On the power of two-way random generators and the impossibility of deterministic poly-space simulation

Borodin, Cook, and Pippenger (Inform. and Control 58 (1983), 96–114) proved that both probabilistic acceptors and transducers working in space S(n) ⩾ log n can be simulated by deterministic machines in O(S(n)²) space. The definition of probabilistic computations uses one-way read-only random tape. Borodin et al. asked: “Is it possible to extend our simulation results to the case of a two-way read-only oracle head?” In the same vein Furst, Lipton, and Stockmeyer (Inform. and Control 64 (1985), 43–51) suggested that it could be a difference between two-way and one-way random tape: “…for space bounded probabilistic computations where the space bound is much less than the length of y, it could matter.” (y denoting the random tape inscription.) In this paper we give a full characterization of two-way random space classes that answers both questions. Karpinski and Verbeek (“There is no Polynomial Deterministic Space Simulation of Probabilistic Space with Two-way Random-Tape Generator,” Inform. and Control 67 (1985), 158–162) proved that there is no polynomial deterministic space simulation of two-way random space without giving any recursive upper bound. The results of this paper solved the open questions of Karpinski and Verbeek, 1985, Karpinski and Verbeek, 1985 and gave full characterization in sense of upper and lower deterministic space classes: they are proved precisely exponentially more powerful than the corresponding one-way classes.