关于主动和被动测试
On active and passive testing
摘要：
给定布尔函数的一个属性，以高概率确定输入函数是否满足该属性或“远”不满足该属性所需的最少查询次数是多少？这是属性测试中的一个基本问题，传统上测试算法允许在整个输入集中选择查询。Balcan，Blais，Blum和Yang最近建议限制测试人员从输入的多项式大小的较小随机子集中进行查询。这种模型被称为主动测试，在极端情况下，当我们可以查询的集合的大小正好是执行的查询数量时，它被称为被动测试。我们证明了被动或主动测试的k-线性函数（即，在Z_2上的n个变量之间的k个变量的总和）需要Theta（k*log n）查询，假设k不是太大。这扩展了情况k=1，（即独裁者的功能），分析了Balcan等。我们也考虑其他类型的功能，包括低次多项式，juntas，和部分对称的功能。我们的方法结合了代数，组合和概率技术，包括Talagrand浓度不等式和Erdos-Rado定理的三角洲系统。
英文摘要：
Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property or is "far" from satisfying it? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have recently suggested to restrict the tester to take its queries from a smaller random subset of polynomial size of the inputs. This model is called active testing, and in the extreme case when the size of the set we can query from is exactly the number of queries performed it is known as passive testing. We prove that passive or active testing of k-linear functions (that is, sums of k variables among n over Z_2) requires Theta(k*log n) queries, assuming k is not too large. This extends the case k=1, (that is, dictator functions), analyzed by Balcan et. al. We also consider other classes of functions including low degree polynomials, juntas, and partially symmetric functions. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdos--Rado theorem on Delta-systems.
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