A RECONSTRUCTION OF ABSTRACT ARGUMENTATION

ADMISSIBLE SEMANTICS INTO DEFAULTS AND ANSWER SETS

PROGRAMMING

Farid Nouioua and Vincent Risch

Aix–Marseille Univ, LSIS – UMR CNRS 6168, Facult´e des Sciences de Saint–J´erˆome

avenue Escadrille Normandie–Niemen, 13397 Marseille cedex 20, France

Keywords:

Abstract argumentation frameworks, Answer-sets, Defaults.

Abstract:

Given a default theory, we ﬁrst show that the justiﬁed extensions of this theory characterize the maximal

conﬂict-free sets of the corresponding abstract argumentation framework such as deﬁned by Dung. We then

show how to specialize justiﬁed extensions in order to represent admissible (and hence preferred and stable)

extensions inside default theories. Relying on the correspondance of justiﬁed extensions with ι-answer sets on

one hand, on the semi-monotonic character of justiﬁed extensions on the other hand, we then show that any

admissible (or preferred) set of arguments of the initial argumentation framework can be directly computed

from the ι-answer sets of the equivalent logic program. Eventually, this allows us to consider the addition of

integrity constraints with whom the admissible sets are ﬁltered from each ι-answer set.

1 INTRODUCTION

Since abstract argumentation frameworks have been

introduced by Phan Minh Dung in his seminal pa-

per (Dung, 1995), several authors have considered

the links with default theories and answer set pro-

gramming (Dung, 1995), (Bondarenko et al., 1997),

(Nieves et al., 2008), (Egly et al., 2010). The whole

of these works proceeds from a common approach

which has successfully stressed, both in defaults and

logic programs, the major role played by the idea of

two conﬂicting informations. In this respect, abstract

argumentation sheds a clear light on how nonmono-

tonicity is at work inside these formalisms. With-

out denying this fact, we propose however to return

to the opposite and, in our opinion, barely explored

question, that is: what default theories and logic pro-

grams can tell us about abstract argumentation frame-

works? Because abstract argumentation frameworks

appear formally to be a fragment of defaults, we es-

pecially would like to investigate how one of the most

basic concepts of argumentation frameworks – ad-

missible sets – is related to the same idea in default

theories. Our motivations are manifold. We expect

to clear up in a more precise way the links among

these various formalisms: especially, while the ques-

tion whether the deﬁnition of arguments should gen-

erally rely on logical criterions is controversial (e.g.

in (Amgoud and Besnard, 2009)), we propose the ba-

sis for a reassessment of abstract argumentation un-

der logic. Although out of the scope of this paper, but

from the same point of view, our work intends to base

the ability for a better understanding on how known

results on preferences, cumulativity, or other logical

properties could be applied to argumentation frame-

works. Eventually, we expect to catch interesting and

powerful methods of computation of arguments from

a direct translation of argumentation frameworks into

logic programming. The last section of the current

paper is a step into this direction. Our paper is orga-

nized as follows: we ﬁrst show that maximal conﬂict-

free sets of arguments correspond strictly to justiﬁed

extensions of a default theory (Łukaszewicz, 1988),

and hence to the ι-answer sets (iota-answer sets) of a

logic program (Gebser et al., 2009). We propose then

a characterization of the admissible sets of arguments

of any abstract argumentation framework obtained

from a default theory via an additional constraint on

justiﬁed extensions of this theory. Relying on the bi-

jection of justiﬁed extensions with ι-answer sets, we

then show that any admissible set of arguments of the

initial argumentation framework can be characterized

via the ι-answer sets of the equivalent logic program.

It becomes then possible to add to any such program

a set of integrity constraints that ﬁlters the admissi-

ble sets of arguments from its ι-answer sets. Since ι-

237

Nouioua F. and Risch V..

A RECONSTRUCTION OF ABSTRACT ARGUMENTATION ADMISSIBLE SEMANTICS INTO DEFAULTS AND ANSWER SETS PROGRAMMING.

DOI: 10.5220/0003717002370242

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 237-242

ISBN: 978-989-8425-95-9

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

answer sets inherit semi-monotonicity (from justiﬁed

extensions), this clearly emphasizes their central role

in a possible incremental treatment of arguments with

logic programs. In the second section below, we re-

call some basic notions about abstract argumentation

frameworks (Dung, 1995) and default theories (Re-

iter, 1980). After reminding (after (Dung, 1995)) how

to extract an argumentation frameworkfrom an equiv-

alent initial default theory, we propose in the third sec-

tion the converse translation, that is how to express an

argumentation framework as a default theory. In the

fourth section we establish a mapping between max-

imal conﬂict-free sets of arguments and justiﬁed ex-

tensions, which allows to further characterize admis-

sible sets of arguments (and hence preferred exten-

sions) as a special kind of justiﬁed extensions. The

ﬁfth section extends this characterization to ι-answer

sets and describes the computation of admissible sets

with help of integrity constraints.

In the following, we will denote atomic elements

by lowercase letters and sets by shift case letters. Fol-

lowing a widespread tradition, greek letters are also

used in deﬁnitions and theorems related to defaults

and answer sets. We will use some of the standard

operations of set theory (∪ for union, \ for set differ-

ence, × for cartesian product, 2

for the power set of

S). The symbols ⊤ and ⊥ denote the usual truth val-

ues, and ¬, ∨, ∧ the usual connectors of propositional

logic.

2 PRELIMINARIES

We brieﬂy recall some basic deﬁnitions, ﬁrst on

argumentation frameworks, then on default theories.

Logic programs will be considered in a further

section.

An argumentation framework is a pair hAR, attacksi

where AR is a set and attacks is a relation over AR,

i.e. attacks ⊆ AR × AR. Each element of AR is

called an argument and a attacks b means that there

is an attack from a to b. Accordingly a is said to

be an attacker of b (thus a is a counterargument for

b). By extension, a set S ⊆ AR attacks an argument

a ∈ AR iff some argument in S attacks a. On the

contrary, S defends a iff for each b ∈ AR, if b attacks a

then S attacks b. In this case, a is also said to be

acceptable with respect to S. The attacks relation

induces a kind of coherence with different degrees

among arguments. First, S ⊆ AR is conﬂict free iff

there are no a and b in S such that a attacks b. Further,

S is said admissible iff S is conﬂict free and defends

all its elements. S is called a complete extension iff

S is an admissible set such that each argument that

S defends is in S. A preferred extension is then a

⊆-maximal admissible subset of AR. Eventually, S

is a stable extension iff S is conﬂict free and attacks

each argument that is not in S.

Example 1. Consider the following argument frame-

work AF1, in which the arrows represent the attack

relation over the arguments a, b, c, d, e, f, g:

a b c d e

The admissible sets are:

0, {a}, {c}, {d}, {a, c},

{a, d}, {d, f}, {a, d, f}. The preferred extensions

are {a, c}, {a, d, f}. The unique stable extension

is {a, d, f}. Remind that, whatever the kind of

extension being under consideration (admissible,

preferred, or stable), it is a subset of a maximal

conﬂict-free set, being here one among {a, c, e},

{a, c, f}, {a, c, g}, {a, d, f}, {a, d, g}, {b, d, f},

{b, d, g}, {b, e}.

Let us now brieﬂy remind some of the principal

notions about default reasoning (Reiter, 1980). A

default is an expression of the form

α:β

...β

where α,

, 1 ≤ i ≤ n, and γ are closed ﬁrst-order sentences

with α being called the prerequisite, β

the justiﬁ-

cations, and γ the conclusion. Considering a set of

defaults D, the functions PREREQ(D), JUST(D),

and CONS(D) refer respectively to all prerequisites,

justiﬁcations, and consequences of the defaults of D.

A default theory ∆ is a pair (W, D) where W is a set of

closed ﬁrst-order sentences, and D is a set of defaults.

Intuitively, the consequence of a default holds if its

prerequisite holds and nothing can prevent the justi-

ﬁcation to hold (i.e. the negation of the justiﬁcation

does not hold). The main consequence of this idea is

captured by the notion of extension. The following

characterizations of R- and J-extensions (respectively

due to (Reiter, 1980) and (Łukaszewicz, 1988)) are

given here after (Risch, 1996). Consider ∆ = (W, D).

A subset D

′

of D is grounded in W iff for all d ∈ D

′

there is a ﬁnite sequence d

, . . . d

of elements

of D

′

such that (1) PREREQ({d

}) ∈ Th(W),

(2) for 1 ≤ i ≤ k − 1, PREREQ({d

i+1

}) ∈

Th(W) ∪ CONS({d

, . . . d

}), and d

= d. Then,

let D

′

be any subset of D; E = Th(W ∪CONS(D

′

)) is:

(1) a J-extension of ∆ iff D

′

is a maximal grounded

subset of D such that for all β ∈ JUST(D

′

), ¬β 6∈ E;

(2) a R-extension of ∆ iff it is a J-extension, and for

each default d ∈ D\ D

′

, d =

α:β

...β

, either α 6∈ E or

ICAART 2012 - International Conference on Agents and Artificial Intelligence

238

¬β

∈ E for some β

. When E = Th(W ∪ CONS(D

′

))

is an extension (either J- or R-), the set D

′

is called

the set of generating defaults of E, and is denoted

by GD(E, ∆). Note that J-extensions have interesting

properties: they always exist, they denote consistant

sets (in the standard case), and R-extensions are a

special case easily characterized among J-extensions.

Moreover, they are semi-monotonic i.e. adding new

defaults to a default theory does not remove the

previous J-extensions of this theory. Note that in the

sequel, since there is no need for the full expressive

power of ﬁrst-order logic, we restrict ourselves to

propositional default theories.

3 TRANSLATING ARGUMENTS

INTO DEFAULTS AND

CONVERSELY

Following (Dung, 1995), let us remind how a default

theory can be expressed as an abstract argumentation

framework. Consider ∆ = (W, D) a default theory

and Λ = {β

, . . . , β

} ⊆ JUST(D). A sentence λ

is said to be a defeasible consequence of ∆ and Λ

(Dung, 1995) if there is a sequence (e

, . . . e

) with

= λ such that, for each e

, 0 ≤ i ≤ n, either (1)

∈ W or (2) e

is a logical consequence of the

preceding members in the sequence, or (3) e

is the

conclusion γ of a default

α:β

...β

whose prerequisite

α is a preceding member in the sequence and whose

justiﬁcations β

belongs to Λ. Λ is said to be a

support for λ with respect to ∆. The theory ∆ is

then interpreted as an argumentation framework

hAR

∆

, attacks

∆

i as follows: (1) AR

∆

= {(Λ, λ) | Λ ⊆

JUST(D), Λ is a support for λ with respect to ∆};

(2) (Λ, λ) attacks

∆

(Λ

′

, λ

′

) iff ¬λ ∈ Λ

′

. Conversely,

let us introduce now a simple translation from any

abstract argumentation framework into the langage of

default theories. We ﬁrst deﬁne a so-called Attackers

function from AR to 2

such that for every a, b in

AR, Attackers(a) = {⊤} ∪ {¬b | b attacks a} In

other word, when an argument in hAR, attacksi is

attacked by no argument, the function associates it

with the “empty” attacker ⊤, otherwise it associates it

with the set of its standard attackers logicaly negated.

Any argumentation framework AF = hAR, attacksi

is then interpreted modularly as a default theory

∆

= (

0, D

) with D

= {

⊤: Attackers(a)

| a ∈ AR}

Note that, while a default with a set of justiﬁcations

restricted to ⊤ will participate in the generation of

a consistant extension, the same default but with an

empty set of justiﬁcations may lead to an inconsistant

set. This represents a degenerated case never consid-

ered in standard default reasoning. Hence, in order

to ensure a standard behaviour, the empty attacker ⊤

is indeed the least element needed in the set of justi-

ﬁcations of the defaults resulting from our translation.

Example 2. (continued) Consider again the argument

framework AF1 given above. The corresponding

default theory is given by ∆

= (

0, D

) with

= {

:⊤

:¬a,¬c

:¬d

:¬c

:¬d,¬g

:¬e

:¬ f

}

Note that, where translating an AF to a default

theory generates a number of defaults equal the

number of arguments in AF, the translation in the

other direction, that is from a default theory to an

AF, generates many more (generaly inﬁnitely many)

arguments. This complexiﬁcation is mainly due

to the fact that we move from full propositional

logics (on the side of defaults) to the simple ﬂat

fragment of propositional atoms, negated or not,

with no operation of deductive closure (on the side

of AF). Let us still point out that, as in the logical

approaches of argumentation (cf. (Besnard and

Hunter, 2008)), the standard translations deﬁned here

leads to deﬁne the arguments with a structure under

the form (support, conclusion). Eventually, note that

the notion of defeasible consequence deﬁned above

allows precisely to express the arguments under this

form by sort of removing the prerequisites from the

initial default theory. This stresses indeed the fact that

going from default theories to argument frameworks

is an abstraction process (which, as noticed, is

intractable in its full generality) while going from

argument frameworks to default theories is a modular

translation toward a fragment of defaults (linear,

since there are as many defaults as arguments and,

for each default, as many justiﬁcations as attackers,

plus one for the empty attacker). Now, what remains

to do regarding this translation is to ensure that

any admissible set of arguments get an equivalent

representation as some sort of default extension.

As shown in (Dung, 1995), there is an exact

correspondence between the R-extensions of a

default theory and the stable extensions of an

abstract argumentation theory. Consider A, a ﬁrst-

order theory, and A

′

⊆ AR

∆

a set of arguments

obtained from a default theory ∆. Deﬁne (1)

arg(A) = {(Λ, λ) ∈ AR

∆

| ∀β ∈ Λ, β ∪ A 6⊢}; (2)

ﬂat(A

′

) = {λ | ∃(Λ, λ) ∈ A

′

}. Let ∆ be a default

theory. Then (lemma 42 and theorem 43 of (Dung,

1995)): (1) Given E any R-extension of ∆, arg(E)

is a stable extension of hAR

∆

, attacks

∆

i; (2) Given

′

any stable extension of hAR

∆

, attacks

∆

i, ﬂat(E

′

)

is an R-extension of ∆. In order to consider into

A RECONSTRUCTION OF ABSTRACT ARGUMENTATION ADMISSIBLE SEMANTICS INTO DEFAULTS AND

ANSWER SETS PROGRAMMING

239

defaults the case of other types of extensions used in

abstract argumentation frameworks, we establish the

following corollary which characterizes the existence

of arguments in AR

∆

regarding a default theory ∆.

Corollary 1. Let ∆ = (W, D) be a default the-

ory and hAR

∆

, attacks

∆

i the corresponding argu-

mentation framework. Then: (1) (Λ, λ) ∈ AR

∆

iff

there exists D

′

⊆ D, D

′

grounded in W such that

Λ = JUST(D

′

) and λ ∈ Th(W ∪ CONS(D

′

)); (2)

ﬂat(AR

∆

) =

′

∈2

′

grounded

Th(W ∪ CONS(D

′

)).

4 A J-EXTENSION BASED

APPROACH OF ADMISSIBLE

EXTENSIONS

Let us consider in deeper way how to get arguments

of AR

∆

from a default theory ∆. Our idea is to map

any subset of applied defaults to a subset of argu-

ments as accurately as possible in the most possible

general way, and hence to relate eventually extensions

of AR

∆

with extensions of ∆. Note that Corollary 1

stresses the crucial role played by the subsets of D in

the constitution of any argument (Λ, λ) of AR

∆

, since

for any such argument there exists D

′

⊆ D such that

Λ = JUST(D

′

) and λ ∈ Th(W ∪ CONS(D

′

)). The last

set is precisely of the form taken by the different kinds

of extensions of ∆ (with possibly different constraints

on it). In other words, we expect to relate different

types of subsets of AR

∆

with some type of extension

of ∆ via JUST(D

′

) (for the supports of the arguments)

and Th(W ∪ CONS(D

′

)) (for the consequences of the

same arguments). To achieve this goal however, the

operator arg deﬁned earlier is too sloppy. Hence we

introduce a more accurate operator, directly deﬁned

on a subset of defaults:

Deﬁnition 1. Given a default theory ∆ = (W, D), and

′

⊆ D, let AR

∆

′

) = {(Λ, λ) ∈ AR

∆

| λ ∈ Th(W ∪

CONS(D

′

))}

Obviously, AR

∆

(D) = AR

∆

. We can now come to

the characterization of conﬂict-free sets of arguments

via a subset D

′

of defaults:

Theorem 1. Given a default theory ∆ = (W, D),

let D

′

⊆ D, D

′

grounded in W and E = Th(W ∪

CONS(D

′

)). Then AR

∆

′

) is conﬂict-free iff ∀β ∈

JUST(D

′

), ¬β 6∈ E.

The two following corollaries show that J-

extensions correspond to conﬂict-free maximal sub-

sets of arguments. More precisely, from the deﬁnition

of J-extensions and theorem 1, we get immediately:

Corollary 2. Let ∆ = (W, D) be a default theory

and hAR

∆

, attacks

∆

i the corresponding argumentation

framework. Let E

∆

be any conﬂict-free ⊆-maximal

subset of AR

∆

. Then ﬂat(E

∆

) is a J-extension of ∆.

Corollary 3. Let ∆ = (W, D) be a default theory

and hAR

∆

, attacks

∆

i the corresponding argumentation

framework. Let E be any J-extension of ∆. Then

∆

(GD(E, ∆)) is a conﬂict-free ⊆-maximal subset

of AR

∆

The question is now to ﬁlter J-extensions in or-

der to represent admissible extensions in default logic.

We do it thank to the following characterization theo-

rem:

Theorem 2. Let ∆ = (W, D) be a default theory

and hAR

∆

, attacks

∆

i the corresponding argumenta-

tion framework. Let {D

i,i∈N

} be any enumeration

of the grounded subsets of D and E(D

) = Th(W ∪

CONS(D

)) for any i ∈ N. For any D

′

⊆ D, AR

∆

′

)

is an admissible set of hAR

∆

, attacks

∆

i iff

(i) there is i ∈ N such that D

′

= D

(ii) there is j ∈ N such that D

′

⊆ D

and E(D

) is a

J-extension

(iii) for any k ∈ N, (∃β ∈ JUST(D

′

), ¬β ∈ E(D

)) ⇒

(∃γ ∈ E(D

′

), ¬γ ∈ JUST(D

))

What is shown here is that in order for a subset

of a J-extension to correspond to an admissible set,

one has to check that if any negation of a justiﬁcation

used to derive this subset can be found in one of

the grounded subsets of D (i.e. some argument is

attacked) then some formula of this grounded subset

will be found negated among the initial justiﬁcations

(i.e. the argument is defended). In other words, in

order to compute any admissible set inside a default

theory (and hence any preferred extension when

considering ⊆-maximal subsets), it is sufﬁcient to

ﬁlter inside the J-extensions. The most interesting

consequence of this result comes from the one-to-one

correspondence between J-extensions and ι-answer

sets, which is the matter of the following section.

Note that in the case where the consequence

(∃γ ∈ E(D

′

), ¬γ ∈ JUST(D

)) of the implication of

(iii) is always true, we characterize stable extensions,

which indeed correspond directly to the R-extensions

via the characterization considered earlier.

5 LINK WITH ι-ANSWER SETS

Put into the context of answer set programming, J-

extensions have been shown by (Delgrande et al.,

2003) to correspond to some way of relaxing answer

ICAART 2012 - International Conference on Agents and Artificial Intelligence

240

sets. This idea was further fully developed in (Gebser

et al., 2009) who deﬁnes the ι-answer sets of a logical

program as the exact counterpart of the J-extensions

of the corresponding default theory. Following (Geb-

ser et al., 2009), remind that a normal logic program

is a ﬁnite set of rules of the form

← p

, . . . , p

, not p

m+1

, . . . , not p

where each p

is an atom. For a rule r, head(r) and

body(r) denote the usual corresponding parts of r,

while body

(r) and body

−

(r) denote respectively the

positive part and the negative part of body(r). This

deﬁnition are extended from a rule to a program Π,

e.g. head(Π) = {head(r) | r ∈ Π}. Eventually, note

that an empty head is similar to ⊥, while an empty

body is similar to ⊤. A program Π is called ba-

sic if body

−

(Π) =

0. Each basic program Π has a

unique ⊆-minimal model, denoted by Cn(Π), that is

the smallest set of atoms closed under the rules of Π.

Let Cn

(Π) = Cn(Π

) = Cn(head(r) ←

body

(r) | r ∈ Π). Considering Π a logic pro-

gram and X a set of atoms, X is an ι-answer

set of Π if X = Cn

(Π

′

) for some maximal

′

⊆ Π such that (1) body

(Π

′

) ⊆ Cn

(Π

′

) and

(2) body

−

(Π

′

) ∩ Cn

(Π

′

) =

0. The ι-answer

sets of a program Π correspond to the justiﬁed

extensions of the default theory given by the fol-

lowing known modular translation: each rule r

of Π yields a default

body

(r):¬|body

−

(r)|

head(r)

(where

| S |= {a | not a ∈ S}), and W =

0. Given Π

′

⊆ Π,

let (1) AR

(Π

′

) = {(body

−

(r), head(r)) | r ∈ Π

′

(2) ﬂat

(Π

′

) = {head(r) | r ∈ Π

′

} Given any argu-

mentation framework AF = hAR, attacksi, from the

translation deﬁned earlier we get a default theory

∆

= (

0, D

). In turn, from the modular translation

deﬁned just above, this default theory yields a logic

program Π

with an empty positive body (i.e.

body

(Π)

0). Clearly, AR

(Π

) = AR, and

for every D

′

⊆ D

there exists Π

′

⊆ Π

such that

∆

′

) = AR

(Π

′

) and ﬂat(AR

∆

′

)) = ﬂat

(Π

′

As a consequence of corollaries 2 and 3 we then get

immediately:

Corollary 4. Let Π be a program with an empty pos-

itive body and AR

(Π) the corresponding argumen-

tation framework. For any Π

′

⊆ Π, AR

(Π

′

) is a ⊆-

maximal conﬂict-free subset of AR

(Π) iff head(Π

′

)

is a ι-answer set of Π.

From theorem 2 we get directly:

Corollary 5. Let Π be a program with an empty pos-

itive body and AR

(Π) the corresponding argumen-

tation framework. Let X

, . . . , X

be a collection of

all the ι-answer sets of Π and AR

(Π

), . . . , AR

(Π

)

the corresponding conﬂict-free maximal subsets of

(Π). For any Π

′

⊆ Π, AR

(Π

′

) is an admis-

sible set of AR

(Π) iff there is i, 1 ≤ i ≤ k, such

that Π

′

⊆ Π

and (∀r

′

∈ Π

′

)(∃r ∈ Π\Π

′

)(body

−

′

)∩

head(r) 6=

0 ⇒ head(Π

′

) ∩ body

−

(r) 6=

0).

Example 3. (continued) Back to AF1, we get the fol-

lowing logic program Π

AF1

: a ← r

: e ← not d, not g

: b ← not a, not c r

: f ← not e

: c ← not d r

: g ← not f

: d ← not c

Eight ι-answer sets are generated, namely X

{a, c, e}, X

= {a, c, f}, X

= {a, c, g}, X

= {a, d, f},

= {a, d, g}, X

= {b, d, f}, X

= {b, d, g},

= {b, e}. For instance, consider especially X

and X

that respectively correspond to the following

two conﬂict-free ⊆-maximal subsets of AR

(Π

AF1

(Π

) = {({}, a), ({not d}, c), ({not d, not g}, e)},

(Π

) = {({}, a), ({not c}, d), ({not f}, g)}, with

= {r

, r

}, Π

= {r

, r

}. Applying corol-

lary 5, it is easy to check that while ({not f}, g)

from AR

(Π

) attacks ({not d, not g}, e) from

(Π

) (that is body

−

) ∩ head(r

) 6=

0),

(Π

) does not defends itself from this attack

(that is head(Π

) ∩ body

−

) =

0). This means that

(Π

) is not admissible and that the argument

({not d, not g}, e) has to be removed in order to get

{({}, a), ({not d}, c)} as an admissible subset of

(Π

AF1

). Similar checks apply to all the ι-answer

sets found here.

Let us now deﬁne the counterpart of an admis-

sible set of arguments inside a logic program:

Deﬁnition 2. Let Π be a logic program and X be a set

of atoms. X is called an admissible answer set of Π iff

there is Π

′

⊆ Π such that X = ﬂat

(Π

′

) and AR

(Π

′

)

is an admissible set of AR

(Π).

Following (Gebser et al., 2009), we can augment

our framework with integrity constraints whose pur-

pose here will be to ﬁlter inside the ι-answer sets the

subsets that are admissible. Remind that an integrity

constraint is a rule c with an empty head, that is

← p

, . . . , p

, not p

m+1

, . . . , not p

After (Gebser et al., 2009), we consider a constraint c

satisﬁed with respect to a set X of atoms if for any rule

r ofΠ, body

−

(c)∩X 6=

0. In order to

eliminate into ι-answer sets the subsets that would not

correspond to admissible sets of AR

(Π) for a given

program Π obtained from an abstract argumentation

framework, let

= {← head(r

′

), body

−

(r) |

r, r

′

∈ Π, head(r) ∈ body

−

′

)}

A RECONSTRUCTION OF ABSTRACT ARGUMENTATION ADMISSIBLE SEMANTICS INTO DEFAULTS AND

ANSWER SETS PROGRAMMING

241

Remark: Just as we label every rule of a program

with a unique natural number, we ﬁnd convenient to

label every constraint from the two rules that yield

it in a program, i.e. in the sequel c

will denote

← head(r

), body

−

) when head(r

) ∈ body

−

Deﬁnition 3. Let Π be a logic program, C

be a set

of integrity constraints and X be a set of atoms. X is

admissible with respect to C

iff X is a subset of an

ι-answer set of Π such that every c ∈ C

is satisﬁed

with respect to X.

Theorem 3. Let Π be a logic program and X be a set

of atoms. Then X is an admissible answer set of Π iff

X is admissible with respect to C

Example 5. (continued) Back to AF1, we add to the

logic program Π

AF1

the set of constraints C

AF1

: ← b c

: ← f, not d, not g

: ← b, not d c

: ← g, not e

: ← e, not c c

: ← e, not f

= {a, c, e} is eliminated by c

since body

) ⊆

while body

−

)∩X

0, and hence c

is not ad-

missibly satisﬁed with respect to X

. On the contrary,

\ {e} is an admissible set of Π

AF1

with respect to

AF1

. Note that any set containing b is immediately

eliminated by c

. Note that in order to automatize

the enumeration of the possible candidates among the

subsets of the X

, it is necessary to relax the deﬁni-

tion of an ι-answer set by removing the condition on

maximality.

6 CONCLUSIONS

In this paper, we have established a characterization

of the admissible semantics deﬁned by Dung inside

defaults and answer-set programming. Although not

surprising, to our opinion these results show a closer

relation of abstract argumentation frameworks with

defaults and answer sets than initially described by

Dung. Notably, and contrary to the approaches used

for instance by (Dung, 1995), (Nieves et al., 2008)

or (Egly et al., 2010), a ﬁrst-order encoding appears

useless for processing abstract argumentation frame-

works with logic programs. Especially, this means

that no grounding is necessary for the logic programs

obtained from our transformation of argumentation

frameworks. Among further perspectives, we are con-

cerned with the ability to extend the current charac-

terization to other different semantics, e.g. obviously

complete extensions, but also the CF2 (Baroni et al.,

2005) or the semi-stable (Caminada, 2006) seman-

tics. Of course, and regarding complexity issues, we

are also concerned with a detailed comparison with

the computation methods proposed in (Nieves et al.,

2008) and (Egly et al., 2010). Finally, note that an-

other direction under way concerns the possibility to

extend the bipolar approach of abstract argumentation

frameworks in order to provide them with the same

expressive power as normal logic programs.

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