自适应线性检验的下限
Lower bounds for adaptive linearity tests
摘要：
线性检验是一种随机算法，它可以通过对函数f的真值表真值表的预言来区分函数f是线性函数还是非线性函数。线性检验首先由Blum、Luby和Rubenfeld（1993年）引入，后来用于PCP定理和其他应用中。线性测试的质量由其正确性c（接受线性函数的概率）、可靠性s（接受非线性函数的概率）和查询复杂度q（查询次数）来描述。研究线性测试是为了降低线性测试的可靠性，同时保持查询复杂度较小（原因之一是为了改进PCP构造）。Samorodnitsky和Trevisan（Samorodnitsky and Trevisan 2000）构建了完全图测试，并证明了没有超级图测试可以比完全图测试更好地执行。后来在（Samorodnitsky和Trevisan 2006）中，他们证明了，除了其他结果，没有非自适应线性测试可以比完全图测试更好地执行。他们的证明使用了高尔斯规范的代数机制。（Ben-Sasson，Harsha和Raskhodova 2005）的结果允许将此下限推广到自适应线性检验。我们还证明了自适应线性检验的最优下界，但我们的证明技术可以说比（Samorodnitsky和Trevisan 2006）中使用的证明技术更简单，更直接。我们还研究，像（Samorodnitsky和Trevisan 2006），二次函数的线性测试的行为。然而，我们并没有分析某些函数的Gowers范数，而是提供了一个更直接的组合证明，研究随机二次函数的线性测试行为。
英文摘要：
Linearity tests are randomized algorithms which have oracle access to the truth table of some function f, and are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP theorem, among other applications. The quality of a linearity test is described by its correctness c - the probability it accepts linear functions, its soundness s - the probability it accepts functions far from linear, and its query complexity q - the number of queries it makes. Linearity tests were studied in order to decrease the soundness of linearity tests, while keeping the query complexity small (for one reason, to improve PCP constructions). Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the Complete Graph Test, and prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove, among other results, that no non-adaptive linearity test can perform better than the Complete Graph Test. Their proof uses the algebraic machinery of the Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to generalize this lower bound also to adaptive linearity tests. We also prove the same optimal lower bound for adaptive linearity test, but our proof technique is arguably simpler and more direct than the one used in (Samorodnitsky and Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the behavior of linearity tests on quadratic functions. However, instead of analyzing the Gowers Norm of certain functions, we provide a more direct combinatorial proof, studying the behavior of linearity tests on random quadratic functions...
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