AM的PCP表征
A PCP Characterization of AM
摘要：
我们介绍了一个2轮随机约束满足问题，并表明其近似版本是完整的（承诺版本）的复杂性类AM。这给出了AM的“PCP特征”，类似于NP的PCP定理。类似的刻画已经给出了更高层次的多项式层次，PSPACE，但是，我们认为，AM的结果可能是特别重要的尝试去随机化这个类。为了测试这一概念，我们提出了一些“随机优化假设”与我们的随机CSP（根据我们的结果）将意味着崩溃的AM结果。不幸的是，这些假设似乎过于强大，我们提出了反对它们的证据。在这个过程中，我们表明，如果NP中的某些语言对大小为2^{Omega（n）}的电路是平均困难的，那么存在一种特别优雅形式的平均困难优化问题。我们所有的证明都使用了一种强大的PCP形式，称为概率可检验的邻近证明，并展示了它们的多功能性。我们还使用已知的结果随机有效的健全性和硬度放大。特别是，我们必须使用的Impagliazzo-Wigderson生成器;我们的分析依赖于最近的一个扩展器行走的Schloff型定理。
英文摘要：
We introduce a 2-round stochastic constraint-satisfaction problem, and show that its approximation version is complete for (the promise version of) the complexity class AM. This gives a `PCP characterization' of AM analogous to the PCP Theorem for NP. Similar characterizations have been given for higher levels of the Polynomial Hierarchy, and for PSPACE; however, we suggest that the result for AM might be of particular significance for attempts to derandomize this class. To test this notion, we pose some `Randomized Optimization Hypotheses' related to our stochastic CSPs that (in light of our result) would imply collapse results for AM. Unfortunately, the hypotheses appear over-strong, and we present evidence against them. In the process we show that, if some language in NP is hard-on-average against circuits of size 2^{Omega(n)}, then there exist hard-on-average optimization problems of a particularly elegant form. All our proofs use a powerful form of PCPs known as Probabilistically Checkable Proofs of Proximity, and demonstrate their versatility. We also use known results on randomness-efficient soundness- and hardness-amplification. In particular, we make essential use of the Impagliazzo-Wigderson generator; our analysis relies on a recent Chernoff-type theorem for expander walks.
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