Aggregative Semantics for Quantitative Bipolar Argumentation Frameworks

Yann Munro; Isabelle Bloch; Marie-Jeanne Lesot

cs.AIMar 6, 2026

Aggregative Semantics for Quantitative Bipolar Argumentation Frameworks

Authors: Yann Munro, Isabelle Bloch, Marie-Jeanne Lesot

Organizations: Sorbonne Université, CNRS, LIP6, F-75005 Paris, France

Abstract

Formal argumentation is being used increasingly in artificial intelligence as an effective and understandable way to model potentially conflicting pieces of information, called arguments, and identify so-called acceptable arguments depending on a chosen semantics. This paper deals with the specific context of Quantitative Bipolar Argumentation Frameworks (QBAF), where arguments have intrinsic weights and can attack or support each other. In this context, we introduce a novel family of gradual semantics, called aggregative semantics. In order to deal with situations in which attackers and supporters do not play a symmetric role, and in contrast to modular semantics, we propose to aggregate attackers and supporters separately. This leads to a three-stage computation, which consists in computing a global weight for attackers and another for supporters, before aggregating these two values with the intrinsic weight of the argument. We discuss the properties that the three aggregation functions should satisfy depending on the context, as well as their relationships with the classical principles for gradual semantics. This discussion is supported by various simple examples, as well as a final example on which five hundred aggregative semantics are tested and compared, illustrating the range of possible behaviours with aggregative semantics. Decomposing the computation into three distinct and interpretable steps leads to a more parametrisable computation: it keeps the bipolarity one step further than what is done in the literature, and it leads to more understandable gradual semantics.

Figures & tables

Panel of Figure 1 — Figure 1: Argumentation graph associated with Example 1 . Plain lines represent the attack relation and dotted lines the support relation. The value beside each argument name is its intrinsic strength, in the range $[0,1]$ .

Name	Symbol	Mathematical expression	Family
Arithmetic mean	$\textup{avg}_{\textup{am}}$	$\textup{avg}_{\textup{am}}(\mathbf{x_{n}})=\displaystyle\frac{1}{n}\sum_{i=1}^{n}x_{i}$	Compromise
Geometric mean	$\textup{avg}_{\textup{gm}}$	$\textup{avg}_{\textup{gm}}(\mathbf{x_{n}})=\displaystyle\left(\prod_{i=1}^{n}x_{i}\right)^{1/n}$	Compromise
Product	$\top_{\textup{prod}}$	$\top_{\textup{prod}}(\mathbf{x_{n}})=\displaystyle\prod_{i=1}^{\phantom{I}n\phantom{I}}x_{i}$	t-norm
Algebraic sum	$\bot_{\textup{prod}}$	$\bot_{\textup{prod}}(\mathbf{x_{n}})=1-\displaystyle\prod_{i=1}^{n}(1-x_{i})$	t-conorm
Minimum	$\min$	$\min(\mathbf{x_{n}})$	t-norm
Maximum	$\max$	$\max(\mathbf{x_{n}})$	t-conorm

Table 1: Example of ten aggregation functions. The four last ones are defined from, but their definition can be extended to

[0,1]^{n}

for all

n\in\mathbb{N}

using associativity.

	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11	P12
$\textup{avg}_{\textup{am}}$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\times$	$\times$	$\times$	$\times$	$\checkmark$	$\checkmark$
$\textup{avg}_{\textup{gm}}$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\times$	$\times$	$\times$	$e_{0}=0$	$\checkmark$	$\checkmark$
$\top_{\textup{prod}}$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\checkmark$	$\times$	$e_{1}=1$	$e_{0}=0$	$\checkmark$	$\checkmark$
$\bot_{\textup{prod}}$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\times$	$\checkmark$	$e_{1}=0$	$e_{0}=1$	$\checkmark$	$\checkmark$
$\min$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$e_{1}=1$	$e_{0}=0$	$\checkmark$	$\checkmark$
$\max$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$e_{1}=0$	$e_{0}=1$	$\checkmark$	$\checkmark$

Table 2: Properties of ten commonly used aggregation functions:

\checkmark

means the property holds and

\times

means it does not.

Panel of Figure 2 — Figure 2: Illustrative example for the commutativity postulate ( P4 ).

Panel of Figure 3 — Figure 3: Illustrative example for the idempotence postulate ( P5 ).

Panel of Figure 4 — Figure 4: Illustrative example for the weakening and reinforcement postulates ( P7-8 ).

Panel of Figure 5 — Figure 5: Illustrative example for the composition postulate ( P11 ).

Panel of Figure 6 — Figure 6: Illustrative example for the composition principle ( P12 ).

Panel of Figure 7 — Figure 7: Illustrative example for the continuity postulate ( P3 ) for $\varphi_{f}$ .

Panel of Figure 8 — Figure 8: Illustrative example of the entanglement of the weakening axiom ( A11 ).

$\varphi_{\mathcal{R}}/\varphi_{\mathcal{S}}$	(P1)	(P2)	(P3)	(P4)	(P5)	(P6)	(P7)	(P8)	(P9)	(P10)	(P11)	(P12)
Sum	$\times$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\times$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\checkmark$
$\bot_{\textup{prod}}$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$	$\times$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$
$\varphi_{f}^{\textup{DF-Quad}}$	$\checkmark$	$\checkmark$	$\checkmark$	-	$\checkmark$	-	$\times$	$\times$	-	$\times$	-	-
$\varphi_{f}^{\textup{Ebs}}$	$\checkmark$	$\checkmark$	$\checkmark$	-	$\checkmark$	-	$\times$	$\times$	-	$\checkmark$	-	-
$\varphi_{f}^{\textup{QE}}$	$\checkmark$	$\checkmark$	$\checkmark$	-	$\checkmark$	-	$\times$	$\times$	-	$\times$	-	-

Table 3: Verified (

\checkmark

), violated (

\times

) and inapplicable (-) postulates for the aggregation operators used in the three commonly used gradual semantics of the literature.

Panel of Figure 9 — Figure 9: Illustrative example analysed in Section 6 .

Figure 10: Distribution of acceptability degrees of argument

a

for the QBAF in Figure 9 .

	$\varphi_{\mathcal{R}}$	$\varphi_{\mathcal{S}}$	$\varphi_{f}$	$\emph{\textup{Deg}}_{A}(i)$	$\emph{\textup{Deg}}_{A}(e)$	$\pi_{\mathcal{R}}(a)$	$\pi_{\mathcal{S}}(a)$	$\emph{\textup{Deg}}_{A}(a)$
(S 1 )	$\bot_{\textup{prod}}$	$\bot_{\textup{prod}}$	$\varphi_{f}^{\textup{DF-Quad}}$	$0.48$	$0.52$	$0.98$	$0.96$	$0.49$
(S 2 )	Sum	Sum	$\varphi_{f}^{\textup{Ebs}}$	$0.48$	$0.53$	$2.03$	$1.97$	$0.49$
(S 3 )	Sum	Sum	$\varphi_{f}^{\textup{QE}}$	$0.48$	$0.52$	$2.02$	$1.98$	$0.50$
(S 4 )	$\textup{avg}_{\textup{am}}$	$\textup{avg}_{\textup{am}}$	$\textup{avg}_{\textup{am}}$	$0.53$	$0.48$	$0.49$	$0.50$	$0.50$
(S 5 )	$\textup{avg}_{\textup{am}}$	$\textup{avg}_{\textup{am}}$	$\top_{\textup{prod}}$	$0.14$	$0.11$	$0.40$	$0.41$	$0.12$
(S 6 )	$\textup{avg}_{\textup{am}}$	$\textup{avg}_{\textup{am}}$	$\bot_{\textup{prod}}$	$0.89$	$0.86$	$0.59$	$0.60$	$0.88$

Table 4: The first three rows are respectively DF-Quad, Ebs and QE.

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