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Kolmogorov Complexity Theory over the Reals [article]

Martin Ziegler, Wouter M. Koolen
2008 arXiv   pre-print
Kolmogorov Complexity constitutes an integral part of computability theory, information theory, and computational complexity theory -- in the discrete setting of bits and Turing machines.  ...  Over real numbers, on the other hand, the BSS-machine (aka real-RAM) has been established as a major model of computation.  ...  Finally we owe to Klaus Meer for pointing us to the seminal work of Montaña and Pardo who first introduced real Kolmogorov Complexity.  ... 
arXiv:0802.2027v2 fatcat:y2wdj23bjjd27fszs5xto5legq

Kolmogorov Complexity Theory over the Reals

Martin Ziegler, Wouter M. Koolen
2008 Electronical Notes in Theoretical Computer Science  
Kolmogorov Complexity constitutes an integral part of computability theory, information theory, and computational complexity theory-in the discrete setting of bits and Turing machines.  ...  Over real numbers, on the other hand, the BSS-machine (aka real-RAM) has been established as a major model of computation.  ...  Finally we owe to Klaus Meer for pointing us to the seminal work of Montaña and Pardo who first introduced real Kolmogorov Complexity.  ... 
doi:10.1016/j.entcs.2008.12.014 fatcat:iulx5fjenzbwxmig2y63r3icnu

On the Kolmogorov complexity of continuous real functions

Amin Farjudian
2013 Annals of Pure and Applied Logic  
An asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions will be presented as well.  ...  Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitely-representable objects-such as rational numbers-used to  ...  Montaña and Pardo [17] and Ziegler and Koolen [25] have studied Kolmogorov complexity over sequences of real numbers based on the theory of real Turing machines.  ... 
doi:10.1016/j.apal.2012.11.003 fatcat:xjky2cr3vrcldlgikehn7uqxz4

Page 10262 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
The paper addresses a question on the Kolmogorov complexity of reals.  ...  Using prefix-free Kolmogorov complexity <j as a reducibil- ity on the set of all reals, it proves that there are continuum many H-degrees in the random reals.  ... 

Complexities Approach to Two Problems In Number Theory [article]

Yang Bai, Xiuli Wang
2016 arXiv   pre-print
problems in the non-logical discipline.Futhermore,we show that resource-bounded Kolmogorov Complexity and computational complexity can at least provide tips or principles to mathematical problems in the  ...  By Kolmogorov Complexity,two number-theoretic problems are solved in different way than before,one problem is Maxim Kontsevich and Don Bernard Zagier's Problem 3 Exhibit at least one number which does  ...  question by Kolmogorov Complexity.  ... 
arXiv:1610.04026v3 fatcat:5udl5eurnzh63c6ysh6b7kta6y

Is Randomness Native to Computer Science? Ten Years After [chapter]

Marie Ferbus-Zanda, Serge Grigorieff
2011 Randomness Through Computation  
I remember that Kolmogorov built the mathematical theory of the so-called Kolmogorov complexity out of this paradox.  ...  Shannon proved that the functions over the reals which are computed by GPACs are exactly the solutions of algebraic differential systems.  ... 
doi:10.1142/9789814327756_0019 fatcat:lxjyg53fzzb2nlqb6i4oofiicy

On the Kolmogorov Complexity of Continuous Real Functions [chapter]

Amin Farjudian
2011 Lecture Notes in Computer Science  
An asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions will be presented as well.  ...  Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitely-representable objects-such as rational numbers-used to  ...  I would also like to thank Douglas Cenzer and Willem Fouché for informing me of their related work on the subject of algorithmic randomness over function spaces.  ... 
doi:10.1007/978-3-642-21875-0_9 fatcat:6pbcuw4ydvbqtox7hkkczztu3q

Page 5977 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
Chapter 2 (“Algorithmic complexity”) is dedicated to the theory of the most general form of Kolmogorov complexity, called in the book the plain Kolmogorov complex- ity.  ...  Kolmogorov complexity is the theory which explains such dis- crepancies and explores the properties and capabilities of objects like the ones above.  ... 

Page 4394 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
Summary: “We consider for a real number a the Kolmogorov complexities of its expansions with respect to different bases.  ...  Philip Ross Watson (Bradford) 2002f:68067 68Q30 Staiger, Ludwig (D-MLU-II; Halle an der Saale) The Kolmogorov complexity of real numbers.  ... 

Computability, Complexity and Randomness (Dagstuhl Seminar 12021)

Veronica Becher, Laurent Bienvenu, Rodney Downey, Elvira Mayordomo, Marc Herbstritt
2012 Dagstuhl Reports  
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  ...  information theory.  ...  Understanding the complexity of the reachability problem is a central concern of computational complexity theory.  ... 
doi:10.4230/dagrep.2.1.19 dblp:journals/dagstuhl-reports/BecherBDM12 fatcat:bkapirz4vfgkzcexiutm33k74q

Is Randomness "Native" to Computer Science? [article]

Marie Ferbus-Zanda
2008 arXiv   pre-print
We survey the Kolmogorov's approach to the notion of randomness through the Kolmogorov complexity theory.  ...  The original motivation of Kolmogorov was to give up a quantitative definition of information. In this theory, an object is randomness in the sense that it has a large information content.  ...  Such Borel sets are, in fact, coded in any inner submodel of set theory. -Solovay random reals over a submodel of set theory are reals outside every measure zero Borel set coded in that submodel.  ... 
arXiv:0801.0289v1 fatcat:7mwcyjmw75fppmgkjpzshxyapi

Review of Algorithmic Randomness and Complexity by Downey and Hirschfeldt

Jason Teutsch
2013 ACM SIGACT News  
In terms of Kolmogorov complexity and computability theory, one might say that highly random sequences contain a lot of information but not much useful information.  ...  Downey and Hirschfeldt examine other topics as well, such as Kummer's Gap Theorem on the Kolmogorov complexity of c.e sets, and his theorem that the set of Kolmogorov random strings are truth-table complete  ... 
doi:10.1145/2447712.2447721 fatcat:ruecbllbf5d6nlwnglakqwb7zy

IS RANDOMNESS "NATIVE" TO COMPUTER SCIENCE? [chapter]

MARIE FERBUS-ZANDA, SERGE GRIGORIEFF
2004 Current Trends in Theoretical Computer Science  
The best reference to the subject is Li-Vitanyi's book [29] (caution: they denote C, K what is here -and in most papers -denoted K, H).  ...  Such Borel sets are, in fact, coded in any inner submodel of set theory. -Solovay random reals over a submodel of set theory are reals outside every measure zero Borel set coded in that submodel.  ...  The Invariance Theorem A: Now, comes the fundamental result of the theory. We shall state it uniquely for Kolmogorov complexity but it also holds for conditional Kolmogorov complexity.  ... 
doi:10.1142/9789812562494_0046 fatcat:cwz6prus4ndwndcmi7zr4mbr7a

On hausdorff and topological dimensions of the kolmogorov complexity of the real line

Jin-yi Cai, Juris Hartmanis
1994 Journal of computer and system sciences (Print)  
We investigate the Kolmogorov complexity of real numbers. Let K be the Kolmogorov complexity function; we determine the Hausdorff dimension and the topological dimension of the graph of K.  ...  Since these dimensions are different, the graph of the Kolmogorov complexity function of the real line forms a fractal in the sense of Mandelbrot.  ...  We thank Professors Allen Back, David Henderson, and Tom Rishel for interesting discussions on dimension theory and analysis.  ... 
doi:10.1016/s0022-0000(05)80073-x fatcat:vgmphpoc6nc4xhwatn3glv2qkq

Computational power of neural networks: a characterization in terms of Kolmogorov complexity

J.L. Balcazar, R. Gavalda, H.T. Siegelmann
1997 IEEE Transactions on Information Theory  
The computational power of recurrent neural networks is shown to depend ultimately on the complexity of the real constants (weights) of the network.  ...  The complexity, or information contents, of the weights is measured by a variant of resource-bounded Kolmogorov complexity, taking into account the time required for constructing the numbers.  ...  ACKNOWLEDGMENT The authors wish to thank E. Sontag for his encouragement and support via many discussions and D. Harel for asking the question that led them to Corollary 5.4.  ... 
doi:10.1109/18.605580 fatcat:bbpev5jiqrh4xhofiaygqtw55q
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