Iterated relative recursive enumerability

Iterated relative recursive enumerability

A result of Soare and Stob asserts that for any non-recursive r.e. set C, there exists a r.e.[C] set A such that A • C is not of r.e. degree. A set Y is called [of] m R E A (m-REA[C]) [degree] iff it is [Turing equivalent to] the result of applying m-many iterated 'hops ' to the empty set (to C), where a hop is any function of the form X ~ X 9 W x . The cited result is the special case m = 0, n...

Iterated Relative Recursive Enummmerability

Recursive Enumerability

One of the fundamental contributions of mathematical logic has been the precise definition and study of algorithms and the closely associated study of recursively enumerable sets. A subset A £ co is recursive (decidable) if there is an algorithm for computing its characteristic function cA and recursively enumerable (r.e.) if there is an algorithm for generating its members. Nonrecursive r.e. s...

Domination, forcing, array nonrecursiveness and relative recursive enumerability

We present some abstract theorems showing how domination properties equivalent to not being GL2 or array recursive can be used to construct sets generic for di¤erent notions of forcing. These theorems are then applied to give simple proofs of several old results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang and Yu [2009] that every degree above any not in GL...

Randomness and Recursive Enumerability

One recursively enumerable real α dominates another one β if there are nondecreasing recursive sequences of rational numbers (a[i] : i ∈ ω) approximating α and (b[i] : i ∈ ω) approximating β and a positive constant C such that for all n, C(α−a[n]) ≥ (β−b[n]). See [Solovay, 1975] and [Chaitin, 1977]. We show every recursively enumerable random real dominates all other recursively enumerable real...