Complexity Classifications for logic-based Argumentation

TL;DR

This paper classifies the computational complexity of key problems in logic-based argumentation within Schaefer's framework, revealing a range from polynomial to high complexity classes depending on the properties of the formulae involved.

Contribution

It provides a comprehensive complexity classification for existence, validity, and relevance problems in logic-based argumentation using algebraic methods within Schaefer's framework.

Findings

Existence of support is polynomial, NP-complete, coNP-complete, or SigP2-complete depending on the language.

Verification problem is either polynomial or DP-complete.

Relevance problem falls into polynomial, NP-complete, or SigP2-complete classes.

Abstract

We consider logic-based argumentation in which an argument is a pair (Fi,al), where the support Fi is a minimal consistent set of formulae taken from a given knowledge base (usually denoted by De) that entails the claim al (a formula). We study the complexity of three central problems in argumentation: the existence of a support Fi ss De, the validity of a support and the relevance problem (given psi is there a support Fi such that psi ss Fi?). When arguments are given in the full language of propositional logic these problems are computationally costly tasks, the validity problem is DP-complete, the others are SigP2-complete. We study these problems in Schaefer's famous framework where the considered propositional formulae are in generalized conjunctive normal form. This means that formulae are conjunctions of constraints build upon a fixed finite set of Boolean relations Ga (the constraint language). We show that according to the properties of this language Ga, deciding whether there exists a support for a claim in a given knowledge base is either polynomial, NP-complete, coNP-complete or SigP2-complete. We present a dichotomous classification, P or DP-complete, for the verification problem and a trichotomous classification for the relevance problem into either polynomial, NP-complete, or SigP2-complete. These last two classifications are obtained by means of algebraic tools.