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New Bounds for the Language Compression Problem
, 2000
"... The CD complexity of a string x is the length of the shortest polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CD A n complexity of all strings x in some set A. The best known upper bound for this problem is 2 log(j ..."
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The CD complexity of a string x is the length of the shortest polynomial time program which accepts only the string x. The language compression problem consists of giving an upper bound on the CD A n complexity of all strings x in some set A. The best known upper bound for this problem is 2 log(jjA n jj) + O(log(n)), due to Buhrman and Fortnow. We show that the constant factor 2 in this bound is optimal. We also give new bounds for a certain kind of random sets R ` f0; 1g n , for which we show an upper bound of log(jjR n jj) + O(log(n)). 1 Introduction Kolmogorov complexity is a notion that measures the amount of regularity in a finite string. It has turned out to be a very useful tool in theoretical computer science. A simple counting argument showing that for each length there exist random strings, i.e. strings with no regularity, has had many applications (see [LV97]). Early in the history of computational complexity resource bounded notions of Kolmogorov complexity were...
Rigorous Mathematical Physics.
, 2008
"... for their friendship, for many invaluable remarks about the issues discussed in this paper... and for their trial of teaching me the style of work of ..."
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for their friendship, for many invaluable remarks about the issues discussed in this paper... and for their trial of teaching me the style of work of
First of all I want to thank:
, 2008
"... for useful discussions and suggestions. They all have no responsibility for any mistake contained in these pages. ..."
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for useful discussions and suggestions. They all have no responsibility for any mistake contained in these pages.
THE DEFINITION OF A RANDOM SEQUENCE OF QUBITS: FROM NONCOMMUTATIVE ALGORITHMIC PROBABILITY THEORY TO QUANTUM ALGORITHMIC INFORMATION THEORY AND BACK
, 2008
"... First of all I want to thank: ..."
MartinLöf Tests Can Help Too
, 1994
"... We argue that MartinLof tests provide useful techniques in proofs involving Kolmogorov complexity. Our examples cover more areas: (1) logical definability; we prove an analogue of the 01 Law for various logics in terms of structures with high Kolmogorov complexity, (2) probabilistic computation: w ..."
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We argue that MartinLof tests provide useful techniques in proofs involving Kolmogorov complexity. Our examples cover more areas: (1) logical definability; we prove an analogue of the 01 Law for various logics in terms of structures with high Kolmogorov complexity, (2) probabilistic computation: we display a relation between the Kolmogorov complexity of the random strings on which a probabilistic algorithm errs and the probability error of the algorithm as well as a tradeoff relation between the error probability, the length of the random bits, the number of provers and the length of the provers' answers in oneround multiprover interactive proof systems for NP, and (3) convergence rates of probability asymptotics; we find a general lower bound on the convergence rate of such probabilities in terms of Kolmogorov complexity and provide a concrete example for the probability that a random graph with n vertices is connected. Keywords: Kolmogorov complexity, MartinLof test, logical de...