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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
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.
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.
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.
On the Quantitative Structure of ...
, 2000
"... We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Comput ..."
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We analyze the quantitative structure of 0 2 . Among other things, we prove that a set is Turing complete if and only if its lower cone is nonnegligible, and that the sets of r.e.degree form a small subset of 0 2 . Mathematical Subject Classification: 03D15, 03D30, 28E15 Keywords: Computable measure theory, Turing degrees, completeness. 1 Introduction We study an eective measure theory suited for the study of 0 2 , the second level of the arithmetical hierarchy (alternatively, the sets computable relative to the halting problem K). This work may be seen as part of the constructivist tradition in mathematics as documented in [6]. The framework for eectivizing measure theory that we employ uses martingales. Martingales were rst applied to the study of random sequences by J. Ville [22]. Recursive martingales were studied in Schnorr [19], and became popular in complexity theory in more recent years through the work of Lutz [14, 15]. Lutz Research supported by a Ma...
Chaitin Omega Numbers and Strong Reducibilities
, 1997
"... 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. 1 Introdu ..."
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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. 1 Introduction Kucera [8] has used Arslanov's completeness criterion 1 to show that all random sets of r.e. Tdegree are in fact Tcomplete. Hence, every Chaitin # number is Tcomplete. In this paper we will strengthen this result by proving that every Chaitin # number is weak truthtable complete. However, no Chaitin # number can be ttcomplete as, because of a result stated by Bennett [1] (see Juedes, Lathrop, and Lutz [9] for a proof), there is no random sequence x such that K # tt x. 2 Notice that in this way we obtain a whole class of natural examples of wttcomplete but not ttcomplete r.e. sets (a fairly complicated construction of such a set was given by Lachlan [10]). # The first has...
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.
DOI: 10.1016/j.ic.2011.10.006
, 2011
"... Author manuscript, published in "Information and Computation (2011)" ..."
Π 0 1 Classes and pseudojump operators
, 2008
"... For a pseudojump V X and a Π 0 1 class P, we consider properties of the set {V X: X ∈ P}. We show that if P is Medvedev complete or if P has positive measure, and ∅ ′ ≤T C, then there exists X ∈ P with V X ≡T C. We examine the consequences when V X is Turing incomparable with V Y for X = Y in P an ..."
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For a pseudojump V X and a Π 0 1 class P, we consider properties of the set {V X: X ∈ P}. We show that if P is Medvedev complete or if P has positive measure, and ∅ ′ ≤T C, then there exists X ∈ P with V X ≡T C. We examine the consequences when V X is Turing incomparable with V Y for X = Y in P and when W X e = W Y e for all X, Y ∈ P. Finally, we give a characterization of the jump in terms of Π 0 1 classes.