  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.
