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

2022
*
arXiv
*
pre-print

The classical

arXiv:2204.08312v1
fatcat:g5y7win7wndgdguem6t5xxzqrq
*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. ...##
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Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

2022

Motivated by this discrepancy, we investigate

doi:10.4230/lipics.icalp.2022.92
fatcat:xnjm6xs7gvfsnbqx2ks3vcrpxy
*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*...##
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Theory and Applications of Probabilistic Kolmogorov Complexity
[article]

2022
*
arXiv
*
pre-print

Since the running

arXiv:2205.14718v1
fatcat:rahzivyparg5jmej4cqvpe44a4
*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 ...##
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How Incomputable Is Kolmogorov Complexity?

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. ...

##
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How incomputable is Kolmogorov complexity?
[article]

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 . ...

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

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. ...

##
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Computational Complexity, Genetic Programming, and Implications
[chapter]

2001
*
Lecture Notes in Computer Science
*

this result does not hold if the random number generator used

doi:10.1007/3-540-45355-5_28
fatcat:2233yj4vfnbvpmq53vtpewcdei
*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. ...##
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Kolmogorov Complexity in Perspective Part I: Information Theory and Randomness
[chapter]

2014
*
Logic, Epistemology, and the Unity of Science
*

Two of the main applications of

doi:10.1007/978-94-017-9217-2_3
fatcat:2my7rmqrwbejhgcio6ek2ljkzy
*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). ...##
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Time-Bounded Kolmogorov Complexity and Solovay Functions

2012
*
Theory of Computing Systems
*

Finally, we investigate the plain

doi:10.1007/s00224-012-9413-4
fatcat:ckfv3fsa75ak7a6yiudifhhuje
*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. ...##
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Time-Bounded Kolmogorov Complexity and Solovay Functions
[chapter]

2009
*
Lecture Notes in Computer Science
*

Finally, we investigate the plain

doi:10.1007/978-3-642-03816-7_34
fatcat:gyqgbfsfh5hzpe2tz4tjezv2o4
*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. ...##
###
Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes
[article]

2010
*
arXiv
*
pre-print

Two of the main applications of

arXiv:1010.3201v1
fatcat:yevwe3gfcbh2tleaq3fliyq66q
*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). ...##
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Complexity distortion theory

2003
*
IEEE Transactions on Information Theory
*

by

doi:10.1109/tit.2002.808135
fatcat:lvwiax43x5as5mjna5vifsd2ny
*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*. ...##
###
Quantum Kolmogorov complexity based on classical descriptions

2001
*
IEEE Transactions on Information Theory
*

Quantum

doi:10.1109/18.945258
fatcat:opbizz2wt5atrak6wqw4yi26sq
*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. ...##
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The Redundancy of a Computable Code on a Noncomputable Distribution
[article]

2009
*
arXiv
*
pre-print

Thus we introduce another characteristic of computable

arXiv:0901.2321v2
fatcat:pdela2toybhfzj5skj3n7hospa
*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 ...##
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An effective Procedure for Speeding up Algorithms
[article]

2001
*
arXiv
*
pre-print

To quantify this statement, the definition of

arXiv:cs/0102018v1
fatcat:mvn3z5p5tnabvhu5id6qyvv6he
*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. ...
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