Daniel Turetsky
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View article: LIMIT COMPLEXITIES, MINIMAL DESCRIPTIONS, AND <i>n</i>-RANDOMNESS
LIMIT COMPLEXITIES, MINIMAL DESCRIPTIONS, AND <i>n</i>-RANDOMNESS Open
Let K denote prefix-free Kolmogorov complexity, and let $K^A$ denote it relative to an oracle A . We show that for any n , $K^{\emptyset ^{(n)}}$ is definable purely in terms of the unrelativized notion K . It was already known that 2-rand…
View article: Normality, Relativization, and Randomness
Normality, Relativization, and Randomness Open
Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of …
View article: Limit Complexities, Minimal Descriptions, and $n$-Randomness
Limit Complexities, Minimal Descriptions, and $n$-Randomness Open
Let $K$ denote prefix-free Kolmogorov Complexity, and $K^A$ denote it relative to an oracle $A$. We show that for any $n$, $K^{\emptyset^{(n)}}$ is definable purely in terms of the unrelativized notion $K$. It was already known that 2-rand…
View article: Computable classifications of continuous, transducer, and regular functions
Computable classifications of continuous, transducer, and regular functions Open
We develop a systematic algorithmic framework that unites global and local classification problems using index sets. We prove that the classification problem for continuous (binary) regular functions among almost everywhere linear, pointwi…
View article: More on bases of uncountable free abelian groups
More on bases of uncountable free abelian groups Open
We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) su…
View article: More on bases of uncountable free abelian groups
More on bases of uncountable free abelian groups Open
We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) su…
View article: Realizing Computably Enumerable Degrees in Separating Classes
Realizing Computably Enumerable Degrees in Separating Classes Open
We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $Π^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c}, \mathbf{0}…
View article: Effectively closed subgroups of the infinite symmetric group
Effectively closed subgroups of the infinite symmetric group Open
We apply methods of computable structure theory to study effectively closed subgroups of . The main result of the paper says that there exists an effectively closed presentation of which is not the automorphism group of any computable str…
View article: Two More Characterizations of K-Triviality
Two More Characterizations of K-Triviality Open
We give two new characterizations of $K$ -triviality. We show that if for all $Y$ such that $\\Omega$ is $Y$ -random, $\\Omega$ is $(Y\\oplusA)$ -random, then $A$ is $K$ -trivial. The other direction was proved by Stephan and Yu, giving us…
View article: Martin-L\"of reducibility and cost functions
Martin-L\"of reducibility and cost functions Open
Martin-L\\"of (ML)-reducibility compares $K$-trivial sets by examining the\nMartin-L\\"of random sequences that compute them. We show that every $K$-trivial\nset is computable from a c.e.\\ set of the same ML-degree. We investigate the\nin…
View article: Martin-Löf reducibility and cost functions
Martin-Löf reducibility and cost functions Open
Martin-Löf (ML)-reducibility compares $K$-trivial sets by examining the Martin-Löf random sequences that compute them. We show that every $K$-trivial set is computable from a c.e.\ set of the same ML-degree. We investigate the interplay be…
View article: TWO MORE CHARACTERIZATIONS OF K-TRIVIALITY
TWO MORE CHARACTERIZATIONS OF K-TRIVIALITY Open
International audience
View article: Partial functions and domination
Partial functions and domination Open
The current work introduces the notion of pdominant sets and studies their recursion-theoretic properties. Here a set A is called pdominant iff there is a partial A-recursive function {\psi} such that for every partial recursive function {…