AI 中文总结
该研究提出超图 Horn 函数中 Horn 模型偏序集的秩-3 表示，结合相关构造证明余原子枚举的复杂性，给出不同秩与频率下的生成算法及复杂度结论。
AI 中文摘要
对于有限超图 H，其关联确定型 Horn CNF 模型的补集恰好是 H 的停止集；因此，其余原子的补集就是包含意义下极小的非空停止集。我们研究从超图关联表示出发的输出敏感型枚举。我们的主要结果是：任意 Horn 模型偏序集的一种表示，其关联大小与归一化 Horn 输入的关联长度呈线性关系。给定 Horn CNF Γ，我们构造一个秩至多为 3 的超图 C(Γ)，其真模型偏序集与 Γ 的模型偏序集呈包含序同构；等价地，每个源模型都有唯一的扩展到真目标模型。因此，Γ 的极大模型与目标余原子一一对应。将该表示与 Kavvadias、Sideri 和 Stavropoulos 的极大模型构造相结合，可证明：即使所有超边的大小为 2 或 3，余原子枚举也不属于 OutputP，除非 P=NP。关联分裂将最大元素频率降至 3，同时保留停止集偏序集；而对大小为 2 的超边进行局部替换，可在最大元素频率至多为 3 的 3-均匀超图中得到相同下界。这些阈值对任意阶枚举而言是条件尖锐的：秩至多为 2 且最大元素频率至多为 2 时，均允许输出线性总时间生成；在频率为 2 的情况下，还存在多项式延迟、多项式空间的算法。相比之下，对于精确元素频率为 2 的 3-均匀超图，余原子扩展已是 NP 完全问题。
英文摘要
For a finite hypergraph H, the complements of the models of its associated definite Horn CNF are exactly the stopping sets of H; hence the complements of its coatoms are the inclusion-minimal nonempty stopping sets. We study their output-sensitive enumeration from the hypergraph incidence representation. Our main result is a representation of arbitrary Horn model posets whose incidence size is linear in the incidence length of the normalized Horn input. Given a Horn CNF $Γ$, we construct a hypergraph $C(Γ)$ of rank at most three whose proper-model poset is inclusion-order isomorphic to the model poset of $Γ$; equivalently, each source model has a unique extension to a proper target model. Thus maximal models of $Γ$ correspond bijectively to target coatoms. Combining this representation with the maximal-model construction of Kavvadias, Sideri, and Stavropoulos shows that coatom enumeration is not in OutputP unless P=NP, even when all hyperedges have size two or three. Incidence splitting reduces maximum element frequency to three while preserving the stopping-set poset, and a local replacement of two-element hyperedges yields the same lower bound for three-uniform hypergraphs of maximum element frequency at most three. These thresholds are conditionally sharp for arbitrary-order enumeration: rank at most two and maximum element frequency at most two both admit output-linear total-time generation; in the frequency-two case, a polynomial-delay, polynomial-space algorithm is also available. In contrast, coatom extension is NP-complete already for three-uniform hypergraphs of exact element frequency two.
