Conceptio › Archive › arXiv CS

arXiv CSopen access

Splitting Argumentation Frameworks with Collective Attacks and Supports

2026 · arxiv_cs

arXiv CS · Papers · License: Open Access · 2026

Open Source ↗Direct PDF ↓

knowledge-representation reasoning

artificial intelligence, reasoning, knowledge representation

Splitting Argumentation Frameworks with Collective Attacks and Supports

arXiv:2604.28112v1 [cs.AI] 30 Apr 2026

Matti Berthold1 , Lydia Blümel1 , Giovanni Buraglio2 , Anna Rapberger3 1 FernUniversität in Hagen, Germany 2 TU Wien, Austria 3 TU Dortmund, Germany {lydia.bluemel, matti.berthold}@fernuni-hagen.de, [email protected], [email protected] Abstract This work proposes novel splitting techniques for argumentation formalisms that incorporate supports between defeasible elements. We base our studies on bipolar set-based argumentation frameworks (BSAFs) which generalize argumentation frameworks with collective attacks (SETAFs), as well as bipolar argumentation frameworks (BAFs), by incorporating both collective attacks and supports. Notably, BSAFs establish a crucial link to structured argumentation as they naturally capture general (potentially non-flat) assumption-based argumentation. The increase in expressiveness calls for diverse forms of splitting. We consider splits over collective attacks (thereby generalizing the recently proposed splitting techniques for SETAFs), splits over collective supports, as well as splits over both collective attacks and supports. We establish suitable splitting schemata and prove their correctness for the most common argumentation semantics.

Introduction

In the field of knowledge representation and reasoning, formal models of argumentation (Baroni et al. 2018) provide computational approaches for representing and reasoning about argumentative scenarios, and have been successfully applied across a wide range of domains such as medical decision-making, law and explainable AI (Atkinson et al. 2017; Cyras et al. 2021; Leofante et al. 2024). In particular, abstract models of argumentation have gained significant popularity due to their flexible and rich modelling capabilities. Starting from Dung’s seminal work (Dung 1995), Argumentation Frameworks (AFs) have become a central formalism for capturing debates and reasoning over them, with the ultimate goal of extracting rational viewpoints. In Dungstyle AFs, arguments are treated as primitive entities, while their internal structure, e.g. the premises supporting their conclusions, is abstracted away. Conflicts between arguments are captured by a binary attack relation. Debates are thus represented via simple directed graphs where nodes are the arguments exchanged and edges keep track of their conflicts. Rational viewpoints are then identified by computing sets of jointly acceptable arguments, so-called extensions. Over the years, numerous generalizations of AFs have been proposed to enrich their simple structure and increase its modelling capabilities (Brewka et al. 2018; Amgoud et al. 2008; Nielsen and Parsons 2006; Modgil 2009). Notably, bipolar argumentation frameworks (BAFs) (Amgoud

et al. 2008; Boella et al. 2010) allow also for the explicit representation of support relations, enabling the modelling of scenarios in which arguments may reinforce one another. In parallel, argumentation frameworks with collective attacks (also called SETAFs) (Nielsen and Parsons 2006) have been introduced to capture situations in which an argument is insufficient to attack another on its own, but can do so jointly with others. More recently, bipolar set-based argumentation frameworks (BSAFs) (Berthold, Rapberger, and Ulbricht 2024) have been proposed as a unifying generalization of both BAFs and SETAFs by combining the accrual of arguments with a notion of support. With this, BSAFs can model scenarios in which arguments ‘join forces’ to support or defeat another argument collectively. In addition, they have been shown to faithfully capture rule-based argumentation formalisms such as general (non-flat) assumption-based argumentation (Berthold, Rapberger, and Ulbricht 2024). As a result, BSAFs offer a simple yet expressive means of representation for complex argumentative scenarios. However, similarly to AFs, BAFs, and SETAFs, bipolar set-based argumentation frameworks exhibit a high computational complexity for most standard reasoning tasks (Berthold, Rapberger, and Ulbricht 2024). In particular, the size of the search space may grow exponentially with the size of the framework, making direct computation of extensions infeasible in practice. To address this challenge, incremental reasoning techniques have been developed that decompose a framework into smaller subframeworks, compute their extensions independently, and then combine the results (Baumann 2011; Liao 2013; Baroni, Giacomin, and Liao 2014; Bengel and Thimm 2025; Blümel et al. 2025; Giacomin, Baroni, and Cerutti 2021). Among these techniques, splitting (Baumann 2011; Baumann et al. 2012; Linsbichler 2014; Buraglio et al. 2024; Buraglio 2025) has proven to be particularly effective in abstract argumentation (Baumann, Brewka, and Wong 2011). The central idea underlying splitting procedures is to compute the extensions of a given framework F by decomposing it into suitable sub-frameworks F1 and F2 and then computing the extensions of F1 and F2 independently. Naturally, this requires suitable modifications of F2 to take into account the acceptance status of the arguments in F1 . Moreover, although computational efficiency is the primary motivation for employing splitting techniques, they also provide

2005; Amgoud et al. 2008; Ulbricht et al. 2024). In particular, they incorporate notions of support and attack among arguments, and generalize both SETAFs and BAFs by modeling collective attacks and supports. Definition 1. A bipolar set-argumentation framework (BSAF) is a tuple F = (A, R, S), where A is a finite set of arguments, R ⊆ 2A × A is the attack relation and S ⊆ 2A × A is the support relation. A SETAF is a BSAF F = (A, R, S) with S = ∅; an AF is a SETAF with |T | = 1 for all (T, h) ∈ R. For an attack (support) (T, h), we call T the tail of the attack (support) and h the head of the attack (support). We call (T, h) a negative link if (T, h) ∈ R, a positive link if (T, h) ∈ S, and simply a link if (T, h) ∈ R ∪ S. If |T | = 1, we write (t, h) instead of ({t}, h), where T = {t}. Furthermore, given BSAFs F = (A, R, S) and F ′ = (A′ , R′ , S ′ ), we define set-operations component-wise; e.g., the union is defined as F ∪ F ′ = (A ∪ A′ , R ∪ R′ , S ∪ S ′ ). In contrast to the broad agreement on the interpretation of attacks, there exist several interpretations of support in the literature (Cohen et al. 2014; Cohen et al. 2018; Polberg 2016). BSAFs adopt a form of deductive support (Boella et al. 2010; Berthold, Rapberger, and Ulbricht 2024). A central notion in this context is the closure of a set. Definition 2. Given BSAF F = (A, R, S) and E ⊆ A, let

a useful tool for the study of argumentation dynamics (Rapberger and Ulbricht 2023; Prakken 2023; Cayrol, de SaintCyr, and Lagasquie-Schiex 2010), as previously computed parts of the framework can be reused when new information is introduced (Baumann 2011; Liao, Jin, and Koons 2011). Splitting has been successfully applied to AFs and generalizations thereof; most crucial in our context, to SETAFs. Nevertheless, despite the increasing demand for efficient approaches that take into account positve links, the development of splitting techniques for argumentation formalisms that feature support relations, particularly in the presence of collective relations, remains largely unaddressed. In this paper, we start to fill this gap by investigating the possibilities and limitations of splitting in AFs that feature both collective attacks and supports. The added expressiveness of BSAFs allows for a richer variety of cuts: a framework may be split along collective attacks, collective supports, or combinations thereof. We introduce the necessary ingredients for splitting BSAFs in a progressive manner, addressing the unique challenges that arise due to the presence of supports. We investigate splitting techniques for attacks and supports separately before we present a unified splitting pipeline that enables the incremental computation of extensions for arbitrary cuts of a BSAF. Our main contributions are as follows. • First, we develop a suitable attack splitting procedure for BSAFs by extending and revising the splitting procedure for SETAFs. We prove the correctness of our splitting schema wrt. the most common argumentation semantics. Notably, we show that splitting works only partially under grounded semantics. Section 3

suppF (E) := E ∪ {h ∈ A | ∃ (T, h) ∈ S : T ⊆ E}. S The closure of E is defined as cl F (E) := i≥1 suppiF (E); Hence, E is closed if clF (E) = E. Definition 3. Given BSAF F = (A, R, S), a set E ⊆ A defends a ∈ A if for each closed attacker E ′ ⊆ A of a, E attacks E ′ ; E defends E ′ if E defends each a ∈ E ′ . We omit the subscript F for cl if clear from context. Let us now head to BSAF semantics. A set E is conflictfree (E ∈ cf(F )) if it does not attack itself; E is admissible (E ∈ adm(F )) if it is conflict-free, closed and defends itself. Definition 4. Let F be a BSAF and let E ∈ adm(F ). • E ∈ com(F ) iff E contains every argument it defends; • E ∈ grd(F ) iff E is ⊆-minimal in com(F ); • E ∈ pref(F ) iff E is ⊆-maximal in adm(F ). • E ∈ stb(F ) iff E attacks each x ∈ A \ E. Given a BSAF F = (A, R, S) and a set of arguments + E ⊆ A, we denote ER := {h | ∃T ⊆ E : (T, h) ∈ R} and ⊕ + the range of E; by ER := ER ∪ ER . We omit R if clear from the context. Given a σ-extension E of F , we say that an argument a ∈ A is: accepted or in if a ∈ E, rejected, + ⊕ defeated or out if a ∈ ER and undecided if a ∈ / ER . Graphically, we depict the attack and the support relations of a BSAF via solid and dashed edges respectively. Example 5. We consider a BSAF F with arguments A = {a, b, c, d, e}, attacks R = {(f, c), ({d, e}, f ), (c, d)}, and support S = {({a, b}, c)}, as depicted below.

• Next, we investigate the splitting of supports. We define suitable modifications that enable splitting wrt. the common argumentation semantics. Notably, for preferred and grounded semantics, the splitting procedure selects only some, and not all, extensions. Section 4 • Finally, we combine the previous techniques and consider arbitrary splits over collective attacks and supports. We prove a general splitting theorem, thereby paving the way for the incremental computation under most of the semantics. The combined splitting inherits the limitations concerning preferred and grounded semantics. Section 5 All proofs are provided in the supplementary material.

Background

We recall BSAFs (Berthold, Rapberger, and Ulbricht 2024) which generalize SETAFs (Nielsen and Parsons 2006), and basics of splitting for SETAFs (Buraglio et al. 2024).

2.1

Bipolar SETAFs

Bipolar argumentation frameworks with collective attacks (BSAFs) (Berthold, Rapberger, and Ulbricht 2024) are a generalization of Dung’s abstract argumentation frameworks (AFs) (Dung 1995). They combine the ideas underlying argumentation frameworks with collective attacks (SETAFs) (Nielsen and Parsons 2006) and bipolar argumentation frameworks (BAFs) (Cayrol and Lagasquie-Schiex

e c

d f

E1′ = {c} on the right-hand side; the attacks and arguments not contained in the reducts are grayed out.

The arguments a and b jointly support c; thus, whenever {a, b} is accepted, the argument c must be true as well. Accepting c is, however, impossible: to defend c against f , both arguments d and e are required; only then, the joint attack ({d, e}, f ) fires. However, accepting both is not possible since c attacks d. Thus, a and b cannot be accepted together. It follows that F has no complete and grounded extension; the admissible sets are ∅, {a}, {b}, {e}, {a, e}, and {b, e}.

2.2

Definition 6. Let SF = (A, R) be a SETAF, SF 1 = (A1 , R1 ) and SF 2 = (A2 , R2 ) two sub-frameworks of SF such that A1 ∩A2 = ∅, A = A1 ∪A2 and R = R1 ∪R2 ∪R3 with R3 ⊆ {(T, h) ∈ R | T ∩ A1 ̸= ∅, T ⊆ A, h ∈ A2 }. The triple (SF 1 , SF 2 , R3 ) is called a splitting of SF and R3 its set of negative links. A link is undecided if no argument in its tail is defeated, but at least one is undecided. Example 7. We consider a splitting (SF 1 , SF 2 , R3 ) of a SETAF SF , as depicted below; SF 1 and SF 2 are left resp. right of the dotted line; the set of shared negative links of SF 1 and SF 2 is given by R3 = {({a, z}, x), ({b, z}, y)}.

x y z

URE31 := {(T, h) ∈ R3 | T ∩ (E1 )+ R = ∅, (T ∩ A1 ) ̸⊆ E1 }. + It holds that (E1 )+ R = (E1 )R1 ∪R3 since A1 is not attacked by links in R2 . Next, we recall the modification which accounts for the effects of the undecided links. Definition 11 (Modification). Let (SF1 , SF2 , R3 ) be a 1 splitting for SETAF SF and E1 ⊆ A1 . Let SF E = 2 E1 E1 E1 (A2 , R2 ) be the (E1 , R3 )-reduct of SF 2 and UR3 the set E1 ⋆ 1 of undecided links wrt. E1 . By modE R3 (SF 2 ) := SF 2 = E (A⋆2 , R2⋆ ) we denote the UR31 -modification (or simply modiE1 ⋆ ⋆ 1 fication) of SF E 2 s.t. A2 := A2 and R2 is given by:

x y

We begin by computing the reduct wrt. E1 . Since R2 = ∅, thus we only deal with the links in R3 . For the attack ({a, z}, x), we have {a, z}∩A1 = {a} ⊆ E1 , {a, z}\A1 ̸= ∅, and {a, z} is not attacked by E1 , thus the attack (z, x) is in the reduct. The attack ({b, z}, y) does not satisfy these conditions, thus it is not part of the reduct. The latter also holds for the reduct for E1′ ; in addition, the attack ({a, z}, x) is defeated since a is defeated. As per the second step, the notion of modification is introduced to account for possibly undecided arguments wrt. E1 . As these may affect other arguments in SF2 via the links, a notion of undecided links is introduce to keep track of such external influences on arguments in (the reduct of) SF2 . Definition 10 (Undecided Links). Given a splitting (SF 1 , SF 2 , R3 ) for a SETAF SF and a set E1 ⊆ A1 we define the set of undecided links wrt. E1 as:

Splitting SETAFs

x y

In their recent work, Buraglio et al. (2024) investigated splitting procedures for SETAFs, extending splitting procedures for AFs (Baumann 2011). A splitting of a SETAF SF = (A, R) separates the framework into two frameworks SF 1 and SF 2 which share attacks directed towards SF 2 .

Buraglio et al. (2024) developed a procedure that enables the incremental computation of the extensions of SF wrt. a given semantics σ. The extensions of SF can be computed as a combination of extensions of SF 1 and (an adjusted version of) SF 2 . The second sub-framework SF 2 gets modified on the basis of the information contained in the extension(s) of SF 1 . Such alteration is tailored to account for the prior accepted and rejected, and undecided arguments. In the literature (Baumann 2011; Buraglio et al. 2024), this is achieved in a two-step procedure, by appealing to the notions of socalled reduct and modification of the sub-framework SF2 . In the first step, the reduct takes care of the arguments in SF2 that are rejected wrt. E1 by deleting them, and modifies the links by projecting the part of the attack to SF2 .

E1 E1 1 R2E1 ∪ {((T ∩ AE 2 ) ∪ {h}, h) | (T, h) ∈ URc , h ∈ A2 }. 3

Taken together, when splitting up a SETAF, we either separate the tail of collective attacks or remove them altogether. Example 12. In Example 9, the link ({b, z}, y) is undecided for both E1 and E1′ ; resulting in the following modifications. c

Definition 8 (Reduct). Let (SF1 , SF2 , R3 ) be a splitting for a SETAF SF . We define the (E1 , R3 )-reduct (or simply reduct) of SF 2 for some extension E1 of SF 1 as the SETAF E1 E1 1 SF E 2 = (A2 , R2 ) where,

y b

A2E1 ={a ∈ A2 | a ∈ / (E1 )+ R3 } and E1 1 R2E1 ={(T, h) ∈ R2 | T ⊆ AE 2 , h ∈ A2 } ∪ {(T ∩ A2 , h) | (T, h) ∈ R3 , T ∩ A1 ⊆ E1 , E1 T ∩ (E1 )+ R3 = ∅, T \ A1 ̸= ∅, h ∈ A2 }

Example 9. We continue Example 7. Below, we depict the reduct wrt. E1 = {a} on the left-hand side, the reduct wrt. 3

x z

x y

Buraglio et al. (2024) proved the correctness of the procedure wrt. the semantics under consideration. They showed that (1) combining extensions of SF 1 and (the altered version of) SF 2 yields extensions of SF , and (2) extensions of SF induce extensions in SF 1 and (the altered version of) SF 2 , when restricted to the respective argument-sets. Theorem 13. Let (SF 1 , SF 2 , R3 ) be a splitting for a SETAF SF = (A, R) with SF 1 = (A1 , R1 ), SF 2 = (A2 , R2 ), and σ ∈ {stb, adm, com, pref, grd}. Below, we E1 1 let SF ⋆2 = modE R3 (SF 2 ).

1. E1 ∈ σ(SF 1 ) and E2 ∈ σ(SF ⋆2 ) implies E1 ∪E2 ∈ σ(SF ). 2. If E ∈ σ(SF ), then E1 = E ∩ A1 ∈ σ(SF 1 ) and E2 = E ∩ A2 ∈ σ(SF2⋆ ).

Example 17. Let us have a closer look into a part of the BSAF from Example 21. Consider the following attack splitting (G1 , G2 , U3 ) of the BSAF G ⊆ F (displayed left).

Example 14. We use the result to compute the admissible extensions of SF from Example 7 incrementally. Both sets E1 and E1′ are admissible in SF 1 . For E1 , {z} is the only (non-empty) admissible extension of SF ⋆2 ; therefore, {a, z} ∈ adm(SF ). For E1′ , the sets {x} and {x, z} are admissible. Using the splitting theorem, we correctly deduce that {c, x}, {c, z} and {c, x, z} are admissible in SF .

Definition 15 (Attack Splitting). Let F = (A, R, S) be a BSAF, F1 = (A1 , R1 , S1 ) and F2 = (A2 , R2 , S2 ) two subframeworks of F such that A1 ∩ A2 = ∅, A = A1 ∪ A2 , S = S1 ∪ S2 , and R = R1 ∪ R2 ∪ R3 with R3 ⊆ {(T, h) ∈ R | T ∩ A1 ̸= ∅, T ⊆ A, h ∈ A2 }. We call the triple (F1 , F2 , R3 ) an attack splitting of F . Moreover, we call R3 the set of negative links wrt. (F1 , F2 , R3 ). Ideally, we would like to transfer the results for SETAFs to the case of attack splitting for BSAFs. There are, however, certain subtleties that need to be taken care of. We illustrate these in the following example. Example 16. Consider the BSAF F below left and attack splitting (F1 , F2 , R3 ); F1 is left and F2 right of the dashed line. Let {d} = E1 ∈ adm(F1 ). We illustrate in the Figure below right how the notions of SETAF reduct and modification apply to the present example. t

t v

y d

w b

y z

The set E = {d, t, w} is admissible in G since d defends t against the closed set {a, b, w}. Note that {a, w} is not closed thus it is not necessary to defend t against ({a, w}, t). We depict G⋆2 wrt. E1 = {d} above on the right-hand side. Observe that {d} is admissible in G1 , but {t} is not admissible in G⋆2 . The issue is that the attack on t is indirectly defeated since its closure is attacked by d. This information, however, is lost when computing the modification G⋆2 . A negative link may be deemed as undecided under Definition 10, but become defeated due to some support in F1 , as the example shows. To account for this issue, we must deal with the negative links first in order to make explicit the information encoded in the support relation. Before we can construct the reduct, we alter F and apply the link-closing procedure, replacing all attacks (T, a) ∈ R3 with (cl(T ), a). Definition 18 (Closed Negative Links). Let F = (A, R, S) be a BSAF, F1 = (A1 , R1 , S1 ) and F2 = (A2 , R2 , S2 ) two sub-frameworks of F such that (F1 , F2 , R3 ) is an attack splitting of F . We define the set of closed negative links as R3c := {(clS (T ), h) | (T, h) ∈ R3 }. As a result, we are able to adapt the SETAF splitting schema while avoiding aforementioned issues concerning the interplay between negative links and supports in SF1 . Example 19. Let (G1 , G2 , U3 ) be as in Example 17. We modify U3 = {({a, w}, t)} as follows: first, we compute the closure of {a, w} and replace the result with the source of the attack, obtaining U3c = {({a, b, w}, t)} (left). Then, the reduct wrt. E1 = {d} makes the attack defeated. Thus, the closed negative link ({a, b, w}, t) is not undecided. We depict the link-closure (G1 , G2 , U3c ) and F2⋆ below.

In this section, we introduce a splitting algorithm for BSAFs. In particular, we take into account the situation where we can split over collective attacks only. Notably, the presence of support relations forces significant changes with respect to the SETAF notions of reduct and modification.

t w

Splitting over Collective Attacks

b d

First, we consider ({a, w}, t). By definition, it is an undecided link, therefore we apply the modification to create the attack ({w, t}, t). However, this makes {t} not admissible. Further, the same happens for the undecided link ({c, w}, v). However, the presence of support ({v, w}, u) makes {v} admissible again by defeating its closed attacker {v, w, u}. Hence, although ({c, w}, v) is truly an undecided link, the SETAF modification is not adequate. Finally, the link (d, x) triggers the elimination of x via the reduct, together with its support, leaving {y, z} as an admissible extension of F2E1 . However, {d, y, z} is not admissible in F since it is not even closed.

t w

b d

Now, {w, t} is admissible in G⋆2 and we retrieve the admissible extension {d, w, t} of G. Observe that closing the negative links preserves the semantics of the original framework. Proposition 20. Let F = (A, R, S) be a BSAF, (T, h) ∈ R an attack and let F ′ = (A, R′ , S) with R′ = (R \ {(T, h)}) ∪ {(cl(T ), h)}. Then σ(F ) = σ(F ′ ) for all σ ∈ {stb, adm, com, pref, grd}. After this preliminary step, we follow the splitting schema and appeal to the notions of reduct and modification. As observed in Example 21, the SETAF reduct cannot be directly inherited in our context. The reason is that the removal of

In what follows, we adapt the splitting schema and show how to solve the aforementioned issues. 4

W3c = W3 = {({c, w}, v)}, i.e., the attack is closed. Since E1 = ∅ is an admissible extension of I1 , we know that the attack ({c, w}, v) is an undecided link in W3c . Via the stanE1 dard SETAF UW c -modification, we would add the set-self 3 attack ({w, v}, v) to obtain I2⋆ (right).

arguments attacked by an extension of F1 imposed by the reduct may interfere with the closed sets in F2 . If defeated arguments in F2 are deleted, all their supporting sets would wrongly be considered closed in the reduct of F2 . Example 21. We consider another problematic part of the BSAF F from Example 21. Let H ⊆ F with a splitting (H1 , H2 , V3 ) as below left. Note that the link in V3 is closed. The only preferred extension of H1 is E1 = {d}. By simply applying the original notion of (E, V3 )-reduct, we obtain the framework H2E1 (right) which has one preferred extension E2 = {y, z}, from which we would erroneously derive that E1 ∪ E2 = {d, y, z} is a preferred extension of H. y y d

E1 ⋆ ⋆ ⋆ ⋆ 1 links wrt. E1 . By modE Rc (F2 ) := F2 = (A2 , R2 , S2 ) 3

we denote the UREc1 -modification (or simply modification) of 3

F2E1 = (A2 , R2E1 , S2 ) s.t. A⋆2 := A2 ∪ {∗0 }, S2⋆ := S2 and R2⋆ is given by: R2E1 ∪ {(∗0 , ∗0 )} ∪ {((T ∩ A2 ) ∪ {∗0 }, h) | (T, h) ∈ URE31 }. Example 26. Let (I1 , I2 , W3 ) as in Example 24 (left). The E1 ⋆ BSAF UW c -modification of I2 is I2 (right); here, v does not 3 defend itself against {w, ∗0 } and is therefore not admissible.

Note that in the updated version of the reduct, no arguments are deleted. On top of that, negative links are projected to their (possibly empty) part in F2 . Example 23. Let (H1 , H2 , V3 ) (left) be as in Example 21. The (E1 , V3c )-reduct consists of the framework H2E1 (right) which has two preferred extensions E2 = {y} and E2′ = {z}, from which we now obtain the two preferred extensions of H, i.e. E = {a, y} and E ′ = {a, z}. x

w c

v ∗0

We are now in the position to state the overall splitting procedure. Given an attack splitting (F1 , F2 , R3 ) for a BSAF F , we proceed as follows. 1. We close the negative links following Definition 18. We obtain a set of new negative closed links R3c . 2. We apply our BSAF reduct following Definition 22. As a result, we obtain two separate frameworks F1 and F2E1 . They have the same arguments A1 and A2 as before; all arguments in F2 that are defeated by attacks in F1 are attacked by the empty set. 3. To account for undecided links, we apply the modification from Definition 25 and preserve undecidedness in F2 . The BSAF F2 has been extended by the self-attacker ∗0 . After these considerations, we are now in possession of all the ingredients to prove the correctness of our splitting schema for BSAFs. As a first step towards it, we ensure that the adjusted version of R-reduct is not in conflict with the notion of closed set of arguments.

y d

Definition 25 (Modification). Let (F1 , F2 , R3 ) be an attack splitting for a BSAF F ; let R3c denote the closed negative links. Let E1 an extension of F1 . Take F2E1 as the (E1 , R3c )-reduct of F2 and UREc1 as the set of undecided

T ∩ (E1 )+ R1 ∪Rc = ∅, T ∩ A1 ⊆ E1 }

A new version of the UREc1 -modification is then needed to 3 deal with scenarios of this kind. Instead of using the target v of the attack to build a new self-attack in F2 , we introduce a self-attacking dummy argument ∗0 that jointly attacks v together with the right-most part of the undecided link. Formally, the BSAF UREc1 -modification is defined as follows.

R2E1 := R2 ∪ {(T ∩ A2 , h) | (T, h) ∈ R3c ,

As previously noticed, the argument v now defend itself against its only closed attacker {w, v, u} by attacking u. Hence, we erroneously conclude that {v} ∈ adm(I).

To account for this issue, we change the reduct as follows: instead of removing arguments defeated by E1 , we let them be attacked by the empty set. This syntactical difference preserves the defeated status of such arguments, while preserving its supports in F2 . In fact, for each defeated argument h which is supported by some set of arguments T in F2 , such an addition works as constraint: not all elements in T can be accepted, otherwise, h should be accepted as well. Definition 22 (R-reduct). Let (F1 , F2 , R3 ) be an attack splitting for a BSAF F ; let R3c denote the closed negative links. We define the (E1 , R3c )-reduct (or simply Rreduct) of F2 for some extension E1 of F1 as the BSAF F2E1 := (A2 , R2E1 , S2 ) with:

It remains to state the final modification to deal with the closed undecided links in R3c . For a given attack splitting (F1 , F2 , R3 ) for a BSAF F and a set E1 of arguments in F1 , we use the notion of undecided links as defined for SETAFs (see Definition 10, but applied to the BSAF splitting (F1 , F2 , R3c ), i.e., after closing the links in R3 . We are now almost ready to apply the modification relative to UREc1 . 3 There is, however, a final subtlety that we need to deal with. Example 24. Let us zoom into another part of the BSAF F from Example 21, as displayed below. We consider the attack splitting (I1 , I2 , W3 ) for the BSAF I ⊆ F (left), with 5

In F , the set {a, d} is admissible since d defends a against the closed attacker {b, c}. In F1 , however, the argument a is defeated by b. The acceptability of a in F1 cannot be accomplished without modifying F1 ; this is, however, not foreseen in the traditional approach to splitting. The underlying issue is that sets in F2 are not closed; as we have demonstrated in the above example, the acceptance status of arguments in F1 may thus depend on those of the arguments in F2 . Intuitively, the support in Example 31 has the effect of directing an attack from F2 to F1 . If we consider shared supports directed in the opposite direction, where the head h of each positive link (T, h) is contained in F1 , all sets in F1 are closed. Therefore, this problem cannot occur in this case. We will therefore focus our efforts on backward support splitting. Definition 32 (Support Splitting). Let F = (A, R, S) be a BSAF, F1 = (A1 , R1 , S1 ) and F2 = (A2 , R2 , S2 ) two subframeworks of F such that A1 ∩ A2 = ∅, A = A1 ∪ A2 , R = R1 ∪ R2 , and S = S1 ∪ S2 ∪ S3 with S3 ⊆ {(T, h) ∈ S | T ∩ A2 ̸= ∅, T ⊆ A, h ∈ A1 }. We call the triple (F1 , F2 , S3 ) support splitting of F . Moreover, we call S3 the set of positive links wrt. (F1 , F2 , S3 ). Example 33. We consider a similar example as before; now, the support is directed in the other direction. The support splitting (F1 , F2 , S3 ) for the BSAF F below consists of F1 (left), F2 (right) and a single shared support S3 = {(c, b)}.

Proposition 27. Let (F1 , F2 , R3 ) be an attack splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ); let R3c denote the closed negative links. Further, let E1 ⊆ A1 , F2E1 the (E1 , R3c )-reduct of F2 , F2⋆ = E1 ⋆ 1 modE Rc (F2 ), and E2 ⊆ A2 . 3

1. If E1 = clF1 (E1 ) and E2 = clF2⋆ (E2 ), then E = (E1 ∪ E2 ) \ {∗0 } = clF (E). 2. If E = clF (E) for some E ⊆ A, then E ∩ A1 = E1 = clF1 (E1 ) and E ∩ A2 = E2 = clF2⋆ (E2 ). From this, we get an attack splitting theorem for BSAFs. Theorem 28. Let (F1 , F2 , R3 ) be an attack splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ); let R3c denote the closed negative links. Let E1 1 σ ∈ {stb, adm, com, pref}. Further, let F2⋆ = modE Rc (F2 ) 3

where F2E1 is the R-reduct wrt. E1 .

1. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2⋆ ), then E1 ∪ E2 ∈ σ(F ). 2. If E ∈ σ(F ), then E1 = E ∩ A1 ∈ σ(F1 ) and E ∩ A2 ∈ σ(F2⋆ ). For grounded semantics, only the first direction of the splitting theorem can be guaranteed, due to a clash between the minimality and completeness conditions. Example 29. Consider the BSAF F with splitting (F1 , F2 , R3 ) as below. The empty set defends d in F , so it is not complete. However, {d} does not defend e, so that E = {a, d, e} is the only grounded extension of F . Note, however, that the empty set is grounded in F1 , making it impossible to reconstruct E. a

Splitting over Collective Supports

We continue our investigation of how splitting translates from SETAFs to full BSAFs by focusing on the support relation. Analogously to attack splitting, we first consider support splitting in an isolated setting; i.e., we split a BSAF F with sub-BSAFs F1 and F2 which share only positive links. In the case of attack splitting, our schema relies on the subsequent evaluation of F1 and (a modification of) F2 for a given BSAF F , unpacking the evaluation of the entire framework in an incremental fashion. Our goal is to find appropriate modifications that enable the modular computation of the semantics if F1 and F2 share positive links. However, a closer investigation of the effects of the supporting links reveals a fundamental issue.

c1 a

b c2

Let E1 = {a}. Since b is defeated, not both c1 and c2 can be accepted in F2 ; otherwise, b would be accepted in F . The example above shows that defeating the head of a shared support induces a constraint. There are two natural options to encode constraints in BSAFs; and we will make use of both of them. Given (T, h) ∈ S3 , the first option is

Example 31. Consider the BSAF F with the subframeworks F1 (left) and F2 (right) and the shared support (b, c) below. a

Note that d defends a against the closed attacker {b, c} as before; however, in the given BSAF, the set {b} is closed as well. Since b is not attacked, a cannot be defended. Our goal is to develop a suitable splitting procedure analogous to the case for splitting attacks. We consider a σextension of F1 and modify F2 so that, for each σ-extension E2 of (the modified version of) F2 , it holds that E1 ∪ E2 is a σ-extension of F . Likewise, each σ-extension E of F should satisfy E ∩ A1 and E ∩ A2 both are σ-extensions of F1 and (the modified version of) F2 , respectively. Given the BSAF from Example 33 and the set E1 = {b}, we can perform the split by simply removing the shared support. The split correctly deduces that {b} and {b, d} are admissible in F . Such an easy solution of simply removing the shared support is, however, not always possible. Example 34. We consider the BSAF F below, with F1 and F2 left resp. right of the dotted line and S3 = {({c1 , c2 }, b)}.

Theorem 30. Let (F1 , F2 , R3 ) be an attack splitting for a BSAF F = (A, R, S) and R3c the set of closed negative links. E1 E1 c 1 Let F2⋆ = modE R3c (F2 ) where F2 is the (E1 , R3 )-reduct. ⋆ If E1 ∈ grd(F1 ) and E2 ∈ grd(F2 ), then E1 ∪E2 ∈ grd(F ).

to add a new argument ∗1 which is attacked by the empty set and supported by T ∩ A2 (type-1-constraint); the second option is to add a new argument ∗2 which is attacked by (T ∩A2 )∪{∗2 } and supported by T ∩A2 (type-2-constraint).

Example 39. Consider the BSAF F with F1 (left) and F2 (right) and the shared support (c, b) below. Let E1 = {a}. a

Example 35. Consider the BSAF from Example 34. We can prevent the joint acceptance of c1 and c2 in F2 by applying one of the following modifications. On the left-hand side, we add an argument that is attacked by the empty set (∗1 ), on the right-hand side, we add an argument (∗2 ) which is the head of the self-attacking set.

d ∗1

In F , cl F ({c}) = {c, b}, is attacked, thus, d is defeated. This is correctly captured by the type-1-modification F2C1 : the set cl F C1 ({c}) = {c, ∗1 } is attacked by the empty set. 2

We consider an example where this construction fails. c1 a

c1 ∗1

∗2

Example 40. Consider the BSAF F with F1 (left) and F2 (right) and the shared support ({c, e}, b) below. Let E1 ∈ {{e}, {a, e}}, then {c, e} ∈ TS3 (E1 ).

We observe that the type-1-constraint (left) defeats all attacks originating from (supersets of) T = {c1 , c2 }. This is because the closure of T is attacked by the empty set.

∗1

We break down which constraint is appropriate in which situation. First, we observe that a shared link (T, h) induces a constraint whenever all arguments in T ∩ A1 are contained in the selected extension E1 , but h ∈ / E1 .

Neither {e, d} nor {a, e, d} are admissible in F . However, in F2C1 , argument d is defended since cl F C1 (c) is attacked.

Definition 36. Let (F1 , F2 , S3 ) be a support splitting for BSAF F , where F1 = (A1 , R1 , S1 ). Let E1 ⊆ A1 . By

Definition 41 (Type-2-Modification). Let (F1 , F2 , S3 ) be a support splitting for a BSAF F , let E1 be an extension of F1 and let DS3 (E1 ) and TS3 (E1 ) denote all sets which are closure-defeated resp. support-incompatible wrt. E1 and S3 . Let TS-D (E1 ) := TS3 (E1 ) \ DS3 (E1 ) We define the type-23 modification of F2 as the BSAF F2C2 := F2 if TS-D (E1 ) = ∅ 3 and, otherwise, F2C2 := (A2 ∪ {∗2 }, R2C2 , S2C2 ) with

We thus require constraints of the second type in this case.

TS3 (E1 ) := {T ⊆ A | ∃h ∈ A1 : (T, h) ∈ S3 , T ∩ A1 ⊆ E1 , h ∈ / E1 } we denote the collection of sets of arguments that are support-incompatible with respect to E1 and S3 . As observed in Example 35, adding a type-1-constraint for a shared link (T, h) by adding ∗1 which is attacked by the empty set and a support from T ∩ A2 implies that all attacks originating from T ∩ A2 are defeated. We can apply this construction therefore only in the very restricted setting in which the set T is already defeated in F1 . We also require T ∩ A1 = ∅ because in all other cases, no outgoing attacks from T exist (recall that F1 and F2 do not share attacks).

R2C2 := R2 ∪ {((T ∩ A2 ) ∪ {∗2 }, ∗2 ) | T ∈ TS-D (E1 )} 3 S2C2 := S2 ∪ {(T ∩ A2 , ∗2 ) | T ∈ TS-D (E1 )} 3 Example 42. We continue Example 40 and apply the type2-modification for E1 ∈ {{e}, {a, e}}. a

b e

Definition 37. Let (F1 , F2 , S3 ) be a support splitting for BSAF F , where F1 = (A1 , R1 , S1 ). Let E1 ⊆ A1 . By DS3 (E1 ) := {T ⊆ A | T ∈ TS3 (E1 ), T ∩ A1 = ∅,

d c

b e

d c

∗1

Now, in F2C2 , the argument c supports ∗2 and d is attacked; thus, no argument can be accepted and the only admissible extension is ∅. This correctly detects the admissible extensions {e} and {a, e} of F . For E1 = {a}, we have {e, c} ∈ / TS3 (E1 ), thus the support is simply removed.

cl F (T ) ∩ (E1 )+ R ̸= ∅} we denote the collection of sets of arguments that are closure-defeated with respect to E1 and S3 . We are ready to state our first modification.

We define the S-reduct as a combination of both modifications. The union of two BSAFs is defined componentwise.

Definition 38 (Type-1-Modification). Let (F1 , F2 , S3 ) be a support splitting for a BSAF F , let E1 be an extension of F1 and let DS3 (E1 ) denote all sets which are closure-defeated wrt. E1 and S3 . We define the type-1-modification of F2 as the BSAF F2C1 := F2 if DS3 (E1 ) = ∅ and, otherwise, F2C1 := (A2 ∪ {∗1 }, R2C1 , S2C1 ) with

Definition 43 (S-reduct). Let (F1 , F2 , S3 ) be a support splitting for a BSAF F , let E1 be an extension of F1 and let DS3 (E1 ) and TS3 (E1 ) denote all sets which are closuredefeated resp. support-incompatible wrt. E1 and S3 . We define the (E1 , S3 )-reduct (or simply S-reduct) of F2 as the E1 E1 E1 C1 C2 1 BSAF F2E1 = (AE 2 , R2 , S2 ) where F2 := F2 ∪ F2 .

R2C1 := R2 ∪ {(∅, ∗1 )}

The splitting procedure preserves admissible, complete and stable semantics.

S2C1 := S2 ∪ {(T ∩ A2 , ∗1 ) | T ∈ DS3 (E1 )}. 7

2. We proceed by constructing the R-reduct F̂2E1 of the attack splitting (F1 , F̂2 , R̂3 ) wrt. E1 , following Definition 22.

Theorem 44. Let (F1 , F2 , S3 ) be a support splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ), and σ ∈ {stb, adm, com}. Further, let F2E1 be the S-reduct of F2 .

1 3. Next, we compute the modification F2⋆ = modE (F̂2E1 ) R̂3 as in Definition 25. The tuple (F1 , F2⋆ , S3 ) is a support splitting for the BSAF (A1 ∪ A⋆2 , A1 ∪ A⋆2 , S).

1. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2E1 ), then E1 ∪(E2 \{∗2 }) ∈ σ(F ). 2. If E ∈ σ(F ), then E1 = E ∩ A1 ∈ σ(F1 ) and (E ∩ A2 ∈ σ(F2E1 ) or (E ∩ A2 ) ∪ {∗2 } ∈ σ(F2E1 )). For preferred and grounded semantics, our procedure constructs only some of the extensions of the entire framework. We illustrate this in the following example. Example 45. Consider the BSAF F and a support splitting (F1 , F2 , S3 ) as depicted below. Let E1 = {a, d}. We compute the reduct F2E1 by adding the dummy argument ∗2 which is supported by {b} and attacked by {b, ∗2 } (depicted right). a d

Definition 48. Let (F1 , F2 , R3 , S3 ) be a splitting for a BSAF F with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ). Let R3c and R2c denote the sets of closed negative links wrt. R3 and R2 , respectively. Define R̂2 := {(T, h) ∈ R2c | T ∩ A1 = ∅} and let F̂2 := (A2 , R̂2 , S2 ) be the updated BSAF. Further, let

a d

4. Finally, we construct the S-reduct F2⊛ of F2⋆ for the support splitting (F1 , F2⋆ , S3 ) wrt. E1 , using Definition 43.

∗2

R̂3 := {(T, h) | (T, h) ∈ R3c ∪ R2c , T ∩ A1 ̸= ∅}.

Let F̂2E1 be the R-reduct of the attack splitting (F1 , F̂2 , R̂3 ) E1 (F̂2E1 ). Finally, we let F2⊛ dewrt. E1 and F2⋆ := modR̂ 3 note the S-reduct for the support splitting (F1 , F2⋆ , S3 ) of the BSAF (A1 ∪ A⋆2 , A1 ∪ A⋆2 , S) wrt. E1 .

In F2E1 , we get pref(F2E1 ) = {∅}, obtaining only {a, d} as preferred extension of the whole framework. We get a splitting theorem for preferred and grounded semantics that considers only the first direction. Theorem 46. Let (F1 , F2 , S3 ) be a support splitting for a BSAF F = (A, R, S), and let F2E1 be the S-reduct of F2 and σ ∈ {grd, pref}. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2E1 ), then E1 ∪ (E2 \ {∗2 }) ∈ σ(F ).

We demonstrate the procedure in the following example. Example 49. We consider a BSAF F with splitting (F1 , F2 , R3 , S3 ), as depicted below. Further, let E1 = {a}. z

Splitting Collective Attacks and Supports

Now that we have the building blocks of our approach, we combine the notions of attack and support splitting and provide results similar to the section before. Definition 47 (Splitting). Let F = (A, R, S) be a BSAF, F1 = (A1 , R1 , S1 ) and F2 = (A2 , R2 , S2 ) two subframeworks of F such that A1 ∩ A2 = ∅, A = A1 ∪ A2 , R = R1 ∪ R2 ∪ R3 , and S = S1 ∪ S2 ∪ S3 with:

d y

b c

First, we proceed by closing the negative links in R2 and R3 , as depicted below. The set R3 contains the single attack (c, x). Since c supports b, the attack ({b, c}, x) is contained in R̂3 . Moreover, we get R̂2 = R2 because x and y do not attack any argument in F2 . Thus we consider the following attack splitting (F1 , F2 , R̂3 ).

R3 ⊆ {(T, h) ∈ R | T ∩ A1 ̸= ∅, T ⊆ A, h ∈ A2 }, S3 ⊆ {(T, h) ∈ S | T ∩ A2 ̸= ∅, T ⊆ A, h ∈ A1 }. We call the 4-tuple (F1 , F2 , R3 , S3 ) a splitting of F . We apply the splits we developed in the previous two sections consecutively. Since attacks (T, h) in F2 can support arguments in F1 , we start our splitting procedure by closing attacks in R2 , before we apply the modifications we developed in Sections 3 and 4. Overall, we perform the following splitting procedure. Given a splitting (F1 , F2 , R3 , S3 ) for a BSAF F and a set E1 ⊆ A1 , we proceed as follows. 1. We close the attacks in R2 ∪ R3 . Note that this may transform R2 attacks into shared attacks between F1 and F2 since the tail of an attack R2 can support an argument in A1 . To account for this, we move the attacks in question into the set R3 ; by F̂2 and R̂3 , we denote the adjusted BSAF and set of common negative links, respectively.

z w

d y

b c

We apply the R-reduct and the modification for the attack splitting (see Section 3). We depict F1 (left) and the result of the procedure, F2⋆ , (right) below. 8

Incremental Computation for Supports While most works in computational argumentation focus on splitting attacks, ADFs constitute a notable exception. In ADFs, each argument has an acceptance condition, capable of expressing positive relations between arguments. Linsbichler (2014) investigated splitting techniques for ADFs, while more recent work has explored serialization (Bengel and Thimm 2025) for abstract dialectical frameworks. The idea behind serialization is based on finding so-called initial models of a given ADF in an iterative way, thereby constructing serialization sequences that allow the incremental computation of its extensions. We notice, however, that even though ADFs have been proven to capture collective attacks (Dvořák, Keshavarzi Zafarghandi, and Woltran 2023), this does not hold for collective supports under the closure condition. From a semantic standpoint, this constitutes a substantial difference with respect to BSAFs. Consequently, existing incremental and splitting results for ADFs cannot be transferred directly to our setting.

∗0 w

d y

b c

Finally, we apply the S-reduct to handle the remaining links. Thus, we obtain F2⊛ (right) adding a type-1-constraint on x and a type-2-constraint on y. z

∗0 w

d y

∗2

∗1 x

b c

Directionality, Modularization, and SCC-recursiveness Further approaches that compute extensions in an incremental manner closely related to splitting techniques include schemes based on strongly connected components (SCCs) (Baroni, Giacomin, and Guida 2005), modularization (Baumann, Brewka, and Ulbricht 2022) and directionality (Baroni and Giacomin 2007). Modularization is directly connected to the splitting schema via a similar notion of reduct (Berthold, Blümel, and Rapberger 2025). The principle of directionality is connected with the possibility of evaluating sub-frameworks independently from the remaining parts, provided that there is no incoming negative link towards them. The relation to attack splitting is thus clear given that we can project extensions of F to those of F1 without losing information. The SCC-recursive property (Baroni, Giacomin, and Guida 2005) of argumentation semantics defines the possibility to retrieve extensions of a given framework by breaking down the computation along its SCCs. Such decomposition ensures that the evaluation of the whole graph can be assessed via a generalized semantics that takes into account previous decisions concerning already evaluated SCCs. Splitting represents a more flexible alternative as it does not dictate that F1 and F2 be strongly connected components, while ensuring that every SCC of the graph is in either of the two.

We are ready to state our main result for splitting BSAFs. Theorem 50. Let (F1 , F2 , R3 , S3 ) be a splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ), and σ ∈ {stb, adm, com}. Let F2⊛ be defined as in Definition 48. 1. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2⊛ ), then E1 ∪ (E2 \ {∗2 }) ∈ σ(F ). 2. If E ∈ σ(F ), then E1 = E ∩ A1 ∈ σ(F1 ) and E ∩ A2 ∈ σ(F2⊛ ) or (E ∩ A2 ) ∪ {∗2 } ∈ σ(F2⊛ ). Example 51. We use the splitting theorem to compute the extensions of BSAF F from Example 49. First, note that the set E1 = {a} is admissible in F1 . Next, we compute the admissible extensions of F2⊛ and obtain the ∅ and {z}. Hence, we retrieve {a} and {a, z} in adm(F ). Note that we inherit the limitations with respect to preferred and grounded semantics. Theorem 52. Let (F1 , F2 , R3 , S3 ) be a splitting for the BSAF F = (A, R, S), let F2⊛ be as in Definition 48 and σ ∈ {grd, pref}. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2⊛ ), then E1 ∪ (E2 \ {∗2 }) ∈ σ(F ).

Related Work

Conclusion

In this paper, we introduced a splitting schema for bipolar set-based argumentation frameworks. Similar to existing literature on splitting for abstract forms of argumentation, we developed a modification-based approach for the incremental computation of extensions of a given BSAF. In particular, after splitting the framework F into two sub-frameworks F1 and F2 , we defined syntactic modifications of F2 that allow one to reconstruct extensions of the original framework from those of the components. By leveraging the richer syntax of BSAFs, we have considered splits over collective attacks, supports, and a combination of both. Importantly, each step of our procedure is computationally feasible and does not

Splitting has a long tradition in the field of knowledge representation and reasoning, in particular, for non-monotonic formalisms. The first splitting procedure has been proposed for logic programs (Lifschitz and Turner 1994) and has received much attention since then, e.g., in the context of answer set programming (Beiser et al. 2024), extended fragments of logic programming (Ben-Eliyahu-Zohary 2025; Cabalar, Fandinno, and del Cerro 2021), for conditional knowledge bases (Heyninck et al. 2023), and for abstract and structured forms of argumentation (Baumann 2011; Baumann, Brewka, and Wong 2011; Buraglio et al. 2024; Linsbichler 2014; Buraglio 2025). 9

Acknowledgments

induce an exponential blow-up in the size of the modified sub-frameworks, making our approach a promising starting point for efficient algorithms. For future work, we plan to extend our splitting techniques to cases where (a certain number of) backward attacks and forward supports are present. In the context of AFs, similar settings have been studied under the term parameterized splitting (Baumann et al. 2012). In addition, we plan to implement our procedure and investigate its benefits for dynamic programming approaches in argumentation. In this regard, observe that for Dung-style AFs, all possible splittings can be computed in linear time by identifying the strongly connected components of the graph (Baumann, Brewka, and Wong 2011; Tarjan 1972). We anticipate that for BSAFs, a similar approach to computing splittings in AFs can be adapted, by treating the support and attack relations separately and by exploiting the so-called primal graph (Dvořák et al. 2024). On top of this, Baumann, Brewka, and Wong (2011) register an average acceleration of 60% for executions with splitting in comparison to an execution without splitting in the context of Dung-style AFs.

This work has been supported by the European Union’s Horizon 2020 research and innovation programme (under grant agreement 101034440) and by the Deutsche Forschungsgemeinschaft (projects “Argumentative Reasoning in Nonsensical Situations”, grant 550735820, and “Explainable Belief Merging”, grant 465447331).

References Amgoud, L.; Cayrol, C.; Lagasquie, M.-C.; and Livet, P. 2008. On bipolarity in argumentation frameworks. International Journal of Intelligent Systems 23:1–32. Atkinson, K.; Baroni, P.; Giacomin, M.; Hunter, A.; Prakken, H.; Reed, C.; Simari, G. R.; Thimm, M.; and Villata, S. 2017. Towards artificial argumentation. AI Magazine 38(3):25–36. Baroni, P., and Giacomin, M. 2007. On principle-based evaluation of extension-based argumentation semantics. Artificial Intelligence 171(10-15):675–700. Baroni, P.; Gabbay, D.; Giacomin, M.; and van der Torre, L., eds. 2018. Handbook of Formal Argumentation. College Publications. Baroni, P.; Giacomin, M.; and Guida, G. 2005. SCCrecursiveness: A general schema for argumentation semantics. Artificial Intelligence 168(1-2):162–210. Baroni, P.; Giacomin, M.; and Liao, B. 2014. On topologyrelated properties of abstract argumentation semantics. A correction and extension to dynamics of argumentation systems: A division-based method. Artificial Intelligence 212:104–115. Baumann, R., and Brewka, G. 2010. Expanding argumentation frameworks: Enforcing and monotonicity results. In Baroni, P.; Cerutti, F.; Giacomin, M.; and Simari, G. R., eds., Proc. COMMA, volume 216 of Frontiers in Artificial Intelligence and Applications, 75–86. IOS Press. Baumann, R.; Brewka, G.; Dvořák, W.; and Woltran, S. 2012. Parameterized splitting: A simple modification-based approach. In Correct Reasoning - Essays on Logic-Based AI in Honour of Vladimir Lifschitz, volume 7265 of LNCS, 57–71. Springer. Baumann, R.; Brewka, G.; and Ulbricht, M. 2022. Shedding new light on the foundations of abstract argumentation: Modularization and weak admissibility. Artificial Intelligence 310:103742. Baumann, R.; Brewka, G.; and Wong, R. 2011. Splitting argumentation frameworks: An empirical evaluation. In Modgil, S.; Oren, N.; and Toni, F., eds., Proceedings of (TAFA11), volume 7132 of Lecture Notes in Computer Science, 17–31. Springer. Baumann, R. 2011. Splitting an argumentation framework. In Proceedings of (LPNMR-11), volume 6645 of LNCS, 40– 53. Springer. Beiser, A.; Hecher, M.; Unalan, K.; and Woltran, S. 2024. Bypassing the ASP bottleneck: Hybrid grounding by splitting and rewriting. In Proceedings of (IJCAI-24), 3250– 3258. ijcai.org.

Crucially, BSAFs are closely related to general (nonflat) ABAFs, as demonstrated by Berthold, Rapberger, and Ulbricht (2024), which is one of the most popular formalisms in the area of structured argumentation. Recent work exploits the close relation of SETAFs and flat ABAFs to translate SETAF splitting results to this fragment of ABA (Buraglio 2025). We thus expect that our results for BSAF splitting can be transferred to general ABA in a similar fashion. Our hope is that this will enhance existing non-flat ABA solvers (Lehtonen et al. 2024). We are furthermore optimistic that our results could provide further valuable insights into structured argumentation dynamics (Rapberger and Ulbricht 2024; Rapberger and Ulbricht 2023; Prakken 2023; Baumann and Brewka 2010; Cayrol, de Saint-Cyr, and Lagasquie-Schiex 2010; Berthold, Rapberger, and Ulbricht 2023) and other popular forms of structured argumentation (Modgil and Prakken 2013). While the present work has focused on the canonical semantics for BSAFs (which capture ABA semantics), the literature on bipolar argumentation provides a wide range of alternative interpretations of support (Cohen et al. 2014; Cohen et al. 2018; Amgoud et al. 2008; Polberg 2016; Boella et al. 2010). Furthermore, (Berthold, Rapberger, and Ulbricht 2024) develop novel BSAF semantics aimed at addressing certain undesirable properties of the canonical approach. Extending established notions of support to the setting of set-supports, as well as developing corresponding splitting techniques for these alternative BSAF semantics, is a promising direction for future research. Lastly, we plan to study the relation of our splitting techniques to the aforementioned principles of directionality, modularity and SCC-recursiveness. For doing so, appropriate notions of these principles in the presence of supports need to be identified, which we deem an interesting direction of future work on its own. As suitable starting points for these studies, we identify the principle-based analyzes for SETAFs (Dvořák et al. 2024) and BAFs (Yu et al. 2023). 10

Cohen, A.; Parsons, S.; Sklar, E. I.; and McBurney, P. 2018. A characterization of types of support between structured arguments and their relationship with support in abstract argumentation. International Journal of Approximate Reasoning 94:76–104. Cyras, K.; Rago, A.; Albini, E.; Baroni, P.; and Toni, F. 2021. Argumentative XAI: A survey. In Zhou, Z., ed., Proceedings of (IJCAI-21), 4392–4399. ijcai.org. Dung, P. M. 1995. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77(2):321–358. Dvořák, W.; König, M.; Ulbricht, M.; and Woltran, S. 2024. Principles and their computational consequences for argumentation frameworks with collective attacks. Journal of Artificial Intelligence Research (JAIR) 79:69–136. Dvořák, W.; Keshavarzi Zafarghandi, A.; and Woltran, S. 2023. Expressiveness of SETAFs and support-free ADFs under 3-valued semantics. Journal of Applied Non-Classical Logics 33(3-4):298–327. Giacomin, M.; Baroni, P.; and Cerutti, F. 2021. Towards a general theory of decomposability in abstract argumentation. In Baroni, P.; Benzmüller, C.; and Wáng, Y. N., eds., Proceedings of (CLAR-21), Lecture Notes in Computer Science, 169–189. Springer. Heyninck, J.; Kern-Isberner, G.; Meyer, T. A.; Haldimann, J. P.; and Beierle, C. 2023. Conditional syntax splitting for non-monotonic inference operators. In Williams, B.; Chen, Y.; and Neville, J., eds., Proceedings of (AAAI-23), 6416– 6424. AAAI Press. Lehtonen, T.; Rapberger, A.; Toni, F.; Ulbricht, M.; and Wallner, J. P. 2024. Instantiations and computational aspects of non-flat assumption-based argumentation. In Proceedings of (IJCAI-24). Leofante, F.; Ayoobi, H.; Dejl, A.; Freedman, G.; Gorur, D.; Jiang, J.; Paulino-Passos, G.; Rago, A.; Rapberger, A.; Russo, F.; Yin, X.; Zhang, D.; and Toni, F. 2024. Contestable AI needs computational argumentation. In Marquis, P.; Ortiz, M.; and Pagnucco, M., eds., Proceedings of (KR24). Liao, B. S.; Jin, L.; and Koons, R. C. 2011. Dynamics of argumentation systems: A division-based method. Artificial Intelligence 175(11):1790–1814. Liao, B. 2013. Toward incremental computation of argumentation semantics: A decomposition-based approach. Annals of Mathematics and Artificial Intelligence 67(3-4):319– 358. Lifschitz, V., and Turner, H. 1994. Splitting a logic program. In Hentenryck, P. V., ed., Proceedings of (ICLP-94), 23–37. MIT Press. Linsbichler, T. 2014. Splitting abstract dialectical frameworks. In Proceedings of (COMMA-14), volume 266 of Frontiers in Artificial Intelligence and Applications, 357– 368. IOS Press. Modgil, S., and Prakken, H. 2013. A general account

Ben-Eliyahu-Zohary, R. 2025. Splitting a disjunctive logic program. Theory and Practice of Logic Programming 25(4):507–521. Bengel, L., and Thimm, M. 2025. Initial models and serialisability in abstract dialectical frameworks. In Proceedings of (IJCAI-25), 4365–4373. ijcai.org. Berthold, M.; Blümel, L.; and Rapberger, A. 2025. On strong and weak admissibility in non-flat assumption-based argumentation. In Proceedings of (KR-25), 137–147. Berthold, M.; Rapberger, A.; and Ulbricht, M. 2023. Forgetting aspects in assumption-based argumentation. In Marquis, P.; Son, T. C.; and Kern-Isberner, G., eds., Proceedings of (KR-23), 86–96. Berthold, M.; Rapberger, A.; and Ulbricht, M. 2024. Capturing non-flat assumption-based argumentation with bipolar setafs. In Marquis, P.; Ortiz, M.; and Pagnucco, M., eds., Proceedings of (KR-24). Blümel, L.; Rapberger, A.; Thimm, M.; and Toni, F. 2025. On independence and scc-recursiveness in assumptionbased argumentation. In Proceedings of (IJCAI-25), 4382– 4390. ijcai.org. Boella, G.; Gabbay, D. M.; van der Torre, L. W. N.; and Villata, S. 2010. Support in abstract argumentation. In Baroni, P.; Cerutti, F.; aspects in assumption-based argumenno Giacomin, M.; and Simari, G. R., eds., Proceedings of (COMMA-10), Frontiers in Artificial Intelligence and Applications, 111–122. IOS Press. Brewka, G.; Ellmauthaler, S.; Strass, H.; Wallner, J. P.; and Woltran, S. 2018. Abstract dialectical frameworks. In Handbook of Formal Argumentation. College Publications. chapter 5, 237–285. also appears in IfCoLog Journal of Logics and their Applications 4(8):2263–2318. Buraglio, G.; Dvorák, W.; König, M.; and Woltran, S. 2024. Splitting argumentation frameworks with collective attacks. In Borg, A.; Ellmauthaler, S.; Mailly, J.; and Niskanen, A., eds., Proceedings of (COMMA-24), volume 3757 of CEUR Workshop Proceedings, 41–55. CEUR-WS.org. Buraglio, G. 2025. Splitting assumption-based argumentation frameworks. In Rapberger, A., and Rudolph, S., eds., Proceedings of (NMR-25), volume 4071 of CEUR Workshop Proceedings, 17–31. CEUR-WS.org. Cabalar, P.; Fandinno, J.; and del Cerro, L. F. 2021. Splitting epistemic logic programs. Theory and Practice of Logic Programming 21(3):296–316. Cayrol, C., and Lagasquie-Schiex, M.-C. 2005. On the acceptability of arguments in bipolar argumentation frameworks. In Godo, L., ed., Proceedings of (ECSQARU-05), volume 3571 of LNCS, 378–389. Springer. Cayrol, C.; de Saint-Cyr, F. D.; and Lagasquie-Schiex, M.C. 2010. Change in abstract argumentation frameworks: Adding an argument. Journal of Artificial Intelligence Research (JAIR) 38:49–84. Cohen, A.; Gottifredi, S.; Garcı́a, A. J.; and Simari, G. R. 2014. A survey of different approaches to support in argumentation systems. The Knowledge Engineering Review 29(5):513–550. 11

of argumentation with preferences. Artificial Intelligence 195:361–397. Modgil, S. 2009. Reasoning about preferences in argumentation frameworks. Artificial Intelligence 173(9-10):901– 934. Nielsen, S. H., and Parsons, S. 2006. A generalization of Dung’s abstract framework for argumentation: Arguing with sets of attacking arguments. In Proceedings of (ArgMAS06), 54–73. Springer. Polberg, S. 2016. Understanding the abstract dialectical framework. In Proceedings of (JELIA-16), Lecture Notes in Computer Science, 430–446. Prakken, H. 2023. Relating abstract and structured accounts of argumentation dynamics: the case of expansions. In Marquis, P.; Son, T. C.; and Kern-Isberner, G., eds., Proceedings of (KR-23), 562–571. Rapberger, A., and Ulbricht, M. 2023. On dynamics in structured argumentation formalisms. Journal of Artificial Intelligence Research (JAIR) 77:563–643. Rapberger, A., and Ulbricht, M. 2024. Repairing assumption-based argumentation frameworks. In Marquis, P.; Ortiz, M.; and Pagnucco, M., eds., Proceedings of (KR24). Tarjan, R. E. 1972. Depth-first search and linear graph algorithms. SIAM Journal on Computing 1(2):146–160. Ulbricht, M.; Potyka, N.; Rapberger, A.; and Toni, F. 2024. Non-flat ABA is an instance of bipolar argumentation. In Wooldridge, M. J.; Dy, J. G.; and Natarajan, S., eds., Proceedings of (AAAI-24), 10723–10731. AAAI Press. Yu, L.; Anaissy, C. A.; Vesic, S.; Li, X.; and van der Torre, L. 2023. A principle-based analysis of bipolar argumentation semantics. In Gaggl, S. A.; Martinez, M. V.; and Ortiz, M., eds., Proceedings of (JELIA-23), volume 14281 of Lecture Notes in Computer Science, 209–224. Springer.

Omitted Proofs of Section 3

with S ⊆ E, d ∈ D in F . But then by definition of R′ there exists an attack either (S, d) or (cl(S), d) ∈ R′ and since E is closed, cl(S) ⊆ E, so E defends h in F ′ (since the closed sets of F and F ′ coincide by Lemma 53), but does not contain h. So E is not complete in F ′ . Contradiction. (grd) Follows directly from com(F ) = com(F ′ ). (pref) Follows directly from adm(F ) = adm(F ′ ). (stb) Let E ∈ stb(F ). Then by the first item E ∈ adm(F ′ ). It is left to show, that E attacks every x ∈ A \ E in F ′ , so let x ∈ A \ E. Since E is stable in F there exists an attack (T, x) ∈ R with T ⊆ E. Therefore an attack either (T, x) or (cl(T ), x) ∈ R′ exists, and since E is closed we have cl(T ) ⊆ E. So E attacks x in F ′ . For the other direction (stb(F ′ ) ⊆ stb(F )) consider that for any attack (S ′ , x) ∈ R′ with S ′ ⊆ E by definition of R′ there exists a subset S ⊆ S ′ ⊆ such that (S, x) ∈ R, so E attacks x in F .

Lemma 53. Let F = (A, R, S) be a BSAF, (T, h) ∈ R and F ′ = (A, R′ , S) with R′ = (R \ {(T, h)}) ∪ {(cl F (T ), h)}. 1. Let E ⊆ A be a closed set. Then E ∈ cf(F ) iff E ∈ cf(F ′ ). 2. Let E ⊆ A. Then E is closed in F iff E is closed in F ′ Proof. 1. Suppose first E is not conflict-free in F , then there exists (T, h) ∈ R with T ⊆ E, h ∈ E. Since E is closed, cl(T ) ⊆ E. So either (T, h) or (cl(T ), h) ∈ R′ is an attack of E on itself in F ′ , so E not conflict-free in F ′ . For the other direction suppose now, E is not conflictfree in F ′ , then there exists (T ′ , h) ∈ R′ , such that T ′ ⊆ E, h ∈ E. By definition of R′ there must exist a subset T ⊆ T ′ such that T ′ = cl(T ) and (T, h) ∈ R. But then also T ⊆ E, h ∈ E, so E is not conflict-free in F . 2. Since argument set and support relation coincide between F and F ′ and closedness is defined solely over the support relation, the closed sets of F and F ′ coincide.

Proposition 20. Let F = (A, R, S) be a BSAF, (T, h) ∈ R an attack and let F ′ = (A, R′ , S) with R′ = (R \ {(T, h)}) ∪ {(cl(T ), h)}. Then σ(F ) = σ(F ′ ) for all σ ∈ {stb, adm, com, pref, grd}.

Proof. (adm) (⊆) First, let E ∈ adm(F ). Then by Lemma 53 E is closed and conflict-free in F ′ . It is left to show that E defends itself against closed attackers in F ′ . Let (T ′ , h) ∈ R′ with h ∈ E, T ′ ⊆ A. Then there exists some T ⊆ T ′ such that (T, h) ∈ R. Now any closed set D containing T ′ also contains T and for any closed set containing T there exists an attack (S, t) ∈ R with S ⊆ E, t ∈ D. By definition of R′ therefore an attack (S, t) or (cl(S), t) ∈ R′ on D exists. Since E is closed, cl(S) ⊆ E, it therefore attacks D in F ′ and thus defends itself. (⊇) Now let E ∈ adm(F ′ ). With the lemma E is closed and conflict-free in F and it is left to show that E defends itself. Let D be a closed set and (T, h) ∈ R with T ⊆ D, h ∈ E. Then either (T, h) or (cl(T ), h) ∈ R′ , so since D is a closed set and T ⊆ D, cl(T ) ⊆ D and D attacks E in F ′ . Since E is admissible in F ′ , there exists an attack (S ′ , t) ∈ R′ with S ′ ⊆ E, t ∈ D and therefore by definition of R′ an attack (S, t) ∈ R with S ⊆ S ′ ⊆ E, t ∈ D. So E defends itself in F . (com) Let E ∈ com(F ). Then by the previous item E ∈ adm(F ′ ). It is left to show that E contains every argument it defends in F ′ . Suppose to the contrary, E defends some h in F ′ and does not contain it. Then for every closed attacker D of h there exists an attack (S ′ , d) ∈ R′ with S ′ ⊆ E, d ∈ D in F ′ . But then by definition of R′ there exists an attack (S, d) ∈ R with S ⊆ S ′ ⊆ E, so E defends h in F (since the closed sets of F and F ′ coincide by Lemma 53, the closed attackers on h coincide in F and F ′ ), but does not contain h. So E is not complete in F . Contradiction. Now let E ∈ com(F ′ ), then E ∈ adm(F ). Suppose again E defends some h in F it does not contain. Then for every closed attacker D of h there exists an attack (S, d) ∈ R

1. If E1 = clF1 (E1 ) and E2 = clF2⋆ (E2 ), then E = (E1 ∪ E2 ) \ {∗0 } = clF (E). 2. If E = clF (E) for some E ⊆ A, then E ∩ A1 = E1 = clF1 (E1 ) and E ∩ A2 = E2 = clF2⋆ (E2 ). Proof. 1. Suppose E1 = clF1 (E1 ) and E2 = clF2⋆ (E2 ). We show that E = (E1 ∪ E2 ) \ {∗0 } is closed. Let a ∈ A s.t. there is (T, a) ∈ S with T ⊆ E. Then, either T ⊆ E1 , T ⊆ E2 , or T has non-empty intersection with both E1 and E2 . We proceed by case distinction. • Case 1: T ⊆ E1 . Then a ∈ E1 since E1 = clF1 (E1 ). • Case 2: T ⊆ E2 . Then a ∈ E2 since E2 = clF2⋆ (E2 ). • Case 3: Suppose T ∩ E1 ̸= ∅ and T ∩ E2 ̸= ∅. Then there is a support (T, a) ∈ S with T ∩ A1 ̸= ∅ and T ∩ A2 ̸= ∅; therefore, (F1 , F2 , R3 ) is not a proper attack splitting since S1 and S2 is not a partition of S. 2. Suppose E = clF (E) for some E ⊆ A, and let E1 = E ∩ A1 and E2 = E ∩ A2 . • We show that E1 is closed: Let a ∈ A1 with (T, a) ∈ S1 . By construction, (T, a) ∈ S, thus a ∈ E. Since S1 and S2 partitions S, we obtain a ∈ E ∩ A1 = E1 . • We show that E2 is closed: Let a ∈ A2 with (T, a) ∈ S2 . Analogous to the first item, we obtain a ∈ E2 since each support is either fully contained in A1 or A2 . Lemma 54. Let (F1 , F2 , R3 ) be a attack splitting for a BSAF F with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ). Let E1 c 1 E1 ⊆ A1 and F2⋆ = modE R3c (F2 ) where R3 is the set of closed attacks wrt. R3 . 1. If E1 ∈ adm(F1 ) and E2 ∈ adm(F2⋆ ), then E1 ∪ E2 ∈ cf(F ). 13

– (T, h) = (T ′ \ A1 , h) for some T ′ ⊆ A such that (T ′ , h) ∈ R3 , T ′ ∩ (E1 )+ R3 = ∅, T ∩ A1 ⊆ E1 , and 1 . (Case 2) h ∈ AE 2 We proceed by case distinction. E1 1 – Case 1. (T, h) ∈ R2 with T ⊆ AE 2 , h ∈ A2 . In this case, we obtain h ∈ / E2 since E ∈ cf(F ) and since R2 ⊆ R. – Case 2. From T = T ′ \A1 ⊆ E2 and T ′ ∩A1 ⊆ E1 we obtain T ′ ⊆ E. We obtain h ∈ / E2 since E ∈ cf(F ) and since R3 ⊆ R. We have shown that E2 ∈ cf(F2E1 ).

2. If E ∈ cf(F ), then E1 = E ∩ A1 ∈ cf(F1 ) and E2 = E ∩ A2 ∈ cf(F2E1 ). Proof. 1. Suppose E1 ∈ adm(F1 ) and E2 ∈ adm(F2⋆ ). Let E = E1 ∪ E2 . We show E ∈ cf(F ). Let (T, h) ∈ R with T ⊆ E. Either (T, h) ∈ R1 , (T, h) ∈ R2 , or (T, h) ∈ R3 . We proceed by case distinction. • Case 1: (T, h) ∈ R1 . Then h ∈ / E since E1 ∈ adm(F1 ). E1 1 • Case 2: (T, h) ∈ R2 . It holds that T ⊆ AE 2 , h ∈ A2 . In this case, we obtain h ∈ / E2 since E2 ∈ adm(F2 ) and since R2 ⊆ R. • Case 3: (T, h) ∈ R3 . Let Tc = cl F (T ). Then either (Tc , h) ∈ UREc1 or (Tc , h) ∈ / UREc1 . 3

Theorem 28. Let (F1 , F2 , R3 ) be an attack splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ); let R3c denote the closed negative links. Let E1 1 σ ∈ {stb, adm, com, pref}. Further, let F2⋆ = modE Rc (F2 )

– Case 3.i: (Tc , h) ∈ UREc1 . By definition of undecided 3 link there is some t ∈ Tc ∩ A1 such that t ∈ + A1 \ (E1 )⊕ R1 . Since Tc ∩ (E1 )R1 ∪R3c = ∅ we have t ∈ A1 \ E1 . Since t ∈ Tc = cl F (T ) and since F1 and F2 have no common supports, it holds that t ∈ cl F1 (T ∩A1 ). We arrived at a contradiction: since E = E1 ∪ E2 , we have T ∩ A1 ⊆ E1 . Since, by assumption, E1 is admissible in F1 it is also closed and therefore contains t. – Case 3.ii: (Tc , h) ∈ / UREc1 . Let Tc = cl F (T ). Then ei-

1. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2⋆ ), then E1 ∪ E2 ∈ σ(F ). 2. If E ∈ σ(F ), then E1 = E ∩ A1 ∈ σ(F1 ) and E ∩ A2 ∈ σ(F2⋆ ). Proof. Closure is proven using Proposition 27; conflictfreeness is proven using Lemma 54. We proceed by proving the statement for each semantics separately.

Stable semantics.

⊕ ther Tc ∪ (E1 )+ R1 ∪R3c ̸= ∅ or Tc ∩ A1 ⊆ (E1 )R1 . Furthermore observe that Tc ∩ A1 ⊆ E1 since T ∩ A1 ⊆ E1 and since F1 and F2 have no common supports. We proceed by case distinction. (a) Suppose Tc ∪ (E1 )+ R1 ∪R3c ̸= ∅. That is, there is some t ∈ Tc which is attacked by E1 via some attack (T ′ , t) in R1 or R3 . (a.1) If t ∈ A1 then (T ′ , t) ∈ R1 and we obtain a contradiction to E1 ∈ adm(F1 ). (a.2) If t ∈ A2 then (T ′ , t) ∈ R3c and t ∈ (E1 )+ R3c . By assumption, T ′ ⊆ E1 (thus, T ′ ∩ A1 ⊆ E1 ) and t ∈ A2 . (a.2.i) Case T ′ ∩ (E1 )+ R3c = ∅. Then, by definition of the R-reduct, R2⋆ contains (∅, t). Since t ∈ E, we obtain contradiction to the conflict-freeness of E2 in F2⋆ . (a.2.ii) Case T ′ ∩ (E1 )+ R3c ̸= ∅. By definition of attack splitting, this is impossible because T ′ ⊆ E1 . + (b) If T ∩A1 ∩ER ̸= ∅, proceed as in (a.1). Otherwise 1 T ∩ A1 ⊆ E1 . Then (T \ A1 , h) ∈ R2E1 . Therefore, h∈ / E2 , and thus h ∈ / E. 2. Suppose E ∈ cf(F ). We show that E1 = E∩A1 ∈ cf(F1 ) E1 1 and E2 = E ∩ AE 2 ∈ cf(F2 ). • E1 ∈ cf(F1 ) since R1 ⊆ R.

1. Suppose E1 ∈ stb(F1 ) and E2 ∈ stb(F2⋆ ). We show that E = E1 ∪ E2 ∈ stb(F ). • E is conflict-free: Note that F2⋆ = F2E1 because UREc1 = 3 ∅. By Lemma 54 and since E1 ∈ stb(F1 ) and E2 ∈ stb(F2E1 ), we obtain E ∈ cf(F ). • E is closed: this follows from Proposition 27. • E attacks all remaining arguments: Let a ∈ A \ E. We proceed by case distinction. – Case 1: a ∈ A1 . Then a ∈ (E1 )+ R1 since E1 ∈ + stb(F1 ), thus a ∈ ER . – Case 2: a ∈ A2 . By hypothesis, we know that a ∈ (E2 )+E1 . Hence, there is a T ⊆ E2 such that (T, a) ∈ R2

R2E1 . We distinguish two cases. + * Case 2.i: (T, a) ∈ R2 . Then, a ∈ (E2 )R2 , from which it immediately follows a ∈ (E)+ R. / R2 . Then, there is a T ⊇ T such * Case 2.ii: (T, a) ∈ that (T ′ , a) ∈ R3 with T ′ ∩ A1 ⊆ E1 . Hence, T ⊆ + + E1 ∪ E2 = E, deriving a ∈ ER c , i.e.a ∈ (E)R . 3

+ In all cases we get a ∈ ER , therefore E ∈ stb(F ). ⊕ 2. Suppose E ∈ stb(F ). It holds that ER = A = A 1 ∪ A2 . • We show that E1 = E ∩ A1 ∈ stb(F1 ). From Lemma 54, we obtain E ∩ A1 ∈ cf(F1 ). Let a ∈ A1 . Since (F1 , F2 , R3 ) is an attack splitting of F , it holds that a is attacked by some T ⊆ A1 , thus (E ∩ A1 )⊕ R1 = A1 and therefore E1 ∈ stb(F1 ).

• We show that E2 ∈ cf(F2E1 ). Let (T, h) ∈ R2E1 with T ⊆ E2 . By definition of R2E1 , one of the following applies: E1 1 – (T, h) ∈ R2 and T ⊆ AE 2 , h ∈ A2 ;

where F2E1 is the R-reduct wrt. E1 .

(Case 1) 14

• We show that E2 = E ∩ A2 ∈ stb(F2⋆ ). By Lemma 54, we obtain E2 ∈ cf(F2E1 ); by Proposition 27, it holds that E2 is closed. It remains to prove that E2 attacks all remaining arguments. Let a ∈ A2 \ E2 . We show a ∈ (E2 )+E1 . Since E ∈

former case, (U, t) ∈ R and thus a is defended against Tc by E. In the latter case, we have U ′ ∩ A1 ⊆ E1 and U = U ′ ∩ A2 ⊆ E2 and thus U ′ ⊆ E; it follows that E defends a in F . In any case, a is defended in F by E. 2. Suppose E ∈ adm(F ). We show that E1 = E ∩ A1 and E2 = E ∩ A2 are admissible in F1 resp. F2⋆ . By Lemma 54, E1 = E ∈ cf(F1 ) and E2 ∈ cf(F2E1 ); by Proposition 27 both sets are closed. We show that E1 and E2 defend themselves in F1 and F2⋆ , respectively, against each attacker. Recall that each attacker in R3c is closed. • E1 defends itself in F1 follows since E defends itself in F and no argument in E1 is attacked by a subset of A2 or defended by E2 . • E2 defends itself in F2⋆ : Consider a closed set Tc ⊆ A2 which attacks E2 . Then there is an attack (T, a) ∈ R2⋆ , T ⊆ Tc and a ∈ E2 . Wlog we assume Tc = cl F2⋆ (T ). Case 1: ∗0 ∈ Tc . By assumption Tc = cl F2⋆ (T ) and since ∗0 is not the head of any support, (T, h) = ((V ∩ A2 ) ∪ {∗0 }, h) for some (V, h) ∈ UR̂E1 . Then there is 3 V ′ ⊆ V such that cl F (V ′ ) = V and (V ′ , h) ∈ R3 . V is a closed set which attacks E in F on h. It holds that V ∩ (E1 )+ R = ∅, thus there is (U, b) ∈ R such such that U ⊆ E and b ∈ V ∩ A2 . It holds that (U, b) ∈ R2 ∪ R3 .

+ + stb(F ) we have a ∈ ER . Thus either a ∈ ER or 2 + a ∈ ER3 . We proceed by case distinction. + – Case 1: a ∈ ER . There exists an attack (T, a) ∈ R2 2

with T ⊆ E2 , and since R2E1 ⊇ R2 , we know (T, a) ∈ R2E1 . Therefore, a ∈ (E2 )+E1 . R2

+ – Case 2: a ∈ ER . There is some (T, a) ∈ R3 with 3 T ⊆ E, i.e., T ∩ A1 ⊆ E1 . Since E is conflict-free in E1 F we have T ∩ (E1 )+ R1 ∪R3 = ∅, thus, in F2 we have E1 1 an attack (T ∩ AE 2 , a). Since T ∩ A2 ⊆ E2 we get + a ∈ (E2 ) E1 . R2

We obtain E2 ∈ stb(F2E1 ) = stb(F2⋆ ). Admissible Semantics. 1. Suppose E1 ∈ adm(F1 ) and E2 ∈ adm(F2⋆ ). Observe that ∗0 ∈ / E2 since ∗0 is self-attacking. We show that E = E1 ∪ E2 ∈ adm(F ). By Lemma 54, we obtain E = cl F (E); moreover, E is closed by Proposition 27. It remains to show that E defends itself against each closed set. Let Tc ⊆ A denote a closed attacker of E. Then there is (T, a) ∈ R with T ⊆ Tc and a ∈ E. Either a ∈ E1 or a ∈ E2 . We proceed by case distinction. • Case 1: a ∈ E1 . Then (T, a) ∈ R1 and Tc ⊆ A1 since A1 and A2 are not linked by supports. We obtain that E defends a since E1 ∈ adm(F1 ). • Case 2: a ∈ E2 . Then (T, a) ∈ R2 ∪ R3c . Case 2.i: (T, a) ∈ R2 . Then T ⊆ A2 and Tc ⊆ A2 . Thus E2 is attacked by Tc in E2 . We obtain that E defends a since E2 ∈ adm(F2⋆ ). Case 2.ii: (T, a) ∈ R3 . Then there is T ′ ⊇ T such that (T ′ , h) ∈ R3c . It holds that cl F (T ) = cl F (T ′ ), and therefore T ′ = Tc . We proceed by case distinction. Case 2.ii.a: T ′ ∩ (E1 )+ R1 ∪Rc ̸= ∅. Then a is defended

– Case 1.i: (U, b) ∈ R2 . Then (U, b) ∈ R2E1 and thus E2 defends itself against Tc in F2⋆ . – Case 1.ii: (U, b) ∈ R3 . Let Uc = cl F (U ) and observe that Uc ⊆ E since E is admissible in F . Further note that Uc ⊆ E2 since no common positive links between F1 and F2 exist. Since Uc ∩ A1 ⊆ E1 , the link is not undecided. Therefore, (U ∩ A2 , b) ∈ R2E1 and thus E2 defends itself against Tc . Case 2: ∗0 ∈ / Tc . In this case, it holds that either (i) 1 (T, a) ∈ R2 or (ii) (T, a) is of the form (T ′ ∩ AE 2 , a), for some attack (T ′ , a) ∈ R3c , T ′ ∩ A1 ⊆ E1 . Note that 1 (T, a) cannot be of the form ((T ′ ∩ AE 2 ) ∪ {∗0 }, a) since we assume ∗0 ∈ / Tc . We proceed by case distinction. – Case 2.i: (T, a) ∈ R2 . Since A1 and A2 have no common support links, it holds that cl F2⋆ (T ) = cl F (T ) = Tc ⊆ A2 . Since a is defended by E in F , there is a counter-attack (U, b) ∈ R2 ∪ R3 s.t. U ⊆ E and b ∈ Tc . Consider the case (U, b) ∈ R2 . It holds that (U, b) ∈ R2E1 and U ⊆ E2 . We obtain that E2 defends a against Tc in F2⋆ . In case (U, b) ∈ R3 we have U ∩A1 ⊆ E1 (since U ⊆ E1 1 E), and hence we get an attack (U ∩ AE 2 , b) ∈ R2 ⋆ which defends a in F2 . ′ 1 – Case 2.ii: (T, a) = (T ′ ∩ AE 2 , a). Here, (T , a) ∈ c ′ ′ R3 and T ∩ A1 ⊆ E1 . By assumption, T is closed 1 in F . As in Case 1, we have Tc = cl F2⋆ (T ′ ∩ AE 2 )= cl F2⋆ (T ) since F1 and F2 are not linked via supports.

by E1 against (T ′ , a), thus also by E. Case 2.ii.b: T ′ ∩ (E1 )+ R1 ∪Rc = ∅. 3

In this case, either (T ′ ∩ A2 , a) ∈ R2⋆ (via the reduct) or ((T ′ ∩ A2 ) ∪ {∗0 }, a) ∈ R2⋆ (via the modification). Recall that a ∈ E2 and, moreover, E2 ∈ adm(F2⋆ ). Observe that cl F2⋆ (T ′ ∩ A2 ) = cl F2⋆ ((T ′ ∩ A2 ) ∪ {∗0 }) since ∗0 is not contained in the tail of any support in S. Thus, in both cases, E2 defends itself (in F2⋆ ) against the closed attacker Tc′ = cl F2⋆ (T ′ ∩ A2 ): there is U ⊆ E2 , t ∈ Tc′ , such that (U, t) ∈ R2⋆ . Furthermore note that Tc′ ⊆ Tc since T ′ ∩ A2 ⊆ T ′ . Therefore, t ∈ Tc . Now, either (a) (U, t) ∈ R2 or (b) there is some (U ′ , t) ∈ R3 with U ′ ⊃ U s.t. U ′ ∩ A1 ⊆ E1 . In the 15

In particular, we have Tc = T ′ ∩ A2 since T ′ is closed in F . The set T ′ attacks E in F . It holds that T ∩ (E1 )+ R = ′ T ∩ (E1 )+ = ∅. Moreover, T ∩ A ⊆ E . Since 1 1 R1 ∪R3 E defends itself against T ′ , there is a counter-attack (U, b) ∈ R2 with b ∈ T ′ ∩ A2 and U ⊆ E. Thus, there is an attack (U ∩ A2 , b) ∈ R2E1 . We therefore obtain that E2 defends a against Tc in F2⋆ .

pref(F ) and E2 ∈ pref(F2⋆ ) but E = E1 ∪ E2 ∈ / pref(F ), then by the previous item E ∈ adm(F ), so there exists a preferred superset E ′ ⊋ E of F . But then either E ′ ∩ A1 = E1′ ⊋ E1 and since E ′ is admissible in F we have E1 ∈ adm(F1 ) so E1 ∈ / pref(F1 ) or E1′ = E1 and E ′ ∩ A2 = E2′ ⊋ E2 with E2 ∈ adm(F2⋆ ), so E2 ∈ / pref(F2⋆ ). Contradiction in both cases. 2. Now suppose E ∈ pref(F ). Then E ∩ A1 = E1 ∈ adm(F1 ) and E ∩ A2 = E2 ∈ adm(F2⋆ ). If there existed some E2′ ⊋ E2 with E2′ ∈ adm(F2⋆ ), then E1 ∪ E2′ ∈ adm(F ) and E ⊊ E1 ∪ E2′ , so E is not preferred. Contradiction. So E2 ∈ pref(F2⋆ ). It is left to show that E1 ∈ pref(F1 ). Suppose there exists some E1† ⊋ E1 , E1† ∈ adm(F1 ). We will show that in this case E2 ∈ adm(F2⋆† ) is admissible in the modification of † the reduct F E1 2 wrt. E1† . We have strictly less undecided

Complete Semantics. 1. Let E1 ∈ com(F1 ), E2 ∈ com(F2⋆ ). Then for E = E1 ∪ E2 we have by the previous item E ∈ adm(F ). It is left to show that E contains everything it defends. So let a ∈ A such that for every closed attacker D on a with some (T, a) ∈ R, T ⊆ D there exists an attack (S, h) from S ⊆ E to h ∈ D. We proceed by case distinction (a ∈ A1 ) Then D ⊆ A1 and therefore (S, h) ∈ R1 , so E1 defends a, so a ∈ E1 ⊆ E, since E1 is complete in F1 . (a ∈ A2 ) Then D ⊆ A2 or D = D1 ∪ D2 with Di ⊆ Ai for i = 1, 2. For any such D let (S, h) be the attack from E to D, then there are two cases: (a) If (S, h) ∈ R2 , then S ⊆ E2 , so E2 defends a in F2⋆ . But then a ∈ E2 , since E2 is complete. (b) If (S, h) ∈ R3 , then, since S ⊆ E, we have S ∩ A1 ⊆ + E1 and, since E is conflict-free, that S ∩ (E1 )R = ∅. By Definition 8 it follows (cl(S) ∩ A2 , h) ∈ R2⋆ , and since E is closed, we have cl(S) ∩ A2 ⊆ E2 so E2 defends a in F2⋆ , so a ∈ E2 . Therefore in any case a ∈ E, so E ∈ com(F ). 2. For the other direction suppose E ∈ com(F ). Then by the previous item E1 = E ∩ A1 is admissible in F1 and E2 = E ∩ A2 is admissible in F2⋆ . Note that this implies that ∗0 ∈ / E2 . It is left to show that both E1 and E2 contain every argument they defend. (a) Let a be defended by E1 in F1 , then for every closed Attacker D on a there exists an attack (S, h) ∈ R1 with S ⊆ E1 , h ∈ D. But then E defends a in F , since S ⊆ E and R1 ⊊ R. So, since E ∈ com(F ) it follows that a ∈ E ∩ A1 = E1 , so E1 ∈ com(F1 ). (b) Let a be defended by E2 in F2⋆ . Then for every closed Attacker D on a there exists an attack (S, h) ∈ R2⋆ with S ⊆ E2 , h ∈ D. If (S, h) ∈ R2 , then (S, h) ∈ R, so E defends a and therefore E ∩ A2 = E2 must contain a, since E ∈ com(F ). Since E is conflict-free, it cannot contain ∗0 , so it cannot be the case that (S ∪ {∗0 }, h) is one of the attacks added in the modification. So if (S, h) ∈ / R2 , there must be an attack (S ∗ , h) ∈ R3c with + S ∗ ∩ (E1 )R1 ∪Rc = ∅, S ∗ ∩ A1 ⊆ E1 and S ∗ ∩ A2 = S. 3 Then S ∗ ⊆ E, so E defends a and therefore, since E is complete, E2 = E ∩ A2 contains a.

E†

†

links, so R2⋆† \ R2 1 ⊆ R2⋆ \ R2E1 and we have RE1 2 = R2E1 ∪ {(T \ A1 , h) | (T, h) ∈ R3 , T ∩ (E1† )+ R1 ∪Rc = 3

∅, T ∩ A1 ⊆ E1† , T ∩ A1 ⊈ E1 , h ∈ A2 } (since E1† is conflict-free, no attacks are lost). Now suppose E2 ∈ / adm(F2⋆† ). Then E2 is either not closed, not conflict-free or does not defend itself. Nothing changed about the arguments or the support, so E2 is E†

closed in F2⋆† . We add only attacks in the set R2 1 = R2E1 ∪ {(T \ A1 , h) | (T, h) ∈ R3 , T ∩ (E1† )+ R3 = † ∅, T ∩ A1 ⊆ E1 , T ∩ A2 ⊈ E1 , h ∈ A2 }. Suppose (T ′ , h) ∈ {(T \ A1 , h) | (T, h) ∈ R3 , T ∩ (E1† )+ R3 = † ∅, T ∩ A1 ⊆ E1 , T ∩ A1 ⊈ E1 , h ∈ A2 } and h ∈ E2 . Then there is an attack (T ′ ∪ {∗0 }, h) ∈ R2⋆ , since the attack (T, h) ∈ R with T ′ = T ∩ A2 is an undecided link wrt. the original reduct F2E1 (because E1† is conflict-free it cannot be a link from defeated arguments). So (T ′ ∪ {∗0 }, h) is an attack on E2 in F2⋆ . Since E2 ∈ adm(F2⋆ ), E2 defends itself against this attack in F2⋆ , but this means E2 attacks itself, since there are no attacks on ∗0 . But E2 is also conflict-free in F2⋆ . Contradiction. So E2 is conflict-free in F2⋆† . It is left to show that E2 defends itself. We can distinguish the following attacks (T ′ , h) on E2 in F2⋆† : E†

• (T ′ , h) ∈ R2 1 . That is, (T ′ , h) ∈ {(T \ A1 , h) | † (T, h) ∈ R3 , T ∩ (E1† )+ R3 = ∅, T ∩ A1 ⊆ E1 , T ∩ A2 ⊈ E1 , h ∈ A2 }. As we just argued, for each such an attack there is a corresponding attack (T ′ ∪{∗0 }, h) ∈ R2⋆ stemming from an undecided link. E2 defends itself against this attack by attacking every closed set containing T ′ ∪ {∗0 } in F2⋆ . Since ∗0 is not supported by anything and only attacked by itself, for each closed set D ∪ {∗0 } the subset D is closed. So since E2 attacks every closed D containing T ′ ∪{∗0 } and does not attack ∗0 , it attacks any closed D containing T ′ . Furthermore, none of the attacks stemming from undecided links are used for this, since ∗0 ∈ / E2 , so E2 also attacks D in F2⋆† .

Preferred Semantics 1. The first direction follows directly from the correspondence results for admissible semantics. Suppose E1 ∈ 16

E†

• (T ′ , h) ∈ / R2 1 . Since the arguments and supports are the same between F2⋆ and F2⋆† , T ′ is contained in the same closed sets D, so E2 attacks every closed set containing T ′ in F2⋆ , since E2 is admissible, so it also does so in F2⋆† , since all attacks going out from E2 are preserved. So E2 defends itself in F ⋆† , so E2 ∈ adm(F2⋆† ). But then E1† ∪ E2 ∈ adm(F ). Contradiction. So E1 ∈ pref(F1 ). Theorem 30. Let (F1 , F2 , R3 ) be an attack splitting for a BSAF F = (A, R, S) and R3c the set of closed negative links. E1 E1 c 1 Let F2⋆ = modE R3c (F2 ) where F2 is the (E1 , R3 )-reduct. If E1 ∈ grd(F1 ) and E2 ∈ grd(F2⋆ ), then E1 ∪E2 ∈ grd(F ). Suppose E1 ∈ grd(F1 ) and E2 ∈ grd(F2⋆ ) but E = E1 ∪ E2 ∈ / grd(F ). Then, by the Theorem 28, E ∈ com(F ). Thus, there exists a grounded subset E ′ ⊊ E of F . But then either E ′ ∩ A1 = E1′ ⊊ E1 and since E ′ is complete in F we have E1 ∈ com(F1 ) so E1 ∈ / com(F1 ) or E1′ = E1 ′ ′ and E ∩ A2 = E2 ⊊ E2 with E2 ∈ com(F2⋆ ), so E2 ∈ / com(F2⋆ ). Contradiction in both cases.

Omitted Proofs of Section 4

Case 3.i: T ∩ A1 = ∅. In this case, T ∈ DS3 (E1 ). By construction of the S-reduct, (T, ∗1 ) ∈ S2⋆ . Since T ⊆ E2 and E2 is closed in F2⋆ , we have ∗1 ∈ E2 . Since (∅, ∗1 ) ∈ R2⋆ , E2 is attacked by the empty set and thus not admissible in F2⋆ ; contradiction to our initial assumption. Case 3.ii: T ∩ A1 ̸= ∅. In this case, T ∈ TS3 (E1 ) \ DS3 (E1 ). By construction of the S-reduct, (T ∩ A2 , ∗2 ) ∈ S2⋆ . Since T ∩ A2 ⊆ E2 and E2 is closed in F2⋆ , we have ∗2 ∈ E2 . Again, we can derive a contradiction to admissibility of E2 in F2⋆ : By construction of the S-reduct, F2⋆ contains the attack ((T ∩ A2 ) ∪ {∗2 }, ∗2 ), thus E2 ∈ / cf(F2⋆ ).

Lemma 55. Let (F1 , F2 , S3 ) be a support splitting for a BSAF F with SF1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ). Let E1 ⊆ A1 and F2E1 the S-reduct of F2 wrt. E1 . 1. If E1 ∈ cf(F1 ) and E2 ∈ cf(F2E1 ), then E1 ∪ (E2 \ {∗2 }) ∈ cf(F ). 2. If E ∈ cf(F ), then E1 = E ∩ A1 ∈ cf(F1 ) and E ∩ A2 ∈ cf(F2E1 ). Proof. We prove the statements separately. 1. Suppose E1 ∈ cf(F1 ) and E2 ∈ cf(F2E1 ). Let E = E1 ∪ (E2 \ ∗2 ). We distinguish two cases: • TS3 (E1 ) = ∅. In this case, we have F2E1 = F2 and thus E2 ∈ cf(F2 ). Therefore, E1 ∪ E2 ∈ cf(F ) follows from the fact that there is no attack from F1 and F2 . • TS3 (E1 ) ̸= ∅. In this case, we have R2E1 ⊇ R2 ∪ {(∅, ∗1 )}, so that E2 ∈ cf(F2 ). Moreover, ∗1 ∈ / E2 because E2 is conflict-free in F2E1 . Furthermore, E2 \ {∗2 } ⊆ E2 and hence (E2 \{∗2 }) ∈ cf(F2 ). Therefore, since there are no attacks between F1 and F2 , we derive that E1 ∪ (E2 \ {∗2 }) ∈ cf(F ). . 2. Suppose E ∈ cf(F ). We show that E1 = E∩A1 ∈ cf(F1 ) and E2 = E ∩ A2 ∈ cf(F2E1 ). • Since R1 ⊆ R, it immediately follows that E1 = E ∩ A1 ∈ cf(F1 ). • We now consider E2 = E ∩ A2 ∈ cf(F2E1 ). Since R2 ⊆ R and (F1 , F2 , S3 ) is a proper support splitting, we know that E ∩ A2 ∈ cf(F2 ). Further, since E ⊆ A, we derive that ∗i ∈ / E ∩ A2 for all i ∈ {1, 2}. Thus, since the reduct only adds attacks towards ∗1 or ∗2 , we know that E ∩ A2 ∈ cf(F2E1 ).

In each case, we have h ∈ E; therefore, E = clF (E). 2. Assume E is admissible in F . Let E1 = E ∩ A1 and E2 = E ∩ A2 . We show that E1 is closed in F1 and E2 is closed in F2E1 . • Let E1 = E ∩ A1 . We prove E1 = clF1 (E1 ). Suppose towards contradiction that E1 supports some a ∈ A1 in F1 , but a ∈ / E1 . Then for some T ⊆ E1 and a ∈ A1 \ E1 we have (T, a) ∈ S1 . Since E1 ⊆ E and S1 ⊆ S, we derive that E supports some a ∈ A \ E in F . Thus, E ̸= clF (E), in contradiction with our hypothesis. • Let E2 = E ∩A2 . We show E2 = clF E1 (E2 ). Towards 2 a contradiction, suppose that there is some T ⊆ E2 E1 such that (T, h) ∈ S2 , but a ∈ / E2 . We proceed by case distinction. – Case 1: (T, h) ∈ S2 . Then T ⊆ E, contradiction to E = cl F (E). – Case 2: h = ∗1 . Then T ∈ DS3 (E1 ). Therefore, there is is some a ∈ A1 with (T, a) ∈ S3 and a ∈ / E1 . We obtain a contradiction to E being closed in F since T ⊆ E but a ∈ / E (as a ∈ / E1 ). – Case 3: h = ∗2 . Then the support is of the form (T, h) = (T ′ ∩ A2 , ∗2 ) for some T ′ ⊇ T with T ′ ∈ TS3 (E1 ). Therefore, there is is some a ∈ A1 with (T ′ , a) ∈ S3 and a ∈ / E1 . It holds that T ′ ⊆ E since T ′ ∩ A1 ⊆ E1 and T = T ′ ∩ A2 ⊆ E1 . Consequently, in F , E supports a but a ∈ / E, contradiction to E = cl F (E).

Lemma 56. Let (F1 , F2 , S3 ) be a support splitting for BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ). Further, let E1 ⊆ A1 , E2 ⊆ A2 , E = E1 ∪ E2 , and let E1 E1 1 F2E1 = (AE 2 , R2 , S2 ) the S-reduct of F2 wrt. E1 . 1. If E1 ∈ adm(F1 ) and E2 ∈ adm(F2E1 ), then E = cl F (E) where E = E1 ∪ (E2 \ {∗2 }). 2. If E ∈ adm(F ), then E ∩ A1 = E1 = clF1 (E1 ) and E2 = clF E1 (E2 ) where E2 = E ∩ A2 . 2

Theorem 44. Let (F1 , F2 , S3 ) be a support splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ), and σ ∈ {stb, adm, com}. Further, let F2E1 be the S-reduct of F2 .

Proof. 1. Let E1 ∈ adm(F1 ) and E2 ∈ adm(F2E1 ). We show that E = E1 ∪ (E2 \ {∗2 }) is closed. Let (T, h) ∈ S with T ⊆ E. It holds that either (T, h) ∈ S1 , (T, h) ∈ S2 , or (T, h) ∈ S3 . We proceed by case distinction. • Case 1: (T, h) ∈ S1 . Then T ⊆ E1 . Since E1 is closed in F1 by assumption, we get h ∈ E1 ⊆ E. • Case 2: (T, h) ∈ S2 . Then T ⊆ E2 . Moreover, S2 ⊆ S2⋆ . Since E2 is closed in F2⋆ , we obtain h ∈ E2 ⊆ E. • Case 3: (T, h) ∈ S3 . It holds that h ∈ A1 by definition of S3 . Recall that T ⊆ E, thus T ∩ A1 ⊆ E1 . Towards a contradiction, suppose h ∈ / E1 . We proceed by case distinction.

1. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2E1 ), then E1 ∪(E2 \{∗2 }) ∈ σ(F ). 2. If E ∈ σ(F ), then E1 = E ∩ A1 ∈ σ(F1 ) and (E ∩ A2 ∈ σ(F2E1 ) or (E ∩ A2 ) ∪ {∗2 } ∈ σ(F2E1 )). Proof. Similarly to Theorem 28, we prove the statements for each semantics. Admissible Semantics. 18

1. Suppose E1 ∈ adm(F1 ) and E2 ∈ adm(F2E1 ). We show E = E1 ∪ E2 \ {∗2 } ∈ adm(F ). • E is closed in F : this follows from Lemma 56. • E is conflict-free: this follows from Lemma 55. • E defends itself against each closed set: Consider a set Tc ⊆ A which is closed in F , i.e., Tc = cl F (Tc ), and attacks E in F . Then there is T ⊆ Tc , h ∈ E, such that (T, h) ∈ R. It holds that either (T, h) ∈ R1 or (T, h) ∈ R2 . – Case 1: (T, h) ∈ R1 . Then h ∈ E1 and T ⊆ A1 . Let T1 = cl F1 (T ). Observe that T1 ⊆ Tc since S1 ⊆ S, moreover, T1 attacks h in F1 . Since E1 is admissible in F1 , it holds that E1 attacks T1 . We obtain E defends itself against Tc . – Case 2: (T, h) ∈ R2 . Then h ∈ E2 and T ⊆ A2 . Let T2 = cl F2 (T ). We first show the following:

itself against T2 . Thus, there is (U, a) ∈ R2 with a ∈ T2 , U ⊆ E2 . We obtain that E defends itself against Tc in F . 2. Suppose E ∈ adm(F ). We show that E1 = E ∩ A1 ∈ adm(F1 ) and E2 = E ∩ A2 ∈ adm(F2E1 ). • Let E1 = E ∩ A1 . E1 is conflict-free by Lemma 55 and closed by Lemma 56. It remains to prove that E1 defends itself. Let Tc be a closed attacker of E1 . Then there is T ⊆ Tc , h ∈ E1 , such that (T, h) ∈ R1 . Observe that cl F (T ) ⊆ A1 since T ⊆ A1 and there is no support (B, a) ∈ S with B ⊆ A1 and a ∈ A2 ; all supports either stay in the respective sets A1 or A2 or go ’the other direction’. By assumption, E defends itself against the closed attacker cl F (T ). Thus, there is an attack (U, a) ∈ R1 with U ⊆ E and a ∈ cl F (T ). By definition of support splitting U ⊆ A1 , thus U ⊆ E1 . We obtain that E1 defends itself against cl F (T ). Since T ⊆ Tc , we obtain cl F (T ) ⊆ Tc . Thus, E1 defends itself against Tc . • Let E2 = E ∩ A2 . E2 is conflict-free by Lemma 55 and closed by Lemma 56. It remains to prove that E2 defends itself. Let Tc denote a closed attacker of E2 in F2E1 . Then there is T ⊆ Tc , h ∈ E2 , such that (T, h) ∈ R2 (observe that (T, h) ∈ / R2E1 \ R2 because h ∈ A2 ). Therefore, T attacks E in F . Due to admissibility of E in F , there is some (U, a) ∈ R with U ⊆ E and a ∈ cl F (T ). Note that a ∈ A. Case 1: (U, a) ∈ R1 . Then U ⊆ A1 and a ∈ A1 . Let us inspect the set cl F (T ): First, we observe that T ⊆ A2 , but cl F (T ) ∩ A1 ̸= ∅. Thus, there is some T ′ ⊆ T such that (T ′ , c) ∈ S3 . Wlog, we can assume that – c∈ / E1 (otherwise, if for all d ∈ (cl F (T ) \ T ) ∩ A1 , it holds that d ∈ E1 , we obtain a ∈ E1 ; then E1 attacks itself via the attack (U, a), contradiction to conflictfreeness of E1 ); and – a ∈ cl F (T ′ ) (otherwise, if there is no T ′′ ⊆ T such that a ∈ cl F (T ′′ ) then a ∈ / cl F (T ), contradiction to our assumption). In summary, it follows that T ′ ∈ DS3 (E1 ): (T ′ , c) ∈ S3 , c ∈ / E1 , T ∩ A1 = ∅, and cl F (T ′ ) is attacked by E1 (via a). Consequently, (T ′ , ∗1 ) ∈ S2E1 and (∅, ∗1 ) ∈ R2E1 . Therefore, the set cl F E1 (T ′ ) is at2 tacked by the empty set. Since T ′ ⊆ T we have cl F E1 (T ′ ) ⊆ cl F E1 (T ) and thus cl F E1 (T ′ ) ⊆ T2 . In

Claim 1. For all a ∈ A2 , a ∈ T2 implies a ∈ Tc . Proof. First, we observe that for all a ∈ A2 , a ∈ T2 iff either a ∈ T or there is (B, a) ∈ S2 with B ⊆ T2 (by construction, the head of each newly introduced support in F2 is contained in {∗1 , ∗2 }). Moreover, for every (B, a) ∈ S2E1 , it holds that {∗1 , ∗2 } ∩ B = ∅. Consequently, a ∈ A2 is supported by T2 in F2E1 implies either a ∈ T or there are supports (B1S, b1 ), . . . , (Bk , bk ) ∈ S2 with bk = a and Bi ⊆ T ∪ j<i bj . In the former case, a ∈ Tc . In the latter case, we observe that this support chain also exists in F , and since T ⊆ Tc , we obtain a ∈ Tc whenever a ∈ T2 . ♢ We proceed by case distinction. We distinguish the cases ∗1 ∈ T2 , ∗2 ∈ T2 , and T2 ∩ {∗1 , ∗2 } = ∅. Case 2.i: ∗1 ∈ T2 . In this case, E2 is defended against T2 since T2 is attacked by the empty set. By construction, there is (T ′ , ∗1 ) ∈ S2E1 with T ′ ⊆ T2 , and T ′ ∈ DS3 (E1 ). Thus it holds that (T ′ , a) ∈ S3 , a ∈ A1 , a ∈ / E1 , and there is b ∈ cl F (T ′ ) such + that b ∈ E1 . Since T ′ ⊆ A2 , we obtain by Claim 1, T ′ ⊆ Tc . Thus, cl F (T ′ ) ⊆ Tc . Consequently, b ∈ Tc , and therefore, E1 ⊆ E defends E against the attack from Tc . Case 2.ii: ∗2 ∈ T2 . Since E2 is admissible in F2E1 , there is some b ∈ T2 such that b ∈ (E2 )+E1 . Two F2

further cases: ′ * Case 2.ii.a: b = ∗2 . Then there is T ∈ TS3 (E1 ) \ ′ ′ DS3 (E1 ) with T ⊆ E2 and ((T ∩A2 )∪{∗2 }, ∗2 ) ∈ R2E1 . Contradiction to conflict-freeness of E1 . * Case 2.ii.b: b ̸= ∗2 . Then b ∈ A2 . There is an attack (U, b) ∈ R2 and U ⊆ E2 (since b ∈ A2 , the attack has not been newly introduced in F2E1 ). By Claim 1, b ∈ Tc . Since U ⊆ E2 ∩ A2 ⊆ E we obtain E defends itself against Tc . Case 2.iii: T2 ∩ {∗1 , ∗2 } = ∅. In this case, T2 ⊆ Tc by Claim 1. Since E2 is admissible in F2 , it defends

other words, E2 defends itself against T2 in F2E1 . Case 2: (U, a) ∈ R2 . Then U ⊆ A2 and a ∈ A2 . Observe that cl F (T )∩A2 ⊆ Tc since S2 ⊆ S2E1 and no set of arguments in A1 supports an argument in A2 . Therefore, U ⊆ E1 attacks a ∈ Tc in F2E1 . We have shown that E2 defends itself against the closed attacker. Stable semantics. 1. Suppose E1 ∈ stb(F1 ) and E2 ∈ stb(F2E1 ). We show that E = E1 ∪ (E2 \ {∗2 }) ∈ stb(F ). 19

• The premisses E1 ∈ com(F1 ) and E2 ∈ com(F2E1 ) directly entail E1 ∈ adm(F1 ) and E2 ∈ adm(F2E1 ), hence E ∈ adm(F ) as shown above. • It remains to be shown that E defends itself in F . – Case 1: TS3 (E1 ) = ∅. Then F2E1 = F2 , and for any (T, h) ∈ S3 : h ∈ E ∨ T ⊊ E. Lets assume E defends a ∈ A \ E in F . If a ∈ A1 then E1 defends a in F1 , contradiction to E1 ∈ com(F1 ). On the other hand, if a ∈ A2 then E2 defends a in F2 , contradiction to E2 ∈ com(F2 ). Thus E defends itself in F . – Case 2: TS3 (E) ̸= ∅. Toward contradiction, we assume there exists an a that is defended by E in F , s.t. a∈ / E. Let T ⊆ A be a minimal closed set that c is defended against by E in F , i.e. ∃T ′ ⊆ T : (T ′ , c) ∈ R and ∃t ∈ T ∃TE ⊆ E : (TE , t) ∈ R. * Case 2i: a ∈ A1 . Then E1 defends a in A1 , which contradicts E1 ∈ com(F1 ). * Case 2ii: a ∈ A2 and cl F2 (T ) ∩ {∗1 , ∗2 } = ∅. Then, cl F2 (T ) attacks a and E2 attacks cl F2 (T ) in F2E1 . Hence E2 is not complete in F2E1 , which contradicts E2 ∈ com(F2E1 ). * Case 2iii: a ∈ A2 and ∗1 ∈ cl F2 (T ). Then T is attacked by E2 in F2 , more specifically, ∅ ⊆ E2 attacks {∗1 } ⊆ cl F2 (T ). Hence E2 is not closed in F2 , which contradicts E2 ∈ com(F2 ). * Case 2iv: a ∈ A2 and ∗2 ∈ cl F2 (T ). Then E does not attack a t ∈ T ∩ A1 , but attacks a t ∈ T ∩ A2 . Hence E2 attacks t ∈ T ∩ A2 , thus E2 is not closed in F2 , contradiction! 2. Suppose E ∈ com(F ). We show that E1 = E ∩ A1 ∈ com(F1 ) and (E ∩ A2 ∈ com(F2E1 ) or (E ∩ A2 ) ∪ {∗2 } ∈ com(F2E1 )). • Utilizing the proof for admissible semantics, we directly conclude E1 = E ∩ A1 ∈ adm(F1 ) and E2 = E ∩A2 ∈ adm(F2E1 ) or (E ∩A2 )∪{∗2 } ∈ adm(F2E1 )). • There are no attacks from A2 towards A1 , neither are there supports from A1 towards A2 . Hence any argument in A1 is defended by E in F iff it is defended by E1 = E ∩ A1 in F1 . Thus E1 defends itself in F1 , therefore E1 = E ∩ A1 ∈ com(F1 ). • It remains to be shown that E2 defends itself in F2E1 . We assume toward a contradiction that there exists a c ∈ A2 \ E2 s.t. c is defended by E2 in F2E1 . – Case 1: TS3 (E) = ∅. Then F2E1 = F2 , and for any (T, h) ∈ S3 : h ∈ E. Thus any argument c ∈ A2 defended by E2 in F2E1 is defended by E in F . The statement c ∈ / E2 entails c ∈ / E. This contradicts the premise that E is complete. – Case 2: TS3 (E) ̸= ∅. Let T ⊆ A2 be a closed set that c is defended against by E2 in F2E1 , i.e. ∃T ′ ⊆ T : (T ′ , c) ∈ R2E1 and ∃t ∈ T ∃TE2 ⊆ E2 : (TE2 , t) ∈ R2E1 . * Case 2i: cl (T \ {∗1 , ∗2 }) ∩ {∗1 , ∗2 } = ∅. Then, by the nature of the construction of F2E1 , T \ {∗1 , ∗2 }

• E is conflict-free: By Lemma 55 and since E1 ∈ stb(F1 ) and E2 ∈ stb(F2E1 ), we obtain E ∈ cf(F ). • E is closed because of Lemma 56 and stb(F ) ⊆ adm(F ) for every BSAF F . • E attacks all remaining arguments: Let a ∈ A \ E. We proceed by case distinction. – Case 1: a ∈ A1 . Then a ∈ (E1 )+ R1 since E1 ∈ + stb(F1 ), thus a ∈ ER . – Case 2: a ∈ A2 . Since E2 ∈ stb(F2E1 ), we derive that a ∈ (E2 )+E1 . By definition of S-reduct, R2E1 R2

augments R2 with an attack towards ∗1 and (possibly) multiple attacks towards ∗2 , i.e. R2E1 ⊆ R2 ∪ {(∅, ∗1 )} ∪ {((T ∩ A2 ) ∪ {∗2 }, ∗2 )} for possibly some T ⊆ A2 . For this, it is immediate to see that no attack towards any a ∈ A2 is added via the S-reduct. Therefore, we derive that a ∈ (E2 )+ R2 and, consequently, + a ∈ ER . + In all cases we get a ∈ ER , therefore E ∈ stb(F ). ⊕ 2. Suppose E ∈ stb(F ). It holds that ER = A = A 1 ∪ A2 .

• We show that E1 = E ∩ A1 ∈ stb(F1 ). From Lemma 55, we obtain E ∩ A1 ∈ cf(F1 ). E1 is closed in F1 because of Lemma 56. Finally, we show that E1 attacks all arguments a ∈ A1 \ E1 . Let a ∈ A1 . Since (F1 , F2 , S3 ) is a support splitting of F , it holds that R is partitioned into R1 and R2 . Thus, a is attacked by some T ⊆ A1 , and (E ∩ A1 )⊕ R1 = A1 . We then derive that E1 ∈ stb(F1 ). • We show that E2 = E ∩ A2 ∈ stb(F2E1 ). By Lemma 55, we obtain E2 ∈ cf(F2E1 ). E2 is closed in F2E1 because of Lemma 56. It remains to prove that E2 attacks all remaining arguments in F2E1 , i.e. (E2 )+E1 = R2

E1 1 AE 2 \E2 . Let a ∈ A2 \E2 . By definition of S-reduct, we can distinguish two cases. – Case 1. a ∈ A2 . For this, notice that R is partitioned into R1 and R2 by definition of support splitting (F1 , F2 , S3 ). Thus, a is attacked by some T ⊆ A2 , i.e. a ∈ (E2 )+ R2 . Further, no attack is removed when computing the reduct F2E1 . Thus, we can derive a ∈ (E2 )+E1 . R2

– Case 2. a ∈ / A2 . Thus, a ∈ {∗1 , ∗2 }. By definition of S-reduct, we know that ∗1 ∈ (E2 )+E1 because R2

∗1 ∈ (∅)+E1 . Consider now the argument ∗2 . If R2

E ∩ A2 attacks ∗2 , then ∗2 ∈ E ∩ A2 in contradiction with conflict-freeness. Therefore, (E ∩ A2 ) ∪ {∗2 } ∈ stb(F2E1 ). Complete Semantics. 1. Suppose E1 ∈ com(F1 ) and E2 ∈ com(F2E1 ). We show E = E1 ∪ E2 \ {∗2 } ∈ com(F ). 20

T ′ ∩ A2 ⊆ E2′ . Since (T ′ ∩ A2 , ∗2 ) ∈ S2E1 , this implies E2′ is not closed. Contradiction to E2′ ∈ adm(F2E1 ). Case 2.i.b: h ∈ A2 . In this case, T ⊆ E2′ and we have a contradiction to the conflict-freeness of E2′ . Case 2.ii: ∗2 ∈ / E2 . Then E2′ ⊋ E2′ , contradiction to E2 ∈ pref(F2 ).

does not support any a ∈ A1 \E1 in F . If T supports an a ∈ E1 , then E attacks a, hence E is not conflictfree, and thus not a complete extension, contradiction. Otherwise, in F , cl (T \ {∗1 , ∗2 }) is closed, attacks c and is attacked by E ⊇ E2 \ {∗2 }. * Case 2ii: ∗1 ∈ cl (T \ {∗1 , ∗2 }). Then by the construction of F2E1 we can infer that there is a subset T ′ ⊆ T \ {∗1 , ∗2 } s.t. T ′ ∈ TS3 (E1 ) \ DS3 (E1 ). Thus ∃b ∈ cl F (T ) : b ∈ E + , meaning that either T is not closed in F , or E defends c against T in F . This contradicts E ∈ com(F ). * Case 2iii: ∗2 ∈ cl (T \ {∗1 , ∗2 }). Then by the construction of F2E1 we can infer that there is a subset T ′ ⊆ T \ {∗1 , ∗2 } s.t. T ′ ∈ DS3 (E1 ). Thus in F , ∃b ∈ cl F (T \ {∗2 }), s.t. E attacks c. We thus infer that E defends c from cl F (T ). This contradicts the premise that E is complete.

Theorem 46. Let (F1 , F2 , S3 ) be a support splitting for a BSAF F = (A, R, S), and let F2E1 be the S-reduct of F2 and σ ∈ {grd, pref}. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2E1 ), then E1 ∪ (E2 \ {∗2 }) ∈ σ(F ). Proof. We prove the statement for each σ ∈ {grd, pref}. Grounded Semantics. Suppose E1 ∈ grd(F1 ) and E2 ∈ grd(F2E1 ). Then E1 ∈ com(F1 ) and E2 ∈ com(F2E1 ), thus by Theorem 44: E = E1 ∪ E2 \ {∗2 } ∈ com(F ). Assuming toward contradiction ∃G ⊊ E : G ∈ com(F ), then ∃G1 ⊊ E1 : G1 ∈ com(F1 ) or ∃G2 ⊊ E2 : G2 ∈ com(F2E1 ). Contradiction! Thus E ∈ grd(F ). Preferred Semantics. Suppose E1 ∈ pref(F1 ) and E2 ∈ pref(F2E1 ). Since E1 and E2 are admissible in F1 and F2E1 , respectively, we obtain E ∈ adm(F ). We show that E = E1 ∪ E2 \ {∗2 } ∈ pref(F ). Towards a contradiction, suppose E ∈ / pref(F ). Then there is E ′ ⊋ E with E ′ ∈ adm(F ). Then E1′ = E ′ ∩ A1 ∈ adm(F1 ) and E2′ = E ′ ∩ A2 ∈ adm(F2E1 ). E ′ is strictly larger than E. Proceed with case distinction. Case 1: E1′ is strictly larger than E1 . This is in contradiction to E1 ∈ pref(F1 ). Case 2: E2′ is strictly larger than E2 \ {∗2 }. Case 2.i: ∗2 ∈ E2 . Let E2′′ = E2 ∪ {∗2 }. If E2′′ ∈ adm(F2E1 ) then E2′′ ⊋ E2′ , contradiction to E2 ∈ pref(F2 ). We show that E2′′ ∈ adm(F2E1 ): E2′′ is closed since E2′ is closed and ∗2 ∈ / U for any (U, b) ∈ S2E1 . E2′′ defends ∗2 ′′ since E2 ⊆ E2 defends ∗2 . E2′′ is conflict-free: Towards a contradiction, suppose there is an attack (T, h) ∈ R2E1 such that T ⊆ E2′′ and h ∈ E2′′ . Case 2.i.a: h = ∗2 . By construction, T has the form ((T ′ ∩ A2 ) ∪ {∗2 }, ∗2 ) for some T ′ ∈ TS3 (E1 ). Thus, 21

E1 and t ∈ A2 . Thus, T ′ ∩ (E1 )+ R3c = ∅. Then, by ⊛ definition of the R-reduct, R2 contains (∅, t). Since t ∈ E, we obtain contradiction to the conflict-freeness of E2 in F2⊛ . (c) If T ∩ A1 ⊆ E1 , we obtain a contradiction to conflict-freeness of E1 . 2. Suppose E ∈ adm(F ). We show that E1 = E ∩ A1 ∈ cf(F1 ) and E2 = E ∩ A2 ∈ cf(F2⊛ ). • E1 ∈ cf(F1 ) since R1 ⊆ R. • We show that E2 ∈ cf(F2⊛ ). Let (T, h) ∈ R2⊛ with T ⊆ E2 . By definition of E2 , we have h ∈ A2 . Thus, by definition of R2⊛ , one of the following applies for (T, h):

Omitted Proofs of Section 5

We recall the notation from Definition 48. Throughout this section, we will make use of the following conventions. Let (F1 , F2 , R3 , S3 ) be a splitting for a BSAF F with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ). Let R3c denote the set of closed attacks wrt. R3 and R2c denote the set of closed attacks wrt. R2 . Let R̂2 = {(T, h) ∈ R2c | T ∩ A1 = ∅} and let F̂2 = (A2 , R̂2 , S2 ) denote the updated BSAF. Further, let R̂3 = {(T, h) | (T, h) ∈ R3c ∪ R2c , T ∩ A1 ̸= ∅}. Let F̂2E1 be the R-reduct of the attack splitting (F1 , F̂2 , R̂3 ) 1 wrt. E1 and F2⋆ = modE (F̂2E1 ). R̂3 ⊛ ⊛ We let F2⊛ = A⊛ denote the S-reduct for the 2 , R2 , S2 support splitting (F1 , F2⋆ , S3 ) wrt. E1 .

– (T, h) ∈ R̂2 and T ⊆ A2 , h ∈ A2 ; (Case 1) – (T, h) = (T ′ \ A1 , h) for some T ′ ⊆ A such that (T ′ , h) ∈ R̂3 , T ′ ∩ (E1 )+ = ∅, T ∩ A1 ⊆ E1 , R1 ∪R̂3 and h ∈ A2 ; (Case 2) – (T, h) = ((T ′ ∩ A2 ) ∪ {∗0 }, h) for some (T ′ , h) ∈ (Case 3) UR̂E1 .

Lemma 57. Let (F1 , F2 , R3 , S3 ) be a splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ). Let F2⊛ be defined as in Definition 48. 1. If E1 ∈ adm(F1 ) and E2 ∈ adm(F2⊛ ), then E = E1 ∪ (E2 \ {∗2 }) ∈ cf(F ). 2. If E ∈ adm(F ), then E1 = E ∩ A1 ∈ cf(F1 ) and E ∩ A2 ∈ cf(F2⊛ ).

We proceed by case distinction. – Case 1. (T, h) ∈ R̂2 with T ⊆ A2 , h ∈ A2 . By definition of R̂2 , we know that (T, h) ∈ R2E1 and T ∩ A1 = ∅. Thus, there is some T ′ ⊆ T ⊆ A2 such that T = clS (T ′ ) and (T ′ , h) ∈ R2 . Thus h ∈ / E2 since E ∈ cf(F ). – Case 2. From T = T ′ \A1 ⊆ E2 and T ′ ∩A1 ⊆ E1 we obtain T ′ ⊆ E. By definition of R̂3 , there is T ′′ ⊆ T ′ such that (T ′′ , h) ∈ R3 or (T ′′ , h) ∈ R2 . We obtain h∈ / E2 since E ∈ cf(F ) and since R3 ⊆ R. – Case 3. It follows that ∗0 ∈ E2 ⊆ E, in contradiction to E ⊆ A. We have shown that E2 ∈ cf(F2⊛ ).

Proof. 1. Let E1 ∈ adm(F1 ) and E2 ∈ adm(F2⊛ ). We show that E = E1 ∪ (E2 \ {∗2 }) ∈ cf(F ). Let (T, h) ∈ R s.t. T ⊆ E. Then either (T, h) ∈ R1 , (T, h) ∈ R2 , or (T, h) ∈ R3 . / E since E1 ∈ • Case 1: (T, h) ∈ R1 . Then h ∈ adm(F1 ). • Case 2: (T, h) ∈ R2 . It holds that T ⊆ A2 , h ∈ A2 . – Case 2i: cl (T ) ⊆ A2 . Then (cl (T ), h) ∈ R2⊛ . – Case 2ii: cl (T ) ∩ A1 ̸= ∅. Then, T ∩ E + = ∅ entails ((cl (T ) \ A1 ), h) ∈ R2⊛ In both cases we have cl (T ) ⊆ E2 and h ∈ / E2 since E2 ∈ adm(F2⊛ ). Thus h ∈ / E. • Case 3: (T, h) ∈ R3 . Let Tc = cl F (T ). Then either (Tc , h) ∈ UREc1 or (Tc , h) ∈ / UREc1 . 3

Lemma 58. Let (F1 , F2 , R3 , S3 ) be a splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ). Let F2⊛ be defined as in Definition 48. 1. If E1 ∈ adm(F1 ) and E2 ∈ adm(F2⊛ ), then E = cl F (E) where E = E1 ∪ (E2 \ {∗2 }). 2. If E ∈ adm(F ), then E ∩ A1 = E1 = clF1 (E1 ) and E2 = clF ⊛ (E2 ) where E2 = E ∩ A2 .

– Case 3.i: (Tc , h) ∈ UREc1 . Then (Tc ∩ A2 ) ∪ {∗2 } at3

tacks h in F2⊛ . Thus, h ∈ / E2 and therefore h ∈ / E. E1 – Case 3.ii: (Tc , h) ∈ / UR c .

Proof. We recall: F1 = (A1 , R1 , S1 ) and F2⊛ = ⊛ ⊛ ⊛ ⊛ ⊛ (A⊛ 2 , R2 , S2 ) with A2 = A2 ∪ {∗0 , ∗1 , ∗2 } and R2 , S2 as defined by the respective modifications for support and attack splitting.

From (Tc , h) ∈ / UREc1 , we get that Tc ∪(E1 )+ R1 ∪R3c ̸= ∅ 3 ⊕ or Tc ∩A1 ⊆ (E1 )R1 . This amounts to: Tc ∩(E1 )+ R1 ̸= ∅ or Tc ∩ (E1 )+ = ̸ ∅ or T ∩ A ⊆ E . c c 1 1 R3

1. E1 ∈ adm(F1 ) and E2 ∈ adm(F2⊛ ). We show that E = E1 ∪ (E2 \ {∗2 }) is closed in F . The proof is similar to the case for support splitting. Let (T, h) ∈ S with T ⊆ E. It holds that either (T, h) ∈ S1 , (T, h) ∈ S2 , or (T, h) ∈ S3 . We proceed by case distinction. • Case 1: (T, h) ∈ S1 . Then T ⊆ E1 . Since E1 is closed in F1 by assumption, we get h ∈ E1 ⊆ E.

We proceed by case distinction. (a) Tc ∩ (E1 )+ R1 ̸= ∅. That is, there is some t ∈ Tc which is attacked by E1 via some attack (T ′ , t) in R1 . Since (T ′ , t) ∈ R1 then t ∈ A1 . We obtain a contradiction to E1 ∈ adm(F1 ). (b) Tc ∩ (E1 )+ R3c ̸= ∅. That is, there is some t ∈ Tc which is attacked by E1 via some attack (T ′ , t) in R3 . ′ Then t ∈ A2 and t ∈ (E1 )+ Rc . By assumption, T ⊆ 3

• Case 2: (T, h) ∈ S2 . Then T ⊆ E2 . Moreover, S2 ⊆ S2⊛ . Since E2 is closed in F2⊛ , we obtain h ∈ E2 ⊆ E. • Case 3: (T, h) ∈ S3 . It holds that h ∈ A1 by definition of S3 . Recall that T ⊆ E, thus T ∩ A1 ⊆ E1 . Towards a contradiction, suppose h ∈ / E1 . We proceed by case distinction. Case 3.i: T ∈ DS3 (E3 ). By construction of the Sreduct, (T, ∗1 ) ∈ S2⊛ . Since T ⊆ E2 and E2 is closed in F2⊛ , we have ∗1 ∈ E2 . Since (∅, ∗1 ) ∈ R2⊛ , E2 is attacked by the empty set and thus not admissible in F2⊛ ; contradiction to our initial assumption. Case 3.ii: T ∈ / DS3 (E3 ). In this case, T ∈ TS3 (E1 ) \ DS3 (E1 ). By construction of the S-reduct, (T ∩ A2 , ∗2 ) ∈ S2⊛ . Since T ∩ A2 ⊆ E2 and E2 is closed in F2⊛ , we have ∗2 ∈ E2 . Again, we can derive a contradiction to admissibility of E2 in F2⊛ : By construction of the S-reduct, F2⊛ contains the attack ((T ∩ A2 ) ∪ {∗2 }, ∗2 ), thus E2 ∈ / cf(F2⊛ ). 2. Let E ∈ adm(F ). We show that E ∩ A1 = E1 = clF1 (E1 ) and E ∩ A2 = E2 = clF ⊛ (E2 ).

Admissible Semantics. 1. Suppose E1 ∈ adm(F1 ) and E2 ∈ adm(F2⊛ ). We show that E = E1 ∪ E2 ∈ adm(F ). By Lemma 57, E is conflict-free in F ; by Lemma 58, E is closed in F . It remains to prove that E defends itself in F . Let Tc denote a closed attacker of E in F . That is, there is T ⊆ Tc with (T, h) ∈ R and h ∈ E. Wlog, we can assume cl F (T ) = Tc . As usual, we have three cases to consider: (T, h) ∈ R1 , (T, h) ∈ R2 , or (T, h) ∈ R3 . In the process of splitting up the attacks, we close all attacks in R2 and R3 ; the resulting sets are denoted by R̂2 and R̂3 ; we let F̂2 denote the updated BSAF. Recall that some of the links that originally stem from R2 may now lie in R̂3 . For our attack (T, h) ∈ R, we thus obtain the following adjusted three cases: (T, h) ∈ R1 , (Tc , h) ∈ R̂2 , or (Tc , h) ∈ R̂3 . We proceed by case distinction. • Case 1: (T, h) ∈ R1 . In this case, Tc ⊆ A1 since T ⊆ A1 and there are no supports (U, b) ∈ S with U ⊆ A1 and b ∈ A2 . We therefore obtain that E1 defends itself against the attack in F1 . Thus, E defends itself against Tc in F . • Case 2: (Tc , h) ∈ R̂2 . It holds that Tc ⊆ A2 and (Tc , h) ∈ R2⊛ . By assumption, E2 is admissible in F2⊛ , thus E2 defends itself against the closed attacker Tc′ = cl F ⊛ (Tc ) in F2⊛ . That is, there is (U, b) ∈ R2⊛ 2 such that U ⊆ E2 and b ∈ Tc′ . We have several cases to consider. We go through the different options for b. Case 2.i: b = ∗0 . Then (U, b) is of the form ({∗0 }, ∗0 ). Since U ⊆ E2 , it follows that E2 contains the selfattacker ∗0 and is therefore not admissible, contradiction to our initial assumption. Case 2.ii: b = ∗1 . Then U = ∅ and ∗1 ∈ Tc′ . Then there is some V ⊆ Tc′ so that (V, ∗1 ) ∈ S2⊛ . By construction of the S-support, V ∈ DS3 (E1 ). Therefore, there is a ∈ A1 such that (V, a) ∈ S3 , V ∩A1 = ∅, a ∈ / E1 , and there is some c ∈ cl F (V ) which is attacked by E1 in F . We argue that cl F (V ) ⊆ Tc : Indeed, by construction of the splitting procedure, the head of all supports that got added to S2⊛ lies in {∗1 , ∗2 } (by definition of the attack splitting procedure, no new supports are added; by definition of the support splitting procedure, all supports lie either in S2 or satisfy the statement above). Thus, V ⊆ Tc and therefore cl F (V ) ⊆ cl F (Tc ) = Tc . Consequently, Tc is attacked by E1 in F . It follows that E attacks Tc in F . Case 2.iii: b = ∗2 . Then (U, b) is of the form ((V ∩ A2 ) ∪ {∗2 }, ∗2 ). Since U ⊆ E2 , it follows that E2 contains ∗2 , attacks itself on ∗2 , and is therefore not admissible, contradiction to our initial assumption. Case 2.iv: b ∈ A2 . We show that b ∈ Tc : it holds that b ∈ Tc′ iff b ∈ T or there is (W, h) ∈ S2⊛ with W ⊆ Tc′ . In the former case, b ∈ Tc ; in the latter case, we have (W, h) ∈ S2 since h ∈ A2 and by construction

• Let E1 = E ∩ A1 and let (T, h) ∈ S1 with T ⊆ E1 . Since T ⊆ E and S1 ⊆ S, we obtain h ∈ E1 . Therefore, E1 = cl F1 (E1 ). • Let E2 = E ∩ A2 and let (T, h) ∈ S2⊛ with T ⊆ E2 . By construction of the S-reduct, either (T, h) ∈ S2 or h = ∗1 or h = ∗2 . Note that splitting attacks (closing links in R3 , constructing the R-reduct and the modification) leaves the supports untouched. Thus, the proof is analogous to the case for support splitting. Towards a contradiction, suppose h ∈ / E2 . This implies h∈ / E. Proceed by case distinction. – Case 1: (T, h) ∈ S2 . Then T ⊆ E, contradiction to E = cl F (E). – Case 2: h = ∗1 . Then T ∈ DS3 (E1 ). Therefore, there is is some a ∈ A1 with (T, a) ∈ S3 and a ∈ / E1 . We obtain a contradiction to E being closed in F since T ⊆ E but a ∈ / E (as a ∈ / E1 ). – Case 3: h = ∗2 . Then the support is of the form (T, h) = (T ′ ∩ A2 , ∗2 ) for some T ′ ⊇ T with T ′ ∈ TS3 (E1 ). Therefore, there is is some a ∈ A1 with (T ′ , a) ∈ S3 and a ∈ / E1 . It holds that T ′ ⊆ E ′ since T ∩ A1 ⊆ E1 and T = T ′ ∩ A2 ⊆ E1 . Consequently, in F , E supports a but a ∈ / E, contradiction to E = cl F (E). Theorem 50. Let (F1 , F2 , R3 , S3 ) be a splitting for a BSAF F = (A, R, S) with F1 = (A1 , R1 , S1 ), F2 = (A2 , R2 , S2 ), and σ ∈ {stb, adm, com}. Let F2⊛ be defined as in Definition 48. 1. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2⊛ ), then E1 ∪ (E2 \ {∗2 }) ∈ σ(F ). 2. If E ∈ σ(F ), then E1 = E ∩ A1 ∈ σ(F1 ) and E ∩ A2 ∈ σ(F2⊛ ) or (E ∩ A2 ) ∪ {∗2 } ∈ σ(F2⊛ ). Proof. We prove the theorem for all considered semantics. 23

R2⊛ with U ⊆ E2 and b ∈ Tc . We consider the different forms the attack may have. By construction of the combined split and since b ∈ A2 , we have three options: Either (a) (U, b) ∈ R̂2 ; or (b) (U, b) is of the form (V \ A1 , b) for some (V, b) ∈ R̂3 , V ∩ (E1 )+ R = ∅ and V ∩ A1 ⊆ E1 ; or (c) (U, b) is of the form (V \ A1 ) ∪ {∗0 }, b) for some (V, b) ∈ R̂3 , V ∩ (E1 )+ R = ∅ and V ∩ (A1 \ E1 ) ̸= ∅. Proceed as in Case 2.iv.a, 2.iv.b, and 2.iv.c, respectively. This concludes the proof. We have shown that E defends itself against Tc , regardless of whether the attack (T, h) with cl F (T ) = Tc stems from R1 , R2 , or R3 . 2. Suppose E ∈ adm(F ). We show that E ∩ A1 ∈ adm(F1 ) and E ∩ A2 ∈ adm(F2⊛ ). By Lemma 58 we know that both sets are closed; moreover, by Lemma 57 they are also conflict-free. It remains to prove that the sets defend itself. • The case for E1 = E ∩ A1 holds since for all (T, h) ∈ R with h ∈ E1 it holds that (T, h) ∈ R1 . Moreover, cl F (T ) = cl F1 (T ) since no positive links go from F1 to F2 . Thus E1 defends itself against all closed attacker in F1 since E is admissible in F , by assumption. • Let E2 = E ∩ A2 . Let Tc be a closed attacker of E2 in F2⊛ such that there is (T, h) ∈ R2⊛ with Tc = cl F ⊛ (T ). 2 We note that h ∈ A2 since h ∈ E2 = E ∩ A2 . Thus, the attack (T, h) cannot stem from constructing the Sreduct (the head of newly introduced attacks is either ∗1 or ∗2 ). Therefore, (T, h) satisfies one of the following:

of the S-support, the head of each novel support either corresponds to ∗1 or ∗2 . Thus, (U, b) ∈ R2⊛ with U ⊆ E2 and b ∈ Tc . We will now consider the different forms the attack may have. By construction of the combined split and since b ∈ A2 , we have three options: Either (a) (U, b) ∈ R̂2 ; or (b) (U, b) is of the form (V \A1 , b) for some (V, b) ∈ R̂3 , V ∩ (E1 )+ R = ∅ and V ∩ A1 ⊆ E1 ; or (c) (U, b) is of the form (V \ A1 ) ∪ {∗0 }, b) for some (V, b) ∈ R̂3 , V ∩ (E1 )+ R = ∅ and V ∩ (A1 \ E1 ) ̸= ∅. – Case 2.iv.a: We have (U, b) ∈ R̂2 . Thus, there is U ′ ⊆ U so that (U, b) ∈ R2 . Therefore, E defends itself against Tc in F since U ⊆ E and b ∈ Tc . – Case 2.iv.b: Suppose (U, b) is of the form (V \ A1 , b) for some (V, b) ∈ R̂3 , V ∩ (E1 )+ R = ∅ and V ∩ A1 ⊆ E1 . Then V ⊆ E since U = V \ A1 ⊆ E2 and V ∩ A1 ⊆ E1 . Thus there is an attack (V, b) ∈ R̂3 with V ⊆ E and b ∈ Tc ; in other words, E defends itself against Tc in F . – Case 2.iv.c: Since U ⊆ E2 , it follows that E2 contains ∗0 , attacks itself on ∗0 , and is therefore not admissible, contradiction to our initial assumption. This concludes Case 2; we have shown that E defends itself against Tc if (Tc , h) ∈ R̂2 . • Case 3: (Tc , h) ∈ R̂3 . By definition of R̂3 , it holds that Tc ∩ A1 ̸= ∅. We proceed by case distinction. – Case 3.i: Tc ∩ (E1 )+ R ̸= ∅. In this case, E defends itself Tc in F . – Case 3.ii: Tc ∩ (E1 )+ R = ∅. In this case, either (a) ⊛ (Tc ∩ A2 , h) ∈ R2 (via the R-reduct) or (b) ((Tc ∩ A2 ) ∪ {∗0 }, h) ∈ R2⊛ (via the modification). Let Tc′ = cl F ⊛ (Tc ∩ A2 ). Observe that

– (T, h) ∈ R̂2 ; (Case 1) ′ ′ – (T, h) = (T \ A1 , h) for some (T , h) ∈ R̂3 , T ′ ∩ ′ (E1 )+ (Case 2) R = ∅ and T ∩ A1 ⊆ E1 ; ′ ′ – (T, h) = (T \ A1 ) ∪ {∗0 }, h) for some (T , h) ∈ R̂3 , ′ T ′ ∩ (E1 )+ (Case 3) R = ∅ and T ∩ (A1 \ E1 ) ̸= ∅. Let us first inspect the relationship between cl F (T ) and Tc . We show that

cl F ⊛ ((Tc ∩ A2 ) ∪ {∗0 }) = cl F ⊛ (Tc ∩ A2 ) ∪ {∗0 } 2

= Tc′ ∪ {∗0 }

Tc \ {∗0 , ∗1 , ∗2 } = cl F (T ).

since ∗0 is not contained in the tail of any support. E2 defends itself against Tc′ (in Case a) resp. Tc′ ∪{∗0 } (in Case b) since it is admissible in F2⊛ . Thus, there is (U, b) ∈ R2⊛ such that U ⊆ E2 and b ∈ Tc′ (in Case a) or b ∈ Tc′ ∪ {∗0 } (in Case b). As in Case 2, we consider the different options for b. * Case 3.ii.I: b = ∗0 . Then (U, b) = ({∗0 }, ∗0 ). As in Case 2.i, we obtain that E2 contains the self-attacker ∗0 and is therefore not admissible, contradiction to our initial assumption. * Case 3.ii.II: b = ∗1 . Analogous to Case 2.ii. We sketch the proof: As in Case 2.ii, we have U = ∅ and ∗1 ∈ Tc′ and thus there is some V ⊆ Tc′ so that (V, ∗1 ) ∈ S2⊛ with V ∈ DS3 (E1 ). Thus E1 attacks cl F (V ). Analogous to Case 2.ii, we can show that cl F (V ) ⊆ Tc . We obtain that E attacks Tc in F . Case 3.ii.III: b = ∗2 . Analogous to Case 2.iii. * Case 3.ii.IV: b ∈ A2 . As in case 2.iv, we can prove * that b ∈ Tc in this case. That is, we have (U, b) ∈

For Case 1, this follows from Tc = cl F ⊛ (T ): It holds 2 that a ∈ cl F ⊛ (T ) iff a ∈ T or there are supports 2 (W1 , a1 ), . . . , (Wk , ak ) ∈ S2⊛ with ak = a and Wi ⊆ S T ∪ j<i aj . Since (T, h) ∈ R̂2 , we have T ⊆ A2 , thus ∗0 ∈ / T . Moreover, since ∗1 and ∗2 do not appear in the tail of any support we can assume that they do not appear in any (Wi , ai ) for i < k. Thus, if a ∈ A2 , then (Wi , ai ) ∈ S2 for all i ≤ k, thus a ∈ cl F (T ); in the latter case, a ∈ {∗1 , ∗2 }. That is, in F2⊛ , T may additionally support some of the newly introduced arguments, thus the statement follows. For Case 2 and 3, we show Tc \ {∗0 , ∗1 , ∗2 } = cl F (T ′ ∩ A2 ): first note that T ′ ∩ A2 = cl F (T ∩ A2 ) since T ′ is closed in F (let (W, a) ∈ S with W ⊆ T ′ ∩ A2 ; then a ∈ cl F (T ′ ); if a ∈ A2 we have a ∈ T ′ ∩ A2 ). Since T = T ′ ∩ A2 we obtain Tc \ {∗0 , ∗1 , ∗2 } = cl F (T ′ ∩ A2 ) = cl F (T ) analogously to the first case. ♢ 24

• There are no attacks from A2 towards A1 , neither are there supports from A1 towards A2 . Hence any argument in A1 is defended by E in F iff it is defended by E1 = E ∩ A1 in F1 . Thus E1 defends itself in F1 , therefore E1 = E ∩ A1 ∈ com(F1 ).

Having settled the relation between cl F (T ) and Tc , we observe that in all cases, E is attacked by a closed set T (T ′ , respectively) in F . By assumption, E defends itself in F . Thus, there is (U, b) ∈ R with U ⊆ E and b ∈ T (b ∈ T ′ , respectively). We furthermore observe that in all cases, b ∈ A2 : in Case 1, this follows since T ⊆ A2 by definition of R̂2 ; in Case 2 and 3, this follows from the condition T ′ ∩ (E1 )+ R = ∅ (suppose b ∈ A1 ; by definition of the splitting tuple, we have (U, b) ∈ R1 ; thus we also have U ⊆ A1 and therefore U ⊆ E1 and thus E1 attacks T ′ ; contradiction). By construction of the combined split, we thus get an attack (Uc , b) ∈ R̂2 ∪ R̂3 with cl F (U ) = Uc with b ∈ T (which amounts to b ∈ T ′ ∩ A2 in Case 2 and 3). As shown above cl F (T ) ⊆ Tc . Below, we distinguish the cases (Uc , b) ∈ R̂2 and (Uc , b) ∈ R̂3 . – Case i: (Uc , b) ∈ R̂2 . Then (Uc , b) ∈ R2⊛ and thus E2 defends itself against Tc in F2⊛ . – Case ii: (Uc , b) ∈ R̂3 . We have Uc ∩ A1 ⊆ E1 and Uc is not attacked by E1 in F (otherwise, E ∈ / adm(F ), contradiction), therefore, R2⊛ contains an attack of the form (Uc ∩ A2 , b). Since Uc ∩ A2 ⊆ E2 and b ∈ T , we have E2 defends itself against Tc in F2⊛ .

• It remains to be shown that E2 defends itself in F2⊛ . We assume toward a contradiction that there exists a ⊛ c ∈ A⊛ 2 \ E2 s.t. c is defended by E2 in F2 . Then for ′ every closed attacker D with (D , c) ∈ R2⊛ , D′ ⊆ D there exists an attack (T, d) ∈ R2⊛ , d ∈ D, T ⊆ E2 . We move on to show that in this case E defends c in F . Note first that c ̸= ∗0 , because ∗0 cannot be defended. If c = ∗1 then E2 attacks the (non-empty) closure of the empty set in F2 , so E does the same in F , since any support by the empty set in S2⊛ is also in S2 , but then E is not conflict-free. Contradiction. So c ̸= ∗1 . If c = ∗2 then E∩A2 ∪{∗2 } is complete. We already have E∩A2 is admissible in F2⊛ , so it is closed and conflict-free and defends itself. Adding ∗2 preserves closedness, since the only supports involving ∗2 point to itself and since we defend ∗2 , defense is maintained. Now, if adding ∗2 would violate conflict-freeness, then there exists a T ⊆ E ∩ A2 such that (T ∪ ∗2 , ∗2 ) ∈ R2⊛ , so there is a support (T ∪ T ′ , h) with h ∈ / E, T ′ ⊆ E1 in F . But then E is not closed. Contradiction. So E ∩ A2 ∪ {∗2 } is indeed conflict-free and therefore complete. Suppose now c ∈ A2 . We have to show, that for every attack (D′ , c) ∈ R we have cl (D) is attacked by E in F . Let (D′ , c) ∈ R. We distinguish the following cases: – (Case 1: (D′ , c) ∈ R3 ). Then after the closure of attacks we have some closed attack (D, c) ∈ R̂3 . If D ∩ E1+ ̸= ∅, then E defends c against D. Otherwise there are two cases: * (Case 1.i D ∩ A1 ⊆ E) Then (D ∩ A2 , c) is a closed attack in F2⊛ . So there exists an attack (T, d) ∈ R2⊛ with T ⊆ E2 , d ∈ D ∩ A2 , since E2 defends c. But for such an attack we have ∗2 ∈ / T , since ∗2 is only part of attacks on itself. T ∩ {∗0 , ∗1 } = ∅, since E is admissible. So T ⊆ A2 , therefore (T, d) ∈ R̂2 ∪ R̂3 , so there exists an attack (T ′ , d) ∈ R with T ′ ⊆ T ⊆ E, so E defends c in F but does not contain it, since c ∈ A2 \ E2 . Contradiction. * (Case 1.ii D ∩ A1 ⊈ E) Then ((D ∩ A2 ) ∪ ∗0 , c) is an attack in F2⊛ .. We can distinguish the following cases: · (Case 1.ii a - clF (D ∩ A2 ) ⊆ A2 · (Case 1.ii b - clF (D ∩ A2 ) ∩ E1+ ̸= ∅) Then E defends c via E1 but does not contain it. Contradiction to E ∈ com(F ). · (Case 1.ii c - clF (D ∩ A2 ) ∩ A1 ̸= ∅, clF (D ∩ A2 ) ∩ E1+ = ∅) Then ((D ∩ A2 ), ∗2 ) ∈ S2⊛ . So clF ⊛ (D ∩ A2 ) = (D ∩ A2 ) ∪ {∗2 } and there is an 2 attack (T, d) from E2 to (D ∩ A2 ) ∪ {∗2 } in F2⊛ . Since any attack on {∗2 } implies the attacking set is not conflict-free and E2 is conflict-free, d ̸= ∗2 .

Complete Semantics. 1. Suppose E1 ∈ com(F1 ) and E2 ∈ com(F2⊛ ). We show that E = E1 ∪ (E2 \ {∗2 }) ∈ com(F ). The premisses E1 ∈ com(F1 ) and E2 ∈ com(F2⊛ ) directly entail E1 ∈ adm(F1 ) and E2 ∈ adm(F2⊛ ), hence E ∈ adm(F ) as shown above. It remains to be shown that E defends itself in F . Assume toward contradiction that there exists an argument d ∈ A \ E that is defended by E in F . We proceed by case distinction. / E1 , we • Case 1: d ∈ A1 . From E1 ∈ com(F1 ) and d ∈ conclude there exists (T1 , d) ∈ R1 , s.t. cl F1 (T1 ) ∩ E1+ = ∅. By the construction of R1 we can infer that (T1 , d) ∈ R and cl F (T1 ) ∩ E + = ∅, therefore d is not defended by E in F . This contradicts our assumption. • Case 2: d ∈ A2 From E2 ∈ com(F2⊛ ) and d ∈ / E2 , we ⊛ conclude there exists (T2 , d) ∈ R2 , s.t. cl F ⊛ (T2 ) ∩ 2 E2+ = ∅. By the construction of R2⊛ we can infer that ∃(T, d) ∈ R2 ∪ R3 s.t. T2 ⊆ (cl F (T ) ∩ A2 ) ∪ {∗0 } holds. Further cl F ⊛ (T2 ) ∩ E2+ = ∅ implies cl F (T ) ∩ E + = ∅. Thus 2 d is not defended by E in F . Contradiction! 2. Suppose E ∈ com(F ). We show that E1 = E ∩ A1 ∈ com(F1 ) and (E ∩ A2 ∈ com(F2⊛ ) or (E ∩ A2 ) ∪ {∗2 } ∈ com(F2⊛ )). • Utilizing the proof for admissible semantics, we directly conclude E1 = E ∩ A1 ∈ adm(F1 ) and E2 = E ∩ A2 ∈ adm(F2⊛ ) or E2 = (E ∩ A2 ) ∪ {∗2 } ∈ adm(F2⊛ ). 25

So (T, d) ∈ R̂2 ∪ R̂3 , but in this case there exists an attack (T ′ , d) ∈ R from E to D, so E defends c in F . Since E does not contain c but E ∈ com(F ), Contradiction. – (Case 2: (D′ , c) ∈ R2 ). Then after the closure of attacks we either have some closed attack (D, c) ∈ R̂3 or (D, c) ∈ R̂2 . For the former refer to Case 1. In the latter case (D, c) ∈ R2⊛ so there exists an attack (T, d) ∈ R2⊛ , d ∈ D, T ⊆ E2 on D in F2⊛ . We have ∗2 , ∗1 ∈ / D, since D is closed in A2 . Furthermore ∗0 ∈ / D, since ∗0 ∈ / A. We also have ∗2 ∈ / T , since ∗2 is only part of attacks on itself. So d ∈ A2 , T ⊆ E2 , so (T, d) ∈ R̂2 is a closed attack from E2 to D, so there exists an attack (T ′ , d) ∈ R, T ′ ⊆ T , so E defends c in F and does not contain it. But E ∈ com(F ), Contradiction.

· Case 2.2.2. a = ∗2 . Then ∗2 ∈ T from the definition of S-reduct. From the fact that (E2 )⊕ = A⊛ 2 R2⊛ and E2 is conflict-free, we derive that either ∗2 ∈ E2 or ∗2 ∈ (E2 )+ . However, the second case is in R2⊛ contradiction with conflict-freeness of E2 . Hence, ∗2 ∈ E2 . + Under each case, we derive that a ∈ ER . Further, ∗1 ∈ / E2 and ∗2 ∈ E2 . Since ∗2 ∈ / A2 , it follows that E1 ∪ (E2 \ {∗2 })⊕ = A. ⊕ + 2. Suppose E ∈ stb(F ), that is ER = E ∪ ER = A. • We first show that E1 = E ∩ A1 ∈ stb(F1 ). From previous lemmata we know that E1 is closed and conflictfree. It remains to prove that E1 attacks every a ∈ A1 \ E1 . Since there is no attack from F2 to F1 by definition of splitting, we know that a ∈ (E)+ R1 by some T ⊆ E ∩ A1 , i.e. a ∈ (E ∩ A1 )+ . R1

Stable Semantics. Notice that under stable semantics, we have UREc1 = ∅, therefore F2⋆ = F̂2E1 and F2⊛ is the S-reduct

• We show that E ∩ A2 ∈ stb(F2⊛ ) or (E ∩ A2 ) ∪ {∗2 } ∈ stb(F2⊛ ). First, consider ∗2 . If ∗2 is not in a stable extension (as in the first disjunct), then it is attacked by it. However, the only way that ∗2 is attacked is via a setself-attack by definition of S-reduct. Hence, E ∩ A2 cannot be stable, because is not conflict-free. Therefore, in the remainder we prove that (E ∩ A2 ) ∪ {∗2 } ∈ stb(F2⊛ ). From Lemma 58 we know that E ∩ A2 is closed in F2⊛ . By definition of the S-reduct, ∗2 does not participate to any support in F2⊛ . Hence, (E ∩ A2 ) ∪ {∗2 } is closed in F2⊛ . From Lemma 57 we know that E ∩ A2 is conflict-free in F2⊛ . Now suppose (E ∩ A2 ) ∪ {∗2 } ∈ / cf(F2⊛ ). ⊛ Then, ((T ∩ A2 ) ∪ {∗2 }, ∗2 ) ∈ R2 for some T ∩ A2 ⊆ E ∩ A2 such that T ∈ TS3 (E1 ) \ DS3 (E1 ). Therefore, we know that T ∩ A1 ⊆ E1 , from which we derive that T ⊆ E. Since E is closed in F from hypothesis and T ∈ TS3 (E1 ), we derive that there is an a ∈ A1 \ E1 such that (T, a) ∈ S3 . This contradicts the fact that E is closed in F . It remains to prove that E ∩ A2 ∪ {∗2 } attacks every argument a ∈ A⊛ 2 \ ((E ∩ A2 ) ∪ {∗2 }). We proceed by case distinction. – Case 1. a ∈ A2 . By hypothesis, we know that a ∈ + ER because E is stable in F . By definition of splitting, there are no attacks from F1 to F2 . Hence, we can distinguish two cases. + * Case 1.1. a ∈ ER2 . Thus, there is an attack (T, a) ∈ R2 such that T ⊆ E ∩A2 . Let Tc = clS (T ). Since E is closed in F by hypothesis, we know that Tc ⊆ E. Hence, when computing the closure R2c , we distinguish two further cases. · Case 1.1.1. Tc ∩ A1 = ∅. Then (Tc , a) ∈ R̂2 . By definition of support and attack reducts, we know + that R̂2 ⊆ R2⊛ , and thus a ∈ ER ⊛.

of F̂2E1 wrt. E1 . 1. Suppose E1 ∈ stb(F1 ) and E2 ∈ stb(F2⊛ ). We show that E = E1 ∪ (E2 \ {∗2 }) ∈ stb(F ). • E is conflict-free: By Lemma 57 and E1 ∈ adm(F1 ) and E2 ∈ adm(F2⊛ ), we obtain E ∈ cf(F ). • E is closed because of Lemma 58 and the fact that stb(F ) ⊆ adm(F ) for every BSAF F . • E attacks every argument a ∈ A \ E. Let a ∈ A \ E. We proceed by case distinction. – Case 1: a ∈ A1 . Then a ∈ (E1 )+ R1 since E1 ∈ + stb(F1 ), thus a ∈ ER . – Case 2: a ∈ A2 . By hypothesis, we know that a ∈ (E2 )+ . Hence, there is a T ⊆ E2 such that (T, a) ∈ R⊛ 2

R2⊛ . Since R2⊛ ⊇ R̂2E1 , we distinguish two further cases. E1 * Case 2.1. (T, a) ∈ R̂2 . By definition of R-reduct, we can distinguish whether (T, a) ∈ R̂2 or not. We proceed by case distinction. · Case 2.1.1. (T, a) ∈ R̂2 . In this case, we know by definition of R̂2 that there is a T ′ ⊆ T ⊆ E2 such that T = clS (T ′ ) and (T ′ , a) ∈ R2 . Hence, a ∈ + (E2 )+ R2 and, consequently, a ∈ ER . · Case 2.1.2. (T, a) ∈ / R̂2 . Then, it means that (T, a) is introduced by the R-reduct. That is, there is a T ′ ⊇ T such that (T ′ , a) ∈ R3c , T ′ ∩ (E1 )+ R1 ∪R3c = ∅ and T ′ ∩ A1 ⊆ E1 . Thus, T ′ ⊆ E and a ∈ (E)+ R3 + and, consequently, a ∈ ER . / R̂2E1 . This means that (T, a) has * Case 2.2. (T, a) ∈ been introduced via the S-reduct. · Case 2.2.1. a = ∗1 . Then T = ∅ by definition of S-reduct. Thus, ∗1 ∈ / E2 since it is admissible. Therefore, ∗1 ∈ /E

· Case 1.1.2. Tc ∩ A1 ̸= ∅. Then (Tc , a) ∈ R̂3 by definition. Further, for all t ∈ Tc ∩ A1 , we know 26

that (T, t) ∈ S3 . Since Tc ⊆ E, we get Tc ∩ A1 ⊆ E ∩ A1 . Then, (Tc \ A1 , a) ∈ R̂2E1 is in the Rreduct, and a ∈ E +E1 . Further, since R̂2E1 ⊆ R2⊛ ,

Grounded Semantics. Suppose E1 ∈ grd(F1 ) and E2 ∈ grd(F2⊛ ). Then E1 ∈ com(F1 ) and E2 ∈ com(F2⊛ ). Therefore, by Theorem 50, we have E = E1 ∪ E2 \ {∗2 } ∈ com(F ). Assuming toward contradiction ∃G ⊊ E : G ∈ com(F ), then (1) ∃G1 ⊊ E1 : G1 ∈ com(F1 ) or (2) ∃G2 ⊊ E2 : G2 ∈ com(F2⊛ ). Case 1: Contradiction to E1 is grounded in F1 . Case 2: We distinguish (i) ∗2 ∈ E2 and (ii) ∗2 ∈ / E2 . In case (i), we are done (then G2 ⊊ E2 or G2 ∪ {∗2 } ⊊ E2 , contradiction). In case (ii), suppose G2 ∪ {∗2 } is grounded in F2⊛ . Then for all attacks in R2⊛ of the form ((T ∩ A2 ) ∪ {∗2 }, ∗2 ), it holds that (G2 )+ ∩ (T ∩ A2 ) ̸= ∅. Since R⊛

R̂2

+ it follows that a ∈ ER ⊛. 2

* Case 1.2. a ∈ ER3 . There is some T ⊆ E such that (T, a) ∈ R3 . Let Tc = clS (T ). Then by definition of R3c , we know that (Tc , a) ∈ R3c . Since E is closed in F by hypothesis, we know that Tc ⊆ E. Further, from E ∈ cf(F ), it follows that Tc ∩ (E1 )+ R1 ∪R3c = ∅. Moreover, Tc ∩ A1 ⊆ E ∩ A1 because Tc ⊆ E. Hence, by definition of R-reduct, there is a (Tc \ + A1 , a) ∈ R2⊛ , and consequently, a ∈ ER ⊛.

G2 ⊊ E2 , it follows that for all attacks in R2⊛ of the form ((T ∩ A2 ) ∪ {∗2 }, ∗2 ), (E2 )+ ∩ (T ∩ A2 ) ̸= ∅. Thus, E2 R⊛

+ In both cases we find a ∈ ER ⊛.

defends ∗2 , contradiction to ∗2 ∈ / E2 . Thus E ∈ grd(F ).

• Case 2. a ∈ / A2 . Thus a ∈ {∗1 }. By definition of Sreduct we know that ∗1 ∈ (E ∩ A2 )+ . R⊛ 2

Theorem 52. Let (F1 , F2 , R3 , S3 ) be a splitting for the BSAF F = (A, R, S), let F2⊛ be as in Definition 48 and σ ∈ {grd, pref}. If E1 ∈ σ(F1 ) and E2 ∈ σ(F2⊛ ), then E1 ∪ (E2 \ {∗2 }) ∈ σ(F ). Proof. We provide a proof for both semantics. Preferred Semantics. Suppose E1 ∈ pref(F1 ) and E2 ∈ pref(F2⊛ ). Since E1 and E2 are admissible in F1 and F2⊛ , respectively, we obtain E ∈ adm(F ). We show that E = E1 ∪ E2 \ {∗2 } ∈ pref(F ). Towards a contradiction, suppose E ∈ / pref(F ). Then there is E ′ ⊋ E with E ′ ∈ adm(F ). Then E1′ = E ′ ∩ A1 ∈ adm(F1 ) and E2′ = E ′ ∩ A2 ∈ adm(F2⊛ ). E ′ is strictly larger than E. Proceed with case distinction. • Case 1: E1′ is strictly larger than E1 . This is in contradiction to E1 ∈ pref(F1 ). • Case 2: E2′ is strictly larger than E2 \ {∗2 }. Case 2.i: ∗2 ∈ E2 . Let E2′′ = E2′ ∪ {∗2 }. If E2′′ ∈ adm(F2⊛ ) then E2′′ ⊋ E2 , contradiction to E2 ∈ pref(F2⊛ ). We show that E2′′ ∈ adm(F2E1 ): E2′′ is closed since E2′ is closed and ∗2 ∈ / U for any (U, b) ∈ S2′ . E2′′ defends ∗2 ′′ since E2 ⊆ E2 defends ∗2 . E2′′ is conflict-free: Towards a contradiction, suppose there is an attack (T, h) ∈ R2⊛ such that T ⊆ E2′′ and h ∈ E2′′ . Case 2.i.a: h = ∗2 . By construction, T has the form ((T ′ ∩ A2 ) ∪ {∗2 }, ∗2 ) for some T ′ ∈ TS3 (E1 ). Thus, T ′ ∩ A2 ⊆ E2′ . Since (T ′ ∩ A2 , ∗2 ) ∈ S2⊛ , this implies E2′ is not closed. Contradiction to E2′ ∈ adm(F2⊛ ). Case 2.i.b: h ∈ A2 . In this case, T ⊆ E2′ and we have a contradiction to the conflict-freeness of E2′ in F2⊛ . Case 2.ii: ∗2 ∈ / E2 . Then E2′ = E2 \ {∗2 } and therefore ′ E2 ⊋ E2 , contradiction to E2 ∈ pref(F2⊛ ). 27

Record · ID 149096 · SHA-256 377f58d56003f32b
Retrieved via Conceptio — every document is proof-bundled with source, license, and retrieval metadata.