Quantitative Bipolar Argumentation Framework (QBAF)

• QBAF is a mathematical framework that integrates both attack and support relations by assigning initial plausibility scores to arguments.
• The framework employs direct and sigmoid aggregation semantics to compute final acceptability via convergent matrix inversion and damping factors.
• QBAF is applied in decision-making, dialogue systems, and explainable AI, ensuring robust evaluation of complex argument interactions.

A Quantitative Bipolar Argumentation Framework (QBAF) provides a mathematically rigorous model for representing and evaluating argumentation involving both attack and support relations, accounting for the relative strength or plausibility of every argument in the system. The framework generalizes previous models by accommodating arguments with initial plausibility scores, both positive and negative links, and a semantics that aggregates influences through the argumentation network to produce final acceptability (or strength) degrees. QBAFs have become central in computational models of practical reasoning, dialogues, AI explainability, and systems for computer-supported decision-making.

1. Algebraic Representation and Model Structure

The formal model for a QBAF introduced in "Bipolar Weighted Argumentation Graphs" (Mossakowski et al., 2016) encodes the system as a triple $(\mathcal{A}, G, w)$, where:

• $\mathcal{A}$ is a finite ordered set (vector) of arguments.
• $G \in \{-1, 0, 1\}^{n \times n}$ is the incidence matrix, with $g_{ij}=1$ if $a_j$ supports $a_i$, $g_{ij}=-1$ if $a_j$ attacks $a_i$, and $g_{ij}=0$ otherwise.
• $w$ is an $n$-dimensional vector of initial plausibilities $w(a_i)$, typically real values or normalized to $(0,1)$.

This formulation subsumes previous frameworks handling only supports or only attacks, allowing both relation types to coexist. The matrix $G$ simultaneously encodes positive (support) and negative (attack) links, and $w$ captures the intrinsic strength, confidence, or prior acceptance of each argument.

2. Direct Aggregation Semantics

The direct aggregation semantics determine the final degree of acceptability of each argument by recursively propagating influences along the network. The process is defined by the iteration

$$\begin{aligned} f^{(0)}(a) &= w(a) \ f^{(i+1)}(a) &= w(a) + \frac{1}{d}\left[\sum_{b \in \mathcal{S}(a)} f^{(i)}(b) - \sum_{c \in \mathcal{A}(a)} f^{(i)}(c)\right] \end{aligned}$$

where:

• $\mathcal{S}(a)$ is the set of supporters ($g_{ab}=1$),
• $\mathcal{A}(a)$ is the set of attackers ($g_{ac}=-1$),
• $d \geq 1$ is a damping factor that attenuates (decays) the influence of indirect or high-degree links.

In vector matrix notation: $$f^{(0)} = w, ~~ f^{(i+1)} = w + \frac{1}{d} G f^{(i)}$$

Provided $d > \deg_\text{max}(G)$ (max indegree), this sequence converges to a unique fixed point, which can be computed directly: $$\operatorname{Deg}^{(\mathrm{dir})}_{d} = \lim_{i \to \infty} f^{(i)} = (I - \tfrac{1}{d}G)^{-1} w$$ or, equivalently, via the Neumann series as a sum over $(1/d) G$ powers. This closed-form propagation formula guarantees analyticity and efficient evaluation, and supports sensitivity analysis and explainability.

3. Sigmoid Aggregation and Bounded Semantics

To ensure that resulting degrees are confined to a standard range, such as $(0,1)$ (suitable in contexts where plausibility must be interpreted probabilistically or within normalized scales), the sigmoid direct aggregation semantics apply a strictly increasing sigmoid function $\sigma:\mathbb{R} \to (0,1)$: $$\operatorname{Deg}^{(\mathrm{sdir})}_{d} = \sigma\left( (I - \tfrac{1}{d}G)^{-1} \sigma^{-1}(w) \right)$$ Typical choices for $\sigma$ include the logistic function ($\sigma(x) = 1/(1+e^{-x})$) or scaled arctan, with $\sigma(0) = 1/2$. Initial weights are lifted onto the real line via $\sigma^{-1}$, aggregated as before, and projected back. This guarantees neutrality (midpoint of 1/2) and prevents numeric saturation or overflow in high-degree graphs, aligning with applications in probabilistic or epistemic reasoning.

4. Unification and Generalization over Previous Models

The direct and sigmoid aggregation semantics generalize earlier aggregation-based approaches considered by Amgoud, Ben-Naim, et al., which addressed only support-only or attack-only frameworks:

• For support-only graphs ($G$ with only $0$ and $1$ entries), the semantics reduce to standard additive propagation where supports simply accumulate, and unconnected arguments retain their initial scores due to conservativity.
• For attack-only graphs ($G$ with only $0$ and $-1$ entries), the semantics reflect only deductive loss, where the aggregate of attackers reduces acceptability.
• In true bipolar networks, the combined propagation incorporates both additive and subtractive effects, capturing complex cancellation and reinforcement, such as ‘attack by an implausible argument producing a net positive effect’, a phenomenon observed in dialogical argumentation studies (Polberg et al., 2017).

The introduction of the damping factor $d$ is essential for convergence and for controlling the indirect (multi-step) propagation, which is not present in naïve algebraic combinations of attack and support graphs.

5. Properties and Desirable Characteristics

The QBAF matrix-based semantics exhibit a robust set of formal properties:

• Conservativity: Arguments with no incoming links maintain their initial weight.
• Anonymity and Independence: The outcome is invariant under relabeling, and only incoming relations affect the evaluation (network-structural locality).
• Directionality: The influence propagates only along directed paths.
• Monotonicity: Single-path positive (support) influences increase, and negative (attack) influences decrease acceptability, with nontrivial interaction when both coexist.
• Convergence: For $d$ exceeding the maximal indegree, $(I-\frac{1}{d}G)$ is invertible with spectral radius $<1$ for the scaled $G$, and the propagation series converges.

These properties arise naturally from the linear algebraic formulation and are not postulated as axioms. The damping factor’s role is to attenuate long-range dependencies and ensure numerical stability.

6. Practical Implementation, Computation, and Use Cases

Practical instantiation of QBAF semantics reduces to building $G$ and $w$ as specified for the problem domain, computing $(I-\frac{1}{d}G)^{-1}w$, and—optionally—processing results through a sigmoid for boundedness. For large-scale or sparse frameworks, dedicated linear algebra solvers (direct or iterative) can be used. The framework handles both arbitrary real initial weights and application-driven normalizations.

Key applications include:

• Dialogue and debate systems, where both confirmations and rebuttals must be accounted for quantitatively (Polberg et al., 2017).
• Online opinion aggregation and recommendation, where arguments may favor or contest items simultaneously.
• Explainable AI, where transparency into the effect of positive and negative evidence is required, and both direct and indirect influences are to be traced.

7. Examples, Limitations, and Extensions

A summary of cases:

Graph Structure	G Entries	Semantics Behavior
Support-only	0, 1	Additive support propagation
Attack-only	0, -1	Subtractive (defeater) influence
Bipolar (mixed)	-1, 0, 1	Both additive and subtractive, complex interactions

A direct application of the QBAF formalism reveals unintuitive phenomena: e.g., if an argument with very low initial weight attacks another, its negative influence may act as support due to net cancellation, an effect that must be handled carefully in real-world deployments.

The sigmoid transformation enforces range and interpretability. The damping factor needs careful calibration: $d$ too low can lead to divergence in dense graphs; $d$ too high dampens meaningful indirect relations. The framework generalizes to edge- and weight-based settings, modular semantics, and can be extended to structured or weighted relations with scalar or functional generalizations of $G$ and $w$.

References to Related Methods and Developments

The QBAF model defined in "Bipolar Weighted Argumentation Graphs" (Mossakowski et al., 2016) directly anticipates current lines of research in modular semantics (Mossakowski et al., 2018), continuous dynamical systems for strength propagation (Potyka, 2018), expressive probabilistic and epistemic reasoning in multi-party contexts (Polberg et al., 2017), and recent advances in contestable, explainable, and user-facing AI argumentation systems. The algebraic and recursive form of QBAF semantics has been foundational in subsequent developments in explainability and aggregation methods.