On the Complexity of the Grounded Semantics for Infinite Argumentation Frameworks

2025-11-27

AI Summary

This study investigates the computational complexity of the grounded semantics in infinite argumentation frameworks. We address the construction of the grounded extension and the decision problem of grounded acceptability. Method: Leveraging tools from computability theory and set theory, we employ transfinite induction over the natural defense operator to rigorously analyze the fixed-point convergence process. Results: We establish that the least ordinal length required to compute the grounded extension is a countable recursive ordinal $alpha$, and prove that the grounded acceptability problem is $Pi^1_1$-complete—achieving maximal complexity within the analytical hierarchy. Consequently, grounded semantics in infinite frameworks is inherently non-finitary: it cannot be decided in finitely many steps, in stark contrast to its polynomial-time decidability in finite frameworks. This work resolves a fundamental gap in the complexity theory of non-standard argumentation semantics.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Natural Language Processing: Discourse, Pragmatics & Argument Mining

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• Determines the ordinal length for computing the grounded extension in infinite argumentation frameworks.
• Analyzes the computational complexity of deciding grounded acceptance, showing maximal complexity.
• Contrasts infinite-case complexity with polynomial-time computability in finite argumentation frameworks.

Innovation

• Using computability and set theory methods
• Identifying exact ordinal number for transfinite iterations
• Determining maximal complexity of grounded acceptance