Relative to a random oracle, NP is not small

Abstract

Resource-bounded measure 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 have p-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 have p-measure zero. The proof exploits the following independence property of algorithmically random sequences: if A is an algorithmically random sequence and a subsequence A0 is chosen by means of a bounded Kolmogorov-Loveland place selection, then the sequence A1 of unselected bits is random relative to A0, i.e., A0 and A1 are 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. © 1996 Academic Press, Inc.