There is no polynomial deterministic space simulation of probabilistic space with a two-way random-tape generator

Abstract

We prove there is no polynomial deterministic space simulation for two-way random-tape probabilistic space (Pr₂SPACE) (as defined in Borodin, A., Cook, S., and Pippenger, N. (1983) Inform. Control 58 113–136) for all functions f: ℕ → ℕ and all α ∈ ℕ, Pr₂SPACE(f(n))DSPACE(f(n)^α). This is the answer to the problem formulated in op cit., whether the deterministic squared-space simulation (for recognizers and transducers) generalizes to the two-way random-tape machine model. We prove, in fact, a stronger result saying that even space-bounded Las Vegas two-way random-tape algorithms (yielding always the correct answer and terminating with probability 1) are exponentially more efficient than the deterministic ones.