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Calibrating randomness
 J. Symbolic Logic
"... 2. Sets, measure, and martingales 4 2.1. Sets and measure 4 2.2. Martingales 5 ..."
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Cited by 59 (34 self)
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2. Sets, measure, and martingales 4 2.1. Sets and measure 4 2.2. Martingales 5
Computational Randomness and Lowness
, 2001
"... . We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [5] on sets tha ..."
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Cited by 26 (1 self)
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. We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [5] on sets that are low for the class of MartinLof random reals. The Cantor space 2 ! is the set of infinite binary sequences; these are called reals and are identified with subsets of !. If oe 2 2 !! , that is, oe is a finite binary sequence, we denote by [oe] the set of reals that extend oe. These form a basis of clopen sets for the usual discrete topology on 2 ! . Write joej for the length of oe 2 2 !! . The Lebesgue measure on 2 ! is defined by stipulating that [oe] = 2 \Gammajoej . With every set U ` 2 !! we associate the open set S oe2U [oe]. When it is convenient, we confuse U with the open set associated to it, in particular we write U for the measure of the open set correspondi...
Some ComputabilityTheoretical Aspects of Reals and Randomness
 the Lect. Notes Log. 18, Assoc. for Symbol. Logic
, 2001
"... We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations. ..."
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Cited by 25 (7 self)
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We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.
On the Autoreducibility of Random Sequences
, 2001
"... A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition ..."
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Cited by 12 (1 self)
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A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truthtableautoreducible.
On Schnorr and computable randomness, martingales, and machines
 Mathematical Logic Quarterly
, 2004
"... examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines. ..."
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Cited by 10 (6 self)
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examine the randomness and triviality of reals using notions arising from martingales and prefixfree machines.
The KolmogorovLoveland stochastic sequences are not closed under selecting subsequences
 Journal of Symbolic Logic
, 2002
"... It is shown that the class of KolmogorovLoveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the KolmogorovLoveland stochastic sequences are closed unde ..."
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Cited by 9 (5 self)
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It is shown that the class of KolmogorovLoveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the KolmogorovLoveland stochastic sequences are closed under selecting subsequences by KolmogorovLoveland selection rules, i.e., by not necessarily monotonic partially computable selection rules. As a corollary, we obtain an easy proof for the previously known result that the KolmogorovLoveland stochastic sequences form a proper subclass of the MisesWaldChurch stochastic sequences.
Approximation representations for ∆2 reals
"... Abstract. We study ∆2 reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of ‘approximation representation ’ and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims ..."
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Cited by 2 (2 self)
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Abstract. We study ∆2 reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of ‘approximation representation ’ and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims at a classification of ∆2 reals based on approximation and it turns out to be quite different than the existing ones (based on information content etc.) 1.
RANDOMNESS – BEYOND LEBESGUE MEASURE
"... Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals wh ..."
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Much of the recent research on algorithmic randomness has focused on randomness for Lebesgue measure. While, from a computability theoretic point of view, the picture remains unchanged if one passes to arbitrary computable measures, interesting phenomena occur if one studies the the set of reals which are random for an arbitrary (continuous) probability measure or a generalized Hausdorff measure on Cantor space. This paper tries to give a survey of some of the research that has been done on randomness for nonLebesgue measures.
Low upper bounds of ideals
"... Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1. ..."
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Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1.