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

We investigate probabilistic space-bounded 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 boundederror randomized log-space machines that use only O(logn log n) random bits and make O(log n) passes over the random tape is in deterministic polynomial-time, then BPTIME(n) ⊆ DTIME(2). Here log n denotes log function applied k times iteratively. This result can be interpreted as follows: If we restrict the number of random bits from O(logn log(n)) to O(logn) in the above statement, then the corresponding set of languages is trivially in P. Further, it can be shown that (proof is given in the main body of the paper) if we restrict the number of passes from O(log n) to O(1), then the set of languages accepted is in P. Thus our result says that an extension of these simulations will lead to a non-trivial and unknown derandomization of BPTIME(n). Motivated by this result, we further investigate the power of multi-pass, probabilistic space-bounded machines and establish additional results.