Relative to a Random Oracle, NP is Not Small

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 6= 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 ∗Much of this author’s research was performed while visiting Iowa State University, supported by National Science Foundation Grant CCR-9157382, with matching funds from Rockwell International and Microware Systems Corporation. †Supported by a grant of the Danish Science Research Council and partially supported by the ESPRIT II Basic Research Actions Program of the European Community under contract No. 7141 (project ALCOM II).