Complexity of Nonemptiness in Control Argumentation Frameworks

Abstract

Control argumentation frameworks (CAFs)  [4] extend the standard model of argumentation framework (AF) due to Dung [6] in a way that takes unquantified uncertainty regarding arguments and attacks into account. Complementing recent work by Skiba et al. [21] for incomplete argumentation frameworks, we study the (nonempty) existence problem for CAFs and fully analyze its computational complexity for the most common semantics.

1 Introduction

Argumentation theory is a key research field within artificial intelligence. In his seminal early work, Dung [6] introduced the central model of abstract argumentation framework, where— neglecting the actual contents of arguments—a discussion is represented as a directed graph (AF) whose vertices are the arguments and whose directed edges represent the attacks among arguments.

Later work established various extensions of this basic model so as to overcome its shortcomings, especially regarding its inability to handle structural uncertainty and incomplete information about the existence of arguments or attacks. One such model—which is central to this paper—proposes control argumentation frameworks (CAF) [4], which feature uncertain arguments and attacks and are intended specifically to model strategic scenarios from the point of view of an agent. The model divides the uncertain elements of an argumentation into a control part and an uncertain part, where the existence of elements in the control part is controlled by the agent, and the existence of elements in the uncertain part is not. This distinguishes CAFs from the related incomplete argumentation frameworks [2], which also feature uncertain arguments and attacks, but which do not specify such a sub-division.

Acceptability of arguments in AFs can be determined using one of several different semantics, where a set of arguments that satisfies a semantics is called an extension. To model various central reasoning tasks in argumentation frameworks formally, a number of decision problems have been introduced and studied. The verification problem asks whether a given set of arguments is an extension; the credulous acceptance and skeptical acceptance problems ask whether a given argument is contained in at least one extension or in all extensions, respectively; and the even more fundamental problem of existence asks whether an extension exists in the first place.

In control argumentation frameworks, the problem of controllability generalizes the credulous and skeptical acceptance problems and asks whether there exists a selection of arguments in the control part (i.e., a control configuration) such that for all selections of elements in the uncertain part (i.e., all completions), a given target set of arguments is a subset of an extension (or of all extensions). This problem was analyzed by Dimopoulos et al. [4] and Niskanen et al. [17].

Extending their work, we tackle natural generalizations of the (nonempty) existence problem for CAFs, which—even though being very fundamental—can be a very hard problem for some semantics. Recall that all other reasoning problems, such as the verification, acceptability, or controllability problems, are based on the existence of extensions. Therefore, it is decisive to know the computational complexity of the (nonempty) existence problem first of all. This enables us to conclude whether the hardness of a variant of verification, acceptability, or controllability comes from the problem itself or is simply an artifact of the underlying (nonempty) existence problem. Thus an analysis of (nonempty) existence problems for control argumentation frameworks is sorely missing and crucially needed.

The existence problem can be trivially solved whenever a semantics guarantees existence of at least one extension, and this is the case for all semantics that allow the empty set to be an extension. This gives rise to consider the nonempty existence (or, nonemptiness, for short) problem, which only accepts nonempty extensions. The existence and nonemptiness problems coincide for semantics that do not allow empty extensions, so we henceforth will focus on the nonemptiness problem only. For CAFs, we define (nonempty) existence problems analogously to the controllability problem studied by Dimopoulos et al. [4] and Niskanen et al. [17]: Does there exist a control configuration such that for all completions, there exists a nonempty set of arguments satisfying a given semantics?

We provide a full analysis of the computational complexity of these generalized nonempty existence problems with respect to the most common Dung semantics. The relationship between the controllability and the nonempty existence problem is the following: A “yes”-instance for controllability is also a “yes”-instance of the nonempty existence problem, and a “no”-instance for nonempty existence is also a “no”-instance of the controllability problem. Table 1 on page 11 in the conclusions gives an overview of our complexity results for this problem regarding the considered semantics, and compares them with those of Skiba et al. [21] for the analogous problems in the related incomplete argumentation frameworks and of Chvátal [3], Modgil and Caminada [16], and Dimopoulos and Torres [5] in standard argumentation frameworks.

This paper is structured as follows. In Sect. 2, we describe the formal models of abstract AFs and its extension to CAFs, and we present a formal definition of the nonemptiness problem in this setting. This is followed by a full analysis of the computational complexity of nonemptiness for CAFs in Sect. 3. In Sect. 4, we summarize our results and point out some related work in this field.

2 Preliminaries

First, we briefly present the model of (abstract) argumentation framework that is due to Dung [6]. We then describe the extension of this model to control argumentation frameworks, as introduced by Dimopoulos et al. [4].

Fig. 1.

Argumentation framework for Example 1

2.1 Abstract Argumentation Frameworks

For a finite set $\mathcal {A}$ of arguments and a set $\mathcal {R}$ of attacks between them, we call the pair $\langle \mathcal {A},\mathcal {R}\rangle$ with $\mathcal {R}\subseteq \mathcal {A}\times \mathcal {A}$ an argumentation framework (AF). Whenever $(a,b) \in \mathcal {R}$, argument a attacks argument b. An argument a is said to be acceptable with respect to a set $S \subseteq \mathcal {A}$ if for all arguments $b \in \mathcal {A}$ attacking a, there is an argument $c \in S$ attacking b; in this case, a is defended by c against its attacker b.

An argumentation framework $\langle \mathcal {A},\mathcal {R}\rangle$ can be depicted by a directed graph with vertex set $\mathcal {A}$ and edge set $\mathcal {R}$.

Example 1

Figure 1 shows the graph representing $AF =\langle \mathcal {A},\mathcal {R}\rangle$ with $\mathcal {A}= \{a,b,c,d,e\}$ and $\mathcal {R}= \{(a,b), (b,e), (c,a), (c,c), (d,a), (d,c), (d,e), (e,d)\}$.

For argumentation frameworks, Dung [6] also introduced the notion of semantics. A set of arguments satisfying the properties of a given semantics is called an extension. We will use the following semantics.

Definition 1

Let $AF =\langle \mathcal {A},\mathcal {R}\rangle$ be an argumentation framework. A set $S \subseteq \mathcal {A}$ is said to be if no arguments in S attack each other. Further, a conflict-free set S is if all arguments in S are acceptable with respect to S; if S is closed under defense (i.e., S contains all arguments that are acceptable with respect to S); if S is the least complete set with respect to set inclusion; if it is a set-maximal admissible set; and if every $b \in \mathcal {A}\setminus S$ is attacked by at least one argument $a \in S$.

Every stable extension is preferred, every preferred extension is complete, and every complete extension is admissible. There always is a conflict-free, admissible, preferred, complete, and grounded extension in every argumentation framework (it is possible that this is just the empty set, though). For the stable semantics, by contrast, existence cannot be taken for granted.

Example 2

Continuing Example 1, the conflict-free sets in our argumentation framework are $\{a,e\}$, $\{b,d\}$, and all their subsets. The only two nonempty admissible sets are $\{d\}$ (since d defends itself against e’s attack) and $\{b,d\}$ (as a’s attack on b is defended by d, and e’s attack on d is defended by b and d). However, no other nonempty, conflict-free set can defend all its arguments: No member of $\{a\}$ or $\{a,e\}$ defends a against c’s attack; $\{e\}$ lacks a defending attack against b’s attack; and $\{b\}$ cannot be defended against a’s attack. Among $\{d\}$ and $\{b,d\}$, only the latter is complete, since d defends b against a’s attack. Further, the empty set is complete, since there are no unattacked arguments—i.e., there are no arguments that are defended by the empty set. Since $\emptyset$ is complete, it is also the (unique) grounded extension. On the other hand, the complete extension $\{b,d\}$ is the only preferred (i.e., maximal admissible) extension. Since d attacks each of a, c, and e, the preferred extension $\{b,d\}$ is also a stable extension (and since there are no other preferred extensions, it is the only stable extension).

We are interested in the existence problem restricted to nonempty extensions (the nonemptiness problem, for short) for the above semantics in control argumentation frameworks. In standard argumentation frameworks, the existence problem was first studied by Dimopoulos and Torres [5] and later on by Dunne and Wooldridge [8]. For each semantics $s \in \{\textsc {CF}, \textsc {AD}, \textsc {CP}, \textsc {GR}, \textsc {PR}, \textsc {ST}\}$, define the following problem:

2.2 Control Argumentation Frameworks

Next, we formally define control argumentation frameworks due to Dimopoulos et al. [4], while adapting some notation from Niskanen et al. [17].

Definition 2

A control argumentation framework (CAF) is a triple $(\mathcal {F},\mathcal {C},\mathcal {U})$ consisting of a fixed part $\mathcal {F}$ of elements known to exist, a control part $\mathcal {C}$ of elements that may or may not exist (and whose existence is controlled by the agent), and an uncertain part $\mathcal {U}$ of elements that may or may not exist (and whose existence is not controlled by the agent):

$$\begin{aligned} \mathcal {F}&=\langle \mathcal {A}_F,\mathcal {R}_F\rangle \text { with } \mathcal {R}_F \subseteq (\mathcal {A}_F \cup \mathcal {A}_U) \times (\mathcal {A}_F \cup \mathcal {A}_U),\\ \mathcal {C}&= \langle \mathcal {A}_C,\mathcal {R}_C\rangle \text { with } \mathcal {R}_C \subseteq (\mathcal {A}_C \times (\mathcal {A}_F \cup \mathcal {A}_C \cup \mathcal {A}_U)) \cup ((\mathcal {A}_F \cup \mathcal {A}_C \cup \mathcal {A}_U) \times \mathcal {A}_C),\\ \mathcal {U}&= \langle \mathcal {A}_U, \mathcal {R}_U \cup \mathcal {R}_U^{\leftrightarrow }\rangle \text { with } \mathcal {R}_U, \mathcal {R}_U^{\leftrightarrow } \subseteq (\mathcal {A}_F \cup \mathcal {A}_U) \times (\mathcal {A}_F \cup \mathcal {A}_U), \end{aligned}$$

where $\mathcal {A}_F$, $\mathcal {A}_C$, and $\mathcal {A}_U$ are pairwise disjoint sets of arguments, and $\mathcal {R}_F$, $\mathcal {R}_C$, $\mathcal {R}_U$, and $\mathcal {R}_U^{\leftrightarrow }$ are pairwise disjoint sets of attacks. While the existence of attacks in $\mathcal {R}_U$ is not known at all, attacks in $\mathcal {R}_U^{\leftrightarrow }$ are known to exist, but their direction is unknown.

For elements in the control part, we always look for only (at least) one configuration to achieve the given goal, while for elements in the uncertain part, we always require all their completions to achieve that goal. Also, the control part always “moves first,” i.e., a configuration of the control part must be chosen before knowing how the uncertain part will be completed.

Example 3

Consider a control argumentation framework with the argument sets $A_F = \{a,c,d\}, A_C = \{b\}, A_U = \{e\}$ and the attack relations $R_F = \{(c,a), (d,a),$ $(d,c)\}$, $R_C = \{(a,b), (b,e)\}$, $R_U = \{(c,c)\}$, and $R_U^\leftrightarrow = \{(d,e),(e,d)\}$. Its graph representation is given in Fig. 2, where we follow the style of display introduced in the original work on CAFs by Dimopoulos et al. [4].

Fig. 2.

Control argumentation framework for Example 3, where control arguments are drawn as solid rectangles, uncertain arguments as dashed rectangles, attacks from $\mathcal {R}_C$ are drawn as boldfaced arrows, uncertain attacks from $\mathcal {R}_U$ are drawn as dotted arrows, and uncertain attacks from $\mathcal {R}_U^\leftrightarrow$ are drawn as doubleheaded dashed arrows.

Configurations of the control part are formally captured by the notion of control configurations, which are used to represent a move of the controlling agent. Here, the arguments from $\mathcal {A}_C$ that the agent wants to use are included in the control configuration, and the arguments that the agent does not want to include are discarded.

Definition 3

A subset $\mathcal {A}_\textit{conf}\subseteq \mathcal {A}_C$ is called a control configuration. $\mathcal {A}_\textit{conf}$ induces a CAF $(\mathcal {F},\langle \mathcal {A}_\textit{conf},\mathcal {R}_\textit{conf}\rangle ,\mathcal {U})$ with $\mathcal {R}_\textit{conf}= \mathcal {R}_C|_{(\mathcal {A}_F\cup \mathcal {A}_U\cup \mathcal {A}_\textit{conf})}$, where the corresponding control move has been made (we define $\mathcal {R}|_{\mathcal {A}} = \mathcal {R}\cap (\mathcal {A}\times \mathcal {A})$). When the CAF is known from context, we may sometimes use the control configuration $\mathcal {A}_\textit{conf}$ to represent its induced CAF.

Now, assuming that a control configuration has been chosen, the remaining uncertainty of the uncertain part is resolved via a completion.

Definition 4

A completion of a CAF $(\mathcal {F},\langle \mathcal {A}_\textit{conf},\mathcal {R}_\textit{conf}\rangle ,\mathcal {U})$ is an argumentation framework $AF^*=(\mathcal {A}^*,\mathcal {R}^*)$ with

$$\begin{aligned} \mathcal {A}_F \cup \mathcal {A}_\textit{conf}&\subseteq \mathcal {A}^*\subseteq \mathcal {A}_F \cup \mathcal {A}_\textit{conf}\cup \mathcal {A}_U, \\ (\mathcal {R}_F \cup \mathcal {R}_\textit{conf})|_{\mathcal {A}^*}&\subseteq \mathcal {R}^*\subseteq (\mathcal {R}_F \cup \mathcal {R}_\textit{conf}\cup \mathcal {R}_U \cup \mathcal {R}_U^\leftrightarrow )|_{\mathcal {A}^*}, \end{aligned}$$

and it is required that $(a,b) \in \mathcal {R}^*$ or $(b,a) \in \mathcal {R}^*$ (or both) for all $(a,b) \in \mathcal {R}_U^\leftrightarrow$.

Example 4

We continue Example 3. Using the control configuration $\{b\}$ and the completion that includes the uncertain argument e and all uncertain attacks (c, c), (d, e), and (e, d) produces the AF of Example 1, with the properties mentioned in Example 2.

2.3 Some Background on Complexity Theory

We will study the computational complexity of the nonemptiness problem for CAFs to be defined in the next section. We assume that the reader is familiar with the needed background on complexity theory, such as the complexity classes of the polynomial hierarchy [15, 22], the zeroth level of which is $\mathrm {P}$, the first level of which consists of $\mathrm {NP}$ and $\mathrm {coNP}$, the second level of which consists of $\Sigma _2^p= \mathrm {NP}^{\mathrm {NP}}$ and $\Pi _2^p= \mathrm {coNP}^{\mathrm {NP}}$, and the third level of which consists of $\Sigma _3^p= \mathrm {NP}^{\mathrm {NP}^{\mathrm {NP}}}$ and $\Pi _2^p= \mathrm {coNP}^{\mathrm {NP}^{\mathrm {NP}}}$. We also assume familiarity with the concepts of hardness and completeness based on polynomial-time many-one reducibility. For more background on computational complexity, we refer the reader to, e.g., the textbooks by Papadimitriou [18] and Rothe [20].

3 Complexity of Nonemptiness in CAFs

For CAFs, we generalize the nonemptiness problem following the pattern of the controllability problem [17]. The generalized problem s -Control-Nonemptiness wraps the $s\text{- }\textsc {Ne}$ problem for standard argumentation frameworks with an outer existential quantifier over control configurations followed by an inner universal quantifier over completions. Using the notation from Definitions 2 and 3, we define this problem as follows for any semantics s.

We now determine the computational complexity of for each semantics $s\in \{\textsc {CF},$ $\textsc {GR},\textsc {AD},\textsc {CP},\textsc {ST},\textsc {PR}\}$. First, we extend a result of Skiba et al. [21] for incomplete argumentation frameworks to CAFs and show that can be solved in polynomial time.

Proposition 1

is in $\mathrm {P}$.

Proof

Let $CAF = (\mathcal {F},\mathcal {C},\mathcal {U})$ be a given control argumentation framework. Consider the control configuration $\mathcal {A}_{ conf }^{\max } = \mathcal {A}_C$ of $CAF$ that includes every control argument in $\mathcal {C}$. If the CAF induced by $\mathcal {A}_{ conf }^{\max }$ has an argument a for which the self-attack (a, a) does not exist in any of the attack relations, then clearly $\{a\}$ is a nonempty conflict-free set in every completion, so . On the other hand, if the CAF induced by $\mathcal {A}_{ conf }^{\max }$ has no such argument without a self-attack, then there cannot be a nonempty conflict-free set in any of its completions. Further, no control configuration other than $\mathcal {A}_{ conf }^{\max }$ can change this fact, since these cannot provide the existence of such an argument. Therefore, it holds that .

The control configuration $\mathcal {A}_{ conf }^{\max }$ and its induced CAF can be created in polynomial time, and checking whether there exists an argument without a self-attack in any of the attack relations can be done in polynomial time, too.    $\square$

Next, we provide a result analogous to that of Skiba et al. [21] for incomplete argumentation frameworks (IAFs) by showing that can also be solved in polynomial time.

Proposition 2

is in $\mathrm {P}$.

Proof

For a instance $(\mathcal {F},\mathcal {C},\mathcal {U})$ with $\mathcal {C}=\langle \mathcal {A}_C,\mathcal {R}_C\rangle$, consider the control configurations $\mathcal {A}_\textit{conf}^\emptyset = \emptyset$ and $\mathcal {A}_\textit{conf}^a = \{a\}$ for each $a\in \mathcal {A}_C$. For each of these $|\mathcal {A}_C|+1$ control configurations, GR-ConNe can be solved by the following algorithm. Recall that a nonempty grounded extension exists in a completion if and only if the completion has an unattacked argument.

Consider a sequence $AF_i$ of completions for control configuration $\mathcal {A}^x_\textit{conf}$ that is constructed as follows. $AF_0$ is the maximal completion that includes all uncertain elements $\mathcal {U}$ and all selected control elements $\mathcal {C}^x_\textit{conf}$:

$$ AF_0= \langle \mathcal {A}_F \cup \mathcal {A}^x_\textit{conf}\cup \mathcal {A}_U, \mathcal {R}_F \cup \mathcal {R}^x_\textit{conf}\cup \mathcal {R}_U \cup \{(a,b), (b,a) \mid (a,b) \in \mathcal {R}_U^\leftrightarrow \}\rangle . $$

Initially, let $i =1$. Define $AF_i$ to be $AF_{i-1}$ but with all arguments removed that originally were in $\mathcal {A}_U$ and are unattacked in $AF_{i-1}$, and with all attacks incident to these arguments removed as well. Repeat this step (incrementing i) until $AF_i = AF_{i-1}$ for some i. Then $AF^*= AF_i$ is our so-called critical completion.

If $AF^*$ has no unattacked arguments, it also has no nonempty grounded extension. If, on the other hand, $AF^*$ has an unattacked argument y, then every completion must have an unattacked argument y: In every completion, either y itself is unattacked, or else the completion must include some arguments $u \subseteq \mathcal {A}_U$ attacking y. The only candidates are removed by the algorithm. These uncertain arguments can only be attacked by other uncertain arguments. Also, at least one of the uncertain arguments needs to be not attacked by any other argument (definite or uncertain); otherwise, the algorithm would not delete such an argument in the first iteration. Hence, one argument must be unattacked in any given completion. Note that excluding any additional uncertain attacks from completions can only make it more likely for unattacked arguments to exist in the completion, so we do not need to consider this separately in the algorithm. Note further that if $AF^*$ has a nonempty grounded extension then it is allowed to answer “yes” for the given GR-ConNe instance.

Clearly, when we get a “yes” answer for at least one of the control configurations, this allows a “yes” answer for the given instance of the GR-ConNe problem, since we have found a control configuration such that for all completions, a nonempty grounded extension exists. On the other hand, if for all control configurations $\mathcal {A}_\textit{conf}^\emptyset$ and $\mathcal {A}_\textit{conf}^a$ with $a\in \mathcal {A}_C$, the answer is “no,” we can deduce a “no” answer for the given instance of GR-ConNe, so we know that none of the control configurations $\mathcal {A}_\textit{conf}^\emptyset$ or $\mathcal {A}_\textit{conf}^a$ can ensure the existence of an unattacked argument for all its completions. The only other control configuration left to be considered are those where multiple control arguments are added simultaneously, and it is clear that these cannot create any new unattacked arguments, since no new arguments are introduced, but only new attacks are possibly included.

It is easy to see that this algorithm runs in polynomial time.    $\square$

Finally, in order to show hardness of the remaining cases of s-CONNE, we provide in Definition 5 a translation from $\Sigma _3$ SAT instances to instances of s-CONNE, where $\Sigma _3$ SAT is the canonical problem for the complexity class $\Sigma _3^p$ in the polynomial hierarchy which asks, for a 3-CNF formula $\varphi$ over a set $X \uplus Y \uplus Z$ of propositional variables,¹ whether $(\exists \tau _X)(\forall \tau _Y)(\exists \tau _Z)[\varphi [\tau _X,\tau _Y,\tau _Z] = \texttt {true}]$, where $\tau _\chi$ denotes a truth assignment over variables in a set $\chi$, and $\varphi [\tau _\chi ]$ is the truth value that $\varphi$ evaluates to under $\tau _\chi$.

Definition 5

Given a $\Sigma _3$ SAT instance $(\varphi ,X,Y,Z)$ with $\varphi = \bigwedge _i c_i$ and $c_i = \bigvee _j \alpha _{i,j}$ for each clause $c_i$, where $\alpha _{i,j}$ are the literals in clause $c_i$, a control argumentation framework $CAF =(\mathcal {F},\mathcal {C},\mathcal {U})$ representing $(\varphi , X,Y,Z)$ is defined by its fixed part $\mathcal {F}$:

$$\begin{aligned} \mathcal {A}_F =~&\{\bar{x}_j \mid x_j \in X\} \cup \{\bar{y}_k \mid y_k \in Y\} \cup \{z_{\ell },\bar{z}_{\ell } \mid z_{\ell } \in Z\} \cup \{c_i \mid c_i \text { in } \varphi \} \cup \{\varphi , d\}, \\\mathcal {R}_F =~&\{(y_k,\bar{y}_k) \mid y_k \in Y\} \cup \{(\bar{z}_{\ell }, z_{\ell }), (z_{\ell }, \bar{z}_{\ell }) \mid z_{\ell } \in Z\} \cup \{(\bar{x}_j,c_i) \mid \lnot x_j \text { in } c_i\} \\&\cup \{(y_k, c_i) \mid y_k \text { in } c_i\} \cup \{(\bar{y}_k,c_i) \mid \lnot y_k \text { in } c_i\} \\&\cup \{(z_{\ell }, c_i) \mid z_{\ell } \text { in } c_i\} \cup \{(\bar{z}_{\ell },c_i) \mid \lnot z_{\ell } \text { in } c_i\} \\&\cup \{(c_i, \varphi ) \mid c_i \in \varphi \} \cup \{(d,a)\mid a\in \{\bar{x}_j,y_k,\bar{y}_k,z_{\ell },\bar{z}_{\ell },c_l\}\} \cup \{(\varphi ,d), (d,d)\}; \end{aligned}$$

its control part $\mathcal {C}$:

$$\begin{aligned} \mathcal {A}_C&= \{{x}_j\mid x_j \in X\},\\ \mathcal {R}_C&= \{(x_j,\bar{x}_j), (d,x_j) \mid x_j \in X\} \cup \{(x_j, c_i) \mid x_j \text { in } c_i\}; \end{aligned}$$

and its uncertain part $\mathcal {U}$:

$$\begin{aligned} \mathcal {A}_U&= \{y_k \mid y_k \in Y\},\\ \mathcal {R}_U&= \mathcal {R}_U^\leftrightarrow = \emptyset . \end{aligned}$$

We call arguments $c_i$ clause arguments and arguments representing literals $\alpha _{i,j}$ literal arguments. A clause argument $c_i$ can be interpreted as “clause $c_i$ is unsatisfied.”

Every assignment $\tau _X$ on X corresponds to a control configuration $\mathcal {A}_{ conf }^{\tau _X}$ for $CAF$ that includes a control argument $x_j$ if and only if $\tau _X(x_j)=\texttt {true}$. Further, given an assignment $\tau _X$ on X and the CAF induced by the control configuration $\mathcal {A}_{ conf }^{\tau _X}$, every assignment $\tau _Y$ on Y corresponds to a completion $AF^{\tau _X,\tau _Y}$ of the CAF induced by the control configuration, where an uncertain argument $y_k$ is included in the completion if and only if $\tau _Y(y_k)=\texttt {true}$. For full assignments $\tau _X$, $\tau _Y$, and $\tau _Z$, we denote by $A^{\tau _X,\tau _Y}[\tau _X, \tau _Y, \tau _Z]$ the corresponding set of arguments in the completion $AF^{\tau _X,\tau _Y}$:

$$\begin{aligned} A^{\tau _X,\tau _Y}[\tau _X, \tau _Y, \tau _Z] =&\{x_j \mid \tau _X(x_j)=\texttt {true}\} \cup \{\bar{x}_j \mid \tau _X(x_j)=\texttt {false}\} \cup ~ \\ {}&\{y_k \mid \tau _Y(y_k)=\texttt {true}\} \cup \{\bar{y}_k \mid \tau _Y(y_k)=\texttt {false}\} \cup ~ \\ {}&\{z_{\ell } \mid \tau _Z(z_{\ell })=\texttt {true}\} \cup \{\bar{z}_{\ell } \mid \tau _Z(z_{\ell })=\texttt {false}\}. \end{aligned}$$

Fig. 3.

CAFs created for the $\Sigma _3$ SAT instances from Example 5 using the translation from Definition 5. The left CAF uses $\varphi = c_1 \wedge c_2 = (x_1 \vee \lnot y_1) \wedge (y_1 \vee \lnot z_1)$ and is a “yes”-instance of s-CONNE for all $s\in \{\textsc {AD},\textsc {ST},\textsc {CP},\textsc {PR}\}$, whereas the right CAF uses $\varphi ' = c_1' \wedge c_2' = (x_1 \vee \lnot z_1) \wedge (y_1)$ and is a “no”-instance of s-CONNE for all $s\in \{\textsc {AD},\textsc {ST},\textsc {CP},\textsc {PR}\}$.

Example 5

Consider the $\Sigma _3$ SAT instance $\varphi = c_1 \wedge c_2 = (x_1 \vee \lnot y_1) \wedge (y_1 \vee \lnot z_1)$, with $X = \{x_1\}$, $Y = \{y_1\}$, and $Z = \{z_1\}$. It is a “yes”-instance of $\Sigma _3$ SAT, since $\varphi [\tau _X,\tau _Y,\tau _Z]=\texttt {true}$ for the assignment $\tau _X$ on X with $\tau _X(x_1)=\texttt {true}$, for any assignment $\tau _Y$ on Y, and for the assignment $\tau _Z$ on Z with $\tau _Z(z_1)=\texttt {false}$. The translation from Definition 5 produces the CAF representation visualized in Fig. 3 (left), where we display control arguments as rectangles, uncertain arguments as dashed rectangles, and elements of $\mathcal {R}_C$ as boldfaced arrows. This CAF is a “yes”-instance of s-CONNE for all $s\in \{\textsc {AD},\textsc {ST},\textsc {CP},\textsc {PR}\}$, since for the control configuration $\{x_1\}$ and for every one of its completions, the set $\{x_1,\hat{y}_1,\bar{z}_1,\varphi \}$ is nonempty and stable (and thus also admissible, complete, and preferred), where $\hat{y}_1=y_1$ if $\tau _Y(y_1)=\texttt {true}$ and $\hat{y}_1=\bar{y}_1$ if $\tau _Y(y_1)=\texttt {false}$.

Now consider a modified $\Sigma _3$ SAT instance $\varphi ' = c_1' \wedge c_2' = (x_1 \vee \lnot z_1) \wedge (y_1)$, where $\lnot y_1$ in the first clause is replaced by $\lnot z_1$ and “$\vee ~\lnot z_1$” is removed from the second clause. This instance is a “no”-instance of $\Sigma _3$ SAT, since for each assignment $\tau _X$ on X, the assignment $\tau _Y$ on Y with $\tau _Y(y_1)=\texttt {false}$, and for each assignment $\tau _Z$ on Z, the second clause is unsatisfied and thus $\varphi '[\tau _X,\tau _Y,\tau _Z]=\texttt {false}$. The corresponding CAF is displayed in Fig. 3 (right). Irrespective of the control configuration used, the completion that excludes argument $y_1$ has no literal argument that can defend $\varphi '$ against the clause argument $c_2'$, so this CAF is a “no”-instance of s-CONNE for all $s\in \{\textsc {AD},\textsc {ST},\textsc {CP},\textsc {PR}\}$.

Next, we observe that the properties shown by Skiba et al. [21] (see their Lemmas 12 and 13) to hold for completions of IAFs created via their Definition 19 do hold for completions of CAFs created via Definition 5 as well. This result follows directly from [21, Lemma 12]: Given a $\Sigma _3$ SAT instance $(\varphi ,X,Y,Z)$ and letting $X'=X\cup Y$ and $Y'=Z$, $(\varphi ,X',Y')$ is a $\Pi _2$ SAT instance that has an IAF representation $IAF$ via [21, Definition 19]. The completion $AF^{\tau _X,\tau _Y}$ considered in the current lemma is the same argumentation framework as the completion $AF^{\tau _{X'}}$ of $IAF$, and the result of [21, Lemma 12] for $AF^{\tau _{X'}}$ applies here.

The first lemma states that the argument $\varphi$ has to be in any nonempty extension satisfying the semantics $s \in \{\textsc {AD}, \textsc {CP}, \textsc {GR}, \textsc {PR}, \textsc {ST}\}$. Otherwise, the argument d is not attacked and has to be in every potential admissible set, which is impossible as this would conflict with the conflict-freeness of every set, as d attacks itself, so d cannot be part of any conflict-free set.

Lemma 1

Given a $\Sigma _3$ SAT instance $(\varphi ,X,Y,Z)$, let $CAF$ be the control argumentation framework created for it according to Definition 5. Let $\tau _{X}$ and $\tau _Y$ be full assignments on X and Y, respectively. In the completion $AF^{\tau _X,\tau _Y}$, the argument $\varphi$ has to be in every nonempty extension that satisfies any of the semantics $s \in \{\textsc {AD}, \textsc {CP}, \textsc {GR}, \textsc {PR}, \textsc {ST}\}$.

Similarly, Lemma 13 by Skiba et al. [21] directly provides the following result for CAFs, which states that if there is a truth assignment for the $\Sigma _3$ SAT instance, there is an extension satisfying the semantics $s \in \{\textsc {AD}, \textsc {CP}, \textsc {GR}, \textsc {PR}, \textsc {ST}\}$.

Lemma 2

Let $(\varphi , X, Y, Z)$ be a $\Sigma _3$ SAT instance and let $\tau _{X}$, $\tau _Y$, and $\tau _{Z}$ be assignments on X, Y, and Z, respectively. Let $CAF$ be the control argumentation framework created by Definition 5 for $(\varphi , X, Y, Z)$. Let $AF^{\tau _X, \tau _Y}$ be its completion corresponding to $\tau _X$ and $\tau _Y$ and let $\mathcal {A}^{\tau _X,\tau _Y}[\tau _X, \tau _Y, \tau _Z]$ be the set of literal arguments corresponding to the total assignment. If $\varphi [\tau _X,\tau _Y,\tau _Z]= \texttt {true}$, then $AF^{\tau _X, \tau _Y}$ has an extension $\mathcal {A}^{\tau _X,\tau _Y}[\tau _X,\tau _Y,\tau _Z] \cup \{\varphi \}$ that is admissible, complete, preferred, and stable.

We are now ready to show $\Sigma _3^p$-completeness of $s\text{- }\textsc {ConNe}$ for the remaining semantics $s\in \{\textsc {AD},\textsc {CP},\textsc {ST},\textsc {PR}\}$.

Theorem 1

s-CONNE is $\Sigma _3^p$-complete for each $s\in \{\textsc {AD},\textsc {CP},\textsc {ST},\textsc {PR}\}$.

Proof

To show membership of the problems $s\text{- }\textsc {ConNe}$ in $\Sigma _3^p$, we look at their quantifier representation. For each semantics $s\in \{\textsc {AD},\textsc {CP},\textsc {ST},\textsc {PR}\}$, the problem can be written as the set of all $CAF$ (using the notation from Definitions 2 and 3) such that $(\exists \mathcal {A}_\textit{conf}\subseteq \mathcal {A}_C) (\forall \text {~completion~}AF^*) [AF^*\in s\text{- }\textsc {Ne}]$, and since $s\text{- }\textsc {Ne}$ is $\mathrm {NP}$-complete for these semantics s [3, 5], this provides a $\Sigma _3^p$ upper bound of $s\text{- }\textsc {ConNe}$. Note that all quantifiers in this representation are polynomially length-bounded in the size of the encoding of the given $CAF$.

To prove $\Sigma _3^p$-hardness, let $(\varphi ,X,Y,Z)$ be a $\Sigma _3$ SAT instance. Let $CAF = (\mathcal {F},\mathcal {C},\mathcal {U})$ be the control argumentation framework created for $(\varphi ,X,Y,Z)$ via Definition 5. We will show that for any $s\in \{\textsc {AD},\textsc {CP},\textsc {PR},\textsc {ST}\}$, $(\varphi ,X,Y,Z)\in$ $\Sigma _3$ SATif and only if $CAF \in s\text{- }\textsc {ConNe}$.

From left to right, assume that $(\varphi ,X,Y,Z)\in$ $\Sigma _3$ SAT, i.e., there exists an assignment $\tau _X$ on X such that for every assignment $\tau _Y$ on Y, there exists an assignment $\tau _Z$ on Z such that $\varphi [\tau _X, \tau _Y, \tau _Z]$ is $\texttt {true}$. We then know from Lemma 2 that, in the completion $AF^{\tau _X, \tau _Y}$ of $CAF$, the set $\mathcal {A}^{\tau _X,\tau _Y}[\tau _X,\tau _Y,\tau _Z] \cup \{\varphi \}$ is a nonempty extension that is admissible, complete, stable, and preferred. Therefore, $CAF \in s\text{- }\textsc {ConNe}$ for $s\in \{\textsc {AD},\textsc {CP},\textsc {PR},\textsc {ST}\}$.

From right to left, assume that, for any $s\in \{\textsc {AD},\textsc {CP},\textsc {PR},\textsc {ST}\}$, $CAF \in s\text{- }\textsc {ConNe}$ holds, i.e., there exists a control configuration such that in all completions, there exists a nonempty s extension $\mathcal {E}$. Every nonempty s extension has to contain $\varphi$; otherwise, d would not be attacked and thus has to be in every potentially admissible set, as it attacks every other argument. However, d also attacks itself, so every extension containing d would not be conflict-free and therefore not admissible. Therefore, no s extension $\mathcal {E}$ can contain any clause argument $c_i$; otherwise, we would create a set that is not conflict-free. Since we have a control configuration all completions of which have a nonempty s extension, these extensions contain a literal argument $v \in X \cup Y \cup Z$ that attacks $c_i$, for each $c_i$. It is obvious that $\mathcal {E}$ can only contain v or $\bar{v}$ and not both. Hence, the extension contains $\varphi$ and several literal arguments, which together are attacking every clause argument. This extension is already seen to be admissible, complete, and preferred, but regarding stability we may have some literals $v_j \in X \cup Y \cup Z$ for which neither argument $v_j$ nor $\bar{v}_j$ is in $\mathcal {E}$. Therefore, either $v_j$ or $\bar{v}_j$ has to be in $\mathcal {E}$. Consequently, we can define a total assignment based on the s extension depending on whether the literal arguments are in or out of $\mathcal {E}$: If a literal argument is in $\mathcal {E}$, we set the corresponding literal to $\texttt {true}$; otherwise we set its negation to $\texttt {true}$. This assignment must be a satisfying assignment, since every clause argument is attacked by $\mathcal {E}$ and, accordingly, every clause is satisfied by the assignment.    $\square$

Finally, let us consider a few special cases of control nonemptiness. The literature on control argumentation frameworks also considers simplified CAFs, which are CAFs where the sets of uncertain elements $\mathcal {A}_U$, $\mathcal {R}_U$, and $\mathcal {R}_U^{\leftrightarrow }$ are empty. Clearly, the problem s-CONNE for simplified CAFs is equivalent to for argument-incomplete argumentation frameworks as defined by Skiba et al. [21], so their complexity results for s-PosNe carry over.

Table 1. Summary of complexity results for s-CONNE for various semantics s, in comparison with the known complexity results for s-Ne, s-PosNe, and s-NecNe. $\mathcal {C}$-c denotes “$\mathcal {C}$-complete” for a complexity class $\mathcal {C}$. New results for s-CONNE are marked by the corresponding theorem or proposition, and results due to previous work are displayed in grey. Results for s-Ne in standard argumentation frameworks are marked by the respective reference (where the result marked by $^*$ is straightforward and not formally proven).

Recently, Mailly [14] introduced the idea of possible controllability, related to the concept of possible problem variants for incomplete argumentation frameworks. A corresponding problem possible control nonemptiness can be defined, which asks, for a given CAF, whether there exists a control configuration such that there is a completion in which a nonempty extension exists. This problem is clearly in $\mathrm {NP}$ via the three collapsing existential quantifiers, and $\mathrm {NP}$ hardness is directly inherited from $s\text{- }\textsc {Ne}$ for $s\in \{\textsc {AD},\textsc {CP},\textsc {PR},\textsc {ST}\}$. For $s\in \{\textsc {CF},\textsc {GR}\}$, the complexity of this variant of nonemptiness remains an open problem.

4 Conclusion

The contribution of this work is the definition of variants of the nonemptiness problem for control argumentation frameworks and a characterization of their computational complexity. Table 1 summarizes our complexity results, also providing a comparison with the corresponding complexity results of Chvátal [3], Modgil and Caminada [16], and Dimopoulos and Torres [5] for the analogous problems in standard argumentation frameworks and of Skiba et al. [21] in incomplete argumentation frameworks.

Comparing the complexity of “necessary nonemptiness” in incomplete argumentation frameworks with that of nonemptiness in control argumentation frameworks, there is a complexity jump from $\Pi _2^p$-hardness to $\Sigma _3^p$-hardness, indicating a significant difference between the two formalisms. Interestingly, the $\Sigma _3^p$-completeness that we determined for $s\text{- }\textsc {ConNe}$ is the same as the complexity of credulous controllability for the same four semantics $s\in \{\textsc {AD},\textsc {CP},\textsc {PR},\textsc {ST}\}$ [17]. Credulous controllability is the problem of deciding whether there is a control configuration, such that for all completions, a given target set of arguments is a subset of at least one s extension. This is a refinement of $s\text{- }\textsc {ConNe}$, since $s\text{- }\textsc {ConNe}$ only requires the existence of at least one nonempty s extension. The coinciding complexities indicate that the hardness of credulous controllability in control argumentation frameworks lies predominantly in making sure that an s extension exists in the first place, while the additional requirement of enforcing that the given target set of arguments is contained in that extension does not add to the asymptotical complexity. In contrast, for the grounded semantics, credulous controllability is $\Sigma _2^p$-hard [17], while $\textsc {GR}\text{- }\textsc {ConNe}$ is in $\mathrm {P}$.

In future work, it would be interesting to extend the complexity analysis of nonemptiness in CAFs to further semantics. Control argumentation frameworks model argumentation by agents strategically. Relatedly, Maher [12] (see also  [13]) introduced a model of strategic argumentation by simulating a game. However, his model does not allow any uncertainty regarding the attacks between arguments. Further, CAFs have some similarities to AF expansion due to Baumann and Brewka [1], where new arguments and new attacks incident to at least one new argument may be added. Opposed to CAFs, the set of new arguments and attacks is not fixed, and new attacks among existing arguments are not allowed. One might also modify the model of control argumentation frameworks by adding weights to the uncertain elements or by following a probabilistic approach (see, e.g., the work of Dunne et al. [7], Li et al. [11], Fazzinga et al. [9], and Gaignier et al. [10]). Such refinements of the model of CAFs may lead to new challenges in future work. One could also consider the approach of Riveret et al. [19] who combine probabilistic argumentation with reinforcement learning techniques.