Gadget构造与结构收敛
Gadget construction and structural convergence
AI总结：
本文通过Gadget构造扩展结构收敛的构造方法，证明其在句子集上保持基本收敛，并给出确保完全一阶收敛的充分条件。
摘要：
Nešetřil和Ossona de Mendez最近提出了图收敛的一种新定义，称为结构收敛。结构收敛框架基于一阶公式的固定片段中逻辑公式满足的概率。选择片段的灵活性允许统一稀疏图和稠密图的经典收敛概念。由于该领域相对年轻，收敛序列的示例范围有限，且已知的构造方法仅有少数几种。我们的目标是通过考虑Gadget构造来扩展构造的多样性。我们证明，当限制在句子集合上时，将Gadget构造应用于基本收敛序列会产生基本收敛序列。另一方面，我们展示了反例，证明在没有额外假设的情况下，推广到完全一阶收敛是不可能的。我们给出了几个不同的充分条件以确保完全收敛。其中一个条件表明，如果被替换的边在原始结构序列中是稠密的，则结果序列是一阶收敛的。
英文摘要：
Nešetřil and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of convergence for sparse and dense graphs. Since the field is relatively young, the range of examples of convergent sequences is limited and only a few methods of construction are known. Our aim is to extend the variety of constructions by considering the gadget construction. We show that, when restricting to the set of sentences, the application of gadget construction on elementarily convergent sequences yields an elementarily convergent sequence. On the other hand, we show counterexamples witnessing that a generalization to the full first-order convergence is not possible without additional assumptions. We give several different sufficient conditions to ensure the full convergence. One of them states that the resulting sequence is first-order convergent if the replaced edges are dense in the original sequence of structures.
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