Abstract Weighted Based Gradual Semantics in Argumentation Theory

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 (Lp,lambda,mu,A)-based gradual semantics and which include well-known semantics, solve all four of the aforementioned problems.

Analytical Solutions for the Inverse Problem within Gradual Semantics

Gradual semantics within abstract argumentation associate a numeric score with every argument in a system, which represents the level of acceptability of this argument, and from which a preference ordering over arguments can be derived. While some semantics operate over standard argumentation frameworks, many utilise a weighted framework, where a numeric initial weight is associated with each argument. Recent work has examined the inverse problem within gradual semantics. Rather than determining a preference ordering given an argumentation framework and a semantics, the inverse problem takes an argumentation framework, a gradual semantics, and a preference ordering as inputs, and identifies what weights are needed to over arguments in the framework to obtain the desired preference ordering. Existing work has attacked the inverse problem numerically, using a root finding algorithm (the bisection method) to identify appropriate initial weights. In this paper we demonstrate that for a class of gradual semantics, an analytical approach can be used to solve the inverse problem. Unlike the current state-of-the-art, such an analytic approach can rapidly find a solution, and is guaranteed to do so. In obtaining this result, we are able to prove several important properties which previous work had posed as conjectures.