Two Sources Are Better than One for Increasing the Kolmogorov Complexity of Infinite Sequences

@article{Zimand2009TwoSA,
  title={Two Sources Are Better than One for Increasing the Kolmogorov Complexity of Infinite Sequences},
  author={Marius Zimand},
  journal={Theory of Computing Systems},
  year={2009},
  volume={46},
  pages={707-722}
}
  • Marius Zimand
  • Published 2009
  • Mathematics, Computer Science
  • Theory of Computing Systems
The randomness rate of an infinite binary sequence is characterized by the sequence of ratios between the Kolmogorov complexity and the length of the initial segments of the sequence. It is known that there is no effective procedure that transforms one input sequence into another sequence with higher randomness rate. By contrast, we display such a uniform effective procedure having as input two independent sequences with positive but arbitrarily small constant randomness rate. Moreover the… Expand
Extracting the Kolmogorov Complexity of Strings and Sequences from Sources with Limited Independence
TLDR
It is shown that from any two strings with sufficiently large Kolmogorov complexity and sufficiently small dependence, one can effectively construct a string that is random even conditioned by any one of the input strings. Expand
Possibilities and impossibilities in Kolmogorov complexity extraction
TLDR
The connection between extractors and Kolmogorov extractors is presented and the basic positive and negative results concerning Kolmogsorov complexity extraction are presented. Expand
Kolmogorov Complexity in Randomness Extraction
TLDR
It is shown that a computable function is an almost randomness extractor if and only if it is a Kolmogorov complexity extractor, thus establishing a fundamental equivalence between two forms of extraction studied in the literature: Kolmogsorov extraction and randomness extraction. Expand
Algorithmically independent sequences
TLDR
It is shown that there is no effective way of producing from an arbitrary sequence with positive constructive Hausdorff dimension two sequences that are independent (even in the weaker type of independence) and have super-logarithmic complexity. Expand
1 6 M ar 2 01 2 Space-Bounded Kolmogorov Extractors ⋆
An extractor is a function that receives some randomness and either “improves” it or produces “new” randomness. There are statistical and algorithmical specifications of this notion. We study anExpand
Extracting Kolmogorov complexity with applications to dimension zero-one laws
TLDR
The extraction procedure for space-bounded complexity is used to establish zero- one laws for the strong dimensions of complexity classes within ESPACE and the unbounded extraction procedure yields a zero-one law for the constructive strong dimensions for Turing degrees. Expand
Extracting information is hard: A Turing degree of non-integral effective Hausdorff dimension
Abstract We construct a Δ 2 0 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimensionExpand
Turing degrees of reals of positive effective packing dimension
TLDR
It is shown that the answer to Jan Reimann's question whether there is a Turing cone of broken dimension is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension, and a minimal degree of effective packing dimension is constructed. Expand
Counting Dependent and Independent Strings
  • Marius Zimand
  • Mathematics, Computer Science
  • Fundam. Informaticae
  • 2014
TLDR
For every n-bit string x with complexity Cx ≥ α + 7 log n, the set of n- bit strings that have α dependency with x has size at least 1/polyn2n-α. Expand
Randomness and Computation
This article examines work seeking to understand randomness using computational tools. The focus here will be how these studies interact with classical mathematics, and progress in the recent decade.Expand
...
1
2
...

References

SHOWING 1-10 OF 27 REFERENCES
Extracting the Kolmogorov Complexity of Strings and Sequences from Sources with Limited Independence
TLDR
It is shown that from any two strings with sufficiently large Kolmogorov complexity and sufficiently small dependence, one can effectively construct a string that is random even conditioned by any one of the input strings. Expand
Increasing Kolmogorov Complexity
TLDR
It is shown that any non-random string of length n can be increased by flipping bits and some strings require $\Omega(\sqrt{n})$ bit flips, and bounds for increasing the complexity of a string by flipping m bits are given. Expand
Extracting randomness using few independent sources
TLDR
This work gives the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2, using a variant of the "stepping-up lemma" used in establishing lower bound on the Ramsey number for hyper-graphs. Expand
Algorithmically independent sequences
TLDR
It is shown that there is no effective way of producing from an arbitrary sequence with positive constructive Hausdorff dimension two sequences that are independent (even in the weaker type of independence) and have super-logarithmic complexity. Expand
Randomness and Computability
This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus onExpand
A Generalization of Chaitin's Halting Probability \Omega and Halting Self-Similar Sets
TLDR
It is shown that the maximum value of the degree of randomness of each point in Euclidean space through its base-two expansion provides the Hausdorff dimension of a self-similar set that is computable in a certain sense. Expand
Algorithmic Information Theory and Kolmogorov Complexity
This document contains lecture notes of an introductory course on Kolmogorov complexity. They cover basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional,Expand
Extracting information is hard: A Turing degree of non-integral effective Hausdorff dimension
Abstract We construct a Δ 2 0 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimensionExpand
Dimension Extractors and Optimal Decompression
  • D. Doty
  • Mathematics, Computer Science
  • Theory of Computing Systems
  • 2007
TLDR
The Kučera-Gács theorem is examined from the perspective of decompression, by showing that every infinite sequence S is Turing reducible to a Martin-Löf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S, which is shown to be the optimal ratio of query bits to computed bits achievable with Turing reductions. Expand
Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws
TLDR
The extraction procedure for space-bounded complexity is used to establish zero-one laws for polynomial-space strong dimension and the exponential-time class E is either minimally complex or maximally complex within ESPACE. Expand
...
1
2
3
...