Partial bi-immunity and NP-completeness

The Turing and many-one completeness notions for NP have been previously separated under measure, genericity, and bi-immunity hypotheses on NP. The proofs of all these results rely on the existence of a language in NP with almost everywhere hardness. In this paper we separate the same NP-completeness notions under a partial bi-immunity hypothesis that is weaker and only yields a language in NP that is hard to solve on most strings. This improves the results of Lutz and Mayordomo (1996), Ambos-Spies and Bentzien (2000), and Pavan and Selman (2002). The proof of this result is a significant departure from previous work.