Abstract Weighted Based Gradual Semantics in Argumentation Theory
Assaf Libman, Nir Oren, Bruno Yun
TL;DR
The paper advances the theory of weighted gradual semantics for argumentation by formulating a unifying scoring-scheme perspective that maps argument weights to acceptability degrees and introduces four inverse-type problems capturing reconstruction, stability, scaling, and continuity properties. It defines a broad class of abstract weighted based gradual semantics and a concrete subfamily, the $(L^p,\lambda,\mu,A)$-based schemes, which guarantee convergence to a unique fixed point and solve the four inverse problems; it also provides explicit inverse formulas for well-known semantics ($\Sigma_{\mathbb{MB}},\Sigma_{\mathbb{CB}},\Sigma_{\mathbb{HC}}$). The paper then presents two new semantics within this framework: a Weighted Euclidean-based semantics and a remote-attacks semantics, both preserving convergence and inverse-solvability. Together, these results enable exact weight reconstruction from acceptability degrees and support applications in preference elicitation, opponent modeling, and argument strategy, while offering a rich set of scalable and interpretable semantics. The work lays groundwork for integrating these semantics with broader evaluation-method frameworks and future empirical validation of the proposed inverse-enabled approaches.
Abstract
Weighted gradual semantics provide an acceptability degree to each argument representing the strength of the argument, computed based on factors including background evidence for the argument, and taking into account interactions between this argument and others. We introduce four important problems linking gradual semantics and acceptability degrees. First, we reexamine the inverse problem, seeking to identify the argument weights of the argumentation framework which lead to a specific final acceptability degree. Second, we ask whether the function mapping between argument weights and acceptability degrees is injective or a homeomorphism onto its image. Third, we ask whether argument weights can be found when preferences, rather than acceptability degrees for arguments are considered. Fourth, we consider the topology of the space of valid acceptability degrees, asking whether "gaps" exist in this space. While different gradual semantics have been proposed in the literature, in this paper, we identify a large family of weighted gradual semantics, called abstract weighted based gradual semantics. These generalise many of the existing semantics while maintaining desirable properties such as convergence to a unique fixed point. We also show that a sub-family of the weighted gradual semantics, called abstract weighted (L^p,λ,μ)-based gradual semantics and which include well-known semantics, solve all four of the aforementioned problems.