伪确定性算法与概率时间结构
Pseudodeterministic Algorithms and the Structure of Probabilistic Time
摘要：
我们将伪确定性算法的研究与$\mathsf{BPTIME}$结构复杂性的两个主要问题联系起来：证明层次定理和证明完整问题的存在。我们的主要贡献可归纳如下。1.我们建立在技术开发，以证明层次定理的概率时间与建议（Fortnow和Santhanam，FOCS 2004），以构建第一个无条件的伪随机生成器的多项式拉伸可计算的伪确定性多项式时间（与一位的建议），是安全的无限往往对多项式时间的计算。作为这一构造的应用，我们得到了关于素数生成和表示复杂性的新结果。2. Oliveira和Santhanam（STOC 2017）无条件地建立了电路接受概率问题（$\mathsf{CAPP}$）的伪确定性算法，该算法在次指数时间内运行，并且在无限多输入长度的电路上的任何可采样分布上具有高概率正确。我们表明，提高这个运行时间或获得的结果，持有每一个大的输入长度将意味着新的时间层次定理的概率时间。此外，我们证明了最坏情况下的多项式时间伪确定性算法$\mathsf{CAPP}$将意味着$\mathsf{BPP}$有完整的问题。3.我们建立了大$\mathsf{rKt}$复杂性字符串的伪确定性构造（Oliveira，ICALP 2019）与概率时间的强层次定理的存在性之间的等价性。更一般地说，这些结果提出了新的方法来设计搜索问题的伪确定性算法和揭示概率时间的结构。
英文摘要：
We connect the study of pseudodeterministic algorithms to two major open problems about the structural complexity of $\mathsf{BPTIME}$: proving hierarchy theorems and showing the existence of complete problems. Our main contributions can be summarised as follows. 1. We build on techniques developed to prove hierarchy theorems for probabilistic time with advice (Fortnow and Santhanam, FOCS 2004) to construct the first unconditional pseudorandom generator of polynomial stretch computable in pseudodeterministic polynomial time (with one bit of advice) that is secure infinitely often against polynomial-time computations. As an application of this construction, we obtain new results about the complexity of generating and representing prime numbers. 2. Oliveira and Santhanam (STOC 2017) established unconditionally that there is a pseudodeterministic algorithm for the Circuit Acceptance Probability Problem ($\mathsf{CAPP}$) that runs in sub-exponential time and is correct with high probability over any samplable distribution on circuits on infinitely many input lengths. We show that improving this running time or obtaining a result that holds for every large input length would imply new time hierarchy theorems for probabilistic time. In addition, we prove that a worst-case polynomial-time pseudodeterministic algorithm for $\mathsf{CAPP}$ would imply that $\mathsf{BPP}$ has complete problems. 3. We establish an equivalence between pseudodeterministic construction of strings of large $\mathsf{rKt}$ complexity (Oliveira, ICALP 2019) and the existence of strong hierarchy theorems for probabilistic time. More generally, these results suggest new approaches for designing pseudodeterministic algorithms for search problems and for unveiling the structure of probabilistic time.
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