Abstract. We introduce and study several notions of computability-theoretic reducibility between subsets of ω that are "robust" in the sense that if only partial information is available about the oracle, then partial information can be recovered about the output. These are motivated by reductions between Π 1 2 principles in the context of reverse mathematics, and also encompasses generic and coarse reducibilities, previously studied by Jockusch and Schupp [JS-2012].

A generic computation of a subset A of N consists of a a computation that correctly computes most of the bits of A, and which never incorrectly computes any bits of A, but which does not necessarily give an answer for every input. The motivation for this concept comes from group theory and complexity theory, but the purely recursion theoretic analysis proves to be interesting, and often counterintuitive. The primary result of this paper is that there are no minimal pairs for generic computability, answering a question of Jockusch and Schupp.

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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. 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.

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.