On the Equivalence Between Logic Programs and Bipolar Argumentation Frameworks

Abstract

In this paper, we extend the connection between argumentation formalisms and logic programs, by showing the equivalence between bipolar argumentation $\beta$-semantics and various 3-valued logic programming semantics. In particular, we show that $\beta$-semi-stable semantics corresponds to L-stable semantics, which has been previously shown not to be expressible by traditional attack-only argumentation frameworks. Besides enhancing our understanding of how logic programs relate to argumentation with support, we provide direct translations between them and their semantics, allowing their interchangeable use.

1 Introduction

Formal argumentation and logic programming are closely related paradigms [7] for representing possibly contradictory, uncertain or incomplete information. In formal argumentation, an argument represents a claim whose acceptability is determined by some criteria (referred to as semantics) that depend on how arguments interact with each other. Dung introduced Abstract Argumentation Frameworks ($AAF$s) in his seminal work [10], defining an $AAF$ by a set of arguments and an attack relation between them. In $AAF$s, an argument is treated as an abstract entity without any internal structure, which means that semantics can only take into account the interactions between arguments. In Normal Logic Programs ($\textit{NLP}$s), a claim is represented by an atom whose evaluation consists in determining whether it can be inferred from logical rules.

Despite expressing information in distinctive perspectives, formal argumentation and logic programming are closely connected. Their connection is a very relevant line of research, as it allows one to view logic programs as argumentation frameworks and vice versa, allowing the most suitable formalism to be employed given a specific application. Besides, algorithms, techniques and semantics developed for a formalism can be adapted to the other. Not surprisingly, equivalences were extensively investigated. There is correspondence between complete and partial stable semantics [16], grounded and well-founded semantics [10], preferred and regular semantics [7], stable (argumentation) and stable (logic programming) semantics [10]. However, there exists a logic programming semantics (L-stable) that cannot be captured by any abstract argumentation semantics [7].

In this paper, we continue this line of research by considering instead Bipolar Argumentation Frameworks ($\textit{BAF}$s [5]), in which additionally there is a support relation between arguments: a positive interaction independent from the attack relation. Although indirectly and not for all semantics, bipolar argumentation can be linked to logic programming by the current literature. In [9], $\textit{BAF}$s are represented by Assumption-Based Argumentation ($\textit{ABA}$) for admissible, preferred and stable semantics and four interpretations of support. As $\textit{ABA}$ is linked to logic programming [8], the transitive nature of these correspondences induces an indirect connection between $\textit{BAF}$s and $\textit{NLP}$s. However, the semi-stable semantics of $\textit{ABA}$ does not correspond in general to the L-stable semantics of $\textit{NLP}$s [8]. A common aspect of these interpretations of support is their asymmetric treatment of attacks and supports. The former are favoured, as an argument is always rejected when it is both attacked and supported by accepted arguments. We instead resort to the interpretation of support from the $\beta$-semantics [1], in which an argument is always accepted if it is supported by an accepted argument.

Several works also relate more expressive argumentation formalisms to logic programming. For instance, abstract dialectical frameworks [2, 6, 14], $\textit{ABA}$ [8], claim-augmented argumentation [13, 15], frameworks with collective attacks [3, 15] and many others [4] are connected to logic programming. The work closest to ours is [13], where the authors relate Bipolar Claim-augmented Argumentation Frameworks ($\textit{BCAF}$s) to $\textit{NLP}$s. They could establish a one-to-one correspondence between $\textit{BCAF}$s and $\textit{NLP}$s in both directions, whilst preserving the main acceptability semantics, including the L-stable semantics. The main distinction between $\textit{BCAF}$s and $\textit{BAF}$s (and hence between their work and ours) is that $\textit{BCAF}$s have additional structure, as each argument is associated to an explicit claim, whereas arguments in a $\textit{BAF}$ are abstract: the $\beta$-semantics depend exclusively on the attack and support relation between arguments. In contrast, the semi-stable semantics of $\textit{BCAF}$s [13] take into account the conclusion of arguments, information not inferred solely from the attack and support relations.

This paper is divided as follows: in Sect. 2, we briefly recall the foundations of Bipolar Argumentation Frameworks ($\textit{BAF}$s) and their $\beta$-semantics, and of Normal Logic Programs ($\textit{NLP}$s) and their 3-valued semantics; in the sequence, we provide a translation from $\textit{NLP}$s to $\textit{BAF}$s and between their semantics; the inverse direction from $\textit{BAF}$s to $\textit{NLP}$s is discussed in Sect. 4; we collect our conclusions in Sect. 5.

2 Preliminaries

2.1 Bipolar Argumentation Frameworks $(\textit{BAF}s)$

In [5], Dung’s Abstract Argumentation Frameworks ($AAF$s) were extended to incorporate an explicit notion of support between arguments, independent from the attack relation. The resulting framework, called $\textit{BAF}$, is defined next:

Definition 1

(BAF [5]). A Bipolar Argumentation Framework ($\textit{BAF}$) is a tuple $(\mathcal {A}, Att ,$ $Sup )$, in which $\mathcal {A}$ is a set of arguments, $Att \subseteq \mathcal {A}\times \mathcal {A}$ is the attack relation and $Sup \subseteq \mathcal {A}\times \mathcal {A}$ is the support relation. For an argument $A \in \mathcal {A}$, we define $Att (A) = \left\{ B \in \mathcal {A}\mid (B, A) \in Att \right\}$ as the set of attackers of A and $Sup (A) = \left\{ B \in \mathcal {A}\mid (B, A) \in Sup \right\}$ as the set of direct supporters of A.

In $AAF$s, the interaction between arguments is specified only by the attack relation. In $\textit{BAF}$s, the novelty is that arguments can also support other arguments. This means that $\textit{BAF}$s $(\mathcal {A}, Att , Sup )$ with $Sup = \emptyset$ amount to (standard Dung) $AAF$s.

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

Definition 2

(Supporters [1]). Let $\mathcal {B} = (\mathcal {A}, Att , Sup )$ be a $\textit{BAF}$ and $A \in \mathcal {A}$ an argument. We define the supporters of A recursively as follows: A is a supporter of A in $\mathcal {B}$; if $A'$ is a supporter of A in $\mathcal {B}$ and $(B,A') \in Sup$, then B is a supporter of A in $\mathcal {B}$. We denote the set of all supporters of A in $\mathcal {B}$ by $\mathfrak {Sup}(A)$.

An acceptance criteria, referred to as semantics, for arguments in a $\textit{BAF}$ is how we determine the accepted or rejected arguments in a framework. Semantics can be equivalently characterised in terms of extensions (sets of accepted arguments) or labellings (mapping arguments to their acceptability) [1]. We will focus on labelling-based semantics as it allows a more direct comparison to logic programming semantics.

Definition 3

(Labelling). Let $\mathcal {B} = (\mathcal {A}, Att , Sup )$ be a $\textit{BAF}$. A labelling (of $\mathcal {B}$) is a total function $\mathcal {L} : \mathcal {A}\rightarrow \left\{ \texttt{in}, \texttt{out}, \texttt{undec}\right\}$.

When $\mathcal {L}$ is a labelling, we write $\texttt{in}(\mathcal {L})$ to denote the set $\left\{ A \mid \mathcal {L}(A) = \texttt{in}\right\}$, $\texttt{out}(\mathcal {L})$ for $\left\{ A \mid \mathcal {L}(A) = \texttt{out}\right\}$ and $\texttt{undec}(\mathcal {L})$ for $\left\{ A \mid \mathcal {L}(A) = \texttt{undec}\right\}$. As a labelling defines a partition among arguments, when convenient we write $\mathcal {L}$ as the triple $(\texttt{in}(\mathcal {L}), \texttt{out}(\mathcal {L}), \texttt{undec}(\mathcal {L}))$. Intuitively, the label $\texttt{in}$ indicates acceptance, the label $\texttt{out}$ indicates rejection and the label $\texttt{undec}$ indicates that the argument is undecided, i.e., neither accepted nor rejected. We can now describe the $\textit{BAF}$ semantics studied in this paper:

Definition 4

(Labelling-based Semantics for $\textit{BAF}$s [1]). Let $\mathcal {B} = (\mathcal {A}, Att ,$ $Sup )$ be a $\textit{BAF}$. A labelling $\mathcal {L}$ is a $\beta$-complete labelling of $\mathcal {B}$ if for any $A \in \mathcal {A}$,

• $\mathcal {L}(A) = \texttt{in}$ if and only if there exists $A' \in \mathfrak {Sup}(A)$ such that for every $B \in Att (A')$, it holds $\mathcal {L}(B) = \texttt{out}$.

• $\mathcal {L}(A) = \texttt{out}$ if and only if for every $A' \in \mathfrak {Sup}(A)$, there exists $B \in Att (A')$ such that $\mathcal {L}(B) = \texttt{in}$.

Additionally, we say a $\beta$-complete labelling $\mathcal {L}$ of $\mathcal {B}$ is $\beta$-grounded (resp. $\beta$-preferred) if $\texttt{in}(\mathcal {L})$ is minimal (resp. maximal) w. r. t. $\subseteq$ among the $\beta$-complete labellings of $\mathcal {B}$; $\beta$-stable if $\texttt{undec}(\mathcal {L}) = \emptyset$; $\beta$-semi-stable if $\texttt{undec}(\mathcal {L})$ is minimal (w. r. t. $\subseteq$) among the $\beta$-complete labellings of $\mathcal {B}$.

Fig. 1.

$\textit{BAF}$

Labelling-based semantics of $\textit{BAF}$s preserve remarking properties of the corresponding labelling-based semantics of $AAF$s [1]: for every $\textit{BAF}$ there is always a uniquely defined $\beta$-complete labelling; every $\beta$-stable labelling is a $\beta$-preferred labelling; and if there exists $\beta$-stable labellings, they coincide with $\beta$-semi-stable and $\beta$-preferred labellings.

Example 1

Consider the $\textit{BAF}$ depicted by the directed graph in Fig. 1, where nodes represent arguments, solid arrows represent attacks, and dashed arrows represent the support relation. We obtain labellings as follows: $\beta$-complete labellings are $\mathcal {L}_1 = (\emptyset , \emptyset , \mathcal {A})$, $\mathcal {L}_2 = (\left\{ A_1, A_4\right\} , \{A_2\}, \{A_3, A_5\})$ and $\mathcal {L}_3 = (\{A_2, A_3,$ $A_4\}, \left\{ A_1\right\} ,\{A_5\})$; $\beta$-grounded labelling is $\mathcal {L}_1$; $\beta$-preferred labellings are $\mathcal {L}_2$ and $\mathcal {L}_3$; there are no $\beta$-stable labellings; and $\beta$-semi-stable labelling is $\mathcal {L}_3$.

2.2 Logic Programs and Semantics

Now, we take a look at Propositional Normal Logic Programs. To delve into their definition and semantics, we will follow the presentation outlined in [7], which draws from the foundation laid out in [12].

Definition 5

A rule r is an expression

$$\begin{aligned} r: c \leftarrow a_1, \ldots , a_m, \mathtt {not\ }b_1, \ldots , \mathtt {not\ }b_n \end{aligned}$$

(1)

where ($m, n \ge 0$); c, each $a_i$ ($1 \le i \le m$) and each $b_j$ ($1 \le j \le n$) are atoms, and $\texttt{not}$ represents negation as failure. A literal is either an atom a (positive literal) or a negated atom $\mathtt {not\ }a$ (negative literal). Given a rule r as above, c is called the head of r, which we denote as $head (r)$, and $body (r) = \left\{ a_1, \ldots , a_m, \mathtt {not\ }b_1, \ldots , \mathtt {not\ }b_n\right\}$ is called the body of r. Further, $body (r)$ is divided in two sets $body ^+(r) = \left\{ a_1, \ldots , a_m\right\}$ and $body ^-(r) = \left\{ \mathtt {not\ }b_1, \ldots , \mathtt {not\ }b_n\right\}$. A fact is a rule where $m = n = 0$. A Normal Logic Program ($\textit{NLP}$) or simply a logic program P is a finite set of rules. If every $r \in P$ has $body ^-(r) = \emptyset$, P is a positive program. The Herbrand Base of P is the set $HB _{P}$ of all atoms appearing in P.

A wide range of $\textit{NLP}$ semantics are based on the 3-valued interpretations of programs [12]:

Definition 6

(3-Valued Herbrand Interpretation [12]). A 3-valued Herbrand Interpretation $\mathcal {I}$ (or simply interpretation) of P is a total function $\mathcal {I}: HB _P \rightarrow \left\{ \textbf{t}, \textbf{f}, \textbf{u}\right\}$.

Given an interpretation $\mathcal {I}$ of P, we write $\textbf{t}(\mathcal {I})$ to denote the set $\{ a \in HB _P \mid$ $\mathcal {I}(a) = \textbf{t}\}$, $\textbf{f}(\mathcal {I})$ for $\left\{ a \in HB _P \mid \mathcal {I}(a) = \textbf{f}\right\}$ and $\textbf{u}(\mathcal {I})$ for $\{ a \in HB _P \mid \mathcal {I}(a) = \textbf{u}\}$. As an interpretation defines a partition among atoms in $HB _P$, when convenient we write $\mathcal {I}$ as the triple $(\textbf{t}(\mathcal {I}), \textbf{f}(\mathcal {I}), \textbf{u}(\mathcal {I}))$. Intuitively, the value $\textbf{t}$ indicates the atom is true in $\mathcal {I}$, the value $\textbf{f}$ indicates the atom is false in $\mathcal {I}$, and the value $\textbf{u}$ indicates the atom is undefined in $\mathcal {I}$, i.e., neither true nor false.

Now we will consider the main semantics for $\textit{NLP}$s. Let $\mathcal {I}$ be a 3-valued Herbrand interpretation of P; the reduct of P with respect to $\mathcal {I}$ (written as $P/\mathcal {I}$) is the $\textit{NLP}$ constructed using the following steps:

• 1. Remove any $a \leftarrow a_1, \ldots , a_m,$ $\mathtt {not\ }b_1, \ldots , \mathtt {not\ }b_n \in P$ such that $\mathcal {I}(b_j) = \textbf{t}$ for some j ($1 \le j \le n$);

• 2. Afterwards, remove any occurrence of $\mathtt {not\ }b_j$ from P such that $\mathcal {I}(b_j) = \textbf{f}$.

• 3. Then, replace any occurrence of $\mathtt {not\ }b_j$ left by a special atom $\textbf{u}$ ($\textbf{u} \not \in HB _P$).

In the above procedure, $\textbf{u}$ is assumed to be an atom not in $HB _{P}$ which is undefined in all interpretations of P (a constant). Note that $P/\mathcal {I}$ is a positive program since all negative literals have been removed. As a consequence, $P/\mathcal {I}$ has a unique least 3-valued model [12] denoted as $\varPsi _P(\mathcal {I})$¹ with minimal $\textbf{t}(\varPsi _P(\mathcal {I}))$ and maximal $\textbf{f}(\varPsi _P(\mathcal {I}))$ (w.r.t. set inclusion) such that, for every $a \in HB _{P}$, (i) $a \in \textbf{t}(\varPsi _P(\mathcal {I}))$ if there is a rule $r^{\prime } \in P/\mathcal {I}$ with $head (r^{\prime }) = a$ and $body ^{+}(r^{\prime }) \subseteq \textbf{t}(\varPsi _P(\mathcal {I}))$; and (ii) $a \in \textbf{f}(\varPsi _P(\mathcal {I}))$ if for every rule $r^{\prime } \in P/\mathcal {I}$ with $head (r^{\prime }) = a$, it holds $body ^{+}(r^{\prime }) \cap \textbf{f}(\varPsi _P(\mathcal {I})) \ne \emptyset$.

We can now describe the $\textit{NLP}$ semantics studied in this paper:

Definition 7

Let P be an $\textit{NLP}$ and $\mathcal {I}$ be an interpretation: $\mathcal {I}$ is a partial stable model of P iff $\varPsi _P(\mathcal {I}) = \mathcal {I}$ [12]; $\mathcal {I}$ is a well-founded (resp. regular) model of P iff $\varPsi _P(\mathcal {I}) = \mathcal {I}$ and $\textbf{t}(\mathcal {I})$ is minimal (resp. maximal) w.r.t. set inclusion among all partial stable models of P [11, 12]; $\mathcal {I}$ is a (2-valued) stable model of P iff $\varPsi _P(\mathcal {I}) = \mathcal {I}$ and $\textbf{u}(\mathcal {I}) = \emptyset$ [12]; $\mathcal {I}$ is an L-stable model of P iff $\varPsi _P(\mathcal {I}) = \mathcal {I}$ and $\textbf{u}(\mathcal {I})$ is minimal w.r.t. set inclusion among all partial stable models of P [11].

Although some of these definitions are not standard in logic programming literature, their equivalence is proved in [7]. This format helps us to relate $\textit{NLP}$ and $\textit{BAF}$ semantics due to the structural similarities between Definition 7 and Definitions 3 and 4. We illustrate these semantics in the following example:

Example 2

Consider the following $\textit{NLP}$ P:

$$ \begin{array}{llcll} r_{0}: & d \leftarrow \texttt{not}\; d & & r_{1}: & c \leftarrow \texttt{not}\; c, \texttt{not}\; d \\ r_{2}: & a \leftarrow \texttt{not}\; b & & r_{3}: & b \leftarrow \texttt{not}\; a \\ r_{4}: & c \leftarrow \texttt{not}\; c, \texttt{not}\; a & & r_{5}: & e \leftarrow \texttt{not}\; e, \texttt{not}\; b \end{array} $$

This program has partial stable models $\mathcal M_1 = (\emptyset , \emptyset , HB _P)$, $\mathcal M_2 = (\left\{ a \right\} , \{ b \}, \{ c,$ $d, e \})$, $\mathcal M_3 = (\left\{ b \right\} , \left\{ a, e \right\} , \left\{ c, d \right\} )$; well-founded model is $\mathcal M_1$; regular models are $\mathcal M_2$ and $\mathcal M_3$; there are no stable models; and L-stable model is $\mathcal M_3$.

3 From $\textit{NLP}$s to $\textit{BAF}$s

In this section, we devise a translation from $\textit{NLP}$s to $\textit{BAF}$s. We wish to show that it preserves the main semantics considered in this paper, including the L-stable semantics. We start by creating arguments for derivations of rules of P, exactly as done in [7].

Definition 8

[7] Let P be an $\textit{NLP}$ and $\mathcal {A}_P$ be the set of arguments recursively defined below.

• If $c \leftarrow \mathtt {not\ }b_1, \ldots , \mathtt {not\ }b_m$ is a rule in P, it is also an argument (say A) with $\texttt{Conc}(A) = c$, $\texttt{Rules}(A) = \{ c \leftarrow \mathtt {not\ }b_1, \ldots , \mathtt {not\ }b_m \}$, $\texttt{Vul}(A) = \{ b_1, \ldots , b_m \}$, and $\texttt{Sub}(A) = \{ A \}$.

• If $c \leftarrow a_1,\ldots ,a_n, \mathtt {not\ }b_1,\ldots , \mathtt {not\ }b_m$ is a rule (say r) in P and for each $a_i$ ($1 \le i \le n$) there exists an argument $A_i$ with $\texttt{Conc}(A_i) = a_i$ and $r \notin \texttt{Rules}(A_i)$, then r derives the argument $c \leftarrow (A_1),\ldots ,(A_n),$ $\mathtt {not\ }b_1,\ldots ,$ $\mathtt {not\ }b_m$ (say A) with $\texttt{Conc}(A) = c$, $\texttt{Rules}(A) = \bigcup _{i=1}^n\texttt{Rules}(A_i) \cup \{ r \}$, $\texttt{Vul}(A) = \bigcup _{i=1}^n\texttt{Vul}(A_i) \cup \{b_1, \ldots , b_m\}$, and $\texttt{Sub}(A) = \{ A \} \cup \bigcup _{i=1}^n\texttt{Sub}(A_i)$.

For any argument A, we say $\texttt{Conc}(A)$ is its conclusion; $\texttt{Rules}(A)$, its set of rules; $\texttt{Vul}(A)$, its set of vulnerabilities; and $\texttt{Sub}(A)$, its set of subarguments. We can intuitively see an argument as a tree-like structure representing a possible derivation of an atom from the rules of a program.

Example 3

Consider the $\textit{NLP}$ P below with rules $\left\{ r_1, \ldots , r_8\right\}$:

$$ \begin{array}{lclclcl} r_1: a & & r_2: b \leftarrow a & & r_3: c \leftarrow \mathtt {not\ }c & & r_7: c \leftarrow f, \mathtt {not\ }g \\ r_4: d \leftarrow b, \mathtt {not\ }a & & r_5: d \leftarrow \mathtt {not\ }c & & r_6: e \leftarrow b, c, \mathtt {not\ }e & & r_8: f \leftarrow c, g \\ \end{array} $$

According to Definition 8, we can construct the following statements from P:

$$ \begin{array}{llcllcll} A_1: & a & & A_2: & b \leftarrow (A_1) & & A_3: & c \leftarrow \texttt{not}\; c\\ A_4: & d \leftarrow (A_2), \texttt{not}\; a & & A_5: & d \leftarrow \texttt{not}\; c & & A_6: & e \leftarrow (A_2), (A_3),\texttt{not}\; e \end{array} $$

Next, we give the conclusions and vulnerabilities of each argument:

The vulnerabilities of an argument A are associated with the negative literals found in the derivation of A. If $\texttt{not}\; b$ is one of them, then b is a vulnerability of A. This means that if b is derived, then $\texttt{Conc}(A)$ cannot be obtained via the derivation represented by A. However, it can still be obtained via other derivations/arguments. For instance, in the program P of Example 3, the derivation of a suffices to prevent the derivation of d via argument $A_4$ (for that reason, $a \in \texttt{Vul}(A_4)$), but we still can derive d via $A_5$. Notice also that there are no arguments with conclusions f and g.

Now we are entitled to determine both the attack and support relations:

Definition 9

Let P be an $\textit{NLP}$ and let A and B be arguments in the sense of Definition 8. We define the attack relation $Att _P$ and the support relation $Sup _P$ as follows: $(A, B) \in Att _P$ iff $\texttt{Conc}(A) \in \texttt{Vul}(B)$; $(A, B) \in Sup _P$ iff $A \ne B$ and $\texttt{Conc}(A) = \texttt{Conc}(B)$.

Our definition of the set $Att _P$ of attacks obtained from an $\textit{NLP}$ P is exactly as it has been characterised in [7]. The novelty here is the definition of $Sup _P$, which in our proposal is determined by arguments with the same conclusion (the condition $A \ne B$ is just to avoid incorporating the redundant case of an argument supporting itself). The failure in guaranteeing the correspondence between $\beta$-semi-stable and L-stable semantics, as discussed in [7], lies in the presence of arguments with distinct acceptabilities representing the derivations of the same atom. As Definition 9 ensures that such arguments support each other, they will have the same acceptability according to the $\beta$-semantics [1]. For the arguments of Example 3, $A_1$ attacks $A_4$; $A_3$ attacks itself, $A_5$ and $A_6$; $A_6$ attacks itself. We also have $A_4$ and $A_5$ supporting each other. Using the thus-defined concepts of arguments, attacks and supports, one can define the Bipolar Argumentation Framework corresponding to a particular $\textit{NLP}$:

Definition 10

Let P be an $\textit{NLP}$. We define its corresponding $\textit{BAF}$ as $\mathcal {B}_P = (\mathcal {A}_P, Att _P, Sup _P)$, where $\mathcal {A}_P$ is the set of arguments in the sense of Definition 8 and $Att _P$ is the attack relation in the sense of Definition 9.

Although the internal structure of arguments constructed from Definition 8 is used to construct $Att _P$ and $Sup _P$, any argument of $\mathcal B_P$ is abstract regarding $\textit{BAF}$ semantics, which depend exclusively on $Att _P$ and $Sup _P$. As an example, the corresponding $\textit{BAF}$ of the $\textit{NLP}$ of Example 3 is depicted in Fig. 2.

Fig. 2.

A $\textit{BAF}$ $\mathcal {B}_P = (\mathcal {A}_P, Att _P, Sup _P)$

Once the $\textit{BAF}$ has been constructed, we show the equivalence between the semantics for P and their counterpart for the corresponding $\textit{BAF}$ $\mathcal {B}_P$. We prove the equivalence results by identifying connections between fundamental notions used in the definition of the semantics for $\textit{NLP}$s and $\textit{BAF}$s. For this purpose, we introduce two functions: $\mathcal {L}2\mathcal {I}_P$ associates an interpretation to each labelling, while $\mathcal {I}2\mathcal {L}_P$ associates a labelling to each interpretation. We then investigate the conditions under which they are each other’s inverses and employ these results to prove the equivalence between the semantics. These functions essentially permit us to treat interpretations and labellings interchangeably.

Definition 11

($\mathcal {L}2\mathcal {I}_P$ and $\mathcal {I}2\mathcal {L}_P$ Functions). Let P be an $\textit{NLP}$, $\mathcal {B}_P = (\mathcal {A}_P,$ $Att _P, Sup _P)$ be its corresponding $\textit{BAF}$, $\mathcal {I} nt$ be the set of all the 3-valued interpretations of P and $\mathcal {L} ab$ be the set of all labellings of $\mathcal {B}_p$.

We introduce a function $\mathcal {L}2\mathcal {I}_P: \mathcal {L} ab \rightarrow \mathcal {I} nt$ where $\mathcal {L}2\mathcal {I}_P(\mathcal {L}) = (T, F, HB _P - (T \cup F))$, in which $T = \{c \in HB _P \mid \exists A \in \texttt{in}(\mathcal {L})\text { with }\texttt{Conc}(A) = c\}$; and $F = \{c \in HB _P \mid \mathcal {L}(A) = \texttt{out}\textit{ for every } A \in \mathcal {A}_P \textit{ such that } \texttt{Conc}(A) = c\}$.

We introduce a function $\mathcal {I}2\mathcal {L}_P: \mathcal {I} nt \rightarrow \mathcal {L} ab$ such that for any $\mathcal I \in \mathcal {I} nt$ and any $A \in \mathcal {A}_P$, $\mathcal {I}2\mathcal {L}_P(\mathcal I)(A) = \texttt{in}$ if $\mathcal I(\texttt{Conc}(A)) = \textbf{t}$; $\mathcal {I}2\mathcal {L}_P(\mathcal I)(A) = \texttt{out}$ if $\mathcal I(\texttt{Conc}(A)) = \textbf{f}$; $\mathcal {I}2\mathcal {L}_P(\mathcal I)(A) = \texttt{undec}$ if $\mathcal I(\texttt{Conc}(A)) = \textbf{u}$.

In general, $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L}))$ is not equal to $\mathcal {L}$. For instance, considering the labelling $\mathcal {L} = (\emptyset , \left\{ A_4\right\} , \left\{ A_1, A_2, A_3, A_5, A_6\right\} )$ and the $\textit{BAF}$ $\mathcal {B}_P$ of Fig. 2, we have that $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L})) = (\emptyset , \emptyset , \left\{ A_1, A_2, A_3, A_4, A_5, A_6\right\} )$. Such an inequality will always occur when two arguments with the same conclusion receive distinct labels. In our example, $A_4$ and $A_5$ have the same conclusion d; $\mathcal {L}(A_4) = \texttt{out}$, but $\mathcal {L}(A_5) = \texttt{undec}$. It holds $\mathcal {L}2\mathcal {I}_P(\mathcal {L})(d) = \textbf{u}$, and consequently, $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L}))(A_4) = \mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L}))(A_5) = \texttt{undec}$, i.e., $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L}))$ is not equal to $\mathcal {L}$. However, if $\mathcal {L}$ is a $\beta$-complete labelling of $\mathcal {B}_P$, arguments with the same conclusion will always have the same label:

Theorem 1

Let $\mathcal {B}_P = (\mathcal {A}_P, Att _P, Sup _P)$ be the corresponding $\textit{BAF}$ of P. If $\mathcal {L}$ is a $\beta$-complete labelling of $\mathcal {B}_P$ and $\texttt{Conc}(A) = \texttt{Conc}(B)$, then $\mathcal {L}(A) = \mathcal {L}(B)$.

Proof

Let $A, B \in \mathcal {A}_P$ with $\texttt{Conc}(A) = \texttt{Conc}(B)$ and $\mathcal {L}$ be a $\beta$-complete labelling of $\mathcal {B}_P$. The result holds trivially for $A = B$. Assume $A \ne B$. By Definition 9, $(A, B) \in Sup _P$ and $(B, A) \in Sup _P$. As $\mathcal {L}$ is $\beta$-complete, $\mathcal {L}(A) = \mathcal {L}(B)$.

Thus, if $\mathcal {L}$ is a $\beta$-complete labelling, we have that $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L})) = \mathcal {L}$:

Theorem 2

Let $\mathcal {B}_P = (\mathcal {A}_P, Att _P, Sup _P)$ be the corresponding $\textit{BAF}$ of P. For any $\beta$-complete labelling $\mathcal {L}$ of $\mathcal {B}_P$, it holds $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L})) = \mathcal {L}$.

Proof

Let $A \in \mathcal {A}_P$. There are 3 cases: $\mathcal {L}(A) = \texttt{in}$ $\Rightarrow$ $\mathcal {L}2\mathcal {I}_P(\mathcal {L})(\texttt{Conc}(A)) = \textbf{t}$ $\Rightarrow$ $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L}))(A) = \texttt{in}$; $\mathcal {L}(A) = \texttt{out}$ $\Rightarrow$ (Theorem 1) $\mathcal {L}(A') = \texttt{out}$ for every $A' \in \mathcal {A}_P$ such that $\texttt{Conc}(A') = \texttt{Conc}(A)$ $\Rightarrow$ $\mathcal {L}2\mathcal {I}_P(\mathcal L)(\texttt{Conc}(A)) = \textbf{f}$ $\Rightarrow$ $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L}))(A) = \texttt{out}$; $\mathcal {L}(A) = \texttt{undec}$ $\Rightarrow$ (Theorem 1) $\mathcal {L}(A') = \texttt{undec}$ for every $A' \in \mathcal {A}_P$ such that $\texttt{Conc}(A') = \texttt{Conc}(A)$ $\Rightarrow$ $\mathcal {L}2\mathcal {I}_P(\mathcal L)(\texttt{Conc}(A)) = \textbf{u}$ $\Rightarrow$ $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L}))(A) = \texttt{undec}$.

In general, $\mathcal {L}2\mathcal {I}_P(\mathcal {I}2\mathcal {L}_P(\mathcal {I}))$ is not equal to $\mathcal {I}$. Consider an atom $c \in HB _P$ that is not the conclusion of any argument in $\mathcal {A}_P$. Then, $\mathcal {L}2\mathcal {I}_P(\mathcal {I}2\mathcal {L}_P(\mathcal {I}))(c) = \textbf{f}$, but $\mathcal {I}$ may assign any truth value to c. However, atom c is evaluated as false in any partial stable model. We obtain the following result:

Theorem 3

Let $\mathcal {B}_P = (\mathcal {A}_P, Att _P)$ be the corresponding $\textit{BAF}$ of P and $\mathcal M$ be a partial stable model of P. It holds that $\mathcal {L}2\mathcal {I}_P(\mathcal {I}2\mathcal {L}_P(\mathcal M)) = \mathcal M$.

Proof

Let $\mathcal M$ be a partial stable model of P and $c \in HB _P$. It suffices to prove the following results:

• $\mathcal M(c) = \textbf{t}$ iff $\mathcal {L}2\mathcal {I}_P(\mathcal {I}2\mathcal {L}_P(\mathcal M))(c) = \textbf{t}$. ($\Rightarrow$) Assume $\mathcal M(c) = \textbf{t}$. As $\varPsi _P(\mathcal M) = \mathcal M$, there exists an argument A with $\texttt{Conc}(A) = c$ such that $\texttt{Vul}(A) \subseteq \textbf{f}(\mathcal M)$. This implies $\mathcal {I}2\mathcal {L}_P(\mathcal M)(A) = \texttt{in}$ and $\mathcal {L}2\mathcal {I}_P(\mathcal {I}2\mathcal {L}_P(\mathcal M))(c) = \textbf{t}$. ($\Leftarrow$) Assume $\mathcal {L}2\mathcal {I}_P(\mathcal {I}2\mathcal {L}_P(\mathcal M))(c) = \textbf{t}$. Then there exists $A \in \mathcal {A}_P$ such that $\texttt{Conc}(A) = c$ and $\mathcal {I}2\mathcal {L}_P(\mathcal M)(A) = \texttt{in}$. From Definition 11, $\mathcal M(c) = \textbf{t}$.

• $\mathcal M(c) = \textbf{f}$ iff $\mathcal {L}2\mathcal {I}_P(\mathcal {I}2\mathcal {L}_P(\mathcal M))(c) = \textbf{f}$, which can be similarly proved.

This means that when restricted to $\beta$-complete labellings and partial stable models, $\mathcal {L}2\mathcal {I}_P$ and $\mathcal {I}2\mathcal {L}_P$ are each other’s inverse. The function $\mathcal {L}2\mathcal {I}_P$ when applied to $\beta$-complete labellings can be characterised as follows:

$$ \mathcal {L}2\mathcal {I}_P(\mathcal {L})(c) = \left\{ \begin{array}{lcl} \textbf{t} & & \textit{if }\exists A \in \mathcal {A}_P \textit{ such that } \texttt{Conc}(A) = c \textit{ and } \mathcal {L}(A) = \texttt{in}\\ \textbf{u}& & \textit{if }\exists A \in \mathcal {A}_P \textit{ such that } \texttt{Conc}(A) = c \textit{ and } \mathcal {L}(A) = \texttt{undec}\\ \textbf{f}& & \textit{otherwise} \end{array} \right. $$

It is clear the relation between a $\beta$-complete labelling $\mathcal {L}$ and $\mathcal {L}2\mathcal {I}_P(\mathcal {L})$ for those atoms $c \in HB _P$ in which there is an argument A with $\texttt{Conc}(A) = c$. The distinction between them is for those atoms $c' \in HB _P$ without a corresponding argument $A'$ with $\texttt{Conc}(A') = c'$. For this case, $\mathcal {L}2\mathcal {I}_P(\mathcal {L})(c') = \textbf{f}$ and $\mathcal {L}$ is not defined for any argument $A'$ with $\texttt{Conc}(A') = c'$.

From Theorems 2 and 3, we can obtain the following result:

Theorem 4

Let $\mathcal {B}_P$ be the corresponding $\textit{BAF}$ of P. It holds $\mathcal {L}$ is a $\beta$-complete labelling of $\mathcal {B}_P$ iff $\mathcal {L}2\mathcal {I}_P(\mathcal {L})$ is a partial stable model of P; and $\mathcal M$ is a partial stable model of P iff $\mathcal {I}2\mathcal {L}_P(\mathcal M)$ is a $\beta$-complete labelling of $\mathcal {B}_P$.

Proof

• 1. If $\mathcal {L}$ is a $\beta$-complete labelling, then $\mathcal {L}2\mathcal {I}_P(\mathcal {L})$ is a partial stable model: Let $\mathcal M = \mathcal {L}2\mathcal {I}_P(\mathcal {L}) = (T, F, U)$. We show $\varPsi _P(\mathcal M) = (T', F', U') = (T, F, U)$:

  • $c \in T$ iff there exists $A \in \mathcal {A}_P$ such that $\texttt{Conc}(A) = c$ and $\mathcal {L}(A) = \texttt{in}$ iff there exists $A' \in \mathfrak {Sup}(A)$ such that $\texttt{Conc}(A') = c$ and for every $B \in Att (A')$, it holds $\mathcal {L}(B) = \texttt{out}$ iff there exists $A' \in \mathcal {A}_P$ such that $\texttt{Conc}(A') = c$ and for every $B \in Att (A')$, it holds $\mathcal {L}(B) = \texttt{out}$ iff there exists $A' \in \mathcal {A}_P$ such that $\texttt{Conc}(A') = c$ and $\texttt{Vul}(A') \subseteq F$ iff $c \in T'$.

  • $c \not \in F$ iff $c \not \in F'$ can be similarly proved.

• 2. If $\mathcal M$ is a partial stable model, then $\mathcal {I}2\mathcal {L}_P(\mathcal M)$ is a complete labelling: Let $\mathcal M = (T, F, U)$ be a partial stable model of P. Then $\varPsi _P(\mathcal M) = (T, F, U)$. Let $A \in \mathcal {A}_P$. We will prove $\mathcal {L} = \mathcal {I}2\mathcal {L}_P(\mathcal M)$ is a $\beta$-complete labelling of $\mathcal {B}_P$:

  • $\mathcal {L}(A) = \texttt{in}$ iff $\texttt{Conc}(A) \in T$ iff there exists $A' \in \mathcal {A}_P$ with $\texttt{Conc}(A') = \texttt{Conc}(A)$ and $\texttt{Vul}(A') \subseteq F$ iff there exists $A' \in \mathfrak {Sup}(A)$ such that for every $B \in Att (A')$, it holds $\mathcal {L}(B) = \texttt{out}$.

  • $\mathcal {L}(A) \ne \texttt{out}$ iff $\texttt{Conc}(A) \ne F$ iff there exists $A' \in \mathcal {A}_P$ with $\texttt{Conc}(A') = \texttt{Conc}(A)$ and $\texttt{Vul}(A') \cap T = \emptyset$ iff there exists $A' \in \mathfrak {Sup}(A)$ such that for every $B \in Att (A')$, it holds $\mathcal {L}(B) \ne \texttt{in}$.

• 3. $\mathcal {L}2\mathcal {I}_P(\mathcal {L})$ is a partial stable model of P $\Rightarrow$ according to item 2 above, $\mathcal {I}2\mathcal {L}_P(\mathcal {L}2\mathcal {I}_P(\mathcal {L}))$ is a $\beta$-complete labelling of $\mathcal {B}_P$ $\Rightarrow$ (via Theorem 2) $\mathcal {L}$ is a $\beta$-complete labelling of $\mathcal {B}_P$.

• 4. $\mathcal {I}2\mathcal {L}_P(\mathcal M)$ is a $\beta$-complete labelling of $\mathcal {B}_P$ $\Rightarrow$ according to item 1 above, $\mathcal {L}2\mathcal {I}_P(\mathcal {I}2\mathcal {L}_P(\mathcal M))$ is a partial stable model of P $\Rightarrow$ (via Theorem 3) $\mathcal M$ is a partial stable model of P.

Theorem 4 is one of the main results of this paper. It plays a central role in ensuring the equivalence between the semantics for $\textit{NLP}$s and their counterpart for $\textit{BAF}$s:

Theorem 5

Let P be an $\textit{NLP}$ and $\mathcal {B}_P = (\mathcal {A}_P, Att _P)$ be the corresponding $\textit{BAF}$. It holds $\mathcal {L}$ is a $\beta$-grounded (resp. $\beta$-preferred, $\beta$-stable, $\beta$-semi-stable) labelling of $\mathcal {B}_P$ iff $\mathcal {L}2\mathcal {I}_P(\mathcal {L})$ is a well-founded (resp. regular, stable, L-stable) model of P.

The following result is a direct consequence of Theorems 3 and 5:

Corollary 1

Let P be an $\textit{NLP}$ and $\mathcal {B}_P = (\mathcal {A}_P, Att _P)$ its corresponding $\textit{BAF}$. $\mathcal M$ is a well-founded (resp. regular, stable, L-stable) model of P iff $\mathcal {I}2\mathcal {L}_P(\mathcal M)$ is a $\beta$-grounded (resp. $\beta$-preferred, $\beta$-stable, $\beta$-semi-stable) labelling of $\mathcal {B}_P$.

Next, we consider the $\textit{NLP}$ exploited by Caminada et al. [7] as a counterexample to show that in general, L-stable models and semi-stable labellings do not coincide with each other in their translation from $\textit{NLP}$s to $AAF$s:

Example 4

Let P and $\mathcal {B}_P$ be the $\textit{NLP}$ and corresponding $\textit{BAF}$ in Fig. 3, where the arguments of $\mathcal B_P$ are

Fig. 3.

$\textit{NLP}$ P and its corresponding $\textit{BAF}$ $\mathcal {B}_P$

Note in $\mathcal {B}_P$ that there is both a mutual attack and a mutual support between arguments $C_1$ and $C_2$. As expected from Theorems 4 and 5, we obtain in Table 1 the equivalence between $\beta$-semantics of $\textit{BAF}$s and 3-valued semantics of $\textit{NLP}$s. We emphasise the coincidence between L-stable models and $\beta$-semi-stable labellings in Table 1 as it does not occur in [7]. In that reference, the corresponding $AAF$ possesses two semi-stable labellings in contrast with the unique L-stable model $\mathcal M_3$ of P.

Table 1. Semantics for P and $\mathcal {B}_P$

Note that if the support relation was ignored in the framework of Fig. 2 as was done in [7], the $\beta$-complete labellings of $\mathcal {B}_P$ would be $\mathcal {L}_1, \mathcal {L}_2', \mathcal {L}_3$, where $\mathcal {L}_2'\!=\! ( \left\{ A \right\} ,\!\left\{ B, C_1 \right\} ,\!\left\{ C_2, D, E \right\} )$, and $\mathcal {L}_2'$ and $\mathcal {L}_3$ would be the semi-stable $\beta$-complete labellings and the correspondence with L-stable models would not sustain anymore.

4 From $\textit{BAF}$s to $\textit{NLP}$s

Now we will provide a translation for the opposite direction, i.e., from $\textit{BAF}$s to $\textit{NLP}$s. As in the previous section, this translation guarantees the equivalence between the semantics for $\textit{NLP}$s and their counterpart for $\textit{BAF}$s.

Definition 12

(Corresponding $\textit{NLP}$). Let $\mathcal B = (\mathcal {A}, Att , Sup )$ be a $\textit{BAF}$. For any $A, B \in \mathcal {A}$, define the rule $r_{A,B}$ such that $head (r_{A,B}) = \mathfrak {Sup}(A)$, $body ^+(r_{A,B}) = \emptyset$ and $body ^-(r_{A,B}) = \{\mathtt {not\ }\mathfrak {Sup}(X) \mid X \in Att (B)\}$. The corresponding $\textit{NLP}$ of $\mathcal B$ is $P_\mathcal {B} = \left\{ r_{A,B} \mid A \in \mathcal {A}, B\in \mathfrak {Sup}(A) \right\}$.

Each argument is encoded in the corresponding $\textit{NLP}$ by its set of supporters, and the Herbrand Base of $P_\mathcal {B}$ is $HB _{P_\mathcal {B}} = \{\mathfrak {Sup}(A) \mid A \in \mathcal {A}\}$. Intuitively, with respect to any partial stable model of $P_\mathcal {B}$, we know $\mathfrak {Sup}(A)$ is true iff there exists $B \in \mathfrak {Sup}(A)$ such that $\mathfrak {Sup}(X)$ is false for every $X \in Att (B)$. This property follows directly from Definition 12, that was conceived to reflect the concept of acceptability in the $\beta$-complete semantics, where an argument $A \in \mathcal {A}$ is accepted iff there exists $B \in \mathfrak {Sup}(A)$ such that X is rejected for every $X \in Att (B)$.

Example 5

Let $\mathcal B = (\mathcal {A}, Att , Sup )$ be the $\textit{BAF}$ of Fig. 4a; its corresponding $\textit{NLP}$ $P_\mathcal {B}$ is shown in Fig. 4b.

Fig. 4.

(a) $\textit{BAF}$ $\mathcal B$ and (b) its corresponding $\textit{NLP}$ $P_\mathcal {B}$

Observe that distinct arguments $A, A'$ are associated to the same atom if $\mathfrak {Sup}(A) = \mathfrak {Sup}(A')$. For instance, the corresponding $\textit{NLP}$ of Example 4b encodes both arguments $B_1$ and $B_2$ as the atom $\mathfrak {Sup}(B_1) = \mathfrak {Sup}(B_2) = \left\{ B_1, B_2, D\right\}$. From the supporters of $B_1$, we obtain the rules $r_{B_1, B_1} = \left\{ B_1, B_2, D\right\} \leftarrow \mathtt {not\ }\left\{ A\right\} ,$ $\mathtt {not\ }\left\{ C\right\}$; $r_{B_1,B_2} = \left\{ B_1, B_2, D\right\} \leftarrow \mathtt {not\ }\left\{ D\right\}$; and $r_{B_1,D} = \left\{ B_1, B_2, D\right\} \leftarrow \mathtt {not\ }\left\{ B_1, B_2, D\right\} , \mathtt {not\ }\left\{ C\right\} , \mathtt {not\ }\left\{ E\right\}$. The intuition is that $B_1$ is accepted iff either (i) A and C are rejected or (ii) D is rejected or (iii) $B_1, B_2, C, D, E$ are rejected. Condition (iii) seems contradictory, but it is similar to what occurs when an argument attacks itself (e.g., argument E is accepted iff E is rejected).

As done in Sect. 3, we want to establish a bijection between (some class of) labellings and interpretations, such that $\beta$-complete labellings correspond to partial stable models. However, we can have more labellings than interpretations.

Example 6

Let $\mathcal B$ be the $\textit{BAF}$ $\mathcal B = (\left\{ A, B, C\right\} , \left\{ (C, B)\right\} , \left\{ (A, B), (B, A)\right\} )$. Its corresponding $\textit{NLP}$ $P_\mathcal {B}$ has 3 rules: $\left\{ A, B\right\} \leftarrow \left\{ C\right\}$ and $\left\{ A, B\right\}$ and $\left\{ C\right\}$. There are $3^{|\mathcal {A}|} = 27$ labellings of $\mathcal B$ and $3^{| HB _{P_\mathcal {B}}|} = 9$ interpretations of $P_\mathcal {B}$. Hence, there is no bijection between labellings of $\mathcal B$ and interpretations of $P_\mathcal {B}$. Notice that a labelling may assign different labels to A and B, whereas an interpretation assigns only one truth value to the atom $\{A, B\}$.

Although there is no bijection between labellings and interpretations for the example above, there is always a bijection if we restrict our translations to labellings that assign the same label to arguments with identical sets of supporters.

Definition 13

Let $\mathcal B = (\mathcal {A}, Att , Sup )$ be a $\textit{BAF}$. A labelling $\mathcal {L}$ of $\mathcal B$ respects $\mathfrak {Sup}$ iff for every $A, B \in \mathcal {A}$ with $\mathfrak {Sup}(A) = \mathfrak {Sup}(B)$ it holds $\mathcal {L}(A) = \mathcal {L}(B)$.

For the $\textit{BAF}$ $\mathcal B$ of Example 6, there are 9 labellings respecting $\mathfrak {Sup}$, since a labelling that respects $\mathfrak {Sup}$ assigns the same label to A and B. The representation of possibly many arguments by the same atom is motivated by the fact that every $\beta$-complete labelling satisfies the following property:

Proposition 1

If $\mathcal {L}$ is a $\beta$-complete labelling of $\mathcal B$, then $\mathcal {L}$ respects $\mathfrak {Sup}$.

By restricting labellings to those that respect $\mathfrak {Sup}$, the relationship between labellings and interpretations is straightforward:

Definition 14

($\mathcal {L}2\mathcal {I}_\mathcal {B}$ and $\mathcal {I}2\mathcal {L}_\mathcal {B}$ Functions). Let $\mathcal B = (\mathcal {A}, Att , Sup )$ be a $\textit{BAF}$ and $P_\mathcal {B}$ be its corresponding $\textit{NLP}$. Denote the set of all labellings of $\mathcal B$ respecting $\mathfrak {Sup}$ as $\mathcal {L} ab ^*$ and the set of all interpretations of $P_\mathcal {B}$ as $\mathcal {I} nt$. We introduce the functions (i) $\mathcal {L}2\mathcal {I}_\mathcal {B}: \mathcal {L} ab ^* \rightarrow \mathcal {I} nt$, in which for every $\mathcal {L} \in \mathcal {L} ab ^*$, $\mathcal {L}2\mathcal {I}_\mathcal {B}(\mathcal {L})(\mathfrak {Sup}(A)) = \textbf{t}$ (resp. $\textbf{f}, \textbf{u}$) if $\mathcal {L}(A) = \texttt{in}$ (resp. $\texttt{out}, \texttt{undec}$); and (ii) $\mathcal {I}2\mathcal {L}_\mathcal {B}: \mathcal {I} nt \rightarrow \mathcal {L} ab ^*$, in which for every $\mathcal M \in \mathcal {I} nt$, $\mathcal {I}2\mathcal {L}_\mathcal {B}(\mathcal M)(A) = \texttt{in}$ (resp. $\texttt{out}, \texttt{undec}$) if $\mathcal M(\mathfrak {Sup}(A)) = \textbf{t}$ (resp. $\textbf{f}, \textbf{u}$).

In contrast with $\mathcal {L}2\mathcal {I}_P$ and $\mathcal {I}2\mathcal {L}_P$, the functions $\mathcal {L}2\mathcal {I}_\mathcal {B}$ and $\mathcal {I}2\mathcal {L}_\mathcal {B}$ are each other’s inverse in the general case:

Theorem 6

Let $\mathcal B = (\mathcal {A}, Att , Sup )$ be a $\textit{BAF}$ and $P_\mathcal {B}$ be its corresponding $\textit{NLP}$. For any labelling $\mathcal {L}$ of $\mathcal B$ respecting $\mathfrak {Sup}$, it holds $\mathcal {I}2\mathcal {L}_\mathcal {B}(\mathcal {L}2\mathcal {I}_\mathcal {B}(\mathcal {L})) = \mathcal {L}$; and for any interpretation $\mathcal {I}$ of $P_\mathcal {B}$, it holds $\mathcal {L}2\mathcal {I}_\mathcal {B}(\mathcal {I}2\mathcal {L}_\mathcal {B}(\mathcal {I})) = \mathcal {I}$.

The next lemma is essential for the correspondence between $\beta$-complete labellings of $\mathcal B$ and partial stable models of $P_\mathcal {B}$. Its proof follows straightforwardly from Definitions 12 and 14.

Lemma 1

Let $\mathcal B = (\mathcal {A}, Att , Sup )$ be a $\textit{BAF}$ and $P_\mathcal {B}$ be its corresponding $\textit{NLP}$. Let $\mathcal {L}$ be a labelling of $\mathcal B$ respecting $\mathfrak {Sup}$ and $\mathcal M$ be an interpretation of $P_\mathcal {B}$. If $\mathcal {L} = \mathcal {I}2\mathcal {L}_\mathcal {B}(\mathcal M)$ or $\mathcal M = \mathcal {L}2\mathcal {I}_\mathcal {B}(\mathcal {L})$, then for any $A \in \mathcal {A}$, (i) $Att (B) \subseteq \texttt{out}(\mathcal {L})$ for some $B \in \mathfrak {Sup}(A)$ iff $\{\mathfrak {Sup}(X) \mid \mathtt {not\ }\mathfrak {Sup}(X) \in body ^-(r)\} \subseteq \textbf{f}(\mathcal M)$ for some $r \in P_\mathcal {B}$ with $head (r) = \mathfrak {Sup}(A)$; and (ii) $Att (B) \cap \texttt{in}(\mathcal {L}) \ne \emptyset$ for every $B \in \mathfrak {Sup}(A)$ iff $\{\mathfrak {Sup}(X) \mid \mathtt {not\ }\mathfrak {Sup}(X) \in body ^-(r)\} \cap \textbf{t}(\mathcal M) \ne \emptyset$ for every rule $r \in P_\mathcal {B}$ with $head (r) = \mathfrak {Sup}(A)$.

From Lemma 1, we obtain a similar result to Theorem 4:

Theorem 7

Let $\mathcal B = (\mathcal {A}, Att , Sup )$ be a $\textit{BAF}$ with corresponding $\textit{NLP}$ $P_\mathcal {B}$, $\mathcal {L}$ be a labelling of $\mathcal B$ respecting $\mathfrak {Sup}$ and $\mathcal M$ be an interpretation of $P_\mathcal {B}$. $\mathcal {L}$ is a $\beta$-complete labelling of $\mathcal B$ iff $\mathcal {L}2\mathcal {I}_\mathcal {B}(\mathcal {L})$ is a partial stable model of $P_\mathcal {B}$; and $\mathcal M$ is a partial stable model of $P_\mathcal {B}$ iff $\mathcal {I}2\mathcal {L}_\mathcal {B}(\mathcal M)$ is a $\beta$-complete labelling of $\mathcal B$.

Proof

Owing to space restrictions, only part of the proof is shown. Assume $\mathcal {L}$ is a $\beta$-complete labelling of $\mathcal B$. It holds $\mathfrak {Sup}(A) \in \textbf{t}(\mathcal {I}) \Longleftrightarrow \mathcal {L}(A) = \texttt{in}\Longleftrightarrow \exists B \in \mathfrak {Sup}(A): Att (B) \subseteq \texttt{out}(\mathcal {L}) \overset{\text {Lemma } 1}{\Longleftrightarrow } \exists r \in P_\mathcal {B}: head (r) = \mathfrak {Sup}(A) \wedge \{\mathfrak {Sup}(X) \mid \mathtt {not\ }\mathfrak {Sup}(X) \in body ^-(r)\} \subseteq \textbf{f}(\mathcal {I}) \Longleftrightarrow \exists r \in P_\mathcal {B}/\mathcal {I}: head (r) = \mathfrak {Sup}(A) \wedge body ^+(r) = \emptyset \Longleftrightarrow \exists r \in P_\mathcal {B}/\mathcal {I}: head (r) = \mathfrak {Sup}(A) \wedge body ^+(r) \subseteq \textbf{t}(\psi _{P_\mathcal {B}}(\mathcal {I})) \Longleftrightarrow \mathfrak {Sup}(A) \in \textbf{t}(\psi _{P_\mathcal {B}}(\mathcal {I}))$. Similarly, we can prove $\mathfrak {Sup}(A) \in \textbf{f}(\mathcal {I})$ iff $\mathfrak {Sup}(A) \in \textbf{f}(\varPsi _{P\mathcal B}(\mathcal {I}))$.

From Theorem 7, we can ensure the equivalence between the semantics for $\textit{BAF}$s and their counterpart for $\textit{NLP}$s:

Theorem 8

Let $\mathcal B$ be a $\textit{BAF}$ and $P_\mathcal {B}$ be its corresponding $\textit{NLP}$. For any labelling $\mathcal {L}$ of $\mathcal B$ respecting $\mathfrak {Sup}$, it holds $\mathcal {L}$ is a $\beta$-grounded (resp. $\beta$-preferred, $\beta$-stable, $\beta$-semi-stable) labelling of $\mathcal B$ iff $\mathcal {L}2\mathcal {I}_\mathcal {B}(\mathcal {L})$ is a well-founded (resp. regular, stable, L-stable) model of $P_\mathcal {B}$.

The following result is a direct consequence of Theorems 6 and 8:

Corollary 2

Let $\mathcal B$ be a $\textit{BAF}$ and $P_\mathcal {B}$ be its corresponding $\textit{NLP}$. $\mathcal M$ is a well-founded (resp. regular, stable, L-stable) model of $P_\mathcal {B}$ iff $\mathcal {I}2\mathcal {L}_\mathcal {B}(\mathcal M)$ is a $\beta$-grounded (resp. $\beta$-preferred, $\beta$-stable, $\beta$-semi-stable) labelling of $\mathcal B$.

Recalling the $\textit{BAF}$ $\mathcal B_P$ of Example 4, its corresponding $\textit{NLP}$ $P_\mathcal {B_P}$ is structurally identical (i.e., equal under renaming) to the $\textit{NLP}$ P of Example 4. We obtain the expected equivalence results related to their semantics (see Table 2).

Table 2. Semantics for $\mathcal B_P$ and $P_\mathcal {B_P}$

5 Conclusions

In this paper, we extend the work of Caminada et al. [7] by considering argumentation frameworks with support under the $\beta$-semantics [1]. We observe that $\beta$-complete, $\beta$-grounded, $\beta$-preferred, $\beta$-stable and $\beta$-semi-stable semantics of Bipolar Argumentation Frameworks ($\textit{BAF}$s) respectively coincide with the partial stable, well-founded, regular, stable and L-stable semantics of Normal Logic Programs ($\textit{NLP}$s). In particular, we note that the inclusion of support in argumentation allowed the connection between $\beta$-semi-stable and L-stable semantics, whereas semi-stable and L-stable semantics did not coincide for attack-only argumentation frameworks [7].

Each formalism encodes knowledge according to its own perspective: argumentation is concerned about arguments and their interaction, whereas $\textit{NLP}$s are interested in deriving atoms from rules. We provide explicit translations between these formalisms and their semantics, allowing them to be used interchangeably and enhancing our understanding about how they relate to each other. Future works include exploring the connection of $\textit{NLP}$s with other extensions of $AAF$s.