201 research outputs found

    Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness

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    We correct Miyabe's proof of van Lambalgen's Theorem for truth-table Schnorr randomness (which we will call uniformly relative Schnorr randomness). An immediate corollary is one direction of van Lambalgen's theorem for Schnorr randomness. It has been claimed in the literature that this corollary (and the analogous result for computable randomness) is a "straightforward modification of the proof of van Lambalgen's Theorem." This is not so, and we point out why. We also point out an error in Miyabe's proof of van Lambalgen's Theorem for truth-table reducible randomness (which we will call uniformly relative computable randomness). While we do not fix the error, we do prove a weaker version of van Lambalgen's Theorem where each half is computably random uniformly relative to the other

    Asymptotic density and the Ershov hierarchy

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    We classify the asymptotic densities of the Δ20\Delta^0_2 sets according to their level in the Ershov hierarchy. In particular, it is shown that for n≥2n \geq 2, a real r∈[0,1]r \in [0,1] is the density of an nn-c.e.\ set if and only if it is a difference of left-Π20\Pi_2^0 reals. Further, we show that the densities of the ω\omega-c.e.\ sets coincide with the densities of the Δ20\Delta^0_2 sets, and there are ω\omega-c.e.\ sets whose density is not the density of an nn-c.e. set for any n∈ωn \in \omega.Comment: To appear in Mathematical Logic Quarterl

    Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes

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    We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume. Part I is dedicated to information theory and the mathematical formalization of randomness based on Kolmogorov complexity. This last application goes back to the 60's and 70's with the work of Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last years.Comment: 40 page

    Algorithmic Randomness for Infinite Time Register Machines

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    A concept of randomness for infinite time register machines (ITRMs), resembling Martin-L\"of-randomness, is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies computability and that an analogue of van Lambalgen's theorem holds

    Calibrating the complexity of Delta 2 sets via their changes

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    The computational complexity of a Delta 2 set will be calibrated by the amount of changes needed for any of its computable approximations. Firstly, we study Martin-Loef random sets, where we quantify the changes of initial segments. Secondly, we look at c.e. sets, where we quantify the overall amount of changes by obedience to cost functions. Finally, we combine the two settings. The discussions lead to three basic principles on how complexity and changes relate
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