Autoreducibility of NP-Complete Sets

We study the polynomial-time autoreducibility of NP-complete sets and obtain separations under strong hypotheses for NP. Assuming there is a p-generic set in NP, we show the following: • For every k ≥ 2, there is a k-T-complete set for NP that is k-T autoreducible, but is not k-tt autoreducible or (k − 1)-T autoreducible. • For every k ≥ 3, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k − 1)-tt autoreducible or (k − 2)-T autoreducible. • There is a tt-complete set for NP that is tt-autoreducible, but is not btt-autoreducible. Under the stronger assumption that there is a p-generic set in NP ∩ coNP, we show: • For every k ≥ 2, there is a k-tt-complete set for NP that is k-tt autoreducible, but is not (k − 1)-T autoreducible. Our proofs are based on constructions from separating NP-completeness notions. For example, the construction of a 2-T-complete set for NP that is not 2-tt-complete also separates 2-Tautoreducibility from 2-tt-autoreducibility.