Results 1 
7 of
7
Mass problems and hyperarithmeticity
, 2006
"... A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of we ..."
Abstract

Cited by 24 (16 self)
 Add to MetaCart
A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α < ω CK 1 let hα be the weak degree of 0 (α) , the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p ∗ be the weak degree
Almost everywhere domination and superhighness
 Mathematical Logic Quarterly
"... Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almost ..."
Abstract

Cited by 17 (9 self)
 Add to MetaCart
Let ω denote the set of natural numbers. For functions f, g: ω → ω, we say that f is dominated by g if f(n) < g(n) for all but finitely many n ∈ ω. We consider the standard “fair coin ” probability measure on the space 2 ω of infinite sequences of 0’s and 1’s. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X ∈ 2 ω, each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of KjosHanssen, KjosHanssen/Miller/Solomon, and others concerning LRreducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0 ′′ is
Mass problems and almost everywhere domination
 Mathematical Logic Quarterly
, 2007
"... We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and MartinLöf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
We examine the concept of almost everywhere domination from the viewpoint of mass problems. Let AED and MLR be the set of reals which are almost everywhere dominating and MartinLöf random, respectively. Let b1, b2, b3 be the degrees of unsolvability of the mass problems associated with the sets AED, MLR×AED, MLR∩AED respectively. Let Pw be the lattice of degrees of unsolvability of mass problems associated with nonempty Π 0 1 subsets of 2 ω. Let 1 and 0 be the top and bottom elements of Pw. We show that inf(b1,1) and inf(b2,1) and inf(b3,1) belong to Pw and that 0 < inf(b1,1) < inf(b2,1) < inf(b3,1) < 1. Under the natural embedding of the recursively enumerable Turing degrees into Pw, we show that inf(b1,1) and inf(b3,1) but not inf(b2,1) are comparable with some recursively enumerable Turing degrees other than 0 and 0 ′. In order to make this paper more selfcontained, we exposit the proofs of some recent theorems due to Hirschfeldt, Miller, Nies, and Stephan.
Mass problems and measuretheoretic regularity
, 2009
"... Research supported by NSF grants DMS0600823 and DMS0652637. ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Research supported by NSF grants DMS0600823 and DMS0652637.
1 Bounded Limit Recursiveness
, 2007
"... Let X be a Turing oracle. A function f(n) issaidtobe boundedly limit recursive in X if it is the limit of an Xrecursive sequence of Xrecursive functions ˜f(n, s) such that the number of times ˜f(n, s) changes is bounded by a recursive function of n. Let us say that X is BLRlow if every function w ..."
Abstract
 Add to MetaCart
Let X be a Turing oracle. A function f(n) issaidtobe boundedly limit recursive in X if it is the limit of an Xrecursive sequence of Xrecursive functions ˜f(n, s) such that the number of times ˜f(n, s) changes is bounded by a recursive function of n. Let us say that X is BLRlow if every function which is boundedly limit recursive in X is boundedly limit recursive in 0. This is a lowness property in the sense of Nies. These notions were introduced by Joshua A. Cole and the speaker in a recently submitted paper on mass problems and hyperarithmeticity. The purpose of this talk is to compare BLRlowness to similar properties which have been considered in the recursiontheoretic literature. Among the properties discussed are: Ktriviality, superlowness, jumptraceability, weak jumptraceability, total ωrecursive enumerability, array recursiveness, array jumprecursiveness, and strong jumptraceability. 2 Definition. If X is a Turing oracle, let BLR(X) betheset of numbertheoretic functions f: ω → ω which are boundedly limit recursive in X. This means that there exist an Xrecursive approximating function ˜ f(n, s) and a recursive bounding function ̂ f(n) such that and for all n. f(n) = lims ˜ f(n, s) {s  ˜ f(n, s) ̸ = ˜ f(n, s +1)}  < ̂ f(n) In particular, BLR(0) = {f  f ≤ wtt 0 ′}. The BLR operator was introduced in Mass problems and hyperarithmeticity, by Joshua A. Cole and Stephen G. Simpson, 20 pages, submitted 2006 to JML. 3 Cole and Simpson used the BLR operator to construct a natural embedding of the hyperarithmetical hierarchy into P w. Namely, we proved that the Muchnik degrees inf(h ∗ α, 1) forα<ωCK 1 are distinct ∈Pw.
Mass Problems and Degrees of Unsolvability
, 2006
"... Recall that E T is the upper semilattice of recursively enumerable Turing degrees. Two basic, classical, unresolved issues concerning E T are: Issue 1: To find a specific, natural, r.e. Turing degree a ∈ E T which is> 0 and < 0 ′. Issue 2: To find a “smallness property ” of an infinite cor.e. set A ..."
Abstract
 Add to MetaCart
Recall that E T is the upper semilattice of recursively enumerable Turing degrees. Two basic, classical, unresolved issues concerning E T are: Issue 1: To find a specific, natural, r.e. Turing degree a ∈ E T which is> 0 and < 0 ′. Issue 2: To find a “smallness property ” of an infinite cor.e. set A ⊆ ω which insures that deg T(A) = a ∈ E T is> 0 and < 0 ′. These unresolved issues go back to Post’s 1944 paper, Recursively enumerable sets of positive integers and their decision problems. Mass Problems to the Rescue! We address Issues 1 and 2 by passing from decision problems to mass problems. 2 Outline of this talk: We embed E T into a slightly larger structure, Pw, which is much better behaved. In the Pw context, we obtain satisfactory, positive answers to Issues 1 and 2. What is this wonderful structure Pw? Briefly, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. In order to explain Pw, we must first explain: • mass problems, • weak degrees, and • nonempty Π 0 1 subsets of 2ω. 3 Mass problems (informal discussion): A “decision problem ” is the problem of deciding whether a given n ∈ ω belongs to a fixed set A ⊆ ω or not. To compare decision problems, we use Turing reducibility. A ≤ T B means that A can be computed using an oracle for B. A “mass problem ” is a problem with a not necessarily unique solution. (By contrast, a “decision problem ” has only one solution.) The “mass problem ” associated with a set P ⊆ ω ω is the “problem ” of computing an element of P. The “solutions ” of P are the elements of P. One mass problem is said to be “reducible” to another if, given any solution of the second problem, we can use it as an oracle to compute a solution of the first problem. 4 Rigorous definition: Let P and Q be subsets of ω ω. We view P and Q as mass problems. We say that P is weakly reducible to Q if (∀Y ∈ Q) (∃X ∈ P) (X ≤ T Y). This is abbreviated P ≤w Q.