Working with Strong Reducibilities above Totally Ω-c.e. and Array Computable Degrees
نویسنده
چکیده
We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allow us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every left-c.e. real of that degree is reducible in a computable Lipschitz way to a random left-c.e. real (an Ω-number).
منابع مشابه
Working with Strong Reducibilities above Totally Ω-c.e. Degrees
We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that ...
متن کاملA Survey of Results on the d-c.e. and n-c.e. Degrees
So a 1-c.e. set is simply a c.e. set, and a 2-c.e. set is a difference of two c.e. sets (also called a d.c.e. set). Putnam actually called the n-c.e. sets “n-trial and error predicates” (and did not require A0 = ∅). On the other hand, Gold [Go65], in a paper published in the same volume of the journal, defined “n-r.e.” to mean Σn (and is sometimes falsely credited with the above definition). Er...
متن کاملEvery nonzero c . e . strongly bounded Turing degree has the anti - cupping property ∗
The strongly bounded Turing reducibilities r = cl (computable Lipschitz reducibility) and r = ibT (identity bounded Turing reducibility) are defined in terms of Turing reductions where the use function is bounded by the identity function up to an additive constant and the identity function, respectively. We show that, for r = ibT, cl, every computably enumerable (c.e.) r-degree a > 0 has the an...
متن کاملTotally < ω-computably enumerable degrees and m-topped degrees
In this paper we will discuss recent work of the authors (Downey, Greenberg and Weber [8] and Downey and Greenberg [6, 7]) devoted to understanding some new naturally definable degree classes which capture the dynamics of various natural constructions arising from disparate areas of classical computability theory. It is quite rare in computability theory to find a single class of degrees which ...
متن کاملNon-isolated quasi-degrees
A set A ⊆ ω is called 2-computably enumerable (2-c.e.), if there are computably enumerable sets A1 and A2 such that A1 −A2 = A. A set A ⊆ ω is quasi-reducible to a set B ⊆ ω (A ≤Q B), if there is a computable function g such that for all x ∈ ω we have x ∈ A if and only if Wg(x) ⊆ B. This reducibility was introduced by Tennenbaum (see [1], p.207) as an example of a reducibility which differs fro...
متن کامل