最难的显式结构
The Hardest Explicit Construction
摘要：
我们研究了显式构造问题的复杂性，其目标是产生一个特定的对象的大小$n$拥有一些伪随机性质的时间多项式在$n$。我们给出了压倒性的证据表明，最初由Kleinberg等人定义的$\bf{APEPP}$，是与对象的显式构造相关联的自然复杂性类，其存在遵循概率方法，通过将各种这样的构造问题放在该类中。然后，我们证明，结果Jeábek证明有界算术，当重新解释为减少搜索问题之间，表明构建一个高电路复杂性的真值表是完整的$\bf{APEPP}$下$\bf{P}^{\bf{NP}}$减少。这说明香农关于硬布尔函数存在性的经典证明实际上是一个$\textit{universal}$概率存在性论证：去随机化他的证明意味着概率方法的一般去随机化。作为推论，我们证明了$\bf{EXP}^{\bf{NP}}$包含电路复杂性语言$2^{n^{Ω(1)}}$当且仅当它包含电路复杂性语言$\frac{2^n}{2n}$。最后，对于几个问题表明在于$\bf{APEPP}$，我们证明直接多项式时间减少明确建设硬真值表。
英文摘要：
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 of objects whose existence follows from the probabilistic method, by placing a variety of such construction problems in this class. We then demonstrate that a result of Jeřábek 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 illustrates that Shannon's classical proof of the existence of hard boolean functions is in fact a $\textit{universal}$ probabilistic existence argument: derandomizing his proof implies a generic derandomization of the probabilistic method. As a corollary, we prove that $\bf{EXP}^{\bf{NP}}$ contains a language of circuit complexity $2^{n^{Ω(1)}}$ if and only if it contains a language of circuit complexity $\frac{2^n}{2n}$. 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.
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