有界度图一阶性质的可测性
On Testability of First-Order Properties in Bounded-Degree Graphs
摘要：
我们研究属性测试的属性是可定义的一阶逻辑（FO）中的有界度图和关系结构模型。我们证明了任何由具有量词前缀$\exists^*\forall^*$的公式定义的FO属性都是可测试的（即，具有恒定查询复杂度的可测试），而存在可由具有不可测试的量词前缀$\forall^*\exists^*$的公式表示的FO属性。在稠密图模型中，一个类似的画面是众所周知的（阿隆，菲舍尔，Krivelevich，Szegedy，Combinatorica 2000），尽管这两个模型的性质非常不同。特别是，我们得到我们的下界的一阶公式，定义了一类有界度扩展，基于锯齿形产品的图形。我们希望这是独立的利益。然后，我们证明了一些一阶属性的可测性，讲同构类型的邻域，包括可测性的$1$ -邻域自由度，和$r$ -邻域自由度下一个温和的假设。
英文摘要：
We study property testing of properties that are definable in first-order logic (FO) in the bounded-degree graph and relational structure models. We show that any FO property that is defined by a formula with quantifier prefix $\exists^*\forall^*$ is testable (i.e., testable with constant query complexity), while there exists an FO property that is expressible by a formula with quantifier prefix $\forall^*\exists^*$ that is not testable. In the dense graph model, a similar picture is long known (Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000), despite the very different nature of the two models. In particular, we obtain our lower bound by a first-order formula that defines a class of bounded-degree expanders, based on zig-zag products of graphs. We expect this to be of independent interest. We then prove testability of some first-order properties that speak about isomorphism types of neighbourhoods, including testability of $1$-neighbourhood-freeness, and $r$-neighbourhood-freeness under a mild assumption on the degrees.
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