Results 1 
9 of
9
Degrees of random sets
, 1991
"... An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrando ..."
Abstract

Cited by 46 (4 self)
 Add to MetaCart
An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrandom and weakly nrandom sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of nrandomness and weak nrandomness including a new definition in terms of a forcing relation analogous to the characterization of ngeneric sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than nrandomness. Chapter III is concerned with intrinsic properties of nrandom sequences. The main results are that an (n + 1)random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that nrandom sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is nrandom then A is nrandom relative to B. It follows that any countable distributive lattice can be embedded
A splitting theorem for the Medvedev and Muchnik lattices
 Mathematical Logic Quarterly
, 2003
"... This is a contribution to the study of the Muchnik and Medvedev lattices of nonempty Π0 1 subsets of 2ω. In both these lattices, any nonminimum element can be split, i.e. it is the nontrivial join of two other elements. In fact, in the Medvedev case, if P>MQ, thenP can be split above Q. Both of t ..."
Abstract

Cited by 34 (1 self)
 Add to MetaCart
This is a contribution to the study of the Muchnik and Medvedev lattices of nonempty Π0 1 subsets of 2ω. In both these lattices, any nonminimum element can be split, i.e. it is the nontrivial join of two other elements. In fact, in the Medvedev case, if P>MQ, thenP can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible ∃theories. 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 25 (1 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 22 (16 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 nonempty 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 nonrecursive functions, and rn is the weak degree of the nrandom 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 onetoone, 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.
On a conjecture of Dobrinen and Simpson concerning almost everywhere domination
, 2005
"... Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the L ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Dobrinen and Simpson [4] introduced the notions of almost everywhere domination and uniform almost everywhere domination to study recursion theoretic analogues of results in set theory concerning domination in generic extensions of transitive models of ZFC and to study regularity properties of the Lebesgue measure on 2ω in reverse mathematics. In this article,
Computability, Complexity, Randomness
, 2008
"... Informally, mass problems are similar to decision problems. The difference is that, while a decision problem has only one solution, a mass problem is allowed to have more than one solution. Many concepts which apply to decision problems apply equally well to mass problems. For instance, a mass probl ..."
Abstract
 Add to MetaCart
Informally, mass problems are similar to decision problems. The difference is that, while a decision problem has only one solution, a mass problem is allowed to have more than one solution. Many concepts which apply to decision problems apply equally well to mass problems. For instance, a mass problem is said to be solvable if it has at least one computable solution. Also, one mass problem is said to be reducible to another mass problem if, given a solution of the second problem, we can use it to find a solution of the first problem. Many unsolvable mathematical problems are most naturally viewed as mass problems rather than decision problems. For example, let CPA be the problem of finding a completion of Peano Arithmetic. A wellknown theorem going back to Gödel and Tarski says that CPA is unsolvable, in the sense that there are no computable completions of Peano Arithmetic. In describing CPA
Cone avoidance and randomness preservation
"... Let X be an infinite sequence of 0’s and 1’s. Let f be a computable function. Recall that X is strongly frandom if and only if the a priori Kolmogorov complexity of each finite initial segment τ of X is bounded below by f(τ) minus a constant. We study the problem of finding a PAcomplete Turing orac ..."
Abstract
 Add to MetaCart
Let X be an infinite sequence of 0’s and 1’s. Let f be a computable function. Recall that X is strongly frandom if and only if the a priori Kolmogorov complexity of each finite initial segment τ of X is bounded below by f(τ) minus a constant. We study the problem of finding a PAcomplete Turing oracle which preserves the strong frandomness of X while avoiding a Turing cone. In the context of this problem, we prove that the cones which cannot always be avoided are precisely the Ktrivial ones. We also prove: (1) If f is convex and X is strongly frandom and Y is MartinLöf random relative to X, then X is strongly frandom relative to Y. (2) X is complex relative to some oracle if and only if X is random with respect to some continuous probability measure.
ALGORITHMIC RANDOMNESS AND MEASURES OF COMPLEXITY
"... Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress s ..."
Abstract
 Add to MetaCart
Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability. 1.