=Paper=
{{Paper
|id=Vol-2777/paper69
|storemode=property
|title=On the Semantics of Recursive Bipolar AFs and Partial Stable Models
|pdfUrl=https://ceur-ws.org/Vol-2777/paper69.pdf
|volume=Vol-2777
|authors=Gianvincenzo Alfano,Sergio Greco,Francesco Parisi,Irina Trubitsyna
|dblpUrl=https://dblp.org/rec/conf/aiia/AlfanoGPT20
}}
==On the Semantics of Recursive Bipolar AFs and Partial Stable Models==
https://ceur-ws.org/Vol-2777/paper69.pdf
On the Semantics of Recursive Bipolar AFs
and Partial Stable Models?
Gianvincenzo Alfano, Sergio Greco, Francesco Parisi, and Irina Trubitsyna
DIMES Department, University of Calabria, Rende, Italy
{g.alfano,greco,fparisi,i.trubitsyna}@dimes.unical.it
Abstract. Extensions of Dung’s abstract Argumentation Framework (AF) in-
clude the general class of Recursive Bipolar AFs (Rec-BAFs), i.e., AFs with
recursive attacks and supports. Although the relationships between AF seman-
tics and Partial Stable Models (PSMs) of logic programs has been deeply investi-
gated, this is not the case for Rec-BAFs. In this paper we explore this relationship,
showing that a Rec-BAF ∆ can be translated into a logic program P∆ so that the
extensions of ∆ under different argumentation semantics coincide with subsets of
the PSMs of P∆ . We provide a logic programming approach that characterizes,
in an elegant and uniform way, the semantics of several AF-based frameworks
which belong to the class of Rec-BAFs. This result allows also to define the se-
mantics for new AF-based frameworks, such as AFs with recursive attacks and
recursive deductive supports.
Keywords: Abstract Argumentation · Partial Stable Models · Semantics.
1 Introduction
Formal argumentation has emerged as one of the important fields in Artificial Intelli-
gence [10, 36]. In particular, Dung’s abstract Argumentation Framework (AF) is a sim-
ple, yet powerful formalism for modelling disputes between two or more agents [22].
An AF consists of a set of arguments and a binary attack relation over the set of ar-
guments that specifies the interactions between arguments: intuitively, if argument a
attacks argument b, then b is acceptable only if a is not. Hence, arguments are abstract
entities whose role is entirely determined by the interactions specified by attacks.
Dung’s framework has been extended in many different ways, including the intro-
duction of new kinds of interactions between arguments and/or attacks. In particular,
the class of Bipolar Argumentation Frameworks (BAFs) is an interesting extension of
the AF which allows for also modelling the support between arguments [32, 37]. Dif-
ferent interpretations of the notion of support have been proposed [15, 18]. Deductive
support [37] is intended to capture the following intuition: if argument a supports argu-
ment b, then the acceptance of a implies the acceptance of b; thus, the non-acceptance
of b implies the non-acceptance of a. On the other hand, necessary support [32, 9] is
interpreted in a dual way [15]: if a supports b, then the acceptance of a is necessary to
?
Copyright c 2020 for this paper by its authors. Use permitted under Creative Commons Li-
cense Attribution 4.0 International (CC BY 4.0).
get the acceptance of b; equivalently, accepting b implies accepting a. An AFN (AF with
Necessities) is a BAF where supports are interpreted as necessities [32]; an AFD (AF
with Deductive supports) is a BAF where supports are interpreted as deductions [37].
Clearly, the way the support is interpreted changes the set of extensions, i.e. the set of
acceptable elements of an argumentation framework.
Further extensions of the Dung framework consider second-order interactions [37],
e.g., attacks to attacks/supports, as well as more general forms of interactions such as
recursive AFs where attacks can be recursively attacked [9, 16] and recursive BAFs,
where attacks/supports can be recursively attacked/supported [27, 17].
Following Dung’s approach, the meaning of recursive AF-based frameworks is still
given by relying on the concept of extension. However, the extensions of an AF with
Recursive Attacks (AFRA) [9] and of an Attack-Support Argumentation Framework
(ASAF) [19, 27] also include the (names of) attacks and supports that intuitively con-
tribute to determine the set of accepted arguments. Particularly, the acceptability of an
attack is related to the acceptability of its source argument: an attack in the AFRA is
defeated even when its source argument is defeated. This is not the case for Recursive
AF (RAF) [16] and Recursive AF with Necessities (RAFN) frameworks [17], which of-
fer a different semantics for recursive AFs and recursive BAFs with necessary supports,
respectively. From a syntax standpoint all the argumentation frameworks mentioned
above can be viewed as (possibly restricted forms of) Recursive Bipolar Argumenta-
tion Frameworks (Rec-BAFs), though semantically different because having different
interpretations of support and different ways of determining the status of attacks.
Recently there has been an increasing interest in studying the relationships between
argumentation frameworks and logic programming (LP). In particular, the semantic
equivalence between complete extensions in AF and 3-valued stable models in LP was
first established in [38]. Then, the relationships of LP with AF have been further studied
in [13], whereas those with Assumption-Based Argumentation [12, 20] have been con-
sidered in [14], and those with Abstract Dialectical Frameworks have been investigated
in [1]. Efficient mappings from AF to Answer Set Programming (i.e. LP with Stable
Model semantics [26]) have been investigated as well [35, 24]. The well-know AF sys-
tem ASPARTIX is implemented by rewriting the input AF into an ASP program and
using an ASP solver to compute extensions. Although the ASPARTIX system allows
also to reason on some extensions of AF, such as Extended AF (EAF) [31] and AFRA,
so far the relationships between LP and more general frameworks extending AF such
as Rec-BAFs has not been adequately studied. Thus, in this paper, we investigate these
relationships by generalizing the work in [13] and providing relationships between LP
and different recently proposed generalizations of the Dung’s framework.
Contributions. The main contributions of this paper are as follows:
– We introduce a general approach for characterizing the extensions of different AF-
based frameworks (e.g. AFRA, RAF, ASAF, RAFN) under several well-known
argumentation semantics in terms of Partial Stable Models (PSMs) of logic pro-
grams. This is achieved by providing a modular definition of the sets of defeated
and acceptable elements (i.e., arguments, attacks and supports) for each AF-based
framework, and by leveraging on the connection between argumentation semantics
and subsets of PSMs. In particular, for any argumentation framework belonging to
the class of Rec-BAFs, our formulation of acceptable elements allow us to easily
and systematically derive a propositional program whose PSMs corresponds to the
extensions (under a given semantics) of the given framework.
– Our approach is used to define new semantics for AFs with recursive attacks and
supports under deductive interpretation of supports, where the status of an attack is
considered independently from the status of its source. We call these frameworks
Recursive Argumentation Framework with Deductive supports (RAFD) and Argu-
mentation Framework with Recursive Attacks and Deductive supports (AFRAD). It
turns out that AFDs are special cases of these frameworks.
Our results can be used i) for better understanding the semantics of several AF-
based frameworks, ii) to easily define new semantics for extended frameworks, and
iii) to provide additional tools for computing stable semantics using answer set solvers
[25] and even other complete-based semantics using classical program rewriting [30]
(see also [35, 24]).
An extended version of this paper can be found in [6]. Proofs are available in [7].
2 Preliminaries
We first recall Dung’s framework and then introduce the general class of recursive bipo-
lar argumentation frameworks.
2.1 Argumentation Frameworks
An abstract Argumentation Framework (AF) is a pair hA, Ωi, where A is a set of argu-
ments and Ω ⊆ A × A is a set of attacks. An AF can be seen as a directed graph, whose
nodes represent arguments and edges represent attacks; an attack (a, b) ∈ Ω from a to
b is represented by a → b.
Different semantics notions have been defined leading to the characterization of col-
lectively acceptable sets of arguments, called extensions [22]. Given an AF ∆ =hA, Ωi
and a set S ⊆ A of arguments, an argument a ∈ A is said to be i) defeated w.r.t. S iff
∃b ∈ S such that (b, a) ∈ Ω, and ii) acceptable w.r.t. S iff for every argument b ∈ A
with (b, a) ∈ Ω, there is c ∈ S such that (c, b) ∈ Ω. The sets of defeated and acceptable
arguments w.r.t. S are defined as follows (where ∆ is understood):
– Def (S) = {a ∈ A | ∃ b ∈ S . (b, a) ∈ Ω};
– Acc(S) = {a ∈ A | ∀ b ∈ A . (b, a) ∈ Ω ⇒ b ∈ Def (S)}.
Given an AF hA, Ωi, a set S ⊆ A of arguments is said to be i) conflict-free iff
S ∩ Def (S) = ∅, and ii) admissible iff it is conflict-free and S ⊆ Acc(S).
Given an AF hA, Ωi, a set S ⊆ A is an extension called:
– complete iff it is conflict-free and S = Acc(S);
– preferred iff it is a maximal (w.r.t. ⊆) complete extension;
– stable iff it is a total preferred extension, i.e. a preferred extension s.t. S∪Def (S) =
A;
– semi-stable iff it is a preferred extension such that S ∪ Def (S) is maximal;
– grounded iff it is the smallest (w.r.t. ⊆) complete extension;
– ideal iff it is the biggest (w.r.t. ⊆) complete extension contained in every preferred
extension.
The set of complete (resp., preferred, stable, semi-stable, grounded, ideal) extensions of
a framework ∆ will be denoted by CO(∆) (resp., PR(∆), ST (∆), SST (∆), GR(∆),
ID(∆)).
Example 1. Let ∆ = hA, Ωi be an AF where A = {a, b, c, d} and Ω = {(a, b), (b, a),
(a, c), (b, c), (c, d), (d, c)}. The set of complete extension is CO(∆) = {∅, {d},
{a, d}, {b, d}}. Consequently, PR(∆) = ST (∆) = SST (∆) = {{a, d}, {b, d}},
GR(∆) = {∅}, ID(∆) = {{d}}.
2.2 Recursive Bipolar Argumentation Frameworks
A Recursive Bipolar Argumentation Framework (Rec-BAF) is a tuple hA, Σ, Π, s, ti,
where A is a set of arguments, Σ is a set of attack names, Π is a set of necessary support
names, s (resp., t) is a function from Σ ∪ Π to A (resp., to A ∪ Σ ∪ Π) mapping
each attack/support to its source (resp., target). In the following, given a set Φ such that
either Φ ⊆ Σ or Φ ⊆ Π, we denote by i) Φ∗ = {(s(γ), t(γ)) | γ ∈ Φ} the set of
pairs connected by an attack/support edge, and ii) Φ+ the transitive closure of Φ. It is
assumed that Π ∗ is acyclic.
Two different semantics have been defined under necessary interpretation of sup-
ports, as recalled in what follows.
Recursive AF with Necessities. The Recursive Argumentation Framework with Ne-
cessities (RAFN) has been proposed in [17]. The semantics combines the RAF inter-
pretation of attacks [16] with that of BAF under the necessity interpretation of sup-
ports (i.e., AFN) [32]. Here we consider a simplified version where supports have a sin-
gle source and the support relation is acyclic. Formally, given an RAFN hA, Σ, Π, s, ti,
X ∈ (A ∪ Σ ∪ Π), a ∈ A, and S ⊆ A ∪ Σ ∪ Π, we say that argument a recursively
attacks X given S (denoted as a attS X) if either (a, X) ∈ (Σ ∩ S)∗ or there exists
b ∈ A such that (a, b) ∈ (Σ ∩ S)∗ and (b, X) ∈ (Π ∩ S)+ .
For any RAFN ∆ and S ⊆ A ∪ Σ ∪ Π, the defeated and acceptable sets (given S)
are defined as follows:
– Def (S) = {X ∈ A ∪ Σ ∪ Π | ∃b ∈ A ∩ S . b attS X};
– Acc(S) = {X ∈ A ∪ Σ ∪ Π | ∀b ∈ A . b attS X ⇒ b ∈ Def (S)}.
Attack-Support AF. The Attack-Support Argumentation Framework (ASAF) has been
proposed in [19, 27]. The semantics combines the AFRA interpretation of attacks [9]
with that of BAF under the necessary interpretation of supports (i.e., AFN). For the sake
of presentation, we consider a slight generalization of ASAF, where attack and support
names are first-class citizens, giving the possibility to represent multiple attacks and
supports from the same source to the same target.1
1
In the original work [19, 27] an ASAF is a tuple hA, Ω, Γ i where A is a set of arguments, Ω
is a set of attacks Ω : A → (A ∪ Ω), and Γ is a set of supports Γ : A → (A ∪ Γ ).
Formally, given an ASAF hA, Σ, Π, s, ti, X ∈ (A ∪ Σ ∪ Π), α ∈ Σ, and S ⊆
A ∪ Σ ∪ Π, we say that i) α (directly or indirectly) attacks X (denoted by α def X)
if either t(α) = X or t(α) = s(X), and ii) α extendedly defeats X given S (denoted
as α defS X) if either α def X or there exists b ∈ A such that t(α) = b and either
(b, X) ∈ (Π ∩ S)+ or (b, s(X)) ∈ (Π ∩ S)+ . For any ASAF ∆ and S ⊆ A ∪ Σ ∪ Π,
the defeated and acceptable sets (given S) are:
– Def (S) = {X ∈ A ∪ Σ ∪ Π | ∃ α ∈ Σ ∩ S . α defS X};
– Acc(S) = {X ∈ A ∪ Σ ∪ Π | ∀α ∈ Σ . α defS X ⇒ α ∈ Def (S)}.
The notions of conflict-free, admissible sets, and the different types of extensions
can be defined in a standard way (see Section 2.1) by considering S ⊆ A ∪ Σ ∪ Π and
by using the definitions of defeated and acceptable sets reported above.
AFs with high-order interactions can be mapped to AFs, though the mapping is not
trivial because extensions also contain attacks and supports. In particular, an equivalent
AF for an ASAF can be obtained by translating it into an AFN [19] that in turns can be
translated into an AF [32] (see also [27]).
Example 2. Consider a situation where we want to decide whether to play tennis on the
basis of some information. Assume we have the following arguments: r (it is raining),
p (play tennis), o (the tennis court is outside), and the logical implications: (α1 ) if it
is raining, then we do not play tennis, and (β1 ) if the tennis court is outside supports,
then implication α1 should hold. This situation can be modelled using the Rec-BAF ∆
shown in Figure 1, where α1 is an attack (denoted by →), and β1 is a support (denoted
by ⇒). Under both ASAF and RAFN semantics CO(∆) = {{o, r, α1 , β1 }}.
AF-based frameworks belonging to the class of Rec-BAFs ASAFs and RAFNs
generalize other AF-based frameworks mentioned in the Introduction. In particular,
(i) an AFRA is an ASAF hA, Σ, Π, s, ti where Π = ∅ 2 ; (ii) an RAF is an RAFN
hA, Σ, Π, s, ti where Π = ∅; and (iii) we can think of an AFN as either an ASAF or
an RAFN hA, Σ, Π, s, ti where function t : Σ ∪ Π → A maps attacks and supports
to arguments only.
It is important to observe that different frameworks extending AF share the same
structure, although they have different semantics. Thus we distinguish between frame-
work and class of frameworks. Two frameworks sharing the same syntax (i.e. the struc-
ture) belong to the same (syntactic) class. For instance, BAF is a syntactic class, whereas
AFN and AFD are two specific frameworks sharing the same BAF syntax; their seman-
tics differ because they interpret supports in different ways. Recursive AF (Rec-AF) is
another syntactic class, where AFs are extended by allowing recursive attacks: AFRA
and RAF are frameworks belonging to this class, which differ only in the determina-
tion of the status of attacks. Finally, ASAF and RAFN are two frameworks belonging
2
For the sake of presentation, we consider a slight generalization of AFRA, where attack names
are first-class citizens, allowing to also represent more than one attack from the same source to
the same target. In the original work an AFRA is a tuple hA, Ωi where A is a set of arguments
and Ω is a set of attacks Ω : A → (A ∪ Ω) [9].
Acronym Framework
AF abstract Argumentation Framework [Dung,1995]
BAF Bipolar AF
AFN AF with Necessities [Nouia and Risch,2011]
AFD AF with Deductive supports [Villata et al.,2012]
Rec-AF Recursive-AF
AFRA AF with Recursive Attacks [Baroni et al.,2011]
RAF Recursive AF [Cayrol et al.,2017]
Rec-BAF Recursive-BAF
ASAF Attack-Support AF [Gottifredi et al.,2018]
RAFN Recursive AF with Necessities [Cayrol et al.,2018]
AFRAD AF with Rec. Attacks and Deductive supports o
RAFD Recursive AF with Deductive supports β1
r α1 p
Table 1: AF-based frameworks considered in the paper.
Syntactic classes are in bold. Fig. 1: Rec-BAF of Example 2
to the general class Rec-BAF presented in this section, consisting in the extension of
BAF with recursive attacks and supports. The differences between ASAF and RAFN
semantics are not in the way they interpret supports (both based on the necessity in-
terpretation), but in a different determination of the status of attacks as they extend
AFRA and RAF, respectively. In Section 3.2 we will present two new frameworks that
also belong to the Rec-BAF class: Recursive Argumentation Framework with Deductive
supports (RAFD) and Argumentation Framework with Recursive Attacks and Deductive
supports (AFRAD). AFDs are special cases of these frameworks.
Thus, the class of Rec-BAF includes the sub-classes BAF and Rec-AF, and thus
the 9 frameworks discussed above, ASAF, RAFN, AFRAD, and RAFD, and their spe-
cializations AF, AFRA, RAF, AFN, and AFD. We use F to denote the set of these 9
frameworks. Table 1 lists the frameworks in F. With a little abuse of notation, we will
use the same symbol ∆ to denote any framework in F.
2.3 Partial Stable Models
In this section, we review the basic concepts which underly the notion of PSMs [34].
A (normal, logic) program is a set of rules of the form A ← B1 ∧ · · · ∧ Bn , with
n ≥ 0, where A is an atom, called head, and B1 ∧ · · · ∧ Bn is a conjunction of lit-
erals, called body. We consider programs without function symbols. Given a program
P , ground(P ) denotes the set of all ground instances of the rules in P . The Herbrand
Base of a program P , i.e. the set of all ground atoms which can be constructed using
predicate and constant symbols occurring in P , is denoted by BP , whereas ¬BP de-
notes the set {¬A | A ∈ BP }. Analogously, for any set S ⊆ BP ∪ ¬BP , ¬S denotes
the set {¬A | A ∈ S}, where ¬¬A = A. Given I ⊆ BP ∪ ¬BP , pos(I) (resp., neg(I))
stands for I ∩ BP (resp., ¬I ∩ BP ). I is consistent if pos(I) ∩ ¬neg(I) = ∅, otherwise
I is inconsistent.
Given a program P , I ⊆ BP ∪¬BP is an interpretation of P if I is consistent. Also,
I is total if pos(I) ∪ neg(I) = BP , partial otherwise. A partial interpretation M of a
program P is a partial model of P if for each ¬A ∈ M every rule in ground(P ) having
as head A contains at least one body literal B such that ¬B ∈ M . Given a program P
and a partial model M , the positive instantiation of P w.r.t. M , denoted by P M , is
obtained from ground(P ) by deleting: (a) each rule containing a negative literal ¬A
such that A ∈ pos(M ); (b) each rule containing a literal B such that neither B nor ¬B
is in M ; (c) all the negative literals in the remaining rules. Clearly, all the rules in P
are definite clauses and hence the minimal Herbrand model of P can be obtained as the
least fixpoint of its immediate consequence operator TP M , denoted by TPωM (∅). For any
partial model M of a logic program P , TPωM (∅) ⊆ M [34].
Let P be a program and M a partial model for P . Then M is (a) founded if
TPωM (∅) = pos(M ); (b) stable if it is founded and it is not a proper subset of any
other founded model. The set of partial stable models of a logic program P , denoted
by PM(P ), define a meet semi-lattice. The well-founded model (denoted by WF(P ))
and the maximal-stable models MS(P )3 , are defined by considering ⊆-minimal and
⊆-maximal elements. The set of (total) stable models (denoted by SM(P )) is obtained
by considering the maximal-stable models which are total, whereas the least-undefined
models (denoted by LM(P )) are obtained by considering the maximal-stable models
with a ⊆-minimal set of undefined atoms (i.e., atoms which are neither true or false).
The max-deterministic model (denoted by MD(P )) is the ⊆-maximal PSM contained
in every maximal-stable model [33, 29].
Example 3. Consider the program P consisting of the following four rules {a ← ¬b;
b ← ¬a; c ← ¬a ∧ ¬b ∧ ¬d; d ← ¬c}. The set of partial stable models of P is
PS(P ) = { ∅, {¬c, d}, {a, ¬b, ¬c, d}, {¬a, b, ¬c, d} }. Consequently, WF(∆) =
{ ∅ }, MD(∆) = { {¬c, d} }, MS = ST (∆) = LS(∆) = {{a, ¬b, ¬c, d}, {¬a, b,
¬c, d}}.
Propositional Programs. Given a set of symbols Λ = {a1 , ..., an }, a (propositional)
program over Λ is a set of |Λ| rules ai ← bodyi (1 ≤ i ≤ n), where every bodyi is
a propositional formula defined over Λ. The semantics of a propositional program P ,
defined over a given alphabet Λ, is given in terms of the set PS(P ) of its Partial Stable
Models (PSMs) that are obtained as follows: i) P is first rewritten into a set of standard
(ground) logic rules P 0 , whose bodies contain conjunction of literals (even by adding
fresh symbols to the alphabet)4 ; ii) next, the set of PSMs of P 0 is computed; iii) finally,
fresh literals added to Λ in the first step are deleted from the models. It is worth noting
that for propositional programs we can assume as Herbrand Base the set of (ground)
atoms occurring in the program.
3 AF-based Frameworks and Partial Stable Models
In this section we present a new way to define the semantics of AF-based frameworks
by considering propositional programs and partial stable models. In order to compare
extensions E of a given framework ∆ (containing acceptable elements) with PSMs of
3
Corresponding to the preferred extensions of [21].
4
A rule a ← (b∨c)∧(d∨e) is rewritten as a ← ¬a1 ∧¬a2 , a1 ← ¬b∧¬c and a2 ← ¬d∧¬e.
a given program P (containing true and false atoms), we denote as Eb = E ∪ {¬a | a ∈
Def (E)} the completion of E. Moreover, for a collection of extensions E, E b denotes
the set {E | E ∈ E}.
b
Observe also that for any framework ∆ and complete extension E for ∆, elements
not occurring in E ∪ Def (E) are said to be undecided (or undefined), whereas for any
program P and PSM M for P , atoms not occurring in pos(M ) ∪ neg(M ) are said to be
undefined. Thus, to compare complete extensions and PSMs it is sufficient to consider
the completion of extensions.
The next proposition states the relationship between the argumentation frameworks
(e.g. AF, BAF, Rec-BAF) and logic programs with partial stable models.
Proposition 1. For any framework ∆ ∈ F and propositional program P , whenever
\ = PS(P ) it holds that PR(∆)
CO(∆) \ = MS(P ), ST \ (∆) = ST (P ), SST\ (∆) =
5
\ \
LM(P ), GR(∆) = WF(P ), and ID(∆) = MD(P ).
The result of Proposition 1 derives from the fact that preferred, stable, semi-stable,
grounded, and ideal extensions are defined by selecting a subset of the complete exten-
sions satisfying given criteria (see Section 2). On the other side, the maximal, stable,
least-undefined, well-founded, and max-deterministic (partial) stable models are ob-
tained by selecting a subset of the PSMs satisfying criteria coinciding with those used
to restrict the set of complete extensions.
Given a framework ∆ and an extension E, for any element a which could occur in
some extension of ∆, the truth value vE (a), or simply v(a) whenever E is understood,
is equal to true if a ∈ E, false if a ∈ Def (E), undec (undecided) otherwise.
Hereafter, we assume that false < undec < true and ¬undec = undec.
The strict relationship between the semantics of AFs (given in terms of subset of
complete extensions) and the semantics of logic programs (given in terms of subset of
PSMs) has been shown in [38, 13]. The relationship is based on the observation that
the meaning of an attack a → b is that the condition v(b) ≤ ¬v(a) must hold. On
the other side, the satisfaction of a logical rule a ← b1 , ..., bn implies that v(a) ≥
min{v(b1 ), ..., v(bn ), true}.
V
Definition 1. Given an AF ∆ = hA, Ωi, we denote as P∆ = {a ← (b,a)∈Ω ¬b | a ∈
A} the propositional program derived from ∆.
The semantics of an AF ∆ can be obtained by considering PSMs of the logic pro-
\ = PS(P∆ ).
gram P∆ . Particularly, for any AF ∆, CO(∆)
In the rest of this section, we show how the semantics defined for frameworks ex-
tending AF can be captured by means of PSMs of logic programs. We propose a general
method that can be applied to all the discussed frameworks, and even to new frameworks
in Section 3.2. Specifically, to model frameworks extending Dung’s framework by logic
programs under PSM semantics, we provide new definitions of defeated and acceptable
sets that, for a given set S, will be denoted by D EF(S) and ACC(S), respectively. These
definitions will be used to derive rules in P∆ , the propositional program for ∆ ∈ F. For
AFs we have that for every set S ⊆ A, D EF(S) = Def (S) and ACC(S) = Acc(S).
5
For the novel frameworks ∆ ∈ {AFRAD, RAFD}, the set CO(∆) of the complete extensions,
and the sets of extensions prescribed by the other semantics, are defined in Section 3.2.
3.1 Recursive BAFs with Necessary Supports
In this section we study the relationship between partial stable models and the seman-
tics of Rec-BAFs. Particularly, we first present results for RAFN semantics, and then
we discuss results for the ASAF framework. We remand to the next section the presen-
tation of two novel semantics for recursive bipolar AFs with deductive interpretation of
supports.
RAFN. We next provide the definitions of defeated and acceptable sets for an RAFN.
Definition 2. For any RAFN hA, Σ, Π, s, ti and set S ⊆ A ∪ Σ ∪ Π, we have that:
– D EF(S) = {X ∈ A ∪ Σ ∪ Π | (∃α ∈ Σ ∩ S . s(α) ∈ S ∧ t(α) = X) ∨
(∃ β ∈ Π ∩ S . s (β) ∈ D EF(S) ∧ t(β) = X) };
– ACC(S) = {X ∈ A ∪ Σ ∪ Π | (∀α ∈ Σ . t(α) = X ⇒ (α ∈ D EF(S) ∨
s(α) ∈ D EF(S))) ∧ (∀β ∈ Π . t(β) = X ⇒ (β ∈ D EF(S) ∨ s(β) ∈ ACC(S))) }.
It is worth noting that D EF(S) and ACC(S) are defined recursively, and whenever
S is a complete extension we obtain the following result.
Theorem 1. Given an RAFN ∆ and an extension S ∈ CO(∆), then Def (S) = D EF(S)
and Acc(S) = ACC(S).
Theorem 1 states that in order to define the semantics for an RAFN ∆ we can use
acceptable sets S = ACC(S). This is captured by the following definition, that shows
how to derive a propositional program from an RAFN from Definition 2.
Definition 3. Given an RAFN ∆ = hA, Σ, Π, s, ti, then P∆ (the propositional pro-
gram derived from ∆) contains, for each X ∈ A ∪ Σ ∪ Π, a rule
^ ^
X← (¬α ∨ ¬s(α)) ∧ (¬β ∨ s(β)).
α∈Σ∧t(α)=X β∈Π∧t(β)=X
Intuitively, starting from the definition of ACC(S) in Definition 2, the rationale of
the above definition is as follows. An element X ∈ A∪Σ ∪Π is true if (i) every attack
α toward X is false or originates from a source s(α) which is false, and (ii) every
support β toward X is false or originates from a source s(β) which is true. These
conditions resemble the conditions (∀α ∈ Σ . t(α) = X ⇒ (α ∈ D EF(S) ∨ s(α) ∈
D EF(S))) and (∀β ∈ Π . t(β) = X ⇒ (β ∈ D EF(S) ∨ s(β) ∈ ACC(S))), respectively,
of Definition 2 after interpreting elements in D EF(S) as false and elements in ACC(S)
as true.
As stated below, the set of complete extensions of an RAFN ∆ coincides with the
set of PSMs of P∆ .
\ = PS(P∆ ).
Theorem 2. For any RAFN ∆, CO(∆)
The previous theorem states that the set of complete extensions of an RAFN ∆ co-
incides with the set of PSMs of the derived logic program P∆ . Consequently, using
Proposition 1, also the others argumentation semantics turns out to be characterized in
terms of subsets of PSMs. Moreover, previous results also apply to restricted frame-
works such as RAF, where Π = ∅, and AFN, where t : Σ ∪ Π → A.
ASAF. We next provide definitions of defeated and acceptable sets for an ASAF. They
will be used in a way similar to that described above in order to obtain a semantically-
equivalent propositional program for ASAFs.
Definition 4. Given an ASAF hA, Σ, Π, s, ti and a set S ⊆ A ∪ Σ ∪ Π, we define:
– D EF(S) = {X ∈ A ∪ Σ ∪ Π | (X ∈ Σ ∧ s(X) ∈ D EF(S)) ∨
(∃α ∈ Σ ∩ S . t(α) = X) ∨
(∃β ∈ Π ∩ S . t(β) = X ∧ s(β) ∈ D EF(S))};
– ACC(S) = {X ∈ A ∪ Σ ∪ Π | (X ∈ Σ ⇒ s(X) ∈ ACC(S)) ∧
(∀α ∈ Σ . t(α)=X ⇒ α ∈ D EF(S)) ∧
(∀β ∈ Π . t(β) = X ⇒ (β ∈ D EF(S) ∨ s(β) ∈ ACC(S)))}.
Analogously to what has been done RAFNs, since it can be shown that for any
complete extension S of an ASAF ∆ it is the case that Acc(S) = ACC(S), the propo-
sitional program for an ASAF ∆ can be derived by looking at the definition ACC(S)
of acceptable elements. In this case, the three conjuncts in the acceptance condition for
an element X will correspond to three (group of) conjuncts, respectively, in the rule for
X of the program P∆ . Specifically, the last two conjuncts in the definition of ACC(S)
can be mapped by reasoning similarly to the RAFN case, whereas the first one entails
a rule’s body conjunction stating that if X is an attack then all of its sources must be
true.
Definition 5. For any ASAF ∆ = hA, Σ, Π, s, ti, P∆ (the propositional program de-
rived from ∆) contains, for each X ∈ A ∪ Σ ∪ Π, a rule of the form
^ ^
X ← ϕ(X) ∧ ¬α ∧ (¬β ∨ s(β))
α∈Σ∧t(α)=X β∈Π∧t(β)=X
where: ϕ(X) = s(X) if X ∈ Σ; otherwise, ϕ(X) = true (recall that ψ ∧ true ≡ ψ).
The set of complete extensions of an ASAF ∆ coincides with the set of PSMs
of the derived logic program P∆ , meaning that using Proposition 1 also the others
argumentation semantics turns out to be characterized in terms of subsets of PSMs.
\ = PS(P∆ ).
Theorem 3. For any ASAF ∆, CO(∆)
Similarly to the case of RAFN, the above results also apply to restricted frameworks
such as AFRA, where Π = ∅, and AFN, where t : Σ ∪ Π → A.
Example 4. Consider a Rec-BAF ∆0 obtained by adding to the Rec-BAF ∆ of Example
2 argument s (it is sunny). Argument r and s attack each other through attacks α2 and
α3 , as shown in Figure 2. As shown below, ∆0 has different extensions when viewed as
an RAFN or an ASAF.
o o
β1 α β1
α
s α3 2 r α1 p s α3 2 r α1 p z
β2
Fig. 2: Rec-BAF of Example 4 Fig. 3: Rec-BAF of Example 6
Under RAFN semantics ∆0 has the following two complete extensions: {o, s, p,
α1 , α2 , α3 , β1 } and {o, r α1 , α2 , α3 , β1 }. Differently, under ASAF semantics ∆0 has
the following complete extensions: {o, s, p, α3 , β1 } and {o, r α1 , α2 , β1 }. Note that
attacks α1 and α2 (resp., α3 ) do not belong to {o, s, p, α3 , β1 } since their source, r,
is defeated. Similarly, α3 does not belong to {o, r α1 , α2 , β1 } since its source, s, is
defeated. The propositional program under the RAFN semantics is the following:
P∆0 = {(o ←), (r ← ¬α3 ∨ ¬s), (p ← ¬α1 ∨ ¬r), (s ← ¬α2 ∨ ¬r), (α1 ← ¬β1 ∨ o),
(α2 ←), (α3 ←), (β1 ←)}, whose set of partial stable models is as follows:
PS(P∆0 ) = {M1 = {o, ¬r, s, p, α1 , α2 , α3 , β1 }, M2 = {o, r, ¬s, ¬p, α1 , α2 , α3 , β1 }}.
Analogously, the propositional program for ∆0 under the ASAF semantics is:
P∆0 = {(o ←), (r ← ¬α3 ), (p ← ¬α1 ), (s ← ¬α2 ), (α1 ← r ∧ (¬β1 ∨ o)),
(α2 ← r), (α3 ← s), (β1 ←)}, whose set of partial stable models is: PS(P∆0 ) =
{M3 = {o, ¬r, s, p, ¬α1 , ¬α2 , α3 , β1 }, M4 = {o, r, ¬s, ¬p, α1 , α2 , ¬α3 , β1 }}. It is
worth noting that M1 (resp., M2 ) differs from M3 (resp., M4 ) in the status of α1 and
α2 (resp., α3 ).
3.2 Recursive BAFs with Deductive Supports
In this section we study two new frameworks both belonging to the Rec-BAF class
and both extending AFD by allowing recursive attacks and deductive supports. The
first one, called Recursive Argumentation Framework with Deductive supports (RAFD),
extends RAF, whereas the second one, called Argumentation Framework with Recursive
Attacks and Deductive supports (AFRAD), extends AFRA. It is again assumed that Π ∗
is acyclic.
As we shall define the semantics by defining directly the sets D EF(S) and ACC(S),
differently from the previous section, we do not have any results regarding the equiva-
lence between the sets Acc(S) and ACC(S) for S = ACC(S).
RAFD. As usual, we first define the sets of defeated and acceptable elements, and then
the propositional logic program for an RAFD.
Definition 6. For any RAFD hA, Σ, Π, s, ti and set S ⊆ A ∪ Σ ∪ Π, we have that:
– D EF(S) = {X ∈ A ∪ Σ ∪ Π | (∃α ∈ Σ ∩ S . t(α) = X ∧ s(α) ∈ S) ∨
(∃β ∈ Π ∩ S . s(β) = X ∧ t (β) ∈ D EF(S)) };
– ACC(S) = {X ∈ A ∪ Σ ∪ Π| (∀α ∈ Σ . t(α) = X ⇒ (α ∈ D EF(S) ∨
s(α) ∈ D EF(S))) ∧ (∀β ∈ Π . s(β) = X ⇒ (β ∈ D EF(S) ∨ t(β) ∈ ACC(S)))}.
The sets of extensions prescribed by the different semantics are based on the defeated
and acceptable sets defined above. That is, given an RAFD ∆ = hA, Σ, Π, s, ti, a
set S ⊆ A ∪ Σ ∪ Π of elements is a complete extension of ∆ iff it is conflict-free
(i.e., S ∩ D EF(S) = ∅) and S = ACC(S). As done for the other frameworks, we use
CO(∆) to denote the set of complete extensions of ∆. Moreover, the set of preferred
(resp., stable, semi-stable, grounded, ideal) extensions is defined in the standard way
(see Section 2.1) by using again D EF(S) and ACC(S).
Using ACC(S), we define the propositional program for an RAFD ∆.
Definition 7. Given an RAFD ∆ = hA, Σ, Π, s, ti, then P∆ (the propositional pro-
gram derived from ∆) contains, for each X ∈ A ∪ Σ ∪ Π, a rule of the form
^ ^
X← (¬α ∨ ¬s(α)) ∧ (¬β ∨ t(β)).
α∈Σ∧t(α)=X β∈Π∧s(β)=X
\ = PS(P∆ ).
Theorem 4. For any RAFD ∆, CO(∆)
Thus, the semantics of an RAFD ∆ can be carried out by using the PSMs of P∆ .
The result also applies to restricted frameworks such as RAF, where Π = ∅, and AFD,
where t : Σ ∪ Π → A.
AFRAD. The following definition formalizes defeated and acceptable sets for an AFRAD.
Definition 8. Given an AFRAD hA, Σ, Π, s, ti and a set S ⊆ A ∪ Σ ∪ Π, we have
– D EF(S) = {X ∈ A ∪ Σ ∪ Π | (X ∈ Σ ∧ s(X) ∈ D EF(S)) ∨
(∃α ∈ Σ ∩ S . t(α) = X) ∨
(∃β ∈ Π ∩ S . s(β) = X ∧ t(β) ∈ D EF(S))};
– ACC(S) = {X ∈ A ∪ Σ ∪ Π |(X ∈ Σ ⇒ s(X) ∈ ACC(S))) ∧
(∀α ∈ Σ . t(α)=X ⇒ α ∈ D EF(S)) ∧
(∀β ∈ Π . s(β) = X ⇒ (β ∈ D EF(S)∨t(β) ∈ ACC(S))))}.
Similarly to what done for RAFDs, the set CO(∆) of complete extensions of an
AFRAD ∆, and the sets of extensions prescribed by the other semantics, are defined by
using the defeated and acceptable sets defined above.
Definition 9. For any AFRAD ∆ = hA, Σ, Π, s, ti, P∆ (the propositional program
derived from ∆) contains, for each X ∈ A ∪ Σ ∪ Π, a rule of the form
^ ^
X ← ϕ(X) ∧ ¬α ∧ (¬β ∨ t(β))
α∈Σ∧t(α)=X β∈Π∧s(β)=X
where: ϕ(X) = s(X) if X ∈ Σ; otherwise, ϕ(X) = true.
\ = PS(P∆ ).
Theorem 5. For any AFRAD ∆, CO(∆)
The result also apply to restricted frameworks such as AFRA and AFD.
Example 5. Consider now the Rec-BAF ∆0 of Example 4 and assume that supports
are interpreted as deductive. The propositional program under RAFD semantics is as
follows: P∆0 = {(o ← ¬β1 ∨α1 ), (r ← ¬α3 ∨¬s), (p ← ¬α1 ∨¬r), (s ← ¬α2 ∨¬r),
(α1 ←), (α2 ←), (α3 ←), (β1 ←)}, whose set of partial stable models is PS(P∆0 ) =
{M1 = {o, ¬r, p, s, α1 , α2 , α3 , β1 }, M2 = {o, r, ¬p, ¬s, α1 , α2 , α3 , β1 }}.
On the other hand, the propositional program for ∆0 under AFRAD semantics is:
P∆0 = {(o ← ¬β1 ∨ α1 ), (r ← ¬α3 ), (p ← ¬α1 ), (s ← ¬α2 ), (α1 ← r), (α2 ←
r), (α3 ← s), (β1 ←)}, whose set of partial stable models is PS(P∆0 ) = {M3 =
{o, ¬r, p, s, ¬α1 , ¬α2 , α3 , β1 }, M4 = {o, r, ¬p, ¬s, α1 , α2 , ¬α3 , β1 } }.
Note that the RAFD (resp., AFRAD) program differs from the RAFN (resp., ASAF)
program given in Example 4 only in rules having as head α1 and o, respectively.
Considering Example 4 and Example 5, we note that the set of PSMs of the propo-
sitional programs for the RAFN and RAFD (resp., ASAF and AFRAD) frameworks
coincide, meaning that these argumentation frameworks have the same sets of exten-
sions. However, in general, RAFNs and RAFDs (resp., ASAFs and AFRADs) prescribe
different sets of extensions as shown in the following example.
Example 6. Consider a Rec-BAF ∆00 obtained by adding to the Rec-BAF ∆0 of Exam-
ple 5 a new argument z and a new support β2 from argument p to z (see Figure 3).
We first revise the propositional programs given in Example 4 to take into account
argument z and support β2 . The propositional program P∆00 under RAFN (resp., ASAF)
semantics is obtained by adding to the propositional program P∆0 under RAFN (resp.,
ASAF) semantics of Example 4 the rules z ← ¬β2 ∨ p and β2 ← (cf. Definitions 3
and 5). Then, the set of partial stable models of P∆00 under RAFN semantics is:
PS(P∆00 ) = {N1 = M1 ∪ {z, β2 }, N2 = M2 ∪ {¬z, β2 }}, where M1 and M2 are the
models given in Example 4 that, as said above, coincide with those of Example 5. The
set of partial stable models of P∆00 under ASAF semantics is as follows: PS(P∆00 ) =
{N3 = M3 ∪ {z, β2 }, N4 = M4 ∪ {¬z, β2 }}, where M3 and M4 are the models given
in Example 4 which, again, coincide with those of Example 5.
Now consider how we have to revise the propositional programs of Example 5 to
take into account the new elements z and β2 . In this case, the propositional program
P∆00 under RAFD (resp., AFRAD) semantics is obtained by adding to the propositional
program P∆0 under RAFD (resp., AFRAD) semantics of Example 5 the rules z ← and
β2 ← and by replacing the rule having as head p with the rule p ← (¬α1 ∨¬r)∧(¬β2 ∨
z) (resp., p ← ¬α1 ∧ (¬β2 ∨ z)). Thus, set of partial stable models of P∆00 under RAFD
semantics is PS(P∆00 ) = {N1 = M1 ∪ {z, β2 }, N20 = M2 ∪ {z, β2 }}, while under
AFRAD semantics we obtain PS(P∆00 ) = {N3 = M3 ∪ {z, β2 }, N40 = M4 ∪ {z, β2 }}.
Hence, here we have that PSM N2 (resp., N4 ) derived from RAFN (resp., ASAF)
∆00 differs from PSM N20 (resp., N40 ) derived from RAFD (resp., AFRAD) ∆00 . Conse-
quently, the sets of the extensions of the four frameworks differ as well.
4 Conclutions and Future Work
By exploring the connection between formal argumentation and logic programming,
we have proposed a simple but general logical framework which is able to capture, in
a systematic and succinct way, the different features of several AF-based frameworks
under different argumentation semantics and interpretation of the support relation. The
proposed approach can be used for better understanding the semantics of extended AF
frameworks (sometimes a bit involved), and is flexible enough for encouraging the study
of other extensions. Our approach can be used to provide additional tools for computing
complete extensions using answer set solvers [25] and classical program rewriting [30,
35, 24]. For instance, ASP-based tools like DLV and Potassco can be used to compute
stable models, and XSB can be used to compute well founded semantics.
Other extensions of the Dung’s framework not explicitly discussed in this paper are
also captured by our technique as they are special cases of some of those studied in this
paper. This is the case of Extended AF (EAF) and hierarchical EAF, which extend AF
by allowing second order and stratified attacks, respectively [31].
Future work will be devoted to further generalize our logical approach in order to
deal also with AF-based framework considering probabilities [23], weights [11], and
preferences [8, 31], and frameworks considering supports with multiple sources [17].
Finally, we plan to investigate incremental techniques tailored at using our approach
to compute extensions of dynamic AF-based frameworks, where the sets of arguments
and interactions change over the time [28, 3, 4, 2, 5].
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