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.
翻译：在财产测试中,近距离不显眼的测试者(POTs)形成一类特别简单的测试算法,其中基本测试可多次进行,可能取决于近距离参数,但基本测试本身独立于近距离参数。Goldreich和Ron[2009年STOC;2009年;SICOMP2011年]在其重要工作中,Goldreich和Ron[STOC2009年;STOC 2009年;SICOMP2011年]表明,允许在约束度模型中进行常阵列近距离不显眼测试的图形属性恰恰是可表现为通用子图自由度(GSF)属性的属性,尽管满足了非调整条件。对于非调整条件是否必要,基本测试是开放的。事实上,将性能称为通用子图显示的自由属性,作为通用分层自由属性(GSF-F-当地属性),他们问所有GSF-当地属性是否不易变异。我们给出否定的答案是,通过显示一个图形属性的属性,这种属性是通用的,尽管是 GOF-F-POF-PA的本地属性。我们通过开发最近的工作证明我们的结果,而没有通过GOF-FS-FS-FS-S-C-FS-C-C-C-C-C-FS-C-C-C-C-C-C-C-C-C-CAR-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C
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