On the equivalence between bipolar argumentation frameworks and frameworks with sets of attacking arguments (SETAFs)

Abstract

Abstract argumentation concerns the construction and evaluation of arguments according to their interactions. In Dung’s abstract argumentation frameworks (AAFs), arguments interact negatively via an attack relation. Since then, a plethora of extensions have been proposed with the aim of expressing other common argument relationships. Two such proposals are bipolar argumentation frameworks (BAFs), in which both attack (negative) and support (positive) relations coexist; and frameworks with sets of attacking arguments (SETAFs), where attacks can be collective, originating from a set of arguments. In this paper, we show equivalences between a specific notion of support ( $β$ -semantics) and of joint attacks in argumentation, by providing direct translations from BAFs and SETAFs (and vice versa) in a one-to-one correspondence between (BAF) $β$ -complete and (SETAF) complete labellings, $β$ -grounded and grounded labellings, $β$ -preferred and preferred labellings, $β$ -stable and stable labellings, and $β$ -semi-stable and semi-stable labellings. Besides semantic equivalences, we show structural (or syntactic) equivalences between BAFs and SETAFs by finding subsets of them for which the proposed translations are each other’s inverse up to isomorphism.

Keywords

bipolar argumentation framework with sets of attacking arguments (SETAF)intertranslatability semantic equivalence syntactic equivalence

1. Introduction

Argumentation is a highly successful paradigm in artificial intelligence and knowledge representation. It concerns constructing and evaluating arguments to decide the acceptability of a claim. A prominent subarea of research deals with arguments abstractly: by determining argument acceptability exclusively from how arguments interact, it is able to model complex patterns of reasoning and is specifically suited for handling uncertain and conflicting information.

In his seminal paper, Dung¹ defines abstract argumentation frameworks ( $AAF$ s) by only a set of arguments and a binary attack relation between them. We can depict such frameworks as directed graphs with nodes representing arguments and edges representing attacks. In Dung’s original proposal, semantics were described in terms of extensions, that is, sets of arguments considered accepted. In this paper, we use equivalent characterizations of semantics via labellings,² which assign a label $i n$ (accepted), $o u t$ (rejected), or $u n d e c$ (undecided) to each argument.

The simplicity of AAFs has inspired many framework extensions for expressing other kinds of argument interactions. Some examples include assumption-based argumentation ( $ABA$ ) frameworks,³ abstract dialectical frameworks ( $ADF$ s),⁴ bipolar argumentation frameworks ( $BAF$ s),⁵ claim-augmented frameworks ( $CAF$ s),⁶ and frameworks with sets of attacking arguments ( $SETAF$ s).⁷

There already exist semantic-preserving translations between many of the aforementioned formalisms, such as between $CAF$ s and $SETAF$ s;^8,9 $SETAF$ s and $ADF$ s;^10–12 $ABA$ frameworks and $CAF$ s;¹³ and $ABA$ frameworks and $SETAF$ s.¹³ An overview is provided by König et al.,¹³ including the relationship between argumentation and the closely related area of logic programming.

The motivation for finding such correspondences lies in the fact that they allow one to select, for each scenario, the most suitable argumentation framework. Moreover, techniques, algorithms, and semantics derived from one formalism can often be adapted to the other via these transformations. It also clarifies which forms of interactions are more (or equally) expressive among those commonly used in argumentation.

Our work follows this line of research, focusing on the connection between $BAF$ s, which allow negative (attack) and positive (support) interactions between arguments, and $SETAF$ s, capable of representing collective attacks. Some connections were already elicited between $BAF$ and logic programming,^14,15 and between logic programming and $SETAF$ .¹⁶ This shows an indirect path connecting $BAF$ s and $SETAF$ s via logic programming. However, there remain questions about whether these connections can be naturally made more direct, and for which class of formalisms these translations can be made invertible up to isomorphism, that is, preserving attack/support relation structures while disregarding argument names.

We study correspondences for the following semantics:

In $SETAF$ s, the basic notion of a joint attack from a set of arguments $S$ to an argument $A$ is that $A$ is rejected if every argument in $S$ is accepted. This criterion gives rise to the labelling-based complete, grounded, preferred, stable, and semi-stable $SETAF$ semantics.¹⁷

In $BAF$ s, attacks and supports between arguments can interact in many ways. Therefore, many semantics for $BAF$ s have been proposed,^18–21 each with its own meaning of support. For translating between $BAF$ s and $SETAF$ s, we consider Alcântara and Cordeiro’s²¹ labelling-based $β$ -complete ( $BAF$ ) semantics and their refinements ( $β$ -grounded, $β$ -preferred, $β$ -stable, and $β$ -semi-stable). The basic notion of support provided by the $β$ -semantics is that an argument $A$ is rejected if every argument supporting $A$ (including $A$ itself) is attacked by an accepted argument.

In both cases, argument $A$ is rejected because of a set of accepted arguments. We choose the $β$ -semantics as its treatment of support aligns naturally with the notion of collective attacks in $SETAF$ s. For finding structural correspondences, we take advantage of a fundamental result: mutually supporting arguments share the same label in the $β$ -semantics. In Section 6, we show that other $BAF$ semantics treat support cycles differently, and that this distinction makes directly applying our strategy unviable. In our work, the correspondences we find involving $BAF$ s are limited to the $β$ -semantics, but similar strategies could help find equivalence results between other formalisms and/or under other semantics.

The main contributions of this work are twofold:

we provide direct translations from $BAF$ s to $SETAF$ s (and vice versa) and their semantics in a one-to-one semantic correspondence between ( $BAF$ ) $β$ -complete and ( $SETAF$ ) complete labellings; $β$ -grounded and grounded labellings; $β$ -preferred and preferred labellings; $β$ -stable and stable labellings; and $β$ -semi-stable and semi-stable labellings; and

we find subclasses of $BAF$ s and $SETAF$ s for which the proposed translations satisfy syntactic equivalences, that is, preserve attack/support relation structures.

This paper is organized as follows: in Section 2, we review both $BAF$ s and $SETAF$ s; we show the translation from $BAF$ s to $SETAF$ s in Section 3, and from $SETAF$ s to $BAF$ s in Section 4; we discuss syntactic correspondences in Section 5; we compare our paper to related work in Section 6; conclusions and future lines of study are presented in Section 7.

2. Preliminaries

2.1. Bipolar argumentation frameworks (BAFs)

$AAF$ s¹ were extended by Amgoud et al.⁵ to incorporate an explicit notion of support between arguments, independent of the attack relation. The resulting framework, called $BAF$ , is defined next:

Definition 1 $BAF$ ⁵

A $BAF$ is a tuple $(A, A t t, S u p)$ , in which $A$ is a finite set of arguments, $A t t \subseteq A \times A$ is the attack relation, and $S u p \subseteq A \times A$ is the support relation. For an argument $A \in A$ , we define $A t t (A) = {B \in A ∣ (B, A) \in A t t}$ and $S u p (A) = {B \in A ∣ (B, A) \in S u p}$ as, respectively, the set of attackers of $A$ and the set of direct supporters of $A$ .

In $AAF$ s, the interaction between arguments is specified only by negative interactions (attacks). In $BAF$ s, the novelty is that a positive interaction (support) can also be specified. A $BAF$ $(A, A t t, S u p)$ in which $S u p = \emptyset$ amounts to a (standard Dung) $AAF$ .

We are interested not only in the direct supporters of an argument but also in the indirect ones. They are indistinguishably called the supporters of an argument:

Definition 2 Supporters²¹

Let $B = (A, A t t, S u p)$ be a $BAF$ and $A \in A$ an argument. We define the supporters of $A$ recursively as follows:

$A$ is a supporter of $A$ in $B$ ;

if $A^{'}$ is a supporter of $A$ in $B$ and $(B, A^{'}) \in S u p$ , then $B$ is a supporter of $A$ in $B$ .

By $S u p (A) = {A^{'} \in A ∣ A^{'} is a supporter of A in B}$ , we denote the set of all supporters of $A$ in $B$ . Additionally, we denote the relation of being a supporter by $S u p = {(A^{'}, A) ∣ A^{'} \in S u p (A)}$ .

Semantics for $BAF$ s can be equivalently characterized in terms of extensions or labellings.²¹ We will focus on labelling-based semantics.

Definition 3 $BAF$ labellings

Let $B = (A, A t t, S u p)$ be a $BAF$ . A labelling (of $B$ ) is a total function $L : A \to {i n, o u t, u n d e c}$ .

When $L$ is a labelling, we write $i n (L)$ to denote the set ${A ∣ L (A) = i n}$ , $o u t (L)$ for ${A ∣ L (A) = o u t}$ , and $u n d e c (L)$ for ${A ∣ L (A) = u n d e c}$ . As a labelling defines a partition among arguments, when convenient we write $L$ as the triple $(i n (L), o u t (L), u n d e c (L))$ . Intuitively, the label $i n$ indicates acceptance, the label $o u t$ indicates rejection, and the label $u n d e c$ indicates that the argument is undecided, that is, neither accepted nor rejected.

When a labelling assigns the same label for arguments with identical sets of supporters, we say it respects $S u p$ .

Definition 4 $L$ respects $S u p$ ²²

Let $B = (A, A t t, S u p)$ be a $BAF$ and $L$ be a labelling of $B$ . We say $L$ respects $S u p$ if $L (A) = L (B)$ for any $A, B \in A$ such that $S u p (A) = S u p (B)$ .

There is no consensus on how exactly supports and attacks should interact with each other and, as a result, many semantics for $BAF$ s have been proposed,^18–21 each with its own interpretation of support. In this paper, we employ Alcântara and Cordeiro’s²¹ $β$ -semantics because its treatment of support aligns naturally with the notion of collective attacks in $SETAF$ s.

Before proceeding to the definition of the $β$ -semantics, we show some intuitions about its notion of support. First, the interpretation of support is deductive: given that $C$ supports $A$ , (i) if $C$ is accepted, then $A$ is accepted; (ii) if $A$ is rejected, then $C$ is rejected. Moreover, the support relation interacts with the acceptance and rejection of arguments as follows:

Argument $A$ is accepted iff some supporter of $A$ is accepted iff every argument supported by $A$ is accepted.

Argument $A$ is rejected iff every supporter of $A$ is rejected iff some argument supported by $A$ is rejected.

This means that each supporter $A^{'}$ of $A$ (recall that $A$ is a supporter of itself) is a sufficient condition for accepting $A$ , and, conversely, for defeating $A$ , one needs to defeat all of its supporters. The attack relation in the $β$ -semantics is weaker than in traditional attack-only $AAF$ s: given $A$ attacks $B$ , it is not always the case that $B$ is rejected when $A$ is accepted, because there might exist other reasons for accepting $B$ , that is, $B$ might be supported by an argument $C$ that is uncontested (or only attacked by rejected arguments). We explore the well-known Tweety example to show that such a weaker notion of attack is sometimes desired, with the support relation representing an exception to a general rule:

Example 5
Consider the following arguments:
$A$ : Tweety cannot fly.

$B$ : Tweety is a bird (and, in general, birds can fly).

$C$ : Tweety is a penguin.
Figure 1 depicts the $BAF$ $B_{1}$ for this scenario. There is a mutual attack between $A$ and $B$ as each argument brings evidence against the other. Additionally, accepting $C$ implies accepting $A$ and $B$ , so $C$ supports both of them. Due to the support from $C$ , neither does accepting $A$ lead to rejecting $B$ , nor does accepting $B$ lead to rejecting $A$ (despite the mutual attack between $A$ and $B$ ). Note that $C$ represents an exception to the general rule that birds usually fly (and if something cannot fly, then it usually is not a bird). Thus, in the unique $β$ -complete labelling of $B_{1}$ , $A$ , $B$ , and $C$ have the label $i n$ .
Figure 1.
Bipolar argumentation framework ( $BAF$ ) $B_{1}$ from Example 5 (Tweety). Nodes represent arguments, solid arrows represent the attack relation, and dashed arrows represent the support relation. In this example, argument $C$ represents an exception to the general rule that birds usually fly.

This deductive interpretation of support is formalized as follows:
Definition 6 Labelling-based $β$ -semantics²¹

Let $B = (A, A t t, S u p)$ be a $BAF$ . A labelling $L$ of $B$ is $β$ -complete if for any $A \in A$ ,

$L (A) = i n$ iff there exists $A^{'} \in S u p (A)$ such that for every $B \in A t t (A^{'})$ , it holds $L (B) = o u t$ ;

$L (A) = o u t$ iff for every $A^{'} \in S u p (A)$ , there exists $B \in A t t (A^{'})$ such that $L (B) = i n$ .

Additionally, we say a $β$ -complete labelling of $B$ is

$β$ -grounded if $i n (L)$ is $\subseteq$ -minimal among the $β$ -complete labellings of $B$ ;

$β$ -preferred if $i n (L)$ is $\subseteq$ -maximal among the $β$ -complete labellings of $B$ ;

$β$ -stable if $u n d e c (L) = \emptyset$ ;

$β$ -semi-stable if $u n d e c (L)$ is $\subseteq$ -minimal among the $β$ -complete labellings of $B$ .

Example 7
Recall that a labelling $L$ can be represented by the triple $(i n (L), o u t (L), u n d e c (L))$ . Consider the $BAF$ $B_{2}$ of Figure 2. From this $BAF$ , we obtain the following labelling-based semantics:
$β$ -complete labellings: $L_{1} = (\emptyset, \emptyset, {A, B, C, D, E})$ , $L_{2} = ({A, D}, {B}, {C, E})$ and $L_{3} = ({B, C, D},$ ${A}, {E})$ .

$β$ -grounded labelling: $L_{1}$ .

$β$ -preferred labellings: $L_{2}$ and $L_{3}$ .

$β$ -stable labellings: None.

$β$ -semi-stable labelling: $L_{3}$ .

Figure 2.
Bipolar argumentation framework ( $BAF$ ) $B_{2}$ from Example 7, where argument $D$ cannot be rejected simply by accepting $E$ (an attacker of $D$ ), as $D$ is supported by $A$ and $B$ .

Moreover, the dual nature of the support relation in the $β$ -semantics allows one to decompose an argument (in a $BAF$ ) by its set of supporters ( $A$ is accepted iff some supporter of $A$ is accepted).
Theorem 8 Alcântara and Cordeiro²¹

If $L$ is a $β$ -complete labelling of $B = (A, A t t, S u p)$ , then for any $A \in A$ ,

$L (A) = i n$ iff $L (A^{'}) = i n$ for some $A^{'} \in S u p (A)$ iff $L (A^{″}) = i n$ for every $A^{″} \in A$ such that $A \in S u p (A^{″})$ ;

$L (A) = o u t$ iff $L (A^{'}) = o u t$ for every $A^{'} \in S u p (A)$ iff $L (A^{″}) = o u t$ for some $A^{″} \in A$ such that $A \in S u p (A^{″})$ .

Alternatively, for any $β$ -complete labelling $L$ , if $B$ is a supporter of $A$ , then $L (B) \leq L (A)$ w.r.t. the ordering $o u t < u n d e c < i n$ . In particular, when $A$ and $B$ support each other, they necessarily share the same label. This means that every $β$ -complete labelling respects $S u p$ :

Theorem 9 Alcântara and Cordeiro²²

If $L$ is a $β$ -complete labelling of $B = (A, A t t, S u p)$ , then $L$ respects $S u p$ .

As arguments with identical sets of supporters always share the same label in $β$ -complete labellings, this allows for a more compact representation of $BAF$ s when translating $BAF$ s to other formalisms.

2.2. Frameworks with sets of attacking arguments ( $SETAF$ s)

Now, we take a look at argumentation frameworks with joint attacks, first proposed by Nielsen and Parsons:⁷

Definition 10 $SETAF$

A framework with sets of attacking arguments ( $S E T A F$ for short) is a pair $A = (A, A t t)$ , in which $A$ is a finite set of arguments and $A t t \subseteq (2^{A} - {\emptyset}) \times A$ is an attack relation such that if $(X, A) \in A t t$ , there is no $X^{'} \subset X$ such that $(X^{'}, A) \in A t t$ , that is, $X$ is a $\subseteq$ -minimal set attacking $A$ . By $A t t (A) = {X \subseteq A ∣ (X, A) \in A t t}$ we mean the set of attackers of $A$ .

We remark that in the original definition,⁷ attacks are not necessarily subset minimal, but Polberg¹⁰^, $^{p135}$ and Dvořák et al.²³ prove that this restriction can be added without changing the semantics considered in this work.

In $AAF$ s, only individual arguments can attack arguments. In $SETAF$ s, the novelty is that sets of two or more arguments can also attack arguments, indicating a joint attack. Notice that a $SETAF$ $(A, A t t)$ with $| X | = 1$ for every $(X, A) \in A t t$ amounts to (standard Dung) $AAF$ s.

Semantics for $SETAF$ s can be equivalently characterized in terms of extensions or labellings.¹⁷ We will focus on labelling-based semantics, as proposed by Flouris and Bikakis:¹⁷

Definition 11 $SETAF$ labellings

Let $A = (A, A t t)$ be a $SETAF$ . A labelling is a total function $L : A \to {i n, o u t, u n d e c}$ .

Similar to $BAF$ labellings, we use the convenient notation of $i n (L) = {A ∣ L (A) = i n}$ , $o u t (L) = {A ∣ L (A) = o u t}$ , and $u n d e c (L) = {A ∣ L (A) = u n d e c}$ , and also $L = (i n (L), o u t (L), u n d e c (L))$ .

Definition 12 Labelling-based $SETAF$ semantics¹⁷

Let $A = (A, A t t)$ be a $SETAF$ . A labelling $L$ of $A$ is complete if for any $A \in A$ ,

$L (A) = i n$ iff for every $X \in A t t (A)$ , there exists $X \in X$ such that $L (X) = o u t$ ;

$L (A) = o u t$ iff there exists $X \in A t t (A)$ such that for every $X \in X$ , it holds $L (X) = i n$ .

Additionally, we say a complete labelling of $A$ is

grounded if $i n (L)$ is $\subseteq$ -minimal among the complete labellings of $A$ ;

preferred if $i n (L)$ is $\subseteq$ -maximal among the complete labellings of $A$ ;

stable if $u n d e c (L) = \emptyset$ ;

semi-stable if $u n d e c (L)$ is $\subseteq$ -minimal among the complete labellings of $A$ .

Example 13
Consider the $SETAF$ of Figure 3. From this $SETAF$ , we obtain the following labelling-based semantics:
Complete labellings $L_{1} = (\emptyset, \emptyset, {A, B, C, D, E})$ , $L_{2} = ({A, D}, {B}, {C, E})$ and $L_{3} = ({B, C, D},$ ${A}, {E})$ .

Grounded labelling: $L_{1}$ .

Preferred labellings: $L_{2}$ and $L_{3}$ .

Stable labellings: None.

Semi-stable labelling: $L_{3}$ .

Figure 3.
Framework with sets of attacking arguments ( $SETAF$ ) $A_{1}$ from Example 13. Nodes represent arguments, and arrows starting at possibly multiple nodes and ending at a single node represent the attack relation. For instance, the arrow starting at nodes $A, B$ and $E$ and ending at $D$ indicates that ${A, B, E}$ is an attacker of $D$ .

As we shall see in Section 3, it is no coincidence that the semantics of the $SETAF$ of Figure 3 and of the $BAF$ of Figure 2 are exactly the same.
2.3. Combinatorics of finite sets

Before proceeding to Section 3, we recall a useful definition from the combinatorics of finite sets²⁴ and contextualize it to argumentation, in which $A$ is a finite set of arguments.

Definition 14 Transversal

Let $H \subseteq 2^{A}$ be a collection of sets of arguments. A set $T \subseteq A$ is a transversal of $H$ if $T$ intersects each element of $H$ , that is, $T \cap H \neq \emptyset$ for any $H \in H$ . A transversal is also called a hitting set, in other works.

Definition 15 $Tr$ operator

Let $H \subseteq 2^{A}$ . The family of $\subseteq$ -minimal transversals of $H$ is denoted by $Tr [H]$ .

One of Berge’s²⁴ results, when applied to the context of argumentation, gives us the following theorem:

Theorem 16 Berge²⁴

Let $H \subseteq 2^{A}$ . If every element of $H$ is $\subseteq$ -minimal (i.e., there is no $X, Y \in H$ such that $X \subset Y$ ), then $Tr [Tr [H]] = H$ .

The $Tr$ operator is essential to our approach of linking $BAF$ s and $SETAF$ s, as we show in the next two sections, in which we, respectively, translate $BAF$ s to $SETAF$ s (Section 3) and $SETAF$ s to $BAF$ s (Section 4). In addition, Theorem 16 guarantees that these translations act as inverses (up to isomorphism) when applied to certain classes of $BAF$ s and $SETAF$ s, as we discuss in Section 5.

3. From BAFs to SETAFs

In this section, we provide a translation from $BAF$ $B$ into its corresponding $SETAF$ $A_{B}$ , such that there is a one-to-one correspondence between, respectively, $β$ -complete, $β$ -grounded, $β$ -preferred, $β$ -stable, and $β$ -semi-stable labellings of $B$ and complete, grounded, preferred, stable, and semi-stable labellings of $A_{B}$ .

3.1. SETAF construction

Recall that arguments with an identical set of supporters are labelled the same according to the $β$ -semantics, that is, for any $β$ -complete labelling $L$ , it holds $L (A) = L (B)$ if $S u p (A) = S u p (B)$ for arguments $A$ and $B$ (Theorem 9). For a more compact representation of arguments in the corresponding $SETAF$ , arguments in $BAF$ $B$ with identical sets of supporters map to the same argument in the corresponding $SETAF$ $A_{B}$ .

For instance, in the $BAF$ from Figure 2, $A$ maps to $S u p (A) = {A}$ , $B$ maps to $S u p (B) = {B}$ , $C$ maps to $S u p (C) = {B, C}$ , $D$ maps to $S u p (D) = {A, B, D}$ , and $E$ maps to $S u p (E) = {E}$ . We introduce a convenient notation for this mapping:

Definition 17 $S$ -projection

Let $B = (A, A t t, S u p)$ be a $BAF$ .

For any set of arguments $S \subseteq A$ , $S_{S} = {S u p (X) ∣ X \in S}$ is the set obtained by replacing each argument $X$ in $S$ by its set of supporters $S u p (X)$ .

With this notation, the set of arguments $A$ from a $BAF$ maps to $A_{S}$ in the corresponding $SETAF$ . Now only the attack relation of the corresponding $SETAF$ remains to be specified.

Each argument $Y \in A_{S}$ in the corresponding $SETAF$ will be a set of arguments ( $Y \subseteq A$ ) in the $BAF$ . More specifically, the set of supporters of some argument $A \in A$ . Thus, we may write $Y = S u p (A)$ for some $A \in A$ . In the corresponding $SETAF$ , an attacker-set $X$ of argument $Y \in A_{S}$ is a set $X \subseteq A_{S}$ , that is, $X$ is a set of elements in $A_{S}$ , so each element of $X$ is a set of arguments in $A$ .

Consider the $BAF$ from Figure 4(a). Argument $F$ has two supporters: $A$ and $F$ itself. In the corresponding $SETAF$ , $F$ will be represented by $S u p (F) = {A, F}$ . We wish to find which sets of arguments in $A_{S}$ should attack $S u p (F)$ . According to the $β$ -semantics, $F$ has two acceptance conditions: $F$ is accepted iff (i) $A$ only has rejected attackers or (ii) $F$ only has rejected attackers. Equivalently, $F$ is accepted iff

(i)
$B$ is rejected, or
(ii)
$E$ and $G$ are rejected.

Figure 4.
Bipolar argumentation framework ( $BAF$ ) $B$ from Example 19 and its corresponding $SETAF$ $A_{B}$ . (a) $B$ , (b) $A_{B}$ .

We can rewrite “(i) or (ii)” as “(1) and (2),” where (1)
$B$ or $E$ are rejected, and
(2)
$B$ or $G$ are rejected.

The conditions (i) and (ii) obtained from the $β$ -semantics are rewritten to formats (1) and (2) by applying the $Tr$ operator, which computes minimal transversals: $Tr [{{B}, {E, G}}] = {{B, E}, {B, G}}$ . Intuitively, ${B, E}$ and ${B, G}$ can be seen as attacker-sets of the $SETAF$ argument $S u p (F)$ representing $F$ , in the sense that for $S u p (F)$ to be accepted, there must be some rejected argument in every attacker-set of $S u p (F)$ . Thus, conditions (1) and (2) are more aligned to the notion of a joint attack in a $SETAF$ . However, $B$ , $E$ , and $G$ are arguments in $A$ but not in $A_{S}$ , and any attacker-set of $S u p (F)$ should be a subset of $A_{S}$ . The attacker-sets of $S u p (F)$ in this case are ${S u p (B), S u p (E)}$ and ${S u p (B), S u p (G)}$ .

This strategy for determining the attacks in the corresponding $SETAF$ is formalized next. Recall that $Y \in A_{S}$ is a subset of $A$ and can be rewritten as $S u p (A)$ for some $A \in A$ .
Definition 18 Corresponding $SETAF$

Let $B = (A, A t t, S u p)$ be a $BAF$ . The corresponding $SETAF$ of $B$ is $A_{B} = (A_{S}, A t t_{B})$ , where $A t t_{B} \subseteq (2^{A_{S}} - \emptyset) \times A_{S}$ is a relation such that $(X, Y) \in A t t_{B}$ iff $X \in Tr [{A t t (A^{'})_{S} ∣ A^{'} \in Y}]$ .

Example 19
For transforming the $BAF$ $B$ (Figure 4(a)) into its corresponding $SETAF$ (Figure 4(b)),
compute $A_{S}$ by finding $S u p (\cdot)$ for each argument of $B$ , that is, $S u p (A) = {A}$ , $S u p (B) = {B}$ , $S u p (C) = {C, G}$ , $S u p (D) = {D}$ , $S u p (E) = {E}$ , $S u p (F) = {A, F}$ , and $S u p (G) = {C, G}$ ;

compute $A t t (\cdot)_{S}$ for each argument of $B$ , that is, $A t t (A)_{S} = {{B}}$ , $A t t (B)_{S} = {{A}}$ , $A t t (C)_{S} = {{B}}$ , $A t t (D)_{S} = \emptyset$ , $A t t (E)_{S} = {{E}}$ , $A t t (F)_{S} = {{E}, {C, G}}$ , and $A t t (G)_{S} = {{C, G}}$ ;

for each argument $Y \in A_{S}$ , find $\subseteq$ -minimal sets $X \subseteq A_{S}$ such that $X \cap A t t (B)_{S} \neq \emptyset$ for every $B \in Y$ : –
for $Y = {A}$ , we have to find $\subseteq$ -minimal $X \subseteq A_{S}$ such that $X \cap A t t (A)_{S} \neq \emptyset$ , that is, such that $X \cap {{B}} \neq \emptyset$ . Hence, $X = {{B}}$ is the only solution.
–
for $Y = {B}$ , we have to find $\subseteq$ -minimal $X \subseteq A_{S}$ such that $X \cap A t t (B)_{S} \neq \emptyset$ , that is, such that $X \cap {{A}} \neq \emptyset$ . Hence, $X = {{A}}$ is the only solution.
–
for $Y = {C, G}$ , we have to find $\subseteq$ -minimal $X \subseteq A_{S}$ such that $X \cap A t t (C)_{S} \neq \emptyset$ and $X \cap A t t (G)_{S} \neq \emptyset$ , that is, such that $X \cap {{B}} \neq \emptyset$ and $X \cap {{C, G}} \neq \emptyset$ . Hence, $X = {{B}, {C, G}}$ is the only solution.
–
for $Y = {D}$ , we have to find $\subseteq$ -minimal $X \subseteq A_{S}$ such that $X \cap A t t (D)_{S} \neq \emptyset$ , that is, such that $X \cap \emptyset \neq \emptyset$ . There is no such $X$ .
–
for $Y = {E}$ , we have to find $\subseteq$ -minimal $X \subseteq A_{S}$ such that $X \cap A t t (E)_{S} \neq \emptyset$ , that is, such that $X \cap {{E}} \neq \emptyset$ . Hence, $X = {{E}}$ is the only solution.
–
for $Y = {A, F}$ , we have to find $\subseteq$ -minimal $X \subseteq A_{S}$ such that $X \cap A t t (A)_{S} \neq \emptyset$ and $X \cap A t t (F)_{S} \neq \emptyset$ , that is, such that $X \cap {{B}} \neq \emptyset$ and $X \cap {{E}, {C, G}} \neq \emptyset$ . Hence, $X = {{B}, {E}}$ and $X = {{B}, {C, G}}$ are the only solutions.

3.2. Equivalence results

In the sequel, we show the equivalence between the semantics of a $BAF$ $B$ and their counterpart for the corresponding SETAF $A_{B}$ . The equivalence results are established by identifying connections between the fundamental concepts that underlie the semantics of $BAF$ s and $SETAF$ s. For this purpose, we introduce two functions:

${B 2 cS}_{B}$ maps labellings of a $BAF$ $B$ to labellings of its corresponding $SETAF$ $A_{B}$ . The acronym ${B 2 cS}_{B}$ stands for “ $BAF$ to corresponding $SETAF$ .”

${cS 2 B}_{B}$ maps labellings of the corresponding $SETAF$ $A_{B}$ to labellings of the $BAF$ $B$ . The acronym ${cS 2 B}_{B}$ stands for “corresponding $SETAF$ to $BAF$ .”

We then investigate the conditions under which these functions act as inverses of each other and employ these results to prove the equivalence between the semantics. Notably, these functions permit us to treat labellings of $BAF$ s and of corresponding $SETAF$ s interchangeably.

Definition 20 ${B 2 cS}_{B}$ and ${cS 2 B}_{B}$ functions

Let $B = (A, A t t, S u p)$ be a $BAF$ , $A_{B} = (A_{S}, A t t_{B})$ be its corresponding $SETAF$ , $L a b_{B}$ be the set of all labellings of $B$ and $L a b_{A_{B}}$ be the set of all labellings of $A_{B}$ . We introduce a function ${B 2 cS}_{B} : L a b_{B} \to L a b_{A_{B}}$ such that ${B 2 cS}_{B} (L) = L^{'}$ , in which

$i n (L^{'}) = {Y \in A_{S} ∣ L (A^{'}) = i n for some A^{'} \in Y}$ ;

$o u t (L^{'}) = {Y \in A_{S} ∣ L (A^{'}) = o u t for every A^{'} \in Y}$ ;

$u n d e c (L^{'}) = {Y \in A_{S} ∣ Y \notin i n (L^{'}) \cup o u t (L^{'})}$ .

We introduce a function ${cS 2 B}_{B} : L a b_{A_{B}} \to L a b_{B}$ such that ${cS 2 B}_{B} (L^{'}) = L$ , in which $L (A) = L^{'} (S u p (A))$ for every $A \in A$ .

The direction from labellings of the corresponding $SETAF$ $A_{B}$ to labellings of the $BAF$ $B$ is straightforward: the label assigned to an argument $A$ in $B$ follows from the label assigned to $S u p (A)$ in $A_{B}$ . In the inverse direction, an argument $Y \in A_{S}$ (consisting of a set of arguments from $A$ ) is accepted in $A_{B}$ if some argument in $Y$ is accepted in $B$ ; is rejected in $A_{B}$ if every argument in $Y$ is rejected in $B$ ; otherwise, $Y$ is undecided in $A_{B}$ .

In general, ${cS 2 B}_{B} ({B 2 cS}_{B} (L))$ is not equal to $L$ . For instance, considering the labelling $L = ({C}, {G}, {A, B, D, E, F})$ of the $BAF$ $B$ of Figure 4(a), we have ${cS 2 B}_{B} ({B 2 cS}_{B} (L)) = ({C, G}, \emptyset, {A, B, D, E, F})$ . Such an inequality will always occur when two arguments with the same set of supporters receive distinct labels. In our example, arguments $C$ and $G$ have the same set of supporters ${C, G}$ , but $L (C) = i n$ and $L (G) = o u t$ . It holds ${B 2 cS}_{B} (L) ({C, G}) = i n$ and thus both $C$ and $G$ are labelled $i n$ by ${cS 2 B}_{B} ({B 2 cS}_{B} (L))$ . However, for $β$ -complete labellings, arguments with the same set of supporters have the same label (Theorem 9). A more general scenario in which the inequality arises is if $L (A) = i n$ and $L (F) = o u t$ , as ${B 2 cS}_{B} (L) ({A, F}) = i n$ and then both $A$ and $F$ are labelled $i n$ by ${cS 2 B}_{B} ({B 2 cS}_{B} (L))$ . This occurs because $A$ is a supporter of $F$ , but $F$ received a smaller label w.r.t. to the ordering $o u t < u n d e c < i n$ . In $β$ -complete labellings, $A^{'} \in S u p (A)$ implies $L (A^{'}) \leq L (A)$ w.r.t. to this ordering (Theorem 8). Thus, for $β$ -complete labellings, we have that ${cS 2 B}_{B} ({B 2 cS}_{B} (L)) = L$ :

Theorem 21
Let $B = (A, A t t, S u p)$ be a $BAF$ and $A_{B} = (A_{S}, A t t_{B})$ be its corresponding $SETAF$ . For any $β$ -complete labelling $L$ of $B$ , it holds ${cS 2 B}_{B} ({B 2 cS}_{B} (L)) = L$ .

In the other direction, ${B 2 cS}_{B} ({cS 2 B}_{B} (L))$ is also not equal to $L$ , in general. For instance, considering the labelling $L = ({{A}}, {{A, F}}, {{B}, {C, G}, {D}, {E}})$ and the $SETAF$ $A_{B}$ of Figure 4(b), we have ${B 2 cS}_{B} ({cS 2 B}_{B} (L)) = ({{A}, {A, F}}, \emptyset, {{B}, {C, G}, {D}, {E}})$ . Such an inequality will always occur when argument $Y$ receives a greater label than $Y^{'}$ (w.r.t. to the ordering $o u t < u n d e c < i n$ ) and $Y \subseteq Y^{'}$ . In our example, arguments ${A}$ and ${A, F}$ satisfy ${A} \subseteq {A, F}$ , but $L ({A}) = i n$ and $L ({A, F}) = o u t$ . The reason for the inequality in this case is that ${cS 2 B}_{B} (L) (A) = i n$ and thus both ${A}$ and ${A, F}$ are labelled $i n$ by ${B 2 cS}_{B} ({cS 2 B}_{B} (L))$ . However, for complete labellings, we have that ${B 2 cS}_{B} ({cS 2 B}_{B} (L)) = L$ :
Theorem 22
Let $B = (A, A t t, S u p)$ be a $BAF$ and $A_{B} = (A_{S}, A t t_{B})$ be its corresponding $SETAF$ . For any complete labelling $L$ of $A_{B}$ , it holds ${B 2 cS}_{B} ({cS 2 B}_{B} (L)) = L$ .

This means that when restricted to $β$ -complete ( $BAF$ ) labellings and complete ( $SETAF$ ) labellings, ${B 2 cS}_{B}$ and ${cS 2 B}_{B}$ are each other’s inverse. From Theorems 21 and 22, we can obtain the following result:
Theorem 23
Let $B = (A, A t t, S u p)$ be a $BAF$ and $A_{B} = (A_{S}, A t t_{B})$ be its corresponding $SETAF$ . It holds
$L$ is a $β$ -complete labelling of $B$ iff ${B 2 cS}_{B} (L)$ is a complete labelling of $A_{B}$ .

$L$ is a complete labelling of $A_{B}$ iff ${cS 2 B}_{B} (L)$ is a $β$ -complete labelling of $B$ .

Theorem 23 is one of the main results of this paper. It plays a central role in ensuring the equivalence between the semantics for $BAF$ s and their counterpart for $SETAF$ s:
Theorem 24
Let $B = (A, A t t, S u p)$ be a $BAF$ and $A_{B} = (A_{S}, A t t_{B})$ be its corresponding $SETAF$ . It holds
$L$ is a $β$ -grounded labelling of $B$ iff ${B 2 cS}_{B} (L)$ is a grounded labelling of $A_{B}$ .

$L$ is a $β$ -preferred labelling of $B$ iff ${B 2 cS}_{B} (L)$ is a preferred labelling of $A_{B}$ .

$L$ is a $β$ -stable labelling of $B$ iff ${B 2 cS}_{B} (L)$ is a stable labelling of $A_{B}$ .

$L$ is a $β$ -semi-stable labelling of $B$ iff ${B 2 cS}_{B} (L)$ is a semi-stable labelling of $A_{B}$ .

The following result is a direct consequence of Theorems 22 and 24:
Corollary 25
Let $B = (A, A t t, S u p)$ be a $BAF$ and $A_{B} = (A_{S}, A t t_{B})$ be its corresponding $SETAF$ . It holds
$L$ is a grounded labelling of $A_{B}$ iff ${cS 2 B}_{B} (L)$ is a $β$ -grounded labelling of $B$ .

$L$ is a preferred labelling of $A_{B}$ iff ${cS 2 B}_{B} (L)$ is a $β$ -preferred labelling of $B$ .

$L$ is a stable labelling of $A_{B}$ iff ${cS 2 B}_{B} (L)$ is a $β$ -stable labelling of $B$ .

$L$ is a semi-stable labelling of $A_{B}$ iff ${cS 2 B}_{B} (L)$ is a $β$ -semi-stable labelling of $B$ .

As expected from Theorems 23 and 24, we obtain in Table 1 the equivalence between $β$ -complete and complete labellings, $β$ -grounded and grounded labellings, $β$ -preferred and preferred labellings, $β$ -stable and stable labellings, and $β$ -semi-stable and semi-stable labellings.

Table 1.
Semantics for $B$ and $A_{B}$ from Example 19.

$β$ -complete labellings of $B$ Complete labellings of $A_{B}$

$\begin{aligned} L_{1} & = ({D}, \emptyset, {A, B, C, E, F, G}) \\ L_{2} & = ({B, D}, {A}, {C, E, F, G}) \\ L_{3} & = ({A, C, D, F, G}, {B}, {E}) \end{aligned}$ $\begin{aligned} L_{1}^{'} & = ({{D}}, \emptyset, {{A}, {B}, {C, G}, {E}, {A, F}}) \\ L_{2}^{'} & = ({{B}, {D}}, {{A}}, {{C, G}, {E}, {A, F}}) \\ L_{3}^{'} & = ({{A}, {C, G}, {D}, {A, F}}, {{B}}, {{E}}) \end{aligned}$

$β$ -grounded labellings of $B$ Grounded labellings of $A_{B}$

$\begin{aligned} L_{1} & = ({D}, \emptyset, {A, B, C, E, F, G}) \end{aligned}$ $\begin{aligned} L_{1}^{'} & = ({{D}}, \emptyset, {{A}, {B}, {C, G}, {E}, {A, F}}) \end{aligned}$

$β$ -preferred labellings of $B$ Preferred labellings of $A_{B}$

$\begin{aligned} L_{2} & = ({B, D}, {A}, {C, E, F, G}) \\ L_{3} & = ({A, C, D, F, G}, {B}, {E}) \end{aligned}$ $\begin{aligned} L_{2}^{'} & = ({{B}, {D}}, {{A}}, {{C, G}, {E}, {A, F}}) \\ L_{3}^{'} & = ({{A}, {C, G}, {D}, {A, F}}, {{B}}, {{E}}) \end{aligned}$

$β$ -stable labellings of $B$ Stable labellings of $A_{B}$

None None

$β$ -semi-stable labellings of $B$ Semi-stable labellings of $A_{B}$

$\begin{aligned} L_{3} & = ({A, C, D, F, G}, {B}, {E}) \end{aligned}$ $\begin{aligned} L_{3}^{'} & = ({{A}, {C, G}, {D}, {A, F}}, {{B}}, {{E}}) \end{aligned}$

4. From SETAFs to BAFs

Now we will provide a translation for the opposite direction, that is, from $SETAF$ s to $BAF$ s. As in the previous section, this translation guarantees the equivalence between the semantics for $SETAF$ s and their counterpart for $BAF$ s.

4.1. BAF construction

Given a $SETAF$ , we obtain arguments and attacks in the corresponding $BAF$ following Caminada et al.’s work:²⁵ each constructed argument is associated with a conclusion and a set of vulnerabilities, and an attack occurs whenever the conclusion of one argument appears among the vulnerabilities of another. Vulnerabilities of arguments and this strategy for determining the attack relation are also studied in the context of semi-structured argumentation.^26,27 However, in our work, arguments remain abstract because conclusions and vulnerabilities are used only as intermediate steps for determining the attack and support relations, from which abstract BAF semantics then evaluate arguments.

According to the complete semantics, an argument $A$ of a $SETAF$ is accepted if every set attacking $A$ has some rejected argument. In this case, we can construct a set of arguments $V$ (standing for vulnerability) with only rejected attackers of $A$ , which is a sufficient condition for accepting $A$ in the $β$ -complete semantics. By this approach, each argument of a $SETAF$ (with possibly multiple vulnerabilities) will be represented by possibly multiple arguments supporting each other in the corresponding $BAF$ (one for each vulnerability). The mutual support is used to link arguments in the corresponding $BAF$ that correspond to the same argument in the $SETAF$ . This is formalized as follows:

Definition 26 Corresponding $BAF$

Let $A = (A, A t t)$ be a $SETAF$ . For any argument $A \in A$ , we define the set of vulnerabilities of $A$ as $V_{A} = Tr [A t t (A)]$ . The corresponding $BAF$ of $A$ is $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ , where

$A_{A} = {(A, V) ∣ A \in A, V \in V_{A}}$ ,

$A t t_{A} = {((A, V), (A^{'}, V^{'})) \in A_{A} \times A_{A} ∣ A \in V^{'}}$ , and

$S u p_{A} = {((A, V), (A, V^{'})) \in A_{A} \times A_{A} ∣ V \neq V^{'}}$ .

The main difficulty in computing the corresponding $BAF$ is finding the set of vulnerabilities for each argument. In contrast, the attack and support relations follow directly from the set of arguments in the corresponding $BAF$ .

Example 27
For transforming the $SETAF$ $A$ (Figure 5(a)) into its corresponding $BAF$ $B_{A}$ (Figure 5(b)),
compute $A_{A}$ by finding $V_{A}$ for each argument $A$ of $B$ : –
For $A$ , we find $\subseteq$ -minimal $V \subseteq A$ such that ${B} \cap V \neq \emptyset$ . Hence, $V = {B}$ is the only solution.
–
For $B$ , we find $\subseteq$ -minimal $V \subseteq A$ such that ${A} \cap V \neq \emptyset$ . Hence, $V = {A}$ is the only solution.
–
For $C$ , we find $\subseteq$ -minimal $V \subseteq A$ such that ${B, C} \cap V \neq \emptyset$ . Hence, $V = {B}$ and $V = {C}$ are the only solutions.
–
For $D$ , we find that $V = \emptyset$ is the $\subseteq$ -minimal set satisfying $X \cap V \neq \emptyset$ for every $X \in A t t (D)$ , as $A t t (D) = \emptyset$ . Hence, $V = \emptyset$ is the only solution.
–
For $E$ , we find $\subseteq$ -minimal $V \subseteq A$ such that ${E} \cap V \neq \emptyset$ . Hence, $V = {E}$ is the only solution.
–
For $F$ , we find $\subseteq$ -minimal $V \subseteq A$ such that ${B, E} \cap V \neq \emptyset$ and ${B, C} \cap V \neq \emptyset$ . Hence, $V = {B}$ and $V = {C, E}$ are the only solutions.

Figure 5.
$SETAF$ $A$ from Example 27 and its corresponding $BAF$ $B_{A}$ . (a) $A$ and (b) $B_{A}$ . $SETAF$ : framework with sets of attacking arguments; $BAF$ : bipolar argumentation framework.

We obtain $A_{A} = {(A, {B}), (B, {A}), (C, {B}), (C, {C}), (D, \emptyset), (E, {E}), (F, {B}),$ $(F, {C, E})}$ .

$A t t_{A}$ and $S u p_{A}$ are then easily obtained from $A_{A}$ .

4.2. Equivalence results

As in Section 3.2, we introduce two functions:

${S 2 cB}_{A}$ maps labellings of a $SETAF$ $A$ to labellings of its corresponding $BAF$ $B_{A}$ . The acronym ${S 2 cB}_{A}$ stands for “ $SETAF$ to corresponding $BAF$ .”

${cB 2 S}_{A}$ maps labellings of the corresponding $BAF$ $B_{A}$ to labellings of the $SETAF$ $A$ . The acronym ${cB 2 S}_{A}$ stands for “corresponding $BAF$ to $SETAF$ .”

We then investigate under which conditions these functions act as inverses of each other and employ these results to prove the equivalence between the semantics. Notably, these functions permit us to treat labellings of $SETAF$ s and of corresponding $BAF$ s interchangeably.

Definition 28 ${S 2 cB}_{A}$ and ${cB 2 S}_{A}$ functions

Let $A = (A, A t t)$ be a $SETAF$ , $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ , $L a b_{A}$ be the set of all labellings of $A$ and $L a b_{B_{A}}$ be the set of all labellings of $B_{A}$ . We introduce a function ${S 2 cB}_{A} : L a b_{A} \to L a b_{B_{A}}$ such that ${S 2 cB}_{A} (L) ((A, V)) = L (A)$ for any $(A, V) \in A_{A}$ , and we define ${cB 2 S}_{A} : L a b_{B_{A}} \to L a b_{A}$ such that ${cB 2 S}_{A} (L) = L^{'}$ , in which

$i n (L^{'}) = {A \in A ∣ L ((A, V)) = i n for some V \in V_{A}}$ ;

$o u t (L^{'}) = {A \in A ∣ L ((A, V)) = o u t for every V \in V_{A}}$ ; and

$u n d e c (L^{'}) = {A \in A ∣ A \notin i n (L^{'}) \cup o u t (L^{'})}$ .

The direction from labellings of a $SETAF$ $A$ to labellings of its corresponding $BAF$ $B_{A}$ is straightforward: the label assigned to an argument $(A, V)$ in $B_{A}$ follows from the label assigned to $A$ in $A$ . In the inverse direction, an argument $A \in A$ is accepted in $A$ if $(A, V)$ is accepted in $B_{A}$ for some vulnerability $V$ of $A$ ; is rejected in $A$ if $(A, V)$ is rejected in $B_{A}$ for every vulnerability $V$ of $A$ ; otherwise, $A$ is undecided in $A$ . Differently from ${B 2 cS}_{B}$ and ${cS 2 B}_{B}$ , the function ${cB 2 S}_{A}$ is a left inverse of ${S 2 cB}_{A}$ , that is, ${cB 2 S}_{A} ({S 2 cB}_{A} (L)) = L$ :

Theorem 29
Let $A = (A, A t t)$ be a $SETAF$ and $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ . For any labelling $L$ of $A$ , it holds ${cB 2 S}_{A} ({S 2 cB}_{A} (L)) = L$ .

However, in general, ${S 2 cB}_{A} ({cB 2 S}_{A} (L))$ is not equal to $L$ . For instance, consider the labelling $L = ({(F, {B})}, {(F, {C, E})}, \dots)$ of the $BAF$ $B_{A}$ of Figure 5(b), where $u n d e c (L) = A_{A} - (i n (L) \cup o u t (L))$ is omitted for convenience. We have that ${cB 2 S}_{A} (L) = ({F}, \emptyset, \dots)$ and ${S 2 cB}_{A} ({cB 2 S}_{A} (L)) = ({(F, {B}), (F, {C, E})}, \emptyset, \dots)$ . Such an inequality will always occur when the labels of $(A, V)$ and $(A, V^{'})$ differ for distinct vulnerabilities $V, V^{'}$ of an argument $A$ . In our example, $i n = L ((F, {B})) \neq L ((F, {C, E})) = o u t$ . It holds ${cB 2 S}_{A} (L) (F) = i n$ and thus both $(F, {B})$ and $(F, {C, E})$ are labelled $i n$ by ${S 2 cB}_{A} ({cB 2 S}_{A} (L))$ . However, as arguments $(A, V)$ and $(A, V^{'})$ mutually support each other, if $L$ respects $S u p$ , we always have $L ((A, V)) = L ((A, V^{'}))$ . Thus, for labellings of $B_{A}$ respecting $S u p$ , we have ${S 2 cB}_{A} ({cB 2 S}_{A} (L)) = L$ .
Theorem 30
Let $A = (A, A t t)$ be a $SETAF$ and $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ . For any labelling $L$ of $B_{A}$ respecting $S u p$ , it holds ${S 2 cB}_{A} ({cB 2 S}_{A} (L)) = L$ .

A similar result to Theorem 23 also holds:
Theorem 31
Let $A = (A, A t t)$ be a $SETAF$ and $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ . It holds
$L$ is a complete labelling of $A$ iff ${S 2 cB}_{A} (L)$ is a $β$ -complete labelling of $B_{A}$ .

$L$ is a $β$ -complete labelling of $B_{A}$ iff ${cB 2 S}_{A} (L)$ is a complete labelling of $A$ .

From Theorem 31, we can ensure the equivalence between the semantics for $SETAF$ s and their counterpart for $BAF$ s:
Theorem 32
Let $A = (A, A t t)$ be a $SETAF$ and $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ . It holds
$L$ is a grounded labelling of $A$ iff ${S 2 cB}_{A} (L)$ is a $β$ -grounded labelling of $B_{A}$ .

$L$ is a preferred labelling of $A$ iff ${S 2 cB}_{A} (L)$ is a $β$ -preferred labelling of $B_{A}$ .

$L$ is a stable labelling of $A$ iff ${S 2 cB}_{A} (L)$ is a $β$ -stable labelling of $B_{A}$ .

$L$ is a semi-stable labelling of $A$ iff ${S 2 cB}_{A} (L)$ is a $β$ -semi-stable labelling of $B_{A}$ .

The following result is a direct consequence of Theorems 30 and 31:
Corollary 33
Let $A = (A, A t t)$ be a $SETAF$ and $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ . It holds
$L$ is a $β$ -grounded labelling of $B_{A}$ iff ${cB 2 S}_{A} (L)$ is a grounded labelling of $A$ .

$L$ is a $β$ -preferred labelling of $B_{A}$ iff ${cB 2 S}_{A} (L)$ is a preferred labelling of $A$ .

$L$ is a $β$ -stable labelling of $B_{A}$ iff ${cB 2 S}_{A} (L)$ is a stable labelling of $A$ .

$L$ is a $β$ -semi-stable labelling of $B_{A}$ iff ${cB 2 S}_{A} (L)$ is a semi-stable labelling of $A$ .

Recalling the $SETAF$ $A$ and its corresponding $BAF$ $B_{A}$ of Example 27, we obtain the expected equivalence results related to their semantics (see Table 2). In the next section, we will find a class of $BAF$ s and $SETAF$ s such that the translation from a $BAF$ to a $SETAF$ (Definition 18) behaves as the inverse (up to isomorphism) of the translation from a $SETAF$ to a $BAF$ (Definition 26).

Table 2.
Semantics for $B$ and $A_{B}$ from Example 27, where $A_{1} = (A, {B})$ , $B_{1} = (B, {A})$ , $C_{1} = (C, {B})$ , $C_{2} = (C, {C}),$ $D_{1} = (D, \emptyset)$ , $E_{1} = (E, {E})$ , $F_{1} = (F, {B}),$ and $F_{2} = (F, {C, E})$ .

Complete labellings of $A$ $β$ -complete labellings of $B_{A}$

$\begin{aligned} L_{1} & = ({D}, \emptyset, {A, B, C, E, F}) \\ L_{2} & = ({B, D}, {A}, {C, E, F}) \\ L_{3} & = ({A, C, D, F}, {B}, {E}) \end{aligned}$ $\begin{aligned} L_{1}^{'} & = ({D_{1}}, \emptyset, {A_{1}, B_{1}, C_{1}, C_{2}, E_{1}, F_{1}, F_{2}}) \\ L_{2}^{'} & = ({B_{1}, D_{1}}, {A_{1}}, {C_{1}, C_{2}, E_{1}, F_{1}, F_{2}}) \\ L_{3}^{'} & = ({A_{1}, C_{1}, C_{2}, D_{1}, F_{1}, F_{2}}, {B_{1}}, {E_{1}}) \end{aligned}$

Grounded labellings of $A$ $β$ -grounded labellings of $B_{A}$

$\begin{aligned} L_{1} & = ({D}, \emptyset, {A, B, C, E, F}) \end{aligned}$ $\begin{aligned} L_{1}^{'} & = ({D_{1}}, \emptyset, {A_{1}, B_{1}, C_{1}, C_{2}, E_{1}, F_{1}, F_{2}}) \end{aligned}$

Preferred labellings of $A$ $β$ -preferred labellings of $B_{A}$

$\begin{aligned} L_{2} & = ({B, D}, {A}, {C, E, F}) \\ L_{3} & = ({A, C, D, F}, {B}, {E}) \end{aligned}$ $\begin{aligned} L_{2}^{'} & = ({B_{1}, D_{1}}, {A_{1}}, {C_{1}, C_{2}, E_{1}, F_{1}, F_{2}}) \\ L_{3}^{'} & = ({A_{1}, C_{1}, C_{2}, D_{1}, F_{1}, F_{2}}, {B_{1}}, {E_{1}}) \end{aligned}$

Stable labellings of $A$ $β$ -stable labellings of $B_{A}$

None None

Semi-stable labellings of $A$ $β$ -semi-stable labellings of $B_{A}$

$\begin{aligned} L_{3} & = ({A, C, D, F}, {B}, {E}) \end{aligned}$ $\begin{aligned} L_{3}^{'} & = ({A_{1}, C_{1}, C_{2}, D_{1}, F_{1}, F_{2}}, {B_{1}}, {E_{1}}) \end{aligned}$

5. On the relation between BAFs and SETAFs

In preceding sections, we have shown how to translate $BAF$ s to $SETAF$ s, and vice versa, while preserving their corresponding semantics. Now we check under which conditions these translations are the inverses (up to isomorphism) of each other, showing that there are also syntactic correspondences between $BAF$ s (under the $β$ -semantics) and $SETAF$ s. We say $BAF$ s $B = (A, A t t, S u p)$ and $B^{'} = (A^{'}, A t t^{'}, S u p^{'})$ are isomorphic if there exists a bijection $f : A \to A^{'}$ such that

${(f (A), f (B)) ∣ (A, B) \in A t t} = A t t^{'}$ , and

${(f (A), f (B)) ∣ (A, B) \in S u p} = S u p^{'}$ .

Similarly, $SETAF$ s $A = (A, A t t)$ and $A^{'} = (A^{'}, A t t^{'})$ are isomorphic if there exists a bijection $f : A \to A^{'}$ such that ${f (X) ∣ X \in X} \in A t t^{'} (f (A))$ iff $X \in A t t (A)$ .

From a $BAF$ $B$ , we obtain its corresponding $SETAF$ $A_{B}$ via Definition 18; from $A_{B}$ , we obtain its corresponding $BAF$ $B_{A_{B}}$ via Definition 26. By following the other direction, from a $SETAF$ $A$ , we obtain its corresponding $BAF$ $B_{A}$ , and from $B_{A}$ , its corresponding $SETAF$ $A_{B_{A}}$ . In this section, we investigate for which class of $BAF$ s we have $B$ is isomorphic to $B_{A_{B}}$ , and for which class of $SETAF$ s we have $A$ is isomorphic to $A_{B_{A}}$ .

5.1. On the isomorphism between $B$ and $B_{A_{B}}$

Initially, we show that, in general, $B$ is not isomorphic to $B_{A_{B}}$ .

Example 34
Let $B = (A, A t t, S u p)$ be the following $BAF$ :

The corresponding $SETAF$ $A_{B}$ has $2$ arguments $($ namely, ${A, B}$ and ${C})$ and no attacks. The corresponding $BAF$ $B_{A_{B}}$ has two arguments: $({A, B}, \emptyset)$ and $({C}, \emptyset))$ . There are no attacks or supports in $B_{A_{B}}$ . Clearly, $B$ and $B_{A_{B}}$ are not isomorphic.

This happened because, in $B$ , we have $A \in S u p (B)$ and $A t t (A) \subset A t t (B)$ , whereas the following property is satisfied for any corresponding $BAF$ :
Proposition 35
Let $A = (A, A t t)$ be a $SETAF$ and $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ . For any $X \in A_{A}$ , there is no $X^{'} \in {S u p}_{A} (X)$ such that $A t t_{A} (X^{'}) \subset A t t_{A} (X)$ .

Proposition 35 shows that any corresponding $BAF$ must necessarily have only minimal attacks. This motivates the definition of a $BAF$ of minimal attacks:
Definition 36 $BAF$ of minimal attacks

We say a $BAF$ $B = (A, A t t, S u p)$ is of minimal attacks if for any $X \in A$ , there is no $X^{'} \in S u p (X)$ such that $A t t (X^{'}) \subset A t t (X)$ .

However, this is not the only decisive factor for determining whether $B$ and $B_{A_{B}}$ are isomorphic.

Example 37
Let $B = (A, A t t, S u p)$ be the $BAF$ shown below:

The corresponding $SETAF$ $A_{B}$ has two arguments $($ namely, ${A}$ and ${A, B})$ and no attacks. The corresponding $BAF$ $B_{A_{B}}$ has two arguments: $({A}, \emptyset)$ and $({A, B}, \emptyset)$ . There are no attacks or supports in $B_{A_{B}}$ . Clearly, $B$ and $B_{A_{B}}$ have a different number of supports and are not isomorphic.

This happened because, in $B$ , we have $A \in S u p (B)$ but $S u p (A) \neq S u p (B)$ , whereas this property is satisfied for any corresponding $BAF$ . A $BAF$ with this property is called an $S - BAF$ :²²
Definition 38 $S - BAF$ ²²

Let $B = (A, A t t, S u p)$ be a $BAF$ . We say $B$ is a $BAF$ of support cliques ( $S - BAF$ ) iff $S u p$ is irreflexive, symmetric, and for any $(A, B), (B, C) \in S u p$ with $A \neq C$ , it holds $(A, C) \in S u p$ .

We now analyze another scenario.

Example 39
Let $B = (A, A t t, S u p)$ be the following $BAF$ :

The corresponding $SETAF$ $A_{B}$ has two arguments $($ namely, ${A, B}$ and ${C})$ and an attack from the singleton set ${{A, B}}$ (with only one argument ${A, B}$ ) toward argument ${C}$ . The corresponding $BAF$ $B_{A_{B}}$ has two arguments: namely, $({A, B}, \emptyset)$ and $({C}, {{A, B}})$ . In $B_{A_{B}}$ , there is only one attack from argument $({A, B}, \emptyset)$ toward argument $({C}, {{A, B}})$ , and there are no supports. We remark that, incidentally, $SETAF$ $A_{B}$ and $BAF$ $B_{A_{B}}$ amount to standard $AAF$ s. Clearly, $B$ and $B_{A_{B}}$ have a different number of arguments and are not isomorphic.

This happened because, in $B$ , we have $A \neq B$ and $S u p (A) = S u p (B)$ and $A t t (A) = A t t (B)$ . Such arguments are redundant and cannot occur in a corresponding $BAF$ . We say such $BAF$ s are redundancy-free:
Definition 40 $RFBAF$ ²²

Let $B = (A, A t t, S u p)$ be a $BAF$ . Arguments $A, B \in A$ are redundant iff $A \neq B$ and $S u p (A) = S u p (B)$ and $A t t (A) = A t t (B)$ . When a $BAF$ $B$ contains redundant arguments, we say $B$ is redundant. Otherwise, $B$ is a redundancy-free $BAF$ ( $RFBAF$ ).

For convenience, when a $BAF$ is both an $S - BAF$ and an $RFBAF$ , we say it is an $S - RFBAF$ :

Definition 41 $S - RFBAF$ ²²

A $BAF$ $B$ is a redundancy-free $S - BAF$ ( $S - RFBAF$ ) if $B$ is both an $S - BAF$ and an $RFBAF$ .

By Definition 26, any corresponding $BAF$ $B_{A}$ is an $S - RFBAF$ .

Proposition 42
Let $A = (A, A t t)$ be a $SETAF$ and $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ . It holds $B_{A}$ is an $S - RFBAF$ .

This is useful, because it means that the isomorphism between $B$ and $B_{A_{B}}$ can only hold if $B$ is an $S - RFBAF$ . Next, we discuss one more criterion necessary for the isomorphism between $B$ and $B_{A_{B}}$ .
Example 43
Let $B = (A, A t t, S u p)$ be the following $BAF$ :

The corresponding $SETAF$ $A_{B}$ is shown below:

It has five arguments: ${A, B}$ , ${C}$ , ${D}$ , ${E}$ , and ${F}$ . There is a joint attack from ${{C}, {D}}$ (two arguments) toward ${A, B}$ (1 argument). For the other attacks, the attacker-set is a singleton: ${{A, B}}$ is the only attacker-set of ${E}$ and ${F}$ .

The corresponding $BAF$ $B_{A_{B}}$ is shown below:

It has six arguments: $A^{'} = ({A, B}, {{C}})$ , $B^{'} = ({A, B}, {{D}})$ , $C^{'} = ({C}, \emptyset)$ , $D^{'} = ({D}, \emptyset)$ , $E^{'} = ({E}, {{A, B}})$ , and $F^{'} = ({F}, {{A, B}})$ . The set notation can be confusing, especially when computing minimal transversals, so we will show how to obtain these arguments in detail.
Let’s compute $Tr [A t t_{A_{B}} ({A, B})]$ . In the $SETAF$ $A_{B}$ , argument ${A, B}$ has one attacker-set (with two arguments): ${{C}, {D}}$ . Let’s denote the attacker-set ${{C}, {D}}$ as $X$ . We have $A t t_{A_{B}} ({A, B}) = {X}$ is a singleton. A minimal transversal $T$ of ${X}$ is a minimal set satisfying $T \cap X \neq \emptyset$ . Specifically, $T \cap {{C}, {D}} \neq \emptyset$ . We find two minimal $T$ satisfying this condition: $T_{0} = {{C}}$ and $T_{1} = {{D}}$ . Hence, in the corresponding $BAF$ $B_{A_{B}}$ we have arguments $({A, B}, T_{0})$ and $({A, B}, T_{1})$ .

Let’s compute $Tr [A t t_{A_{B}} ({E})]$ . In the $SETAF$ $A_{B}$ , argument ${E}$ has one attacker-set (with one argument): ${{A, B}}$ . Let’s denote the attacker-set ${{A, B}}$ as $X$ . We have $A t t_{A_{B}} ({E}) = {X}$ is a singleton. A minimal transversal $T$ of ${X}$ is a minimal set satisfying $T \cap X \neq \emptyset$ . Specifically, $T \cap {{A, B}} \neq \emptyset$ . We find only one minimal $T$ satisfying this condition: $T = {{A, B}}$ . Hence, in the corresponding $BAF$ $B_{A_{B}}$ we have an argument $({E}, {{A, B}})$ .

Similarly, we find an argument $({F}, {{A, B}})$ in the corresponding $BAF$ $B_{A_{B}}$ .

Let’s compute $Tr [A t t_{A_{B}} ({C})]$ . In the $SETAF$ $A_{B}$ , argument ${C}$ has no attacker-sets. We have $A t t_{A_{B}} ({C}) = \emptyset$ . Also, $Tr [\emptyset] = {\emptyset}$ , because any $T$ satisfies $\forall H \in \emptyset, T \cap H \neq \emptyset$ , and the empty set is the minimal $T$ satisfying this condition. Hence, we find the argument $({C}, \emptyset)$ in the corresponding $BAF$ $B_{A_{B}}$ .

Similarly, we find an argument $({D}, \emptyset)$ .

We found six arguments in the corresponding $BAF$ $B_{A_{B}}$ . However, both arguments $A^{'}$ and $B^{'}$ attack the same arguments: namely, $E^{'}$ and $F^{'}$ . Hence, there are four attacks in $B_{A_{B}}$ , whereas $B$ only has two attacks. Clearly, $B$ and $B_{A_{B}}$ are not isomorphic.

This happened because, in $B$ , we have $S u p (A) = S u p (B)$ , but $A$ and $B$ do not attack the same arguments. This motivates the definition of a $BAF$ of support-guided attacks:
Definition 44 $BAF$ of support-guided attacks

Let $B = (A, A t t, S u p)$ be a $BAF$ . If $A$ and $B$ attack the same arguments for any $A, B \in A$ with $S u p (A) = S u p (B)$ , then we say $B$ is of support-guided attacks.

By Definition 26, any corresponding $BAF$ is of support-guided attacks.

Proposition 45
Let $A = (A, A t t)$ be a $SETAF$ and $B_{A} = (A_{A}, A t t_{A}, S u p_{A})$ be its corresponding $BAF$ . It holds $B_{A}$ is of support-guided attacks.

This is useful, because it means that the isomorphism between $B$ and $B_{A_{B}}$ can only hold if $B$ is a $BAF$ of support-guided attacks. Now, we prove that the isomorphism holds precisely for the class of $S - RFBAF$ s of minimal and support-guided attacks. For convenience, we will give this class a name.
Definition 46 $S^{*} - RFBAF$

Let $B = (A, A t t, S u p)$ be a $BAF$ . We say $B$ is an $S^{*} - RFBAF$ if $B$ is an $S - RFBAF$ of minimal and support-guided attacks.

Theorem 47
Let $B$ be a $BAF$ , $A_{B}$ be its corresponding $SETAF$ , and $B_{A_{B}}$ be the corresponding $BAF$ of $A_{B}$ . It holds $B$ and $B_{A_{B}}$ are isomorphic iff $B$ is a $S^{} - RFBAF$ .

As expected from Theorem 47, $BAF$ s $B$ and $B_{A_{B}}$ from Figure 6 are isomorphic, as $B$ is an $S^{} - RFBAF$ .

Figure 6.
(a) $BAF$ $B$ , (b) its corresponding $SETAF$ $A_{B}$ , and (c) the corresponding $BAF$ $B_{A_{B}}$ of $A_{B}$ , where $A^{'} = {A, B}$ , $C^{'} = {C, F}$ , $D^{'} = {D}$ and $E^{'} = {E}$ ; $A_{1}^{'} = (A^{'}, {D^{'}})$ , $A_{2}^{'} = (A^{'}, {E^{'}})$ , $C_{1}^{'} = (C^{'}, {A^{'}})$ , $C_{2}^{'} = (C^{'}, {C^{'}, E^{'}})$ , $D_{1}^{'} = (D^{'}, \emptyset)$ and $E_{1}^{'} = (E^{'}, {E^{'}})$ . (a) $BAF$ $B$ , (b) corresponding $SETAF$ $A_{B}$ , and (c) corresponding $BAF$ $B_{A_{B}}$ . $BAF$ : bipolar argumentation framework; $SETAF$ : framework with sets of attacking arguments.
5.2. On the isomorphism between $A$ and $A_{B_{A}}$

As opposed to the isomorphism between $BAF$ s, we can ensure that each $SETAF$ $A$ is isomorphic to $A_{B_{A}}$ without any additional conditions on the $SETAF$ $A$ (as in Definition 10). We remark that we add the minimality restriction to the original definition of $SETAF$ from Nielsen and Parsons⁷ without changing the semantics considered in our work.^10,23 However, the isomorphism between $A$ and $A_{B_{A}}$ only holds for frameworks without non-minimal attacks.

Theorem 48
Let $A$ be a $SETAF$ , $B_{A}$ be its corresponding $BAF$ , and $A_{B_{A}}$ be the corresponding $SETAF$ of $B_{A}$ . It holds $A$ and $A_{B_{A}}$ are isomorphic.

As expected from Theorem 48, $SETAF$ s $A$ and $A_{B_{A}}$ of Figure 7 are isomorphic.

Figure 7.
(a) $SETAF$ $A$ , (b) its corresponding $BAF$ $B_{A}$ and (c) the corresponding $SETAF$ $A_{B_{A}}$ of $B_{A}$ , where $A_{1} = (A, {D})$ , $A_{2} = (A, {E})$ , $C_{1} = (C, {A})$ , $C_{2} = (C, {C, E})$ , $D_{1} = (D, \emptyset)$ and $E_{1} = (E, {E})$ ; $A^{'} = {A_{1}, A_{2}}$ , $C^{'} = {C_{1}, C_{2}}$ , $D^{'} = {D_{1}}$ and $E^{'} = {E_{1}}$ . $SETAF$ : framework with sets of attacking arguments; $BAF$ : bipolar argumentation framework.

We conclude that when restricted to $S^{} - RFBAF$ s, the translation from $BAF$ s to $SETAF$ s (Definition 18) is the inverse up to isomorphism of the translation from $SETAF$ s to $BAF$ s (Definition 26). This means that $SETAF$ s and $S^{} - RFBAF$ s have a deeper connection than $SETAF$ s and $BAF$ s, as in the former there is not only a semantic correspondence, but also a syntactic (or structural) one.
6. Related works

Finding intertranslatability results for argumentation formalisms is a very common research topic in formal argumentation. Many links leverage earlier work connecting argumentation and logic programming, which goes back to Dung’s seminal paper.¹ Some results include semantic correspondences relating normal logic programs ( $NLP$ s) to $AAF$ s,²⁵ $ABA$ frameworks,^28,29 $ADF$ s,^30,31 $BAF$ s,^18,22 claim-augmented argumentation frameworks ( $CAF$ s),¹³ claim-augmented BAFs,¹⁴ and $SETAF$ s.^13,16 König et al.¹³ show semantic and syntactic correspondences between $NLP$ s and $SETAF$ s for $SETAF$ extension-based complete, grounded, preferred, and stable semantics. Alcântara et al.¹⁶ highlight that these equivalence results also apply to $SETAF$ labelling-based semantics, including semi-stable labellings.

Similar to our work, several studies explore the links between different argumentation formalisms. We discuss and contrast them with our own approach, starting with works involving $SETAF$ s and $ADF$ s. With regard to $ADF$ s, they are capable of encoding many kinds of interactions between arguments, so it is natural to expect they might generalize $BAF$ s under the $β$ -semantics. Our work confirms that this is true due to previous results from the literature. Before discussing this, we review some concepts of $ADF$ s.

In an $ADF$ $D = (S, L, C)$ , $S$ is a set of statements (called arguments here), $L \subseteq S \times S$ is a set of links, and $C$ is a set of acceptance conditions (one for each argument). The acceptance condition of $s \in S$ is $C_{s} \in C$ , which maps each $R \subseteq p a r (s)$ to either $t$ (true) or $f$ (false), where $p a r (s) = {t \in S ∣ (t, s) \in L}$ is the set of arguments linked to $s$ . A link $(r, s) \in L$ is attacking when there is no $R \subseteq p a r (s)$ such that $C_{s} (R) = f$ and $C_{s} (R \cup {r}) = t$ ; is supporting when there is no $R \subseteq p a r (s)$ such that $C_{s} (R) = t$ and $C_{s} (R \cup {r}) = f$ ; and is redundant when it is both attacking and supporting. The class of $ADF$ s with only attacking links is called Attacking $ADF$ s ( $AD F^{+}$ s).³¹

Now we discuss why $ADF$ s can capture the notion of support represented by the $β$ -semantics. Polberg¹⁰ translates $ADF$ s to $SETAF$ s, and $SETAF$ s to $ADF$ s—more specifically, $AD F^{+}$ s—in a one-to-one correspondence between $ADF$ extensions and $SETAF$ extensions for the complete, grounded, preferred, and stable semantics. Unlike our work, syntactic correspondences and the semi-stable semantics are not considered in their work. These aspects were considered in Alcântara and Sá’s¹¹ work, in which they translate $AD F^{+} s$ to $SETAF$ s while preserving the complete, grounded, preferred, stable, and semi-stable semantics. They also demonstrate that this translation is the inverse of the one proposed by Polberg¹⁰ from $SETAF$ s to $ADF$ s when restricted to $AD F^{+} s$ without redundant links. This means $AD F^{+} s$ without redundant links and $SETAF$ s are linked both semantically and syntactically. Together with our work, this means that the interpretation of support from the $β$ -semantics is suitably represented by $AD F^{+} s$ without redundant links.

Other works involving $SETAF$ s are those of Dvořák et al.¹² and König et al.:¹³

Dvořák et al.¹² translate $SETAF$ s to $ADF$ s—more specifically, to $SETAF$ -like $ADF$ s ( $SETADF$ s). Then, $SETADF$ s can be translated to $SETAF$ s and also to the class of Support-Free $ADF$ s. They find a one-to-one correspondence between $SETAF$ labellings and $ADF$ labellings for the complete, grounded, preferred, and stable semantics.

König et al.¹³ translate $SETAF$ s to $ABA$ frameworks (and vice versa) in a one-to-one correspondence between $SETAF$ extensions and $ABA$ extensions for the complete, grounded, preferred, and stable semantics.

In both works, the semi-stable semantics is not considered. Similar to our work, König et al.¹³ find syntactic correspondences by demonstrating that any $SETAF$ $A$ coincides with the corresponding $SETAF$ $A_{D_{A}}$ of the corresponding $ABA$ $D_{A}$ of $SETAF$ $A$ . Such syntactic correspondence is even stronger than ours, as they obtain an equality between $A$ and $A_{D_{A}}$ , whereas in our work, we obtain an isomorphism between $A$ and $A_{B_{A}}$ .

As the $β$ -semantics generalizes $AAF$ semantics (complete, grounded, preferred, stable, and semi-stable), the following connections between $SETAF$ s and $AAF$ s are noteworthy:

Flouris and Bikakis¹⁷ translate $SETAF$ s to $AAF$ s, in a one-to-one correspondence between $SETAF$ extensions and $AAF$ extensions for the complete, grounded, preferred, and stable semantics. As each $AAF$ can be naturally seen as a $SETAF$ , the direction from $AAF$ s to $SETAF$ s is trivial.

Caminada et al.³² investigate the relation between extension-based and labelling-based node and arrow semantics for $AAF$ s and $SETAF$ s. In particular, they show that $SETAF$ arrow labellings correspond to node labellings of the corresponding $AAF$ obtained via an inside-out translation, in which arrows of the $SETAF$ become nodes of the corresponding $AAF$ . Notably, the semi-stable semantics is preserved in this translation.

Compared to Flouris and Bikakis’s¹⁷ approach, the advantage of our work is that the semantic correspondences also hold for the semi-stable semantics. Our translation from $BAF$ s to $SETAF$ s is not trivial (as that from $AAF$ s to $SETAF$ s), but this allows us to study when our proposed translations behave as each other’s inverse (up to isomorphism). Caminada et al.’s³² work helps visualize where the difficulty lies in preserving the semi-stable semantics: their solution is to consider arrow labellings of a $SETAF$ and node labellings of the corresponding $AAF$ . In contrast, when translating $SETAF$ s to $BAF$ s under the $β$ -semantics, we obtain correspondences for the semi-stable semantics even between node labellings of a $SETAF$ and of its corresponding $BAF$ , without the need to change the target of minimization (labels of nodes/arrows).

We now focus on works involving $CAF$ s—more specifically, well-formed $CAF$ s, in which arguments with the same claim attack the same arguments.

Dvořák et al.^8,9 translate well-formed $CAF$ s to $SETAF$ s (and vice versa) while preserving the preferred, stable, and claim-based semi-stable semantics, but not the inherited semi-stable semantics.

Rapberger³³ translates well-formed $CAF$ s to $SETAF$ s (and vice versa) in a one-to-one syntactic correspondence and preserving the complete, grounded, preferred, and stable semantics. To also capture equivalences to the semi-stable semantics, she introduces hybrid semantics (h-semantics) for $CAF$ s. In particular, the h-semi-stable semantics of $CAF$ s corresponds to the semi-stable semantics of $SETAF$ s.

In these translations from well-formed $CAF$ s to $SETAF$ s,^8,9,33 each argument in the corresponding $SETAF$ is a claim of an argument in the original $CAF$ . This aligns with our approach of using the $S$ -projection (Definition 17) to obtain the set of arguments $A_{S}$ of the corresponding $SETAF$ (Definition 18). The set of supporters $S u p (A)$ (Definition 2) of an argument $A$ in a $BAF$ takes the role of $c l (A)$ (the claim of $A$ ) in a $CAF$ . This strategy allows us to obtain syntactic correspondences involving $SETAF$ s and $BAF$ s under the $β$ -semantics (Section 5), which was similarly shown by Rapberger³³ for well-formed $CAF$ s. In her work, such as ours, the attack relation is obtained by computing minimal hitting sets.

In contrast, Dvořák et al.^8,9 determine attacks by constructing for each claim an attack-formula in disjunctive normal form. In that case, as illustrated by Figure 8, the corresponding $SETAF$ of a $CAF$ might have non-minimal attacks. Their translation from $CAF$ s is restricted to well-formed $CAF$ s: $c_{1}$ and $c_{2}$ must attack the same arguments. Well-formed $CAF$ s are similar to $BAF$ s with support-guided attacks (Definition 44), as arguments with the same claim (the same set of supporters, in the case of $BAF$ s) attack the same arguments. The distinction is that our translation applies to any $BAF$ , and not just to $BAF$ s of support-guided attacks. In Rapberger’s³³ translation, the resulting $SETAF$ does not have the non-minimal attack $({b, c}, c)$ , similarly to our $BAF$ -based approach depicted in Figure 9. However, our translation works even if we remove attacks such as $(C_{1}, C_{1})$ and $(C_{2}, C_{2})$ from the $BAF$ , which would entail the same corresponding $SETAF$ . Contrastively, if we remove the attacks $(c_{1}, c_{1})$ and $(c_{2}, c_{2})$ from the $CAF$ of Figure 8, we would obtain a $CAF$ that is not well-formed, disrespecting an essential premise of her translation. As the hybrid semantics³³ and claim-based semantics⁸ for $CAF$ s capture the semi-stable semantics for $SETAF$ s, our work reveals that the $β$ -semantics is a suitable interpretation of support for representing these (claim-based) notions in $BAF$ s.

Figure 8.

Example taken from the work of Dvořák et al.⁹ A well-formed $CAF$ is translated into a $SETAF$ . In their work, they follow the original definition of $SETAF$ s by Nielsen and Parsons,⁷ where attacks are not required to be minimal. (a) Well-formed $CAF$ . Arguments $c_{1}$ and $c_{2}$ have the same claim $c$ and (b) corresponding $SETAF$ (allowed to have non-minimal attacks). $CAF$ : claim-augmented argumentation framework; $SETAF$ : framework with sets of attacking arguments.

Figure 9.

The $BAF$ associated with the $CAF$ from Figure 8 is translated into a $SETAF$ . (a) $BAF$ representing the $CAF$ from Figure 8. As arguments do not have explicit claims in a $BAF$ , we connect arguments $C_{1}$ and $C_{2}$ by the support relation, and (b) Corresponding $SETAF$ , where $A^{'} = {A}$ , $B^{'} = {B}$ , and $C^{'} = {C_{1}, C_{2}}$ . $BAF$ : bipolar argumentation framework; $CAF$ : claim-augmented argumentation framework; $SETAF$ : framework with sets of attacking arguments.

Some of the translations discussed so far do not encompass every instance of a formalism in some of the translation directions: Alcântara and Sá’s¹¹ work is restricted to $AD F^{+} s$ ; Dvořák et al.’s¹² work, to $SETADF$ s; Rapberger,³³ Dvořák et al.’s,^8,9 and König et al.’s¹³ work, to well-formed $CAF$ s. We recall that although our isomorphism results presented in Section 5 hold only for a subclass of $BAF$ s (and any $SETAF$ s), the translations defined in Sections 3 and 4 are applicable to any $BAF$ and $SETAF$ , respectively.

By studying translations involving $BAF$ s, our work serves as an additional motivation for the recently proposed²¹ $β$ -semantics for $BAF$ s (among so many interpretations of support), as it is closely connected to well-known formalisms and their semantics. We now discuss why the equivalence results hold for the $β$ -semantics, but do not directly extend to other $BAF$ semantics.

Theorem 9 guarantees that all arguments in the same support cycle share the same label in a $β$ -complete labelling. Such arguments can be labelled as accepted, rejected, or undecided. In contrast, other works treat support cycles very differently. In some of them, $BAF$ s are assumed to be acyclic with respect to the support relation.^5,34 In others, even when a support cycle is allowed, every argument in a support cycle is always rejected, because such cycles are seen as a form of circular reasoning.^18,35–37

As an example of how we employ support cycles in $BAF$ s to encode the $SETAF$ semantics, consider the $SETAF$ $A$ depicted in Figure 10. It has an argument $A$ attacked by two disjoint attacker-sets $X = {X_{1}, X_{2}}$ and $Y = {Y_{1}, Y_{2}}$ . In a complete labelling $L$ of $A$ , it holds

$L (A) = i n$ iff (i) $L (X) = o u t$ for some $X \in X$ and (ii) $L (Y) = o u t$ for some $Y \in Y$ ;

$L (A) = o u t$ iff (iii) $L (X) = i n$ for every $X \in X$ or (iv) $L (Y) = i n$ for every $Y \in Y$ .

Figure 10.

How support cycles in the corresponding $BAF$ encode collective attacks. (a) $SETAF$ $A$ where argument $A$ is attacked by two disjoint attacker-sets and (b) corresponding $BAF$ $B_{A}$ uses support to connect arguments representing $A$ . $BAF$ : bipolar argumentation framework; $SETAF$ : framework with sets of attacking arguments.

In our strategy, these conditions are “rewritten” to a format more suitable for the $β$ -semantics. Argument $A$ is accepted iff both (i) and (ii) are true. It is rejected iff (iii) or (iv) is true. We can rewrite “(i) and (ii)” as “(1) or (2) or (3) or (4),” where (1)

$L (X_{1}) = o u t$ and $L (Y_{1}) = o u t$ ;

(2)

$L (X_{1}) = o u t$ and $L (Y_{2}) = o u t$ ;

(3)

$L (X_{2}) = o u t$ and $L (Y_{1}) = o u t$ ;

(4)

$L (X_{2}) = o u t$ and $L (Y_{2}) = o u t$ .

The conditions above can be represented as the sets $V_{1} = {X_{1}, Y_{1}}$ , $V_{2} = {X_{1}, Y_{2}}$ , $V_{3} = {X_{2}, Y_{1}}$ , and $V_{4} = {X_{2}, Y_{2}}$ , each representing a “vulnerability set” for $A$ . For instance, $V_{1}$ indicates that $A$ is accepted if $X_{1}$ and $Y_{1}$ are rejected. In the example, $A$ is accepted in $A$ when there exists $V \in {V_{1}, V_{2}, V_{3}, V_{4}}$ such that all arguments in $V$ are rejected. We find that the $β$ -semantics is suitable for this representation by using cycles of support. In our approach, the pairs $(A, V_{1})$ , $(A, V_{2})$ , $(A, V_{3})$ , and $(A, V_{4})$ are arguments in the corresponding $BAF$ , each representing one scenario in which $A$ is accepted: for instance, $(A, V_{1})$ indicates that $A$ is accepted if $X_{1}$ and $Y_{1}$ are rejected. The support relation connects pairs when they refer to the same argument (of the $SETAF$ ), which is the case for the pairs $(A, V_{i})$ for $i \in {1, 2, 3, 4}$ . Hence, these four pairs are in a support cycle. In the concrete example of Figure 10, arguments $X_{1}, X_{2}, X_{3}, X_{4}$ are always accepted because they are not attacked. However, $SETAF$ $A$ can be seen as a fragment of a larger and arbitrary $SETAF$ in which arguments $X_{1}, X_{2}, Y_{1}, Y_{2}$ could be attacked, illustrating the flexibility of our strategy. The decomposition of “(i) and (ii)” into a disjunction “(1) or (2) or (3) or (4)” is viable due to how support cycles are handled in the $β$ -semantics, and is not applicable to existing interpretations of support which reject any argument in any support cycle. Our strategy does not extend to these semantics, and it is an open question whether and how one can define syntactic correspondences between $BAF$ s and $SETAF$ s under these semantics.

7. Conclusion and future works

In this paper, we investigate the connections between $BAF$ s and $SETAF$ s, for, respectively, Alcântara and Cordeiro’s²¹ labelling-based $β$ -semantics and Flouris and Bikakis’s¹⁷ labelling-based acceptability semantics. We provide a mapping between $BAF$ s and $SETAF$ s (and vice versa), and also between their labellings, such that the $β$ -complete, $β$ -grounded, $β$ -preferred, $β$ -stable, and $β$ -semi-stable semantics for $BAF$ s correspond to, respectively, the complete, grounded, preferred, stable, and semi-stable semantics for $SETAF$ s.

Our translation from $SETAF$ s to $BAF$ s offers a key advantage over the translation from $SETAF$ s to $AAF$ s presented by Flouris and Bikakis.¹⁷ In their approach, the semi-stable semantics for $SETAF$ s do not correspond in general to the semi-stable semantics for $AAF$ s, whereas our approach captures the equivalence between semi-stable labellings of $SETAF$ s and $β$ -semi-stable labellings of $BAF$ s. Moreover, we guarantee additional results regarding the syntactic equivalence between $BAF$ s and $SETAF$ s. For any $SETAF$ $A$ , we can obtain its corresponding $BAF$ $B_{A}$ and the corresponding $SETAF$ $A_{B_{A}}$ such that $A$ and $A_{B_{A}}$ are isomorphic. This means that, disregarding the names of arguments, from a corresponding $BAF$ $B^{'}$ , we can recover the original $SETAF$ which originated $B^{'}$ . In the other direction, from $BAF$ $B$ we obtain the corresponding $SETAF$ $A_{B}$ and the corresponding $BAF$ $B_{A_{B}}$ . We find that the class of $BAF$ s for which the isomorphism between $B$ and $B_{A_{B}}$ holds consists of redundancy-free $BAF$ s of support cliques and of minimal and support-guided attacks ( $S^{*} - RFBAF$ s). This means that, disregarding the names of arguments, from a corresponding $SETAF$ $A^{'}$ originated from some $S^{*} - RFBAF$ , we can recover the original $BAF$ $B$ which originated $A^{'}$ . Hence, when compared to the relationship between $SETAF$ s and $AAF$ s, the relationship between $SETAF$ s and $BAF$ s is demonstrably more robust. It extends beyond semantics to encompass structural aspects.

Regarding the significance and potential impact of our results, we emphasize that by enlightening the connections between $BAF$ s and $SETAF$ s, many insights, semantics, and techniques naturally developed for the former may be applied to the latter, and vice versa. For instance, from an algorithm that transforms large $SETAF$ s into smaller $SETAF$ s we can derive an algorithm that transforms large $BAF$ s to smaller $BAF$ s via the translations presented in this paper. Moreover, the correspondence between support and joint attacks allows one to equivalently use the most suitable interaction given a particular application.

Natural ramifications of this work include establishing semantic and syntactic equivalence for argumentation frameworks with other kinds of interaction between arguments, including other notions of support. Equivalence between formalisms/semantics can also be studied from the perspective of realizability^38,39 that focuses on whether there are $BAF$ s/ $SETAF$ s given a set of labellings as their semantics. In addition, a possible line of research is to investigate how the specific notion of support from the $β$ -semantics interacts with joint attacks. For example, $BAF$ s could be augmented with joint supports, by allowing sets of two or more arguments to support other arguments; and $SETAF$ s could be augmented with ordinary or joint supports.

Supplemental Material

sj-pdf-1-arg-10.1177_19462174261442070 - Supplemental material for On the equivalence between bipolar argumentation frameworks and frameworks with sets of attacking arguments (SETAFs)

Supplemental material, sj-pdf-1-arg-10.1177_19462174261442070 for On the equivalence between bipolar argumentation frameworks and frameworks with sets of attacking arguments (SETAFs) by Renan Cordeiro and João Alcântara in Argument & Computation

Footnotes

Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP).

ORCID iDs

Renan Cordeiro

João Alcântara

Ethical considerations

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (grant number 130974/2024-2) and Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP) (grant number BMD-0008-00739.01.07/24).

Declaration of conflicting interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability

Not applicable.

Supplemental Material

Supplemental material for this article, including the proofs from Sections 3, 4, and 5, is available online.

References

Dung

. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif Intell 1995; 77: 321–357.

Caminada

Gabbay

. A logical account of formal argumentation. Studia Log 2009; 93: 109–145.

Bondarenko

Dung

Kowalski

, et al. An abstract, argumentation-theoretic approach to default reasoning. Artif Intell 1997; 93: 63–101.

Brewka

Woltran

. Abstract dialectical frameworks. In: Proceedings of the twelfth international conference on principles of knowledge representation and reasoning (KR'10). pp.102–111. Toronto, Ontario, Canada. May 9–13, 2010. Washington, DC: AAAI Press.

Amgoud

Cayrol

Lagasquie-Schiex

, et al. On bipolarity in argumentation frameworks. Int J Intell Syst 2008; 23: 1062–1093.

Dvořák

Woltran

. Complexity of abstract argumentation under a claim-centric view. Artif Intell 2020; 285: 103290.

Nielsen

Parsons

. A generalization of Dung’s abstract framework for argumentation: arguing with sets of attacking arguments. In: International workshop on argumentation in multi-agent systems, 2006, pp.54–73. ArgMAS 2006, Hakodate, Japan, May 8, 2006. Springer Berlin, Heidelberg.

Dvořák

Rapberger

Woltran

. Argumentation semantics under a claim-centric view: properties, expressiveness and relation to SETAFs. In: Proceedings of the international conference on principles of knowledge representation and reasoning, vol. 17, 2020, pp.341–350. Rhodes, Greece. September 12–18, 2020. Washington, DC: IJCAI Organization.

Dvořák

Rapberger

Woltran

. On the relation between claim-augmented argumentation frameworks and collective attacks. In: ECAI 2020, 2020, pp.721–728. Santiago de Compostela, Spain. August 29–September 8, 2020. Amsterdam, the Netherlands: IOS Press.

10.

Polberg

. Developing the abstract dialectical framework. PhD Thesis, Technische Universität Wien, 2017.

11.

Alcântara

Sá

. Equivalence results between SETAF and attacking abstract dialectical frameworks. In: Proceedings NMR, 2021, pp.139–148. Hanoi, Vietnam. November 3–5, 2021. Bonn, Germany: CEUR-WS.org, Gesellschaft für Informatik e.V. (GI).

12.

Dvořák

Keshavarzi Zafarghandi

Woltran

. Expressiveness of SETAFs and support-free ADFs under 3-valued semantics. J Appl Non-Class Log 2023; 33: 298–327.

13.

König

Rapberger

Ulbricht

. Just a matter of perspective. In: Computational models of argument (eds Toni F, Polberg S, Booth R, Caminada M and Kido H), 2022, pp.212–223. Amsterdam, the Netherlands: IOS Press.

14.

Rocha

VHN

Cozman

. Bipolar argumentation frameworks with explicit conclusions: connecting argumentation and logic programming. In: NMR, vol. 3197, 2022, pp.49–60. August 7–9, 2022, Bonn, Germany: Haifa, Israel. CEUR-WS.org, Gesellschaft für Informatik e.V. (GI).

15.

Cordeiro

Alcântara

. On the equivalence between logic programs and bipolar argumentation frameworks. In: Brazilian conference on intelligent systems, 2024, pp.238–253. November 17 to 21, 2024. Belém do Pará, Brazil. Cham, Switzerland: Springer Cham.

16.

Alcântara

Cordeiro

Sá

. On the equivalence between logic programming and SETAF. Theory Pract Logic Program 2024; 24: 1208–1236.

17.

Flouris

Bikakis

. A comprehensive study of argumentation frameworks with sets of attacking arguments. Int J Approx Reason 2019; 109: 55–86.

18.

Nouioua

Risch

. Argumentation frameworks with necessities. In: Scalable uncertainty management: 5th international conference, SUM 2011, Dayton, OH, USA, October 10–13, 2011. Proceedings 5, 2011, pp.163–176. Berlin, Heidelberg: Springer-Verlag.

19.

Cayrol

Lagasquie-Schiex

. Bipolarity in argumentation graphs: Towards a better understanding. Int J Approx Reason 2013; 54: 876–899.

20.

Gabbay

. Logical foundations for bipolar and tripolar argumentation networks: preliminary results. J Log Comput 2016; 26: 247–292.

21.

Alcântara

Cordeiro

. Bipolar argumentation frameworks with a dual relation between defeat and defence. J Log Comput 2024; 35: exae006.

22.

Alcântara

Cordeiro

. On the equivalence between logic programs and bipolar argumentation frameworks. J Artif Intell Res 2025; 84: 1–43.

23.

Dvořák

Rapberger

Woltran

. On the different types of collective attacks in abstract argumentation: equivalence results for setafs. J Log Comput 2020; 30: 1063–1107.

24.

Berge

. Hypergraphs: Combinatorics of finite sets. vol. 45. North-Holland, Amsterdam: Elsevier, 1984.

25.

Caminada

Sá

Alcântara

, et al. On the equivalence between logic programming semantics and argumentation semantics. Int J Approx Reason 2015; 58: 87–111.

26.

Rapberger

Ulbricht

. On dynamics in structured argumentation formalisms. J Artif Intell Res 2023; 77: 563–643.

27.

Prakken

. Relating abstract and structured accounts of argumentation dynamics: the case of expansions. In: Proceedings of the international conference on principles of knowledge representation and reasoning, vol. 19. 2023, pp.562–571. Rhodes, Greece. September 2-8, 2023. California, USA: IJCAI Organization.

28.

Caminada

Schulz

. On the equivalence between assumption-based argumentation and logic programming. J Artif Intell Res 2017; 60: 779–825.

29.

Sá

Alcântara

. Assumption-based argumentation is logic programming with projection. In: European conference on symbolic and quantitative approaches with uncertainty, 2021, pp.173–186. Prague, Czech Republic, September 21–24, 2021. Cham, Switzerland: Springer Cham.

30.

Strass

. Approximating operators and semantics for abstract dialectical frameworks. Artif Intell 2013; 205: 39–70.

31.

Alcântara

Sá

Acosta-Guadarrama

. On the equivalence between abstract dialectical frameworks and logic programs. Theory Pract Log Program 2019; 19: 941–956.

32.

Martin

König

Rapberger

, et al. Attack semantics and collective attacks revisited. Argum Comput 2024; 16(2): 151–211. https://doi.org/10.3233/AAC-230011

33.

Rapberger

. Unpacking the argument: a claim-centric view on abstract argumentation. PhD Thesis, Technische Universität Wien, 2023.

34.

Gottifredi

Cohen

Garcia

, et al. Characterizing acceptability semantics of argumentation frameworks with recursive attack and support relations. Artif Intell 2018; 262: 336–368.

35.

Nouioua

Boutouhami

. Argumentation frameworks with necessities and their relationship with logic programs. Argum Comput 2023; 14: 17–58.

36.

Lagasquie-Schiex

. Handling support cycles and collective interactions in the logical encoding of higher-order bipolar argumentation frameworks. J Log Comput 2023; 33: 289–318.

37.

Alfano

Greco

Parisi

, et al. Cyclic supports in recursive bipolar argumentation frameworks: semantics and lp mapping. Theory Pract Log Program 2024; 24: 921–941.

38.

Pührer

. Realizability of three-valued semantics for abstract dialectical frameworks. In: Proceedings of the 24th international conference on artificial intelligence, 2015, pp.3171–3177. Buenos Aires, Argentina, 25–31 July 2015. Palo Alto, California, USA: AAAI Press / International Joint Conferences on Artificial Intelligence.

39.

Strass

. Expressiveness of two-valued semantics for abstract dialectical frameworks. J Artif Intell Res 2015; 54: 193–231.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.33 MB

0.00 MB

$β$ -complete labellings of $B$	Complete labellings of $A_{B}$
$\begin{aligned} L_{1} & = ({D}, \emptyset, {A, B, C, E, F, G}) \\ L_{2} & = ({B, D}, {A}, {C, E, F, G}) \\ L_{3} & = ({A, C, D, F, G}, {B}, {E}) \end{aligned}$	$\begin{aligned} L_{1}^{'} & = ({{D}}, \emptyset, {{A}, {B}, {C, G}, {E}, {A, F}}) \\ L_{2}^{'} & = ({{B}, {D}}, {{A}}, {{C, G}, {E}, {A, F}}) \\ L_{3}^{'} & = ({{A}, {C, G}, {D}, {A, F}}, {{B}}, {{E}}) \end{aligned}$
$β$ -grounded labellings of $B$	Grounded labellings of $A_{B}$
$\begin{aligned} L_{1} & = ({D}, \emptyset, {A, B, C, E, F, G}) \end{aligned}$	$\begin{aligned} L_{1}^{'} & = ({{D}}, \emptyset, {{A}, {B}, {C, G}, {E}, {A, F}}) \end{aligned}$
$β$ -preferred labellings of $B$	Preferred labellings of $A_{B}$
$\begin{aligned} L_{2} & = ({B, D}, {A}, {C, E, F, G}) \\ L_{3} & = ({A, C, D, F, G}, {B}, {E}) \end{aligned}$	$\begin{aligned} L_{2}^{'} & = ({{B}, {D}}, {{A}}, {{C, G}, {E}, {A, F}}) \\ L_{3}^{'} & = ({{A}, {C, G}, {D}, {A, F}}, {{B}}, {{E}}) \end{aligned}$
$β$ -stable labellings of $B$	Stable labellings of $A_{B}$
None	None
$β$ -semi-stable labellings of $B$	Semi-stable labellings of $A_{B}$
$\begin{aligned} L_{3} & = ({A, C, D, F, G}, {B}, {E}) \end{aligned}$	$\begin{aligned} L_{3}^{'} & = ({{A}, {C, G}, {D}, {A, F}}, {{B}}, {{E}}) \end{aligned}$

Complete labellings of $A$	$β$ -complete labellings of $B_{A}$
$\begin{aligned} L_{1} & = ({D}, \emptyset, {A, B, C, E, F}) \\ L_{2} & = ({B, D}, {A}, {C, E, F}) \\ L_{3} & = ({A, C, D, F}, {B}, {E}) \end{aligned}$	$\begin{aligned} L_{1}^{'} & = ({D_{1}}, \emptyset, {A_{1}, B_{1}, C_{1}, C_{2}, E_{1}, F_{1}, F_{2}}) \\ L_{2}^{'} & = ({B_{1}, D_{1}}, {A_{1}}, {C_{1}, C_{2}, E_{1}, F_{1}, F_{2}}) \\ L_{3}^{'} & = ({A_{1}, C_{1}, C_{2}, D_{1}, F_{1}, F_{2}}, {B_{1}}, {E_{1}}) \end{aligned}$
Grounded labellings of $A$	$β$ -grounded labellings of $B_{A}$
$\begin{aligned} L_{1} & = ({D}, \emptyset, {A, B, C, E, F}) \end{aligned}$	$\begin{aligned} L_{1}^{'} & = ({D_{1}}, \emptyset, {A_{1}, B_{1}, C_{1}, C_{2}, E_{1}, F_{1}, F_{2}}) \end{aligned}$
Preferred labellings of $A$	$β$ -preferred labellings of $B_{A}$
$\begin{aligned} L_{2} & = ({B, D}, {A}, {C, E, F}) \\ L_{3} & = ({A, C, D, F}, {B}, {E}) \end{aligned}$	$\begin{aligned} L_{2}^{'} & = ({B_{1}, D_{1}}, {A_{1}}, {C_{1}, C_{2}, E_{1}, F_{1}, F_{2}}) \\ L_{3}^{'} & = ({A_{1}, C_{1}, C_{2}, D_{1}, F_{1}, F_{2}}, {B_{1}}, {E_{1}}) \end{aligned}$
Stable labellings of $A$	$β$ -stable labellings of $B_{A}$
None	None
Semi-stable labellings of $A$	$β$ -semi-stable labellings of $B_{A}$
$\begin{aligned} L_{3} & = ({A, C, D, F}, {B}, {E}) \end{aligned}$	$\begin{aligned} L_{3}^{'} & = ({A_{1}, C_{1}, C_{2}, D_{1}, F_{1}, F_{2}}, {B_{1}}, {E_{1}}) \end{aligned}$

On the equivalence between bipolar argumentation frameworks and frameworks with sets of attacking arguments (SETAFs)

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. Bipolar argumentation frameworks (BAFs)

Definition 1 BAF 5

Definition 2 Supporters 21

Definition 3 BAF labellings

Definition 4 L respects S u p 22

Theorem 9 Alcântara and Cordeiro 22

2.2. Frameworks with sets of attacking arguments ( SETAF s)

Definition 10 SETAF

Definition 11 SETAF labellings

Definition 12 Labelling-based SETAF semantics 17

Definition 14 Transversal

Definition 15 Tr operator

Theorem 16 Berge 24

3. From BAFs to SETAFs

3.1. SETAF construction

Definition 17 S -projection

Definition 20 B 2 cS B and cS 2 B B functions

4.1. BAF construction

Definition 26 Corresponding BAF

Definition 28 S 2 cB A and cB 2 S A functions

5.1. On the isomorphism between B and B A B

Definition 41 S - RFBAF 22

Supplemental Material

sj-pdf-1-arg-10.1177_19462174261442070 - Supplemental material for On the equivalence between bipolar argumentation frameworks and frameworks with sets of attacking arguments (SETAFs)

Footnotes

Acknowledgements

ORCID iDs

Ethical considerations

Consent to participate

Consent for publication

Funding

Declaration of conflicting interest

Data availability

Supplemental Material

References

Supplementary Material

Definition 1 $BAF$ ⁵

Definition 2 Supporters²¹

Definition 3 $BAF$ labellings

Definition 4 $L$ respects $S u p$ ²²

Theorem 9 Alcântara and Cordeiro²²

2.2. Frameworks with sets of attacking arguments ( $SETAF$ s)

Definition 10 $SETAF$

Definition 11 $SETAF$ labellings

Definition 12 Labelling-based $SETAF$ semantics¹⁷

Definition 15 $Tr$ operator

Theorem 16 Berge²⁴

Definition 17 $S$ -projection

Definition 20 ${B 2 cS}_{B}$ and ${cS 2 B}_{B}$ functions

Definition 26 Corresponding $BAF$

Definition 28 ${S 2 cB}_{A}$ and ${cB 2 S}_{A}$ functions

5.1. On the isomorphism between $B$ and $B_{A_{B}}$

Definition 41 $S - RFBAF$ ²²