The Generic Degrees of Density-1 Sets, and a Characterization of the Hyperarithmetic Reals

Abstract

A generic computation of a subset A of N is a computation which correctly computes most of the bits of A, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory, where it has been noticed that frequently, it is more important to know how difficult a type of problem is in the general case than how difficult it is in the worst case. When we study this concept from a recursion theoretic point of view, to create a transitive relationship, we are forced to consider oracles that sometimes fail to give answers when asked questions. Unfortunately, this makes working in the generic degrees quite difficult. Indeed, we show that generic reduction is Π 1 1 −complete. To help avoid this difficulty, we work with the generic degrees of density-1 reals. We demonstrate how an understanding of these degrees leads to a greater understanding of the overall structure of the generic degrees, and we also use these density-1 sets to provide a new a characterization of the hyperartithmetical Turing degrees.Proof. Any generic oracle for R(A) can be used uniformly to compute A, and so, in particular, to generically compute A.The study of how far down the Turing degrees go is, in some sense, the study of the quasi-minimal generic degrees.Definition 2.2. A nontrivial generic degree, b, is quasi-minimal if for every Turing degree a, if b ≥ g R(a), then a = 0.

Asymptotic Density and the Theory of Computability: A Partial Survey

“…The class of such sets coincides with the class of ∆ 1 1 sets, which are those sets which can be defined by a prenex formula of second-order arithmetic with all set quantifiers universal and also by a prenex formula with all set quantifiers existential. 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.…”

A Minimal Pair in the Generic Degrees

“…We now give the proof, followed by a few comments on coarse computability, further work of Igusa [8], and the notions of dense and effective dense computability studied in [1].…”

“…The sets X j built in the proof of Theorem 7 have density 1, which means that they are coarsely computable, and is also interesting in light of work of Igusa [8]: Astor, Hirschfeldt, and Jockusch [1] showed that the upper cone above any nontrivial (nonuniform or uniform) generic degree has measure 0, so a minimal generic degree would necessarily be half of a minimal pair. It is open whether there are minimal generic degrees, but Igusa [8] showed that the generic degree of a density-1 set cannot be minimal. Interestingly, he also showed that if a uniform generic degree does not bound a nontrivial uniform generic degree containing a density-1 set, then it is half of a minimal pair, but again it is not known whether such degrees exist.…”

Density-1-Bounding and Quasiminimality in the Generic Degrees

“…The results in this paper are motivated in large part by the following question, posed by the second author as Question 3 in [8].…”

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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.

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