Filters








5,797 Hits in 5.5 sec

Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity [article]

Zhenjian Lu, Igor C. Oliveira, Marius Zimand
2022 arXiv   pre-print
The classical coding theorem in Kolmogorov complexity states that if an n-bit string x is sampled with probability δ by an algorithm with prefix-free domain then K(x) ≤log(1/δ) + O(1).  ...  As in previous work, our coding theorem is efficient in the sense that it provides a polynomial-time probabilistic algorithm that, when given x, the code of the sampler, and δ, it outputs, with probability  ...  Zimand was supported in part by the National Science Foundation through grant CCF 1811729. Z. Lu and I.C.  ... 
arXiv:2204.08312v1 fatcat:g5y7win7wndgdguem6t5xxzqrq

Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

Zhenjian Lu, Igor C. Oliveira, Marius Zimand, Mikołaj Bojańczyk, Emanuela Merelli, David P. Woodruff
2022
Motivated by this discrepancy, we investigate optimal coding theorems in the time-bounded setting.  ...  The classical coding theorem in Kolmogorov complexity states that if an n-bit string x is sampled with probability δ by an algorithm with prefix-free domain then 𝖪(x) ≤ log(1/δ) + O(1).  ...  and log A L P 2 0 2 2 92:4 Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity 2.1 A Tighter Efficient Coding Theorem Our first result addresses the question posed in [31, Problem 37]. ▶ Theorem  ... 
doi:10.4230/lipics.icalp.2022.92 fatcat:xnjm6xs7gvfsnbqx2ks3vcrpxy

Theory and Applications of Probabilistic Kolmogorov Complexity [article]

Zhenjian Lu, Igor C. Oliveira
2022 arXiv   pre-print
Since the running time of algorithms is a key resource in these fields, it is crucial in the corresponding arguments to consider time-bounded variants of Kolmogorov complexity.  ...  The theory of Kolmogorov complexity is well understood, but many useful results and properties of Kolmogorov complexity are not known to hold in time-bounded settings.  ...  To this point, we have mentioned the existence of an optimal coding theorem for time-unbounded Kolmogorov complexity and an optimal conditional coding theorem for Kt (in fact, the conditional result holds  ... 
arXiv:2205.14718v1 fatcat:rahzivyparg5jmej4cqvpe44a4

How Incomputable Is Kolmogorov Complexity?

Paul M.B. Vitányi
2020 Entropy  
Kolmogorov complexity is the length of the ultimately compressed version of a file (i.e., anything which can be put in a computer).  ...  We discuss the incomputability of Kolmogorov complexity, which formal loopholes this leaves us with, recent approaches to compute or approximate Kolmogorov complexity, which approaches are problematic,  ...  Problematic Use of the Coding Theorem Fix an optimal universal prefix Turing machine U.  ... 
doi:10.3390/e22040408 pmid:33286182 fatcat:imcanhdpnval5i5dj5wejyrj2m

How incomputable is Kolmogorov complexity? [article]

Paul Vitanyi
2020 arXiv   pre-print
Kolmogorov complexity is the length of the ultimately compressed version of a file (that is, anything which can be put in a computer).  ...  We discuss the incomputabilty of Kolmogorov complexity, which formal loopholes this leaves us, recent approaches to compute or approximate Kolmogorov complexity, which approaches are problematic and which  ...  PROBLEMATIC USE OF THE CODING THEOREM Fix an optimal universal prefix Turing machine U .  ... 
arXiv:2002.07674v2 fatcat:u6kz3wkdsrccjlcyvpio6mn6dq

The cost of offline binary search tree algorithms and the complexity of the request sequence

Jussi Kujala, Tapio Elomaa
2008 Theoretical Computer Science  
In this paper we lower bound the cost of an optimal offline binary search tree using the Kolmogorov complexity of the request sequence. We obtain several applications for this result.  ...  Third, the expected cost of splay trees is within a constant factor of the expected cost of an optimal offline binary search tree algorithm in a subset of Markov chains.  ...  Acknowledgment This work has been in part supported by the Academy of Finland.  ... 
doi:10.1016/j.tcs.2007.12.015 fatcat:27dk2gehqfce3hzvefpowmtpyq

Computational Complexity, Genetic Programming, and Implications [chapter]

Bart Rylander, Terry Soule, James Foster
2001 Lecture Notes in Computer Science  
this result does not hold if the random number generator used in the evolution is a true random source; and 3) an optimization problem being solved with a GP will have a complexity that can be bounded  ...  This paper presents proofs that show that 1) a program that is the output of a GP or any inductive process has complexity that can be bounded by the Kolmogorov complexity of the originating program; 2)  ...  Acknowledgments This research was funded in part by NSF EPS0080935, NIH F33GM20122-01, and NSA MDA 904-98-C-A894.  ... 
doi:10.1007/3-540-45355-5_28 fatcat:2233yj4vfnbvpmq53vtpewcdei

Kolmogorov Complexity in Perspective Part I: Information Theory and Randomness [chapter]

Marie Ferbus-Zanda, Serge Grigorieff
2014 Logic, Epistemology, and the Unity of Science  
Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume.  ...  We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity.  ...  Kolmogorov complexity and undecidability 2.1 K is unbounded Let K = K V : O → N where V : {0, 1} * → O is optimal (cf. Theorem §1.10).  ... 
doi:10.1007/978-94-017-9217-2_3 fatcat:2my7rmqrwbejhgcio6ek2ljkzy

Time-Bounded Kolmogorov Complexity and Solovay Functions

Rupert Hölzl, Thorsten Kräling, Wolfgang Merkle
2012 Theory of Computing Systems  
Finally, we investigate the plain Kolmogorov complexity C and its time-bounded variant C t of initial segments of computably enumerable sets.  ...  A Solovay function is a computable upper bound g for prefixfree Kolmogorov complexity K that is nontrivial in the sense that g agrees with K, up to some additive constant, on infinitely many places n.  ...  In order to define plain and prefix-free Kolmogorov complexity, we fix additively optimal oracle Turing machines V and U, where U has prefixfree domain.  ... 
doi:10.1007/s00224-012-9413-4 fatcat:ckfv3fsa75ak7a6yiudifhhuje

Time-Bounded Kolmogorov Complexity and Solovay Functions [chapter]

Rupert Hölzl, Thorsten Kräling, Wolfgang Merkle
2009 Lecture Notes in Computer Science  
Finally, we investigate the plain Kolmogorov complexity C and its time-bounded variant C t of initial segments of computably enumerable sets.  ...  A Solovay function is a computable upper bound g for prefixfree Kolmogorov complexity K that is nontrivial in the sense that g agrees with K, up to some additive constant, on infinitely many places n.  ...  In order to define plain and prefix-free Kolmogorov complexity, we fix additively optimal oracle Turing machines V and U, where U has prefixfree domain.  ... 
doi:10.1007/978-3-642-03816-7_34 fatcat:gyqgbfsfh5hzpe2tz4tjezv2o4

Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes [article]

Marie Ferbus-Zanda
2010 arXiv   pre-print
Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume.  ...  We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity.  ...  Kolmogorov complexity and undecidability 2.1 K is unbounded Let K = K V : O → N where V : {0, 1} * → O is optimal (cf. Theorem §1.10).  ... 
arXiv:1010.3201v1 fatcat:yevwe3gfcbh2tleaq3fliyq66q

Complexity distortion theory

D.M. Sow, A. Eleftheriadis
2003 IEEE Transactions on Information Theory  
by Kolmogorov in Kolmogorov complexity theory and its extension to lossy source coding, CDT.  ...  Using this model, the mathematical framework for examining the efficiency of coding schemes is the algorithmic or Kolmogorov complexity.  ...  The CDF is introduced in a similar manner. In a -ball centered around this time, let be the sequence in with the smallest Kolmogorov complexity.  ... 
doi:10.1109/tit.2002.808135 fatcat:lvwiax43x5as5mjna5vifsd2ny

Quantum Kolmogorov complexity based on classical descriptions

P.M.B. Vitanyi
2001 IEEE Transactions on Information Theory  
Quantum Kolmogorov complexity coincides with the classical Kolmogorov complexity on the classical domain.  ...  Quantum Kolmogorov complexity is upper bounded and can be effectively approximated from above under certain conditions. With high probability a quantum object is incompressible.  ...  The Shannon-Fano code gives an expected codeword length close to the entropy, and, by Shannon's Noiseless Coding Theorem, it possesses the optimal expected codeword length.  ... 
doi:10.1109/18.945258 fatcat:opbizz2wt5atrak6wqw4yi26sq

The Redundancy of a Computable Code on a Noncomputable Distribution [article]

Łukasz Dębowski
2009 arXiv   pre-print
Thus we introduce another characteristic of computable codes: The catch-up time is the length of data for which the code length drops below the Kolmogorov complexity plus the prescribed margin.  ...  Some codes may have smaller catch-up times than Bayesian codes.  ...  The research reported in this work was supported in part by the IST Program of the European Community, under the PASCAL II Network of Excellence, IST-2002-506778, and done on the author's leave from the  ... 
arXiv:0901.2321v2 fatcat:pdela2toybhfzj5skj3n7hospa

An effective Procedure for Speeding up Algorithms [article]

Marcus Hutter
2001 arXiv   pre-print
To quantify this statement, the definition of Kolmogorov complexity is extended, and two new natural measures for the complexity of a function are defined.  ...  Blum's speed-up theorem is avoided by taking into account only programs for which a correctness proof exists.  ...  For complex approximation problems, for instance, where no good and fast time bound exists, M p * is still optimal, but in this case, only apart from a large multiplicative factor.  ... 
arXiv:cs/0102018v1 fatcat:mvn3z5p5tnabvhu5id6qyvv6he
« Previous Showing results 1 — 15 out of 5,797 results