Almost Weakly 2-Generic Sets

There is a family of questions in relativized complexity theory|weak analogs of the Friedberg Jump-Inversion Theorem|that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper deenes aw2-generic sets, i.e., sets which meet every dense set of strings that is r.e. in some incomplete r.e. set. Aw2-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any set X there is an aw2-generic set G such that NP G \ co-NP G 6 6 P GX. (On the other hand, if G is 1-generic, then NP G \ co-NP G P GSAT , where SAT is the NP-complete Satissability problem 6].) This result runs counter to the fact that most nite extension constructions in complexity theory can be made eeective. These results imply that any nite extension construction that ensures any of the Friedberg analogs must be noneeective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of aw2-generic sets diier radically from those of 1-generic sets: every degree above 0 0 contains an aw2-generic set; no aw2-generic set exists below any incomplete r.e. set; there is an aw2-generic set which is the join of two Turing equivalent aw2-generic sets. Finally, a result of Shore is presented 30] which states that every degree above 0 0 is the jump of an aw2-generic degree.