AbstractFocusing on property testing tasks that have query complexity that is independent of the size of the tested object (i.e., depends on the proximity parameter only), we prove the existence of a rich hierarchy of the corresponding complexity classes. That is, for essentially any function $$q : (0, 1] \rightarrow \mathbb{N}$$q:(0,1]→N, we prove the existence of properties for which $$\epsilon$$ϵ-testing has query complexity $$\Theta(q(\Theta(\epsilon)))$$Θ(q(Θ(ϵ))). Such results are proved in three standard domains that are often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs.These results complement hierarchy theorems of Goldreich, Krivelevich, Newman, and Rozenberg (Computational Complexity, 2012), which refer to the dependence of the query complexity on the size of the tested object, and focus on the case that the proximity parameter is set to some small positive constant. We actually combine both flavors and get tight results on the query complexity of testing when allowing the query complexity to depend on both the size of the object and the proximity parameter.

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用于在大小不明确的查询复杂性中测试属性的层次定理

Abstract 针对查询复杂度与被测对象大小无关（即仅依赖于邻近参数）的属性测试任务，我们证明了相应复杂度类的丰富层次结构的存在。也就是说，对于基本上任何函数 $$q : (0, 1] \rightarrow \mathbb{N}$$q:(0,1]→N，我们证明 $$\epsilon$$ϵ -testing 具有查询复杂性 $$\Theta(q(\Theta(\epsilon)))$$Θ(q(Θ(ϵ)))。这样的结果在属性测试中经常考虑的三个标准域中得到证明： 泛型函数、描述（密集）图的邻接谓词和描述有界度图的关联函数。这些结果补充了 Goldreich、Krivelevich、Newman 和 Rozenberg 的层次定理（计算复杂性，2012），指的是查询复杂度对被测对象大小的依赖性，重点关注接近度参数设置为某个小的正常数的情况。当允许查询复杂性取决于对象的大小和邻近度参数时，我们实际上结合了两种风格并在测试的查询复杂性上获得了严格的结果。