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 = 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 ...

• Type: Conference
• Year: 1994
• Where: COCO
• Authors: Steven M. Kautz, Peter Bro Miltersen
• Keywords: Algorithmically Random Sequence, Algorithms, Classical Measure Theory, COCO 1994, Resource-bounded Measure