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Using random sets as oracles

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by Denis R. Hirschfeldt , Frank Stephan , André Nies
Citations:32 - 13 self
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@ARTICLE{Hirschfeldt_usingrandom,
    author = {Denis R. Hirschfeldt and Frank Stephan and André Nies},
    title = {Using random sets as oracles},
    journal = {},
    year = {}
}

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Abstract

Let R be a notion of algorithmic randomness for individual subsets of N. We say B is a base for R randomness if there is a Z �T B such that Z is R random relative to B. We show that the bases for 1-randomness are exactly the K-trivial sets and discuss several consequences of this result. We also show that the bases for computable randomness include every ∆ 0 2 set that is not diagonally noncomputable, but no set of PA-degree. As a consequence, we conclude that an n-c.e. set is a base for computable randomness iff it is Turing incomplete. 1

Citations

411 Recursively Enumerable Sets and Degrees - Soare - 1987
294 The definition of random sequences - Martin-Löf
283 Classical Recursion Theory - Odifreddi - 1999
274 A theory of program size formally identical to information theory - Chaitin - 1975
264 Algorithmic Information Theory - Chaitin - 1987
133 Zufälligkeit und Wahrscheinlichkeit - Schnorr - 1971
123 Algorithmic randomness and complexity - Downey, Hirschfeldt - 2010
103 Π 0 1 classes and degrees of theories - Soare - 1972
93 An Introduction to Kolmogorov Complexity - Li, Vitanyi - 1997
68 Computability and Randomness - Nies - 2009
67 Lowness properties and randomness - Nies - 2005
58 Degrees of unsolvability - Sacks - 1963
55 Random sequences - Lambalgen - 1987
54 Draft of a paper (or series of papers) on Chaitin’s work. Unpublished notes - Solovay - 1975
38 Every sequence is reducible to a random one - Gács - 1986
34 Computability by probabilistic machines - Leeuw, Moore, et al. - 1956
32 Degrees of Unsolvability - Shoenfield - 1971
20 Degrees of unsolvability: a survey of results - Simpson - 1977
20 Relativizing Chaitin’s halting probability - Downey, Hirschfeldt, et al. - 2005
16 Π 0 1-classes and complete extensions of PA, in Recursion Theory Week (Oberwolfach - Measure - 1984
15 Recursively enumerable sets modulo iterated jumps and extensions of Arslanov’s completeness criterion - Jockusch, Lerman, et al. - 1989
15 and André Nies. Randomness and computability: open questions - Miller
12 The Degrees of Hyperimmune Sets, Zeitschrift fur Mathematische Logik und Grundlagen der - Miller, Martin - 1968
9 Calibrating randomness, The Bulletin of Symbolic Logic - Downey, Hirschfeldt, et al. - 2004
7 André Nies. Relativizing Chaitin’s halting probability - Downey, Hirschfeldt, et al.
7 Some theorems on the algorithmic approach to probability theory and information theory - LEVIN - 1971
6 Randomness and computability: open questions, The Bulletin Symbolic Logic 12 - Miller, Nies - 2006
5 and Nenad Mihailović, On the construction of effective random sets - Merkle
3 André Nies, and Sebastiaan A. Terwijn, Calibrating randomness, The Bulletin of Symbolic Logic 12 - Downey, Hirschfeldt - 2006
3 Turing incomparability in Scott sets, to appear - Kucera, Slaman
3 Martin-L"of random and PA-complete sets, Logic Colloquium 2002 - Stephan - 2002
3 Soare, \Pi 01 classes and degrees of theories - Robert - 1972
3 and Nenad Mihailovi'c, On the construction of effectively random sets - Merkle
3 Computability and Randomness, to appear. [22] Piergiorgio Odifreddi - Nies - 1989
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