Density-1-Bounding and Quasiminimality in the Generic Degrees

Abstract

We consider the question "Is every nonzero generic degree a density-1-bounding generic degree?" By previous results [8] either resolution of this question would answer an open question concerning the structure of the generic degrees: A positive result would prove that there are no minimal generic degrees, and a negative result would prove that there exist minimal pairs in the generic degrees.We consider several techniques for showing that the answer might be positive, and use those techniques to prove that a wide class of assumptions is sufficient to prove density-1-bounding.We also consider a historic difficulty in constructing a potential counterexample: By previous results [7] any generic degree that is not density-1-bounding must be quasiminimal, so in particular, any construction of a non-density-1-bounding generic degree must use a method that is able to construct a quasiminimal generic degree. However, all previously known examples of quasiminimal sets are also density-1, and so trivially density-1-bounding. We provide several examples of non-density-1 sets that are quasiminimal.Using cofinite and mod-finite reducibility, we extend our results to the uniform coarse degrees, and to the nonuniform generic degrees. We define all of the above terms, and we provide independent motivation for the study of each of them.Combined with a concurrently written paper of Hirschfeldt, Jockusch, Kuyper, and Schupp [4], this paper provides a characterization of the level of randomness required to ensure quasiminimality in the uniform and nonuniform coarse and generic degrees.

Dense computability, upper cones, and minimal pairs

“…As mentioned above, Cholak, Hirschfeldt, and Igusa (see [3]) showed that 1generic sets are quasiminimal for nonuniform generic reducibility, as are weakly 2-random sets. By forcing an effective dense oracle in place of their generic oracle, their methods also apply to nonuniform effective dense reducibility.…”

“…Unfortunately, this is not the case for dense reducibility, as there seems to be no good analog to the existence of τ in that proof. Indeed, the aforementioned proofs in [10] and [3] for the uniform case also seem to fail for dense reducibility. Thus we do not know whether defining dense reducibility using dense oracles and Turing functionals would yield the same notions.…”

Asymptotic Density and the Theory of Computability: A Partial Survey

“…For the uniform coarse degrees, this result was strengthened by independently motivated work by Cholak and Igusa [6] .…”

“…For the uniform coarse degrees, this result was strengthened by independently motivated work by Cholak and Igusa [6] . Astor, Hirschfeldt and Jockusch [3] introduced "dense computability" as a weakening of both generic and coarse computability.…”

“…Igusa [18] proves the following striking characterization. Cholak and Igusa [6] consider the question of whether or not every nonzero generic degree bounds a non-zero density-1 generic degree. By the results of [18] a positive answer would show that there are no minimal generic degrees and a negative answer would show that there are minimal pairs in the generic degrees.…”