Relative to a Random Oracle, NP Is Not Small

AbstractResource-boundedmeasure as originated by Lutz is an extension of classical measure theory which provides a probabilistic means of describing the relative sizes of complexity classes. Lutz has proposed the hypothesis that NP does not havep-measure zero, meaning loosely that NP contains a non-negligible subset of exponential time. This hypothesis implies a strong separation of P from NP and is supported by a growing body of plausible consequences which are not known to follow from the weaker assertion P≠NP. It is shown in this paper that relative to a random oracle, NP does not havep-measure zero. The proof exploits the followingindependenceproperty of algorithmically random sequences: ifAis an algorithmically random sequence and a subsequenceA0is chosen by means of abounded Kolmogorov– Loveland place selection, then the sequenceA1of unselected bits is random relative toA0, i.e.,A0andA1are independent. A bounded Kolmogorov–Loveland place selection is a very general type of recursive selection rule which may be interpreted as the sequence of oracle queries of a time-bounded Turing machine, so the methods used may be applicable to other questions involving random oracles