Recent works have explored the use of counting queries coupled with Description Logic ontologies. The answer to such a query in a model of a knowledge base is either an integer or $\infty$, and its spectrum is the set of its answers over all models. While it is unclear how to compute and manipulate such a set in general, we identify a class of counting queries whose spectra can be effectively represented. Focusing on atomic counting queries, we pinpoint the possible shapes of a spectrum over $\mathcal{ALCIF}$ ontologies: they are essentially the subsets of $\mathbb{N} \cup \{ \infty \}$ closed under addition. For most sublogics of $\mathcal{ALCIF}$, we show that possible spectra enjoy simpler shapes, being $[ m, \infty ]$ or variations thereof. To obtain our results, we refine constructions used for finite model reasoning and notably rely on a cycle-reversion technique for the Horn fragment of $\mathcal{ALCIF}$. We also study the data complexity of computing the proposed effective representation and establish the $\mathsf{FP}^{\mathsf{NP}[\log]}$-completeness of this task under several settings.

翻译：近期研究探索了计数查询与描述逻辑本体的结合使用。此类查询在知识库模型中的答案要么是一个整数，要么是$\infty$，其谱系是所有模型上答案的集合。尽管通常不清楚如何计算和操作这样的集合，但我们识别出了一类谱系可以有效表示的计数查询。聚焦于原子计数查询，我们精确刻画了在$\mathcal{ALCIF}$本体上谱系的可能形态：它们本质上是$\mathbb{N} \cup \{ \infty \}$中在加法下封闭的子集。对于$\mathcal{ALCIF}$的大多数子逻辑，我们证明了可能的谱系具有更简单的形态，即$[ m, \infty ]$或其变体。为获得这些结果，我们改进了用于有限模型推理的构造方法，并特别依赖于$\mathcal{ALCIF}$的Horn片段中的循环反转技术。我们还研究了计算所提出的有效表示的数据复杂度，并在多种设定下确定了该任务的$\mathsf{FP}^{\mathsf{NP}[\log]}$完全性。