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A Unified Framework for Representation and Development of Dialectical Proof

Procedures in Argumentation

P. M. Dung

Department of Computer Science

Asian Institute of Technology

dung@cs.ait.ac.th

P. M. Thang

thangfm@ait.ac.th

Abstract

We present an unified methodology for represen-

tation and development of dialectical proof proce-

dures in abstract argumentation based on the no-

tions of legal environmentsand dispute derivations.

A legal environment specifies the legal moves of

the dispute parties while a dispute derivation de-

scribes the procedure structure. A key insight of

this paper is that the opponent moves determine

the soundness of a dispute while the completeness

of a dispute procedure depends on the proponent

moves.

1 Introduction

Argumentation is a form of reasoning, that could be viewed

as a dispute resolution, in which the participants present their

arguments to establish, defend or attack certain propositions.

The probably most abstract among the well-known

formalisms for argumentation is the abstract argumenta-

tion framework [Dung, 1995b] (or abstract argumenta-

tion for short) consisting simply of a set of ”atomic” ar-

guments together with a binary relation representing at-

tack relation between them.

on dialectical proof procedures for abstract argumenta-

tion [Cayrol,Doutre,Mengin, 2003; Dunne,Bench-Capon,

2003; Modgil,Caminada, 2009; Vreeswijk,Prakken, 2000;

Jakobovits,Vermeir, 1999; Vreeswijk, 2006; Verheij, 2007].

Though clearly related, their formal and precise relationship

remains unexplored. We address this problem by providing

an unified framework for representing dialectical proof pro-

ceduresbased onthe notionsof legalenviromentsanddispute

derivation trees. A key insight of this paper is that the oppo-

nent moves determine the soundness of a dispute while the

completeness depends on the proponent moves.

There is extensive research

2 Abstract Argumentation: Preliminaries

Anabstract argumentationframework[Dung,1995b]is a pair

AF = (A,att), whereA is a set ofarguments,andattis a bi-

nary relation over A representing the attack relation between

the arguments (att ⊆ A × A) with (A,B) ∈ att meaning A

attacks B. For simplicity, we restrict ourself on frameworks

with finite sets of arguments. A set S of arguments attacks

an argument A if some argument in S attacks A; S attacks

another set S?if S attacks some argument in S?.

A set S of argumentsis conflict-free iff it does not attack it-

self. Argument A is acceptable with respect to S iff S attacks

each argument attacking A. S is admissible iff S is conflict-

free and each argument in S is acceptable with respect to S. S

is a preferred extension iff S is maximally (wrt set inclusion)

admissible.

The semantics of argumentation could also be charac-

terized by a fixpoint theory of the characteristic function

F(S) = {A ∈ A | A is acceptable wrt S}. It is easy to

see that S is admissible iff S is conflict free and S ⊆ F(S).

As F is monotonic , it follows that S is a preferred extension

iff S is a maximal fixpoint of F. The least fixed point of F is

defined as the grounded extension.

An argument A is said to be credulously accepted iff it is

contained in at least one preferred extension and groundedly

accepted iff it is contained in the grounded extension.

Given an argument B ∈ A, AttackBand AttackedBde-

note the set of arguments attacking B, i.e. AttackB= {A |

(A,B) ∈ att}, and attacked by B, i.e. AttackedB = {A |

(B,A) ∈ att}.

3 Dispute derivations

In a dispute, the proponent starts by putting forward an ini-

tial argument and then the proponent and opponent alternate

in attacking each other’s previous arguments. The proponent

wins if the opponent runs out of arguments to make a move.

A dispute is represented by a dispute derivation in which dis-

pute derivation trees are successively constructed by expand-

ing the previous one by adding children to some frontier (or

leaf) nodes following some legal move functions for the dis-

pute parties. The initial partial derivationtree consists of only

the root.

Definition 3.1 A dispute derivation tree (or simply deriva-

tion tree for short) T for an argumentA is definedas follows:

1. Every node of T is labeled by an argument and is as-

signed the status of proponent node or opponent node,

but not both. The status of a child node is different from

that of its parent. Arguments labeling children nodes at-

tack arguments labeling their parent node.

2. The root is a proponent node labeled by A.

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For example, various derivation trees are given in figure 1.

Abusing the notation, for a node N labeled by A, we often

refer to AttackAby AttackN.

In the following, we introduce the notion of legal envi-

ronments generalizing the well-known notions of legal move

functions in the literature.

Definition 3.2

nent (resp opponent) legal move function if F assigns

to pairs (N,T) of a frontier opponent (resp. proponent)

node N in a derivation tree T an argument in (resp. a

subset of) AttackN. If F(N,T) is not defined, we write

F(N,T) = ⊥.

• A pair of proponent and opponent legal move functions

Φ = (ΦP,ΦO) is called a legal environment.

• A partial function F is called a propo-

Examplesof legalmovefunctionsare giveninfigure1. For

simplicity, if it is clear from context, we often write Φ(N,T)

for ΦP(N,T) if N is a opponent node, and for ΦO(N,T) if

N is an proponent node. We often refer to the leaf nodes in a

derivation tree as frontier nodes to indicate that they are the

ones to be expanded later.

Definition 3.3 Given a legal environment Φ = (ΦP,ΦO)

and derivation trees T,T?. We say that T?is obtained from

T by expansion of a frontier (leaf) node N in T wrt Φ if

Φ(N,T) is defined and following condition holds:

• If N is an opponentnode then a child node labeled by an

argument selected in ΦP(N,T) is added to N

• If N is proponent node then a set of child nodes labeled

by argumentsin ΦO(N,T) are addedto N. There is one-

onecorrespondencebetweenthechildrennodesofNand

the arguments in ΦO(N,T)

In our model of dispute, when the opponent moves against

a proponent argument, he delivers all attacks that are legal at

once. This assumption simplifies the formal framework with-

out affecting its generality. We discuss alternative strategies

later.

Definition 3.4

ment Φ = (ΦP,ΦO) is a sequence T0,...,Ti,...,Tnof

derivation trees where T0consists of a single node and

Ti+1is obtained from Tiby expansion of a frontier node

N in Ti.

• A dispute derivation T0,...,Tnterminates if for each

frontier node N of the final tree Tn, Φ(N,Tn) = ∅

• A terminating dispute derivation is said to be successful

if all the frontier nodes of the final tree are proponent

nodes.

• A dispute derivation wrt legal environ-

The sequence T0,T1,T2in figure 1 is an example of a suc-

cessful dispute derivation.

For a derivation tree T, PRO(T), OPP(T) denote respec-

tively the sets of arguments labeling the proponent or oppo-

nent nodes in T. Arguments in PRO(T) or OPP(T) are often

called respectively proponent or opponent arguments.

Definition 3.5

is no argument labeling both a proponent node and an

opponent node.

1. A derivation tree T is consistent if there

Figure 1:

2. A derivation tree is successful wrt legal environment Φ

if it is the final tree of a successful dispute derivationwrt

Φ.

Consider a frontier proponent node N in a derivation tree

T such that AttackN∩ PRO(T) ?= ∅. This implies imme-

diately that the set of proponent arguments is not conflict-

free as one of them attacks the argument labeling N. We cer-

tainly do not want to expand T at this node, i.e. ΦO(N,T)

should not be defined. Further in general, we want to con-

sider only those legal environments guaranteeing that an ex-

pansionofconsistent derivationtrees leadsto consistentones.

This requirement imposes that for frontier opponent node N,

ΦP(N,T) ∩ OPP(T) = ∅.

• A legal environment Φ is eligible iff fol-

lowing conditions are satisfied:

Definition 3.6

1. For each derivation tree T, and each frontier pro-

ponent node N of T, ΦO(N,T) = ⊥ if AttackN∩

PRO(T) ?= ∅.

2. If N is an opponent node then ΦP(N,T)

AttackN\ OPP(T)

• An eligible legal environment is said to be fully defined

if the ”if” in the condition(1) aboveis replacedby ”iff”.

⊆

Examples of eligible legal environmentsare givenin figure

2.

Figure 2:

It is not difficult to show that the eligibility of legal envi-

ronment guarantees that only consistent derivation trees are

generated in a dispute derivation.

Lemma 3.1 Let Φ = (ΦP,ΦO) be an eligible legal environ-

ment and T0,...,Tnbe a dispute derivation wrt Φ. Then for

each i, Tiis consistent.

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Proof By induction and left to the readers. ?

Definition 3.7 Let Φ be a legal environment.

1. Φ is said to be credulously (resp. groundedly) sound if

the sets of proponentargumentsof the successful deriva-

tion trees wrt Φ are admissible (resp. admissible subsets

of the grounded extension).

2. Φ is said to be credulously (resp. groundedly) complete

if for each credulously (resp. groundedly ) accepted ar-

gument there is a successful derivation tree wrt Φ whose

initial argument is A.

3. Φ is said to be terminating if there exists no infinite dis-

pute derivation wrt Φ.

4 Legal Environments for Credulous

Semantics

To ensure admissibility of proponentarguments, the legal en-

vironment should guarantee that all possible attacks against

proponentargumentsare accounted for. This is the case if for

each proponent node N, ΦO(N,T) = AttackN. But if some

arguments in AttackNhave been defeated by the proponent

arguments then deploying them will not help the opponent

winning the case. The opponent also does not need to de-

ploy arguments already deployed by itself in previous steps.

This insight is formalized in the following theorem in which

a general class of credulouslysound legal environmentsis in-

troduced.

Theorem 4.1 Let Φ = (ΦP,ΦO) be an eligible legal envi-

ronment such that for each derivation tree T, each proponent

node N in T

AttackN \ (AttackedPRO(T)∪ OPP(T)) ⊆ ΦO(N,T)

whenever ΦO(N,T) is defined.

Then Φ is credulously sound .

Proof Let T0,...,Tnbe a successful derivation wrt Φ. First

we want to show that PRO(Tn) counterattacks every attack

against it. Let A be an argument attacking an argument B ∈

PRO(Tn). Let N be a proponent node labelled by B. There

are two cases:

1. N is a leaf node in Tn.

From AttackN \ (AttackedPRO(T)∪ OPP(Tn)) ⊆

ΦO(N,Tn), it follows AttackN\ (AttackedPRO(Tn)∪

OPP(T)) = ∅, i.e. AttackN ⊆ (AttackedPRO(Tn)∪

OPP(Tn)). Hence A

OPP(Tn)).As all leaf nodes in Tn are proponent

nodes, it follows that OPP(Tn) ⊆ AttackedPRO(Tn).

A is hence attacked by PRO(Tn).

Hence ΦO(N,Tn) = ∅.

∈

(AttackedPRO(Tn) ∪

2. N is an internal nodeN in Tn. Let Tibe the tree at which

the opponentattacks N and expandsTiintoTi+1. If A ∈

ΦO(N,Ti) then A labels one of the opponentchild of N.

As Tnis a successful tree, A hasa proponentchildin Tn.

Hence A is attacked by PRO(Tn). If A ?∈ ΦO(N,Ti),

then A ∈ (AttackedPRO(Tn)∪ OPP(Tn)). As each

opponent node in Tnhas a proponent child, it follows

that A is attacked by PRO(Tn).

It remainsto show that PRO(Tn) is conflict-free. Suppose

PRO(Tn) is not conflict-free. An argument A ∈ PRO(Tn)

is said to be controversial if A attacks or is attacked by some

argument in PRO(Tn). Hence the set of controversial argu-

ments in PRO(Tn) is not empty.

We first show that no such argument labels a leaf node in

Tn. Suppose the contrary that a controversial argument A la-

bels a leaf node N in Tn. As PRO(Tn) counterattacks every

attack against it, it follows that there is some proponentargu-

ment in Tnattacking A. Hence AttackN∩ PRO(Tn) ?= ∅.

Therefore ΦO(N,Tn) = ⊥ contradiction to the successful-

ness of Tn.

Therefore every controversial argument labels an internal

node in Tn. Let m be the maximum number such that Tmis

expanded at a proponent node N (by the opponent) labelled

bycontroversialargumentA. Furtherlet B ∈ PRO(Tn) such

thatB attacks A.Fromthedefinitionofm, it followsthatB la-

belsaproponentnodeinTm. HenceAttackN∩PRO(Tm) ?=

∅. Therefore ΦO(N,Tn) = ⊥ contradiction to the success-

fulness of Tn.

Therefore the set of controversial arguments is empty.

Contradiction. Hence PRO(Tn) is conflict-free.

?

Theorem 4.1 states formally that the soundness of a le-

gal environment will not be affected if filtering mecha-

nisms are deployed to filter out some or all arguments in

AttackedPRO(T)∪ OPP(T). An important insight gained

from this theorem is that the soundness of a legal environ-

ment depends only on the legal moves of the opponent. The

following theorem shows that the legal moves of the propo-

nent determine the completeness.

Theorem 4.2 Let Φ = (ΦP,ΦO) be an eligible, credulously

sound and terminating legal environment. Further suppose

that Φ is fully defined and for each derivation tree T, each

opponent node N in T

AttackN\ Conf(T) ⊆ ΦP(N,T)

where Conf(T) consists of all self-attacking arguments and

all arguments attacking or being attacked by arguments in

PRO(T).

Then Φ is credulously complete.

Proof Let A be a credulously accepted argument and S be

an admissible set of arguments and A ∈ S. We construct by

inductiononi a successfulderivationT0,...,Ti,...,Tnsuch

that A labels the root of the trees and PRO(Ti) ⊆ S.

If Tiis successful then we are done, the theorem is proved.

Suppose now that Tiis not successful. Select a frontier node

N of Ti.

SupposeNis aopponentnode. Thenfromtheadmissibility

of S and PRO(Ti) ⊆ S, it follows that AttackN∩ S ?= ∅.

Because S is conflict-free and PRO(Ti) ⊆ S, it follows that

S∩Conf(Ti) = ∅. Therefore(AttackN\Conf(Ti))∩S ?=

∅. Hence there is an argument B ∈ S ∩ ΦP(N,Ti). Select B

and expand Tiby adding a child to N labelled by B.

Suppose N is a proponent node. As the argument labelling

N belongs to S and S is admisisble and PRO(Ti) ⊆ S, it

follows that AttackN∩ PRO(Ti) = ∅. Sinc Φ is fully de-

fined, it follows that ΦO(N,Ti) ?= ⊥. If ΦO(N,Ti) ?= ∅ then

expand Tiat N.

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As Φ is terminating, the process must stop sometimes. It

follows that all the frontier nodes are proponent nodes. The

theorem is proved.?

The following theorem shows that one can guarantee the

termination of dispute by not allowing the opponent to repeat

his moves.

Theorem 4.3 Let Φ = (ΦP,ΦO) be an eligible legal envi-

ronment such that

ΦO(N,T) ⊆ AttackN\ AOPP(N,T)

where AOPP(N,T) contains all arguments labeling op-

ponent nodes on the path from the root to N.

Then Φ is terminating.

Proof

ΦO(N,T) ⊆ AttackN\ AOPP(N,T). Let T0,...,Ti,...

be a nonterminating derivation. Let T be the limit of this

sequence of trees. T is hence infinite. There is then an

infinite path in T. There are hence infinite opponent nodes

on this path. Since the set of arguments is finite, there

are many opponent nodes labelled by the same argument.

Contradition.?

As we have discussed, a key insight of theorems 4.1, 4.2

is that the credulous soundness of a legal environment is de-

termined by opponent legal moves while its completeness de-

pends on the legal moves of the proponent.

is best illustrated by the special case where ΦO(N,T) =

AttackNwhenever it is defined. It turns out that in this case,

the legal environment is even groundedly sound.

Suppose

Φ

is notterminating and

This insight

Lemma 4.1 Let Φ be an eligible legal environment such that

ΦO(N,T) = AttackNwhenever ΦO(N,T) is defined. Then

Φ is groundedly sound.

Proof Let Φ?be a legal environment such that Φ?

AttackNwheneverit is defined and Φ?

OPP(T). It is easy to see that each successful derivationwrt

Φ is a successful derivation wrt Φ?. Let T be a successful

derivation tree wrt Φ?. It is easy to show by induction on the

hight of T that PRO(T) is a subset of the grounded exten-

sion. ?

O(N,T) =

P(N,T) = AttackN\

5 Legal Environments for Grounded

Semantics

We are now turning our attention to legal environments for

groundedsemantics. Lemma 4.1 points out that the opponent

could do nothing wrong if he/she keeps bringing up all pos-

sible counter-evidences to whatever the proponent says. The

legal environment is groundedly sound independent of what-

ever the proponent does. The best the proponent could do

in such cases is to make sure that it can win if the initial ar-

gument is groundedly accepted, i.e. to ensure that its legal

move function guarantees completeness. This is the case if

ΦP(N.T) = AttackN\ OPP(T). But this environment is

not terminating. It turns out that to ensure both termination

and completeness,the proponentshouldnot repeat arguments

put forward by itself previously. Though the legal environ-

ment just described is groundedly sound, complete and ter-

minating, it could be quite inefficient as the opponent could

Figure 3:

repeat many arguments that have been defeated before as the

following example illustrates.

Consider the derivation tree T in figure 3. It would be

wasteful if argument D at node N should be defeated again.

We hence expect from an efficient legal environment that

ΦO(N,T) = ∅. But in contrast to the credulous case, not

all previous opponent arguments could be filtered out as il-

lustrated by derivation tree T’ in figure 3. If the opponent

argument B is filtered out at N i.e. ΦO(N,T) = ∅ then T?

would be a successful derivation tree though A is not ground-

edly accepted.

Theexamplessuggestthatwe shouldallowthe opponentto

filter out previous opponent arguments except those lying on

the path from the root to the considered node, i.e. AttackN\

(OPP(T) \ AOPP(N,T)) ⊆ ΦO(N,T). Unfortunately,

this constraintis not strongenoughto guaranteethe grounded

soundness of the legal environment as the following example

shows:

Figure 4:

Note that when an argument labels exactly one node in a

derivation tree, we identify the argument and the node for

simplicity.

In figure 4,

AOPP(E,T)

{C},OPP(T) = {B,C}. Hence AttackE\ (OPP(T) \

AOPP(E,T)) = ∅. AOPP(D,T) = {C},AttackD =

{B}. Hence AttackD\ (OPP(T) \ AOPP(D,T)) = ∅.

HenceΦO(E,T) = ΦO(D,T) = ∅ satisfies theconstraint. T

would be a successful derivation tree though A is not ground-

edly accepted. The reason is that B,E,C,D,B is a cycle

=

{B},AttackE

=

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in the argumentation framework and this cycle makes the fil-

tering of B,C groundedly unsound. It turns out filtering is

groundedly sound if it avoids filtering out previous opponent

arguments lying in cycles like B,E,C,D,B. This motivates

the following definition.

Definition 5.1 Let T be a derivation tree.

1. Let N be a proponent node and M be an opponent node

and A be the argument of M. M is an adopted child of N

in T if A attacks the argument of N and N has no child

labelled by A1.

2. A generalized path in T is a sequence of nodes

N0,...,Nk, k ≥ 1 such that Ni+1is either a child or

an adopted child of Ni.

A generalized path is a cycle if the first and last nodes co-

incide.

Infigure3, Eis anadoptedchildofN inTandthesequence

B,N,E is a generalized path. B,A,B is a cycle in T?while T

contains no cycle. In figure 4, the sequence B,E,C,D,B is a

cycle in T.

Definition 5.2 Let N be a frontier proponent node in T, and

A be an argument attacking N. A is said to be a taboo for

filtering out by opponent at N if there is an cycle N0,...,Nn

in T such that N = Niand A labels Ni+1for some i.

The set of taboos for filtering out at N by opponent in T is

denoted by TABOO(N,T).

In figure 3, TABOO(N,T) = ∅. Hence it is possible

to filter out the argument E attacking D at node N. In con-

trast, TABOO(N,T?) = {B}. Hence it is not possible

to filter out B at N. T?is hence not a successful derivation

tree while T is. Similarly, in figure 4, TABOO(E,T) =

{C},TABOO(D,T) = {B}.

Theorem 5.1 Let Φ = (ΦP,ΦO) be an eligible legal envi-

ronment such that for each derivation tree T, each frontier

proponent node N labeled by A,

AttackN\ (OPP(T) \ TABOO(N,T)) ⊆ ΦO(N,T)

whenever ΦO(N,T) is defined.

Then Φ is groundedly sound.

Proof Let T0,...,Tnbe a successful dispute derivation wrt

Φ. A cycle is said to be internal if it does not contain any

frontier node. We prove by induction that for each i,there is

no internal cycle in Ti.

Suppose that there is no internal cycle in Tibut some in-

ternal cycles in Ti+1. Let Cyc = N0,...,Nnbe an internal

cycle in Ti+1labelled by A0,...,An. As there is no internal

cycle in Ti, Cyc must contain a frontier node in Ti. Since

Cyc is an internal cycle in Ti+1, Cyc contains exactly one

frontier node in Tithat is the frontier node N at which Tiis

expanded. Let N = Nk. If N is an opponent node in Tithen

Nk+1is a child of N. Hence Nk+1is a frontier node of Ti+1.

Contradiction to the assumption that Cyc is an internal cycle

in Ti+1.

Hence N must be a proponent node in Ti. As Cyc is an

internal cycle, Nk+1is not a frontier node in Ti+1. Hence

1Note that we do not define adopted children for opponent nodes

Nk+1isanadoptedchildofNinTiandTi+1. Fromdefinition

5.2, it follows that Ak+1∈ TABOO(N,Ti).

As Nk+1 is an adopted child of N also in Ti+1, there

is no child of N in Ti+1 labelled by Ai+1.

Ai+1 ?∈ ΦO(N,Ti). From the properties of Φ, it follows

that Ai+1 ?∈ AttackN \ (OPP(Ti) \ TABOO(N,Ti)).

Therefore Ai+1 ∈ (OPP(T) \ TABOO(N,Ti)). Hence

Ai+1?∈ TABOO(N,Ti). Contradiction.

We prove now that there is no cycle in Tn.

that there is a cycle Cyc in Tn. As Tnhas no internal cy-

cle, Cyc must contains a frontier node, say N. Therefore

TABOO(N,Tn) ?= ∅. Hence ΦO(N,Tn) ?= ∅. Contra-

diction.

We prove now that for all nodes M in Tn, if M is a propo-

nent (resp opponent) node then the argument labelling M is

groundedlyaccepted(resp rejected,i.e. attacked bya ground-

edly accepted argument). The proof is by induction on the

rank of the nodes in Tndefined as follows..

The rank of nodes without any kind of children is 0. The

rank of a node M is equal 1 plus the maximum of the ranks of

the children or adopted children of M.

We show by induction that the nodes of even (resp odd)

ranks are proponent (resp. opponent) nodes. Obviously that

nodes of rank 0 are proponent nodes. Suppose now that the

property holds for all nodes of rank ≤ i. Let M be a node of

rank i + 1. As one of the children or adopted children of M

is of rank i, M is a proponent (resp opponent)node if i is odd

(resp even) from the induction hypothesis.

We could easily prove by induction now that nodes of

even (resp odd) ranks are groundedlyaccepted (resp rejected)

based on the following property:

Let N be proponent node in Tn. Then AttackN = {A| A

labels a child or an adopted child of N}.

Let Tibe the tree at which N is selected for expansion.

Suppose A is an argument attacking N and A does not la-

bel a child of N. Hence A ?∈ ΦO(N,Ti). Therefore A ∈

(OPP(Ti) \ TABOO(N,Ti)). Hence there is an opponent

node M in Tilabeled by A. As A does not label a child of N,

M is an adopted child of N. ?

It follows

Suppose

Theorem 5.2 Let Φ = (ΦP,ΦO) be an eligible and ground-

edly sound legal environment such that for each derivation

tree T, each frontier opponent node N in T

AttackN\ (Conf(T) ∪ APRO(N,T)) ⊆ ΦP(N,T)

⊆ AttackN\ (OPP(T) ∪ APRO(N,T))

where APRO(N,T) contains all proponent arguments lying

on the path from the root to N.

Then Φ is groundedly complete and terminating

Proof For groundedly accepted (resp rejected) argument A,

define rank(A) to be the smallest number i such that A ∈

Fi(∅) (resp. A is attacked by Fi(∅)). We prove the theorem

by induction on rank(A).

Let A be a groundedly accepted argument. A successful

derivation T0,...,Tnof A is constructed by induction as fol-

lows:

If Tk is obtained from Tk−1 by expanding an opponent

node then select an argument belonging to Fj(∅) where j is

the rank of the argument labelling the node that is expanded.

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It is not difficult to see that constructing Ti’s that way will

produce a successful derivation wrt Φ. ?

6 Discussion and Conclusion

In [Cayrol,Doutre,Mengin, 2003], two proof procedures

called φ1,φ2are given for credulous semantics. φ1 corre-

sponds to a sequentialization of a legal environment with

Φ1,O(N,T) = AttackN\ (AttackedPRO(T)∪ OPP(T))

and Φ1,P(N,T)=

AttackN \ Conf(T).

sponds to Φ2,O(N,T) = AttackN \ AttackedAPRO(N,T)

and Φ2,P(N,T) =

AttackN \ Conf(N,T)2.

gil,Caminada, 2009] gives also several proof procedures for

credulous semantics. They are somewhat similar to the

φ2 procedure above. For example, one of them corre-

sponds to the legal environment: Φ3,O(N,T) = AttackN\

AOPP(N,T) and Φ3,P(N,T) = AttackN\ Conf(N,T).

While φ1is sound, completeand terminating,φ2,Φ3may not

beeligibleandhencearesoundonlyiftheirsuccessfulderiva-

tion trees are consistent. But no condition is given under

what conditions is the consistency of the successful deriva-

tion trees is guaranteed. [Modgil,Caminada, 2009]gives also

proof procedures for grounded semantics.

corresponds to the sound and complete and terminating le-

gal environment: ΦO(N,T) = AttackN and ΦP(N,T) =

AttackN\ APRO(N,T).

In general, the opponentdoes not need to unleash all of his

attacks at once. By dropping the ”one-one-correspodence”

condition in the second bullet of the definition 3.3 and

requires an extra condition in theorem 4.1 and 5.1 that

ΦO(N,T) does not contain any arguments labelling children

of N and allowing the opponent to attack at all proponent

nodes as long as there is an option for him there, we obtain

new dispute procedures with the same characteristics.

A possibleapplicationof theinsightthat thesoundnessofa

dispute is determined by the opponent moves while its com-

pleteness by the proponent ones, is a modular methodology

for designing dispute procedures. The design consists of four

tasks: ensuringthe eligibility of the legal environment,ensur-

ing the soundness by looking at opponent moves and guaran-

teeing completeness by looking at proponent moves and en-

forcing terminating by forbidding one of the participant not

to repeat his moves.

A possible interesting application of this paper lies in the

design of proof procedures for logic-based argumentation

systems [Dung,Kowalski,Toni, 2006; Dung,Mancarella,Toni

, 2007; Eshghi,Kowalski, 1989; Kakas,Toni, 1999; Dung,

1995a]bytranslatingautomotacicallyproofproceduresinab-

stract argumentationinto ones for the latter systems.

φ2 corre-

[Mod-

The basic one

7 Acknowledgements

We thank the referees for constructive comments and criti-

cisms. This work was partially funded by the Sixth Frame-

work IST program of the European Commission under the

035200 ARGUGRID project.

2Conf(N,T) consists of all arguments that selfattack or attacked

or being attacked by proponent arguments on the path from the root

to N

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