that if

R

sup

is redu edto a singleton

(a, b)

, Case 1 and Case 2of Denition 7do not apply. In this ase, the atta k

(a, c)

is added in AS

BAS

i

(b, c) ∈ R

att

(this is a supported atta k) and the atta k

(c, a)

isadded inAS

BAS

i

(c, b) ∈ R

att

(thisisa mediatedatta k). TurningBASinto AS

BAS

enables to onsiderthe semanti s dened byDung. Moreover,the rststep leading to addnew atta ks, itfallswithin worksabout dynami s of AS.

3 Dynami s in argumentation systems

When studying argumentation dynami s, an important issue is to save omputation, that is to reuse as far as possible previous omputations arried out in the original argumentation system. This issue hasbeen extensively dis ussed in

[

Bisquert et al., 2013 ℄

with the following methodology: Atypologyof hangeoperationshasbeenproposedandtheimpa tofea h hange operationon the omputation of theextensions hasbeen studied. So,the work of

[Bisquert et al.,2013

℄

isparti ularly suitable forour purposeandeasily adaptable. 1

Inthis paper, following Example 1, we use the hange operations orresponding to eitherthe addition ofan argument and theintera tions (onlyatta ks) involving it, or the addition ofsome intera tions:

Def. 8 (Addition in an AS) Let AS

= hA, Ri

. Two hange operations are onsidered: 1. Let

z

be an argumentand

I

z

be aset ofintera tionss.t.

I

z

⊆ (A × {z}) ∪ ({z} × A)

. Adding

z

and

I

z

is a hange operation, denoted by

⊕

z

I

z

, providing a new system s.t.:

⊕

z

I

z

hA, Ri =

hA ∪ {z}, R ∪ I

z

i

.

2. Let

I

be a set of intera tions s.t.

I ⊆ (A × A)

and

I ∩ R = ∅

. Adding

I

is a hange operation, denoted by

⊕

I

, providinga new system s.t.:

⊕

I

hA, Ri = hA, R ∪ Ii

.

The systemresultingof a hange, denoted byAS

′

_{= hA}

′

_{, R}

′

_i

,willberepresented by thegraph

G

′

. In ea h ase, given a semanti s, the set of extensions of AS (resp. AS

′

) is denoted by

E

(resp.

E

′

),with

E

1

, . . . ,

E

n

(resp.

E

′

1

, . . . ,

E

n

′

) standing for theextensions. We onsider thesame semanti s before andafter the hange.

The impa t of a hange operation has been studied in

[Bisquert et al.,

2013℄

through the notion of hange property that an be seen as a set of pairs

(G, G

′

₎

, where

G

and

G

′

are argu-mentation graphs. Here we justre allsome ofthese properties.

1

Otherworks ouldbe onsideredforaddressingtheissueofin remental omputationinadynami ontext. [Baroni

et al., 2014℄

for instan epresentsa more general approa hdealing with modularity inabstra t argu-mentation,basedonthepartitionof anargumentationframeworkinintera tingsubframeworks. However, the appli ationtoourpurposeisnotstraightforwardandrequiresfurtherinvestigation.

of an AS that are aused by a hange operation. For thatpurpose, a partition based on three possible ases of evolutionof thesetof extensions,hasbeen dened in

[Bisquert et al.,

2013℄:

•

theextensive ase, inwhi h the number ofextensions in reases,

•

therestri tive ase, inwhi hthe numberof extensionsde reases,

•

the onstant ase, inwhi hthenumber ofextensionsremains thesame.

Forea h ase, numeroussub- ases areproposedanddenoted byaletter (

e

for theextensive ase,

r

fortherestri tive aseand

c

for the onstant ase) subs riptedbytheexpression

γ

− γ

′

, where

γ

(resp.

γ

′

) des ribesthe set of extensions before (resp. after) the hange. Thus

γ

and

γ

′

an be:

• ∅

: theset ofextensions isempty,

• 1e

: the setofextensions isredu ed to oneemptyextension,

• 1ne

: theset ofextensions isredu ed to onenon-empty extension,

• k

(resp.

j

): the setof extensions ontains

k

(resp.

j

) extensionss.t.

1 < k

(resp.

1 < j < k

: notethatthe symbol

j

is usedonlyifthesymbol

k

belongs also totheexpression

γ

− γ

′

). Forinstan e, thenotation

e

∅

−1ne

meansthatthe hange in reasesthenumberofextensions (so it is an extensive ase), with no initial extension (

∅

) and one non-empty nal extension (

1ne

).

Nevertheless, some spe ial sub- ases of the onstant ase are denoted by another method sin etheyarebasedonnotionsdistin t fromtheemptinessorthenumberoftheextensions; for these sub- ases, the subs ript is repla ed by a qualier. For instan e, the - onservative ase des ribesthe ase where theextensionsremainun hanged after the hange.

Here is the formal denition of these hanges. First, we study the ase in whi h a hange in reases (resp. de reases)the numberofextensions, alledextensive (resp. restri tive) hange. Def. 9 (Extensive and Restri tive hanges) The hange from

G

to

G

′

is extensive (resp. restri tive) i

|E| < |E

′

_|

(resp.

|E| > |E

′

_|

). 2 The sub- ases of extensive hanges from

G

to

G

′

are: 1.

e

∅

−1ne

i

|E| = 0

and

|E

′

_{| = 1}

, with

E

′

_{6= ∅}

. 2.

e

∅

−k

i

|E| < |E

′

_|

,

|E| = 0

and

|E

′

_{| > 1}

. 3.

e

1e−k

i

|E| < |E

′

_|

and

|E| = 1

,with

E = ∅

. 4.

e

1ne−k

i

|E| < |E

′

_|

and

|E| = 1

, with

E 6= ∅

. 5.

e

j−k

i

1 < |E| < |E

′

_|

.

The sub- ases of restri tive hanges from

G

to

G

′

are: 1.

r

1ne−∅

i

|E| = 1

, with

E 6= ∅

, and

|E

′

_{| = 0}

. 2.

r

k−∅

i

|E| > |E

′

_|

,

|E| > 1

and

|E

′

_{| = 0}

. 3.

r

k−1e

i

|E| > |E

′

_|

and

|E

′

_{| = 1}

, with

E

′

_{= ∅}

. 4.

r

k−1ne

i

|E| > |E

′

_|

and

|E

′

_{| = 1}

, with

E

′

_{6= ∅}

. 5.

r

k−j

i

1 < |E

′

_{| < |E|}

.

The onstant hange orresponds to the ase where the number of extensions remains un- hanged whilein lusion relations mayexist between extensionsof

G

and extensionsof

G

′

. Def. 10 (Constant hange) The hangefrom

G

to

G

′

is onstanti

|E| = |E

′

_|

. Thesub- ases of onstant hangesfrom

G

to

G

′

are: 1. - onservative i

E

= E

′

. 2.

c

1e−1ne

i

E

= {{}}

and

E

′

_{= {E}

′

_}

,with

E

′

_{6= ∅}

. 3.

c

1ne−1e

i

E

= {E}

, with

E 6= ∅

and

E

′

_{= {{}}}

. 2

4. -expansive i

E

6= ∅

and

|E| = |E

′

_|

and

∀E

i

∈ E, ∃E

′

j

∈ E

′

, ∅

6= E

i

⊂ E

j

′

and

∀E

′

j

∈

E

′

_,

_∃E

_i

_{∈ E, ∅ 6= E}

_i

_{⊂ E}

′

j

.

5. -narrowing i

E

6= ∅

and

|E| = |E

′

_|

and

∀E

i

∈ E, ∃E

′

j

∈ E

′

, ∅

6= E

j

′

⊂ E

i

and

∀E

′

j

∈

E

′

_,

_∃E

_i

_{∈ E, ∅ 6= E}

′

j

⊂ E

i

. 6. -altering i

|E| = |E

′

_|

and it is neither - onservative, nor

c

1e−1ne

, nor

c

1ne−1e

, nor -expansive, nor -narrowing.

Def.10.1, 10.2, 10.3 and 10.6 are fairly straightforward. Def.10.4 states that a -expansive hange is a hange where all the extensions of

G

, whi h are initially not empty, are in reased by some arguments. A -narrowing hange, a ording to Def.10.5, is a hange where all the extensionsof

G

areredu ed bysome argumentswithout be oming empty.

Ex.1( ont'd) Inthis example, all theagents alwayspropose onstant hanges, sin e theywant totake a de isionwithout ambiguity.

Properties about the a eptability of a set of arguments A hange an also have an impa tonthea eptabilityofsetsofarguments. Forinstan e,inadialog,itwouldbeinteresting to knowwhetherthe addition(ortheremoval)ofan argument modies thea eptabilityofthe argumentspreviouslya epted. Wesaymonotonyfrom

G

to

G

′

 wheneveryargumenta epted before the hange isstill a epted afterthe hange, i.e.,noa epted argument islost and there isa (notne essarilystri t) expansion ofa eptability.

3

Def. 11 (Simple expansive monotony) The hange from

G

to

G

′

satises the property of simple expansive monotony i

∀E

i

∈ E, ∃E

′

j

∈ E

′

,

E

i

⊆ E

j

′

. Note that [Bisquert et al., 2013℄

des ribes many other properties su h as, for instan e, a property ofenfor ement thatwould beinteresting for

J

1

inExample 1 inorder to obtainthe a eptability ofArgument

a

.

4 Denition of a hange operation taking into a ount support Firstof all,itshouldbenotedthat turningBAS

= hA, R

att

, R

sup

i

into its dedu tive asso iated DungsystemAS

BAS

orrespondstotheupdateofaspe i system,AS

= hA, R

att

i

,theredu tion of BAS to its dire t atta ks (see Figure 1). The next step is to allow for updating a BAS. So Def.8 isgeneralized: BAS

= hA, R

att

, R

sup

i

AS

= hA, R

att

i

AS BAS

= hA, D

Rsup

R

att

i

redu tionofBAS to itsdire tatta ks translation

(Def.7)

hangebyadditionof atta ks(Def. 8.2)

Figure1: The translationof BASinto AS BAS

isan update Def. 12 (Addition in a BAS) Let BAS

= hA, R

att

, R

sup

i

. Two hange operations are on-sidered:

3

Ase ond ase, referredas monotonyfrom

G

′

to

G

,has beendes ribedin [

Bisquertet al.,2013 ℄

. It isnot usedinthispaper.

1. Let

z

be an argument,

Ia

z

be a set of atta ks on erning

z

and

Is

z

be a set of supports on erning

z

(

Is

z

∪ Ia

z

is denoted by

I

z

). We assume that

I

z

⊆ (A × {z}) ∪ ({z} × A)

. Adding

z

and

I

z

is a hange operation, denoted by

⊕

z

(Ia,Is)

, providing a new bipolar system s.t.:

⊕

z

(Ia,Is)

hA, R

att

, R

sup

i = hA ∪ {z}, R

att

∪ Ia

z

, R

sup

∪ Is

z

i

.

2. Let

Ia

be a set of atta ksand

Is

be a set of supports (

Is ∪ Ia

is denoted by

I

). We assume that

I ⊆ (A × A)

and

I ∩ (R

att

∪ R

sup

) = ∅

.

Adding

I

is a hange operation, denoted by

⊕

(Ia,Is)

, providing a new bipolar system s.t.:

⊕

(Ia,Is)

hA, R

att

, R

sup

i = hA, R

att

∪ Ia, R

sup

∪ Isi

. The system resulting ofa hange isdenoted by BAS

′

_{= hA}

′

_{, R}

att

′

_{, R}

sup

′

_i

andits dedu tive asso- iated Dung AS is denoted by AS

BAS′ .

Due to la k of pla e, in this paper, we only study the ase orresponding to Def-inition 12.1. As we onsider dedu tive support and from Denitions 12 and 7, the following onsequen e obviously holds:

Conseq. 1 Let BAS

= hA, R

att

, R

sup

i

. Let

⊕

z

(Ia,Is)

be a hange operation on BAS produ ing BAS

′

. AS BAS′

= hA ∪ {z}, D

R

sup

∪Is

z

R

_att∪Ia

z

i

.

Due to theabove result, it seemsnatural to study theupdateof BAS by omparing AS BAS andAS

BAS′

. However,itisnot always possibleto identify aunique hangeonAS BAS

,asdened in Denition8,thatprodu esAS

BAS′

. Indeed,theaddition ofanargument withintera tions in BAS anindu ethe additionin

D

Rsup

R

att

ofnewatta ks between argumentsof

A

asshownbythe following example:

Ex. 3 Let BAS

= h{a, b}, ∅, ∅i

, let us apply on BAS the hange

⊕

z

(Ia,Is)

with

Ia

z

= {(a, z)}

and

Is

z

= {(b, z)}

; inthis ase, followingDef.12.1and7, AS

BAS′

ontains thenew atta k

(a, b)

that does not on ern

z

.

Another example shows thatthis problemalso existsevenif

Ia

z

= ∅

: Ex. 4 Consider BAS

= h{a, b, c}, {(c, a)}, ∅i

, andapplyon BASthe hange

⊕

z

(Ia,Is)

with

Ia

z

=

∅

and

Is

z

= {(b, z), (z, c)}

; in this ase, following Def. 12.1 and 7, AS BAS′

ontains the new atta k

(b, a)

that does not on ern

z

.

So, if we add an argument

z

with at least one support in BAS, the hange of AS BAS

into AS

BAS′

annot always be expressed using either Def. 8.1 (sin e atta ks are added that do not on ern

z

),or Def.8.2(sin etheargument

z

isadded). Thelinksbetween thedierent systems areillustrated by Figure2.

The di ulties pointedbyExamples 3and 4 suggest to onsidertwo parti ular ases. The rst one on erns a BAS withonly one supportfrom

z

to

a

,

z

being unatta ked. In this ase, Denition 7obviouslyimplies that

z

hasin AS

BAS

exa tly thesame role as

a

inAS: Prop. 3 Let BAS

= hA, R

att

, R

sup

i

with

R

sup

= {(z, a)}

and

z

is not atta ked in BAS. The following properties hold:

•

if

a

is unatta ked in BAS then

z

is unatta ked in AS BAS

(no dire t atta k, no dire t or indu tive supported or mediated atta k on

z

);

•

if

a

isatta ked by

b

inBASthen

z

isatta ked by

b

inAS BAS

(thisisamediated atta kon

z

);

•

if

a

atta ks

b

in BASthen

z

atta ks

b

in AS BAS

(this isa supported atta k).

•

if

a

isdefended by

c

against

b

inBASthen

z

isdefended by

c

against

b

in AS BAS

(the defen e of a dire t atta k on

a

an be used forthe defen e of the mediated atta k on

z

).

BAS

= hA, R

att

, R

sup

i

BAS

′

_{= hA ∪ {z},}

R

att

∪ Ia

z

, R

sup

∪

Is

z

i

AS

= hA, R

att

i

AS BAS

= hA, D

Rsup

R

att

i

AS BAS′

= hA ∪ {z}, D

Rsup

∪Is

z

R

_att∪Ia

z

i

hangeofBAS (Def.12.1) redu tionofBAS hangebyadditionof atta ks(Def. 8.2) translation (Def.7) translation (Def.7) hangenot apturedbyDef.8 hangenot aptured

byDef.8.

Figure2: Linksbetween thedierent systems BASredu edto an AS BAS

′

AS BAS′ additionofone argument

withonesupport(Def. 12.1)

translation (Def.7) additionofoneargument

withatta ks( hange aptured byDef.8.1)

Figure 3: Linksbetween systemsifthere isno supportinBAS

•

if

c

isdefendedby

b

against

a

in BASthen

c

isdefended by

b

against

z

inAS BAS

(amediated atta k an be used as a defen e against a supported atta k).

Ase ondparti ular ase on erns aBASwithonlyone supportonanunatta kedargument. Inthis ase, Denition 7obviouslyimpliesthat thesetof atta ks remains un hanged:

Prop. 4 Let BAS

= hA, R

att

, R

sup

i

with

R

sup

= {(a, z)}

and

z

unatta ked by BAS . Then

D

Rsup

R

att

= R

att .

Moreover, in these parti ular ases, following Denition 12.1, Propositions 3 and 4, the addition of one argument involved in only one support in BAS annot add atta ks between argumentsof

A

and preservesa eptability:

Prop. 5 Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. 4 Let

⊕

z

(Ia,Is)

be a hange operation dened on BAS with

Ia

z

= ∅

,

|Is

z

| = 1

and produ ing BAS

′

.

• ∀x, y ∈ A

,s.t.

y

does notatta k

x

in BASthen there isno atta k from

y

to

x

in AS BAS′

.

• ∀y ∈ A

, if

y

isunatta ked in BASthen it remainsunatta ked in AS BAS′

.

•

Consider

F

(resp.

F

′

) the hara teristi fun tion of AS (resp. AS BAS′

).

∀S ⊆ A

,

F(S) ⊆

F

′

_(S)

.

Thus, onsidering a BAS redu ed to an AS (i.e. without any support), if we add only one argument withone support,thelinks between thedierent systemsare given by Figure3.

So we are able to hara terize the addition of a support by an addition of atta ks. In the next se tion,westudy this simplied hange operation.

5 Chara terizing the addition of an argument and a support InSe tion5.1(resp.Se tion5.2),wegivesomeresultsaboutthe hara terizationoftheaddition of asupported (resp. supporting)argument inaBAS.

4

Inthis ase,BASisredu edtoanAS.SoBAS,itsredu tionASandAS BAS

updatedwith

z

and AS BAS

′

before hange after hange thesupport

(a, z)

c

b

a

z

/ /

c

b

a

z

/ /

{a, c}

is the grounded, pre-ferred and stableextension

{a, c, z}

is the grounded, pre-ferred and sta-bleextension The hangeis -expansive

c

a

z

/ /

c

a

z

/ /

∅

is the grounded ex-tension;

{a}

and

{c}

are the preferred and stable extensions

{z}

is the grounded ex-tension;

{a, z}

and

{c, z}

are the preferred and stable extensions The hangeis -expansive(preferred,

stable)or

c

1e−1ne

(grounded)

b

c

a

z

/ / /

b

c

a

z

/ / /

∅

is the grounded and preferred ex-tensions; there is no stable extension

{z}

is the grounded and preferred ex-tensions; there is no stable extension

The hangeis -expansive(preferred), or

c

1e−1ne

(grounded),

or - onservative(stable) Table 1: Addition ofa supportedargument inanAS

5.1 Case of an added supported argument

Inthis ase,asadire tappli ation ofProposition4,weprovethattheupdateofaBASwithout supports hasadedu tive asso iated Dung ASthat orrespondsto theaddition of an argument withoutintera tion into the initial BAS.

Prop. 6 Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

be a hange operation dened on BAS with

Ia

z

= ∅

and

Is

z

= {(a, z)}

andprodu ing BAS

′

. AS BAS′

= ⊕

z

∅

hA, R

att

i

. Due toProposition 6,Denitions8.1and 12.1, we have:

Prop. 7 Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

be a hange operation dened on BAS with

Ia

z

= ∅

and

Is

z

= {(a, z)}

andprodu ing BAS

′

.

•

Let

s

beasemanti sbelongingto

{

grounded, preferred, stable

}

.

E

isanextensionofASunder

s

i

E

′

_{= E ∪ {z}}

is an extensionof AS BAS′

under

s

.

•

There isno stable extension in AS ithere isno stable extension in AS BAS′

. And an obvious onsequen eof Proposition 7is:

Conseq. 2 The hange

⊕

z

(∅,{(a,z)})

is onlyeither -expansive, or

c

1e−1ne

, or - onservative. In the last ase, the onlypossibilityis

E

= E

′

_{= ∅}

. Some examplesof this hange aregiven inTable1.

5.2 Case of an added supporting argument Inthis ase, the existen eof y lesis preserved asshown by:

Prop. 8 Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

be a hange operation dened on BAS with

Ia

z

= ∅

and

Is

z

= {(z, a)}

andprodu ing BAS

′

.

•

If

a

belongs to a y le of atta ks in BASthen

z

belongs to a new y le of atta ks in AS BAS′

and thelength of both y les isthe same.

•

If

a

does not belong toa y le of atta ksin BASthen there is no y le of atta ks in AS BAS′

involving

z

.

This result is proven using Denitions 5 to 7 and by redu tio ad absurdum for the se ond item.

Following Denition7 and Proposition 3,we an hara terize theimpa t of this hange for stablesemanti s:

Prop. 9 Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

be a hange operation dened on BAS with

Ia

z

= ∅

and

Is

z

= {(z, a)}

and produ ing BAS

′

. Let

E

be a stable extension of AS:

•

if

a

6∈ E

then

E

is a stable extension of AS BAS′

;

•

if

a

∈ E

then

E ∪ {z}

is a stable extension of AS BAS′

.

And more generally,the simpleexpansive monotony ofthe hange operation an beproven: Prop. 10 Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

s

be a semanti s belonging to

{

grounded, preferred, stable

}

. Let

⊕

z

(Ia,Is)

be a hange operation dened on BASwith

Ia

z

= ∅

and

Is

z

= {(z, a)}

andprodu ing BAS

′

.

∀E

extension of AS under

s

,

∃E

′

an extension of AS BAS′ under

s

s.t.

E ⊆ E

′

.

This result is proven using Denition 3, Propositions 3 and 5, by indu tion on the har-a teristi fun tion for the grounded semanti s, showing that

E

is admissible in AS

BAS′

for the preferredsemanti s and following Proposition 9 forthestablesemanti s.

An obvious onsequen e ofthe two previous results is: Conseq. 3 The hange

⊕

z

(∅,{(z,a)})

annot be restri tive, nor -narrowing, nor -altering, nor

c

1ne−1e

.

Some examplesof this hange aregiven inTable2.

6 Con lusion and future works

Thispaperpresentspreliminaryworkabout hange forabstra tbipolarargumentationsystems, i.e. where there exist two kinds of intera tion, atta ks and supports. The entral idea is to take advantageoftwo kindsofpreviousworks,worksaboutdynami s inargumentation systems (AS) and works about bipolar argumentation systems (BAS). Indeed, it has been shown that a BAS an be turned into a standard Dung's AS by adding appropriate atta ks. Our main ontribution is to show how theaddition of one argument together withone support involving it (and without any atta k) impa ts the extensions of the resulting system. In this parti ular ase, we have learly identied theatta ks that must be added and we have obtained spe i propertieswhi henableto hara terize this hange. These hara terizationsreneand omplete theresultspresentedin

[

Bisquertetal.,2013 ℄

that annotbeuseddire tlyfor hara terizingthe impa t of these new atta ks (the onditions used in

[Bisquert et al.,

2013℄

are too strong with regard toour aseand thus they annotbesatised here). Ourwork isofparti ular interestin a multiagent ontext ifwe do not want to re ompute the extensions whena agent gives a new argument thatsupports (orissupported by) analready existingargument.

updatedwith

z

and AS BAS

′

before hange after hange thesupport

(z, a)

z

a

z

a

{a}

is the grounded, pre-ferred and stableextension

{a, z}

is the grounded, pre-ferred and stableextension The hange is -expansive

z

a

/

z

a

/ / /

∅

is the grounded and preferred exten-sion; there isno stableextension

{z}

is the grounded, pre-ferred and stable extension The hange is

c

1e−1ne

(grounded,

preferred)or

e

∅−1ne

(stable)

z

a

b

/

z

a

b

/ /

{b}

is the grounded, pre-ferred and stableextension

{b}

is the grounded, pre-ferred and stableextension The hange is - onservative

z

a

b

c

d

/ / / / / /

z

a

b

c

d

/ / / / / / / / /

∅

is the grounded and preferred exten-sion; there isno stableextension

∅

is the grounded ex-tension;

{z, c}

and

{z, d}

are the preferred and stable extensions

The hangeis - onservative (grounded) or

e

1e−k

(preferred),or

e

∅−k

(stable)

z

a

b

/ / /

z

a

b

/ / / / / / /

∅

is the grounded exten-sion;

{b}

is the preferred and stableextension

∅

is the grounded ex-tension;

{b}

and

{z}

are the preferred and stable extensions

The hangeis - onservative (grounded) or

e

1ne−k

(preferred,stable)

z

a

b

c

/ / / / / / /

z

a

b

c

/ / / / / / / / / / / / /

∅

is the grounded ex-tension;

{b}

and

{c}

are the preferred and stable extensions

∅

is the grounded ex-tension;

{b}

,

{c}

and

{z}

are the preferred and stable extensions

The hangeis - onservative (grounded) or

e

j−k

(preferred,stable) Table2: Addition ofa supporting argument inanAS

port),we thinkthatthey an begeneralized onsidering thattheadditionofasetofarguments with intera tions an be viewed asa sequen e of simple additions. Nevertheless, in order to a hievethisgeneralization,there aretwo issuestobesolved: (1) hara terize theaddition ofan argumentwithatta ks(aswasdone forAS;resultsgivenin

[

Bisquertetal.,2013 ℄

willbeuseful) and (2) study the addition of intera tions (this operation has been dened in

[Bisquert et al., 2013℄

for ASand inour paperforBAS but not ompletelystudied). Thislast study ould also givea wayfor omputingdire tly theAS

BAS

ofa BAS.It will be thesubje tof futureworks. Moreover, our work on erns only a spe ial variant of support, the dedu tive one. Using the duality between ne essary and dedu tive supports, our results an be easily translated for ne essary support. However, it remains to adapt them to the ase of a generalized support(a supportfrom aset ofargumentsto an argument asproposedby

[ Nouioua,2013 ℄ ). Referen es [Amgoud and Cayrol, 2002℄

L.AmgoudandC.Cayrol. Areasoningmodelbasedonthe produ -tion ofa eptable arguments. Annals of Mathemati s andArti ial Intelligen e, 34:197216, 2002.

[Amgoud et al.,

2000℄

L.Amgoud, N.Maudet, andS.Parsons. Modellingdialoguesusing argu-mentation. InPro . of ICMAS, pages3138, 2000.

[

Amgoudet al.,2008 ℄

L.Amgoud,C.Cayrol,M-C.Lagasquie-S hiex,andP.Livet. On bipolar-ityinargumentation frameworks. InternationalJournalofIntelligentSystems,23:10621093, 2008.

[Baroni etal.,

2011℄

P. Baroni,M. Caminada, andM. Gia omin. An introdu tion to argumen-tation semanti s. The knowledge engineeringreview, 26 (4):365410, 2011.

[Baroni etal.,

2014℄

P. Baroni, G. Boella, F. Cerutti, M. Gia omin, L. van der Torre, and S.Villata.Ontheinput/outputbehaviorofargumentationframeworks.Arti ialIntelligen e, 217:144197, 2014.

[Bisquert et al.,

2013℄

P. Bisquert, C. Cayrol, F. Dupin de Saint Cyr Bannay, and M-C. Lagasquie-S hiex. Chara terizing hange in abstra t argumentation systems. In E. Fermé, D. Gabbay, and G. Simari, editors, Trends in Belief Revision andArgumentation Dynami s, volume 48 of Studiesin Logi , pages75102. College Publi ations, 2013.

[Boella etal.,

2012℄

G. Boella, D. M. Gabbay, L. van der Torre, and S. Villata. Modelling defeasible and prioritized supportinbipolar argumentation. Annals of Mathemati s and AI, 66:163197, 2012.

[

Cayroland Lagasquie-S hiex, 2005 ℄

C.CayrolandM-C.Lagasquie-S hiex. Onthe a eptabil-ityofargumentsinbipolarargumentationframeworks. InPro .ofECSQARU,pages378389. Springer-Verlag, 2005.

[Cayrol

and Lagasquie-S hiex, 2010℄

C. Cayrol and M-C. Lagasquie-S hiex. Coalitions of ar-guments: a tool for handling bipolar argumentation frameworks. International Journal of IntelligentSystems,25:83109, 2010.

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A Proofs

Conseq.1: Let BAS

= hA, R

att

, R

sup

i

. Let

⊕

z

(Ia,Is)

bea hange operation onBAS pro-du ingBAS

′

. AS BAS

′

= hA ∪ {z}, D

R

sup

∪Is

z

R

att

∪Ia

z

i

.

✷

Proof of Conseq.1: ByDenition 12.1, BAS

′

= hA ∪ {z}, R

att

∪ Ia

z

, R

sup

∪ Is

z

i

. Then, following Denition7,AS BAS

′

= hA ∪ {z}, D

R

sup

∪Is

z

R

att

∪Ia

z

i

.

✷

Prop.5: Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

be a hange operation denedonBASwith

Ia

z

= ∅

,

|Is

z

| = 1

andprodu ingBAS

′

.

•

Forall

x, y

∈ A

,s.t.

y

doesnotatta k

x

in BASthenthere isnoatta kfrom

y

to

x

in AS

BAS

′

.

•

Forall

y

∈ A

, if

y

isunatta kedinBAS thenitremainsunatta kedinAS BAS

′

.

•

Consider

F

(resp.

F

′

) the hara teristi fun tion of AS (resp. AS BAS

′

).

∀S ⊆ A

,

F (S) ⊆ F

′

_(S)

.

✷

Proof ofProp.5:

•

TherstitemisprovenusingDenition5toDenition7: weknowthatalltheatta ksin

D

R

sup

∪Is

z

R

att

∪Ia

z

areprodu ed using

R

att

and

R

sup

∪ Is

z

(either dire tly, orindu tivelyby buildingthesupported or mediatedatta ks);andweassumethat

R

sup

= ∅

and

Is

z

is redu edto onesupport(either

(z, a)

or

(a, z)

),sotheonlysupport on erns

z

thatisnotin

A

;so,followingDenition12.1,Proposition3and Proposition4,thesetofatta ksbetweenargumentsof

A

remainun hangedin AS

BAS

′

.

•

These onditemistriviallydedu edfrom therstone.

•

For the third item, onsider

F

(resp.

F

′

) the hara teristi fun tion of AS (resp. AS BAS

′

). Let

x

∈ F (S)

s.t.

x

is atta ked in AS BAS

′

. Either

x

is atta ked in AS BAS

′

byonly argumentsof

A

and then following thepreviousitems,

x

is defendedby

S

in AS

BAS

′

; or

x

is also atta kedin AS BAS

′

by

z

and then

x

wasalso atta kedby

a

in AS (followingDenition 7) and defendedby

S

in AS andin AS

BAS

′

(followingProposition3).

✷

Prop.6: LetBAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

bea hangeoperation de-nedonBASwith

Ia

z

= ∅

and

Is

z

= {(a, z)}

andprodu ingBAS

′

. AS BAS

′

= ⊕

z

∅

hA, R

att

i

.

✷

Proofof Prop.6: ByDenition 12.1,BAS

′

_{= hA ∪ {z}, R}

att

,

{(a, z)}i

. Inthis ase,following Proposi-tion4,theset ofd-atta ksexa tly orrespondsto

R

att . ThenAS BAS

′

= hA ∪ {z}, R

att

i

andtrivally orrespondsto

⊕

z

∅

hA, R

att

i

(seeDenition 8.1).

✷

Prop.7: Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

be a hange operation denedonBASwith

Ia

z

= ∅

and

Is

z

= {(a, z)}

andprodu ingBAS

′

.

•

Let

s

beasemanti sbelongingto

{

grounded,preferred,stable

}

.

E

isanextensionofAS under

s

i

E

′

_{= E ∪ {z}}

isanextensionofAS BAS

′

under

s

.

•

Thereisnostableextensionin ASithereisnostableextensioninAS BAS

′

.

✷

•

Following Proposition 6, AS BAS

′

= ⊕

z

∅

hA, R

att

i

. So, following Denition 8.1, AS BAS

′

= hA ∪

{z}, R

att

i

. Sin e

z

is involved in no atta k,

z

must be added to any (grounded, preferred, stable) extensionofBAS

=

AS andnootherargumentisae ted.

•

Itfollowsdire tlyfromthepreviousitem by ontraposition. Note thatthis pointmakessense only for stable semanti s. Note also that

A

is not emptysin e there exists at least the argument

a

∈ A

thatsupports

z

.

✷

Prop.8: Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

be a hange operation denedonBASwith

Ia

z

= ∅

and

Is

z

= {(z, a)}

andprodu ingBAS

′

.

•

If

a

belongs to a y le of atta ksin BAS then

z

belongs to anew y le of atta ks in AS

BAS

′

andthelengthofboth y lesisthesame.

•

If

a

does not belong to a y le of atta ks in BAS then there is no y le of atta ks in AS

BAS

′

involving

z

.

✷

ProofofProp.8: Consider

z

6∈ A

and

a

∈ A

s.t.

z

supports

a

. LetAS BAS

′

bethededu tiveasso iated Dung ASofBAS

′

•

If

a

belongstoa y leofatta ksinBASthen

∃n ≥ 1

s.t.

a

= a

1

R

att

a

2

R

att

. . . R

att

a

n

R

att

a

1

= a

;so onsideringDenition5andDenition6,thereexistasupportedatta k

(z, a

2

)

andamediatedatta k

(a

n

, z)

in AS BAS

′

; moreoversin eatta ksin BAS arealso atta ksin BAS

′

andremain inAS BAS

′

(see Denition7),

z

belongstothe y leofatta ks

(z, a

2

, . . . , a

n

, z)

inAS

BAS

′

;moreoverthelengthofthis y leinAS

BAS

′

is equalstothelengthofthe y le ontaining

a

inBAS .

•

Proof byredu tio ad absurdum: if

z

belongs to a y le ofatta ks

(z, a

1

, . . . , a

n

, z)

in AS BAS

′

, then onsidering than

z

6∈ A

and thefa t that

D

R

sup

∪Is

z

R

att

∪Ia

z

is built with

R

sup

= Ia

z

= ∅

, we andedu e that theatta ks

(z, a

1

)

and

(a

n

, z)

arenew atta ksgeneratedby thesupport

(z, a)

, whereas(due to Proposition 6) the otheratta ks in the y le belong to

R

att

; moreoverthe atta k

(z, a

1

)

an appear only if there exists

x

s.t.

z

supports

x

and

xR

att

a

1

; similarly the atta k

(a

n

, z)

an appear only if there exists

y

s.t.

z

supports

y

and

a

n

R

att

y

; knowingthat there is onlyone support added to BAS ,

x

= y = a

;sothereexists inBASasequen e

aR

att

a

1

R

att

. . . R

att

a

n

R

att

a

; thismeansthat

a

belongs toa y leofatta ksin BASandthatisin ontradi tionwiththeassumption.

✷

Conseq.2: The hange

⊕

z

(∅,{(a,z)})

isonlyeither -expansive,or

c

1e−1ne

,or - onservative. Inthelast ase,theonlypossibilityis

E

= E

′

_{= ∅}

.

✷

Proof of Conseq.2: FollowingProposition 7and the denitions of hange properties, ifthere exists at least oneextension before the hange, it isobviousthat the hange is -expansive or

c

1e−1ne

(sin e at ea hextension

E

of BAS orresponds anextension of BAS

′

that stri ly ontains

E

). And, following Proposition7,ifthereisnoextensionbeforethe hange(thisispossibleonlywithstablesemanti s)then thereisalsonoextensionafterthe hange( - onservative hange).

✷

Prop.9: Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

⊕

z

(Ia,Is)

be a hange operation dened onBAS with

Ia

z

= ∅

and

Is

z

= {(z, a)}

andprodu ing BAS

′

. Let

E

be astable extensionofAS:

•

if

a

6∈ E

then

E

isastableextensionofAS BAS

′

•

if

a

∈ E

then

E ∪ {z}

isastableextensionofAS BAS

′

.

✷

Proofof Prop.9: Let

E

beastableextension.

E

stable inASmeansthat

E

is oni tfreeinAS and

E

atta ks

A

\ E

.

•

Considerthe asewhen

a

6∈ E

. As

E

is oni tfreeinAS,duetoProposition5,

E

remains oni tfree in AS

BAS

′

. Then

a

isatta kedby anargument

x

of

E

. FollowingProposition3,

z

isalso atta kedby

x

inAS BAS

′

andso

E

atta ks

A

∪ {z} \ E

;thatimpliesthat

E

is astableextensionofAS BAS

′

.

•

Considerthe asewhen

a

∈ E

.

E ∪ {z}

atta ks

A

\ (E ∪ {z})

. Weshowby redu tioad absurdum that

E ∪ {z}

is oni tfree;weassumethatthereisanargument

x

∈ E

su hthateither

x

atta ks

z

,or

z

atta ks

x

;in the rst ase,followingDenition 7,there existsin AS anatta kfrom

x

to

a

,so

E

is not oni tfree;andin these ond ase,on eagainfollowingDenition 7,thereexistsin ASanatta k from

a

to

x

,so

E

isnot oni tfree;in ea h ase,thereisa ontradi tion. Thus

E ∪ {z}

is oni tfree inAS

BAS

′

anditis astableextensionofAS BAS

′

.

✷

Prop.10: Let BAS

= hA, R

att

, R

sup

i

s.t.

R

sup

= ∅

. Let

s

be asemanti sbelonging to

{

grounded, preferred, stable

}

. Let

⊕

z

(Ia,Is)

be a hange operation dened on BAS with

Ia

z

= ∅

and

Is

z

= {(z, a)}

andprodu ing BAS

′

.

∀E

extensionof ASunder

s

,

∃E

′

anextensionofAS BAS

′

under

s

s.t.

E ⊆ E

′

.

✷

Proof ofProp.10:

•

Grounded semanti s

F

(resp.

F

′

)denotesthe hara teristi fun tion ofAS (resp. AS BAS

′

). Let provebyindu tionon

i

≥ 1

that

∀i ≥ 1

,

F

i

_{(∅) ⊆ F}

′i

_(∅)

. The ase

i

= 1

istrivial,followingProposition5.

Assume that

F

i

_{(∅) ⊆ F}

′i

_(∅)

. Take

S

= F

i

_(∅)

. From Proposition5,

F (S) ⊆ F

′

_(S)

. So

F

i+1

_{(∅) ⊆}

F

′

_(F

i

_(∅))

. As

F

′

ismonotoni andusingtheindu tiveassumption,wehave

F

′

_(F

i

_{(∅)) ⊆ F}

′

_(F

′i

_{(∅)) =}

F

′i+1

_(∅)

. So

∀i ≥ 1

,

F

i

_{(∅) ⊆ F}

′i

_(∅)

. Hen e,

E ⊆ E

′

.

•

Preferred semanti sItissu ienttoshowthat ea hpreferredextension

E

ofASisadmissiblein AS

BAS

′

. Let

E

beapreferred extensionof AS.

E

is oni tfreein AS and soitis also oni tfreein AS

BAS

′

( f Proposition 5). Assume that

y

∈ E

is atta kedby

x

in AS BAS

′

. Two ases are possible: either

x

∈ A

or

x

= z

.

If

x

∈ A

, theatta k

(x, y)

is alreadyin AS andsin e

E

isadmissible in ASthere exists

e

∈ E

s.t.

e

atta ks

x

inAS. So

e

defends

y

inAS

BAS

′

. If

x

= z

,then theatta k

(z, y)

in AS

BAS

′

isgenerated using theatta k

(a, y)

in AS andthe support

(z, a)

. Sin e

E

is admissible,there exists

e

∈ E

s.t.

e

atta ks

a

in ASand, followingProposition 3,

e

defends

y

against

x

= z

inAS

BAS

′

. Inea h ase,

E

defends

y

against

x

in AS

BAS

′

. Thus

E

isadmissible in AS BAS

′

and so in ludedin a preferredextensionofAS BAS

′

.

•

Stablesemanti sTriviallyfollowsProposition9

✷

Conseq.3: The hange

⊕

z

(∅,{(z,a)})

annot be restri tive, nor -narrowing, nor -altering,

nor

c

1ne−1e

.

✷

inanextensionofBAS

′

(sothe hange annotbe -narrowing,nor -altering). Moreover,thenumberof extensions annotbede reased(sothe hange annotberestri tive)andanemptyextension annotbe appeared(so the hange annotbe

c

1ne−1e

).

✷