On the Complexity of the Grounded Semantics for Infinite Argumentation Frameworks

TL;DR

This paper analyzes the computational complexity of the grounded semantics in infinite argumentation frameworks, revealing it requires transfinite iterations and is maximally complex, unlike the finite case.

Contribution

It precisely characterizes the ordinal length of the grounded extension process and establishes the maximal complexity of deciding grounded acceptance in infinite frameworks.

Findings

Grounded extension iteration length corresponds to a specific ordinal.

Deciding grounded acceptance in infinite frameworks is maximally complex.

Finite frameworks allow polynomial-time computation of grounded extension.

Abstract

Argumentation frameworks, consisting of arguments and an attack relation representing conflicts, are fundamental for formally studying reasoning under conflicting information. We use methods from mathematical logic, specifically computability and set theory, to analyze the grounded extension, a widely-used model of maximally skeptical reasoning, defined as the least fixed-point of a natural defense operator. Without additional constraints, finding this fixed-point requires transfinite iterations. We identify the exact ordinal number corresponding to the length of this iterative process and determine the complexity of deciding grounded acceptance, showing it to be maximally complex. This shows a marked distinction from the finite case where the grounded extension is polynomial-time computable, thus simpler than other reasoning problems explored in formal argumentation.