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Splitting Assumption-Based Argumentation Frameworks

Splitting Assumption-Based Argumentation Frameworks

arXiv:2604.27964v1 [cs.AI] 30 Apr 2026

Giovanni Buraglio , Wolfgang Dvořák , Stefan Woltran Institute of Logic and Computation, TU Wien, Austria {giovanni.buraglio, wolfgang.dvorak, stefan.woltran}@tuwien.ac.at,

Abstract Assumption-Based Argumentation (ABA) is a wellestablished formalism for modelling and reasoning over debates, with a wide range of applications. However, the high computational complexity of core reasoning tasks in ABA poses a significant challenge for its applicability. This issue is further aggravated when ABA frameworks (ABAFs) are instantiated into graph-based argumentation formalisms, such as Dung’s Argumentation Frameworks (AFs) and Argumentation Frameworks with Collective Attacks (SETAFs). In knowledge representation and reasoning, a key strategy to address computational intractability is to optimise reasoning over a given knowledge base through divide-and-conquer algorithms. A paradigmatic example of this approach is splitting, where extensions of a given framework are computed incrementally, by restricting the search space to sub-frameworks only, and then combining the obtained results. This approach has been successfully applied to AFs, for which also a parametrised version has been introduced under stable semantics. However, the exponential growth produced by the instantiation might undermine the usefulness of splitting on the argument graphs induced by ABAFs. To address this issue, our work investigates the concept of splitting on the knowledge base rather than on its graph-based instantiation. Furthermore, we generalise splitting to its parametrised version for ABAFs.

1

Introduction

Computational models of argumentation in AI (Gabbay et al. 2021) offer formal approaches to represent and reason over debates involving conflicting or uncertain information (Carrera and Iglesias 2015; Dimopoulos, Mailly, and Moraitis 2019; Fan and Toni 2012; Gao et al. 2016; Hadoux, Hunter, and Polberg 2023; Toni 2013). Assumption-Based Argumentation (ABA) (Bondarenko et al. 1997) captures argumentative scenarios by means of so-called ABA frameworks (ABAFs), or ABA knowledge bases, consisting of a set of defeasible sentences (assumptions) and inference rules. Argumentative reasoning can be performed in ABA following two different approaches. The direct approach, typically employed by ABA solvers (Lehtonen, Wallner, and Järvisalo 2021a; Lehtonen, Wallner, and Järvisalo 2021b), allows to reason over the ABA knowledge base itself through semantics defined at the level of assumptions. In contrast, the indirect approach (Lehtonen et al. 2024) realises reasoning in

a two-step process: first an argument graph comprising arguments and their relations is generated from the ABAF, by means of the so-called instantiation procedure; then, semantics from abstract argumentation are applied to the obtained graph in order to find acceptable sets of arguments, along with the assumptions supporting them. Although ABA is a well-established formalism for nonmonotonic reasoning, with applications in medical decisionmaking, explainable AI, and causal discovery (Fan et al. 2014; Fan 2018; Russo, Rapberger, and Toni 2024), the high computational complexity of core reasoning tasks in ABA poses a significant challenge for its deployment in practice (Cyras, Heinrich, and Toni 2021). This issue is further aggravated when ABAFs are instantiated into argument graphs, such as Dung’s Argumentation Frameworks (AFs) (Dung 1995) or Argumentation Frameworks with Collective Attacks (SETAFs) (Nielsen and Parsons 2006), as the instantiation can be computationally expensive and may result in exponentially large graphs (Lehtonen et al. 2023). In the context of non-monotonic reasoning, one prominent strategy to address computational intractability is to optimise reasoning over a given knowledge base through divide-and-conquer algorithms. A paradigmatic example of this approach is splitting, originally developed for answerset programming (Lifschitz and Turner 1994) and later adapted to other nonmonotonic formalisms, e.g. default theories (Turner 1996) and recently Abstract Argumentation (Baumann 2011; Baumann et al. 2012; Linsbichler 2014; Baumann, Brewka, and Wong 2011; Liao 2013; Baroni, Giacomin, and Liao 2014). This approach focuses on incrementally computing the extensions of a given argumentation framework by means of the extension of its sub-frameworks, thereby avoiding to consider the entire solution-space of the original framework. Despite the successful application in graph-based argumentation (Baumann, Brewka, and Wong 2011), splitting, so far, has been neglected for rule-based argumentation systems like Assumption-Based Argumentation. A straightforward attempt to apply splitting in ABA would be to first instantiate the ABAF to a corresponding Dung’s AF and then perform AF splitting. However, the exponential number of auxiliary arguments introduced during instantiation might invalidate the usefulness of splitting. To overcome this limitation, we instead consider the instantiation of ABAFs as

SETAFs, which yield more concise graphical representations (König, Rapberger, and Ulbricht 2022). Thus, we first develop a splitting scheme for SETAFs, enabling incremental computation for the indirect approach of reasoning. This method has the advantage of being easily applicable to any formalism that can be instantiated as a SETAF. However, its drawback is that we have to instantiate the knowledge base, which comes with a computational cost, and we cannot even use the splitting to speed up the instantiation step. To address this issue, we present a splitting schema that operates directly on ABA knowledge bases. To this end, this paper makes the following contributions:

A sentence q ∈ L is tree-derivable from S ⊆ A and rules R ⊆ R, denoted by S ⊢R q, if there is a finite rooted labelled tree T where: the root of T is labelled with q; the set of labels for the leaves of T is equal to S or S ∪{⊤}; and for every inner node v of T there is a rule r ∈ R such that v is labelled with head(r), and every successor of v is labelled with a ∈ body(r) or ⊤ if body(r) = ∅. We sometimes write S ⊢ q instead of S ⊢R q if it does not cause confusion. Moreover, we call T hD (S) = {p ∈ L | S ⊢ p} the theory of S w.r.t. the ABAF D. Throughout the paper, we assume that ABAFs do not contain dummy rules, whose body is not derivable from any set of assumptions.

• Towards ABA splitting, we generalise existing notions of splitting from AFs to SETAFs (Section 3) for complete, stable, preferred and grounded semantics.

Definition 2. Let D = (L, R, A, ) be an ABAF. A set S ⊆ A attacks T ⊆ A if S ′ ⊢ a for some S ′ ⊆ S and a ∈ T . A set S is conflict-free in an ABAF D (S ∈ cf(D)) if it does not attack itself; S defends T iff it attacks each attacker of T ; S is closed iff S ⊢ a implies a ∈ S; S is admissible (S ∈ adm(D)) if it is conflict-free and defends itself.

• We then introduce the notion of ABA splitting in Section 4, along with the syntactic adjustments required to establish a splitting theorem, which we prove under complete, stable, preferred and grounded semantics. • In Section 5 we extend our results to the more general notion of parameterised splitting (Baumann et al. 2012), showing that a splitting theorem holds for ABAFs under stable semantics. • In Section 6 we address computing splittings for SETAFs and ABAFs, by reducing it to a graph splitting problem and reusing dedicated algorithms for AFs. • In Section 7 we take a closer look at the relationship between the splitting algorithm for ABAFs and the instantiation procedure into SETAFs. • Finally, after reviewing related work (Section 8), we conclude with a summary and outline directions for future research (Section 9).

2

Preliminaries

Assumption-Based Argumentation We recall here the basic concepts of assumption-based argumentation (ABA) (Cyras et al. 2018). Debates are represented by means of so-called ABA Frameworks (ABAFs), which consist of a deductive system (L, R), where L is a set of sentences, and R is a set of rules over L. A rule r ∈ R has the form a0 ← a1 , . . . , an with ai ∈ L, body(r) = {a1 , . . . , an } and head(r) = a0 . Definition 1. An ABAF is a tuple (L, R, A, ), where (L, R) is a deductive system, A ⊆ L a set of assumptions, and : A → L is a total mapping, called contrary function. For a set of assumptions, S ⊆ A we use S to indicate the set of contraries of S. Conversely, we define the partial function α : L → A assigning an assumption to its contrary b ∈ A such that α(b) = a if b = a. This generalises to sets of contraries as before. For a set of rules R, we fix head(R) = {head(r) | r ∈ R}, body(R) = {body(r) | r ∈ R}. Further, we use atom(S) = {p ∈ L | p ∈ S ∨ α(p) ∈ S ∨ p ∈ S}. In what follows, we read atom(p) as atom({p}). For a rule r ∈ R, we say that r is: a fact if body(r) = ∅; a loop-rule if a = head(r) and a ∈ body(r).

We say a set S of assumptions attacks an assumption a if S attacks the singleton {a}. Moreover, for a set B ⊆ A we say that S attacks B if S attacks some b ∈ B. We use + SR = {a ∈ A | S ⊢R a} and define the range of S w.r.t. ⊕ + R ⊆ R as SR = S ∪ SR . In this paper, we assume ABAFs to be flat, unless specified otherwise. We call an ABAF flat if every set S of assumptions is closed, and non-flat otherwise. We next recall definitions for grounded, complete, preferred, and stable ABA semantics (abbr. grd, com, pref, stb). Definition 3. Let D be an ABAF and let S ∈ adm(D). S ∈ com(D) iff S contains every assumption set it defends; S ∈ grd(D) iff S is ⊆-minimal in com(D); S ∈ pref(D) iff S is ⊆-maximal in com(D); S ∈ stb(D) iff S attacks each {x} ⊆ A \ S. We call σ(D) the set of σ-extensions of D. SETAF Instantiation Reasoning in ABAFs is often performed on graphs induced from the instantiation procedure. Due to their popularity and simple structure, this step has traditionally been performed by means of Dung’s AFs (Dung 1995; Bondarenko et al. 1997; Cyras et al. 2018), i.e., directed graphs where nodes and their (binary) relation are interpreted as arguments and attacks among them. However, this representation does not capture the possibility of having multiple assumptions attacking another directly, which might result in a large number of auxiliary arguments. Thus, in recent years, a hyper-graph representation (König, Rapberger, and Ulbricht 2022) of ABAFs, based on argumentation frameworks with collective attacks (SETAFs) (Nielsen and Parsons 2006), has become increasingly popular (Buraglio et al. 2024b; Russo, Rapberger, and Toni 2024; Berthold, Rapberger, and Ulbricht 2024; Dimopoulos et al. 2024). Definition 4. A SETAF is a pair SF = (A, R) where A is a finite set of arguments, and R ⊆ 2A × A is the attack relation. For an attack (T, h) ∈ R we call T the tail and h the head of the attack. We write (t, h) to denote the set-attack ({t}, h). For S ⊆ A, we say S attacks an argument a ∈ A if there is an attack (T, a) ∈ R with T ⊆ S. Moreover, for a set B ⊆ A we say that S attacks B if S attacks some b ∈ B.

+ We use SR = {a ∈ A | S attacks a} and define the range ⊕ + of S w.r.t. R as SR = S ∪ SR . By AFs we refer the class of SETAFs where all attacks (T, h) ∈ R are such that |T | = 1.

For all semantics under our considerations, an ABAF D = (L, R, A, ) can be instantiated as the (equivalent) SETAF SFD = (AD , RD ) where AD = A and (S, a) ∈ RD iff S ⊢ a (König, Rapberger, and Ulbricht 2022). Example 1. For the ABAF D = (L, R, A, ) (left) and its corresponding SETAF SFD (right), it holds that {a, w, z} ∈ pref(D) = pref(SFD ). A = {a, b, v, w, x, y, z} y z w x L = A ∪ A ∪ {p} v R = {b ← b, p ← a, v ← a, a b x ← p, w, y ← x, y ← b, z} Notice that such a mapping is many-to-one, i.e., several ABAFs correspond to the same SETAF. This is because the SETAF instantiation only remembers the attacks between assumptions but forgets all non-assumptions. In the above example, we lose the sentence p when instantiating the derivation {a, w} ⊢ x build via x ← p, w and p ← a into ({a, w}, x). On the other hand, SETAFs can be seen — syntactically — as a fragment of flat ABAFs, by modelling each attack (S, a) via a dedicated rule a ← S. Splitting of AFs We now recall the splitting approach for AFs (Baumann 2011). A splitting identifies two subframeworks F1 and F2 separated by a set of attacks going from F1 to F2 . Then, the information contained in an extension of F1 is propagated, computing the so-called reduct of F2 accordingly. Definition 5. Let F = (A, R) be an AF, F1 = (A1 , R1 ) and F2 = (A2 , R2 ) two sub-frameworks of F s.t. A1 ∩ A2 = ∅, A = A1 ∪ A2 and R = R1 ∪ R2 ∪ R3 with R3 ⊆ A1 × A2 . The triple (F1 , F2 , R3 ) is called a splitting of F . For such a splitting and a set E ⊆ A1 , the (E, R3 )-reduct is the AF + AF ′ = (A′ , R′ ) with A′ = A2 \ ER and R′ = R2 ∩ 3 ′ ′ (A × A ). Moreover, the set of undecided arguments w.r.t. ⊕ E ⊆ A1 is UE = A1 \ ER . 1 The reduct is designed to take care of the arguments attacked by the set E. To account for the propagation of undecided arguments w.r.t. E, a further modification is needed: self-attacks are propagated from F1 to arguments in F2 . Definition 6. Let (F1 , F2 , R3 ) be a splitting for an AF F and E an extension of F1 . Moreover, take F2′ = (A′2 , R2′ ) as the (E, R3 )-reduct of F2 and UE as the set of undecided arguments w.r.t. E. The (UE , R3 )-modification of F2 is: ′ ′ ′ 3 modR UE (F2 ) = (A2 , R2 ∪{(b, b) | ∃a ∈ UE : (a, b) ∈ R3 }).

Using these definitions, Baumann (2011) has shown that it is possible to split the AF and compute the extensions for each sub-framework incrementally such that their combination yields extensions of the original framework. Theorem 1. (Baumann 2011) Let (F1 , F2 , R3 ) be a splitting for an AF F = (A, R) and σ ∈ {adm, stb, com, pref, grd}.

′ 3 1. If E1 ∈ σ(F1 ) and E2 ∈ σ(modR UE (F2 )), then E1 ∪E2 ∈ σ(F ). 2. If E ∈ σ(F ), then E ∩ A1 ∈ σ(F1 ) and E ∩ A2 ∈ ′ 3 σ(modR UE (F2 )).

Later, this idea has been generalised by relaxing the strict separation requirement, which significantly narrows the applicability of splitting, introducing so-called parametrised splitting (Baumann et al. 2012). Instead of demanding that the first part is completely unaffected by the second, it allows some forms of interaction. This generalisation is captured by the notion of quasi-splitting, where arguments in F1 may be externally attacked by arguments in F2 . The goal is to preserve correctness while broadening the applicability of splitting. This is achieved by enriching F1 with meta-information that encodes facts about potential influences (e.g. attacks) from the second sub-framework. In particular, for each externally attacked argument a, a fresh argument a′ is added to F1 along with a symmetric attack on a, enforcing a choice between a and a′ in F1 . Then, F2 is modified accordingly: the previous choices are propagated in the second sub-framework via the reduct as well as additional nodes and attacks. Stable extensions of the entire AF are then recovered by composing compatible solutions from the two modified sub-frameworks.

3

Splitting Collective Attacks

Towards a splitting approach for ABA we first introduce a splitting scheme for argumentation frameworks with collective attacks (SETAFs). First, due to their correspondence with ABAFs this paves the way for the splitting scheme on the ABA knowledge base that we will introduce in the next second. Second, this provides a divide-and-conquer methodology to enhance existing solvers for SETAFs (Dvořák, Greßler, and Woltran 2018; Greßler, Dvořák, and Woltran 2024), and thus allows for splitting when solving ABAFs via instantiations to SETAFs. We introduce a notion of splitting for SETAFs that generalises the one for Dung-style AFs. Definition 7. Let SF = (A, R) be a SETAF, SF1 = (A1 , R1 ) and SF2 = (A2 , R2 ) two sub-frameworks of SF such that A1 ∩A2 = ∅, A = A1 ∪A
2 and R = R1 ∪R2 ∪R3 with R3 ⊆ (2A1 \ {∅}) ∪ 2A2 × A2 . We call a splitting of SF the triple (SF1 , SF2 , R3 ). Moreover, we call R3 the set of links w.r.t. (SF1 , SF2 , R3 ) and say that a link is undecided if no argument in its tail is defeated (i.e. attacked by an extension), but at least one is undecided. As for AFs, the general idea is to compute extensions of SF as a combination of extensions of SF1 and SF2 . Due to the links from SF1 to SF2 we have to modify SF2 according to the extension(s) of SF1 to account for the prior accepted and rejected arguments. Following Baumann (2011), we introduce the notions of reduct and modification, in application to the second part (that is, SF2 ) of the original SETAF. Intuitively, the reduct takes care of the arguments in SF2 that are already defeated by E1 by deleting them. Moreover, the reduct removes an attack in R2 if one of the arguments in the tail or the head is defeated by E1 . The attacks of R3

are first filtered and the simplified by the reduct in order to fit to the new argument set. An attack is neglected if either a tail or the head argument is defeated by E1 , all the tail arguments are in A1 , or there is a tail arguments in A1 that is not accepted by E1 . The remaining attacks are simplified by removing the arguments of A1 from the tail of the attack. Definition 8 (Reduct). Let (SF1 , SF2 , R3 ) be a splitting for a SETAF SF . We define the (E1 , R3 )-reduct (or simply reduct) of SF2 for some extension E1 of SF1 as the SETAF SF2′ = (A′2 , R2′ ) with: A′2 ={a ∈ A2 | a ∈ / (E1 )+ R3 }; R2′ ={(T, h) ∈ R2 | T ⊆ A′2 , h ∈ A′2 } ∪ {(T \ A1 , h) | (T, h) ∈ R3 , T \ A1 ̸= ∅, ′ T ∩ A1 ⊆ E1 , T ∩ (E1 )+ R3 = ∅, h ∈ A2 }. When dealing with undecidedness, what guides our intuition towards a certain modification is not the status of the arguments in SF1 , but rather the status of the links. Hence, we slightly tweak the original definition and base our notion solely on the undecided links. Definition 9 (Undecided Links). Given a splitting (SF1 , SF2 , R3 ) for a SETAF SF and an extension E1 ∈ SF1 we define the set of undecided links w.r.t. E1 as:

URE31 = {(T, h) ∈ R3 | T ∩ (E1 )+ R1 ∪R3 = ∅, ∃t ∈ T : t ∈ A1 \ (E1 )⊕ R1 }. In what follows, we define the modification, which is applied on the reduct, and accounts for the effects of the undecided links. In particular, for each undecided link (S, t) ∈ R3 we add to F2′ a (set-)self-attack from t (together with the F2′ -part of the attack) to itself. Definition 10 (Modification). Let (SF1 , SF2 , R3 ) be a splitting for a SETAF SF and E1 an extension of SF1 . Take SF2′ as the (E1 , R3 )-reduct of SF2 and URE31 as the set of ′ 1 undecided links w.r.t. E1 . We denote with modE R3 (SF2 ) = SF2⋆ = (A⋆2 , R2⋆ ) the URE31 -modification (or simply modification) of SF2′ s.t. A⋆2 = A′2 and R2⋆ is given by: R2′ ∪ {((T ∩ A′2 ) ∪ {h}, h) | (T, h) ∈ URE31 , h ∈ A′2 }. Before we present the splitting theorem we illustrate Definitions 8–10 in the following example. Example 2. Consider the SETAF SF of Example 1 that separates arguments in A1 = {a, b} from those in A2 = {v, w, x, y, z}. We select the preferred extension E1 = {a} of SF1 , and compute the reduct SF2′ . This is obtained by: (i) deleting v, as it is attacked by E1 ; (ii) projecting ({a, w}, x) to ({w}, x) since a ∈ E1 ; and (iii) neglecting the attack ({b, z}, y) because b ∈ / E1 . Subsequently, we construct the modification SF2⋆ . For this, case (iii) is crucial: the attack ({b, z}, y) is an undecided link since b is not in the range of E1 and {b, z} is not attacked by it. Thus, we introduce set-self-attack ({y, z}, y) to SF2′ . w

y

x

z

v

SF2′ a

w

z

v

SF2⋆ b

y

x a

b

Finally, we obtain E2 = {w, z} as the preferred extension of SF2⋆ , retrieving E = E1 ∪ E2 = {a, w, z} as a preferred extension of SF . Having these notions at hand, we now establish the adequacy of the splitting technique for SETAFs. We start by establishing that (a) conflict-freeness of the sub-frameworks SF1 and SF2 carries over to the whole SETAF SF , and (b) conflict-free sets of SF induce conflict-free subsets in SF1 and SF2′ . Proposition 1. Let (SF1 , SF2 , R3 ) be a splitting for a SETAF SF = (A, R) with SF1 = (A1 , R1 ) and SF2 = (A2 , R2 ). Let SF2′ = SF2E1 . ′ 1 1. If E1 ∈ cf(SF1 ) and E2 ∈ cf(modE R3 (SF2 )), then E1 ∪ E2 ∈ cf(SF ). 2. If E ∈ cf(SF ), then E1 = E ∩ A1 ∈ cf(SF1 ) and E ∩ A2 ∈ cf(SF2′ ). Finally, we are ready to characterize the splitting algorithm by generalising the splitting theorem for SETAFs under the standard Dung semantics. Theorem 2. Let (SF1 , SF2 , R3 ) be a splitting for a SETAF SF = (A, R) with SF1 = (A1 , R1 ), SF2 = (A2 , R2 ), and σ ∈ {stb, adm, com, pref, grd}. Further, let SF2⋆ = ′ 1 modE R3 (SF2 ).

1. If E1 ∈ σ(SF1 ) and E2 ∈ σ(SF2⋆ ), then E1 ∪ E2 ∈ σ(SF ). 2. If E ∈ σ(SF ), then E1 = E ∩ A1 ∈ σ(SF1 ) and E ∩ A2 ∈ σ(SF2⋆ ). While the existing instantiation procedure from ABA frameworks to SETAFs provides a foundation for defining splitting, attempting to directly replicate the SETAF-style idea of splitting among assumptions fails to yield a natural notion of splitting. This disconnect stems from a fundamental structural difference: in SETAFs, attacks are primitive, whereas in ABA, they are derived from the underlying deductive system (L, R). As a result, naively mimicking SETAF-style splitting in ABA would require (i) arbitrarily partitioning the assumption set into A1 and A2 , and (ii) computing attacks as derivations from assumptions in A1 to those in A2 . However, splitting should be possible solely by inspecting the knowledge base at hand. Moreover, while instantiating ABAFs into SETAFs has been shown useful in specific contexts (Russo, Rapberger, and Toni 2024; Buraglio et al. 2024b), this approach comes with a critical drawback: it can yield an exponential growth in the number of collective attacks generated, thus increasing in input size. This inefficiency motivates many ABA solvers to operate directly on ABAFs rather than relying on their abstract representations. Therefore, to enable an efficient form of splitting, we propose a dedicated splitting algorithm tailored to the syntactic structure of ABAFs.

4

Splitting in Assumption-Based Argumentation

In this section we present splitting results for ABAFs. The rule-set of an ABAF is split into a bottom and a top part

whenever no assumption occurs in the bottom part whose contrary is derived by some rule in the top. This ensures that the assumptions in the bottom can be evaluated independently of what can be deduced by inspecting the top part. We capture this intuition via the notion of splitting set: Definition 11. Given an ABAF D = (L, R, A, ), a set S ⊆ L is a splitting set (or simply a splitting) of D if S = atom(S) and for all r ∈ R, head(r) ∈ S implies body(r) ⊆ S. A splitting set partitions the deductive system into two sub-systems (L1 ,R1 ) and (L2 ,R2 ), called the ‘bottom’ and ‘top’. In particular, we have (i) L1 = S and R1 = {r ∈ R | head(r) ∈ S} and (ii) L2 = L \ S and R2 = {r ∈ R | head(r) ∈ / S}. These induce respectively D1 = (L1 , R1 , A1 , 1 ) and D2 = (L2 , R2 , A2 , 2 ) with Ai = Li ∩ A and i defined over Ai . Example 3. Consider again the ABAF D = (L, R, A, ) from Example 1, with assumptions A = {a, b, v, w, x, y, z}, sentences L = A ∪ A ∪ {p}, and rules R as follows: b←b

p←a v←a

x ← w, p

y←x

y ← b, z

Take the set S = {a, b, a, b, p}. It can be easily checked that S is a splitting set of D, through which we obtain two subsystems (L1 , R1 ) (bottom) and (L2 ,R2 ) (top) such that: • L1 = S and L2 = L \ S; • R1 = {b ← b, p ← a} and R2 = R \ R1 . Notice that some atoms contained in L1 (but not in L2 ) may occur in the body of some rule in R2 (a, b and p in Example 3). This intermediate mismatch will be resolved later by the notion of reduct. Moreover, their occurrences in R2 does not affect the acceptance status of such atoms. In fact, a first sanity check, we observe that our notion of splitting prevents building attacks from assumptions of D2 towards assumptions of D1 using top-rules in R2 . This is ensured by the fact that contraries of assumptions occurring in the bottom part are not derived via rules in the top part (via construction of R2 ). As a result, assumptions in A1 are attacked only via rules in R1 by assumptions in A1 . Thus, no attack generated from A2 (by means of rules in R2 ) is directed towards A1 . Proposition 2. Let D be an ABAF and S a set of literals that splits D into D1 and D2 . For every derivation T ⊢R a with a ∈ A1 , it holds that R ⊆ R1 and T ⊆ A1 . The attacks of the bottom part can be extended in a conservative way: whatever happens in the second sub-framework does not affect the acceptability of assumptions in D1 . Thus, to compute incrementally an extension of an ABAF D, we can first select an extension E of D1 and later modify D2 according to the information contained in E. Consequently, we can evaluate the modified framework D2 and augment its extensions with E. Again, we follow the approach of Baumann and appeal to the notions of reduct and modification in application to D2 via a two-step process. First, we propagate all the information we get from a σ-extension E of D1 to ensure that rules in contrast with E are removed. The outcome is called the E-reduct of D2 .

Definition 12. Let D = (L, R, A, ) be an ABAF, S a splitting set of D into D1 and D2 and E a σ-extension of D1 . 2 ) the E-reduct (or simply We call D2E = (L2 , RE 2 , A2 , E reduct) of D2 , where R2 is obtained by deleting: • each rule r ∈ R2 with body(r) ∩ S ̸⊆ Th D1 (E); • all literals in Th D1 (E) from the remaining rules. As we anticipated, the rule-set RE 2 now contains all and only those atoms occurring in L2 . Therefore, the reduct can be evaluated in complete isolation from D1 . In the second step, we modify the reduct to propagate the information about assumptions (or their contraries) which are not contained in Th D1 (E). We call these assumptions undecided, as they are not in E nor their contrary is derivable from it (i.e. are not attacked by E). Then, the set of undecided assumptions of D1 w.r.t. E is UAD1 (E) = {a ∈ A1 | a ∈ / E and a ∈ / Th D1 (E)}. Since their status can be transmitted to other assumptions via rules, we define the concept of undecided theory of D1 , capturing all sentences derived from some undecided (but no defeated) assumptions. Definition 13. Let D = (L, R, A, ) be an ABAF and E ∈ σ(D). The undecided theory UTD (E) of D w.r.t. E is the set of sentences p ∈ L for which there is a T ⊆ A such that (i) T ⊢ p, (ii) T ∩ UAD (E) ̸= ∅, and (iii) T ∩ T hD (E) = ∅. Rules in D2 whose bodies contain elements of UTD1 (E) might carry over undecidedness from D1 . However, this scenario could be overwritten by the presence of incompatible + sentences w.r.t. E, captured by ISD1 (E) = T hD1 (ER )∪E, 1 + R a, R ⊆ R }. Hence, a set where ER = {a ∈ A | E ⊢ 1 1 1 of sentences from D1 will carry undecidedness to sentences in D2 if and only if (i) none of its elements is incompatible and (ii) at least one of its elements is in the undecided theory w.r.t. the previously selected extension. This concept mirrors the notion of undecided links for SETAFs. We are now in the position to formally define the modification of D2E . First, we expand the set of sentences with a fresh assumption xu and corresponding contrary. Further, we introduce (i) a loop-rule for xu and (ii) a modified version of every rule with some undecided (but no incompatible) sentence in the body. In particular, we expand their body with xu , after projecting to L2 . Definition 14. Let D be an ABAF, S a set that splits D into two sub-frameworks D1 and D2 and E an extension of D1 . Further, let D2′ be the E-reduct of D2 . We use ⋆ ⋆ ⋆ ′ ⋆ ⋆ ) to denote the EmodE D1 (D2 ) = D2 = (L2 , R2 , A2 , modification (or simply modification) of D2′ where D2⋆ = D2′ if UAD1 (E) = ∅, and otherwise: L⋆2 = L2 ∪ {xu , xu }; R⋆2 = R′2 ∪ {xu ← xu } ∪ {head(r) ← (body(r) ∩ L2 ) ∪ {xu } | r ∈ R2 , body(r) ∩ ISD1 (E) = ∅, body(r) ∩ UTD1 (E) ̸= ∅}. Example 4. Consider again the ABAF D = (L, R, A, ) from Example 3 and splitting set S = {a, b, a, b, p}. We know that {a} ∈ pref(D1 ). Therefore, the {a}-reduct of D2 {a} {a} {a} is D2 = (L2 , R2 , A2 , 2 ), where the rule-set R2 is: v←a

x ← w, p

y←x

y ← b, z

Moreover, UAD1 ({a}) = {b} is the set of undecided assumptions and U TD1 ({a}) = {b, b}. We then compute the modification D2⋆ w.r.t. {a} by expanding the set of sentences {a} L2 with {xu , xu } and the rule-set R2 above such that: v←

x←w

y←x

y ← xu , z

xu ← xu

Since {w, z} ∈ pref(D2⋆ ), we get {a, w, z} as in Example 2. We can now prove that our procedure preserves conflictfree sets under incremental computation as well as projection to sub-frameworks, as for SETAFs in Section 3. Proposition 3. Let S ⊆ L be a splitting set of an ABAF D E1 1 into D1 and D2 . Moreover, let D2⋆ = modE D1 (D2 ). 1. If E1 ∈ cf(D1 ) and E2 ∈ cf(D2⋆ ), then E1 ∪ E2 ∈ cf(D). 2. If E ∈ cf(D), then E ∩ A1 ∈ cf(D1 ) and E ∩ A2 ∈ cf(D2E ). We then show that splitting is adequate with respect to stable, admissible, complete, preferred and grounded semantics. Due to space constraints we include proof details only for admissible semantics, being prototypical for the others. Theorem 3. Let S be a splitting set for an ABAF D into D1 and D2 and σ = {stb, adm, com, pref, grd}. Further, let E1 ⋆ 1 modE D1 (D2 ) = D2 . 1. If E1 ∈ σ(D1 ) and E2 ∈ σ(D2⋆ ), then E1 ∪ E2 ∈ σ(D). 2. If E ∈ σ(D), then E ∩ A1 ∈ σ(D1 ) and E ∩ A2 ∈ σ(D2⋆ ). Proof. (admissible). (1.) Since admissibility implies conflict-freeness from Proposition 3, we know that E = E1 ∪E2 ∈ cf(D). Thus we only need to show that E defends itself in D, i.e. for all a ∈ E, if T ⊢ a, then T ′ ⊢ t for some t ∈ T and T ′ ⊆ E. If a ∈ E1 , we know that a is defended by E1 in A1 from hypothesis. Thus, from Proposition 2, we can deduce that E1 ∈ adm(D). Consider now an assumption a ∈ E2 and some T ⊆ A such that T ⊢R a and R ⊆ R. If T ∩ T hD1 (E1 ) ̸= ∅, then E1 defends a against T in D. If T ∩ T hD1 (E1 ) = ∅, this means that T ⊆ A2 and T ⊢ a in D2′ (a is attacked in the reduct) or T ∪ xu ⊢R a in D2⋆ (a is attacked in the modification). In both cases, since E2 is conflict-free and defends a in D2⋆ , there is a T ′ ⊆ E2 such that T ′ ⊢ t with t ∈ T . We distinguish two cases: either (i) T ′ ⊢ t already in D2 , in which case a is defended by E in D, or (ii) there is some T ′′ ⊃ T ′ such that T ′′ ⊢ t in D and T ′′ ∩ A1 ⊆ E1 . Thus, since T ⊆ E1 ∪ E2 , a is defended by E1 ∪ E2 in D. In any case a is defended in D by E. (2.) By Proposition 3, we get E1 = E ∩ A1 ∈ cf(D1 ) and E2 = E ∩ A2 ∈ cf(D2′ ). First, we know that E1 = E ∩ A1 ∈ adm(D1 ) because E defends itself in D and E ∩ A1 is not attacked by a subset of A2 (Proposition 2). It remains to prove that E2 = E ∩ A2 ∈ adm(D2⋆ ). Take an assumption a ∈ E2 such that T ⊢ a in D2⋆ . Each such derivation corresponds to exactly one derivation T ′ ⊢R a with R ⊆ R. There are two cases: either (i) T ′ = T ⊆ A2 and R ⊆ R2 or (ii) T ′ ⊃ T \ {xu } where T ′ \ T ⊆ E1 (assumptions deleted from simplified rules in the reduct). From both (i) and (ii) we deduce that T ′ ∩ T hD1 (E1 ) = ∅: for (i) because it would entail T ̸⊆ A2 ; for (ii) because

otherwise T ̸⊢ a in D2⋆ . Nonetheless, since E defends a in D, in case (i) there is a counter-attack T ′′ ⊢R2 t such that T ′′ ⊆ E and t ∈ (T \ {a}). In case (ii), the same holds but t ∈ (T ′ \ {a, xu }). If T ′′ ∩ A1 = ∅, we know that {t} ⊆ A′2 and together with the fact that T ′′ ⊆ E, we derive T ′′ ⊆ E ∩ A′2 = E2 . Hence, T ′′ defends a from T in D2⋆ . If T ′′ ∩A1 ̸= ∅, then T ′′ ∩A1 ⊆ E1 . Therefore, from T ′′ ⊆ E ′ we get T ′′ ∩ A′2 ⊢R2 t, which defends a against T in D2⋆ . Thus a is always defended in D2⋆ , as desired. The above results guarantees the correctness of our divide-and-conquer algorithm. However, as anticipated in the introduction, splitting is only possible when certain structural properties are met. For graph-based frameworks, it is required that all attacks between the two subframeworks share the same direction. For this reason, the presence of a single attack can invalidate the possibility of splitting (e.g. when the AF in question has the form of a Strongly Connected Component (SCC)). In the case of ABA, the same can happen when an assumption b in the splitting set S is attacked by assumptions outside of it. In such cases, splitting is impossible, as the bottom part of ABAF is not independent from the top. Thus, the acceptability of b cannot be ensured by projecting to the bottom. Similarly to graph-based frameworks, the possibility to make use of a splitting algorithm may be hindered, even in cases where only one out of many assumptions in S is externally attacked. To overcome such issue, we consider in the following section a generalised version of ABA splitting.

5

Parametrised Splitting

We now introduce a more general version of splitting for ABAFs, called parametrised splitting inspired by Baumann et al. (2012). This relaxes the structural constraint for the application of splitting, potentially allowing a fixed number of assumptions in the bottom part to be attacked from assumptions in the top. Precisely, in contrast with the previous notion of splitting, we allow some contraries of assumptions occurring in bodies of R1 to appear as the heads of rules in R2 . The number of these assumptions then represents a measure of how far we are from obtaining a splitting of ABA knowledge base. The concept of a splitting set is then generalised accordingly as follows: Definition 15. For any ABAF D = (L, R, A, ), a set S ⊆ L is called a quasi-splitting of D if S = atom(S) and for all r ∈ R, head(r) ∈ S implies body(r) \ A ⊆ S. Let VS← = {b ∈ A \ S | ∃r, r′ ∈ R : b ∈ body(r) ∩ A, head(r) ∈ S, head(r′ ) = b, r ̸= r′ }. We call S: • k-splitting of D, if |VS← | = k; • (proper) splitting of D, if |VS← | = 0. As before, the rule-set is split into a bottom and top part, depending on the rule-head respectively being or not in S. As a result, VS← is the set of assumptions in the bottom whose contrary is derived in the top. We call VS← the set of vulnerabilities with respect to S, since it contains assumptions that are attacked by S. Whenever |VS← | ̸= 0, there are some heads in R2 whose corresponding assumption may

appear in bodies of R1 . Therefore, the notion of splitting of Definition 11 corresponds to a 0-splitting. To account for elements of VS← , the ABAFs D1 and D2 induced by the chosen splitting set are constructed in a slightly different way than before. In particular, we fix D1 and D2 as before, but let L1 = S ∪ VS← ∪ VS← . Moreover, since contraries in VS← may be derived by top rules, the status of their corresponding assumptions in the bottom depends on rules in the top. Consequently D1 cannot be evaluated in complete isolation from the rest, in contrast with proper splitting. For computing extension of the sub-framework D1 , we first need to modify the ABAF. First, we modify the rules by removing body-atoms not in L1 . Indeed, these atoms occur in L2 and are unattacked in D, therefore they can be disregarded when evaluating D1 . Further, we proceed by adding: (i) a fresh assumption b′ (and its contrary b′ ) for each b ∈ VS← ; (ii) rules that encode the choice for or against the presence of each assumption b ∈ VS← in the extension. In this way, we store at the object level the meta-information regarding our choices on each b ∈ VS← . Definition 16. Let D = (L, R, A, ) be an ABAF, S ⊆ L be a quasi-splitting of D inducing the sub-frameworks D1 and D2 . Moreover, let VS← be the set of vulnerabilities of D1 with respect to S and (R1 )↓L1 = {head(r) ← body(r) ∩ L1 | r ∈ R1 }. From D1 we construct the ABAF ⌞D1 ⌟ = (⌞L1 ⌟, ⌞R1 ⌟, ⌞A1 ⌟, ) by letting: • ⌞L1 ⌟ = L1 ∪ {b′ , b′ | b ∈ VS← }; • ⌞R1 ⌟ = (R1 )↓L1 ∪ {b ← b′ , b′ ← b | b ∈ VS← }. Intuitively, the additional rules allow us to choose whether we want to accept an extension E of ⌞D1 ⌟ containing b or one that does not. After this choice, we can safely compute the E-reduct of D2 , as for proper splitting. In this way, we propagate the meta-information to which we committed by means of our choice. A further modification of D2 is now needed to ensure our hypothesis regarding b: we add a factrule b ← or a loop-rule b ← b, depending on whether the previously chosen extension E contains b or b′ . These represent a type of (positive and negative) constraints in ABA. Definition 17. Let D = (L, R, A, ) be an ABAF, S a quasi-splitting of D into D1 and D2 . Moreover, let VS← be the set of vulnerabilities with respect to S and D2E the E-reduct of D2 for some E ∈ σ(⌞D1 ⌟). We denote with 2 ⌜D2E ⌝ = (L2 , ⌜RE ) the ABAF such that: 2 ⌝, A2 , E ← ′ ⌜RE 2 ⌝ = R2 ∪ {b ←| b ∈ E ∩ VS } ∪ {b ← b | b ∈ E}.

Notice that such a modification can make ⌜D2E ⌝ non-flat, as cl(∅) = {b | b ∈ E ∩ VS← }. For stable semantics, however, this does not result in a higher complexity for the same reasoning tasks (Cyras, Heinrich, and Toni 2021). Example 5. Consider the ABAF D = (L, R, A, ) where A = {a, b, c, d}, L = A ∪ A ∪ {p}, and rule-set R: b←a

d←b

a ← p, c

p←b

First, E = {b, c} and E ′ = {a, c, d} are stable extensions in D. Now let S = {a, a, d, d, p} be a quasi-splitting of D

and VS← = {b} the set of vulnerabilities w.r.t. S. We get ⌞L1 ⌟ = S ∪ {b} ∪ {b} ∪ {b′ , b′ } and ⌞R1 ⌟ such that: d←b

a ← p, c

p←b

b′ ← b

b ← b′

We derive two stable extensions E1 = {b} and E1′ = {b , a, d}. Now consider D2 with L2 = L \ S = {b, c, b, c}. 1 For the former we get ⌜D2E1 ⌝ with ⌜RE 2 ⌝ = ∅ ∪ {b ←} from which we derive E2 = {b, c} as a stable extension. For E′ 1 the latter we get ⌜D2 1 ⌝ with ⌜RE 2 ⌝ = {b ←} ∪ {b ← b} from which we derive E2′ = {c} as a stable extension. We then obtain E = (E1 ∩ S) ∪ E2 and E ′ = (E1′ ∩ S) ∪ E2′ . ′

Theorem 4. Let D = (L, R, A, ) be an ABAF and S ⊆ L a quasi-splitting inducing the sub-frameworks D1 and D2 . 1. If E1 ∈ stb(⌞D1 ⌟) and E2 ∈ stb(⌜D2E1 ⌝), then (E1 ∩ S) ∪ E2 ∈ stb(D). 2. If E ∈ stb(D), then there is a set X ⊆ {a′ | a ∈ VS← } s.t. E1 = (E ∩ S) ∪ X ∈ stb(⌞D1 ⌟) and E2 = E ∩ A2 ∈ stb(⌜D2E1 ⌝). Proof. In what follows, let D2′ = (L2 , R′2 , A2 , 2 ) be the reduct of D2 w.r.t. E1 = (E ∩ S) ∪ X and E = E1 ∪ E2 . (1.) To prove the statement we need to show (E1 ∩ S) ∪ E2 ∈ cf(D) and ((E1 ∩ S) ∪ E2 )⊕ R = A. We start with conflict-freeness. Since E1 ∈ cf(⌞D1 ⌟), then E1 ∩ S ∈ cf(⌞D1 ⌟) (less assumptions) and E1 ∩ S ∈ cf(D1 ) (less attacks). We show that E1 ∩ S ∈ cf(D). Assume towards contradiction that E1 ∩S ⊢R a for some a ∈ E1 ∩S and R ⊆ R. Since E1 ∩ S ∈ cf(D1 ) and R2 = {r ∈ R | head(r) ∈ / S}, it must be that E1 ∩ S ⊢R2 p where R2 = R ∩ R2 for some p ∈ body(r′ ) and rule r′ ∈ R1 = R ∩ R1 . Since p ∈ L2 , we distinguish two cases: if p ∈ VS← , then it is derivable from some b′ in ⌞D1 ⌟, and r′ fires; if p ∈ / VS← , then it gets removed from the body of r′ in ⌞D1 ⌟, which in turn fires. ′ Thus, in both cases we have that E1 ∩ S ⊢R1 a where R1′ ⊆ ⌞R1 ⌟. Contradiction. Consider now E2 ∈ stb(⌜D2′ ⌝). Since being stable implies conflict-freeness we immediately get E2 ∈ cf(⌜D2′ ⌝). Again, since R′2 ⊆ ⌜R′2 ⌝, we obtain E2 ∈ cf(D2′ ). Furthermore, Proposition 3 for proper splittings, together with E1 ∩ S ∈ cf(D1 ) and E2 ∈ cf(D2′ ), entail (E1 ∩ S) ∪ E2 ̸⊢R a for any a ∈ E2 and R ⊆ R. It only remains to consider possible attacks from E2 to E1 ∩ S in D. Suppose that there are T ⊆ E2 and a ∈ E1 ∩ S such that T ⊢R a for some R ⊆ R. First, notice that since T ⊆ E2 , we get body(R) ∩ S = ∅ ⊆ T hD1 (E1 ), and thus R ⊆ R′2 . Moreover, a ∈ VS← so that ⌜R′2 ⌝ = R′2 ∪ {a ←}. Therefore, T ⊢R a and T ⊢R a for some R ⊆ ⌜R′2 ⌝, i.e. E2 is either not conflict-free or not closed in ⌜D2′ ⌝. We now show that ((E1 ∩ S) ∪ E2 )⊕ R = A. Towards contradiction, consider an assumption a ∈ / ((E1 ∩ S) ∪ E2 )⊕ R . Assume a ∈ S. By hypothesis, E1 ∈ stb(⌞D1 ⌟), i.e. either a ∈ E1 or E1 ⊢R a for some R ⊆ ⌞R1 ⌟. From our assumption, we get a ∈ / (E1 ∩ S)⊕ / E1 ∩ S and (ii) R , that is (i) a ∈ R E1 ∩ S ̸⊢ a for any R ⊆ R. If (i) holds, we immediately derive E1 ⊢R a for some R ⊆ ⌞R1 ⌟. Consider now our assumption (ii). Because a ∈ S we know that every rule of R is contained in R1 . For the same reason such rules

are in ⌞R1 ⌟ (b ∈ / VS← ). Therefore, E1 ∩ S ̸⊢R a for any R ⊆ ⌞R1 ⌟ in contradiction with our hypothesis. Assume now a ∈ A \ S. By hypothesis we know either a ∈ E2 or E2 ⊢R a for some R ⊆ ⌜R′2 ⌝. From the assumption, we get a ∈ / (E2 )⊕ / E2 and (ii) E2 ̸⊢R a for any R , that is (i) a ∈ R ⊆ R. As before, from (i) and our hypothesis we derive E2 ⊢R a must hold for some R ⊆ ⌜R′2 ⌝. If a ∈ VS← , there are two possibilities: a ∈ E1 \ S or a ∈ / E1 \ S. In the first scenario, ⌜R′2 ⌝ = R′2 ∪ {a ←}. Again, E2 ⊢R a and E2 ⊢R a for some R ⊆ ⌜R′2 ⌝, in contradiction with the fact that E2 is a stable extension of ⌜D2′ ⌝. If a ∈ / E1 \ S, then a′ ∈ E1 , which means ⌜R′2 ⌝ = R′2 ∪ {a ← a}. Since a∈ / E2 , the loop-rule a ← a is not in R, therefore R ⊆ R′2 . Thus, for each rule r′ ∈ R′2 there is a rule r ∈ R2 such that body(r) ⊆ body(r) ∪ T hD1 (E1 ). Hence, it follows directly that (E1 ∩ S) ∪ E2 ⊢R a for some R ⊆ R1 ∪ R2 = R. If a∈ / VS← , then a ∈ / A1 . If E2 ⊢R a for some R ⊆ ⌜R′2 ⌝, it is not because {a ← a} ⊆ ⌜R′2 ⌝. Thus R ⊆ R′2 . As before, for each rule r′ ∈ R′2 there is exactly one rule r ∈ R2 such that body(r) ⊆ body(r′ ) ∪ T hD1 (E1 ∩ S). As a result, in D it holds that (E1 ∩ S) ∪ E2 ⊢ a. Contradiction. (2.) First we get E ∈ cf(D) and thus E ∩ S ∈ cf(D1 ) (less attacks). Now let B = A1 \ (E ∩ S)⊕ R . Since E ∈ stb(D), it attacks every other assumption. Hence, we can infer that assumptions in B are contained in A1 and attacked + by E ∩ A2 in D, that is B ⊆ ER \ (E ∩ S)⊕ R = (E ∩ + A2 )R . Therefore there is a rule r ∈ R2 with head(r) = b for each b ∈ B, meaning that B = VS← . Now let X = {b′ | b ∈ B}. Thus ⌜R′2 ⌝ contains a pair of rule {b ← b′ , b′ ← b} for each b ∈ B. Consequently, conflict-freeness of (E ∩ S) ∪ X is ensured since b ∈ / E ∩ S for all b ∈ B. Moreover, X attacks every b ∈ B in ⌞D1 ⌟, making (E ∩ S) ∪ X stable. It now remains to show E2 = E ∩ A2 ∈ stb(⌜D2′ ⌝). As before, we know that E ∩ A2 ∈ cf(D2 ) since E is conflict-free in D (less assumptions), and E ∩ A2 ∈ cf(D2′ ) because T hD2′ (E ∩A2 ) ⊆ T hD2 (E ∩A2 ) (less rules and attacks). Consider now the modified framework ⌜D2′ ⌝ wrt (E ∩ S) ∪ X. By construction, E ∩ A2 ∈ / cf(⌜D2′ ⌝) only if b ∈ B ∩ (E ∩ A2 ). Recall that B ⊆ (E ∩ A2 )+ R . Thus, E ∩ A2 ∈ / cf(D). By contradiction, we derive that E ∩ A2 is conflict free in ⌜D2′ ⌝. We now show that E2 ⊢R a for all a ∈ A2 \ E2 and some R ⊆ ⌜R′2 ⌝. Towards contradiction, we assume there is an a ∈ A2 \ E2 such that E2 ̸⊢R a, i.e. a ∈ / T h⌜D2′ ⌝ (E2 ). Therefore, since a ∈ / E2 , we get a∈ / T hD2′ (E2 ). Hence, before the reduct is applied, it holds that (E ∩ S) ∪ E2 ̸⊢R a with R ⊆ R2 . Since no rule r ∈ R1 is such that head(r) = a, we derive (E ∩ S) ∪ E2 ̸⊢R a in D, contradicting our hypothesis. Finally, we ensure that cl(E2 ) = E2 in ⌜D2′ ⌝. Assume the contrary holds. Since D2′ is flat, that means {a ←} ⊆ ⌜R′2 ⌝ and a ∈ / E2 . These facts respectively entail a ∈ E1 and E2 ⊢ a, against the conflict-freeness of E. Thus, E2 is conflict-free, closed and attacks every other assumption.

6

Computing the Splitting Set

In order to apply the introduced splitting schemes, we need to first compute a splitting of the ABAF and SETAF respec-

tively. In this section, we show that this can be reduced to the same graph problem as for computing a splitting of an AFs, and we can therefore reuse existing algorithms (Baumann, Brewka, and Wong 2011) for the computation of our splittings. First, for a SETAF SF we consider its primal graph, which draws an egde from argument a to b if there is an attack (T, b) ∈ SF with a ∈ T . For ABA splittings, we build a Dependency Graph GD of the ABAF to depict the influences among the sentences in L (Blümel et al. 2025; Rapberger, Ulbricht, and Wallner 2022). That is, for an ABAF D = (L, R, A, ) the dependency graph GD = (VD , ED ) is constructed as follows: we take a node a ∈ VD for each atom in a ∈ L; for every rule r ∈ R we take: • an edge (a, b) ∈ ED iff head(r) = b and a ∈ body(r), • two symmetric edges (a, b) and (b, a) between each assumption node a ∈ A and its contrary node b = a. The first condition makes sure that influences among sentences encoded via rules are mirrored by the edges. Moreover, the second condition ensures that S = atom(S) for any splitting set S when we later compute the SCCs. Example 6. Recall the ABAF D of Example 3. Its corresponding dependency graph GD = (VD , ED ) is: w

w

x

x

v

v

p a

z

y

z

y

a

b

b

We now have a simple directed graph, whose splittings correspond to the splittings of the ABAF, SETAF respectively. Thus, we can apply the strategy of Baumann, Brewka, and Wong (2011) to obtain a splitting set. In particular, after running Tarjan’s algorithm on GD to compute the SCCs, we contract every SCC to a single vertex leading to an acyclic graph G◦D , where nodes are the SCCs (highlighted in Example 6) of the original dependency graph. Example 7. We continue Example 6 and construct G◦D . {w, w}

{x, x} p

{a, a}

{y, y}

{z, z}

{v, v} {b, b}

As a result, each splitting set of D corresponds to exactly one splitting of G◦D . Further, Baumann, Brewka, and Wong (2011) focus on splitting that divides G◦D as evenly as possible in two parts with respect to the total number of nodes (Baumann, Brewka, and Wong 2011, Section 3.2, Algorithm 2). In this sense, a desirable splittings for G◦D in Example 7 could be S = {a, a, b, b, p, v, v}. Notice that the size of the dependency graph GD is linear in the size of the ABAF (as |VD | = |L|) and the SCCs can be computed in linear time (Tarjan 1972). In addition, the present approach can also be employed to compute quasisplittings. Once the dependency graph has been obtained, we contract only those pairs of nodes comprising an assumption and its contrary. Subsequently, we can apply the same

techniques used for Dung-style AFs (Baumann et al. 2012), although these are more involved than those required for standard splitting. This strategy relies on the computation of minimum cuts in directed graphs using existing polynomialtime algorithms (Hao and Orlin 1994).

7

Relating ABA and SETAF Splittings

In this section, we compare splitting schemes for SETAFs and ABAFs. First, notice that for an ABAF D, every splitting set yields a splitting on the corresponding SETAF SFD . Conversely, several ABA splitting sets may correspond to the same SETAF splitting. Observation 1. Let D = (L, R, A, ) be an ABAF and SFD = (AD , RD ) the corresponding SETAF. Further, let S be a splitting set of D into D1 and D2 . Then there is a splitting (SF1 , SF2 , R3 ) of SFD such that the assumptions in S are exactly the arguments of SF1 . Vice versa, for a SETAF SF and a splitting (SF1 , SF2 , R3 ), there are several splitting sets S1 , . . . , Sn for the ABAF DSF such that the arguments of SF1 coincide with the assumptions in Si . This opens the possibility of applying splitting on ABAFs before or after the instantiation into the corresponding SETAFs. To illustrate this point, consider the following. Example 8. Consider the ABAF D = (L, R, A, ) (left) and its corresponding SETAF SFD (right). A = {a, b, c} a L = A ∪ A ∪ {p, q, s} c R = {c ← s, q, s ← b, b q ← p, p ← a, a ← a} The corresponding SETAF is SFD = (AD , RD ) where AD = A and RD = {(a, a), ({a, b}, c)}. By appropriately combining {a, a, b, b} with sentences in {p, q, s} we get multiple splitting sets of D corresponding to a single SETAF splitting with R3 = {({a, b}, c)}. S1 = {a, b, a, b} a S2 = S1 ∪ {p} c S3 = S1 ∪ {s} b S4 = S1 ∪ {p, q, s} Moreover, selecting {b} as preferred extension of D1 and SF1 , we add, respectively, the fresh assumption xu attacking itself and c, and the self-attack (c, c) to SF2 . This results in ∅ being the only preferred extension of the D2⋆ and SF2⋆ . As the previous example showcases, both strategies allow for the same splittings and, indeed, compute the same expected result. When combining splitting with an instantiation based solver, the above considerations open different possibilities of how and when to split throughout the reasoning process. A first strategy, when instantiation is feasible, is to generate the SETAF and apply the splitting schema at the level of the graph. Alternatively, when this is not possible, one can first split the ABAF D into D1 and D2 , then instantiate them into SETAFs SFD1 and SFD2 and apply the SETAF splitting algorithm, i.e. modify SFD2 into (SFD2 )⋆ and compute extensions to be merged. This might allow one to utilise the SETAF splitting algorithm in cases where the argument graph of the entire ABAF is too hard to compute.

8

Related Work

We hereby clarify the relation of our splitting approach to SCC-recursiveness (Dvořák et al. 2024; Blümel et al. 2025), and we compare our notion of splitting to existing ones for logic programs (LPs) (Lifschitz and Turner 1994) and abstract dialectical frameworks (ADFs) (Linsbichler 2014), and syntax splitting (Parikh 1999). In the incremental computation approach induced by the SCC-recursive property, one computes the extensions in subframeworks, and ultimately combines the thereby computed extension parts (as in the splitting approach). In contrast to splittings however, this is restricted to subframeworks that make up strongly connected components w.r.t. the primal graph of the SETAF or the ABAF’s dependency graph (Dvořák et al. 2024; Blümel et al. 2025). Indeed, given the primal graph of a SETAF (resp. an ABAF’s dependency graph), our approach corresponds to splitting between arguments (resp. sentences) such that the involved attacks (positive edges) have the same direction. However, splitting is more general in this regard, as the subframeworks do not have to be SCCs. Finally, SCC-recursiveness relies on a generalised semantics to deal with the decisions of prior parts of the framework, in contrast to the syntactic manipulationbased approach of splitting. This allows splitting to be implemented on top of existing ABA and SETAF solvers. It has been shown that ABA captures normal logic programs with negation-as-failure under several semantics (Caminada and Schulz 2017). In particular, partial stable, well-founded and regular models in logic programming correspond to complete, grounded and preferred extensions in ABA. The translation works by turning every negative (resp. positive) literals of an LP into an assumption (resp. contrary), and vice versa. By means of this simple transformation, our splitting algorithm can be adapted in the context of logic programming to obtain a divide-and-conquer approach beyond stable semantics, thus generalising the original result (Lifschitz and Turner 1994). As a by-product, we are then able to split normal logic programs under semantics based on partially stable, well-founded and regular models. ADFs (Brewka et al. 2018) are an expressive argumentation formalism, where each argument is associated with a propositional formula over arguments as variables as an acceptance condition. It is well-known that SETAFs can be interpreted as a special kind of ADFs with acceptance conditions in the form of a conjunction of disjunctive clauses of negated literals (Dvořák, Keshavarzi Zafarghandi, and Woltran 2023). That is, in principle we can apply ADF splitting to SETAFs. However, it is not clear that following the ADF approach the modified second framework again is of the desired (SETAF-like) form and whether one can avoid certain overheads in the simpler case of SETAFs. Upon closer inspection and with minor syntactic manipulation the ADF approach in the special case of SETAF-like frameworks is similar to introducing an artificial self-attacking argument that behaves similarly to Definition 14. However, such a trick is not needed for SETAFs in our case. Parikh (1999) introduced the concept of syntax splitting to characterize belief sets that contain independent pieces of information. In this setting, a syntax splitting consists

of a partition of the knowledge base into independent components, where each independent piece of information can be represented using only one part of these partition. In this paper, we have considered a notion of splitting which is more general, where the different parts of the knowledge base may not be independent from each other. For this, we provided suitable modifications to account for such interdependencies.

9

Conclusion and Future Work

In this paper, we have presented a modification-based approach to splitting assumption-based argumentation frameworks. First, we have introduced a splitting schema for SETAFs, to make splitting available on ABAFs after instantiation. Leveraging on the close connection between SETAFs and ABAFs, we have thus proposed a way to split when reasoning in ABA is performed indirectly via the corresponding argument graph. Moreover, to overcome the requirement of an instantiation and its associated costs, we have introduced a splitting schema that works directly on ABA knowledge bases. For both approaches, we have shown that extensions of a given ABAF can be obtained incrementally from its sub-frameworks, by means of simple syntactic modifications. Conversely, we can project an arbitrary extension of the whole framework to its subframeworks. Since this is bound to the specific structure of the underlying ABAF, we have considered a more general variant of splitting called parametrised splitting inspired by Baumann et al. (2012). Moreover, it is easy to see that each of the steps involved can be carried out efficiently and implemented on top of common ABA (or SETAF) solvers. Indeed, the splitting techniques introduced in the paper require only syntactic modifications to the sub-frameworks, which do not lead to an exponential increase in size (and may, in some cases, even reduce it). In particular, through the reduct and modification, we introduce at most one additional rule and one additional assumption with respect to the original sub-framework. In the case of parametrised splitting, the number of additional rules and assumptions is bounded by 2k for a k-splitting. Therefore, an obvious next step is to implement our algorithm and perform an experimental evaluation in the spirit of Baumann, Brewka, and Wong (2011), where executions with splitting result in an average acceleration of 60% in comparison to executions without splitting. In particular, we believe that parametrised splitting could be helpful in the context of the recently proposed Argumentative Causal Discovery (Russo, Rapberger, and Toni 2024). In fact, this framework faces a major challenge in terms of its scalability, as it exhibits suboptimal performance on larger instances.

Acknowledgments This paper is an extended and revised version of two workshop papers that were presented in SAFA 2024 (Buraglio et al. 2024a) and NMR 2025 (Buraglio 2025). We are grateful to the anonymous reviewers for their valuable comments and suggestions on a preliminary version of this paper. Moreover, the authors would like to thank Matthias König for his

contributions to the paper presented at SAFA 2024. This work has been supported by the Austrian Science Fund (FWF) under grant 10.55776/COE122 and by the European Union’s Horizon 2020 research and innovation programme (under grant agreement 101034440).

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A

Omitted Proofs

Proposition 1. Let (SF1 , SF2 , R3 ) be a splitting for a SETAF SF = (A, R) with SF1 = (A1 , R1 ) and SF2 = (A2 , R2 ). Let SF2′ = SF2E1 . ′ 1 1. If E1 ∈ cf(SF1 ) and E2 ∈ cf(modE R3 (SF2 )), then E1 ∪ E2 ∈ cf(SF ). 2. If E ∈ cf(SF ), then E1 = E ∩ A1 ∈ cf(SF1 ) and E ∩ A2 ∈ cf(SF2′ ).

Proof. (1.) We need to show for each (T, h) ∈ R1 ∪ R2 ∪ R3 that T ∪ {h} ̸⊆ E = E1 ∪ E2 . Let SF2′ = (A′2 , R2′ ) and SF2⋆ = (A⋆2 , R2⋆ ). If (T, h) ∈ R1 we immediately get T ∪ {h} ̸⊆ E, since we know E1 is conflict-free in SF1 . For (T, h) ∈ R2 there are two cases: either (a) the attack is removed when we construct the reduct or (b) the attack remains, i.e., (T, h) in SF2⋆ . Case (a) happens if some a ∈ T ∪ {h} is attacked by E1 , i.e., (T ∪ {h}) ∩ (E1 )+ R1 ∪R3 ̸= ∅. Then at least one argument a ∈ T ∪ {h} of the attack does not occur in the modification (i.e., (T ∪ {h}) ̸⊆ A⋆2 ), and since we assume E2 ∈ cf(SF2⋆ ) we know E2 ⊆ A⋆2 . Hence we obtain T ∪ {h} ̸⊆ E. For case (b) we get from E2 ∈ cf(SF ⋆ ) that at least one argument a ∈ T ∪ {h} is not in E2 , which also means T ∪ {h} ̸⊆ E. Finally, for (T, h) ∈ R3 we again consider two cases: (a) T ∩ A1 ⊆ E1 , and (b) T ∩ A1 ̸⊆ E1 . For case (a) we either have T ⊆ A1 in which case h ∈ (E1 )+ / E (since then R3 and we obtain h ∈ h∈ / A′2 while we know E2 ⊆ A′2 ), or if T ̸⊆ A1 we get an attack (T ∩A′2 , h) ∈ R2⋆ (if otherwise T ∩A′2 = ∅ this means we removed some a ∈ T ∩ A2 when constructing the reduct, which means a ∈ / E2 and consequently a ∈ / E), which since E2 ∈ cf(SF2⋆ ) either means T ∩ A′2 ̸⊆ E2 or h ∈ / E2′ , both give us T ∪ {h} ̸⊆ E. For case (b) we have T ∩ A1 ̸⊆ E1 , which means T ∪ {h} ̸⊆ E. (2.) Suppose now that E ∈ cf(SF ). From this we derive that E ∩ A1 ∈ cf(SF1 ) because every subset of a conflictfree set is also conflict-free. We now show that E ∩ A2 ∈ cf(SF2′ ). Given that E ∈ cf(SF ), then for all T ⊆ E ∩ A1 and a ∈ E ∩ A2 , we have (T, a) ∈ / R3 . Hence, no argument in E is deleted going from SF2 to the reduct SF2′ . Thus, we conclude that E ∩ A2 ⊆ A′2 . Moreover, by E ∈ cf(SF ) we know for each (T, h) ∈ R2 that T ∪ {h} ̸⊆ E which carries over to SF2′ , since the attacks in R2 may be removed, but are never changed. Finally, whenever for a link (T, h) ∈ R3 with T ∩ A1 ⊆ E we add an attack (T ∩ A′2 , h) ∈ R2′ when constructing the reduct, we also obtain (T ∩ A′2 ) ∪ {h} ̸⊆ E since otherwise T ∪{h} ⊆ E. Therefore, E ∩A2 ∈ cf(SF2′ ) concluding the proof. Theorem 2. Let (SF1 , SF2 , R3 ) be a splitting for a SETAF SF = (A, R) with SF1 = (A1 , R1 ), SF2 = (A2 , R2 ), and σ ∈ {stb, adm, com, pref, grd}. Further, let SF2⋆ = ′ 1 modE R3 (SF2 ). 1. If E1 ∈ σ(SF1 ) and E2 ∈ σ(SF2⋆ ), then E1 ∪ E2 ∈ σ(SF ). 2. If E ∈ σ(SF ), then E1 = E ∩ A1 ∈ σ(SF1 ) and E ∩ A2 ∈ σ(SF2⋆ ).

Proof. (stable). (1.) From Proposition 1 together with the assumptions that E1 ∈ stb(SF1 ) and E2 ∈ stb(SF2⋆ ), we know that E = E1 ∪ E2 ∈ cf(SF ). Let a ∈ A \ E, we + show that a ∈ ER . If a ∈ A1 by E1 ∈ stb(SF1 ) we get a ∈ + + (E1 )R1 , which immediately gives us a ∈ ER . If a ∈ A2 , ′ ⋆ ′ ⋆ either a ∈ A = A or a ∈ / A = A . If a ∈ / A′ = A⋆ this + can only be because a ∈ (E1 )+ R3 which gives us a ∈ ER . If ′ ⋆ ⋆ a ∈ A = A then by E2 ∈ stb(SF ) we get a ∈ (E2 )+ R2⋆ . ⋆ Hence, either E2 defeats (in SF2 ) a via some (T, a) ∈ R2 + (in which case a ∈ ER ) or via some other (T, a) ∈ R2⋆ \ R2 , which can only be the remaining part of an attack from R3 . Clearly in this case a ∈ / T (as then (T, a) would not defeat a), so we know (T, a) is not constructed from an undecided link in URE31 (which do not occur in stable semantics as A1 = E1 ∪ (E1 )+ R1 ). Instead, we must have obtained (T, a) while constructing the reduct, i.e., there is an attack (T ′ , a) ∈ R3 with T ′ ⊃ T , and T ′ ∩ A1 ⊆ E1 . From this we get T ′ ⊆ E, + + and consequently a ∈ ER . In all cases we get a ∈ ER , 3 + which means A = E ∪ E , i.e., E ∈ stb(SF ). (2.) Assume E ∈ stb(SF ). From this we know that ⊕ ER = A = A1 ∪ A2 . We first prove that E1 = E ∩ A1 ∈ stb(SF1 ). From Proposition 1 we know E ∩ A1 ∈ cf(SF1 ). Since (SF1 , SF2 , R3 ) is a splitting of SF we know that the only attacks towards arguments in A1 are from R1 , so we immediately get (E ∩ A1 )⊕ R1 = A1 , i.e., E1 ∈ stb(SF1 ). We know turn to prove E2 = E ∩ A2 ∈ stb(SF2⋆ ). First, notice that SF2⋆ = SF2′ because URE31 = ∅. From Proposition 1 we again obtain E2 ∈ cf(SF2′ ). Let a ∈ A′2 \ E2 . We + show a ∈ (E2 )+ R′ . Since E ∈ stb(SF ) we know a ∈ ER 2

+ + which means (a) a ∈ ER or (b) a ∈ ER . In case (a) 2 3 we have an attack (T, a) ∈ R2 with T ⊆ E2 , and since E2 ⊆ A′2 (which is because E2 ∈ cf(SF ′ )) and also a ∈ A′2 by assumption we know (T, a) ∈ R2′ , i.e., a ∈ (E2 )+ R2′ . If (b) is the case we know that there is some (T, a) ∈ R3 with T ⊆ E, i.e., T ∩ A1 ⊆ E1 . Clearly since E is conflict-free ′ in SF we have T ∩ (E1 )+ R1 ∪R3 = ∅ which means in SF2 ′ ′ we have an attack (T ∩ A2 , a). Since T ∩ A2 ⊆ E2 we get + ′ a ∈ (E2 )+ R2′ . In both cases we get E2 ∪ (E2 )R2′ = A2 , i.e., E2 ∈ stb(SF2′ ) = stb(SF2⋆ ). (admissible). (1.) Since admissibility implies conflictfreeness, we know from Proposition 1 that E = E1 ∪ E2 ∈ cf(SF ). We need to show that E defends itself in SF , i.e. for all a ∈ E, if (T, a) ∈ R1 ∪ R2 ∪ R3 , then (T ′ , t) ∈ R1 ∪ R2 ∪ R3 for T ′ ⊆ E and t ∈ T . Consider an argument a ∈ E1 . E1 defends a from each attack in R1 towards a since E1 ∈ adm(SF1 ). Therefore, E1 ∈ adm(SF ). Consider now an argument a ∈ E2 and an arbitrary attack (T, a) ∈ R2 ∪ R3 towards a. If T ∩ (E1 )+ R1 ∪R3 ̸= ∅ we know a is defended (in SF ) by E1 against (T, a) and we are done, hence, we proceed with the assumption T ∩(E1 )+ R1 ∪R3 = ∅. This means that either (T ∩ A′2 , a) ∈ R2⋆ (via the reduct) or ((T ∩ A′2 ) ∪ {a}, a) ∈ R2⋆ (via the modification). Since a ∈ E2 and E2 ∈ adm(SF2⋆ ) we know there is a counterattack in R2⋆ which defends a. Even in case ((T ∩ A′2 ) ∪ {a}, a) ∈ R2⋆ this counter-attack cannot be against a since

this violates conflict-freeness of E2 in SF2⋆ . Hence, there is some (S, t) ∈ R2⋆ s.t. S ⊆ E2 and t ∈ T ∩ A′2 with t ∈ / S. Hence, either (a) (S, t) ∈ R2 in which case a is defended by E in SF or (b) there is some (S ′ , t) ∈ R3 with S ′ ⊃ S s.t. S ′ ∩ A1 ⊆ E1 , in which case a is defended (in SF ) by E via the attack (S ′ , t) since then S ′ ⊆ E1 ∪ E2 . In any case we showed that a is defended in SF by E, i.e., E ∈ adm(SF ). (2.) By Proposition 1 we get E1 = E ∩ A1 ∈ cf(SF1 ) and E2 = E ∩ A2 ∈ cf(SF2′ ). Since E is defends itself in SF we get E ∩ A1 ∈ adm(SF1 ) because (SF1 , SF2 , R3 ) is a splitting of SF , i.e. no argument in E ∩ A1 is attacked by a subset of A2 or defended by E ∩ A2 . That is, in SF1 every attack towards an argument in E∩A1 is countered by E∩A1 . It remains to show that E ∩ A2 ∈ adm(SF2⋆ ). Consider now an argument a ∈ E2 and an arbitrary attack (T, a) ∈ R2⋆ against a. This attack (T, a) either corresponds to an attack (T, a) ∈ R2 or (T ′ , a) ∈ R3 with T ′ ⊃ T \ {a} (which accounts for both the case of addition in the reduct and the modification). In both cases we have that T ∩ (E1 )+ R1 ∪R3 = ∅ (or T ′ ∩ (E1 )+ = ∅, resp.) as otherwise (T, a) would R1 ∪R3 not be in R2⋆ . However, since a is defended by E in SF , there is a counter-attack (S, t) ∈ R2 ∪ R3 s.t. S ⊆ E and t ∈ (T \ {a}) (or t ∈ (T ′ \ {a}), resp.). If (S, t) ∈ R2 then from S ⊆ E and E2 ⊆ A′2 (which we get from E2 ∈ cf(SF2′ ) via Proposition 1) and the fact that then (S, t) ∈ R2′ since S ∪ {t} ⊆ A′2 we get that E2 defends a via (S, t) against (T, a) in SF2⋆ . If (S, t) ∈ R3 since S ⊆ E we have S ∩ A1 ⊆ E1 , and hence we get an attack (S ∩ A′2 , t) ∈ R2′ which again defends a against (T, a) in SF2⋆ . Hence, in every case a is defended in SF2⋆ , i.e., E2 ∈ adm(SF2⋆ ). (complete). (1.) Given statement 1 of admissible semantics proven above, we only need to show that a ∈ E1 ∪ E2 for all a ∈ A defended by E1 ∪ E2 in SF . Assume towards contradiction that there is an a ∈ (A1 ∪ A2 ) \ (E1 ∪ E2 ) defended by E1 ∪ E2 . From E1 ∈ com(SF1 ), we know that a ∈ / A1 \ E1 . Hence, a ∈ A2 \ E2 and, because (SF1 , SF2 , R3 ) is a splitting and E1 ∪ E2 ∈ cf(SF ), we obtain a ∈ A′2 \ E2 . Indeed, if a ∈ (E1 )+ R3 , then E1 ∪ E2 defends a from an attack of E1 , which is against conflictfreeness of E1 ∪ E2 . Consider now possible attacks scenarios towards a: if a is not attacked, then it would be in every complete extension, hence E2 ∈ / com(SF2⋆ ). If a is attacked by some set of arguments T , then (T, a) ∈ R2 or (T, a) ∈ R3 . We show that both cases lead to a contradiction. Consider now (T, a) ∈ R2 . Again, in this case we distinguish three attack scenarios: 1. (T, a) ∈ R2 with T ∩ (E1 )+ R1 ∪R3 ̸= ∅. Since a is defended by E1 ∪ E2 , such attacks are countered by E1 via a link. Given that (T, a) ∈ / R2⋆ (eliminated by the reduct), a is vacuously defended by E2 in SF2⋆ . Thus, E2 ∈ / com(SF2⋆ ). 2. (T, a) ∈ R2 \ {(a, a)} with T ∩ (E1 )+ R1 ∪R3 = ∅. This means there is a counter-attack (S, t) ∈ R2 ∪ R3 with S ⊆ E1 ∪ E2 . Given that the reduct and modification do not eliminate such attacks, a is defended by E2 in SF2⋆ . Thus, E2 ∈ / com(SF2⋆ ). 3. (T, a) = (a, a) ∈ R2 with T ∩ (E1 )+ R1 ∪R3 = ∅. From

SETAF Fundamental Lemma (Nielsen and Parsons 2006) and the assumption that a is defended by E1 ∪ E2 , we get that E1 ∪E2 ∪{a} is an admissible (and thus conflict-free) extension in SF . Since a is a self-attacking argument, we derive a contradiction. All of the above derive a contradiction. Therefore, we now consider the case where (T, a) ∈ R3 . Since E1 ∪E2 defends a, we know that for some t ∈ T ∩ A1 , (E1 , t) ∈ R1 or for some t ∈ T ∩ A2 and S ⊆ E1 ∪ E2 , (S, t) ∈ R3 . In the first case, the reduct of SF2 does not contain (T ∩ A′2 , a) ′ because T ∩ (E1 )+ R1 ̸= ∅. Hence, a is unattacked in R2 . For + + the same reason, and given that (E1 )R1 ∪R3 ⊇ (E1 )R1 , we also know that T ∩(E1 )+ / URE31 and R1 ∪R3 ̸= ∅. Thus, (T, a) ∈ ⋆ a is unattacked in R2 . Again, a is vacuously defended by E2 in SF2⋆ and E2 ∈ / com(SF2⋆ ). Contradiction. Consider now the case where (S, t) ∈ R3 for some S ⊆ E1 ∪ E2 and t ∈ ′ T ∩ A2 . if S ⊆ E1 , then T ∩ (E1 )+ / R3 ̸= ∅ and (T ∩ A2 , a) ∈ ′ ′ ⋆ R2 . For the same reason as before, (T ∩A2 , a) ∈ / R2 . Hence, a is vacuously defended by E2 in SF2⋆ and E2 ∈ / com(SF2⋆ ). If S ̸⊆ E1 , then (S ∩ A′2 , t) ∈ R2′ because S ∩ A′2 ̸= ∅, t ∈ A′2 , S ∩ A1 ⊆ E1 and S ∩ (E1 )+ R1 ∪R3 = ∅. We derive directly that (S ∩ A′2 , t) ∈ R2⋆ since the modification does not delete attacks. Hence, E2 defends a in SF2⋆ . Finally, this contradicts our hypothesis that E2 ∈ com(SF2⋆ ), concluding the proof. (2.) Admissibility of E1 = E ∩ A1 and E2 = E ∩ A2 has been shown above. We need to show that for all a defended by E1 in SF1 and by E2 in SF2⋆ , we have a ∈ E1 and a ∈ E2 respectively. Let us consider E1 first. Towards contradiction, assume there is an a ∈ A1 \ E1 such that a is defended by E1 . This implies that a is such that a ∈ A1 ∪ A2 \ E and a is defended by E, in contradiction with the completeness of E in SF . Consider now E2 . As before, we need to show that there is no a ∈ A′2 \ E2 such that E2 defends a in SF2⋆ . Again, assume that there is such an a ∈ A′2 \ E2 . As before, we consider possible attack scenarios towards a. If a does not receive any attack in SF2⋆ , then for every attack (T, a) ∈ R2 ∪R3 it holds that T ∩(E1 )+ R1 ∪R3 ̸= ∅ ((T, a) was eliminated by the reduct). Hence, E1 defends a in SF , in contradiction with E ∈ com(SF ). Assume now that a receives an attack in SF2⋆ , then for all (T, a) ∈ R2⋆ we have (S, t) ∈ R2⋆ for some S ⊆ E2 and t ∈ T , as we assume that E2 defends a in R2⋆ . Note that t ∈ / S, as otherwise (S, t) would not counter the attack (T, a) which we assumed. But we know (S, t) ∈ R2⋆ corresponds to some attack in R2 ∪R3 . If (S, t) ∈ R2 we have that E defends a in SF (via (S, t)), a contradiction to E ∈ com(SF ). If on the other hand there is some (S ′ , t) ∈ R3 with S ′ ⊃ S we know that also S ′ ∩ A1 ⊆ E1 , as otherwise we would have t ∈ S (if (S, t) was introduced in via the modification), which we already ruled out. Finally, the attack (T, a) is either in R2 or corresponds to some attack (T ′ , a) ∈ R3 with T ′ ⊆ T \ {a}, in both cases SF defends a via (S, t) or (S ′ , t) against the attack. Hence, we derive a contradiction to E ∈ com(SF ). As every possible way a could be defended by E2 in SF2⋆ but not in E2 leads to a contradiction, this cannot be the case, hence, E2 ∈ com(SF2⋆ ).

(preferred). (1) From statement 1 for admissible semantics above, we derive that E1 ∪ E2 ∈ adm(SF ). Moreover, from hypothesis we have that there is no S1 ∈ adm(SF1 ) such that S1 ⊃ E1 and no S2 ∈ adm(SF2⋆ ) such that S2 ⊃ E2 . We need to prove that there is no S ∈ adm(SF ) such that S ⊃ E = E1 ∪ E2 . Towards contradiction, suppose there is such an S. Then S1 = S ∩ A1 ⊃ E1 or S2 = S ∩ A2 ⊃ E2 . Consider the first case. Since E1 ∈ pref(SF1 ) by hypothesis, it must hold that S1 ∈ / adm(SF1 ). However, this is in contradiction with statement 2 shown above for the admissible semantics (i.e. if S ∈ adm(SF ) and (SF1 , SF2 , R3 ) is a splitting for SF , then S ∩ A1 ∈ adm(SF1 )). Consider now the case where S2 ⊃ E2 . We can assume S ∩ A1 = E1 , as otherwise we derive a contradiction as above. Similarly to the case before, it must hold that S2 ∈ / adm(SF2⋆ ). Again, since we assumed S ∈ adm(SF ), then it must hold that S2 ∈ adm(SF2⋆ ) (statement 2 of admissible semantics). Both directions lead to a contradiction, hence there is no S ∈ adm(SF ) such that S ⊃ E1 ∪ E2 . Thus we conclude that E1 ∪ E2 ∈ pref(SF ). (2.) By hypothesis, we have E ∈ pref(SF ) and hence, there is no S ∈ adm(SF ) such that S ⊃ E. Moreover, by statement 2 of admissible semantics, we get E ∩ A1 ∈ adm(SF1 ) and E ∩ A2 ∈ adm(SF2⋆ ). Consider now E ∩ A1 . By directionality of preferred semantics (Dvořák et al. 2024) we obtain E ∩ A1 ∈ pref(SF1 ). For E ∩ A2 , assume now there is an S2 ∈ adm(SF2⋆ ) such that S2 ⊃ E ∩ A2 . For statement 1 of admissible semantics, (E ∩ A1 ) ∪ S2 is admissible in SF which contradicts the maximality of E. This conclude the proof. (grounded). (1) Since the grounded extension is also complete, we only need to show that E1 ∪ E2 is the minimal complete extension in SF . Suppose the contrary is true: there is a set S ∈ com(SF ) such that S ⊂ E1 ∪ E2 . Hence, S1 = S ∩ A1 ⊂ E1 or S2 = S ∩ A2 ⊂ E2 . Consider the first case. From the statement 2 of complete semantics above, we derive that S1 ∈ com(SF1 ). But this contradicts our hypothesis that E1 ∈ grd(SF1 ), since S1 would be the ⊆-minimal complete extension of SF1 . Hence, S1 = E1 . For the second case, we can assume S ∩ A1 = E1 (as otherwise we derive a contradiction via the first case) and we deduce again from statement 2 of complete semantics that S2 ∈ com(SF2⋆ ). However, by hypothesis we have that E2 = grd(SF2⋆ ), which is incompatible with that fact that S2 ⊂ E2 . Both cases lead to a contradiction, hence we derive that E1 ∪ E2 = grd(SF ). (2) By hypothesis, we have E ∈ grd(SF ) and hence, there is no S ∈ adm(SF ) such that S ⊂ E. Moreover, by statement 2 of complete semantics, we get E ∩ A1 ∈ com(SF1 ) and E ∩ A2 ∈ com(SF2⋆ ). Consider now E ∩ A1 . By directionality of grounded semantics (Dvořák et al. 2024) we obtain E ∩ A1 ∈ grd(SF1 ). For E ∩ A2 , assume now there is an S2 ∈ com(SF2⋆ ) such that S2 ⊂ E ∩ A2 . For statement 1 of complete semantics, (E ∩ A1 ) ∪ S2 is complete in SF which contradicts the minimality of E. This conclude the proof. Proposition 2. Let D be an ABAF and S a set of literals that splits D into D1 and D2 . For every derivation T ⊢R a

with a ∈ A1 , it holds that R ⊆ R1 and T ⊆ A1 . Proof. The statement follows from Definition 11, by induction on the depth k of the tree rooted in a with leaves T ∪ ⊤. By definition, there is a finite tree rooted in a and leaves T or ⊤. For k = 1, we have one rule r : a ← T where T = {b1 , . . . , bm }. Since a ∈ A1 and S = atom(S), we know a ∈ S. Hence, T ⊆ S and r ∈ R1 by definition of splitting. Assume the statement holds for depth k. We show that at depth k ′ = k + 1, R ⊆ R1 and T ⊆ A1 . At depth k there is at some rules {r1 , . . . , rk } ⊂ R with ri : head(ri ) ← Bi (1 ≤ i ≤ k). Again, since atom(head(ri )) ⊆ S and S = atom(S), we get Bi ⊆ S Sk and ri ∈ R1 for all i. Thus i=1 Bi ⊆ T , concluding T ⊆ A1 . Proposition 3. Let S ⊆ L be a splitting set of an ABAF D E1 1 into D1 and D2 . Moreover, let D2⋆ = modE D1 (D2 ). 1. If E1 ∈ cf(D1 ) and E2 ∈ cf(D2⋆ ), then E1 ∪ E2 ∈ cf(D). 2. If E ∈ cf(D), then E ∩ A1 ∈ cf(D1 ) and E ∩ A2 ∈ cf(D2E ). Proof. For notational convenience, let E = E1 ∪ E2 and let D2′ = (L2 , R′2 , A2 , 2 ) be the reduct of D2 w.r.t. E1 = + E ∩ A1 . Moreover, we adapt SR = {a | S ⊢R a} and ⊕ + SR = S ∪ SR from SETAFs. (1.) To prove the statement we need to show that there is no a ∈ E1 ∪E2 and and R ∈ R such that E1 ∪E2 ⊢R a. Towards contradiction, assume there is indeed such an a. Thus either (i) a ∈ E1 or (ii) a ∈ E2 . Assume (i) is true, that is ∃a ∈ E1 such that E1 ∪E2 ⊢R a and R ∈ R. From Proposition 2, we know that E2 = ∅ and R ⊆ R1 . Thus, E1 ⊢R a, in contradiction with E1 ∈ cf(D1 ). Assume now that (ii) is true, i.e. ∃a ∈ E2 and R ∈ R such that E1 ∪ E2 ⊢R a. Hence, there is a tree-derivation τ from E1 ∪E2 ∪{⊤} rooted in a and a non-empty set of rules R2 = R∩R2 . For each rule r ∈ R2 , there are three possible outcomes when computing D2⋆ : (a) r does not get removed when computing the reduct; (b) r gets removed and later added in the modification; (c) r gets removed for good. Assume (a) is the case. If a rule r is not removed when computing the reduct, it is modified into a rule r′ ∈ R′2 such that body(r′ ) = body(r) \ T hD1 (E1 ) and head(r′ ) = head(r). Thus, body(r′ ) consists of elements E1 of E2 or atoms derivable⋆from it. Therefore, E2 ⊢R2 a and consequently E2 ⊢R2 a (more rules). Finally, we get E∈ / cf(D2⋆ ), contradicting our hypothesis. Assume now (b) is the case. By definition of derivation, this means that E2 derives a in D2⋆ only if xu ∈ E2 . However, this contradicts conflict-freeness of E2 in the modification. Finally, consider case (c). Since r gets removed, but not added in the modification, we infer that body(r) ∩ ISD1 (E1 ) ̸= ∅. Hence, either E1 ∩ body(r) ̸= ∅ or T hD1 ((E1 )+ R1 ) ∩ body(r) ̸= ∅. However, since E1 ∈ cf(D1 ), this means that either r is a dummy rule or that ∃b ∈ body(r) ∩ A1 ̸⊆ E1 . Thus, in both cases E1 ∪ E2 ̸⊢R a, contradicting our assumption. (2.) Suppose now that E ∈ cf(D). From this we derive that E ∩ A1 ∈ cf(D1 ) (subset of a conflict-free set). We now show that E ∩ A2 ∈ cf(D2′ ). Towards contradiction, assume

E ∩ A2 ∈ / cf(D2′ ). There is an a ∈ E ∩ A2 such that E ∩ R′2 A2 ⊢ a. By definition of reduct, we know that each r′ ∈ R′2 is obtained from a corresponding rule r ∈ R2 such that body(r) ⊆ body(r′ )∪T hD1 (E∩A1 ). Therefore, (E∩A1 )∪ (E∩A2 ) ⊢R1 ∪R2 a. By definition of splitting, we know that R = R1 ∪ R2 and E = (E ∩ A1 ) ∪ (E ∩ A2 ), deriving E ⊢R a, and finally E ∈ / cf(D). Contradiction. We borrow from SETAFs the notions of projection, influence and the related principle of Directionality (Dvořák et al. 2024), and apply them to ABA. This will later be used to prove the splitting theorem for ABAFs. Definition 18 (projection). Let D = (L, R, A, ) be an ABAF with deductive system (L, R) and S ⊆ L a set of sentences such that S = atom(S). We define the projection D↓S of D on S as the ABAF induced by the deductive system (S, R↓S ) with R↓S = {r ∈ R | head(r) ∈ S, body(r) ⊆ S}. Further, D↓S has for assumption set S ∩ A and contrary function S the restriction of on S (i.e. aS = a for all a ∈ S). Definition 19 (influence). Let D = (L, R, A, ) be an ABAF. An assumption a ∈ A influences b ∈ A if there is a derivation S ⊢ b where a ∈ S. Moreover, a set U ⊆ A is uninfluenced in D if no a ∈ A \ U influences any b ∈ U . We denote the set of uninfluenced sets as US(D). Principle 1 (Directionality). A semantics σ satisfies directionality if for all ABAFs D and every U ∈ US(D) it holds that σ(D↓U ) = {E ∩ U | E ∈ σ(D)}. Observation 2. Due to existing results of semantic equivalence between ABAFs and SETAFs, we can trivially infer that for any flat ABAF D and semantics σ ∈ {cf, adm, com, pref, grd}, σ satisfies Directionality. Specifically, the notions of projection corresponds to the SETAF counterpart. In fact, for a set S ⊆ L, we have SFD↓S = (SFD )↓A where A is the set of arguments corresponding to A ∩ S. Likewise, the notion of influence in ABA is captured by the homonymous SETAF notion since a ∈ A influences b ∈ A when the corresponding argument a ∈ AD participates to an attack towards b ∈ AD . Theorem 3. Let S be a splitting set for an ABAF D into D1 and D2 and σ = {stb, adm, com, pref, grd}. Further, let E1 ⋆ 1 modE D1 (D2 ) = D2 . 1. If E1 ∈ σ(D1 ) and E2 ∈ σ(D2⋆ ), then E1 ∪ E2 ∈ σ(D). 2. If E ∈ σ(D), then E ∩ A1 ∈ σ(D1 ) and E ∩ A2 ∈ σ(D2⋆ ). Proof. In what follows, we prove 1. and 2. for each semantics. (stable). (1.) First notice that since E1 ∈ stb(D1 ), E1 we have UD = ∅, and consequently A′2 = A⋆2 . (1.) 1 From Proposition 3 together with the hypotheses that E1 ∈ stb(D1 ) and E2 ∈ stb(D2⋆ ), we know that E1 ∪ E2 ∈ cf(D). + Thus, for any a ∈ A \ E, we show that a ∈ ER , i.e. E ⊢R a for some R ⊆ R. We proceed by cases. Let a ∈ A1 . From hypothesis we know that E1 ⊢R1 a for some R1 ⊆ R1

+ which immediately implies a ∈ ER . Let a ∈ A2 . From 1 R2 a for some R2 ⊆ R′2 . hypothesis, we know that E2 ⊢ ′ ′ Thus, for each rule r ∈ R2 there is a rule r ∈ R2 such that body(r) ⊆ body(r) ∪ T hD1 (E1 ). Hence, it follows directly that E1 ∪ E2 = E ⊢R a for some R ⊆ R1 ∪ R2 = R. ⊕ (2.) Assume E ∈ stb(D). From this we know that ER = A = A1 ∪ A2 . We first prove that E1 = E ∩ A1 ∈ stb(D1 ). From Proposition 3 we know E ∩ A1 ∈ cf(D1 ). Moreover, from Proposition 2, we know that any set of assumptions which is not entirely contained in A1 attacks a ∈ A1 via rules in R1 , therefore we get E∩A1 ⊢R1 a for all a ∈ A1 \E for some R1 ⊆ R1 . Hence, E1 ∈ stb(D1 ). We know turn to prove E2 = E ∩ A2 ∈ stb(D2′ ). We know conflict-freeness holds from Proposition 3. Hence, we only need to show that ′ for every a ∈ A′2 \ E2 , E2 ⊢R2 a for some R2′ ⊆ R′2 . Since E ⊢R a in D, we have two possibilities: (a) E1 = ∅ or (b) E1 ̸= ∅. If (a) holds, we get R ⊆ R2 and E = E2 ⊢R a where E2 ⊆ A′2 and a ∈ A′2 . Thus, E2 ⊢R a holds for some R ⊆ R′2 . If (b) holds, E1 ∪ E2 ⊢R a in D. Therefore, each rule r ∈ R ∩ R2 has a corresponding rule r′ ∈ R′2 such that body(r′ ) = body(r) \ T hD1 (E1 ). Since E1 ∪ E2 ∈ cf(D) by hypothesis, we know that T hD1 (E1 ) ∩ E2 = ∅. Hence, ′ (E \ E1 ) ⊢R2 a where R2′ ⊆ R′2 . In both cases we have ′ ′ E2 ∪ (E2 )+ R′2 = A2 , concluding E2 ∈ stb(D2 ). (complete). (1.) Given statement 1 of admissible semantics proven above, we only need to show that a ∈ E1 ∪ E2 for all a ∈ A defended by E1 ∪ E2 in D. Assume towards contradiction that there is an a ∈ (A1 ∪ A2 ) \ (E1 ∪ E2 ) defended by E1 ∪ E2 . From E1 ∈ com(D1 ), we know that a∈ / A1 \ E1 . Hence, a ∈ A2 \ E2 . Indeed, if E1 ⊢ a, then E1 ∪ E2 defends a from an attack of E1 , which is against conflict-freeness of E1 ∪ E2 . Consider now possible attacks scenarios towards a: if a is not attacked, then it would be in every complete extension, hence E2 ∈ / com(D2⋆ ). Consider now the case where a is attacked by some set of assumptions T , i.e. T ⊢R a. If T ⊆ A2 , by definition of splitting set we have R ⊆ R2 . In this case we distinguish two attack scenarios:

1. T ̸= {a}. Since E1 ∪ E2 defends a, there is an S ⊆ ′ E1 ∪ E2 such that S ⊢R t for some t ∈ T and R′ ⊆ R. By effect of the reduct, no rule r ∈ R′ ∩ R2 gets removed. Otherwise, there is a rule r ∈ R′ ∩ R2 and a sentence p ∈ body(r) ∩ L1 such that p ∈ / T hD1 (E1 ). ′ This entails S ̸⊢R t. Thus, for each such r we have 1 r′ : head(r) ← body(r) \ T hD1 (E1 ) in RE 2 . ThereE1 fore, S \ E1 ⊢ t in D2 . Moreover, since no rule is removed, no further attacks are added towards a by the modification. We conclude that E2 defends a in D2⋆ , i.e. E2 ∈ / com(D2⋆ ). Contradiction. 2. T = {a}. From the Fundamental Lemma (Cyras et al. 2018) (Thm. 2.13) and the hypothesis that a is defended by E1 ∪ E2 , we get that E1 ∪ E2 ∪ {a} is an admissible (and thus conflict-free) extension in D. Since a is a selfattacking argument, we derive a contradiction. If T ̸⊆ A2 , since E1 ∪ E2 defends a, we know that for some t ∈ T ∩A1 , E1 ⊢ t or for some t ∈ T ∩A2 and S ⊆ E1 ∪E2 ,

′

S ⊢R t. In the first case, every rule r ∈ R2 whose body contains t (or a sentence derivable from it) gets removed by the reduct. This is guaranteed because t ∈ / T hD1 (E1 ) given that E1 is conflict-free. Therefore, we conclude that a is not derivable in D2E1 (a is unattacked). Further, each rule r removed under the reduct is not reintroduced via the modification, since t ∈ (E1 )+ R1 from hypothesis, and therefore body(r) ∩ ISD1 (E1 ) ̸= ∅. As a result, a is not derivable in D2⋆ , meaning that a is vacuously defended by E2 , making E2 not complete. Contradiction. Consider now the second case. It can be easily derived that E2 defends a in D2⋆ through similar considerations case (1) above. (2.) E1 = E ∩ A1 and E2 = E ∩ A2 has been shown to be admissible sets above. We need to show that for all a defended by E1 in D1 and by E2 in D2⋆ , we have a ∈ E1 and a ∈ E2 respectively. Let us consider E1 first. Towards contradiction, assume there is an a ∈ A1 \ E1 such that a is defended by E1 . This directly implies that a ∈ A1 ∪ A2 \ E and a is defended by E, in contradiction with the completeness of E in D. Consider now E2 . As before, we need to show that there is no a ∈ A2 \ E2 such that E2 defends a in D2⋆ . Again, assume that there is such an a ∈ A2 \ E2 . As before, we consider possible attack scenarios towards a. If a does not receive any attack in D2⋆ , then for every T ⊆ A and R ⊆ R2 such that T ⊢R a in D, then for some r ∈ R it holds that body(r) ∩ ISD1 (E1 ) ̸= ∅ (r was eliminated by the reduct and not reintroduced). Hence, E1 defends a in D, in contradiction with E ∈ com(D). Assume now that a receives an attack in D2⋆ , then for all T ⊆ A2 and R ∈ R⋆2 such that T ⊢R a, there is an S ⊆ E2 and R′ ∈ R⋆2 such ′ that S ⊢R t with t ∈ T (E2 defends a in D2⋆ ). Note that ′ {t, xu } ̸⊆ S, as E2 ∈ cf(D2⋆ ). But we know S ⊢R t cor′′ responds to some derivation S ′ ⊢R t with R′′ ⊆ R and S ′ ⊇ S. If S ′ = S, we know that E defends a in D, contradicting the hypothesis that E ∈ com(D). If on the other hand S ′ ⊃ S, we know that also S ′ ∩ A1 ⊆ E1 , as otherwise some rule r ∈ R′′ would be removed by the reduct. Finally, for the attack T ⊢R a, we have two distinct possibilities: either R ⊆ R2 (the derivation is unaffected by reduct and modification) or it corresponds to some derivation T ′ ⊢ a with T ′ ∩ A1 ̸= ∅ and T ′ ⊆ T \ {xu }. In both cases D defends a via S or S ′ against the attack. Hence, we derive a contradiction to E ∈ com(D). Every possible way in which a could be defended by E2 in D2⋆ , but not in E2 , leads to a contradiction. Hence, E2 ∈ com(D2⋆ ). The proof for preferred and grounded semantics is constructed in the same way as the one for SETAFs, by replacing arguments with assumptions and appealing to Observation 2 above. In particular, observe that for the splitting set S, we have D↓S = D1 .