向下Löwenheim-Skolem性质的有限证明
A Finitary Analogue of the Downward Löwenheim-Skolem Property
摘要：
我们提出了一个模型理论的有限结构的性质，可以被看作是一个有限的模拟研究向下Löwenheim-Skolem属性从经典模型理论。我们称这个性质为 * $\mathcal{L}$ -等价有界子结构性质 *，记为$\mathcal{L}$ - $\mathsf{EBSP}$，其中$\mathcal{L}$是FO或MSO。直觉$\mathcal{L}$ - $\mathsf{EBSP}$指出，一个大的有限结构包含一个小的“逻辑相似”的子结构，其中逻辑相似性意味着相对于具有给定量词嵌套深度的$\mathcal{L}$的句子的不可解释性。事实证明，这个简单陈述的性质是计算机科学中各种感兴趣的类别所享有的：例子包括各种偏序集，如单词，树（无序，有序或排名）和嵌套单词的正则语言，以及各种图类，如上图，有界树深度的图类，有界灌木深度的图类和$n$ -partite上图。此外，$\mathcal{L}$ - $\mathsf{EBSP}$保持保存在通过使用无量词转换方案可实现的操作从以上生成的类中。我们表明，自然树表示的结构，所有上述类承认，小的和逻辑上类似的子结构的一个大的结构可以计算在时间线性的表示的大小，给线性时间固定参数易处理（FPT）。用于检查大型结构的$\mathcal{L}$可定义属性的算法。最后，我们提出了一个加强的$\mathcal{L}$ - $\mathsf{EBSP}$，断言“逻辑自相似性在所有尺度”的一个合适的概念的规模。我们称之为“逻辑分形”属性，并证明上面提到的大多数类确实是逻辑分形。
英文摘要：
We present a model-theoretic property of finite structures, that can be seen to be a finitary analogue of the well-studied downward Löwenheim-Skolem property from classical model theory. We call this property as the *$\mathcal{L}$-equivalent bounded substructure property*, denoted $\mathcal{L}$-$\mathsf{EBSP}$, where $\mathcal{L}$ is either FO or MSO. Intuitively $\mathcal{L}$-$\mathsf{EBSP}$ states that a large finite structure contains a small "logically similar" substructure, where logical similarity means indistinguishability with respect to sentences of $\mathcal{L}$ having a given quantifier nesting depth. It turns out that this simply stated property is enjoyed by a variety of classes of interest in computer science: examples include various classes of posets, such as regular languages of words, trees (unordered, ordered or ranked) and nested words, and various classes of graphs, such as cographs, graph classes of bounded tree-depth, those of bounded shrub-depth and $n$-partite cographs. Further, $\mathcal{L}$-$\mathsf{EBSP}$ remains preserved in the classes generated from the above by operations that are implementable using quantifier-free translation schemes. We show that for natural tree representations for structures that all the aforementioned classes admit, the small and logically similar substructure of a large structure can be computed in time linear in the size of the representation, giving linear time fixed parameter tractable (f.p.t.) algorithms for checking $\mathcal{L}$ definable properties of the large structure. We conclude by presenting a strengthening of $\mathcal{L}$-$\mathsf{EBSP}$, that asserts "logical self-similarity at all scales" for a suitable notion of scale. We call this the *logical fractal* property and show that most of the classes mentioned above are indeed, logical fractals.
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