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 conjec
ture holds relative to any sp-generic oracle, i.e., for any symmetric
perfect generic set A, all NPA-complete sets are polynomial-time isomo
rphic relative to A. Prior to this work, there were no known oracles r
elative to which the isomorphism conjecture held. As part of the proof
that the isomorphism conjecture holds relative to symmetric perfect g
eneric sets, it is also shown that P-A=FewP(A) for any symmetric perfe
ct generic A.