Randomness, relativization and Turing degrees

@article{Nies2005RandomnessRA,
  title={Randomness, relativization and Turing degrees},
  author={Andr{\'e} Nies and Frank Stephan and Sebastiaan Terwijn},
  journal={Journal of Symbolic Logic},
  year={2005},
  volume={70},
  pages={515 - 535}
}
Abstract We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅(n − 1). We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) ≥ ∣x∣ − c. The ‘only if’ direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove… 
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References

SHOWING 1-10 OF 64 REFERENCES
Lowness properties and randomness
Abstract The set A is low for (Martin-Lof) randomness if each random set is already random relative to A . A is K -trivial if the prefix complexity K of each initial segment of A is minimal, namely ∀
The Kolmogorov complexity of random reals
TLDR
There are uncountably many K-degrees of random reals by proving that μ({β : β⩽ K α})=0 and as a corollary to the proof there is no largest K-degree.
On the Construction of Effective Random Sets
TLDR
The mentioned results on self- and autoreducibility complement work of Ebert, Merkle, and Vollmer, from which it follows that no Martin-Lof random set is Turing-autoreducible and that no rec-random set is truth-table autoreachable.
Trivial Reals
TLDR
TheConstruction of an H-trivial computably enumerable set is an easy, injury-free construction of an incomplete computablyumerable set and is related to other classes of “highly nonrandom” reals that have been previously studied.
Every 2-random real is Kolmogorov random
TLDR
It is proved that 2-random reals are Kolmogorov random, which provides a natural characterization of 2- randomness in terms of plain complexity.
Recursively Enumerable Sets Modulo Iterated Jumps and Extensions of Arslanov's Completeness Criterion
TLDR
In this paper the main concern is to study the degrees of functions with no fixed points, and considers both fixed points in the strict sense above and fixed points modulo various equivalence relations on recursively enumerable sets.
Computational randomness and lowness*
TLDR
It is proved that there are uncountably many sets that are low for the class of Schnorr random reals and it is shown that they all have Turing degree incomparable to 0′.
The Definition of Random Sequences
TLDR
It is shown that the random elements as defined by Kolmogorov possess all conceivable statistical properties of randomness and can equivalently be considered as the elements which withstand a certain universal stochasticity test.
On Relative Randomness
  • A. Kucera
  • Mathematics, Computer Science
    Ann. Pure Appl. Log.
  • 1993
TLDR
The aim of the paper is to answer a question whether there can be a nonrecursive set A having the property that there is a set B such that B is 1-random relative to A and simultaneously A is recursive in B.
Complexity Oscillations in Infinite Binary Sequences
We shall consider finite and infinite binary sequences obtained by tossing an ideal coin, failure and success being represented by 0 and 1, respectively. Let sn=x 1 + x 2 + ... + x , be the frequency
...
1
2
3
4
5
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