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Computability and randomness
- A. Nies
- Computer Science
This book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.
Lowness properties and randomness
- A. Nies
- 20 October 2005
Randomness, relativization and Turing degrees
It is shown that the notions of Martin-Löfrandomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree.
Using random sets as oracles
It is shown that the bases for 1‐randomness are exactly the K‐trivial sets, and several consequences of this result are discussed.
Randomness and Computability: Open Questions
Automatic structures: richness and limitations
- B. Khoussainov, A. Nies, S. Rubin, F. Stephan
- MathematicsProceedings of the 19th Annual IEEE Symposium on…
- 13 July 2004
It is proven that the free Abelian group of infinite rank and many Fraisse limits do not have automatic presentations, and the complexity of the isomorphism problem for the class of all automatic structures is /spl Sigma//sub 1//sup 1/-complete.
Randomness and Differentiability
It is shown that a real z is ML random if and only if every computable function of bounded variation is differentiable at z, and similarly for absolutely continuous functions.
Relativizing Chaitin's Halting Probability
A comparison is drawn between the jump operator from computability theory and this Omega operator, which is a natural uniform way of producing an A-random real for every A ∈ 2ω, and many other interesting properties of Omega operators.
Demuth randomness and computational complexity
An analogy between cardinal characteristics and highness properties of oracles
We present an analogy between cardinal characteristics from set theory and highness properties from computability theory, which specify a sense in which a Turing oracle is computationally strong.…