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A Constructive Version of Birkhoff’s Ergodic Theorem for MartinLöf Random Points,” arXiv: 1007.5249 [math.DS
"... Abstract. We prove the effective version of Birkhoff’s ergodic theorem for MartinLöf random points and effectively open sets, improving the results previously obtained in this direction (in particular those of V. Vyugin, Nandakumar and Hoyrup, Rojas). The proof consists of two steps. First, we pr ..."
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Abstract. We prove the effective version of Birkhoff’s ergodic theorem for MartinLöf random points and effectively open sets, improving the results previously obtained in this direction (in particular those of V. Vyugin, Nandakumar and Hoyrup, Rojas). The proof consists of two steps. First, we prove a generalization of Kučera’s theorem that is a special case of effective ergodic theorem: a trajectory of a computable ergodic mapping that starts from a random point cannot remain inside an effectively open set of measure less than 1. Second, we show that the full statement of the effective ergodic theorem can be reduced to this special case. Both steps use the statement of classical ergodic theorem but not its proof, so we get a new simple proof of the effective ergodic theorem (with weaker assumptions than before). This result was recently obtained independently by Franklin, Greenberg, Miller and Ng. 1
Π 0 1 CLASSES WITH COMPLEX ELEMENTS.
"... Abstract. An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π 0 1 class P contains a complex element if and only if it contains a wttcover for the Cantor set. That is, if and only if for every real ..."
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Abstract. An infinite binary sequence is complex if the Kolmogorov complexity of its initial segments is bounded below by a computable function. We prove that a Π 0 1 class P contains a complex element if and only if it contains a wttcover for the Cantor set. That is, if and only if for every real Y there is an X in the P such that X �wtt Y. We show that this is also equivalent to the Π 0 1 class’s being large in some sense. We give an example of how this result can be used in the study of scattered linear orders. §1. Introduction. There has been interest in the literature over many years in studying various notions of the size of subclasses of 2 ω. In this paper we have tried to generalise and consolidate some of these ideas. We investigate a notion of size that has appeared independently in [1] and [5], namely the notion of a computable perfect class (computably growing in [5] and nonuphi in [1]). It is
Vitali’s theorem and WWKL
 Archive for Mathematical Logic
"... Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several wellknown measuretheoretic propositions ..."
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Abstract. Continuing the investigations of X. Yu and others, we study the role of set existence axioms in classical Lebesgue measure theory. We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA0. We show that several wellknown measuretheoretic propositions including the Vitali Covering Theorem are equivalent to WWKL over RCA0. 1.
WORKING WITH STRONG REDUCIBILITIES ABOVE TOTALLY ωC.E. DEGREES
"... Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω ..."
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Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ωc.e. iff every set in it is weak truthtable reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every leftc.e. real of that degree is reducible in a computable Lipschitz way to a random leftc.e. real (an Ωnumber). 1.
Algorithmic Randomness and Computability
"... Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. ..."
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Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.
Indifferent Sets
, 2008
"... We define the notion of indifferent set with respect to a given class of {0, 1}sequences. Roughly, for a set A in the class, a set of natural numbers I is indifferent for A with respect to the class if it does not matter how we change A at the positions in I: the new sequence continues to be in the ..."
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We define the notion of indifferent set with respect to a given class of {0, 1}sequences. Roughly, for a set A in the class, a set of natural numbers I is indifferent for A with respect to the class if it does not matter how we change A at the positions in I: the new sequence continues to be in the given class. We are especially interested in studying those sets that are indifferent with respect to classes containing different types of stochastic sequences. For the class of MartinLöf random sequences, we show that every random sequence has an infinite indifferent set and that there is no universal indifferent set. We show that indifferent sets must be sparse, in fact sparse enough to decide the halting problem. We prove the existence of coc.e. indifferent sets, including a coc.e. set that is indifferent for every 2random sequence with respect to the class of random sequences. For the class of absolutely normal numbers, we show that there are computable indifferent sets with respect to that class and we conclude that there is an absolutely normal real number in every nontrivial manyone degree.
Some results on effective randomness (Preliminary version September 2003)
"... Classification: Computational and structural complexity (random sequences, effective measure). Abstract. We investigate the characterizations of effective randomness in terms of MartinLöf covers and martingales. First, we address a question of AmbosSpies and Kučera [1], who asked for a characteriz ..."
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Classification: Computational and structural complexity (random sequences, effective measure). Abstract. We investigate the characterizations of effective randomness in terms of MartinLöf covers and martingales. First, we address a question of AmbosSpies and Kučera [1], who asked for a characterization of computable randomness in terms of covers. We argue that computable randomness randomness can be characterized in term of MartinLöf covers and effective mass distributions on Cantor space. Second, we show that the class of MartinLöf random sets coincides with the class of sets of reals that are random with respect to computable martingale processes. This improves on results of Hitchcock and Lutz [14], who showed that the later class is contained in the class of MartinLöf random sets and is a strict superset of the class of recrandom sets. Third, we analyze the sequence of measures of sets in a universal MartinLöf test. Kučera and Slaman [17] showed that any set which appears as the component of a universal MartinLöf test has measure which is MartinLöf random. Further, since the sets in a MartinLöf test are uniformly computably enumerable, so is their sequence of measures. We prove an an exact converse and hence a characterization. We show that if α0, α1,... is a uniformly computably enumerable sequence such that for each i, αi is MartinLöf random and less than 2 −i, then there is a universal MartinLöf test M0, M1,... such that for each i, Mi has measure αi. 1
Some Results on Effective Randomness
, 2003
"... We investigate the characterizations of effective randomness in terms of MartinLöf covers and martingales. First, we address a question of AmbosSpies and Kucera [1], who asked for a characterization of computable randomness in terms of covers. We argue that computable... ..."
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We investigate the characterizations of effective randomness in terms of MartinLöf covers and martingales. First, we address a question of AmbosSpies and Kucera [1], who asked for a characterization of computable randomness in terms of covers. We argue that computable...
Chaitin Numbers and Strong Reducibilities 1
"... Abstract: We prove that any Chaitin number (i.e., the halting probability of a universal selfdelimiting Turing machine) is wttcomplete, but not ttcomplete. In this way we obtain a whole class of natural examples of wttcomplete but not ttcomplete r.e. sets. The proof is direct and elementary. Ke ..."
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Abstract: We prove that any Chaitin number (i.e., the halting probability of a universal selfdelimiting Turing machine) is wttcomplete, but not ttcomplete. In this way we obtain a whole class of natural examples of wttcomplete but not ttcomplete r.e. sets. The proof is direct and elementary. Key Words: Chaitin number, wttcomplete r.e. set, ttcomplete r.e. set Category: F.1