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.

翻译：论辩框架由论证及表示冲突的攻击关系构成，是形式化研究冲突信息下推理的基础工具。本文运用数理逻辑方法，特别是可计算性与集合论，分析基础扩展——一种广泛应用的极大怀疑推理模型，其定义为自然防御算子的最小不动点。在无额外约束条件下，求解该不动点需进行超限迭代。我们精确确定了该迭代过程长度对应的序数，并判定基础接受性决策的复杂性，证明其具有最大复杂度。这表明与有限情形存在显著差异：在有限论辩框架中，基础扩展可在多项式时间内计算，因而比形式论辩中探讨的其他推理问题更为简单。