# Analytical Solutions for the Inverse Problem within Gradual Semantics

arXiv: https://arxiv.org/abs/2203.01201 · HTML: https://mlsanity.com/paper/2203.01201/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 12 references that are in the map.
- First posted on arXiv 2 Mar 2022.

## Builds on (most cited first, 3 of 3)

- [A Comparative Study of Ranking-based Semantics for Abstract Argumentation](https://mlsanity.com/paper/1602.01059/) (2016)
- [Argument Ranking with Categoriser Function](https://mlsanity.com/paper/1406.3877/) (2014)
- [The Inverse Problem for Argumentation Gradual Semantics](https://mlsanity.com/paper/2202.00294/) (2022)

## Built on by (most cited first, 3 of 3)

- [Inferring Attack Relations for Gradual Semantics](https://mlsanity.com/paper/2211.16118/) (2022)
- [Eliciting Rational Initial Weights in Gradual Argumentation](https://mlsanity.com/paper/2502.07452/) (2025)
- [On the Existence of an Inverse Solution for Preference-Based Reductions in Argumentation](https://mlsanity.com/paper/2604.22958/) (2026)

## Abstract (arXiv)

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.