We prove that every randomized Boolean function admits a supersimulator: a randomized polynomial-size circuit whose output on random inputs cannot be efficiently distinguished from reality with constant advantage, even by polynomially larger distinguishers. Our result builds on the landmark complexity-theoretic regularity lemma of Trevisan, Tulsiani and Vadhan (2009), which, in contrast, provides a simulator that fools smaller distinguishers. We circumvent lower bounds for the simulator size by letting the distinguisher size bound vary with the target function, while remaining below an absolute upper bound independent of the target function. This dependence on the target function arises naturally from our use of an iteration technique originating in the graph regularity literature. The simulators provided by the regularity lemma and recent refinements thereof, known as multiaccurate and multicalibrated predictors, respectively, as per Hebert-Johnson et al. (2018), have previously been shown to have myriad applications in complexity theory, cryptography, learning theory, and beyond. We first show that a recent multicalibration-based characterization of the computational indistinguishability of product distributions actually requires only (calibrated) multiaccuracy. We then show that supersimulators yield an even tighter result in this application domain, closing a complexity gap present in prior versions of the characterization.

翻译：我们证明每个随机布尔函数均存在一个超级模拟器：一个随机化的多项式规模电路，其随机输入上的输出无法以恒定优势被高效区分于真实结果，即使对抗规模多项式的区分器。该结果建立在 Trevisan、Tulsiani 和 Vadhan（2009）的标志性复杂性理论正则性引理基础上，然而该引理提供的是能欺骗较小规模区分器的模拟器。我们通过让区分器规模界随目标函数变化，同时保持低于独立于目标函数的绝对上界，从而规避了模拟器规模的下界限制。这种对目标函数的依赖性源于我们使用源自图正则性文献的迭代技术。正则性引理及其后续改进（即 Hebert-Johnson 等人（2018）提出的多精度与多校准预测器）所提供的模拟器先前已被证明在复杂性理论、密码学、学习理论等领域具有广泛的应用。我们首先证明，近期基于多校准的乘积分布计算不可区分性刻画实际上仅需（校准后的）多精度即可实现。随后展示超级模拟器在该应用领域中能产生更紧密的结果，填补了先前版本刻画中存在的复杂性差距。