4 research outputs found
List Approximation for Increasing Kolmogorov Complexity
It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. But is it possible to construct a few strings, not longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that on input a string x of length n returns a list with O(n^2) many strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than n. We obtain similar results for other parameters, including a polynomial-time construction
Approximations of Kolmogorov Complexity
In this paper we show that the approximating the Kolmogorov complexity of a
set of numbers is equivalent to having common information with the halting
sequence. The more precise the approximations are, and the greater the number
of approximations, the more information is shared with the halting sequence. An
encoding of the 2^N unique numbers and their Kolmogorov complexities contains
at least >N mutual information with the halting sequence. We also provide a
generalization of the "Sets have Simple Members" theorem to conditional
complexity.Comment: 7 pages, 2 figure
In Memoriam, Solomon Marcus
This book commemorates Solomon Marcus’s fifth death anniversary with a selection of articles in mathematics, theoretical computer science, and physics written by authors who work in Marcus’s research fields, some of whom have been influenced by his results and/or have collaborated with him
List Approximation for Increasing Kolmogorov Complexity
It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. However, is it possible to construct a few strings, no longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that takes as input a string x of length n and returns a list with O(n2) strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than n−loglogn−O(1). We also present an algorithm that obtains a list of quasi-polynomial size in which each element can be produced in polynomial time