A polynomial-time construction of a hitting set for read-once branching programs of width 3

Abstract

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 ε > 5/6. We announced this result at conferences almost 10 years ago, including only proof sketches, which motivated a plenty 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.

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“…The class of small-depth Boolean circuits (AC 0 ) is a class of central interest in unconditional derandomization, and has been the subject of intensive research in this area over the past 30 years [6,49,57,58,54,53,45,72,74,13,62,21,46,29,2,1,69,51,31,35,34,73,33,70,36,24]. This highly successful line of work on derandomizing AC 0 has generated a wealth of ideas and techniques that have become mainstays in the field of pseudorandomness.…”