On sparse oracles separating feasible complexity classes

Abstract

This article clarifies which oracles separate NP from P and which do not. In essence, we are changing our research paradigm from the study of which problems can be relativized in two conflicting ways to the study and characterization of the class of oracles achieving a specified relativation. Results of this type have the potential to yield deeper insights into the nature of relativation problems and focus our attention on new and interesting classes of languages. A complete and transparent characterization of oracles that separate NP from P would resolve the long-standing P = ?NP question. Here we settle a central case. We fully characterize the sparse oracles separating NP from P in worlds where P = NP. These separating oracles are exactly the non-self-printable sets. Equivalently, they are the sets of high self-referential Kolmogorov complexity. We prove related results about co-NP and PSPACE.