  Assume that for every derandomization result for logspace algorithms, there
is a pseudorandom generator strong enough to nearly recover the derandomization
by iterating over all seeds and taking a majority vote. We prove under a
precise version of this assumption that $\mathbf{BPL} \subseteq \bigcap_{\alpha
> 0} \mathbf{DSPACE}(\log^{1 + \alpha} n)$.
  We strengthen the theorem to an equivalence by considering two
generalizations of the concept of a pseudorandom generator against logspace. A
targeted pseudorandom generator against logspace takes as input a short uniform
random seed and a finite automaton; it outputs a long bitstring that looks
random to that particular automaton. A simulation advice generator for logspace
stretches a small uniform random seed into a long advice string; the
requirement is that there is some logspace algorithm that, given a finite
automaton and this advice string, simulates the automaton reading a long
uniform random input. We prove that $\bigcap_{\alpha > 0}
\mathbf{promise\mbox{-}BPSPACE}(\log^{1 + \alpha} n) = \bigcap_{\alpha > 0}
\mathbf{promise\mbox{-}DSPACE}(\log^{1 + \alpha} n)$ if and only if for every
targeted pseudorandom generator against logspace, there is a simulation advice
generator for logspace with similar parameters.
  Finally, we observe that in a certain uniform setting (namely, if we only
worry about sequences of automata that can be generated in logspace), targeted
pseudorandom generators against logspace can be transformed into simulation
advice generators with similar parameters.
