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SET THEORY
The study of sets is important and thus popular in the business and economic world for three major reasons: Basic understanding of concepts in sets and set algebra provides a form of logical languageExpand
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Randomness, relativization and Turing degrees
TLDR
We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) ≥ ∣x∣ − c. Expand
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Using random sets as oracles
AbstractLet R be a notion of algorithmic randomness for individual subsets of ℕ. A set 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 basesExpand
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Hausdorff dimension in exponential time
TLDR
In this paper we investigate effective versions of Hausdorff dimension which have been recently introduced by Lutz. Expand
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Trivial Reals
TLDR
We show that there are noncomputable reals α such that H(α n) 6 H(1n) + O(1) is prefix-free Kolmogorov complexity. Expand
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Lowness for the Class of Schnorr Random Reals
TLDR
We answer a question of Ambos-Spies and Kucera in the affirmative. Expand
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Language Learning from Texts: Mindchanges, Limited Memory, and Monotonicity
TLDR
We explore language learning in the limit under various constraints on the number of mindchanges, memory, and monotonicity. Expand
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Kolmogorov Complexity and the Recursion Theorem
TLDR
We introduce the concepts of complex and autocomplex sets, where a set A is complex if there is a recursive, nondecreasing and unbounded lower bound on the Kolmogorov complexity of the prefixes (of the characteristic sequence) of A, and Autocomplex is defined likewise with recursive replaced by A-recursive. Expand
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Schnorr trivial sets and truth-table reducibility
TLDR
We give several characterizations of Schnorr trivial sets, including a new lowness notion for Schnorr trivials based on truth-table reducibility. Expand
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