We investigate the complexity of explicit construction problems, where the goal is to produce a particular object of size $n$ possessing some pseudorandom property in time polynomial in $n$. We give overwhelming evidence that ${\bf APEPP}$, defined originally by Kleinberg et al., is the natural complexity class associated with explicit constructions for objects whose existence follows from the probabilistic method, by proving that a host of well-studied explicit construction problems lie in this class. We then observe that a result of Je\v{r}\'{a}bek on provability in Bounded Arithmetic, when reinterpreted as a reduction between search problems, shows that constructing a truth table of high circuit complexity is complete for ${\bf APEPP}$ under ${\bf P}^{\bf NP}$ reductions. This demonstrates that constructing a hard truth table is a universal explicit construction problem in a concrete sense. This result in fact gives a precise algorithmic characterization of proving $2^{\Omega(n)}$ circuit lower bounds for ${\bf E}^{\bf NP}$: the complete problem for ${\bf APEPP}$ has a ${\bf P}^{\bf NP}$ algorithm if and only if such a lower bound holds. Together with our proof that pseudorandom generators can be constructed in ${\bf APEPP}$, this also yields a self-contained and significantly simplified proof of the celebrated result of Impagliazzo and Wigderson that worst-case-hard truth tables can be used to derandomize algorithms (although the conclusion is weaker as our derandomization requires an ${\bf NP}$ oracle). As another corollary of this completeness result, we show that ${\bf E}^{\bf NP}$ contains a language of circuit complexity $2^{\Omega(n)}$ if and only if it contains a language of circuit complexity $\frac{2^n}{3n}$. Finally, for several of the problems shown to lie in ${\bf APEPP}$, we demonstrate direct polynomial time reductions to the explicit construction of hard truth tables.

中文翻译：

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最难的显式构造

我们研究了显式构造问题的复杂性，其目标是生成一个大小为 $n$ 的特定对象，该对象在 $n$ 的时间多项式中具有一些伪随机属性。我们提供了压倒性的证据，证明 ${\bf APEPP}$ 最初由 Kleinberg 等人定义，是与存在遵循概率方法的对象的显式构造相关联的自然复杂性类，通过证明大量经过充分研究的显式施工问题出在这一类。然后我们观察到 Je\v{r}\'{a}bek 关于有界算术中的可证明性的结果，当重新解释为搜索问题之间的减少时，表明构建高电路复杂性的真值表对于 ${\ bf APEPP}$ 低于 ${\bf P}^{\bf NP}$ 减少。这表明构建硬真值表是一个具体意义上的普遍显式构建问题。这个结果实际上给出了证明 $2^{\Omega(n)}$ 电路下界 ${\bf E}^{\bf NP}$ 的精确算法特征：${\bf APEPP}$ 的完整问题有一个 ${\bf P}^{\bf NP}$ 算法当且仅当这样的下界成立。连同我们证明伪随机生成器可以在 ${\bf APEPP}$ 中构造的证明一起，这也产生了一个独立且显着简化的证明，证明了 Impagliazzo 和 Wigderson 的著名结果，即最坏情况硬真值表可用于去随机化算法（尽管结论较弱，因为我们的去随机化需要 ${\bf NP}$ oracle）。作为这种完整性结果的另一个推论，我们证明 ${\bf E}^{\bf NP}$ 包含电路复杂性 $2^{\Omega(n)}$ 的语言当且仅当它包含电路复杂性 $\frac{2^n {3n}$。最后，对于 ${\bf APEPP}$ 中显示的几个问题，我们证明了显式构造硬真值表的直接多项式时间减少。