The Isomorphism Conjecture Holds Relative to an Oracle

Abstract

The authors introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the Berman--Hartmanis isomorphism conjecture holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set $A$, all ${\bf NP}^A$-complete sets are polynomial-time isomorphic relative to $A$. Prior to this work, there were no known oracles relative to which the isomorphism conjecture held. As part of the proof that the isomorphism conjecture holds relative to symmetric perfect generic sets, it is also shown that ${\bf P^A} {\bf =} {\bf FewP^A}$ for any symmetric perfect generic $A$.