201 research outputs found
Van Lambalgen's Theorem for uniformly relative Schnorr and computable randomness
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
We classify the asymptotic densities of the sets according to
their level in the Ershov hierarchy. In particular, it is shown that for , a real is the density of an -c.e.\ set if and only if
it is a difference of left- reals. Further, we show that the densities
of the -c.e.\ sets coincide with the densities of the
sets, and there are -c.e.\ sets whose density is not the density of an
-c.e. set for any .Comment: To appear in Mathematical Logic Quarterl
Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes
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
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
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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