The online manipulation-resilient testing model, proposed by Kalemaj, Raskhodnikova and Varma (ITCS 2022 and Theory of Computing 2023), studies property testing in situations where access to the input degrades continuously and adversarially. Specifically, after each query made by the tester is answered, the adversary can intervene and either erase or corrupt $t$ data points. In this work, we investigate a more nuanced version of the online model in order to overcome old and new impossibility results for the original model. We start by presenting an optimal tester for linearity and a lower bound for low-degree testing of Boolean functions in the original model. We overcome the lower bound by allowing batch queries, where the tester gets a group of queries answered between manipulations of the data. Our batch size is small enough so that function values for a single batch on their own give no information about whether the function is of low degree. Finally, to overcome the impossibility results of Kalemaj et al. for sortedness and the Lipschitz property of sequences, we extend the model to include $t<1$, i.e., adversaries that make less than one erasure per query. For sortedness, we characterize the rate of erasures for which online testing can be performed, exhibiting a sharp transition from optimal query complexity to impossibility of testability (with any number of queries). Our online tester works for a general class of local properties of sequences. One feature of our results is that we get new (and in some cases, simpler) optimal algorithms for several properties in the standard property testing model.

翻译：在线操控鲁棒性测试模型由Kalemaj、Raskhodnikova和Varma提出（ITCS 2022与《计算理论》2023），研究在输入访问被连续且对抗性降级情况下的属性测试问题。具体而言，测试者每次查询得到答复后，对抗者可以进行干预，擦除或篡改$t$个数据点。本文针对原始模型中的新旧不可能性结果，探索该在线模型的更精细版本。我们首先在原始模型中给出线性性质的最优测试器，并给出布尔函数低次测试的下界。通过允许批量查询（即测试者在每次数据操控间隔内完成一组查询），我们克服了这一下界。我们的批量大小足够小，使得单个批次的函数值本身无法提供函数是否为低次的相关信息。最后，为克服Kalemaj等人在序列的有序性和Lipschitz性质方面的不可能性结果，我们将模型扩展至$t<1$，即每个查询对应少于一次擦除的对抗者。针对有序性，我们刻画了可执行在线测试的擦除速率，展现了从最优查询复杂度到测试不可行性（使用任意数量查询）的尖锐转变。我们的在线测试器适用于序列局部性质的一般类别。研究结果的一个特点是：在标准属性测试模型中，我们对多种性质获得了新的（且在部分情形下更简洁的）最优算法。