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 an FO 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 use our class of FO definable bounded-degree expanders to answer a long-standing open problem for proximity-oblivious testers (POTs). POTs are a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter. In their seminal work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that are constant-query proximity-oblivious testable in the bounded-degree model are precisely the properties that can be expressed as a generalised subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. We give a negative answer by showing that our property is a GSF property which is propagating. Hence in particular, our property does not admit a POT. For this result we establish a new connection between FO properties and GSF-local properties via neighbourhood profiles.

翻译：我们研究了在有界度图与关系结构模型中可用一阶逻辑（FO）定义的性质的性质测试问题。我们证明，任何由量词前缀$\exists^*\forall^*$公式定义的FO性质都是可测试的（即具有常数查询复杂度），而存在一个由量词前缀$\forall^*\exists^*$公式表达的FO性质是不可测试的。在稠密图模型中，尽管两种模型的性质截然不同，但类似结论早已被确立（Alon, Fischer, Krivelevich, Szegedy, Combinatorica 2000）。特别地，我们通过一个定义基于图的zig-zag积的有界度展开图类别的FO公式得到了下界结果，这预期具有独立研究价值。随后，我们利用所构造的FO可定义有界度展开图类，解决了邻近无关测试器（POT）领域一个长期未决的开放问题。POT是一类极其简单的测试算法，其中基础测试的执行次数可能依赖于邻近参数，但基础测试本身与邻近参数无关。在他们的开创性工作中，Goldreich和Ron [STOC 2009; SICOMP 2011]证明，在有界度模型中具有常数查询复杂度的邻近无关可测试图性质，正是那些可表示为满足非传播条件的广义子图无关（GSF）性质。非传播条件是否必要仍为开放问题。我们通过证明所构造的性质是一个传播性GSF性质给出了否定答案，因此该性质特别地不具备POT。为获得此结果，我们通过邻域剖面建立了FO性质与GSF-局部性质之间的新联系。