# On sparse oracles separating feasible complexity classes

## Abstract

This note 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 relativization. Results of this type have the potential to yield deeper insights into the nature of relativization 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. In this note, we settle a central case. We fully characterize the sparse oracles separating NP from P in worlds where P=NP. We display related results about coNP, E, NE, coNE, and PSPACE.

## Keywords

Computation Path Kolmogorov Complexity Advice Function Oracle Query Polynomial Hierarchy## Preview

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