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Feasible reductions to kolmogorov-loveland stochastic sequences

Jack H. Lutz, David L. Schweizer
1999 Theoretical Computer Science  
For every binary sequence A, there is an infinite binary sequence S such that A <ft S and S is stochastic in the sense of Kolmogorov and Loveland.  ...  Acknowledgement The second author gratefUlly acknowledges the hospitality of Dan Ashlock and the Iowa State University Department of Mathematics, where he was a visitor when this research was conducted  ...  The main theorem also implies that the class RAND of all random oracles cannot be replaced by the class KL-STOCH of all Kolmogorov-Loveland stochastic oracles in some known characterizations of complexity  ... 
doi:10.1016/s0304-3975(99)00041-9 fatcat:c4vufjke5nfehg75paujnnlgli

An Algorithmic Complexity Interpretation of Lin's Third Law of Information Theory

Joel Ratsaby
2008 Entropy  
Instead of static entropy we assert that the Kolmogorov complexity of a static structure such as a solid is the proper measure of disorder (or chaoticity).  ...  Through the inequality that relates stochasticity and chaoticity of random binary sequences we maintain that Lin's notion of stability corresponds to the stability of the frequency of 1s in the selected  ...  When subsequences selected by such a selection rule pass the unbiasness test they are called Kolmogorov-Loveland stochastic (KL-stochastic for short).  ... 
doi:10.3390/entropy-e10010006 fatcat:c2zsot7amjhobfoqae5ombhg4e

The complexity of stochastic sequences

Wolfgang Merkle
2008 Journal of computer and system sciences (Print)  
We review and slightly strengthen known results on the Kolmogorov complexity of prefixes of effectively random sequences.  ...  Furthermore, we demonstrate that there is no Mises-Wald-Church stochastic sequence such that all non-empty prefixes of the sequence have Kolmogorov complexity O(log n).  ...  Acknowledgments We thank Klaus Ambos-Spies, Nicolai Vereshchagin, and Paul Vitányi for helpful discussions and Jack Lutz, Alexander Shen, and the anonymous referees for their comments.  ... 
doi:10.1016/j.jcss.2007.06.018 fatcat:mx3t5w5s25ejvduduooqg72574

Kolmogorov and mathematical logic

Vladimir A. Uspensky
1992 Journal of Symbolic Logic (JSL)  
In my life, in my personal experience, there were three such men, and one of them was Andrei Nikolaevich Kolmogorov. I was lucky enough to be his immediate pupil.  ...  Kolmogorov was not only one of the greatest mathematicians of the twentieth century. By the width of his scientific interests and results he reminds one of the titans of the Renaissance.  ...  So the sequences that are random in von Mises', Church's and Kolmogorov's sense should be called respectively "stochastic", "Church stochastic" and "Kolmogorov stochastic".  ... 
doi:10.2307/2275276 fatcat:j5x5hcf4nrfz7jsk62mx2sp264

Randomness [article]

Paul M.B. Vitanyi
2001 arXiv   pre-print
to Kolmogorov Complexity and Its Applications" (M.  ...  Li and P. Vitanyi), 2nd Ed., Springer-Verlag, 1997.  ...  Kolmogorov and V.A. Uspensky, SIAM J. Theory Probab. Appl., 32(1987), 387-412]. N. Kolmogorov [Sankhyā, Ser. A, 25(1963), 369-376] and Donald William Loveland (1934-) [Z. Math. Logik Grundl.  ... 
arXiv:math/0110086v2 fatcat:jqstqwkyvrgh3onpviqq3k4iny

Page 5099 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
All infinite sequences R random in the sense of Martin-Lof have a property called Kolmogorov-Loveland (K-L) stochasticity: if a subsequence A of R is chosen according to a (very broadly defined) “admissible  ...  Feasible reductions to Kolmogoroy-Loveland stochastic sequences. (English summary) Theoret. Comput. Sci. 225 (1999), no. 1-2, 185-194.  ... 

Page 2001 of Mathematical Reviews Vol. 58, Issue 3 [page]

1979 Mathematical Reviews  
Kolmogorov, D. Loveland, P. Martin-Lof and R. J.  ...  Fine, Terrence L. 58 #13240 A computational complexity viewpoint on the stability of relative frequency and on stochastic independence.  ... 

Algorithms and Randomness

A. N. Kolmogorov, V. A. Uspenskii
1988 Theory of Probability and its Applications  
The review articles [30] and [31] refer to Church stochastic sequences as "Mises-Church random sequences" and to Kolmogorov stochastic sequences as "Mises-Kolmogorov-Loveland random sequences."  ...  It is not even known whether any subsequence formed by applying a Kolmogorov admissible rule to a Kolmogorov stochastic sequence is itself a Kolmogorov stochastic.  ... 
doi:10.1137/1132060 fatcat:ttnurlwgerfrpgpbssi5ghiu7q

Mathematical Foundations of Randomness [chapter]

Abhijit Dasgupta
2011 Philosophy of Statistics  
Bandyopadhyay and the anonymous referee for several useful suggestions.  ...  Thus Kolmogorov-Loveland randomness is strictly stronger than Kolmogorov-Loveland stochasticity, partial computable randomness is strictly stronger than Mises-Wald-Church stochasticity, and computable  ...  Kolmogorov-Loveland Partial Computably Computably Random (KLR) Random (PCR) Random (CR) Place selection Kolmogorov-Loveland Mises-Wald-Church Church Stochastic (KLS) Stochastic (MWCS) Stochastic  ... 
doi:10.1016/b978-0-444-51862-0.50021-6 fatcat:7pik7bt4hzcureaxtkt4mjcdia

Book Review: Kolmogorov complexity and algorithmic randomness

J. Maurice Rojas
2019 Bulletin of the American Mathematical Society  
Acknowledgments I am indebted to Manuel Blum, Qi Cheng, Eviatar Procaccia, Salil Vadhan, and David Zuckerman for their patience with, and answers to, my many naive questions.  ...  Kolmogorov complexity.  ...  The account of Kolmogorov complexity and randomness in [SUV] is masterful.  ... 
doi:10.1090/bull/1676 fatcat:qrtd3szbrnfvrgdwuzz5lszxua

Kolmogorov–Loveland randomness and stochasticity

Wolfgang Merkle, Joseph S. Miller, André Nies, Jan Reimann, Frank Stephan
2006 Annals of Pure and Applied Logic  
The concept is named after Kolmogorov (9) and Loveland (14) , who studied non-monotonic selection rules to define accordant stochasticity concepts, which we will describe later.  ...  nondecreasing, and unbounded function g and almost all n, the prefix of A of length n has prefix-free Kolmogorov complexity at least n − g(n).  ...  Acknowledgments We are grateful to Klaus Ambos-Spies, Rod Downey, Antonín Kučera, Stephen Lempp, Jack Lutz, Boris Ryabko, and Ted Slaman for helpful discussion. Bibliography  ... 
doi:10.1016/j.apal.2005.06.011 fatcat:4de2qsox65eb5cx2svnguy2wxy

Kolmogorov-Loveland Randomness and Stochasticity [chapter]

Wolfgang Merkle, Joseph Miller, André Nies, Jan Reimann, Frank Stephan
2005 Lecture Notes in Computer Science  
nondecreasing, and unbounded function g and almost all n, the prefix of A Introduction of length n has prefix-free Kolmogorov complexity of at least n − g(n).  ...  by a factor of α < 1 with respect to prefix-free Kolmogorov complexity.  ...  Acknowledgments We are grateful to Klaus Ambos-Spies, Rod Downey, Antonín Kučera, Stephen Lempp, Jack Lutz, Boris Ryabko, and Ted Slaman for helpful discussion.  ... 
doi:10.1007/978-3-540-31856-9_35 fatcat:xhqu5agth5ewxd7wps34pzcmfq

Andrei Nikolaevich Kolmogorov, 25 April 1903 - 20 October 1987

David George Kendall
1991 Biographical Memoirs of Fellows of the Royal Society  
It was in the Kolmogorov home at Tunoshna that ANK spent his earliest years.  ...  Andrei Nikolaevich is always known to us by the family name of his maternal grandfather Yakov Stepanovich Kolmogorov, a leading member of the Uglich nobility.  ...  The portrait photograph shows Kolmogorov lecturing his Moscow schoolchildren. R e f e r e n c e s  ... 
doi:10.1098/rsbm.1991.0015 fatcat:7t3rg7l2cre3pdyinufdyoiahu

The dimensions of individual strings and sequences

Jack H. Lutz
2003 Information and Computation  
The Kolmogorov complexity of a string is proven to be the product of its length and its dimension.  ...  This gives a new characterization of algorithmic information and a new proof of Mayordomo's recent theorem stating that the dimension of a sequence is the limit inferior of the average Kolmogorov complexity  ...  of his and Ryabko's earlier work on Hausdorff dimension, Kolmogorov complexity, and martingales.  ... 
doi:10.1016/s0890-5401(03)00187-1 fatcat:na6axs36qzchdo2eojhtgkr7t4

Random scattering of bits by prediction [article]

Joel Ratsaby
2010 arXiv   pre-print
Bad learners generate mistake sequences that are atypically complex or diverge stochastically from a purely random Bernoulli sequence.  ...  We obtain estimates of their error, algorithmic complexity and divergence from a purely random Bernoulli sequence.  ...  When subsequences selected by such a selection rule pass the unbiasedness test they are called Kolmogorov-Loveland stochastic (KL-stochastic for short).  ... 
arXiv:0909.3648v2 fatcat:m6sckhxd5ngidoftu2qxfp2dq4
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