正则ROBP及BPL之外计算模型的SC去随机化

中文摘要 AI 辅助

我们研究了正则只读分支程序（ROBPs）以及BPL之外计算模型的SC去随机化。对于长度为$n$、宽度为$w$且具有多个接受节点的正则ROBPs，我们取得了三项结果。1. 当$n \le w$时，我们展示了一个空间为$O(\log^2 n+\log w)$、误差为$1/\text{poly}(nw)$的SC去随机化。2. 当$n \ge w$时，我们展示了一个空间为$O(\log n \log w)$、误差为$1/\text{poly}(w)$的SC去随机化。3. 当$w=O(\log n)$时，我们展示了一个最优的$O(\log n)$空间去随机化，误差为$1/\text{poly}(w)$。我们进一步证明了BPL的两个超集可以在SC中计算。1. 对于具有双向访问随机带的概率对数空间图灵机，我们证明如果随机带的每个条目最多被访问常数次，则可以在SC中近似计算。2. 对于具有多项式大小栈的概率对数空间图灵机，即概率对数空间辅助下推机（AuxPDMs），我们证明如果压栈/弹栈/空闲栈操作的时间不依赖于随机性，则可以在SC中近似计算。第一个模型是Impagliazzo、Nisan和Wigderson（STOC'94）考虑的读多重性模型，他们证明了他们的INW生成器可以欺骗此类计算。对于第二个模型，我们指出它包含Apers和Edenhofer（CCC'25）考虑的用于区分BQL和BPL的候选语言。

英文摘要

We study SC derandomizations for regular read-once branching programs (ROBPs) and computation models beyond BPL. For regular ROBPs with length $n$, width $w$, and multiple accept nodes, we attain the following results. 1. When $n \le w$, we show an SC derandomization with space $O(\log^2 n+\log w)$ and two-sided error $1/\text{poly}(nw)$. 2. When $n \ge w$, we show an SC derandomization with space $O(\log n \log w)$ and two-sided error $1/\text{poly}(w)$. In addition, when $w=O(\log n)$, we attain an optimal $O(\log n)$ space derandomization with two-sided error $1/\text{poly}(w)$. 3. When $w \le 2^{O(\sqrt{\log n})}$, we show that reachability of regular ROBPs (i.e. derandmization of one-sided unbounded small error ROBPs) can be computed in SC. We also show that when regular ROBPs are powering, optimal deterministic logspace can be attained for computing reachability and for bounded two-sided error derandomization, when $w = n^{1/c}$ for some constant $c$. We further show that two super sets of BPL can be computed in SC. 1. For probabilistic logspace TMs with a two-way access random tape, we show that it can be approximated in SC if each entry of the random tape is accessed for at most a constant number of times. 2. For probabilistic logspace TMs with a polynomial size stack, i.e. probabilistic logspace Auxiliary Push-down Machines (AuxPDMs), we show that it can be approximated in SC if the timings of push/pop/idle stack operations do not depend on the randomness. The first model is the read-multiplicity model considered by Impagliazzo, Nisan, Wigderson (STOC'94), in which they show that their INW generator can fool such computations. For the second model, we indicate that it contains candidate languages separating BQL from BPL considered by Apers and Edenhofer (CCC'25).

发表机构

• Peking University（北京大学）

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