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
Copyright
c
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
d
2
A
× 2
A
based on is defined as follows : X
1
, X
2
A, X
1
d
X
2
iff x
2
X
2
\ X
1
, x
1
X
1
\ X
2
such that x
1
> x
2
.
A rich PAF is a PAF equipped with a refinement
relation
d
used to select the best extensions.
Definition 3. A Rich PAF is a tuple τ = hA, R, ,
d
i
where hA, R, ≥i is a PAF and
d
2
A
× 2
A
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
d
is used to
select the best elements of Ψ : Max(Ψ,
d
) = {ψ
Ψ| 6 ψ
Ψ s.t. ψ
d
ψ and not (ψ
d
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
= {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
1
. 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
F
Γ
= h, Atti where is the set of the clus-
ters constructed from all the arguments of A
( = {C
a
|a A}) and Att is an attack relation defined
by : C
a
Att C
b
iff x C
a
, y C
b
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.
1
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
F
Γ
= h, Atti the corresponding meta AF. If S is a sta-
ble (resp. naive) extension of Γ then E = {C
a
|a S}
is a stable (resp. naive) extension of F
Γ
. Inversely,
if E = {C
a
1
, . . .C
a
n
|a
i
A,C
a
i
} is a stable (resp.
naive) extension of F
Γ
then S = {a
1
, . . . a
n
} 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
Γ
= h, Atti such
that = {C
a
,C
b
,C
c
,C
d
} with: C
a
= {a}, C
b
= {b, a},
C
c
= {c}, C
d
= {d, c} and Att is depicted in figure
1-(2). F
Γ
has two naive extensions : {C
a
,C
b
} and
{C
c
,C
d
} whose flattened forms {a, b} and {c, d} are
the naive extensions of Γ. Only {C
c
,C
d
} is a stable
extension of F
Γ
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
a
. 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,
d
i
where h, Atti is the meta AF corresponding to
hA, R, Ni and
d
× is the democratic relation
based on (i.e. C
1
,C
2
,C
1
d
C
2
iff x
2
C
2
\ C
1
, x
1
C
1
\ C
2
such that x
1
> x
2
). If E is a
stable (naive) extension of Λ then S = {a|C
a
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
= {a, e},C
b
=
{b},C
c
= { c, e},C
d
= { d},C
e
= {e}} and the
attack relation Att is defined by : C
b
Att C
a
,
C
b
Att C
c
, C
b
Att C
e
, C
d
Att C
a
, C
d
Att C
c
,
C
d
Att C
e
. Figure 2-(2) depicts the meta PAF be-
fore the reparation of Att. The democratic relation
d
× based on is defined by : C
a
d
C
a
,
C
a
d
C
b
, C
a
d
C
e
, C
b
d
C
b
, C
c
d
C
c
, C
c
d
C
d
, C
c
d
C
e
, C
d
d
C
d
, C
e
d
C
e
.
2. The strict version >
d
of the relation
d
is de-
fined by : C
a
>
d
C
b
, C
a
>
d
C
e
, C
c
>
d
C
d
,
C
c
>
d
C
e
. Thus, the critical attacks are C
b
Att C
a
and C
d
Att C
c
. These attacks are then inversed and
we obtain the repaired attack relation Def defined
as follows : C
a
Def C
b
, C
b
Def C
c
, C
b
Def C
e
,
C
c
Def C
d
, C
d
Def C
a
, C
d
Def C
e
. The resulting
meta AF h, Defi is depicted in figure 2-(3).
3. The naive (and stable) extensions of h, De fi are
{C
a
= {a, e},C
c
= {c, e},C
e
= {e}} and {C
b
=
{b},C
d
= {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
d
.
Definition 13. Let Σ = hA, R, N,≥i be a preference-
based AFN and Λ = h, Att,
d
i be the corre-
sponding meta PAF. We define the refinement
relation 2
× 2
as the democratic relation
based on
d
, i.e., ξ
1
, ξ
2
, ξ
1
ξ
2
iff c
2
ξ
2
\ ξ
1
, c
1
ξ
1
\ ξ
2
such that c
1
>
d
c
2
, (i.e. c
1
d
c
2
and not(c
2
d
c
1
)).
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,
d
, i where h, Att,
d
i
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,
d
, 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,
d
i.
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
a
,C
c
,C
e
} and {C
b
,C
d
}. It is not diffi-
cult to check that the comparison between these two
extensions w.r.t to the refinement relation gives :
{C
a
,C
c
,C
e
} {C
b
,C
d
} and we have not {C
b
,C
d
}
{C
a
,C
c
,C
e
}. Thus the only stable extension of the
corresponding meta Rich-PAF is : {C
a
,C
c
,C
e
} 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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