On Probabilistic Space-Bounded Machines with Multiple Access to Random Tape

Abstract

We investigate probabilistic space-bounded Turing machines that are allowed to make multiple passes over the random tape. As our main contribution, we establish a connection between derandomization of such probabilistic space-bounded classes to the derandomization of probabilistic time-bounded classes. Our main result is the following.

• For some integer $k>0$, if all the languages accepted by bounded-error randomized log-space machines that use $O(\log n \log ^{(k+3)} n)$ random bits and make $O(\log ^{(k)} n)$ passes over the random tape is in deterministic polynomial-time, then $\mathrm{ BPTIME}(n) \subseteq \mathrm{ DTIME}(2^{o(n)})$. Here $\log ^{(k)} n$ denotes $\log$ function applied k times iteratively.

This result can be interpreted as follows: If we restrict the number of random bits to $O(\log n)$ for the above randomized machines, then the corresponding set of languages is trivially known to be in P. Further, it can be shown that (proof is given in the main body of the paper) if we instead restrict the number of passes to only O(1) for the above randomized machines, then the set of languages accepted is in P. Thus our result implies that any non-trivial extension of these simulations will lead to a non-trivial and unknown derandomization of $\mathrm{ BPTIME}(n)$. Motivated by this result, we further investigate the power of multi-pass, probabilistic space-bounded machines and establish additional results.

D. Mandal and A. Pavan—Research Supported in part by NSF grants 0916797, 1421163.

N.V. Vinodchandran—Research Supported in part by NSF grants 0916525, 1422668.