Analytical Solutions for the Inverse Problem within Gradual Semantics

TL;DR

This paper introduces an analytical method to solve the inverse problem in gradual argumentation semantics, providing rapid and guaranteed solutions, and proves several conjectured properties.

Contribution

It presents a novel analytical approach for the inverse problem in gradual semantics, improving speed and certainty over existing numerical methods.

Findings

Analytical solutions can rapidly solve the inverse problem.

The approach guarantees to find a solution when it exists.

Several properties of the inverse problem are formally proven.

Abstract

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.