Promise问题遭遇伪决定论
Promise Problems Meet Pseudodeterminism
摘要：
接受概率估计问题（APEP）是对布尔电路的接受概率进行加法近似。这个问题允许一个概率近似方案。一个中心问题是，我们是否可以设计一个伪确定性的近似算法这个问题：概率多项式时间算法，输出一个具有高概率的规范近似。最近，它表明，这样的算法将意味着每一个近似算法可以伪确定性（迪克森，Pavan，Vinodchandran; ITCS 2021）。这项工作的主要概念性贡献是建立一个伪确定性算法APEP的存在从根本上连接到概率承诺类和相应的标准复杂性类之间的关系。特别是，我们证明了以下等价性：PromiseBPP中的每个承诺问题在BPP中都有一个解决方案，当且仅当APEP有一个伪确定性算法。基于这种直觉，我们表明，APEP的伪确定性算法可以揭示在复杂性理论，如电路下界，概率层次定理，和多伪确定性的几个中心议题。
英文摘要：
The Acceptance Probability Estimation Problem (APEP) is to additively approximate the acceptance probability of a Boolean circuit. This problem admits a probabilistic approximation scheme. A central question is whether we can design a pseudodeterministic approximation algorithm for this problem: a probabilistic polynomial-time algorithm that outputs a canonical approximation with high probability. Recently, it was shown that such an algorithm would imply that every approximation algorithm can be made pseudodeterministic (Dixon, Pavan, Vinodchandran; ITCS 2021). The main conceptual contribution of this work is to establish that the existence of a pseudodeterministic algorithm for APEP is fundamentally connected to the relationship between probabilistic promise classes and the corresponding standard complexity classes. In particular, we show the following equivalence: every promise problem in PromiseBPP has a solution in BPP if and only if APEP has a pseudodeterministic algorithm. Based on this intuition, we show that pseudodeterministic algorithms for APEP can shed light on a few central topics in complexity theory such as circuit lowerbounds, probabilistic hierarchy theorems, and multi-pseudodeterminism.
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