Generic oracles and oracle classes

Abstract

In this paper, we examine various complexity issues relative to an oracle for a generic set in order to determine which are the more "natural" conjectures for these issues. Generic oracle results should be viewed as parallels to random oracle results, as in [BG]; the two are in many ways related, but, as we shall exhibit, not equivalent. Looking at computation relative to a generic oracle is in some ways a better reflection of computation without an oracle; for example, whereas adding a random oracle allows a deterministic polynomial-time machine to solve any problem in BPP, adding a generic oracle will not help solve any recursive problem faster than it could be solved without an oracle. Generic sets were first introduced by Cohen as a tool for proving independence results in set theory [Co]. Their recursion theoretic properties have also been explored in depth; for example, see [J] and [Ku2]. Some related work using forcing and/or generic sets as tools in oracle constructions can be found in [Ku3], [Do], [P], and [A-SFH]. However, this is to our knowledge the first knowledge the first thorough examination of complexity relative to a generic Oracle.