Recently, an interest in constructing pseudorandom or hitting set generators for restricted branching programs has increased, which is motivated by the fundamental issue of derandomizing space-bounded computations. Such constructions have been known only in the case of width 2 and in very restricted cases of bounded width. In this paper, we characterize the hitting sets for read-once branching programs of width 3 by a so-called richness condition. Namely, we show that such sets hit the class of read-once conjunctions of DNF and CNF (i.e. the weak richness). Moreover, we prove that any rich set extended with all strings within Hamming distance of 3 is a hitting set for read-once branching programs of width 3. Then, we show that any almost $O(\log n)$-wise independent set satisfies the richness condition. By using such a set due to Alon et al. (1992) our result provides an explicit polynomial-time construction of a hitting set for read-once branching programs of width 3 with acceptance probability $\varepsilon>5/6$. We announced this result at conferences more than ten years ago, including only proof sketches, which motivated a number of subsequent results on pseudorandom generators for restricted read-once branching programs. This paper contains our original detailed proof that has not been published yet.
翻译:最近,对为限制分支方案建造假随机或撞击固定发源器的兴趣有所增加,其动机是空间限制计算中解密的根本问题。这种构造只有在宽度2和宽度限制极有限的情况下才为人所知。在本文中,我们将宽度3的重新分支方案的触碰装置定性为所谓的丰富性条件。也就是说,我们显示,这些装置击中了DNF和CNF的阅读连接(即弱富度)的类别。此外,我们证明,在3号Hamming距离内所有字符串扩展的任何富集都是宽度3号的触发器。然后,我们显示,任何接近O(log n) 和明智的独立设置都符合丰富性条件。我们的结果是,使用与Alon et al.(1992年) 的这种设置,我们的结果为宽度3号的重新连接程序提供了明确的多时制式设置的打击装置,接受概率为$\vareplon>5/6$。我们在10年前的限值会议上宣布了这个结果,而现在的原始版本是一张硬文件的原始版本。