Logical depth and physical complexity
A half-century survey on The Universal Turing Machine
An almost machine-independent theory of program-length complexity, sophistication, and induction
Information Sciences: an International Journal
The universal Turing machine (2nd ed.)
On the Length of Programs for Computing Finite Binary Sequences: statistical considerations
Journal of the ACM (JACM)
A Kolmogorov complexity characterization of constructive Hausdorff dimension
Information Processing Letters
On the algorithmic complexity of the trajectories of points in dynamical systems
On the algorithmic complexity of the trajectories of points in dynamical systems
Dimension in Complexity Classes
SIAM Journal on Computing
Kolmogorov complexity estimation and application for information system security
Kolmogorov complexity estimation and application for information system security
Computational depth: concept and applications
Theoretical Computer Science - Foundations of computation theory (FCT 2003)
Low-Depth Witnesses are Easy to Find
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Effective Strong Dimension in Algorithmic Information and Computational Complexity
SIAM Journal on Computing
An Introduction to Kolmogorov Complexity and Its Applications
An Introduction to Kolmogorov Complexity and Its Applications
Theory of Computing Systems
Worst-Case Running Times for Average-Case Algorithms
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Depth as Randomness Deficiency
Theory of Computing Systems - Special Issue: Computation and Logic in the Real World; Guest Editors: S. Barry Cooper, Elvira Mayordomo and Andrea Sorbi
IEEE Transactions on Information Theory
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We revisit the notion of computational depth and sophistication for infinite sequences and study the density of the sets of deep and sophisticated infinite sequences. Koppel defined the sophistication of an object as the length of the shortest total program that, given some data as input, produces the object and such that the sum of the size of the program with the size of the data is as concise as the smallest description of the object. However, the notion of sophistication is not properly defined for all sequences. We propose a new definition of sophistication for infinite sequences as the limit of the ratio of the sophistication of the initial segments and its length. We prove that the set of sequences with sophistication equal to zero has Lebesgue measure one and that the set of sophisticated sequences is dense, when the sophistication is, respectively, defined either using lim inf or lim sup. Antunes, Fortnow, van Melkebeek and Vinodchandran captured the notion of useful information by computational depth, the difference between time bounded and unbounded Kolmogorov complexities. We show that the set of deep infinite sequences is dense. We also prove that sophistication and depth for infinite sequences are distinct information measures.