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Kolmogorov Complexity in Randomness Extraction

2011
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ACM Transactions on Computation Theory
*

We clarify the role of

doi:10.1145/2003685.2003686
fatcat:h23gnhro4nacrpvpvnylf633bq
*Kolmogorov**complexity**in*the area of*randomness**extraction*. ... studied*in*the literature:*Kolmogorov**extraction*and*randomness**extraction*. ... This restriction allows us to effectively cycle through all nice*Kolmogorov**Complexity**in**Randomness**Extraction*1:5 distributions. ...##
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Possibilities and impossibilities in Kolmogorov complexity extraction
[article]

2012
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arXiv
*
pre-print

We present the connection between extractors and

arXiv:1104.0872v2
fatcat:n74urvnpnvfaxkyu66eixdn4ta
*Kolmogorov*extractors and the basic positive and negative results concerning*Kolmogorov**complexity**extraction*. ...*Randomness**extraction*is the process of constructing a source of*randomness*of high quality from one or several sources of*randomness*of lower quality. ...*In*Section 3 we discuss*Kolmogorov**complexity**extraction*from finite strings (setting 2), and*in*Section 4 we discuss*Kolmogorov**complexity**extraction*from infinite sequences (setting 3). ...##
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Kolmogorov complexity: Recent research in Moscow
[chapter]

1996
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Lecture Notes in Computer Science
*

Let us say only that most basic questions connected with

doi:10.1007/3-540-61550-4_145
fatcat:fpraul7ur5dgrecoheujjwx3um
*Kolmogorov**complexity*were solved*in*seventies; the main achievement, probably, was the definition of*randomness*for individual*random*sequence ... Introduction The*Kolmogorov**complexity*theory emerged*in*sixties; main definitions were independently found by Ray Solomonoff, A.N.*Kolmogorov*and G. Chaitin. ...##
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Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov extractors
[article]

2011
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arXiv
*
pre-print

We prove a strong Symmetry of Information relation for

arXiv:1103.5669v1
fatcat:fh45vyhumfgs5pioak5qpdqrim
*random*strings (*in*the sense of*Kolmogorov**complexity*) and establish tight bounds on the amount on nonuniformity that is necessary for*extracting*... ) amount of advice regarding the source is not enough for*extracting*a string with*randomness*rate 1 from a source string with constant*random*rate, \omega(1) amount of advice is. ...*In*the second model, a source is a string x ∈ {0, 1} n and its*randomness*is given by its*Kolmogorov**complexity*C(x). ...##
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Nonuniform Kolmogorov extractors
[article]

2012
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arXiv
*
pre-print

We establish tight bounds on the amount on nonuniformity that is necessary for

arXiv:1204.6696v1
fatcat:swfpfse3ungrbi57u7vnshyh5a
*extracting*a string with*randomness*rate 1 from a single source of*randomness*with lower*randomness*rate. ... More precisely, as instantiations of more general results, we show that while O(1) amount of advice regarding the source is not enough for*extracting*a string with*randomness*rate 1 from a source string ... It is desirable to have m ≈ k (i.e., to*extract*all, or almost all, of the*randomness**in*the source). The problem of*randomness**extraction*has been modeled*in*two ways. ...##
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Impossibility of Independence Amplification in Kolmogorov Complexity Theory
[chapter]

2010
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Lecture Notes in Computer Science
*

The paper studies

doi:10.1007/978-3-642-15155-2_61
fatcat:5f6ywkzmlrcpzbvdgc675nb6x4
*randomness**extraction*from sources with bounded independence and the issue of independence amplification of sources, using the framework of*Kolmogorov**complexity*. ... The dependency of strings x and y is dep(x,y) = {C(x) - C(x | y), C(y) - C(y| x)}, where C(·) denotes the*Kolmogorov**complexity*. ... It shows that there exists a string with relatively high*Kolmogorov**complexity*, so that all functions computable with a given amount of advice fail to*extract*its*randomness*. ...##
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Algorithmic Information Theory Using Kolmogorov Complexity

2012
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Journal of Applied & Computational Mathematics
*

Citation: Meng NK (2012) Algorithmic Information Theory Using

doi:10.4172/2168-9679.1000e106
fatcat:nsj356iwmnh5tpdezt2cjq5me4
*Kolmogorov**Complexity*. J Applied Computat Mathemat 1:e106. ... The main issue*in*translating theory to application is the fact that*Kolmogorov**complexity*is non-computable. ... This information will clearly pass any reasonable statistical test for*randomness*, but its stochastic nature prevents any useful information to be*extracted*. ...##
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Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws
[chapter]

2006
*
Lecture Notes in Computer Science
*

We apply recent results on

doi:10.1007/11786986_30
fatcat:y2yfcqfdufgr5fxde6fygl5kua
*extracting**randomness*from independent sources to "*extract*"*Kolmogorov**complexity*. ... This result holds for both classical and space-bounded*Kolmogorov**complexity*. ... This result about*extracting**Kolmogorov*-*randomness*also holds for polynomial-space bounded*Kolmogorov**complexity*. We apply this to obtain some zero-one laws for the dimensions of*complexity*classes. ...##
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Extracting Kolmogorov complexity with applications to dimension zero-one laws

2011
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Information and Computation
*

We apply recent results on

doi:10.1016/j.ic.2010.09.006
fatcat:7btnxoslzvhpnkdwgwu5fr47zm
*extracting**randomness*from independent sources to "*extract*"*Kolmogorov**complexity*. ... This result holds for both classical and space-bounded*Kolmogorov**complexity*. ... This result about*extracting**Kolmogorov*-*randomness*also holds for polynomial-space bounded*Kolmogorov**complexity*. We apply this to obtain some zero-one laws for the dimensions of*complexity*classes. ...##
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Computability, Complexity and Randomness (Dagstuhl Seminar 12021)

2012
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Dagstuhl Reports
*

The Dagstuhl seminar 12021 "Computability,

doi:10.4230/dagrep.2.1.19
dblp:journals/dagstuhl-reports/BecherBDM12
fatcat:bkapirz4vfgkzcexiutm33k74q
*Complexity*and*Randomness*" was aimed to meet people and ideas*in*these areas to share new results and discuss open problems. ... Research on the notions of information and*randomness*has drawn on methods and ideas from computability theory and cumputational*complexity*, as well as core mathematical subjects like measure theory and ... Recently it has been shown that the converse direction also holds and*Kolmogorov**extraction*is*in*fact equivalent to*randomness**extraction*. ...##
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Symmetry of Information: A Closer Look
[chapter]

2012
*
Lecture Notes in Computer Science
*

It was shown

doi:10.1007/978-3-642-27654-5_18
fatcat:dh22yxt6wbcntdtpkduudy3zgi
*in*[Zim11] that*in*the case of strings with simple*complexity*(that is the*Kolmogorov**complexity*of their*Kolmogorov**complexity*is small), the relevant information can be packed*in*a very ... This result implies a van Lambalgentype theorem for finite strings and plain*complexity*: If x is c-*random*and y is c-*random*relative to x, then xy is O(c)-*random*. ...*complexity**extraction*and*randomness**extraction*. ...##
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Symmetry of Information: A Closer Look
[article]

2012
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arXiv
*
pre-print

It was shown

arXiv:1206.5184v1
fatcat:2hlc625npvhg3igx5wwdfdnglm
*in*[Zim11] that*in*the case of strings with simple*complexity*(that is the*Kolmogorov**complexity*of their*Kolmogorov**complexity*is small), the relevant information can be packed*in*a very ... This result implies a van Lambalgen- type theorem for finite strings and plain*complexity*: If x is c-*random*and y is c-*random*relative to x, then xy is O(c)-*random*. ...*complexity**extraction*and*randomness**extraction*. ...##
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27 Open Problems in Kolmogorov Complexity
[article]

2022
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arXiv
*
pre-print

The paper proposes open problems

arXiv:2203.15109v1
fatcat:2mjciipal5amvlhld3w6u2dzv4
*in*classical*Kolmogorov**complexity*. Each problem is presented with background information and thus the article also surveys some recent studies*in*the area. ... references*in*the surveys [37, 49] ), and recently*in*the framework of*Kolmogorov**complexity*[45] ). ... Martin-Löf defined*randomness**in*1966 for infinite sequences;*in*1970s Schnorr and Levin established connections between*randomness*and*complexity*and now the theory of algorithmic*randomness*is a well ...##
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Nearly optimal language compression using extractors
[chapter]

1998
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Lecture Notes in Computer Science
*

This extends work of Sipser Sip83] and Buhrman and Fortnow BF97]. { We use extractors to

doi:10.1007/bfb0028551
fatcat:yypx2ijjtva3nbsxbsd3xwhscy
*extract*the*randomness*of strings. ... We show two sets of results applying the theory of extractors to resource-bounded*Kolmogorov**complexity*: { Most strings*in*easy sets have nearly optimal polynomial-time CD*complexity*. ... Randomly*extracting*CD*complexity*Another trade-o we obtain to save a log D term is to choose a counterpart y to a string x*in*a set*in*P at*random*. ...##
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Symmetry of Information and Bounds on Nonuniform Randomness Extraction via Kolmogorov Extractors

2011
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2011 IEEE 26th Annual Conference on Computational Complexity
*

Symmetry of Inf. and bounds on nonuniform

doi:10.1109/ccc.2011.21
dblp:conf/coco/Zimand11
fatcat:tda47lty35gmxi7m6jmry2ywli
*randomness*extractors Marius Zimand (Towson U.) ... Row v is bad if the number of A-cells*in*the v -th row is > ∆ |A| M N 1 . · · · · · u 1 u 2 · · · v ♠ ♠ ♠ ♠ ♠ · · · u N Number of bad rows is ≤ K (otherwise E would not be (K , D, ∆)-balanced). ... A combinatorial object equivalent to a*Kolmogorov*extractor. We also want E computable by poly-size circuits. Solution: Derandomization. ...
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