One bit of advice

Abstract

The results in this paper show that coNP is contained in NP with 1 bit of advice (denoted NP/1) if and only if the Polynomial Hierarchy (PH) collapses to DP, the second level of the Boolean Hierarchy (BH). Previous work showed that BH ⊆ DP ⇒ coNP ⊆ NP/poly. The stronger assumption that PH ⊆ DP in the new result allows the length of the advice function to be reduced to a single bit and also makes the converse true. The one-bit case can be generalized to any constant k: PH ⊆ BH2k ⇔ coNP ⊆ NP/k where BH2k denotes the 2k-th level of BH and NP/k denotes the class NP with k-bit advice functions. © Springer-Verlag Berlin Heidelberg 2003.