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Kolmogorov Complexity in Randomness Extraction

John M. Hitchcock, A. Pavan, N. V. Vinodchandran
2011 ACM Transactions on Computation Theory  
We clarify the role of 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.  ... 
doi:10.1145/2003685.2003686 fatcat:h23gnhro4nacrpvpvnylf633bq

Possibilities and impossibilities in Kolmogorov complexity extraction [article]

Marius Zimand
2012 arXiv   pre-print
We present the connection between extractors and 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).  ... 
arXiv:1104.0872v2 fatcat:n74urvnpnvfaxkyu66eixdn4ta

Kolmogorov complexity: Recent research in Moscow [chapter]

Vladimir A. Uspensky
1996 Lecture Notes in Computer Science  
Let us say only that most basic questions connected with 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.  ... 
doi:10.1007/3-540-61550-4_145 fatcat:fpraul7ur5dgrecoheujjwx3um

Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov extractors [article]

Marius Zimand
2011 arXiv   pre-print
We prove a strong Symmetry of Information relation for 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).  ... 
arXiv:1103.5669v1 fatcat:fh45vyhumfgs5pioak5qpdqrim

Nonuniform Kolmogorov extractors [article]

Marius Zimand
2012 arXiv   pre-print
We establish tight bounds on the amount on nonuniformity that is necessary for 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.  ... 
arXiv:1204.6696v1 fatcat:swfpfse3ungrbi57u7vnshyh5a

Impossibility of Independence Amplification in Kolmogorov Complexity Theory [chapter]

Marius Zimand
2010 Lecture Notes in Computer Science  
The paper studies 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.  ... 
doi:10.1007/978-3-642-15155-2_61 fatcat:5f6ywkzmlrcpzbvdgc675nb6x4

Algorithmic Information Theory Using Kolmogorov Complexity

Ng Keng Meng
2012 Journal of Applied & Computational Mathematics  
Citation: Meng NK (2012) Algorithmic Information Theory Using 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.  ... 
doi:10.4172/2168-9679.1000e106 fatcat:nsj356iwmnh5tpdezt2cjq5me4

Extracting Kolmogorov Complexity with Applications to Dimension Zero-One Laws [chapter]

Lance Fortnow, John M. Hitchcock, A. Pavan, N. V. Vinodchandran, Fengming Wang
2006 Lecture Notes in Computer Science  
We apply recent results on 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.  ... 
doi:10.1007/11786986_30 fatcat:y2yfcqfdufgr5fxde6fygl5kua

Extracting Kolmogorov complexity with applications to dimension zero-one laws

Lance Fortnow, John M. Hitchcock, A. Pavan, N.V. Vinodchandran, Fengming Wang
2011 Information and Computation  
We apply recent results on 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.  ... 
doi:10.1016/j.ic.2010.09.006 fatcat:7btnxoslzvhpnkdwgwu5fr47zm

Computability, Complexity and Randomness (Dagstuhl Seminar 12021)

Veronica Becher, Laurent Bienvenu, Rodney Downey, Elvira Mayordomo, Marc Herbstritt
2012 Dagstuhl Reports  
The Dagstuhl seminar 12021 "Computability, 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.  ... 
doi:10.4230/dagrep.2.1.19 dblp:journals/dagstuhl-reports/BecherBDM12 fatcat:bkapirz4vfgkzcexiutm33k74q

Symmetry of Information: A Closer Look [chapter]

Marius Zimand
2012 Lecture Notes in Computer Science  
It was shown 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.  ... 
doi:10.1007/978-3-642-27654-5_18 fatcat:dh22yxt6wbcntdtpkduudy3zgi

Symmetry of Information: A Closer Look [article]

Marius Zimand
2012 arXiv   pre-print
It was shown 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.  ... 
arXiv:1206.5184v1 fatcat:2hlc625npvhg3igx5wwdfdnglm

27 Open Problems in Kolmogorov Complexity [article]

Andrei Romashchenko, Alexander Shen, Marius Zimand
2022 arXiv   pre-print
The paper proposes open problems 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  ... 
arXiv:2203.15109v1 fatcat:2mjciipal5amvlhld3w6u2dzv4

Nearly optimal language compression using extractors [chapter]

Lance Fortnow, Sophie Laplante
1998 Lecture Notes in Computer Science  
This extends work of Sipser Sip83] and Buhrman and Fortnow BF97]. { We use extractors to 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.  ... 
doi:10.1007/bfb0028551 fatcat:yypx2ijjtva3nbsxbsd3xwhscy

Symmetry of Information and Bounds on Nonuniform Randomness Extraction via Kolmogorov Extractors

Marius Zimand
2011 2011 IEEE 26th Annual Conference on Computational Complexity  
Symmetry of Inf. and bounds on nonuniform 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.  ... 
doi:10.1109/ccc.2011.21 dblp:conf/coco/Zimand11 fatcat:tda47lty35gmxi7m6jmry2ywli
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