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Kolmogorov-Loveland Randomness and Stochasticity
[chapter]

2005
*
Lecture Notes in Computer Science
*

An infinite binary sequence X is

doi:10.1007/978-3-540-31856-9_35
fatcat:xhqu5agth5ewxd7wps34pzcmfq
*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. ...##
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Kolmogorov–Loveland randomness and stochasticity

2006
*
Annals of Pure and Applied Logic
*

The concept is named after

doi:10.1016/j.apal.2005.06.011
fatcat:4de2qsox65eb5cx2svnguy2wxy
*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 ...##
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Feasible reductions to kolmogorov-loveland stochastic sequences

1999
*
Theoretical Computer Science
*

For every binary sequence A, there is an infinite binary sequence S such that A <ft S

doi:10.1016/s0304-3975(99)00041-9
fatcat:c4vufjke5nfehg75paujnnlgli
*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*. ...##
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How much randomness is needed for statistics?

2014
*
Annals of Pure and Applied Logic
*

In this paper, we prove that this result no longer holds for other notions of

doi:10.1016/j.apal.2014.04.014
fatcat:os3jxjd6cndq7hcntfsqyhx3wi
*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". ...##
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How Much Randomness Is Needed for Statistics?
[chapter]

2012
*
Lecture Notes in Computer Science
*

In this paper, we prove that this result no longer holds for other notions of

doi:10.1007/978-3-642-30870-3_40
fatcat:i7ezgutmsnfnpn5m2coc7owmsq
*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". ...##
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An Algorithmic Complexity Interpretation of Lin's Third Law of Information Theory

2008
*
Entropy
*

Through the inequality that relates

doi:10.3390/entropy-e10010006
fatcat:c2zsot7amjhobfoqae5ombhg4e
*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). ...##
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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*”. ...##
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Polynomial clone reducibility

2013
*
Archive for Mathematical Logic
*

We also show that the same result holds if Kurtz

doi:10.1007/s00153-013-0351-x
fatcat:yjlq4ooxhnd4fdu324pcndufuu
*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*. ...##
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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.” ...

##
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The complexity of stochastic sequences

2008
*
Journal of computer and system sciences (Print)
*

We review

doi:10.1016/j.jcss.2007.06.018
fatcat:mx3t5w5s25ejvduduooqg72574
*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. ...##
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Algorithms and Randomness

1988
*
Theory of Probability and its Applications
*

The review articles [30]

doi:10.1137/1132060
fatcat:ttnurlwgerfrpgpbssi5ghiu7q
*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 ...##
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Mathematical Foundations of Randomness
[chapter]

2011
*
Philosophy of Statistics
*

Bandyopadhyay

doi:10.1016/b978-0-444-51862-0.50021-6
fatcat:7pik7bt4hzcureaxtkt4mjcdia
*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*...##
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Randomness
[article]

2001
*
arXiv
*
pre-print

to

arXiv:math/0110086v2
fatcat:jqstqwkyvrgh3onpviqq3k4iny
*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*. ...##
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Independence Properties of Algorithmically Random Sequences
[article]

2003
*
arXiv
*
pre-print

In this paper we show that if A is an algorithmically

arXiv:cs/0301013v1
fatcat:uszu3cdeevc6lpvsblkkbdlufa
*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. ...##
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Kolmogorov and mathematical logic

1992
*
Journal of Symbolic Logic (JSL)
*

In my life, in my personal experience, there were three such men,

doi:10.2307/2275276
fatcat:j5x5hcf4nrfz7jsk62mx2sp264
*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*". ...
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