HANDLING PREFERENCES IN ARGUMENTATION

FRAMEWORKS WITH NECESSITIES

Imane Boudhar, Farid Nouioua and Vincent Risch

LSIS - UMR CNRS 6168, Avenue Escadrille Normandie Niemen, 13397, Marseille Cedex 20, France

Keywords:

Abstract argumentation, Argummentation frameworks with necessities, Preferences, Acceptability semantics.

Abstract:

Argumentation theory is a promising reasoning model which is more and more used to solve various key

problems in artiﬁcial intelligence. Most of the developments in this domain are based on extended versions of

Dung argumentation frameworks (AFs). In this paper, we propose an argumentation model that extends Dung

AFs by two additional aspects : a necessity relation that represents a particular positive interaction between

arguments and a preference relation that allows to represent arguments that do not have the same strength.

1 MOTIVATION

In argumentation theory, handling preferences is mo-

tivated by the fact that in real contexts, arguments are

often different in strength. Regardless of the source

and nature of the information about preferences, in

Dung style model, a main concern in the different pro-

posed approaches lies in solving possible conﬂicts be-

tween preferences and attacks. Intuitively, the prob-

lematic case is that of critical attacks arising when

an argument attacks another one while the former

is less preferred than the second. Most of existing

approaches of preference-based argumentation like

(Amgoud and Cayrol, 2002), (Bench-Capon, 2003)

and (Modgil, 2009) suggest to merely removethe crit-

ical attacks. A main drawback of these approaches

is the possibility to tolerate extensions that are not

conﬂict-free with respect to the initial attack relations.

To overcome this limit, the approach in (Amgoud and

Vesic, 2010) (Amgoud and Vesic, 2011), that we will

call here the repairing-based approach, suggets to in-

verse the direction of any critical attack. The under-

pining idea is to keep the incompatibility between the

arguments involved in the attack while respecting the

explicit information about their preferences.

On the other hand, some works have been de-

voted to extend Dung’s model in order to represent the

idea of support as a positive interaction between ar-

guments. (Cayrol and Lagasquie-Schiex, 2005) pro-

poses the bipolar argumentation frameworks (BAFs)

by adding an explicit support relation to Dung AFs.

In (Cayrol and Lagasquie-Schiex, 2010) methods to

turn BAFs into Dung meta AFs are proposed. A

main drawback of this approach is that the new

proposed semantics do not guarantee admissibility.

(Boella et al., 2010) introduces the so-called deduc-

tive supports and proposes a meta framework which

ensures admissibility of extensions. (Brewka and

Woltran, 2010) proposes abstract dialectical frame-

works (ADFs), a powerful generalization of Dung

AFs to formalize the concept of proof standards. The

acceptability semantics are redeﬁned by adapting the

Gelfond/Lifshitz reduct used in logic programs (LPs).

(Nouioua and Risch, 2011) starts from the idea that

the exact meaning of the support is essential to deter-

mine its possible interactions with the attack relation.

It considers the case where a supports b means that a

is necessary for b. This specialization allows to gener-

alize the acceptability semantics in a natural way that

ensures admissibility. The aim of this paper is to ex-

tend Dung AFs to take into account both necessity and

preference relations between arguments. To do so, a

ﬁrst concern will be to understand hownecessities and

preferences should interact. Then, on the light of this

understanding, the repairing-based approach will be

adapted to the case of AFs with necessities (AFNs).

Section 2 represents a background that recalls the

main ideas of the preference-based AFs (we present

namely the repairing-based approach) as well as the

argumentation frameworks with necessities (AFNs).

In section 3 we present a new method to construct

a Dung meta AF from an AFN. We show in sec-

tion 4 how to use this new method to generalize the

repairing-based approach to the case of preference-

based AFNs. In section 5, we conclude and discuss

some perspectives of future work.

340

Boudhar I., Nouioua F. and Risch V..

HANDLING PREFERENCES IN ARGUMENTATION FRAMEWORKS WITH NECESSITIES.

DOI: 10.5220/0003746103400345

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 340-345

ISBN: 978-989-8425-95-9

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2 BACKGROUND

2.1 Preferences in Dung’s AFs : The

Repairing-based Approach

A Dung AF (Dung, 1995) is a pair F = hA, Ri where

A is a set of arguments and R is a binary attack re-

lation over A. A set S ⊆ A attacks an argument b iff

there is a ∈ S such that a R b. S is conﬂict-free iff

there is no a, b ∈ S such that a R b. The ⊆-maximal

conﬂict-free subsets of A are called naive extensions

(Bondarenko et al., 1997) and represent a ﬁrst man-

ner to construct sets of acceptable arguments. Many

other acceptability semantics have been proposed in

(Dung, 1995). We focus in this paper on one of them,

the stable semantics : S is a stable extension iff S is

conﬂict-free and ∀a ∈ A\S, S R a.

We mean here by repairing-based approach the

works presented in (Amgoud and Vesic, 2010) (Am-

goud and Vesic, 2011) which renew and extend the

initial approach proposed in (Amgoud and Cayrol,

2002) for preference-based AFs in order to overcome

a common limit of most of existing approaches,

which is the possibility to obtain extensions that

are not conﬂict-free. To do so, the repairing-based

approach inverses the direction of critical attacks

instead of removing them. Formally, this version of

preference-based AFs is deﬁned as follows :

Deﬁnition 1. A preference-based AF (PAF) is a

tuple Λ = hA, R, ≥i where hA, Ri is a Dung AF and

≥⊆ A × A is a preorder. The stable extensions of Λ

are the stable extensions of the repaired framework

hA, Atti where Att = {(a, b)|a R b and not (b >

a)} ∪ {(b, a)|a R b and (b > a)}.

In addition to repairing attacks, a second role of

the preference relation is to compare subsets of A.

Deﬁnition 2. Let S be a set of objects and ≥⊆ S × S

be a preorder. The democratic relation ≥

⊆ 2

× 2

based on ≥ is deﬁned as follows : ∀X

, X

⊆ A, X

≥

iff ∀x

∈ X

\ X

, ∃x

∈ X

\ X

such that x

> x

A rich PAF is a PAF equipped with a reﬁnement

relation ≥

used to select the best extensions.

Deﬁnition 3. A Rich PAF is a tuple τ = hA, R, ≥, ≥

where hA, R, ≥i is a PAF and ≥

⊆ 2

× 2

is the

democratic relation based on ≥ called a reﬁnement

relation. Let Ψ be the set of stable extensions of the

PAF hA, R, ≥i. The reﬁnement relation ≥

is used to

select the best elements of Ψ : Max(Ψ, ≥

) = {ψ ∈

Ψ| 6 ∃ψ

′

∈ Ψ s.t. ψ

′

≥

ψ and not (ψ ≥

′

)}.

2.2 AFs with Necessities

The AFNs (Nouioua and Risch, 2011) extend Dung

AFs by a support relation having the meaning of

necessity. Let us present brieﬂy their main ideas.

Deﬁnition 4. An AFN is a tuple Γ = hA, R, Ni

where A is a set of arguments, R is a binary attack

relation and N is a binary irreﬂexive and transitive

relation, called the necessity relation. For two argu-

ments a, b ∈ A, a N b means that a is necessary for

b, i.e. if b is accepted then a must have been accepted.

The irreﬂexive and transitive nature of N excludes

any risk to have a cycle of necessities. Indeed, such

cycles are undesirable because they correspond to a

kind of fallacy (begging the question). Notice that

one may easily generalize the following results to an

arbitrary necessity relation, by just ﬁltering out the

extensions containing cycles of necessities. Let us

now deﬁne the key notions of coherence and strong

coherence used in redeﬁning the extensionsfor AFNs.

Deﬁnition 5. Let Γ = hA, R, Ni be an AFN and

S ⊆ A. S is coherent iff S is closed under N

−1

, i.e.

∀a ∈ S, ∀b ∈ A, if b N a then b ∈ S. S is strongly

coherent iff it is coherent and conﬂict-free (w.r.t R).

Let us now deﬁne the naive and stable extensions :

Deﬁnition 6. Let Γ = hA, R, Ni be an AFN and

S ⊆ A. S is a naive extension of Γ iff S is a ⊆-

maximal strongly coherent subset of A. S is a stable

extension of Γ iff S is strongly coherent and (∀a ∈

A\ S) either S R a or ((∃b ∈ A\ S) such that b N a).

A ﬁrst couple of results that hold for AFNs are

given by the following propositions 1 and 2 :

Proposition 1. Naive extensions of an AFN are

independent from the directions of attack links.

Proposition 2. Any stable extension of an AFN is a

naive extension. The inverse is not true.

Example 1. Consider the AFN Γ = hA, R, Ni depicted

in ﬁgure 1-(1) (attacks are represented by continuous

arcs and necessities by dashed arcs). The strong co-

herent sets are : {a}, {a, b}, {c} and {c, d}. Among

them {a, b} and {c, d} are the naive extensions. {c, d}

is also stable because A\ {c, d} = {a, b} and we have

{c, d} R a and a N b but a ∈ A \ {c, d}. However,

{a, b} is not stable because A\ {a, b} = {c, d} and we

have neither {a, b} R c nor x N c for any x ∈ {c, d}.

HANDLING PREFERENCES IN ARGUMENTATION FRAMEWORKS WITH NECESSITIES

341

Figure 1: (1) An AFN, (2) The corresponding meta AF.

3 AFNs AS META AFs

In this section we present a new approach to turn

any AFN into a meta Dung AF so that the usual

Dung acceptability semantics may be applied. A

similar approach has been proposed in (Cayrol

and Lagasquie-Schiex, 2010) for BAFs where the

so-called coalitions of arguments are used as meta

arguments. Intuitively, a coalition of arguments

is a ⊆-maximal conﬂict-free subset of arguments

connected with the support relation. For example, the

system of ﬁgure 1-(1) has two coalitions : {a, b} and

{c, d}. Each of them is the unique element of a naive

and stable extension of the meta AF. But in the result

we expect, only {c, d} must be stable. To obtain

this result, we propose a new method to build new

coalitions of arguments that we call here clusters.

Intuitively, each argument gives rise to a cluster that

contains all arguments that are necessary for it.

Deﬁnition 7. Let Γ = hA, R, Ni be an AFN and an

argument a ∈ A. the cluster corresponding to a is

deﬁned by : C

= {a} ∪ {b | b N a}.

Unlike (Cayrol and Lagasquie-Schiex, 2010), the

deﬁnition of clusters takes into account the direction

of the necessity arcs and it is not required that a

cluster is conﬂict-free

. Then, a cluster attacks

another if the former contains at least an argument

that attacks (w.r.t R) an argument of the second :

Deﬁnition 8. Let Γ = hA,R, Ni be an AFN.

The Dung meta AF corresponding to Γ is

= h∆, Atti where ∆ is the set of the clus-

ters constructed from all the arguments of A

(∆ = {C

|a ∈ A}) and Att is an attack relation deﬁned

by : C

Att C

iff ∃x ∈ C

, ∃y ∈ C

such that x R y.

The traditional acceptability semantics are then

applied on the meta AF. The ﬂattening of the re-

sulting extensions gives the extensions under the

same semantics of the original AFN. Proposition

3 formalizes this result for stable and naive semantics.

Conﬂictual clusters lead to self attacked meta argu-

ments that do not belong to any extension.

Proposition 3. Let Γ = hA, R, Ni be an AFN and

= h∆, Atti the corresponding meta AF. If S is a sta-

ble (resp. naive) extension of Γ then E = {C

|a ∈ S}

is a stable (resp. naive) extension of F

. Inversely,

if E = {C

, . . .C

∈ A,C

∈ ∆} is a stable (resp.

naive) extension of F

then S = {a

, . . . a

} is a stable

(resp. naive) extension of Γ.

Example 1 (continued). The meta AF correspond-

ing to the AFN Γ of ﬁgure 1-(1) is F

= h∆, Atti such

that ∆ = {C

} with: C

= {a}, C

= {b, a},

= {c}, C

= {d, c} and Att is depicted in ﬁgure

1-(2). F

has two naive extensions : {C

} and

} whose ﬂattened forms {a, b} and {c, d} are

the naive extensions of Γ. Only {C

} is a stable

extension of F

and it corresponds to the only stable

extension of Γ : {c, d}.

4 PREFERENCES IN AFNs

We are now ready to analyze what happen when we

put together the necessity relation and the information

about preferences in a same framework. In particular

we will give a generalization of the repairing-based

approach to preference-based AFNs and for that

purpose we will use the Dung meta Framework

corresponding to an AFN that we discussed in the

previous section. A preference-based AFN is deﬁned

simply by adding a preference relation to an AFN :

Deﬁnition 9. A preference-based AFN is deﬁned by

Σ = hA, R, N, ≥i where Γ = hA, R, Ni is an AFN and

≥⊆ A × A is a preference relation : ≥ is a (partial or

total) preorder over the elements of A.

Handling preferences within Dung AFs is based

on the idea that the very meaning of an attack hides

an implicit preference of the attacker over the attacked

argument. Additional information about explicit pref-

erences is then treated by solving the possible con-

ﬂicts arising from these two kinds of preferences,

either by removing or inversing the critical attacks.

Now, the question is to knowwhat kind of interactions

results from preferences and necessities and what are

the appropriate treatments to capture them. A ﬁrst

idea that comes to mind is to see if the case of neces-

sities can be handled in a similar manner as attacks,

i.e., if a necessity relation hides a preference that may

contradict an explicit preference. The answer is neg-

ative because in general we can ﬁnd cases where an

argument a is necessary for an argument b while b is

considered as preferred to a and other cases where a

ICAART 2012 - International Conference on Agents and Artificial Intelligence

342

is necessary for b and a is preferred to b.

To propose a method for handling preferences in

AFNs, let us turn to the meaning of a necessity re-

lation. To accept an argument a, we have to accept

all its necessary arguments. But since these necessary

arguments may be more or less preferred than a, the

initial preference of a becomes only a gross prefer-

ence and its effective preference will depend on the

preferences of all its necessary arguments. In other

words, the effective preference of an argument a will

correspond to the set of the initial preferences of all its

necessary arguments in addition to its proper initial

preference, i.e., the preferences of all the arguments

of the cluster C

. Then, the interaction between ne-

cessities and preferences will be captured by means

of a new preference relation induced from the initial

one and deﬁned on sets of arguments.

4.1 Using a Meta PAF

From the previous analysis, the ﬁrst idea is to turn a

preference-based AFN into a meta PAF deﬁned by the

meta AF which corresponds to the AFN (without the

preference relation) in addition to a new preference

relation deﬁned on the set of clusters. Different meth-

ods have been proposed in the literature for the use

of a preference relation on single objects to induce a

preference relation on sets of these objects. Among

them we can ﬁnd the democratic and the elitist

relations. Unlike the elitist relation which privileges

minimal sets (if A ⊂ B then A ≥ B) the democratic

relation privileges the maximal sets (if A ⊂ B then

A ≥ B). This represents an intuitive motivation for

our choice to use the democratic relation to compare

our clusters, since our aim will be to compute (naive

and stable) extensions that are maximal sets verifying

some conditions. The following deﬁnition describes

how to turn a preference-based AFN into a meta PAF.

Deﬁnition 10. We turn any preference-based AFN

Σ = hA, R, N, ≥i into the meta PAF Λ = h∆, Att, ≥

where h∆, Atti is the meta AF corresponding to

hA, R, Ni and ≥

⊆ ∆ × ∆ is the democratic relation

based on ≥ (i.e. ∀C

⊆ ∆,C

≥

iff ∀x

∈

\ C

, ∃x

∈ C

\ C

such that x

> x

). If E is a

stable (naive) extension of Λ then S = {a|C

∈ E} is

a stable (resp. naive) extension of Σ.

It is worth noticing that preferences play com-

pletely different roles when interacting with attacks

and with necessities. Indeed, when the necessity

relation is absent, preferences are directly used

to repair the attack relation. However when the

necessity relation is present we start ﬁrst by us-

ing it to revise the preferences given initially as

input in the framework. This corresponds also to

a kind of reparation but here, it is the preference

relation which is repaired using the information

about necessities and not vice versa as in the case of

attacks. Then, the revised preferences are used to re-

pair the attack relation (between the clusters) as usual.

Example 2. Consider Σ = hA, R, N, ≥i where the

AFN hA, R, Ni is illustrated in ﬁgure 2-(1) and the

preference relation is deﬁned by : a ≥ b, c ≥ d.

Figure 2: (1) A preference-based AFN, (2) The meta PAF

before repairing (3) The meta PAF after repairing.

Following deﬁnition 10, we apply the following steps:

1. The set of clusters is ∆ = {C

= {a, e},C

{b},C

= { c, e},C

= { d},C

= {e}} and the

attack relation Att is deﬁned by : C

Att C

, C

Att C

, C

Att C

, C

Att C

. Figure 2-(2) depicts the meta PAF be-

fore the reparation of Att. The democratic relation

≥

⊆ ∆ × ∆ based on ≥ is deﬁned by : C

≥

, C

≥

, C

≥

, C

≥

, C

≥

, C

≥

, C

≥

, C

≥

2. The strict version >

of the relation ≥

is de-

ﬁned by : C

, C

. Thus, the critical attacks are C

Att C

and C

Att C

. These attacks are then inversed and

we obtain the repaired attack relation Def deﬁned

as follows : C

Def C

, C

Def C

, C

Def C

, C

Def C

, C

Def C

. The resulting

meta AF h∆, Defi is depicted in ﬁgure 2-(3).

3. The naive (and stable) extensions of h∆, De fi are

= {a, e},C

= {c, e},C

= {e}} and {C

{b},C

= {d}}. We deduce then that the naive

and stable extensions of Σ are : {a, c, e}, {b, d}.

Now, let us give some properties for the exten-

sions of preference-based AFNs. The ﬁrst result is

that preference-based AFNs represent a proper gen-

eralization of both AFNs and PAFs. Indeed, when

the preference relation is reduced just to the reﬂexive

relation, the extensions of the preference-based AFN

coincide with that of the corresponding AFN (propo-

sition 4) and when the necessity relation is absent, we

obtain the same results of PAFs (proposition 5):

HANDLING PREFERENCES IN ARGUMENTATION FRAMEWORKS WITH NECESSITIES

343

Proposition 4. Let Σ = hA, R, N,≥i be a preference-

based AFN where ≥= {(a, a)|a ∈ A}. The stable

(resp. naive) extensions of Σ coincide with the stable

(resp. naive) extensions of the AFN Γ = hA, R, Ni.

Proposition 5. Let Σ = hA, R, N,≥i be a preference-

based AFN where N =

0 then, the stable (resp. naive)

extensions of Σ coincide with the stable (resp. naive)

extensions of the PAF Λ = hA, R, ≥ i.

The results of propositions 1 and 2 continue

to hold for preference-based AFNs. Moreover, is

expressed in terms of the preference relation :

Proposition 6. Let Σ = hA, R, N,≥i be a preference-

based AFN. Naive extensions of Σ are independent

from the preference relation ≥ and correspond to the

naive extensions of the simple AFN Γ = hA, R, Ni.

Proposition 7. Let Σ = hA, R, N,≥i be a preference-

based AFN. Any stable extension of Σ is a naive

extension of Σ.

The following interesting corollary determines in

some sense the role of preferences in an AFN.

Corollary 1. Adding or updating preferences in a

AFN affects the selection function of stable exten-

sions among naive extensions that remain unchanged.

4.2 Using a Meta Rich-PAF

As pointed out in (Amgoud and Vesic, 2010) (Am-

goud and Vesic, 2011), a further role of the preference

relation consists in inducing a reﬁnement relation to

compare sets of arguments. This allows to compare

the extensions obtained under a given semantics.

Following the same principle, we associate to a

preference-based AFN a meta Rich-PAF which adds

to the meta PAF deﬁned in the previous section a

reﬁnement relation deﬁned on sets of clusters. We

use the democratic relation based on ≥

Deﬁnition 13. Let Σ = hA, R, N,≥i be a preference-

based AFN and Λ = h∆, Att, ≥

i be the corre-

sponding meta PAF. We deﬁne the reﬁnement

relation ⊲ ⊆ 2

∆

× 2

∆

as the democratic relation

based on ≥

, i.e., ∀ξ

, ξ

⊆ ∆, ξ

⊲ ξ

iff ∀c

∈

\ ξ

, ∃c

∈ ξ

\ ξ

such that c

, (i.e. c

≥

and not(c

≥

)).

Once the reﬁnement relation is deﬁned, it is easy

to deﬁne the meta Rich-PAF corresponding to a

preference-based AFN as follows:

Deﬁnition 14. Let Σ = hA, R, N, ≥i be a preference-

based AFN. We deﬁne the corresponding meta

Rich-PAF by τ = h∆, Att, ≥

, ⊲i where h∆, Att, ≥

is the corresponding meta PAF and ⊲ is a reﬁnement

relation (in the sense of deﬁnition 13).

Now, among the stable extensions of the meta

PAF, only the maximal ones w.r.t the reﬁnement

relation ⊲ are chosen as extensions of τ.

Deﬁnition 15. Let Σ = hA, R, N, ≥i be a preference-

based AFN and τ = h∆, Att, ≥

, ⊲i be the corre-

sponding meta Rich-PAF. The stable extensions of

Σ seen as a rich PAF (we will call them rich-stable

extensions) are the elements of Max(Ψ, ⊲) (the

maximal elements of Ψ with respect to ⊲) where Ψ is

the set of ﬂattened forms of the stable extensions of

the meta PAF Λ = h∆, Att, ≥

Notice that we have not distinguished also the

rich-naive extensions ( i.e., the maximal naive exten-

sions w.r.t reﬁnement relation ⊲) because they simply

coincide with the rich-stable extensions.

Example 2 (continued). Let us take again the

preference-based AFN Σ of example 2. We have

seen that the corresponding meta PAF has two ex-

tensions : {C

} and {C

}. It is not difﬁ-

cult to check that the comparison between these two

extensions w.r.t to the reﬁnement relation ⊲ gives :

} ⊲ {C

} and we have not {C

} ⊲

}. Thus the only stable extension of the

corresponding meta Rich-PAF is : {C

} and

{a, c, e} is then the unique rich-stable extension of Σ.

5 DISCUSSION

This paper has shown how to handle information

about preferences in a kind of bipolar Dung style

framework where the support relation has the partic-

ular meaning of necessity. The precise meaning of

the support relation allowed to specify how informa-

tion about preferences should be taken into account.

The main idea in this context was to distinguish in

a sense two levels in representing preference. The

ﬁrst level is the input level corresponding to the in-

put preference relation. The second level which is

the effective one, considers that the effective prefer-

ence of an argument depends on the preferences of

all the arguments it requires, since accepting an ar-

gument imposes to accept all its necessary arguments

regardless their quality. Based on this analysis, the

paper proposed an extension of different results of the

ICAART 2012 - International Conference on Agents and Artificial Intelligence

344

repairing-based approach to the case of AFNs.

The ideas developed in this work remain valid in

the context of Dung style argumentation frameworks

where no assumption is made on the structure of ar-

guments. However, an entire body of work on argu-

mentation is based on structured arguments and a va-

riety of attack relations. This body includes abstract

argumentation systems (Vreeswijk, 1997), defeasible

logic programming (Simari and Loui, 1992) (Gar-

cia and Simari, 2004), defeasible logic (G. Governa-

tori and Billington, 2004), logical-based argumenta-

tion (Besnard and Hunter, 2008), logic-programming

based argumentation system (Prakken and Sartor,

1997) and recently the ASPIC system (Caminada and

Amgoud, 2007), (Prakken, 2010), (Prakken, 2011).

In all these approaches arguments are structured and

represent deductive or defeasible inferences. Thus

the notion of support is already present in such ap-

proaches as an internal mechanism in the argument it-

self. It would be interesting to study the possible links

between these kinds of supports and our necessity re-

lation. We want in particular to check if our necessity

relation can be seen as an abstraction of these kinds

of supports and if it is the case, to deﬁne methods al-

lowing to see the argumentation approaches based on

structured arguments as instantiations of AFNs. Also,

working on arguments with structures may lead to re-

vise some of the basic hypotheses of the present work.

For example, it may limit the cases where the in-

teraction between preferences and attacks is handled

simply by inversing the directions of critical attacks.

Consequently, in presence of necessities, even if we

keep the idea that the effective preference of an argu-

ment depends on the preferences of all their required

arguments, the handling of the resulting attacks be-

tween clusters of arguments would require a revision

that takes into account the structures of arguments.

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