Results 1 - 10
of
22
Low for random reals and positive-measure domination
- Proceedings of the American Mathematical Society, 2007. Preprint
, 2005
"... Abstract. The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1. ..."
Abstract
-
Cited by 18 (0 self)
- Add to MetaCart
Abstract. The low for random reals are characterized topologically, as well as in terms of domination of Turing functionals on a set of positive measure. 1.
Uniform almost everywhere domination
- Journal of Symbolic Logic
, 2006
"... ABSTRACT. We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed t ..."
Abstract
-
Cited by 17 (0 self)
- Add to MetaCart
ABSTRACT. We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the prooftheoretic strength of the regularity of Lebesgue measure for G δ sets. Our constructions essentially settle the reverse mathematical classification of this principle. 1.
An extension of the recursively enumerable Turing degrees
- Journal of the London Mathematical Society
, 2006
"... Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overco ..."
Abstract
-
Cited by 16 (13 self)
- Add to MetaCart
Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. Although RT is known to be structurally rich, a major source of frustration is that no specific, natural degrees in RT have been discovered, except the bottom and top degrees, 0 and 0 ′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (also known as Muchnik degrees) of mass problems associated with non-empty Π 0 1 subsets of 2ω. It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here, d is the weak degree of the diagonally non-recursive functions, and rn is the weak degree of the n-random reals. It is known that r1 can be characterized as the maximum weak degree ofaΠ 0 1 subset of 2ω of positive measure. We now show that inf(r2, 1) can be characterized as the maximum weak degree of a Π 0 1 subset of 2ω, the Turing upward closure of which is of positive measure. We exhibit a natural embedding of RT into Pw which is one-to-one, preserves the semilattice structure of RT, carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2, 1) inPw. 1.
Almost everywhere domination and superhighness
- MATHEMATICAL LOGIC QUARTERLY
, 2007
"... 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)
Abstract
-
Cited by 11 (6 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 Kjos-Hanssen, Kjos-Hanssen/Miller/Solomon, and others concerning LR-reducibility 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 truth-table computable from B ′ , the Turing jump of B.
Mass problems and almost everywhere domination
- Mathematical Logic Quarterly
"... Mathematical Logic Quarterly, 53, 2007, pp. 483–492. 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 Martin-Löf random, respectively. Let b1, b2, b3 be the degrees of unsolvabil ..."
Abstract
-
Cited by 11 (6 self)
- Add to MetaCart
Mathematical Logic Quarterly, 53, 2007, pp. 483–492. 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 Martin-Löf 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.Let1 and 0 be the top and bottom elements of Pw. We show that inf(b1, 1) andinf(b2, 1) andinf(b3, 1) belongtoPw 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, weshow that inf(b1, 1) andinf(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 self-contained, we exposit the proofs of some recent
Some fundamental issues concerning degrees of unsolvability
- In [6], 2005. Preprint
, 2007
"... Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a ..."
Abstract
-
Cited by 8 (7 self)
- Add to MetaCart
Recall that RT is the upper semilattice of recursively enumerable Turing degrees. We consider two fundamental, classical, unresolved issues concerning RT. The first issue is to find a specific, natural, recursively enumerable Turing degree a ∈ RT which is> 0 and < 0 ′. The second issue is to find a “smallness property ” of an infinite, co-recursively enumerable set A ⊆ ω which ensures that the Turing degree deg T (A) = a ∈ RT is> 0 and < 0 ′. In order to address these issues, we embed RT into a slightly larger degree structure, Pw, which is much better behaved. Namely, Pw is the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of 2 ω. We define a specific, natural embedding of RT into Pw, and we present some recent and new research results.
Randomness, lowness and degrees
- J. of Symbolic Logic
, 2006
"... Abstract. We say that A ≤LR B if every B-random number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally a ..."
Abstract
-
Cited by 6 (2 self)
- Add to MetaCart
Abstract. We say that A ≤LR B if every B-random number is Arandom. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e. restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α is not GL2 the LR degree of α bounds 2 ℵ0 degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. 1.
A cappable almost everywhere dominating computably enumerable degree
- Electronic Notes in Theoretical Computer Science
, 2007
"... Abstract. We show that there exists an almost everywhere (a.e.) dominating computably enumerable (c.e.) degree which is half of a minimal pair. 1. ..."
Abstract
-
Cited by 5 (4 self)
- Add to MetaCart
Abstract. We show that there exists an almost everywhere (a.e.) dominating computably enumerable (c.e.) degree which is half of a minimal pair. 1.
RANDOMNESS NOTIONS AND PARTIAL RELATIVIZATION
"... Abstract. We study weak 2 randomness, weak randomness relative to ∅ ′ and Schnorr randomness relative to ∅ ′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the Martin-Löf random reals relative to A. We discuss the connections ..."
Abstract
-
Cited by 3 (0 self)
- Add to MetaCart
Abstract. We study weak 2 randomness, weak randomness relative to ∅ ′ and Schnorr randomness relative to ∅ ′. One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the Martin-Löf random reals relative to A. We discuss the connections with LR-reducibility and also study the reducibility associated with weak 2-randomness. 1.
Mass problems and measure-theoretic regularity
, 2009
"... Research supported by NSF grants DMS-0600823 and DMS-0652637. ..."
Abstract
-
Cited by 3 (3 self)
- Add to MetaCart
Research supported by NSF grants DMS-0600823 and DMS-0652637.
