一句话：刻画平面与有界度限制下一阶有限谱的表达力：包含 NE 的重要片段，但并非全部 NE 集合。
有限谱是一阶句子所有有限模型大小的集合，已知全部有限谱构成的类等于复杂度类 NE。本文考察对模型加以限制后得到的谱：一是要求模型作为图是平面的，二是要求模型中每个元素的度有界。作者证明，这类受限谱仍然相当丰富：NE 的重要片段都可以表示为平面谱或有界度谱。与此同时，文章给出非平凡的上界，说明并非 NE 中的所有集合都能由平面或有界度模型的一阶句子得到。该结果从描述复杂度角度刻画了平面性与有界度限制下一阶逻辑的表达能力。
The finite spectrum of a first-order sentence is the set of positive integers that are the sizes of its models. The class of finite spectra is known to be the same as the complexity class NE. We consider the spectra obtained by limiting models to be either planar (in the graph-theoretic sense) or by bounding the degree of elements. We show that the class of such spectra is still surprisingly rich by establishing that significant fragments of NE are included among them. At the same time, we establish non-trivial upper bounds showing that not all sets in NE are obtained as planar or bounded-degree spectra.Comment: 21 pages. Accepted for publication in Logical Methods in Computer Science