Acceptance in Incomplete Argumentation Frameworks

Abstract argumentation frameworks (AFs), originally proposed by Dung, constitute a central formal model for the study of computational aspects of argumentation in AI. Credulous and skeptical acceptance of arguments in a given AF are well-studied problems both in terms of theoretical analysis—especially computational complexity—and the development of practical decision procedures for the problems. However, AFs make the assumption that all attacks between arguments are certain (i.e., present attacks are known to exist, and missing attacks are known to not exist), which can in various settings be a restrictive assumption. A generalization of AFs to incomplete AFs was recently proposed as a formalism that allows the representation of both uncertain attacks and uncertain arguments in AFs. In this article, we explore the impact of allowing for modeling such uncertainties in AFs on the computational complexity of natural generalizations of acceptance problems to incomplete AFs under various central AF semantics. Complementing the complexity-theoretic analysis, we also develop the ﬁrst practical decision procedures for all of the NP-hard variants of acceptance in incomplete AFs. In terms of complexity analysis, we establish a full complexity landscape, showing that depending on the variant of acceptance and property/semantics, the complexity of acceptance in incomplete AFs ranges from polynomial-time decidable to completeness for Σ p 3 . In terms of algorithms, we show through an extensive empirical evaluation that an implementation of the proposed decision procedures, based on boolean satisﬁ-ability (SAT) solving, is effective in deciding variants of acceptance under uncertainties. We also establish conditions for what type of atomic changes are guaranteed to be redundant from the perspective of preserving extensions of completions of incomplete AFs, and show that the results allow for considerably improving the empirical efﬁciency of the proposed SAT-based counterexample-guided abstraction reﬁnement algorithms for acceptance in incomplete AFs for problem variants with complexity beyond NP.


Introduction
The study of computational aspects of argumentation is a topical area of artificial intelligence research. With strong connections to other forms of nonmonotonic reasoning, abstract argumentation frameworks [41] (AFs) provide a central formal model for the study of argumentation in AI. Argumentation frameworks take the form of directed graphs, where the nodes represent abstract arguments, and directed edges form an attack relation between arguments. 5 While originally AFs make the assumption that all attacks between arguments are certain, in various settings such an assumption turns out to be restrictive. In an answer to bypass this restriction, a generalization of AFs to incomplete argumentation frameworks (IAFs) was recently proposed [16,17], bringing together earlier-proposed ideas of a generalization of AFs to partial argumentation frameworks [33,14] (integrating uncertainty about the existence of attacks into AFs) and the integration of uncertainty about the existence of arguments into AFs [18]. This 10 article contributes to the study of incomplete argumentation frameworks from different computational perspectives. In particular, we establish the computational complexity of central reasoning problems and their variants in the context of IAFs under various argumentation semantics, and develop practical decision procedures for NP-hard reasoning tasks in IAFs.
A key motivation behind incomplete argumentation frameworks is that they allow for representing unquantified probability of existence. Uncertainty is inherently present in argumentative scenarios. In particular, real arguments as presented by people in dialogues are usually enthymemes [56,3], which represent explicitly only a part of the underlying knowledge (i.e., the premises and the claim of the argument). In discussions one can typically assume a common knowledge base which one can use to construct arguments in a partially implicit manner. However, assump- 20 tions on the common knowledge base can be different for different agents, which means that the AFs constructed by the agents may have different arguments and attacks [63]. The ASPIC + framework [66] and the encoding proposed by Wyner et al. [84] are two examples of specific instantiation methods that could translate uncertainty in an underlying knowledge base to structural uncertainty represented by an incomplete argumentation framework (see Section 8 on related work for more details). 25 Additionally, incomplete argumentation frameworks allow for modeling different types of real-world application scenarios. As an example, consider several agents, each with their own AF representing their subjective view, and the problem of merging these AFs [33] (again, see Section 8 for more examples on interesting application scenarios). While the AFs may coincide in terms of the presence of some arguments and attacks (which can be considered definite), it is likely that agents partially disagree on the existence of specific arguments or attacks, or that specific 30 arguments or attacks are simply not represented in the AFs of some agents. Such arguments and attacks may be considered uncertain. By merging the agents' AFs into a single incomplete argumentation framework by taking the union of all arguments and attacks and specifying elements as definite if they occur in each AF, and otherwise as uncertain, the resulting single incomplete AF allows for reasoning about, e.g., whether some or all agents find an argument acceptable. For more details about various approaches to collective acceptability and specific methods for 35 structural aggregation of AFs, we refer to the forthcoming handbook chapter by Baumeister et al. [13].
An incomplete argumentation framework can be seen as a representation of a set of possible worlds, called completions, each of which is a standard argumentation framework that shares all definite elements of the incomplete framework and where each of its uncertain elements is either included or excluded. Existing criteria for argumentation frameworks can then be generalized to incomplete argumentation frameworks by either asking whether they 40 are satisfied possibly (in at least one completion) or necessarily (in all completions), i.e., whether the uncertainty either can or must be resolved in a way that satisfies the conditions of the given criterion. The answer may help with decisions in strategic scenarios, where the uncertainty represents possible moves. In scenarios where uncertainty represents missing information, the preliminary answer may be sufficient for the task at hand, removing the need to actually resolve the uncertainty. 45 In this article, we focus on the central reasoning problems of credulous and skeptical acceptance for incomplete argumentation frameworks. Credulous and skeptical acceptance are today well-understood when it comes to (standard) argumentation frameworks (i.e., AFs without any uncertainties on the existence of arguments and attacks). In terms of standard AFs, acceptance consists of asking-parameterized by a semantics and for a given argumentation framework and an argument in that framework-either whether that argument is in at least one extension (for credulous accep- 50 tance) or in all extensions (for skeptical acceptance) of the framework with respect to the semantics. Generalizing acceptance in a natural way to IAFs, we focus on variants of the following four problem combinations.
• Possible Credulous Acceptance (PCA): Is there any way to accept the given argument?
• Necessary Credulous Acceptance (NCA): 55 Is the given argument in at least one extension, regardless of how the uncertainty is resolved?
• Possible Skeptical Acceptance (PSA): Can the uncertainty be resolved in such a way that the given argument is in all extensions?
• Necessary Skeptical Acceptance (NSA): Is the given argument absolutely guaranteed to be accepted? 60 A "no" answer to PCA indicates that the target argument is a hopeless case and will never be accepted, while a "yes" answer to NSA guarantees that it will be accepted under all circumstances. If the goal is to have the target argument credulously accepted, then a "yes" answer to NCA ensures that this goal is satisfied, whatever completion or extension is chosen. If the goal is to have the target skeptically accepted, then the answer to PSA indicates whether that goal can be achieved by choosing the right completion. All of these answers provide information that is valuable even when results, our SAT-based algorithms cover all of the considered variants of acceptance in incomplete AFs and semantics considered in our complexity analysis. In particular, for the problem variants that turn out to be complete for the first level of the polynomial hierarchy, generalizing SAT encodings of AF semantics [20] to cover acceptance in incomplete AFs, we present direct SAT encodings which allow for deciding acceptance with a single call to a SAT solver. For those problem variants that turn out to be complete for the second or third level of the polynomial hierarchy, 100 we develop SAT-based counterexample-guided abstraction refinement procedures, making incremental use of a SAT solver to decide acceptance in an iterative manner. We also present results from an extensive empirical study of our implementation of all of the SAT-based algorithms presented, showing the effectiveness of the approach. Bridging theory and practice, extending on earlier results on the persistence of extensions under atomic changes to AFs [77], we also establish conditions for what type of atomic changes are guaranteed to be redundant from the per-105 spective of preserving extensions of completions of IAFs. While of interest on their own, this analysis proves central as a basis of SAT-based counterexample-guided abstraction refinement (CEGAR) algorithms for IAFs. In particular, we show empirically that the more in-depth analysis of atomic changes to the uncertain part under which an extension persists gives noticeably stronger refinements for the SAT-based CEGAR algorithms for deciding acceptance in incomplete AFs, evidenced in practice by noticeably improved runtimes.  • A is preferred (PR) if A is a set-maximal admissible set.
A set of arguments that satisfies the conditions of a semantics is called an extension of the argumentation framework with respect to that semantics. Every stable extension is preferred, every preferred extension is complete, every 155 complete extension is admissible, and every admissible set is conflict-free. Further, the unique grounded extension is complete, and coincides with the intersection of all complete extensions. There are argumentation frameworks that have no stable extension, while the other semantics defined above guarantee the existence of at least one extension. {a, c, f }, which thus is the only preferred extension of AF. AF has no stable extension, because no conflict-free set has a chance to attack the self-attacking argument g. Both complete extensions of this example are displayed in Figure 2 using the labeling representation by Caminada [26], where arguments are labeled IN if they are in the extension, labeled OUT if they are attacked by IN 170 arguments, and labeled UNDEC otherwise.
We investigate problems concerning the acceptability of individual arguments in an argumentation framework. The most established notion of acceptability for single arguments proposed by Dunne and Bench-Capon [43] is derived from their membership in extensions. For a given s ∈ {CF, AD, CP, GR, PR, ST}, an argument a can only be considered acceptable if a is contained in at least one s extension (called credulous acceptance), and a is ultimately accepted 175 if it is contained in all s extensions (called skeptical acceptance). The following decision problems formalize these notions.
The question of the s-SKEPTICAL-ACCEPTANCE problem is equivalent to asking whether each set of arguments that does not include a is not s in A , R . This alternative formulation moves the verification part of the problem (which checks whether a set of arguments is an extension) from the scope of the quantifier to the predicate after the quantifier, which will allow us to directly derive upper complexity bounds later.
The SKEPTICAL-ACCEPTANCE problem exhibits two types of special behavior that we need to address. Firstly, the answer to s-SKEPTICAL-ACCEPTANCE is trivially "no" if the empty set always satisfies s-this is the case for 185 s ∈ {CF, AD} among the properties/semantics that we use in this article. More meaningful results can be obtained by excluding the empty set for acceptance problems. Therefore, we restrict our investigation to nonempty conflict-free sets (denoted CF / 0 ) and nonempty admissible sets (denoted AD / 0 ), for which the SKEPTICAL-ACCEPTANCE problem is nontrivial, while the CREDULOUS-ACCEPTANCE problem remains unaffected by this change.
Secondly, if an argumentation framework AF has no s extension, then no argument is credulously accepted in AF 190 with respect to s, but at the same time each argument is skeptically accepted. This behavior is due to the convention that a universal quantifier over an empty set (here, the set of s extensions) defaults to true, but it might be counterintuitive to call an argument skeptically accepted when it is in fact never accepted. This situation can occur for any semantics s that does not guarantee the existence of an extension-in our work, this is the case for the CF / 0 , AD / 0 , and ST semantics. Dunne and Wooldridge [44] propose a refined version of s-SA for semantics s that do not guarantee the 195 existence of an extension. The refined problem additionally requires the existence of at least one s extension in order to give a "yes" answer. We call this problem s-EXISTENCE-AND-SKEPTICAL-ACCEPTANCE (s-EXSA), since it is the intersection of s-EXISTENCE (asking whether there exist an s extension of the given argumentation framework) and s-SKEPTICAL-ACCEPTANCE.

Given:
An argumentation framework A , R and an argument a ∈ A . Question: Is there at least one s extension in A , R and is a in all s extensions of A , R ?

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The EXSA problem can equivalently be represented as the intersection of credulous and skeptical acceptance. For any semantics s that guarantees the existence of at least one extension in every argumentation framework, it is apparent that s-EXSA and s-SA are equivalent. We will therefore investigate s-EXSA for s ∈ {CF / 0 , AD / 0 , ST} only. Since AF has no stable extension, it holds that (AF, arg) ∈ ST-SA for all arguments arg ∈ A , but on the other hand, (AF, arg) ST-EXSA for any argument arg ∈ A .
The following relations hold between the CA and SA problems.

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Observation 5. AD-CA = AD / 0 -CA = CP-CA = PR-CA, since for every argumentation framework, the union of all its (nonempty) admissible sets, the union of all its complete extensions, and the union of all its preferred extensions is the same.
Observation 6. GR-CA = GR-SA, since every argumentation framework has a single unique grounded extension.
Observation 7. GR-SA = CP-SA, since the intersection of all complete extensions in an argumentation framework is 215 its grounded extension.
We obtain GR-CA = GR-SA = CP-SA from the combination of Observations 6 and 7.

Incomplete Argumentation Frameworks
An incomplete argumentation framework A , A ? , R, R ? splits both the set of arguments and the set of attacks into two disjoint parts, the definite part (A and R) and the uncertain part (A ? and R ? ), where both R and R ? are 220 subsets of (A ∪ A ? )×(A ∪ A ? ). For uncertain elements (members of A ? or R ? ), it is not known whether they are part of the argumentation-they might be added or removed in the future, or the uncertainty may just represent the limited knowledge of some agent about those elements. Definite arguments (elements of A ) are known to exist,  while definite attacks (elements of R) exist if and only if both incident arguments exist, too. To account for this, we call attacks in R that are incident to at least one uncertain argument conditionally definite, since these attacks may 225 vanish alongside an incident uncertain argument. On the other hand, attacks in R that are incident to only definite arguments are called definite. If A ? = / 0, we have a (purely) attack-incomplete argumentation framework [33]; for R ? = / 0, a (purely) argument-incomplete argumentation framework; and A ? = R ? = / 0 yields standard argumentation frameworks without uncertainty. An attack-incomplete argumentation framework may be abbreviated as A , R, R ? and an argument-incomplete argumentation framework as A , A ? , R .

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The scope of the target argument a is restricted to a ∈ A instead of allowing a ∈ A ∪ A ? in all our problem definitions, since allowing uncertain arguments as target does not produce any additional interesting cases. Specifically, for the necessary problem variants, all instances with a ∈ A ? are trivial "no" instances. On the other hand, for each possible problem variant s-P ∈ {s-PCA, s-PSA, s-PEXSA}) and any instance ( can be reduced to an equivalent formulation where a ∈ A .  semantics  4a  4b  4c  4d  4e  4f  4g  4h  preferred {a We make a few observations: • (IAF, d) PR-PCA, since no completion has d in any of its preferred extensions.
• (IAF, b) ∈ PR-PSA, because there are completions that have b in all of their preferred extensions. But on the other hand, (IAF, b) PR-NCA, since there are also completions that have b in none of their preferred extensions.

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• (IAF, f ) ∈ ST-NSA, because every completion either has no stable extension (resulting in a trivial "yes" answer for every ST-SA query) or has f in all of its stable extensions. On the other hand, (IAF, f ) ST-NEXSA, since not every completion has a stable extension. Table 1: Overview of previously known and new complexity results for credulous and skeptical acceptance problems, where previously known  results are ascribed to the respective source (references are numbers in brackets), and new results are labeled with the respective theorem number. Results marked with an asterisk ( * ) are straight-forward. Results for EXSA problem variants (in parentheses) are given only when they differ from the corresponding result for SA. For any complexity class C , C -c. stands as a shorthand for "C -complete." Furthermore, "trivial" means that the answer is always "no," while "∈ P" implies the existence of a

Complexity Results
In this section, we will fully characterize the computational complexity of all variants of the CA and SA problems 280 defined above for incomplete argumentation frameworks. Our results (together with previously known results) are summarized in Table 1.
We assume the reader to be familiar with the needed notions from computational complexity theory, such as the concepts of hardness and completeness based on polynomial-time many-one reducibility as well as the classes of the polynomial hierarchy due to Meyer and Stockmeyer [65,80] and of the boolean hierarchy studied early on by Cai 285 et al. [25] and Köbler et al. [57]. In particular, we will consider the well-known classes capturing deterministic and nondeterministic polynomial time, P and NP, the class DP due to Papadimitriou and Yannakakis [76] (which belongs to the second level of the boolean hierarchy and has the canonical complete problem 3-SAT-UNSAT, defined as the intersection of 3-SAT and 3-UNSAT; for the latter two problems, see Table 2 on page 12), and the classes Σ p 2 , Π p 2 , and Σ p 3 from the second and third levels of the polynomial hierarchy with their canonical complete problems 290 Σ 2 SAT, Π 2 SAT, and Σ 3 SAT (again, see Table 2 on page 12 for their definitions). More background on computational complexity can be found, for instance, in the textbooks by Papadimitriou [75] and Rothe [78]. The computational complexity of the original CA and SA problems for conflict-freeness, admissibility, and the stable, complete, grounded, and preferred semantics ranges from P membership to Π p 2 -completeness and was investigated by Dimopoulos and Torres [38], Dunne and Bench-Capon [43], Coste-Marquis et al. [34], and Dunne and 295 Wooldridge [44] (again, see Table 1). In the remainder of this section, we will present the proofs for our complexity results.

Upper Bounds
We start with P membership results for all problem variants that use the CF / 0 semantics. Proposition 11. The following problems are all in P. Proof. Whenever (a, a) ∈ R, any set containing a cannot be conflict-free, and whenever (a, a) R, {a} is a nonempty conflict-free set that contains a. We can infer that  Figure 5: Example for the completions AF pos and AF nec used in the proof of Proposition 11. Argument a is credulously and skeptically accepted in AF pos , and therefore possibly credulously and possibly skeptically accepted in IAF. Argument a is not credulously and not skeptically accepted in AF nec , and therefore not necessarily credulously and not necessarily skeptically accepted in IAF.
is given in Figure 5. AF pos can be constructed in polynomial time and we have that (IAF, a) is a "yes" instance of CF / 0 -PCA, CF / 0 -PSA, and CF / 0 -PEXSA, respectively, if and only if (AF pos , a) is a "yes" instance of the corresponding base problem CF / 0 -CA, CF / 0 -SA, and CF / 0 -EXSA, respectively. The implication from the base problem to its possible generalization is trivial. For the other direction of the equivalence, we cover the problems CF / 0 -CA, CF / 0 -SA, and CF / 0 -EXSA individually.

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• If (AF pos , a) CF / 0 -CA, then there is no conflict-free set in AF pos that contains a. This can only be the case if (a, a) ∈ R pos , which by construction of R pos means that (a, a) ∈ R is a definite attack. This implies that a cannot be in a conflict-free set in any completion, so (IAF, a) CF / 0 -PCA.
• If (AF pos , a) CF / 0 -SA, then there is a nonempty conflict-free set E ⊆ A pos that does not contain a. This can only be the case if (b, b) R pos for some b a in A pos . Since all possible self-attacks of arguments other than 320 a are included in R pos and since b ∈ A pos = A is a definite argument, this means that b ∈ A * and (b, b) R * for any completion A * , R * . Therefore, {b} is a conflict-free set not containing a in every completion, so (IAF, a) CF / 0 -PSA.
• If (AF pos , a) CF / 0 -EXSA, then there is no conflict-free set in AF pos that contains a, or there is a nonempty conflict-free set E ⊆ A pos that does not contain a. The previous proofs provide that (IAF, a) CF / 0 -PCA or 325 that (IAF, a) CF / 0 -PSA, and therefore (IAF, a) cannot be in the intersection CF / 0 -PEXSA of these problems. Analogously, for an instance (IAF, a) with IAF = A , A ? , R, R ? of CF / 0 -NCA, CF / 0 -NSA, or CF / 0 -NEXSA, we construct a single completion AF nec = A nec , R nec with A nec = A ∪A ? and R nec = R ∪ {(a, a)} ∩ R ? . Again, an example is given in Figure 5. AF nec can be constructed in polynomial time and we have that (IAF, a) is a "yes" instance of CF / 0 -NCA, CF / 0 -NSA, and CF / 0 -NEXSA, respectively, if and only if (AF nec , a) is a "yes" instance of 330 the corresponding base problem CF / 0 -CA, CF / 0 -SA, and CF / 0 -EXSA, respectively. Here, the implication from the necessary generalization to the base problem is trivial. For the other direction of the equivalence, we again cover the problems CF / 0 -CA, CF / 0 -SA, and CF / 0 -EXSA individually. • If (AF nec , a) ∈ CF / 0 -CA, then there is a nonempty conflict-free set E ⊆ A nec with a ∈ E . This can only be the case if (a, a) R nec . Due to the construction of AF nec , we know that (a, a) R * for any completion 335 A * , R * , and since a is definite, this means that {a} is a nonempty conflict-free set in every completion, so (IAF, a) ∈ CF / 0 -NCA.
• If (AF nec , a) ∈ CF / 0 -SA, then there is no nonempty conflict-free set E ⊆ A nec with a E . This can only be the case if (b, b) ∈ R nec for all arguments b ∈ A nec with b a. Due to the fact that A nec = A ∪ A ? contains all possible arguments, and because R nec contains only definite self-attacks for arguments b a, we can conclude 340 that (b, b) ∈ R * for all arguments b ∈ A * with b a holds for every completion A * , R * . Thus there can be no nonempty conflict-free set without a in any completion, and therefore (IAF, a) ∈ CF / 0 -NSA. In Proposition 12, we derive upper bounds for all remaining acceptance problem variants from their respective quantifier representations. In Section 3.2, we will prove matching lower bounds.  3. AD / 0 -EXSA is in DP.
5. For s ∈ {AD / 0 , ST}, s-PSA and s-PEXSA are in Σ p 2 . 6. PR-PSA is in Σ p 3 . Proof. All acceptance problem variants can be represented using a sequence of polynomially length-bounded ex-355 istential or universal quantifiers (which we will simply call "existential quantifiers" or "universal quantifiers" further on) followed by a predicate that corresponds to the s-VERIFICATION (s-VER) problem for the respective semantics. An instance (AF, E ) of s-VER-where AF = A , R is an argumentation framework and E ⊆ A is a set of arguments in AF-is a "yes" instance if and only if E is an s extension of AF. It is known [41] that PR-VER is coNP-complete and s-VER is in P for all other semantics used in the current article (clearly, the complexity of CF / 0 -VER and CF-VER 360 is the same, as is that of AD / 0 -VER and AD-VER). The quantifier representations for s-CA and s-SA are as follows: For all semantics s for which s-VER is in P, this representation immediately provides an NP upper bound for s-CA and a coNP upper bound for s-SA. In particular, we have AD / 0 -SA ∈ coNP. The s-EXSA problem is the intersection of s-CA and s-SA and thus has a DP upper bound. In particular, AD / 0 -EXSA is in DP. The possible (respectively, necessary) generalization of the acceptance problems for incomplete argumentation frameworks adds an existential (respectively, universal) quantifier over completions to the representation: The representations of s-PSA, s-NSA, s-PEXSA, and s-NEXSA only differ in that they use s-SA and s-EXSA, 365 respectively, instead of s-CA in the inner predicate.
For semantics s with s-VER ∈ P, these representations allow to infer the following bounds. For s-PSA and s-PEXSA, we get a Σ p 2 upper bound, since their quantifier representation has an existential quantifier followed by a, respectively, coNP and DP predicate. Similarly, we obtain a Π p 2 upper bound for s-NCA and s-NEXSA, since their quantifier representation has a universal quantifier followed by, respectively, an NP and a DP predicate. Since two 370 subsequent existential or two subsequent universal quantifiers can be collapsed into one such quantifier, we get an NP upper bound for s-PCA and a coNP upper bound for s-NSA. For the preferred semantics, where the inner predicate is a coNP instead of a P predicate, we obtain a This completes the proof. K

Lower Bounds
The possible and necessary variants of acceptance problems are true generalizations of their base problem. Whenever the uncertain elements in an instance are empty (i.e., A ? = / 0 and R ? = / 0), both variants collapse to their base 385 problem. Therefore, the possible and necessary variants inherit all lower complexity bounds from their base problems. In some cases, these bounds match the corresponding upper bounds from Section 3.1 and provide tight characterizations. We collect these cases in Corollary 13.
For all remaining cases, we obtain tight lower bounds by reducing from different quantified satisfiability (QSAT) problems. The QSAT problems that we require are defined in Table 2. We use X, Y , and Z to denote pairwise disjoint sets of propositional variables, and ϕ to represent a boolean formula in conjunctive normal form-i.e., a conjunction of disjunctive clauses of literals-with at most three literals per clause (3-CNF). τ X with τ X : X → {true, false} 395 denotes a truth assignment on a set X of propositional variables, and for a formula ϕ over X, ϕ[τ X ] denotes the truth value that ϕ evaluates to when applying assignment τ X to ϕ.
Each QSAT problem variant is complete for a different class in the polynomial hierarchy (last column of Table 2). We use a translation from QSAT instances to instances of acceptance problems for incomplete argumentation frameworks that is based on the standard translation by Dimopoulos and Torres [38], which we extend to incorpo-400 rate uncertainty about arguments or attacks. Our translation is given in Definition 14. It allows to create either a purely argument-incomplete argumentation framework (where R ? = / 0) or a purely attack-incomplete argumentation framework (where A ? = / 0), which allows the hardness results obtained to hold even in these special cases.
Definition 14. Let (ϕ, X,Y ) be an instance of Σ 2 SAT or Π 2 SAT, and let (ϕ, X) (with Y = / 0) be an instance of 3-SAT or 3-UNSAT. Let ϕ = i c i and c i = j α j for each clause c i in ϕ, where the α j are the literals over X ∪Y that occur in clause c i . We define a set of arguments A and a set of attacks R as follows:

An attack-incomplete argumentation framework representation of the given QSAT instance is given by
Alternatively, an argument-incomplete argumentation framework 405 representation of the given QSAT instance is given by A , We call arguments x i , y i ,x i , andȳ i literal arguments and argumentsc i clause arguments. We call two arguments a and a counterparts of each other. The unattacked arguments g i are the grounded arguments which enforce the acceptability of either x i orx i via the completion. Arguments ϕ andφ will be used as the target arguments in acceptance problems and are called target arguments. Figure 6: Visualization of either the argument-incomplete argumentation framework (not including the bottom framed part) or attack-incomplete argumentation framework (not including the top framed part) created by Definition 14 for the clauses c 1 = x 1 ∨ ¬y 1 and c 2 = y 1 ∨ ¬y 2 given in Example 15.
Both the argument-incomplete argumentation framework and the attack-incomplete argumentation framework created for (ϕ, X,Y ) by the translation of Definition 14 are displayed in Figure 6.
For an incomplete argumentation framework IAF created according to Definition 14, we associate a given truth as- true} for the argument-incomplete encoding, and G[τ S ] = {g i | x i ∈ X} for the attack-incomplete encoding.
In Lemma 16, we prove that both constructions behave similarly and can, in effect, be used interchangeably.
Lemma 16. Let (ϕ, X,Y ) be a QSAT instance, let A , R, R ? or A , A ? , R be an incomplete argumentation framework created for it according to Definition 14, and let τ X be an assignment on X. In the completion AF τ X , A τ X [τ X ] is a subset of the grounded extension and therefore a subset of all complete extensions.
Proof. Each argument g i is always unattacked and therefore clearly in the grounded extension of every completion 425 that contains g i . Each argumentx i with τ X (x i ) = true is attacked by the corresponding argument g i , which thus defends the counterpart x i ofx i , so these x i are included in the grounded extension. All argumentsx j for which τ X (x j ) = false remain unattacked, since either g i A τ X (argument-incomplete variant) or (g i ,x i ) R τ X (attackincomplete variant), and are thus included in the grounded extension. K Example 17. Recall the QSAT instance from Example 15. The set X = {x 1 } allows two assignments, τ 1 X with The completion AF τ 1 X of IAF includes the uncertain argument g 1 (in case of the argument-incomplete version) or the uncertain attack (g 1 ,x 1 ) (in case of the attack-incomplete version), while the completion AF τ 2 X excludes g 1 or (g 1 ,x 1 ), respectively. In both incarnations of AF τ 1 X , the grounded extension is for the argument-incomplete version.

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Lemma 18 shows a crucial correspondence between assignments in a QSAT instance and sets of arguments in the respective incomplete argumentation framework.
Lemma 18. Given a QSAT instance (ϕ, X,Y ) and full assignments τ X and τ Y (τ Y only if applicable). Let IAF be an incomplete AF created for (ϕ, X,Y ) following Definition 14, let AF τ X be its completion corresponding to τ X , and let A τ X [τ X , τ Y ] be the set of literal arguments and grounded arguments corresponding to the total assignment.
is admissible, complete, preferred, and stable. Figure 7: Visualization of the correspondence between truth assignments and completions plus extensions for the formula ϕ = (x 1 ∨ ¬y 1 ) ∧ (y 1 ∨ ¬y 2 ) and the assignments τ 1 X , τ 2 X , and τ Y given in Example 15.
{ϕ} is admissible, complete, preferred, and stable in AF τ X , and for Y = / 0 also grounded.
preferred, and stable in AF τ X , and for Y = / 0 also grounded.
It is easy to see from Definition 14 that E is conflictfree, since there are no attacks between literal arguments for distinct variables, ϕ, or the g i . Further, E attacks each argument in A τ X \ E . Argumentφ is attacked by ϕ ∈ E . Each literal argument from X that does not occur in E is either excluded from the completion, attacked by its counterpart, or attacked by some g i in E . Each literal argument from Y that is not in E is attacked by its counterpart in E . For each clause argumentc i , we know by assumption that 450 the corresponding clause c i in ϕ is satisfied by the total assignment, since ϕ[τ X , τ Y ] = true. Since c i is satisfied, at least one literal in c i must be satisfied. By construction of E we know that at least one literal argument corresponding to a literal in c i is in E , and by construction of IAF, this argument attacks the clause argumentc i . In total, this means that all clause arguments are attacked by E , and we proved that E is stable in AF τ X . Since E is stable, it is also preferred, complete, and admissible. For Y = / 0, the set A τ X [τ X ], which is a subset of the grounded extension by 455 Lemma 16, already attacks all clause arguments and thus defends ϕ, there is at least one clause c i in ϕ that is not satisfied by the total assignment, so none of the literals in c i is 460 satisfied. These literals correspond to literal arguments in IAF, which are the only arguments in IAF that attack the clause argumentc i . By construction of E , we know that none of these arguments are in E , so E does not attack c i and thusc i ∈ C. We now show that E is stable in AF τ X . Again, E is clearly conflict-free. All literal arguments from X that do not occur in E are again either excluded from the completion, attacked by some g i , or attacked by their counterpart in E . Each literal argument from Y that is not in E is attacked by its counterpart in E . Each clause 465 argument that is not in C is attacked by some d ∈ A τ X [τ X , τ Y ] due to the definition of C. Finally, argument ϕ is attacked by all arguments in C ⊆ E , of which there is at least one since C / 0. Since E is stable, it is also preferred, complete, and admissible. For Y = / 0, the set A τ X [τ X ], which is a subset of the grounded extension due to Lemma 16, already attacks all clause arguments in A τ X \C and thus defends all arguments in C, which in turn defendφ, so E is the grounded extension of AF τ X .
{ϕ} is admissible, complete, preferred, and stable in AF τ 1 X . The completion and this set of arguments are displayed in Figure 7a.
corresponding to the assignments τ 2 X and τ Y is {x 1 , y 1 ,ȳ 2 , g 1 } for the attack-incomplete encoding and {x 1 , y 1 ,ȳ 2 } for the argument-incomplete encoding. By Lemma 18, we know that The completion and this set of arguments are displayed in Figure 7b. 480 We have now completed our preparations for the hardness proofs of the remaining problems.
, so by Lemma 18 there exists a completion of the corresponding incomplete argumentation framework IAF where ϕ is in the grounded extension, and we have (IAF, ϕ) ∈ 485 GR-PCA.
, soφ is in the grounded extension of all completions of the corresponding incomplete argumentation framework IAF, so ϕ cannot be in the grounded extension of any completion, and we have (IAF, ϕ) GR-PCA. K Together with Observation 9, the following corollary follows immediately.

495
If (ϕ, X) 3-UNSAT, we have that (∃τ X )[ϕ[τ X ] = true], so there exists a completion of the corresponding incomplete argumentation framework IAF where ϕ is in the grounded extension, soφ cannot be in the grounded extensions of all completions, and we have (IAF,φ) GR-NCA. K Again, Observation 9 immediately gives the following corollary.
This means that ϕ cannot be a member of any admissible set in that completion-and therefore neither in a complete, stable, or preferred set-so (IAF, ϕ) s-NCA for s ∈ {AD / 0 , CP, ST, PR}. K Figure 8: PR-PSA instance created from clauses c 1 = x 1 ∨ ¬y 1 ∨ ¬z 1 and c 2 = y 1 ∨ ¬y 2 ∨ z 1 using the construction of Definition 26. A slight modification that uses c 2 = y 1 ∨ ¬y 2 instead of c 2 can be obtained by excluding the dotted attack (z 1 ,c 2 ).
There clearly can be no stable extension other than these, so To prove Σ p 3 -hardness of PR-PSA, we extend the translation of Definition 14 to incorporate the third set Z of variables that occurs in a Σ 3 SAT instance (ϕ, X,Y, Z). This translation follows the same idea as that used by Dunne 520 and Bench-Capon [43] to show Π p 2 -hardness of PR-SA. Definition 26. Let (ϕ, X,Y, Z) be an instance of Σ 3 SAT. Let IAF = A , A ? , R , R ? be an incomplete argumentation framework created for ϕ, X, and Y according to Definition 14. We extend IAF to IAF by adding literal arguments and corresponding attacks against clause arguments for all literals in Z, and by lettingφ attack itself and every literal argument from Z. Formally, Figure 8 displays a graph representation of the incomplete argumentation framework created for this instance of Σ 3 SAT: For τ X (x 1 ) = true, any assignment τ Y on {y 1 , y 2 }, and for some τ Y and a corresponding τ Z , so ϕ is skeptically preferred.

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Lemma 28. Given a Σ 3 SAT instance (ϕ, X,Y, Z) and an assignment τ X on X, let IAF be an incomplete argumentation framework created for (ϕ, X,Y, Z) following Definition 26, and let AF τ X be its completion corresponding to τ X . AF τ X has a preferred extension that does not contain ϕ if and only if ( and let τ Z be any assignment on Z. Since ϕ[τ X , τ Y , τ Z ] = false, there is a clause c k in ϕ that is not satisfied by the total assignment given by τ X , τ Y , and τ Z , which means that argument ϕ is not defended against the respective clause argumentc k by of the literal arguments is admissible. This admissible set can only be enlarged by adding appropriate clause arguments, namely to the set This set is a preferred set that does not contain ϕ, which proves the claim.

540
For the other direction, assume that Let τ Y be any assignment on Y and let τ Z be an assignment on Z that satisfies ϕ[τ X , τ Y , τ Z ] = true. We prove that every preferred extension of AF τ X contains ϕ. For the sake of contradiction, assume that there is a preferred extension E of AF τ X with ϕ E . Then, there must be an argument in E that attacks ϕ, or there must be a clause argumentc k against which ϕ is not defended by E . Since the only arguments that attack ϕ are the clause arguments, and since E must be conflict-free, the first condition is a 545 special case of the second and we may assume that there is a clause argumentc k that is unattacked by E . We know the following about E : • We can again apply Lemma 16 and know that A τ X [τ X ] is a subset of the grounded extension and therefore contained in every preferred extension, in particular, • It is clear that every preferred extension of AF τ X contains exactly one of the arguments y i orȳ i for each variable • We know that E cannot contain any arguments representing Z variables, since these would require ϕ ∈ E to defend them againstφ.
We can summarize that E must consist of A τ X [τ X , τ Y ] (for some τ Y ) and at least one clause argument that is unattacked by , which means 555 that, for the fixed given assignment τ X on X and for every assignment τ Y on Y , we can find an assignment τ Z on Z such that A τ X [τ X , τ Y , τ Z ] attacks all clause arguments. E cannot defend clause arguments against the attacks by Z arguments, so E cannot contain any clause arguments. But since A τ X [τ X , τ Y , τ Z ] attacks all clause arguments, it defends ϕ, which makes A τ X [τ X , τ Y , τ Z ] ∪ {ϕ} a stable extension, contradicting the assumption that E was preferred. K We are now ready to prove Σ p 3 -hardness of PR-PSA.
Let IAF be the incomplete argumentation framework corresponding to (ϕ, X,Y, Z), let τ X be any assignment on X, and let AF τ X be the completion of IAF representing τ X . By Lemma 28, we know that AF τ X has a preferred extension that does not contain ϕ, so (IAF, ϕ) PR-PSA. K We turn now to the problems AD / 0 -SA, AD / 0 -PSA, and AD / 0 -PEXSA, for which the generic translation of 570 Definition 14 must be adapted, too. All completions of the incomplete argumentation frameworks generated by the translation of Definition 14 have admissible sets consisting of only grounded arguments g i and certain literal arguments, so the target arguments ϕ andφ will never be skeptically accepted with respect to the AD / 0 semantics. Thus, for the problems AD / 0 -SA, AD / 0 -PSA, and AD / 0 -PEXSA, we use the adapted translation given in Definition 30, in whichφ additionally attacks all literal arguments and grounded arguments. This does not impair admissible sets 575 that contain ϕ, but has the effect that argumentφ is a member of all nonempty admissible sets unless formula ϕ is satisfiable.  Lemma 32. Given a QSAT instance (ϕ, X,Y ) and an assignment τ X on X. Let IAF be an incomplete argumentation framework created for (ϕ, X,Y ) following Definition 30, and let AF τ X be its completion corresponding to τ X . 585 1. For any assignment τ Y on Y , let A τ X [τ X , τ Y ] be the set of literal arguments and grounded arguments corresponding to the total assignment. If 2. Let C be a set of arguments containingφ and any number of-but at least one-clause argumentc i . C is admissible in AF τ X .
3. If ϕ[τ X , τ Y ] = false for all assignments τ Y on Y , then the sets C as defined in the previous Part 2 are the only 590 nonempty admissible sets in AF τ X .
Proof. For Part 1, the proof of Lemma 18 applies here as well. The only difference here are the additional attacks by argumentφ, which are defended by ϕ. Now assume that ϕ[τ X , τ Y ] = false (Part 2). Let C be a set containingφ and any positive number of clause argumentsc i . C is clearly conflict-free in all completions of IAF. Also, C has (conditionally) definite attacks against 595 all arguments in A ∪ A ? \ C:φ has (conditionally) definite attacks against all grounded arguments g i and against all literal arguments, and each clause argumentc i has a definite attack against ϕ. Thus C is necessarily stable in IAF and, in particular, admissible in AF τ X .
Finally, assume that ϕ[τ X , τ Y ] = false for the fixed assignment τ X on X and for all assignments τ Y on Y (Part 3). We show that the sets C are the only nonempty admissible sets in AF τ X . For the sake of contradiction, assume that 600 there is a nonempty admissible set A in AF τ X that is not one of the sets C.
• Ifφ ∈ A, since A is assumed to be conflict-free, then the only possibility is A = {φ}. However, {φ} cannot defend itself against ϕ and is thus not admissible.
• Ifφ A, butc i ∈ A for some clause argumentc i , then the literal arguments corresponding to the negations of the literals in clause c i must be in A, too, to defendc i against its attackers. These, in turn, must be defended 605 against the attack byφ, which can only be achieved by having ϕ ∈ A. However, sincec i attacks ϕ and A is conflict-free, this is a contradiction. So, we cannot havec i ∈ A whenφ A.
• If ϕ ∈ A, then A must contain the literal arguments that defend ϕ against all clause arguments. However, such a set of literal arguments would correspond to a satisfying assignment for the formula ϕ, which cannot exist for the fixed assignment τ X , due to our assumption that ϕ[τ X , τ Y ] = false for all assignments τ Y on Y .

610
• If ϕ A, the only remaining possibility is that A consists of only literal arguments and/or grounded arguments g i . But this set cannot defend itself againstφ and is thus not admissible.
This concludes the proof. Proof. We reduce , so by Lemma 32, there is a completion AF τ X of the corresponding incomplete argumentation framework IAF where every nonempty admissible set containsφ, so (IAF,φ) ∈ AD / 0 -PSA and , so for all completions of the corresponding incomplete argumentation framework IAF, there is some τ Y such that the set A τ X [τ X , τ Y ] ∪ {ϕ} 620 is admissible, and thus, not every nonempty admissible set in AF τ X containsφ. Therefore, (IAF,φ) AD / 0 -PSA and (IAF,φ) AD / 0 -PEXSA. If we fix X = / 0, then the Σ 2 SAT instance (ϕ, / 0,Y ) corresponds to a 3-UNSAT instance (ϕ,Y ). As a special case of the previous proof, we obtain a reduction from 3-UNSAT to AD / 0 -SA, which directly provides coNP-hardness of AD / 0 -SA. Lemma 35 pinpoints the behavior of the argumentation frameworks generated by Definition 34.

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Lemma 35. Given a QSAT instance (ϕ, X,Y ) and an assignment τ X on X. Let IAF be an incomplete argumentation framework created for (ϕ, X,Y ) following Definition 34, and let AF τ X be its completion corresponding to τ X .
1. For any assignment τ Y on Y , let A τ X [τ X , τ Y ] be the set of literal arguments and grounded arguments corresponding to the total assignment. AF τ X (Part 2). Let A ⊆ A τ X be a nonempty admissible set.
• A cannot contain the self-attacking argumentφ.
• Ifc i ∈ A for some clause argumentc i , then the literal arguments corresponding to the negations of the literals in clause c i must be in A, too, to defendc i against its attackers. These, in turn, must be defended against the attack byφ, which can only be achieved by having ϕ ∈ A. However, sincec i attacks ϕ and A is conflict-free, this is a 645 contradiction. So, we cannot havec i ∈ A.
• If ϕ ∈ A, then A must contain the literal arguments that defend ϕ against all clause arguments. Such a set of literal arguments corresponds to a satisfying assignment for the formula ϕ, so we have = false for all assignments τ Y , then such a set cannot exist for our fixed assignment τ X on X.

650
• If ϕ A, the only remaining possibility is that A consists of only literal arguments and/or grounded arguments g i . But this set cannot defend itself againstφ and is thus not admissible.
We showed that only sets of the form A τ X [τ X , τ Y ] ∪ {ϕ} can be nonempty admissible sets in AF τ X . Since every stable extension of AF τ X is also a nonempty admissible set, we also know the same for the stable semantics. This concludes the proof.
, so by Lemma 35, there is a completion AF τ X of the corresponding incomplete argumentation framework IAF that has no nonempty admissible set and no stable extension, so (IAF, ϕ) AD / 0 -NEXSA and (IAF, ϕ) ST-NEXSA. K We have covered the complexity of all possible and necessary generalizations of acceptance problems in incomplete argumentation frameworks. What is left to classify is the new problem AD / 0 -EXSA for standard argumentation 665 frameworks, which, just like its sibling ST-EXSA, is hard for the class DP of the boolean hierarchy, which lies between the first and second level of the polynomial hierarchy. To prove DP-hardness of AD / 0 -EXSA, we reduce from the canonical DP-hard problem 3-SAT-UNSAT, which is the intersection of 3-SAT and 3-UNSAT. An instance (ϕ 1 , X 1 , ϕ 2 , X 2 ) consists of a 3-CNF formula ϕ 1 on a set X 1 of variables and a 3-CNF formula ϕ 2 on a set X 2 of variables, and the question is whether (ϕ 1 , X 1 ) ∈ 3-SAT and (ϕ 2 , X 2 ) ∈ 3-UNSAT. Our reduction bears some similarities 670 to the one used by Dunne and Wooldridge [44] in their proof that ST-EXSA is DP-hard. The increased difficulty of our reduction is due to avoiding unwanted AD / 0 sets that do not include the target argument, which is not accomplished by (and not necessary for) the reduction by Dunne and Wooldridge.
Lemma 39. Let (ϕ 1 , X 1 , ϕ 2 , X 2 ) be an instance of 3-SAT-UNSAT, and let A , R be the argumentation framework created for it by Definition 37.
• Firstly, the self-attacking argumentsφ 1 and ψ can clearly never be members of admissible sets.

• The literal arguments {x 2
i ,x 2 i | x 2 i ∈ X 2 } need to be defended against the attack byφ 2 in order to appear in 695 admissible sets-however, the only argument that could provide this defending attack is ϕ 2 , and we know that no admissible set can defend ϕ 2 against all clause argumentsc 2 i , since τ X 2 is unsatisfiable. Therefore, neither ϕ 2 nor any literal argument in {x 2 i ,x 2 i | x 2 i ∈ X 2 } can be in an admissible set.
• A set of arguments containing a clause argumentc 2 i must also defendc 2 i against the attacks by literal arguments. The only candidates for this are other literal arguments in {x 2 i ,x 2 i | x 2 i ∈ X 2 } orφ 2 , and since we ruled out literal 700 arguments in the previous step, the only option is that the admissible set must containφ 2 .
• Literal arguments in {x 1 i ,x 1 i | x 1 i ∈ X 1 } must be defended against ψ in order to be members of admissible sets, which can only be done byφ 2 .
• Again, a set of arguments containing a clause argumentc 1 i must defendc 1 i against the attacks by literal arguments. The only candidates for this are other literal arguments in For the latter, we 705 know that ψ can never occur in admissible sets, and for the former, we determined that this also requiresφ 2 to be in the admissible set.
• A set of arguments containing argument ϕ 1 must defend it against all clause argumentsc 1 i , which can only be done by literal arguments in {x 1 i ,x 1 i | x 1 i ∈ X 1 }, which in turn requireφ 2 to be in the admissible set, too. This completes the proof. Proof. We reduce from 3-SAT-UNSAT. Let (ϕ 1 , X 1 , ϕ 2 , X 2 ) be a 3-SAT-UNSAT instance.

Preservation of Extensions
In this section, we describe atomic changes to completions (adding or removing a single uncertain argument or 720 attack) that are guaranteed to be redundant from the point of view of preserving a given s extension of the completion. As we will show later in Section 6-7, these observations will prove to be crucial in designing efficient algorithms for the acceptance problems with complexity beyond NP (assuming, as is widely believed, that the polynomial hierarchy does not collapse to NP). Related to the observations presented in this section, there is a substantial amount of research on the dynamic aspects of AFs (see [39] for an overview), in particular on the impact of the addition and removal of 725 arguments and attacks on the semantics, which we overview in Section 8. The preservation of stable and preferred extensions when removing and adding attacks was first studied by Rienstra et al. [77] (in terms of labelings); our results for these semantics naturally coincide, and we further extend the results to (nonempty) admissible sets. For an incomplete argumentation framework IAF, a completion AF * = A * , R * of IAF, a semantics s, and an extension E ∈ s(AF * ), we denote IN We begin by considering adding an argument a ∈ A ? \A * to the completion. If there is a definite attack (b, a) ∈ R with b ∈ IN(E ), then any attacks by a against arguments in E would be defended by E in the modified completion, which ensures that E stays an extension, both under nonempty admissible and stable semantics. Proof. Clearly, the set E ⊆ A * is still conflict-free in AF .
• If s = AD / 0 , we know that every attack (c, d) ∈ R * against E with c ∈ A * and d ∈ E is defended by E since E ∈ AD / 0 (AF * ), and the only new attacks against E that might have been added in AF are attacks by a, which 740 are defended by b ∈ E by assumption. Therefore, E ∈ AD / 0 (AF ).
• If s = ST, since E ∈ ST(AF * ), for each c ∈ A * \ E there exists an attack (d, c) ∈ R * with d ∈ E . Also, by assumption (b, a) ∈ R and b ∈ E , and thus E ∈ ST(AF ).
This completes the proof. K We continue by considering removing an argument a ∈ A ? ∩ A * from the completion. In particular, removing 745 arguments (along with incident attacks) that are members of OUT(E ) or of UNDEC(E ) has no effect on the current extension E . Proof. Again, the set E ⊆ A * is still conflict-free in AF .

750
• If s = AD / 0 , since a E and E ∈ AD / 0 (AF * ), E still defends all its members in AF and thus E ∈ AD / 0 (AF ).
• If s = ST, since E ∈ ST(AF * ), for each c ∈ A * \ E there exists an attack (d, c) ∈ R * with d ∈ E . Since a E , these attacks still exist in AF . Thus E ∈ ST(AF ).
This completes the proof. K We continue with an illustrative example on an application of Proposition 42. Next, we identify redundant attacks. Removing an uncertain attack (b, a) ∈ R ? ∩ R * with the source b ∈ OUT(E ) has no effect on the extension E , and neither does removing an attack with the target a ∈ IN(E ). Further, the fact that E is admissible is not changed by removing attacks between UNDEC(E ) arguments or from UNDEC(E ) to OUT(E ) arguments (when E is stable, UNDEC(E ) = / 0).

Proof. Clearly, the set E ⊆
, each attack (a , c) ∈ R * where c ∈ E and a ∈ A * is countered by an attack (b , a ) ∈ R * with b ∈ E . This holds in AF , since by assumption b E , and hence the counterattack is not 770 removed. Thus E ∈ AD / 0 (AF ).
• s = ST: Since E ∈ ST(AF * ), for each a ∈ A * \ E there is an attack (b , a ) ∈ R * with b ∈ E . This holds in AF , since by assumption b E . Thus E ∈ ST(AF ).
Suppose a OUT(E ), that is, there is no attack (e, a) ∈ R * with e ∈ E .
• s = AD / 0 : Since E ∈ AD / 0 (AF * ), each attack (a , c) ∈ R * where c ∈ E and a ∈ A * is countered by an attack If for some attack we would have a = a, this would be a contradiction with the assumption that there is no attack (e, a) ∈ R * with e ∈ E . That is, the counterattacks are not removed, and hence, E ∈ AD / 0 (AF ). 23 • s = ST: Since E ∈ ST(AF * ), for each a ∈ A * \ E there is an attack (b , a ) ∈ R * with b ∈ E . Again, if we would have a = a, this would be a contradiction with the assumption that there is no attack (e, a) ∈ R * with e ∈ E . Thus E ∈ ST(AF ).
This completes the proof. K Finally, we consider adding uncertain attacks to the completion. Adding an uncertain attack (b, a) ∈ R ? \ R * is only relevant if the source b OUT(E ) and if the target a ∈ IN(E ), assuming b, a ∈ A * . This is due to the fact that attacks by OUT(E ) arguments are defended and thus inactivated by E , and that adding any further attacks against 785 OUT(E ) arguments has no effect on E . The admissibility of E is not changed by adding attacks between UNDEC(E ) arguments or from IN(E ) arguments to UNDEC(E ) arguments. Note that we assume b, a ∈ A * here, since if b A * or a A * , we would first need to add the argument in order to add the attack, and we only consider atomic changes.
Proof. The set E ⊆ A * is still conflict-free in AF . Suppose b ∈ OUT(E ), that is, there exists an attack (e, b) ∈ R * with e ∈ E .
• s = AD / 0 : Since E ∈ AD / 0 (AF * ), each attack (a , c) ∈ R * where c ∈ E and a ∈ A * is countered by an attack (b , a ) ∈ R * with b ∈ E . If a ∈ E , by assumption there exists an attack (e, b) ∈ R * with e ∈ E , so the attack 795 (b, a) is countered as well. Hence, E ∈ AD / 0 (AF ).
• s = AD / 0 : Since E ∈ AD / 0 (AF * ), each attack (a , c) ∈ R * where c ∈ E and a ∈ A * is countered by an attack 800 (b , a ) ∈ R * with b ∈ E . Since by assumption a E , adding the attack (b, a) has no effect on the admissibility of E . Hence, E ∈ AD / 0 (AF ).
• s = ST: Since E ∈ ST(AF * ), for each a ∈ A * \ E there is an attack (b , a ) ∈ R * with b ∈ E . This holds in AF , and hence E ∈ ST(AF ).
This completes the proof. K

805
We continue with an illustrative example on an application of Proposition 45. In addition to admissible and stable semantics, we make use of similar results for preferred semantics, under which the picture is slightly different as the conditions on undecided arguments are more restrictive when removing arguments and changing the attack structure. Consider first adding an argument in A ? \ A * . If it is definitely attacked by an argument in the preferred extension, adding it will not affect the extension being preferred, exactly as in 815 Proposition 41 for admissible and stable semantics. Proof. Since now E ∈ AD(F * ), due to Proposition 41, E ∈ AD(AF ), and it remains to show that there is no 820 E ∈ AD(AF ) with E ⊃ E . Suppose on the contrary that such a set E exists. Since now a IN(E ) as otherwise E would not be conflict-free, by Proposition 42, E ∈ AD(AF * ), contradicting the assumption that E ∈ PR(AF * ). This completes the proof. K If E is a preferred extension, removing an argument a ∈ A ? ∩ A * with a ∈ UNDEC(E ) does not necessarily preserve E , in contrast to Proposition 42. 1 On the other hand, the observation in Proposition 42 regarding arguments 825 in OUT(E ) does still hold for preferred semantics.
Proof. Since E ∈ AD(AF * ), via Proposition 42, E ∈ AD(AF ). It remains to show that there is no E ∈ AD(AF ) with E ⊃ E . Suppose on the contrary that such a set E exists. Since a ∈ OUT(E ), there exists b ∈ E with (b, a) ∈ R * . The 830 conditions of Proposition 41 now hold for the incomplete argumentation framework IAF = A * , {a}, R * , R ? | A * ∪{a} and the completion AF . Since E ∈ AD(AF ), this results in E ∈ AD(AF * ), which contradicts our assumption E ∈ PR(AF * ). This completes the proof. K Regarding removing and adding attacks, the results by Rienstra et al. [77] for preferred semantics are directly applicable in our context. The key difference to admissible and stable semantics is that removing and adding attacks 835 between undecided arguments and adding attacks from the extension to undecided arguments may result in a larger admissible extension. We summarize these results in the following propositions.  Regarding atomic changes to the set of attacks, Table 3 summarizes the allowed changes.

Encoding into SAT
We begin the algorithmic part of this work by providing encodings in boolean satisfiability (SAT) for the problem variants with first-level complexity. The encodings extend the standard encodings for argumentation semantics [20] to incomplete AFs, essentially conditioning relevant parts of the formulas on the existence of an argument or an attack.

855
Given an incomplete argumentation framework IAF and a semantics s, we will present propositional formulas ϕ s (IAF) such that completions of the incomplete AF and s extensions of the completion are in a one-to-one correspondence with models of ϕ s (IAF). These encodings will then form the basis for the SAT-based algorithms deciding the various Table 3: Overview of the atomic changes on the set of attacks that do not affect an extension E of a completion, based on Propositions 44 and 45 for AD / 0 and ST and on Propositions 49 and 50 for PR. Here, the symbol means that the change has no effect under all semantics AD / 0 ,ST, and PR, () means that the change has no effect under semantics AD / 0 or ST but may have an effect under PR, and means that the change may have an effect under all semantics AD / 0 ,ST, and PR.
forms of acceptance in incomplete argumentation frameworks, including the NP-and coNP-complete problems which are solved using a single SAT solver call, as well as problems beyond NP for which we develop counterexample-guided 860 approaches in Section 6. Consider an input IAF = A , A ? , R, R ? . We use variables x a and y a for all a ∈ A ∪ A ? and r a,b for all (a, b) ∈ R ∪ R ? , with the following interpretations: where AF * = A * , R * is a completion of IAF defined by the y * and r * , * variables. In other words, the x * variables encode an extension of the completion encoded by the y * and r * , * variables. Now, the formula encodes conflict-free sets in a completion of IAF, expressing that if two arguments and an attack between them are present in the completion, one cannot include both of the arguments in the extension. To express IAF semantics, we use additional variables z a for each a ∈ A ∪ A ? , binding their value to the rest of the variables via equivalences Now z a is assigned to true if and only if a is attacked by the extension encoded by the x * variables in the completion encoded by the y * and r * , * variables. Using this and the encoding for conflict-free sets, encodes admissible sets by expressing that, for each argument and for each attack on the argument, if the argument is included in the extension of the completion, the attacker must be attacked by the extension. That is, the extension defends itself. Nonempty admissible semantics can now be encoded as The encoding for complete semantics is stating that a complete extension is admissible, and for each argument, if the argument is defended by the extension of the completion (i.e., each attacker of the argument is attacked by the extension if the attack and the incident arguments are present in the completion), it is also included in the extension. We express stable semantics with which states that a stable extension is conflict-free, and for each argument, if the argument exists in the completion and is not included in the extension, it must be attacked by the extension. Finally, for the grounded semantics we adapt the encoding due to Niskanen et al. [73], using additionally variables l a n for each a ∈ A ∪ A ? and for each integer n ∈ {1, . . . , |A ∪ A ? |/2 }, with the interpretation that l a n = trueif and only if a ∈ F n AF * ( / 0). The upper bound on n is explained by the fact that for AFs with |A | arguments, the maximum number of iterations of the characteristic function to the empty set until the fixpoint is |A |/2 . This upper bound is realized by an AF which consists of a directed path of arguments. The constraints ϕ a n (IAF) = l a n ↔ y a ∧ (b,a)∈R∪R ?
simulate the application of the characteristic function on a completion of the input incomplete AF, and finally x a ↔ |A ∪A ? |/2 n=1 l a n ∧ |A ∪A ? |/2 n=1 ϕ a n (IAF) requires that an argument is included if and only if it is included in F n AF * ( / 0) for some n, hence capturing the grounded extension of the completion AF * .

870
Finally, to encode valid completions, we observe that y a is always true for all definite arguments, and likewise r a,b is true for all definite attacks. 2 Further, if an uncertain argument is not included in the completion, it cannot be accepted, and all incident attacks are not included either. This information is encoded as The latter conjunction is in fact redundant, given that the ϕ s (IAF) formulas are conditioned on the existence of an attack and incident arguments. However, as binarý clauses, these can be beneficial for a SAT solver during search.
The key properties of the encodings are summarized in the following proposition.

880
In this section we present a SAT-based approach for argument acceptance in incomplete argumentation frameworks. For problems on the first level of the polynomial hierarchy, a single call to a SAT solver suffices using the encodings presented in Section 5. However, for problems beyond NP, there presumably exists no polynomial-time reduction to SAT. Hence, we develop algorithms based on counterexample-guided abstraction refinement (CEGAR), originally proposed in the context of model checking [30,31]. In SAT-based CEGAR, an abstraction of the problem 885 at hand-an overapproximation of the original problem that is in NP-is used to query a SAT solver for a candidate solution to the problem. Then, a SAT solver is again used to check whether the candidate solution is an actual solution to the problem by asking it for a counterexample. If one is found, the abstraction is refined by ruling out the counterexample. Then the algorithm proceeds iteratively until the solution of the original problem is found.
For the rest of this section, let IAF = A , A ? , R, R ? be an incomplete AF and a ∈ A be the query argument.

Credulous Acceptance
We start with possible and necessary credulous acceptance. First of all, we note that due to Observation 5, possible and necessary credulous acceptance coincides for semantics AD, AD / 0 , CP, and PR. Hence, to cover all semantics and both variants of credulous acceptance, it suffices to consider the problems s-PCA and s-NCA for s ∈ {AD, ST, GR}.
Possible credulous acceptance. Recall that the possible variant s-PCA is NP-complete for all s ∈ {AD, ST, GR}, and due to Proposition 51, is decided by satisfying the input formula Note that a satisfying truth assignment also yields a completion of the input IAF and an s extension of the completion 895 containing the query argument.
Necessary credulous acceptance. The necessary variant s-NCA is Π p 2 -complete for s ∈ {AD, ST} and coNP-complete for s = GR. Due to Observations 6 and 7, GR-NCA = GR-NSA = CP-NSA, so necessary credulous acceptance under grounded semantics can be decided by using the procedure for necessary skeptical acceptance under complete semantics, which we cover in Section 6.2.

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We are left with the problems AD-NCA and ST-NCA, for which we develop a CEGAR procedure, presented as pseudocode in Algorithm 1. Based on Π p 2 -completeness, the idea is to look for a counterexample, that is, a completion (sat, τ) ← SAT(ϕ) 14: if sat = false break 15: AF * ← EXTRACT(τ) 16: ϕ ← ϕ∧ REFINE(IAF, AF * ) 17: if SAT(ϕ \ ϕ ST (IAF)) = true return reject 18: else return accept of IAF which has no extension containing the query argument a. We start by solving an abstraction of the problem, given in this case by the formula (line 1) If ABSTRACTION s (IAF, ¬x a ) is satisfiable, we get a truth assignment τ from which we can extract a completion AF * = EXTRACT(τ) (lines 3 and 5) of the input IAF with EXTRACT(τ) = A * , R * , where along with an s extension E ∈ s(AF * ) that does not include a. We still do not know whether there is some other s extension containing a, so we check whether a is credulously accepted under s in AF * , which can be done with a single SAT call on the formula (line 6) encodes the completion AF * currently under consideration. Adding these unit clauses essentially reduces the encoding ϕ s (IAF) for incomplete AFs to the respective encoding for standard AFs. If CHECK s (IAF, AF * , a) is unsatisfiable, there is no extension containing a in AF * , so we successfully obtained a counterexample to necessary credulous acceptance of a, and can reject (line 7). If, however, the formula CHECK s (IAF, AF * , a) is satisfiable, our counterexample is not valid for the problem instance at hand and we need to refine the abstraction by adding the clause to the original SAT instance (line 8), which tells the solver that we need to find another completion of IAF, and iterate until we either find a counterexample to necessary credulous acceptance, or reach unsatisfiability. If ABSTRACTION s (IAF, ¬x a ) is unsatisfiable, Proposition 51 implies that a is necessarily skeptically accepted. For s = AD, this implies that a is also necessarily credulously accepted, so we accept (line 10). For s = ST, it is not necessarily the case that a is necessarily credulously accepted, since it might be the case that there are completions 905 which have no stable extension (recall that an argument is by definition skeptically accepted if the AF has no extension). In other words, the abstraction initialized with ABSTRACTION ST (IAF, ¬x a ) only considers those AFs that have a stable extension not containing a.
To resolve this issue, we iteratively call a SAT solver with the input formula (ϕ \ {¬x a }) ∪ {x a } consisting of ϕ ? (IAF)∧ϕ ST (IAF)∧x a and additional refinement clauses, refining the formula similarly via REFINE(IAF, AF * ) using 910 the completion AF * extracted from a satisfying assignment, until reaching unsatifiability (lines [11][12][13][14][15][16]. Essentially, this procedure rules out all completions that possess a stable extension. Finally, we check using ϕ \ ϕ ST (IAF) whether there still are completions; if so, we reject the query, since these completions have no stable extension, and otherwise we accept it (lines 17-18).

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We continue by presenting procedures for deciding (both variants of) possible and necessary skeptical acceptance. As explained in Section 6.1, due to Observations 6 and 7, it suffices to consider CP-NSA in order to decide necessary (credulous and skeptical) acceptance under grounded semantics. In addition, since possible (credulous and skeptical) acceptance under grounded semantics coincides with CP-PSA, CP-PSA can be solved using the procedure for GR-PCA described in Section 6.1. Finally, s-PSA and s-PEXSA coincide for semantics under which existence of an 920 extension is guaranteed. That is, it suffices to consider s-PEXSA, s-PSA, s-NSA, and s-NEXSA for s ∈ {AD / 0 , ST}, PR-PSA, CP-NSA, and PR-NSA to cover all semantics and all variants of skeptical acceptance.
Necessary skeptical acceptance. Recall that s-NSA is coNP-complete under s ∈ {AD / 0 , ST, CP} and via Proposition 51 can be decided by checking whether the formula is unsatisfiable. Note that if this formula is satisfiable, we obtain a truth assignment from which we can extract a completion of the input IAF and an s extension of the completion not containing the query argument.
Necessary existence and skeptical acceptance. Possible existence and skeptical acceptance. We continue with s-PEXSA under s ∈ {AD / 0 , ST}, which is Σ p 2complete. This hints at the idea of looking for a witness completion in which the query argument is skeptically 930 accepted, i.e., included in every s extension. The CEGAR procedure is presented in pseudocode as Algorithm 2.
In this algorithm, an abstraction is now formed via the formula (line 1) If ABSTRACTION s (IAF, x a ) is unsatisfiable, we can reject the query (line 4) since then by Proposition 51 a is not even possibly credulously accepted. If ABSTRACTION s (IAF, x a ) is satisfiable, we extract the current completion AF * = EXTRACT(τ) = A * , R * from the satisfying truth assignment τ (line 5), where A * and R * are as defined on page 29. We proceed by checking whether a is skeptically accepted under s in AF * . This is accomplished via a SAT call with the input formula (line 6) where COMPLETION(IAF, AF * ) is defined on page 29.
If CHECK s (IAF, AF * , ¬x a ) is unsatisfiable, this proves that we have successfully found a witness completion for skeptical acceptance of a, and can accept (line 7). If the formula CHECK s (IAF, AF * , ¬x a ) is satisfiable, we have shown that there exists an extension E ∈ s(AF * ) not containing the query argument, so we refine the abstraction by adding 935 the clause REFINE(IAF, AF * ) (line 8), as defined on page 29, excluding the current completion. Then, we continue iteratively until the abstraction becomes unsatisfiable or we find a witnessing completion to possible existence and skeptical acceptance.
Possible skeptical acceptance. For s-PSA under s ∈ {AD / 0 , ST}, we can use the same procedure as for s-PEXSA, with the following exception. If the abstraction becomes unsatisfiable, we rule out all completions that have an s 940 extension, and check whether there still are completions without an s extension, as described in the procedure for ST-NCA in Section 6.1. If there are completions left, this implies possible skeptical acceptance, and hence we accept, otherwise we reject.
Necessary skeptical acceptance under preferred semantics. Consider now the PR-NSA problem, which is Π p 2complete, implying that we are again looking for a counterexample. The source of hardness is the Π p 2 -completeness if sat = false return accept 8: ϕ ← ϕ∧ REFINE(IAF, AF * ) of standard skeptical acceptance under preferred semantics [43]. Motivated by this, we develop a CEGAR procedure similar to the SAT-based AF solver CEGARTIX [46]. We describe the algorithm briefly due to the similarities of the approaches. Our abstraction is based on the complete semantics via If the abstraction is unsatisfiable, we accept the argument. A satisfying truth assignment τ yields us a completion AF * = EXTRACT(τ) = A * , R * and a complete extension not containing a. Then, we iteratively subset-945 maximize the complete extension under the constraint that a is still not included, and finally check whether including a yields a larger complete extension. If not, we have successfully obtained a counterexample, that is, a completion where a is not skeptically accepted under preferred semantics, and we reject. If yes, we refine the abstraction via REFINE(IAF, AF * ) ∨ b∈A * \E x b , where E is the subset-maximal complete extension we found, ruling out all subsets of it for that particular completion, and continue iteratively.

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Possible skeptical acceptance under preferred semantics. The only problem complete for a class in the third level of the polynomial hierarchy is the Σ p 3 -complete PR-PSA. Reflecting the computational complexity, we are looking for a witness, that is, a completion where a is skeptically accepted under preferred semantics. For the CEGAR algorithm, we initialize the abstraction using the complete semantics via If the abstraction is unsatisfiable, we reject the query argument. A satisfying truth assignment corresponds to a completion AF * and a complete extension containing the query a. Then, our goal is to check whether a is skeptically accepted under preferred semantics. This is accomplished by first checking whether a is skeptically accepted in the current completion under complete semantics via If this formula is unsatisfiable, we can safely accept, as skeptical acceptance under complete semantics implies skeptical acceptance under preferred semantics. If it is satisfiable, we get a complete extension not containing a, and enter the same subset-maximization procedure as in the algorithm for PR-NSA. Finally, we check whether including a into the subset-maximal complete extension not containing a yields a larger complete extension. If this is not possible, we refine the abstraction via REFINE(IAF, AF * ) and continue iteratively, since a is not skeptically accepted under 955 preferred semantics in AF * , which is witnessed by the preferred extension not containing a. If this is possible, we add the clause COMPLETION(IAF, AF * ) → b∈A * \E x b , where E is the maximal complete extension, ruling out all subsets for that completion, and continue.
Finally, we remark that although our algorithm for AD / 0 -PEXSA and ST-PEXSA covers the second-level incomplete variant, it also allows for solving the DP-complete problems AD / 0 -EXSA and ST-EXSA. In this case, the 960 CEGAR algorithm is bound to work within the resource limits of DP, i.e., terminating after one iteration. This is due to the fact that if we find a counterexample to the solution of the abstraction, the refinement clause is empty because there are no incomplete elements, causing immediate termination.

Strong Refinements
Recall the CEGAR algorithms for s-NCA (Algorithm 1) and s-PEXSA (Algorithm 2) for s ∈ {AD / 0 , ST}. Before 965 refining the abstraction, in both algorithms we have obtained a counterexample extension E , which either contains (for s-NCA) or does not contain (for s-PEXSA) the query argument. In both cases, however, we would ideally like to exclude all completions which still possess the counterexample E as an s extension, since these completions would only cause more undesired iterations in the CEGAR algorithm. Due to Propositions 41-45 in Section 4, the following strong refinement is also valid, and excludes certain other completions which admit the counterexample extension: Thus in Algorithms 1 and 2 we may replace REFINE(IAF, AF * ) with REFINE s (IAF, AF * , E ) while maintaining correctness. Since in the procedure described for PR-NSA on page 30 the counterexample extension E is also an admissible set containing the query, here we may replace REFINE(IAF, The situation is different in the procedure for PR-PSA described on page 31. Here we refine the abstraction via 975 adding the clause REFINE(IAF, AF * ) if we find a preferred extension not containing the query. Thus we instead make use of Propositions 47-50 yielding the following strong refinement: We will show in Section 7 that strong refinements are crucial for solving these problems in practice.

Experiments
We continue by an overview of results from an empirical evaluation of the scalability of the approaches to accep-980 tance problems in incomplete AFs described in Section 6. Our implementation, TAEYDENNAE, 3 uses Glucose 4.1 [4] as the underlying SAT solver, and is available in open source: https://bitbucket.org/andreasniskanen/taeydennae .
In the implementation, we employ fully incremental SAT solving, making use of the assumptions interface of Glucose.

985
We generated incomplete AFs based on the benchmarks used in the 2nd International Competition on Computational Models of Argumentation (ICCMA 2017) [53] as follows. For each AF, we select a query argument uniformly at random from the set of arguments. For each probability p ∈ {0.05, 0.1, 0.15, 0.2}, we generated three incomplete AFs: one where each argument (except for the query) is uncertain with probability p, one where each attack is uncertain with probability p, and one where each argument and attack is uncertain with probability p. We used the ICCMA

Results
For the NP-complete problems AD-PCA (= CP-PCA = PR-PCA) and ST-PCA, and the coNP-complete problems AD / 0 -NSA, ST-NSA, and CP-NSA (= GR-NSA = GR-NCA), the mean run times (with timeouts included as the timeout limit of 900 seconds) are visualized in Figure 11 for different values of p and argument-incompleteness or 1000 attack-incompleteness. Interestingly, the empirical hardness of the instances does not increase as incompleteness is increased; in fact, for problems other than CP-NSA, we are able to solve the corresponding incomplete instance sets faster on average than for p = 0, which is exactly the original ICCMA benchmark set (i.e., "normal" acceptance problems in standard AFs), especially when introducing attack-incompleteness. We hypothesize this to be due to the fact that by making certain elements uncertain, there are more possibilities to have a satisfiable SAT instance-this is witnessed by the fact that as p is increased, the number of accept answers increases considerably for PCA problems (e.g., from 35% of solved instances with p = 0 to 71% with p = 0.2 for AD-PCA), and decreases for NSA problems (e.g., from 56% of solved instances with p = 0 to 21% with p = 0.2 for ST-NSA). The only exception is CP-NSA which is, for our approach and for this set of instances, surprisingly fast to solve, with mean running times around 10 seconds for each choice of parameters despite coNP-completeness of the task for incomplete AFs (as opposed to polynomial-time computability for standard AFs). This can be explained by the fact that the number of accept answers, i.e., unsatisfiable SAT instances, is very low (about between 6% and 8%) for all choices of p.
The Σ p 2 -complete problems AD / 0 -PEXSA and ST-PEXSA are solved via the CEGAR approach. The mean run times (with timeouts included as 900 seconds) are shown in Figure 12. We observe that, in contrast to the NP encodings, introducing uncertainty makes the instances significantly harder to solve. We suspect this to be due to the 1015 fact that the number of potential completions to guess and check is exponential in the number of uncertain elements. However, the strong refinements described in Section 6.3 resulting from the analysis presented in Section 4, significantly speed up the CEGAR approach. Mean run times when using the trivial refinement are considerably higher than when using the strong refinement, especially for AD / 0 -PEXSA, to the extent that the strong refinements are central in making the CEGAR approach viable for deciding acceptance in incomplete AFs. This is also evident from Figure 14 (top left and top right), which shows the running times for each instance using the trivial refinement plotted on the x-axis and for the strong refinement on the y-axis. While a major part of the instances takes almost the exact same amount of time to solve, the effect of strong refinements is witnessed by the large number of timeouts for the trivial refinement on the right-hand side of each plot for instances that are solved within the timeout limit using the strong refinement.

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The Π p 2 -complete problems AD-NCA and ST-NCA are also solved employing CEGAR, with the mean running times (with timeouts included) shown in Figure 13. Here we also observe that the empirical hardness of instances increases significantly when increasing incompleteness, both in terms of increasing p and in terms of introducing both argument-and attack-incompleteness. Again, in the worst case we need to check for an exponential number of completions with respect to the number of uncertain elements, which we hypothesize to be the cause of this behavior.

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Employing strong refinements is essential also for these problems, as further shown in Figure 14 (bottom left and bottom right).
Finally, mean run times for the Σ p 3 -complete problem PR-PSA are shown in Figure 15 (left). We observe that, similarly to the problems complete for the second level, introducing uncertainty both via increasing p and introducing both argument-and attack-incompleteness increases the mean running times. Perhaps surprisingly, the scale of the 1035 mean run times is also similar despite the gap in theoretical complexity. Clearly, also in this case employing strong refinements is crucial for solving instances of this problem efficiently, as is evident from significantly smaller run times on average, as well as the large number of timeouts shown in Figure 15 (right).

Empirical Comparison with Enumeration-based Acceptance
The SAT-based algorithms presented in this article are the first ones proposed in the context of incomplete AFs, 1040 barring direct empirical runtimes comparison with other approaches. However, in principle the acceptance problems considered in this work can also be decided with a more simple algorithmic approach which enumerates all com-pletions of an input IAF and, for each completion, calls a standard AF solver for the respective standard acceptance problem. For obtaining a baseline for an empirical scalability comparison, we implemented this enumeration approach based on the state-of-the-art AF solver µ -TOKSIA [68]. For a "possible" variant, the enumeration algorithm outputs "yes" as soon as the AF solver outputs "yes." Dually, for a "necessary" variant, the algorithm outputs "no" as soon as the AF solver outputs "no." Otherwise, it enumerates through all completions of the input IAF. If this is the case, note that, for an input IAF with a large number of uncertain elements (and thus completions), this algorithm cannot be expected to terminate in a reasonable amount of time, as the number of completions becomes simply too large.
The results of a comparison of the empirical performance of our SAT-based algorithms and the simple SAT-based 1050 enumeration approach are summarized in Table 4. For the first-level problems AD-PCA, ST-PCA, AD / 0 -NSA, ST-NSA, and CP-NSA, the direct SAT-based approach considerably outperforms the enumeration-based approach, with the enumeration-based approach exhibiting at least seven times as many timeouts as the direct one. For the Σ p 2complete problems AD / 0 -PEXSA and ST-PEXSA, the SAT-based CEGAR algorithm produces approximately five times fewer timeouts than the enumeration algorithm, and is thus again clearly the superior approach. The picture is 1055 less drastic but still clear for the Π p 2 -complete problems AD-NCA and ST-NCA, for which the enumeration-based method produces more timeouts than the direct one by a factor of 2.4 and 1.5, respectively. We hypothesize this to be due to a large ratio of "no" answers to "yes" answers. For the Σ p 3 -complete PR-PSA, the SAT-based CEGAR again clearly outperforms the enumeration algorithm which exhibits 3.7 times as many timeouts. All in all, the enumerationbased approach turned out not to be competitive with the more intricate SAT-based algorithms developed in this work. 1060

Related Work
Finally, we review works related from different angles to the work presented in this article.

Instantiation of Incomplete AFs
Various formalisms have been proposed in the literature that allow to instantiate abstract argumentation frameworks from structured data-typically consisting of literals in some formal language, inference rules that relate these literals, preference relations over different parts of the data, and possibly more auxiliary information. When there is structural uncertainty about elements in the underlying data, this may translate to structural uncertainty about arguments or attacks in the abstract argumentation framework that represents it, and thus produce incomplete argumentation frameworks. We give a few examples (for formal definitions, please refer to the original papers).
• In value-based argumentation frameworks [19], when we have two arguments a and b with associated values 1070 val(a) and val(b), and if it is known that the preference between the two values is val(b) ≥ val(a), but it is not known whether val(b) > val(a) or val(b) = val(a), then this results in an uncertain attack (a, b).
• A similar situation can arise when using the ASPIC + model [66]: Consider two rebutting arguments a and b that have two rules r a and r b as their respective top rules, with preference r b ≥ r a , but it is unknown whether r b > r a or r b = r a . This may result in the rebutting attack (a, b) being uncertain.

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• As another example, consider an ASPIC + argument a with top rule r. If there is structural uncertainty about r in the underlying knowledge base, then this can translate to a being uncertain in the resulting argumentation framework.
• The same happens when using the method of Wyner et al. [84] to instantiate argumentation frameworks: uncertainty about the existence of a rule r in the knowledge base produces an uncertain argument r in the argumentation framework representation. Wyner et al.'s method also allows uncertain attacks: Uncertainty about whether an inference rule r is strict or defeasible in the knowledge base produces an uncertain attack (c, r), where c is the abstract argument that represents the negation of r's head literal.

IAFs and Other Generalizations of AFs
First, while this work studies "possible" and "necessary" variants of the acceptance problem in IAFs, Baumeister et al. [16,17,18] have studied "possible" and "necessary" variants of the verification problem and Skiba et al. [79] "possible" and "necessary" variants of the existence problem in IAFs in terms of their complexity. Some other generalizations of abstract AFs exist in the literature that also aim at modeling uncertainty or dynamics of argumentation in a similar fashion as IAFs. One of them is the model of probabilistic argumentation frameworks (PrAFs) [60]. A PrAF is an AF with an associated probability distribution both over the set of arguments and over the set of attacks. Both IAFs and PrAFs allow to represent structural uncertainty in AFs, and both share an assumption of independence, i.e., that the existence of an uncertain element is independent of that of the other uncertain elements. However, PrAFs contain more information than IAFs, which, as a positive consequence, allows to represent uncertainty more precisely when there is detailed information about probabilities. On the other hand, they overdefine cases where the likelihood of existence of the uncertain elements is not known exactly. When this difference is disregarded, PrAFs and IAFs can be mapped to each other by the following procedure: For any IAF, a corresponding probabilistic argumentation framework PrAF can be obtained by setting the probability P PrAF (x) = 1 for all definite elements x ∈ A ∪ R and setting 0 < P PrAF (y) < 1 for all uncertain elements y ∈ A ? ∪ R ? . However, this representation indicates a level of precision that may not be justified, since the incomplete AF does not provide information about how likely the existence of its uncertain elements is. Conversely, any probabilistic argumentation framework PrAF can be represented as an IAF by including all arguments and attacks x with P PrAF (x) = 1 as definite elements, and all elements y with 0 < P PrAF (y) < 1 as uncertain elements. This is a lossy representation, since the different probabilities of the uncertain elements are not preserved. Given any pair of IAF and PrAF corresponding to each other, the possible verification problem s-INCPV 4 for IAF corresponds to the question of whether the target set of arguments has probability greater than zero to be an s extension of PrAF, and the necessary verification problem s-INCNV 5 for IAF corresponds to the question of whether the target set of arguments has probability 1 to be an s extension of PrAF: in purely argument-incomplete argumentation frameworks can be represented by CAFs using their control-part, while necessary problem variants in incomplete argumentation frameworks can be represented by CAFs using their uncertain-part. Although CAFs include an additional symmetric attack relation in the uncertain part, where only the direction-not the existence-of an attack is uncertain, the complexity results of this article also hold for incomplete AFs augmented with this form of uncertainty. Furthermore, the SAT encodings presented in this article are also easily 1105 adapted to this formalism. In fact, the results of this work have subsequently (after the writing of this article) proven to be useful for the complexity analysis of and algorithms for the so-called controllability problem in CAFs [70,64].

Dynamics of Argumentation Frameworks
The PCA problem in incomplete argumentation frameworks is related to extension enforcement [9,35], where, given an argumentation framework and a subset of its arguments, the task is to determine how the attack relation and/or 1110 the set of arguments of the argumentation framework can be modified in a smallest possible way so that the given set becomes part of an extension. Instances for acceptance problems in incomplete argumentation frameworks and for enforcement problems coincide when the incomplete argumentation framework has only uncertain attacks and no uncertain arguments, and when the enforcement instance allows only changes to the attack relation and its given subset is a singleton. However, enforcement aims at finding a smallest possible change to the argumentation framework, 1115 which is not the aim in deciding acceptance in incomplete argumentation frameworks. On the other hand, the question of whether acceptance of a target argument can at all be achieved is trivially true in most variants of enforcement, while this is the key question for possible-credulous acceptance problems in incomplete argumentation frameworks. Furthermore, recent work on maximizing goals achievement (in terms of argument labels) while minimizing the number of actions (additions and deletions of arguments) [32] is also related to the PCA problem in argument-1120 incomplete argumentation frameworks. While in PCA we look at the acceptance of a specific target argument, which can be seen as a single IN goal, the aforementioned problem has several target arguments expressed as both IN and OUT goals. A key difference is also that we do not consider optimizing over the presence or absence of uncertain arguments, which can be seen as actions in the aforementioned problem.
Closely related to the analysis presented in Section 4, Cayrol et al. [28] study the problem of adding an argument and incident attacks, specifically by studying necessary and sufficient conditions for satisfying different properties, also defining the atomic changes (adding and removing arguments and attacks) we study in the context of incomplete AFs. However, their focus is on grounded and preferred semantics, whereas we consider admissible and stable semantics. Semantical change when removing an argument along with incident attacks was studied by Bisquert et al. [22] under preferred, stable, and grounded semantics, also focusing on the satisfaction of properties of extensions.

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The preservation of the grounded extension was studied by Boella et al. [23,24] when removing arguments and attacks, or adding attacks; again, we focus on admissible and stable extensions. Finally, the work of Rienstra et al. [77] focuses on the preservation of grounded, complete, preferred, stable, and semi-stable labelings under changes to the attack structure, and their results for the stable and preferred semantics coincide with ours. Incremental algorithms for dynamic argumentation frameworks, where in addition to an AF a change or a sequence of changes to the attack structure is provided, and the task is to answer a query for all AFs defined by the sequence of changes, have been recently studied both in the context of computing extensions [54, 55,1] and for acceptance problems [2]. These algorithms build on the division-based method [61,7] which divides the updated AF into affected and unaffected parts, and expand this method by, e.g., taking into account an initial extension of the first AF. In dynamic AFs, the sequence of changes is provided as input, hence the number of AFs considered is linear with 1140 respect to the size of the input. In contrast, in incomplete AFs the number of completions is exponential with respect to the input size. However, via casting a dynamic AF to an attack-incomplete AF by setting all attacks amenable to change as uncertain attacks, a "no" answer to s-PCA implies that the answer is "no" also for all AFs defined by the dynamic AF. Similarly, a "yes" answer to s-NSA implies that the answer is "yes" for all AFs of the dynamic AF. In a sense, this work on reasoning in incomplete AFs provides a shortcut for acceptance problems in dynamic AFs.

1145
The problem of adding an argument along with incident attacks is also related to expansions [11,9,8], where sets of new arguments along with incident attacks are added to an AF. Likewise, removing arguments has been studied in the form of deletions [10]. There may be potential for using further theoretical results of expansion, deletion, and update equivalence in order to strengthen the refinement in the CEGAR algorithm. However, equivalence is a considerably more general concept, as it concerns preserving all extensions of a given AF, whereas in our approach we are interested in preserving just the counterexample. In addition, recent work on preprocessing argumentation frameworks via so-called replacement patterns [45], based on a parameterized notion of equivalence [12], may prove to be useful in the context of algorithms for incomplete AFs, as well. In particular, it would be interesting to study an adaptation of this form of equivalence for incomplete AFs, where changes are allowed only on the uncertain part.

SAT-Based Approaches to Argumentation
In terms of practical systems for reasoning in argumentation frameworks, the SAT-based approaches developed in this work continue the successful line of work on applying SAT-based approaches to reasoning over various computational models of argumentation. These include argument acceptance problems and extension enumeration in standard AFs [46,58,50,29], other generalization of AFs such as abstract dialectical frameworks [62], as well as various forms of optimization problems underlying argumentations dynamics, including enforcement, adjustment, repairment, syn-1160 thesis, reasoning in dynamic AFs, and goals achievement [83,72,73,59,74,67,32]. The strong refinements we develop have subsequently (after the writing of this article) shown to be applicable in the context of CEGAR algorithms for second-level complete variants of enforcement and synthesis [69]. Similar ideas have also been considered in the specific context of the NP-complete problem of extension enforcement under grounded semantics [73].

1165
The recently proposed notion of incomplete argumentation frameworks generalizes Dung's standard abstract argumentation frameworks by allowing for modeling uncertain attacks and arguments. In contrast to standard AFs, the computational complexity of variants of acceptance problems in the context of incomplete AFs has not been thoroughly established to-date. Furthermore, the introduction of incomplete AFs raises the challenge of developing practical decision procedures for reasoning about acceptance under uncertainties. In this article, we address both of 1170 these challenges. In particular, we proposed natural generalizations of credulous and skeptical acceptance in AFs to incomplete AFs, giving rise to several variants for both of the two reasoning modes.
By establishing a full complexity landscape of acceptance in AFs for the variants under various central argumentation semantics (see Table 1 on page 9), we showed that acceptance in incomplete AFs is most often hard for the first level of the polynomial hierarchy, and can reach completeness for this hierarchy's second or even third level in 1175 some cases. Motivated by the success of SAT-based practical decision procedures developed for reasoning in standard AFs, we proposed SAT-based algorithms for all of the variants of acceptance in incomplete AFs covered by our complexity analysis. We showed through an empirical evaluation that for NP-complete variants of acceptance, reasoning in incomplete AFs turns out to be at least as efficient in practice as reasoning about acceptance in standard AFs, and that the CEGAR approaches we developed for the variants with beyond-NP complexity also scale to instances of 1180 reasonable size.
While the complexity results provided in this article cover several central argumentation semantics, the analysis could be extended to cover even further semantics such as ideal [40], semi-stable [27], and stage [82] semantics. Exploring SCC-recursive semantics [6] in the context of incomplete AFs is also a possible interesting direction for future work. Analyzing the complexity of acceptance in structural or distance-based subclasses of incomplete AFs 1185 would also be of interest, in analogy with related analyses provided earlier in the context of standard AFs [34,42,48,47,46]. Further, it would also be interesting to extend both the complexity analysis and the decision procedures presented in this article to other related formalisms that allow to represent unquantified uncertainty, such as control argumentation frameworks. In terms of potential improvements to the empirical performance of the proposed decision procedures, further analysis on possible ways of obtaining even stronger refinements than those obtained through the 1190 presented analysis on the persistence of extensions under change may be fruitful.