Logic-based argumentation is a well-established formalism modelling nonmonotonic reasoning. It has been playing a major role in AI for decades, now. Informally, a set of formulas is the support for a given claim if it is consistent, subset-minimal, and implies the claim. In such a case, the pair of the support and the claim together is called an argument. In this paper, we study the propositional variants of the following three computational tasks studied in argumentation: ARG (exists a support for a given claim with respect to a given set of formulas), ARG-Check (is a given set a support for a given claim), and ARG-Rel (similarly as ARG plus requiring an additionally given formula to be contained in the support). ARG-Check is complete for the complexity class DP, and the other two problems are known to be complete for the second level of the polynomial hierarchy (Parson et al., J. Log. Comput., 2003) and, accordingly, are highly intractable. Analyzing the reason for this intractability, we perform a two-dimensional classification: first, we consider all possible propositional fragments of the problem within Schaefer's framework (STOC 1978), and then study different parameterizations for each of the fragment. We identify a list of reasonable structural parameters (size of the claim, support, knowledge-base) that are connected to the aforementioned decision problems. Eventually, we thoroughly draw a fine border of parameterized intractability for each of the problems showing where the problems are fixed-parameter tractable and when this exactly stops. Surprisingly, several cases are of very high intractability (paraNP and beyond).

中文翻译：

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Schaefer框架中基于逻辑的参数的参数化复杂性

基于逻辑的论证是一种公认​​的形式主义，用于建模非单调推理。数十年来，它一直在AI中扮演重要角色。非正式地，如果给定的权利要求是一致的，最小的子集并暗含该权利要求，则一组公式是对给定权利要求的支持。在这种情况下，一对支持和索赔一起称为论点。在本文中，我们研究了在论证中研究的以下三个计算任务的命题变式：ARG（针对给定的一组公式对给定的要求提供支持），ARG-Check（对a的给定集提供了支持）给定的权利要求）和ARG-Rel（类似于ARG，另外要求在载体中包含一个额外给定的公式）。对于复杂度等级DP，ARG-Check已完成，已知其他两个问题对于多项式层次结构的第二层是完整的（Parson等人，J。Log。Comput。，2003年），因此非常棘手。分析这种难解性的原因，我们进行了二维分类：首先，在Schaefer的框架内（STOC 1978）考虑问题的所有可能命题片段，然后为每个片段研究不同的参数化。我们确定与上述决策问题相关的合理结构参数（索赔的大小，支持，知识库）的列表。最终，我们针对每个问题彻底画出了参数化的难处理性的细微边界，显示出问题在哪里是固定参数易处理的，以及何时确切停止。出奇，