Bases for Weighted Gradual Semantics and Inverse Problems in Argumentation Theory

Abstract

Weighted gradual semantics provide an acceptability degree to each argument representing its final strength, computed based on factors including the argument's background evidence, and taking into account interactions between the argument and others.

We introduce five important problems linking gradual semantics and acceptability degrees. First, we re-examine the inverse problem, seeking to identify each argument's initial weights within 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 one-to-one. Third, we ask if this mapping is a homeomorphism so that small perturbations in weights lead to small perturbation in acceptability degrees and vice versa. Fourth, we ask whether argument weights can be found when preferences, rather than acceptability degrees for arguments are considered. Last, we consider the geometry 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, and building on the geometry of the acceptability degree space, we identify a large family of weighted gradual semantics which contains many of the existing well-known semantics while maintaining desirable properties such as convergence to a unique fixed point and solving all five aforementioned problems.