Computationally Limited Randomness.
Matei David, Phuong M. Nguyen, Periklis A. Papakonstantinou, Anastasios Sidiropoulos · 2011
Abstract: The starting point of this work is the basic question of whether there exists a formal and meaningful way to limit the computational power that a time bounded randomized Turing Machine can employ on its randomness. We attack this question using a fascinating connection between space and time bounded machines given by Cook [Coo71]: a Turing Machine S running in space s with access to an unbounded stack is equiv-alent to a Turing Machine T running in time 2O(s). We extend S with access to a read-only tape containing 2O(s) uniform random bits, and a usual error regime: one-sided or two-sided, and bounded or unbounded. We study the effect of placing a bound p on the number of passes S is allowed on its random tape. It follows from Cook’s results that: • If p = 1 (one-way access) and the error is one-sided unbounded, S is equivalent to deterministic T. • If p = ∞ (unrestricted access), S is equivalent to randomized T (with the same error). As our first two contributions, we completely resolve the case of unbounded error. We show that we cannot meaningfully interpolate between deterministic and randomized T by increasing p: • If p = 1 and the error is two-sided unbounded, S is still equivalent to deterministic T.