Saving Queries with Randomness

摘要

In this paper, we investigate the power of randomness to save a query to an NP-complete set. We show that the PSAT ∥ [k] ≤pm-complete language randomly reduces to a language in PSAT ∥ [k − 1] with a one-sided error probability of 1/⌈k/2⌉ or a two sided-error probability of 1/(k+1). Furthermore, we prove that these probability bounds are tight; i.e., they cannot be improved by 1/poly, unless PH collapses. We also obtain tight performance bounds for randomized reductions between nearby classes in the Boolean and bounded query hierarchies. These bounds provide probability thresholds for completeness under randomized reductions in these classes. Using these thresholds, we show that certain languages in the Boolean hierarchy which are not ≤pm-complete in some relativized worlds, nevertheless inherit many of the hardness properties associated with the ≤pm-complete languages. Finally, we explore the relationship between randomization and functions that are computable using bounded queries to SAT. For any function h(n) = O(log n), we show that there is a function f computable using h(n) nonadaptive queries to SAT, which cannot be computed correctly with probability 1/2 + 1/poly by any randomized machine which makes less than h(n) adaptive queries to any oracle, unless PH collapses.