Quantitative Bipolar Argumentation Frameworks

• Quantitative Bipolar Argumentation Frameworks (QBAFs) are models that represent both supportive and attacking relations with real-valued weights to compute nuanced acceptability degrees.
• They employ a damped, iterative propagation method, using matrix-based updates and convergence guarantees, to balance direct and indirect influences.
• QBAFs generalize classical attack-only and support-only models, offering robust applications in decision support, explainable AI, and multi-criteria analysis.

Quantitative Bipolar Argumentation Frameworks (QBAFs) are mathematical and computational models that extend classical argumentation frameworks to explicitly represent both positive (supportive) and negative (attacking) relationships, and to allow fine-grained, real-valued weights on arguments and their interactions. The central objective is to provide semantics that aggregate these influences quantitatively, yielding a real-valued or normalized "acceptability degree" for each argument. QBAFs unify and generalize both "support-only" and "attack-only" weighted argumentation models, facilitating nuanced representation and analysis of argumentative structures in decision support, recommender systems, explainable AI, and beyond.

1. Mathematical Structure and Representation

A QBAF is typically represented as a triple or quadruple ⟨𝒜, G, w⟩ (or ⟨𝒜, ℛ⁻, ℛ⁺, τ⟩), where:

• 𝒜 is a finite, ordered list of arguments (nodes).
• G is an incidence matrix with entries g₍ᵢⱼ₎ ∈ {–1, 0, 1}, where g₍ᵢⱼ₎ = 1 denotes that argument aⱼ supports aᵢ, and g₍ᵢⱼ₎ = –1 denotes that aⱼ attacks aᵢ.
• w (or τ) is an initial plausibility or base score vector in ℝ^|𝒜| or [0,1]^|𝒜| assigning a weight to each argument.

This structure enables both direct and indirect (transitive) propagation of influence across potentially complex argumentation graphs, accommodating mixed-sign interaction (both attacks and supports), as well as potential cycles and high indegree (Mossakowski et al., 2016).

2. Quantitative Semantics and Aggregation Mechanisms

The core challenge in QBAFs lies in defining semantics that aggregate—and balance—the often-conflicting effects of multiple (possibly chained) attackers and supporters. The principal approach, called direct aggregation semantics, is formulated as a damped, iterative propagation process:

• Base case: f^0(a) = w(a) for each argument.
• Iterative update: f^i(a) = w(a) + (1/d) × [Σ_supporters f^i–1(b) – Σ_attackers f^i–1(c)] Or in matrix notation: f^i = w + (1/d) G f^i–1

Assuming the sequence converges (guaranteed when the damping factor d ≥ indegree(G)), the acceptability degrees are:

$$\operatorname{Deg}_{\text{dir}}^{(G,d)}(w) = \lim_{i\to\infty} f^{(i)} = \sum_{i=0}^\infty \left( \frac{1}{d} G\right)^i w = \left( I - \frac{1}{d} G \right)^{-1} w$$

This formula aggregates positive (supportive) and negative (attacking) influences, capturing both direct and indirect effect propagation with exponentially diminishing impact for longer paths (Mossakowski et al., 2016).

To ensure compatibility with prior frameworks constraining weights to [0,1], a sigmoid function σ is introduced (e.g., σ(x) = 1/(1+e^–x)). The acceptability degrees for bounded weights are thus:

$$\operatorname{Deg}_{\text{sdir}}^{(G,d)}(w) = \sigma \left( \left( I - \frac{1}{d} G \right)^{-1} \sigma^{-1}(w) \right)$$

Here, the “neutral” value is ½, signifying a more intuitive division between acceptance and rejection than the 0-neutrality in attack-only or support-only semantics.

Generalization:

QBAFs strictly generalize support- and attack-only frameworks. With G restricted to {0,1} semantics revert to additive support graphs; restricting G to {0,–1} recovers purely attacking models. The mixed-sign structure enables the treatment of “reverse impact,” where a supporter with a negative plausibility can, in effect, function as an attacker (Mossakowski et al., 2016).

3. Modular and Iterative Semantics

Subsequent work on modular semantics (Mossakowski et al., 2018) established that acceptability computation should be partitioned into:

• An aggregation function (α), e.g., sum, sum-pos, or top, which collects all influences from an argument’s direct predecessors.
• An influence function (ι), which integrates the aggregated influence with the argument’s own initial weight.

The overall fixpoint equation for all arguments D is:

$D = ((G, D), w)$

where the ith component is D_i = ι(α(G_i, D), w_i).

Sum-based and top-based aggregators:

• “Sum” (α_sum) is α_sum(g, d) = g * d, i.e., a regular dot product.
• “Top” focuses only on the single most influential attacker/supporter to guarantee convergence in arbitrarily connected graphs.

The modular decomposition facilitates in-depth analysis of desirable properties (e.g., anonymity, reinforcement, stability, neutrality) and enables theoretical convergence analysis (Mossakowski et al., 2018).

Convergence guarantees:

• All well-behaved modular semantics converge for acyclic graphs.
• For sum-based aggregation, convergence on cyclic graphs is ensured when indegree(G) < 1/m, where m is the maximal local Lipschitz constant of the influence function.
• Top-based and certain damped aggregators yield convergence for any graph topology, enabling robust application to practical, cyclic settings (Mossakowski et al., 2018).

4. Comparison to Classical and Probabilistic Frameworks

Traditional Dung-style argumentation models (attack-only, binary acceptance) are insufficient for capturing complex real-world dialogues, which feature graded agreement and both support and attack relations. Empirical studies reveal that users apply nuanced, multi-level belief ratings and often identify ambiguous, non-binary relations between statements. Bipolar frameworks, expressible as triples (A, R, S) or even (A, R, S, D) for “dependent” links, more closely model such perceptions (Polberg et al., 2017).

QBAFs are structurally and computationally richer than these classical frameworks, quantifying continuous degrees of acceptability and allowing the seamless, axiomatically driven integration of positive and negative edges. When combined with probabilistic argumentation (e.g., epistemic distributions over arguments), QBAFs further support the representation of user-specific, granular belief states not attainable in the extension-based approach.

5. Axiomatic Properties and Expressiveness

QBAFs are characterized by axiomatizable properties ensuring intuitive interpretability and computational soundness:

• Conservativity: If an argument is isolated (no attackers or supporters), its degree equals its initial plausibility: if Att(a) = Sup(a) = ∅ then Deg(a) = w(a).
• Anonymity, Independence: Acceptability depends only on network structure and initial weights.
• Parent Monotony: Increasing a supporter’s plausibility cannot decrease the acceptability of the supported argument; increasing an attacker’s plausibility cannot increase that of the target.
• Neutrality and Reverse Impact: Removal or neutralization of an argument with neutral degree does not affect others; negative plausibility in a supporter acts as an attack.
• Reinforcement, Impact, Continuity: Systematic strengthening of supporters has coherent, predictable effects.

These axioms formalize robustness, transparency, and stability, establishing QBAFs as a sound basis for applications requiring accountable quantitative reasoning (Mossakowski et al., 2016).

6. Applications, Algorithmic Considerations, and Limitations

Applications:

QBAFs have been deployed in decision support systems, dialogical agents, recommendation systems, social media analysis, and explainable AI, where both acceptability degrees and their provenance (i.e., which attacks/supports drove the outcome) are essential.

Algorithmic aspects:

• Direct aggregation semantics can be computed via matrix inverses for moderate-sized graphs (when d > indegree(G)); iterative schemes are used in large or streaming settings.
• When using sigmoid normalization, comparisons to bounded-weight systems become meaningful and facilitate integration with machine learning architectures (e.g., neural network structures that encode both positive and negative “argument” connections) (Potyka, 2020).

Limitations and trade-offs:

• Sum-based aggregation may diverge in cyclic graphs of high indegree; appropriate damping or restriction to top-based aggregators is essential for convergence (Mossakowski et al., 2018).
• Introduction of negative plausibility can counterintuitively convert supporters to attackers, demanding careful calibration in knowledge engineering.
• Real-world deployment often necessitates parameter selection (e.g., damping factor) based on the specific argumentation graph's topology and expected inference depth.

7. Summary Table: Key Mathematical Formulations in QBAFs

Concept	Formula / Definition	Comment
Incidence Matrix	G∈{–1, 0, 1}^	𝒜
Iterative Update	f^i = w + (1/d) G f^i–1	Damped propagation, d ≥ indegree(G)
Limit (Direct Aggregation)	Deg_dir^G,d(w) = lim_{i→∞} f^i = ∑_{i=0}^∞ (1/d G)^i w	Series converges if d > indegree(G)
Fixed Point Equation	D = w + (1/d) G D ⇒ D = (I – (1/d)G)^–1 w	Unique solution when d > indegree(G)
Sigmoid Aggregation	Deg_sdir^G,d(w) = σ( (I – (1/d)G)^–1 σ^–1(w) )	σ(x)=1/(1+e^–x), normalization to (0,1)
Modular Semantics	acc = ι( α(g, d), w )	Aggregation/influence modularity
Top-based Aggregation	α_top(g, d) = top(g, d) * d	Strongest attacker/supporter only

This modular, iterative, and axiomatic treatment of weighted bipolar argumentation graphs—rooted in transparent matrix and vector computations—comprehensively characterizes how to reconcile complex, multi-relation argument structures into interpretable, convergent, and application-ready quantitative acceptability degrees, generalizing and extending the theory and practice of computational argumentation (Mossakowski et al., 2016).