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Kolmogorov-Loveland Randomness and Stochasticity [chapter]

Wolfgang Merkle, Joseph Miller, André Nies, Jan Reimann, Frank Stephan
2005 Lecture Notes in Computer Science  
An infinite binary sequence X is Kolmogorov-Loveland (or KL) random if there is no computable non-monotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while  ...  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).  ...  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

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.  ...  An infinite binary sequence against which no computable non-monotonic betting strategy succeeds is called Kolmogorov-Loveland random, or KL-random, for short.  ...  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

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  ...  and Komogorov-Loveland stochasticity.  ... 
doi:10.1016/s0304-3975(99)00041-9 fatcat:c4vufjke5nfehg75paujnnlgli

How much randomness is needed for statistics?

Bjørn Kjos-Hanssen, Antoine Taveneaux, Neil Thapen
2014 Annals of Pure and Applied Logic  
In this paper, we prove that this result no longer holds for other notions of randomness, namely computable randomness and stochasticity.  ...  The first author showed in 2010 that in the particular case where the notion of randomness considered is Martin-Löf randomness and the measure λ is a Bernoulli measure, classical randomness and Hippocratic  ...  Without his help and the university's support this paper would never exist. Taveneaux's research has been helped by a travel grant of the "Fondation Sciences Mathématiques de Paris".  ... 
doi:10.1016/j.apal.2014.04.014 fatcat:os3jxjd6cndq7hcntfsqyhx3wi

How Much Randomness Is Needed for Statistics? [chapter]

Bjørn Kjos-Hanssen, Antoine Taveneaux, Neil Thapen
2012 Lecture Notes in Computer Science  
In this paper, we prove that this result no longer holds for other notions of randomness, namely computable randomness and stochasticity.  ...  The first author showed in 2010 that in the particular case where the notion of randomness considered is Martin-Löf randomness and the measure λ is a Bernoulli measure, classical randomness and Hippocratic  ...  Without his help and the university's support this paper would never exist. Taveneaux's research has been helped by a travel grant of the "Fondation Sciences Mathématiques de Paris".  ... 
doi:10.1007/978-3-642-30870-3_40 fatcat:i7ezgutmsnfnpn5m2coc7owmsq

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

Joel Ratsaby
2008 Entropy  
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  ...  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).  ...  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

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  ...  It follows that there are sequences that are K-L stochastic but also strongly deep in the sense of Bennett, and thus are computationally “very far from random”.  ... 

Polynomial clone reducibility

Quinn Culver
2013 Archive for Mathematical Logic  
We also show that the same result holds if Kurtz random is replaced by Kolmogorov-Loveland stochastic.  ...  We show that if A is Kurtz random and C1 C2 are distinct polynomial clones, then there is a B that is C1-reducible to A but not C2-reducible to A.  ...  Then we achieve the same result via Kolmogorov-Loveland (KL) stochastic sequences, the class of which is incomparable with the class of Kurtz randoms.  ... 
doi:10.1007/s00153-013-0351-x fatcat:yjlq4ooxhnd4fdu324pcndufuu

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

1979 Mathematical Reviews  
Kolmogorov, D. Loveland, P. Martin-Lof and R. J.  ...  The vehicle for both these argu- ments will be a CC notion, especially as developed by Kolmogorov and Martin-Léf.”  ... 

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.  ...  This implies a sharp bound for the complexity of the prefixes of Mises-Wald-Church stochastic and of partial-recursively random sequences.  ...  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

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."  ...  Church stochastic sequences do not have this important property of randomness" Loveland constructed an example of a Church stochastic sequence in 10] which ceases to be Church stochastic after a certain  ... 
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

Randomness [article]

Paul M.B. Vitanyi
2001 arXiv   pre-print
to Kolmogorov Complexity and Its Applications" (M.  ...  Here we present in a single essay a combination and completion of the several aspects of the problem of randomness of individual objects which of necessity occur scattered in our texbook "An Introduction  ...  associated with stochastic randomness.  ... 
arXiv:math/0110086v2 fatcat:jqstqwkyvrgh3onpviqq3k4iny

Independence Properties of Algorithmically Random Sequences [article]

S. M. Kautz
2003 arXiv   pre-print
In this paper we show that if A is an algorithmically random sequence, A_0 is selected from A via a bounded Kolmogorov-Loveland selection rule, and A_1 denotes the sequence of nonselected bits of A, then  ...  A bounded Kolmogorov-Loveland selection rule is an adaptive strategy for recursively selecting a subsequence of an infinite binary sequence; such a subsequence may be interpreted as the query sequence  ...  Kolmogorov and (independently) Loveland (see [21] ) offered the generalization of a selection rule given in Definition 4.1 below.  ... 
arXiv:cs/0301013v1 fatcat:uszu3cdeevc6lpvsblkkbdlufa

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
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