A Bayesian approach to forward and inverse abstract argumentation problems

Abstract

This paper studies a fundamental mechanism by which conflicts between arguments are drawn from sentiments regarding acceptability of the arguments. Given sets of arguments, an inverse abstract argumentation problem seeks attack relations between arguments such that acceptability semantics interprets each argument in the sets of arguments as being acceptable in each of the attack relations. It is an inverse problem of the traditional problem we refer to as the forward abstract argumentation problem. Given an attack relation, the forward abstract argumentation problem seeks sets of arguments such that acceptability semantics interprets each argument in the sets of arguments as being acceptable in the attack relation. We give a probabilistic model of argumentation-theoretic inference. It is a generative model formalising the process by which acceptability semantics interprets acceptability of arguments in a given attack relation. We show that it gives a broad view of solutions to the forward and inverse abstract argumentation problems. Specifically, solutions to the inverse and forward abstract argumentation problems are shown to be equivalent to a maximum likelihood estimate and maximum likelihood prediction, respectively, which are both available with the generative model. In addition, they are shown to be special cases of the posterior distribution and the evidence, respectively, which are both obtained by probabilistic inference on the generative model. We report an experiment result and application example of the generative model in the inverse problems.

Keywords:

• Abstract argumentation frameworks
• acceptability semantics
• inverse problems
• generative models
• Bayesian statistics
• machine learning

Acknowledgments

The authors are grateful to Martin Caminada for valuable discussion. This work was supported by JSPS KAKENHI Grant Number 18K11428.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 $X\propto Y$ denotes ‘X is proportional to Y’ and thus there is a constant K such that X = KY.

2 For the sake of simplicity, m also represents a set of two arguments. $At{t}_m$ in this case represents the existence of a symmetric attack relation between the arguments in m.

3 $|X|$ denotes the cardinality of set X.

4 See Appendix for the textual contents of the 10 arguments.