In Property Testing, proximity-oblivious testers (POTs) form 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 allow constant-query proximity-oblivious testing 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. Indeed, calling properties expressible as a generalised subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by exhibiting a property of graphs that is GSF-local and propagating. Hence in particular, our property does not admit a POT, despite being GSF-local. We prove our result by exploiting a recent work of the authors which constructed a first-order (FO) property that is not testable [SODA 2021], and a new connection between FO properties and GSF-local properties via neighbourhood profiles.

中文翻译：

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GSF局部性不足以进行邻近度测试

在“性能测试”中，接近度忽略测试器（POT）构成了一类特别简单的测试算法，其中，一次基本测试执行了多次，这可能取决于接近度参数，但基本测试本身与接近度参数无关。Goldreich和Ron在他们的开创性工作中[STOC 2009; [SICOMP 2011]显示，在有界度模型中允许进行常量查询的接近性遗忘测试的图属性正是可以表示为满足非传播条件的广义子图自由度（GSF）属性的属性。是否需要非传播条件则保持开放状态。的确，他们称可表示为广义子图自由度属性的GSF局部属性表示的属性，他们询问所有GSF局部属性是否都是非传播的。我们通过展示GSF局部图和传播图的性质来给出否定答案。因此，特别是，尽管我们的物业位于GSF本地，但我们不接受POT。我们通过利用作者的最新工作来证明我们的结果，这些工作构建了不可测试的一阶（FO）属性[SODA 2021]，以及通过邻域轮廓在FO属性和GSF局部属性之间建立了新的联系。