In a recent work, Chen, Hoza, Lyu, Tal and Wu (FOCS 2023) showed an improved error reduction framework for the derandomization of regular read-once branching programs (ROBPs). Their result is based on a clever modification to the inverse Laplacian perspective of space-bounded derandomization, which was originally introduced by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford and Vadhan (FOCS 2020). In this work, we give an alternative error reduction framework for regular ROBPs. Our new framework is based on a binary recursive formula from the work of Chattopadhyay and Liao (CCC 2020), that they used to construct weighted pseudorandom generators (WPRGs) for general ROBPs. Based on our new error reduction framework, we give alternative proofs to the following results for regular ROBPs of length $n$ and width $w$, both of which were proved in the work of Chen et al. using their error reduction: $\bullet$ There is a WPRG with error $\varepsilon$ that has seed length $\tilde{O}(\log(n)(\sqrt{\log(1/\varepsilon)}+\log(w))+\log(1/\varepsilon)).$ $\bullet$ There is a (non-black-box) deterministic algorithm which estimates the expectation of any such program within error $\pm\varepsilon$ with space complexity $\tilde{O}(\log(nw)\cdot\log\log(1/\varepsilon)).$ (This was first proved in the work of Ahmadinejad et al., but the proof by Chen et al. is simpler.) Because of the binary recursive nature of our new framework, both of our proofs are based on a straightforward induction that is arguably simpler than the Laplacian-based proof in the work of Chen et al.

翻译：在近期的工作中，Chen、Hoza、Lyu、Tal与Wu（FOCS 2023）针对正则只读分支程序（ROBPs）的去随机化问题提出了改进的误差约化框架。该结果基于对空间有界去随机化中逆拉普拉斯视角（最初由Ahmadinejad、Kelner、Murtagh、Peebles、Sidford与Vadhan在FOCS 2020中引入）的巧妙修正。本文为正则ROBPs给出一种替代性误差约化框架。该新框架基于Chattopadhyay与Liao（CCC 2020）工作中用于为一般ROBPs构造加权伪随机生成器（WPRGs）的二元递归公式。基于新框架，我们为长度$n$、宽度$w$的正则ROBPs给出以下结果的替代性证明——这些结果均已在Chen等人的工作中通过其误差约化方法证明： $\bullet$ 存在一个误差为$\varepsilon$的WPRG，其种子长度为$\tilde{O}(\log(n)(\sqrt{\log(1/\varepsilon)}+\log(w))+\log(1/\varepsilon)).$ $\bullet$ 存在一种（非黑盒）确定性算法，可在空间复杂度$\tilde{O}(\log(nw)\cdot\log\log(1/\varepsilon))$内以误差$\pm\varepsilon$估计任意此类程序的期望值。（该结果最初由Ahmadinejad等人证明，但Chen等人的证明更为简洁。） 由于新框架的二元递归特性，两个证明均基于直截了当的归纳法，其简洁度可论证优于Chen等人工作中基于拉普拉斯算子的证明。