Space-bounded hierarchies and probabilistic computations

Abstract

We study three aspects of the power of space-bounded probabilistic Turing machines. First, we give a simple alternative proof of Simon's result that space-bounded probabilistic complexity classes are closed under complement. Second, we demonstrate that any language recognizable by an alternating Turing machine in log n space with a constant number of alternations (the log n space “alternation hierarch”) also can be recognized by a log n spacebounded probabilistic Turing machine with small error probability; this is a generalization of Gill's result that any language in NSPACE (log n) can be recognized by such a machine. Third, we give a new definition of space-bounded oracle machines, and use it to define a space-bounded “oracle hierarchy” analogous to the original definition of the polynomial time hierarchy. Unlike its polynomial time analogue, the entire log n space “alternation hierarchy” is contained in the second level of the log n space “oracle hierarchy.” However, the entire log n space “oracle hierarchy”is still contained in bounded-error probabilistic space log n.