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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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Cited by 9 (2 self)
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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...
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: ..."
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.
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