Bipolar Set-Based Argumentation Frameworks

• BSAFs are abstract argumentation frameworks that enable collective attacks and supports, extending Dung AFs, SETAFs, and BAFs.
• They introduce support closure semantics that reshape defense and extension properties, addressing limitations of traditional attack-only models.
• BSAFs integrate computational, gradual, and aggregation methods to provide a comprehensive abstraction for structured and non-flat assumption-based argumentation.

Abstract Bipolar Set-Based Argumentation Frameworks (BSAFs) are abstract argumentation formalisms in which both attacks and supports may be collective: a finite set of arguments can jointly attack, or jointly support, a single argument. In the standard presentation, a BSAF is a triple $F=(A,R,S)$, where $A$ is a finite set of arguments, $R \subseteq 2^A \times A$ is a collective attack relation, and $S \subseteq 2^A \times A$ is a collective support relation. This places BSAFs at the intersection of bipolar argumentation and set-based interaction: they strictly subsume Dung-style abstract argumentation frameworks, frameworks with sets of attacking arguments, and singleton-tail bipolar argumentation frameworks, while also providing a natural abstraction of general, potentially non-flat, assumption-based argumentation (Berthold et al., 30 Apr 2026, Berthold et al., 15 Aug 2025). Empirical work on dialogical argumentation has also supported the need for bipolar formalisms, reporting that defense by attack chains cannot account for all positive links recognized by participants and that users often explicitly identify support relations (Polberg et al., 2017).

1. Formal definition and position in the argumentation landscape

A BSAF is defined as

$F = (A, R, S),$

where $A$ is finite, $R \subseteq 2^A \times A$, and $S \subseteq 2^A \times A$. Each $(T,h)\in R$ is a collective attack from tail $T \subseteq A$ to head $A$0, and each $A$1 is a collective support. For singleton tails, the notation $A$2 abbreviates $A$3 (Berthold et al., 30 Apr 2026).

BSAFs generalize several well-known abstract formalisms. A SETAF is obtained when $A$4, so only collective attacks remain. A Dung AF is a SETAF in which every attack tail is a singleton. A bipolar argumentation framework (BAF) is recovered when both attacks and supports have singleton tails (Berthold et al., 30 Apr 2026).

Formalism	Attacks	Supports
Dung AF	singleton tails	none
SETAF	set tails	none
BAF	singleton tails	singleton tails
BSAF	set tails	set tails

This hierarchy matters technically because collective support and collective attack interact with closure, defense, and extension construction in ways that do not arise in AFs or ordinary BAFs. A plausible implication is that BSAFs are best understood not as a minor variant of BAFs, but as a proper abstract setting for reasoning about joint justificatory and defeating structures.

2. Support closure, defense, and extension-based semantics

The defining semantic operation in BSAFs is closure under support. Given $A$5 and $A$6,

$A$7

and

$A$8

A set $A$9 is closed iff $R \subseteq 2^A \times A$0 (Berthold et al., 15 Aug 2025, Berthold et al., 30 Apr 2026).

Defense is defined relative to closed attackers. In this setting, $R \subseteq 2^A \times A$1 defends an argument $R \subseteq 2^A \times A$2 if for every closed set that attacks $R \subseteq 2^A \times A$3, $R \subseteq 2^A \times A$4 attacks that attacker. On this basis, the usual Dung-style semantics are reformulated for collective attacks and supports: $R \subseteq 2^A \times A$5 is conflict-free if it does not attack itself; admissible if it is conflict-free, closed, and defends itself; complete if it is admissible and contains every argument it defends; grounded if it is $R \subseteq 2^A \times A$6-minimal among complete extensions; preferred if it is $R \subseteq 2^A \times A$7-maximal among admissible extensions; and stable if it is admissible and attacks every argument outside $R \subseteq 2^A \times A$8 (Berthold et al., 30 Apr 2026).

The interaction of collective support and collective attack can destroy extension existence properties familiar from AFs. One example uses arguments $R \subseteq 2^A \times A$9, a collective support $S \subseteq 2^A \times A$0, attacks $S \subseteq 2^A \times A$1, $S \subseteq 2^A \times A$2, and $S \subseteq 2^A \times A$3. In that configuration, accepting both $S \subseteq 2^A \times A$4 and $S \subseteq 2^A \times A$5 forces $S \subseteq 2^A \times A$6 by closure; defending $S \subseteq 2^A \times A$7 requires accepting both $S \subseteq 2^A \times A$8 and $S \subseteq 2^A \times A$9; but $F = (A, R, S),$0 attacks $F = (A, R, S),$1. The reported consequence is that no complete nor grounded extension exists (Berthold et al., 30 Apr 2026). This illustrates a common misconception: support in BSAFs is not merely an “extra positive edge,” because support closure can generate obligations that alter admissibility itself.

3. BSAFs as abstractions of assumption-based argumentation

A central line of work uses BSAFs as an abstract representation of assumption-based argumentation (ABA), especially non-flat ABA, where assumptions may themselves be derivable. For an ABA framework $F = (A, R, S),$2, the corresponding BSAF is

$F = (A, R, S),$3

with

$F = (A, R, S),$4

and

$F = (A, R, S),$5

For classical admissibility-based semantics,

$F = (A, R, S),$6

for $F = (A, R, S),$7 (Berthold et al., 15 Aug 2025). The abstraction eliminates rules, ordinary atoms, and argument trees, and retains only collective attack and support between assumptions. This is why recent work describes BSAFs as establishing a crucial link to structured argumentation and as naturally capturing general, potentially non-flat, ABA (Berthold et al., 30 Apr 2026, Berthold et al., 15 Aug 2025).

This setting has also been used to study alternatives to standard admissibility. Strong admissibility is defined recursively so that every accepted argument must be defended by a strictly smaller strongly defended subset, thereby avoiding circular justification. Weak admissibility is defined through a reduct $F = (A, R, S),$8, where defense is tested only against attackers that remain viable in the reduct. Both notions extend to preferred, complete, and grounded variants (Berthold et al., 15 Aug 2025).

A key structural result is modularization. For classical, strong, and weak admissibility semantics, the following implication holds: $F = (A, R, S),$9 This supports decomposition of reasoning into a solved part $A$0 and a residual part $A$1 (Berthold et al., 15 Aug 2025). At the same time, no admissibility-based variant is reported to satisfy all desirable rationality conditions in general non-flat ABA/BSAF settings. In particular, properties such as the Fundamental Lemma, Unique Maximum, and Unique Relative Maximum can fail, and attempts to repair them through $A$2-closure improve strong admissibility but break modularization for weak admissibility (Berthold et al., 15 Aug 2025). The resulting picture is not one of a single canonical semantics, but of a controlled trade-off between anti-circularity, liberality, and compositionality.

4. Computational properties, algorithms, and splitting

Complexity and algorithmics for BSAFs have largely been analyzed through translations to bipolar ABA. One reported result is that the standard reasoning tasks for admissible, preferred, and set-stable semantics have the same complexity as in abstract argumentation, and that these results apply directly to BSAFs via polynomial transformations (Karamlou et al., 2019).

Semantics	Representative decision complexity	Extension enumeration
Admissible	$A$3: NP-complete; $A$4: P; $A$5: NP-complete	$A$6
Preferred	$A$7: coNP-complete; $A$8: $A$9-complete	$R \subseteq 2^A \times A$0
Set-Stable	$R \subseteq 2^A \times A$1: NP-complete; $R \subseteq 2^A \times A$2: P; $R \subseteq 2^A \times A$3: coNP-complete	$R \subseteq 2^A \times A$4

The same work presents backtracking-driven algorithms for enumerating admissible, preferred, and set-stable extensions, proves soundness and completeness, and reports scalability on instances with up to $R \subseteq 2^A \times A$5 sentences and $R \subseteq 2^A \times A$6K rules, with acceptable runtimes under $R \subseteq 2^A \times A$7 seconds (Karamlou et al., 2019).

A more recent development is splitting for BSAFs. Because BSAFs combine collective attacks, collective supports, and support closure, decomposition is more intricate than in AFs or SETAFs. Three splitting modes are studied: splits over collective attacks, splits over collective supports, and splits over both. For attack splitting, cross-part attacks are first closed under support,

$R \subseteq 2^A \times A$8

before reduct construction. For support splitting, auxiliary arguments such as $R \subseteq 2^A \times A$9 and $S \subseteq 2^A \times A$0 are introduced as constraint gadgets to block combinations that would force unsupported closure across the split. Correctness is established for the most common semantics; for $S \subseteq 2^A \times A$1, extensions can be reconstructed from subframework extensions in both directions, whereas for grounded and preferred semantics only the direction from split solutions to full solutions is generally guaranteed (Berthold et al., 30 Apr 2026). This is a sharp indication that support closure is not merely a local phenomenon: it shapes the very possibility of modular computation.

5. Quantitative and gradual semantics

BSAFs have also become a substrate for gradual reasoning. In this line, a quantitative BSAF takes the form

$S \subseteq 2^A \times A$2

where $S \subseteq 2^A \times A$3 assigns initial strengths. The semantics are modular and iterative, combining a set aggregation function, an aggregation function over incoming attacks and supports, and an influence function updating initial strengths (Rapberger et al., 14 Jul 2025). The recursion is

$S \subseteq 2^A \times A$4

with final assignment

$S \subseteq 2^A \times A$5

Typical set aggregation functions include product and minimum; typical aggregation functions include sum and product; typical influence functions include a linear update and a quadratic-energy update (Rapberger et al., 14 Jul 2025).

This framework explicitly generalizes modular semantics for quantitative bipolar argumentation. In the earlier modular account of bipolar weighted argumentation graphs, acceptability is computed as a composition of aggregation and influence,

$S \subseteq 2^A \times A$6

and general convergence results show that all well-behaved modular semantics converge for acyclic graphs, whereas no sum-based semantics converges for all graphs (Mossakowski et al., 2018). The BSAF-based gradual semantics inherit this modular design but add a set-aggregation layer to accommodate collective support and attack (Rapberger et al., 14 Jul 2025).

The reported properties include balance, monotonicity, anonymity, independence, and directionality. Convergence is guaranteed under conditions involving maximal supporter/attacker set size $S \subseteq 2^A \times A$7, maximal indegree $S \subseteq 2^A \times A$8, and influence parameter $S \subseteq 2^A \times A$9; one given criterion is $(T,h)\in R$0 (Rapberger et al., 14 Jul 2025). An experimental study on $(T,h)\in R$1 randomly generated ABA frameworks reports that the BSAF-based gradual semantics converged in about $(T,h)\in R$2 of non-timeout cases, whereas an argument-based QBAF baseline converged in about $(T,h)\in R$3, with the BSAF approach also converging faster and showing greater robustness to initialization and structural variation (Rapberger et al., 14 Jul 2025).

Set-based quantitative explanation has been extended further through set contribution functions, which quantify how a set of arguments jointly contributes to the final strength of a topic argument. Removal-based, intrinsic removal-based, gradient-based, and Shapley value-based set contribution functions have been defined, together with principles such as weak quantitative contribution existence, consistency, and monotonicity (Naudot et al., 18 Sep 2025). This suggests that BSAFs are not only representational devices for collective support and attack, but also natural carriers for group-level attribution in explainable quantitative argumentation.

6. Collective aggregation, logical encodings, and higher-order generalizations

A distinct research direction concerns aggregation of multiple agents’ views and the logical encoding of richer set-based frameworks. For bipolar argumentation frameworks with shared arguments and attacks but different support relations, aggregation rules have been studied under social-choice axioms such as unanimity, groundedness, neutrality, and independence. Essential Constraint and Closedness are preserved by any grounded rule; conflict-freeness, safety, d-admissibility, s-admissibility, and some acceptability claims under preferred semantics are preserved by unanimity; but preserving preferred or stable extension status under those axioms forces dictatorship when $(T,h)\in R$4 (Chen, 2021). The same work states that its methodology and results directly inform the design and analysis of BSAFs, especially in multi-agent settings. A plausible implication is that collective rationality for BSAFs will face analogous impossibility trade-offs whenever global extension-based properties are required to survive aggregation.

BSAFs also sit inside a broader family of generalized abstract frameworks. Higher-order Argumentation Frameworks with Supports (HAFS) explicitly allow attacks and supports to be both sources and targets of interactions, thereby strictly generalizing BSAFs. HAFS support extension-based semantics, adjacent complete labelling semantics in $(T,h)\in R$5, and numerical equational semantics over $(T,h)\in R$6, together with normal encodings into Łukasiewicz three-valued propositional logic and fuzzy propositional logic systems (Tang, 29 Dec 2025). Closely related work on argumentation frameworks with set attackers proposes higher-order set AFs (HSAFs), translates complete semantics to $(T,h)\in R$7, and translates equational semantics to $(T,h)\in R$8, establishing model equivalence in each case (Tang et al., 11 Apr 2025). These frameworks move beyond ordinary BSAFs by allowing attacks on attacks and other higher-order structures, but they preserve the central set-based insight: group interaction can be treated as a first-class formal object rather than encoded indirectly.

Taken together, these developments place BSAFs in a distinctive position. They are abstract enough to serve as a clean interface between structured and unstructured argumentation, expressive enough to capture collective support and attack, and sufficiently rich to support extension-based, gradual, computational, and logical investigations. At the same time, the literature consistently reports nontrivial trade-offs: closure can obstruct extension existence, admissibility variants distribute desirable properties unevenly, extension enumeration is not output-polynomial, and collective aggregation of rich semantics runs into Arrow-style impossibility results (Berthold et al., 30 Apr 2026, Berthold et al., 15 Aug 2025, Karamlou et al., 2019, Chen, 2021). In this sense, BSAFs function less as a settled endpoint than as a central abstraction around which several active strands of argumentation theory are now organized.