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Extracting Kolmogorov complexity with applications to dimension zeroone laws
 IN PROCEEDINGS OF THE 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING
, 2006
"... We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y ..."
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Cited by 20 (4 self)
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We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y) ? (1 \Gamma ffl)jyj. This result holds for both classical and spacebounded Kolmogorov complexity. We use the extraction procedure for spacebounded complexity to establish zeroone laws for polynomialspace strong dimension. Our results include: (i) If Dimpspace(E) ? 0, then Dimpspace(E=O(1)) = 1. (ii) Dim(E=O(1) j ESPACE) is either 0 or 1. (iii) Dim(E=poly j ESPACE) is either 0 or 1. In other words,
Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences
, 2007
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EXTRACTING THE KOLMOGOROV COMPLEXITY OF STRINGS AND SEQUENCES FROM SOURCES WITH LIMITED INDEPENDENCE
"... An infinite binary sequence has randomness rate σ if, for almost every n, the Kolmogorov complexity of its prefix of length n is at least σn. It is known that for every rational σ ∈ (0, 1), on one hand, there exists sequences with randomness rate σ that can not be effectively transformed into a sequ ..."
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Cited by 8 (5 self)
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An infinite binary sequence has randomness rate σ if, for almost every n, the Kolmogorov complexity of its prefix of length n is at least σn. It is known that for every rational σ ∈ (0, 1), on one hand, there exists sequences with randomness rate σ that can not be effectively transformed into a sequence with randomness rate higher than σ and, on the other hand, any two independent sequences with randomness rate σ can be transformed into a sequence with randomness rate higher than σ. We show that the latter result holds even if the two input sequences have linear dependency (which, informally speaking, means that all prefixes of length n of the two sequences have in common a constant fraction of their information). The similar problem is studied for finite strings. 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.
Constructive dimension and Turing degrees
"... This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) ..."
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Cited by 6 (0 self)
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This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) − ɛ, for arbitrary ɛ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1 − ɛ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dimH(S)/dimP(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truthtable and bounded Turing reductions differ in their ability to extract dimension.
TURING DEGREES OF REALS OF POSITIVE EFFECTIVE PACKING DIMENSION
"... Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show tha ..."
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Cited by 3 (2 self)
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Abstract. A relatively longstanding question in algorithmic randomness is Jan Reimann’s question whether there is a Turing cone of broken dimension. That is, is there a real A such that {B: B ≤T A} contains no 1random real, yet contains elements of nonzero effective Hausdorff Dimension? We show that the answer is affirmative if Hausdorff dimension is replaced by its inner analogue packing dimension. We construct a minimal degree of effective packing dimension 1. This leads us to examine the Turing degrees of reals with positive effective packing dimension. Unlike effective Hausdorff dimension, this is a notion of complexity which is shared by both random and sufficiently generic reals. We provide a characterization of the c.e. array noncomputable degrees in terms of effective packing dimension. 1.
Algorithmically Independent Sequences
, 2008
"... Two objects are independent if they do not affect each other. Independence is wellunderstood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite bi ..."
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Cited by 2 (2 self)
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Two objects are independent if they do not affect each other. Independence is wellunderstood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite binary sequences and studies their properties. Our two proposed notions of independence have some of the intuitive properties that one naturally expects. For example, for every sequence x, the set of sequences that are independent (in the weaker of the two senses) with x has measure one. For both notions of independence we investigate to what extent pairs of independent sequences, can be effectively constructed via Turing reductions (from one or more input sequences). In this respect, we prove several impossibility results. For example, 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 superlogarithmic complexity. Finally, a few conjectures and open questions are discussed.
Algorithmically Independent Sequences
, 802
"... Two objects are independent if they do not affect each other. Independence is wellunderstood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite bi ..."
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Two objects are independent if they do not affect each other. Independence is wellunderstood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite binary sequences and studies their properties. Our two proposed notions of independence have some of the intuitive properties that one naturally expects. For example, for every sequence x, the set of sequences that are independent (in the weaker of the two senses) with x has measure one. For both notions of independence we investigate to what extent pairs of independent sequences, can be effectively constructed via Turing reductions (from one or more input sequences). In this respect, we prove several impossibility results. For example, 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 superlogarithmic complexity. Finally, a few conjectures and open questions are discussed. 1
Université de Provence
, 2009
"... This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) ..."
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This paper examines the constructive Hausdorff and packing dimensions of Turing degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is Turing equivalent to a sequence R with dimH(R) ≥ (dimH(S)/dimP(S)) − ǫ, for arbitrary ǫ> 0. Furthermore, if dimP(S)> 0, then dimP(R) ≥ 1 − ǫ. The reduction thus serves as a randomness extractor that increases the algorithmic randomness of S, as measured by constructive dimension. A number of applications of this result shed new light on the constructive dimensions of Turing degrees. A lower bound of dimH(S)/dimP(S) is shown to hold for the Turing degree of any sequence S. A new proof is given of a previouslyknown zeroone law for the constructive packing dimension of Turing degrees. It is also shown that, for any regular sequence S (that is, dimH(S) = dimP(S)) such that dimH(S)> 0, the Turing degree of S has constructive Hausdorff and packing dimension equal to 1. Finally, it is shown that no single Turing reduction can be a universal constructive Hausdorff dimension extractor, and that bounded Turing reductions cannot extract constructive Hausdorff dimension. We also exhibit sequences on which weak truthtable and bounded Turing reductions differ in their ability to extract dimension. 1
www.stacsconf.org FOREWORD
"... The interest in STACS has remained at a high level over the past years. The STACS 2009 call for papers led to over 280 submissions from 41 countries. Each paper was assigned to three program committee members. The program committee held a twoweek electronic meeting at the beginning of November and ..."
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The interest in STACS has remained at a high level over the past years. The STACS 2009 call for papers led to over 280 submissions from 41 countries. Each paper was assigned to three program committee members. The program committee held a twoweek electronic meeting at the beginning of November and selected 54 papers. As cochairs of the program committee, we would like to sincerely thank its members and the many external referees for their valuable work. The overall very high quality of the submissions made the selection a difficult task. We would like to express our thanks to the three invited speakers, Monika Henzinger, JeanÉric Pin and Nicole Schweikardt, for their contributions to the proceedings. Special thanks are due to A. Voronkov for his EasyChair software (www.easychair.org). Moreover we would like to thank Sonja Lauer for preparing the conference proceedings and continuous help throughout the conference organization. For the second time this year’s STACS proceedings are published in electronic form. A printed version was also available at the conference, with ISBN 9783939897095. The electronic proceedings are available through several portals, and in particular through HAL and