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Gradual semantics within abstract argumentation associate a numeric scorewith every argument in a system, which represents the level of acceptability ofthis argument, and from which a preference ordering over arguments can bederived. While some semantics operate over standard argumentation frameworks,many utilise a weighted framework, where a numeric initial weight is associatedwith each argument. Recent work has examined the inverse problem within gradualsemantics. Rather than determining a preference ordering given an argumentationframework and a semantics, the inverse problem takes an argumentationframework, a gradual semantics, and a preference ordering as inputs, andidentifies what weights are needed to over arguments in the framework to obtainthe desired preference ordering. Existing work has attacked the inverse problemnumerically, using a root finding algorithm (the bisection method) to identifyappropriate initial weights. In this paper we demonstrate that for a class ofgradual semantics, an analytical approach can be used to solve the inverseproblem. Unlike the current state-of-the-art, such an analytic approach canrapidly 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 posedas conjectures.

Analytical Solutions for the Inverse Problem within Gradual Semantics