Related papers: Around Kolmogorov complexity: basic notions and re…
We explore the relationship between causality, symmetry, and compression. We build on and generalize the known connection between learning and compression to a setting where causal models are not identifiable. We propose a framework where…
The theory of random real numbers is exceedingly well-developed, and fascinating from many points of view. It is also quite challenging mathematically. The present notes are intended as no more than a gateway to the larger theory. They…
For any quantum algorithm given by a path in the space of unitary operators we define the computational complexity as the typical computational time associated with the path. This time is defined using a quantum time estimator associated…
The main goal of this article is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result of Vereshchagin saying that $\limsup_n C(x|n)$ equals $C^{0'}(x)$. Then we use the…
The algorithmic Markov condition states that the most likely causal direction between two random variables X and Y can be identified as that direction with the lowest Kolmogorov complexity. Due to the halting problem, however, this notion…
The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space,…
Information theory is a mathematical theory of learning with deep connections with topics as diverse as artificial intelligence, statistical physics, and biological evolution. Many primers on information theory paint a broad picture with…
In this paper we prove estimates on the behaviour of the Kolmogorov-Sinai entropy relative to a partition for randomly perturbed dynamical systems. Our estimates use the entropy for the unperturbed system and are obtained using the notion…
This Thesis explores the notion of Krylov complexity as a probe of quantum chaos and as a candidate for holographic complexity. The first Part is devoted to presenting the fundamental notions required to conduct research in this area.…
Normalized information distance (NID) uses the theoretical notion of Kolmogorov complexity, which for practical purposes is approximated by the length of the compressed version of the file involved, using a real-world compression program.…
We revisit the fundamentals of Circuit Complexity and the nature of efficient computation from a fresh perspective. We present a framework for understanding Circuit Complexity through the lens of Information Theory with analogies to results…
The paper deals with the order statistics and empirical mathematical expectation (which is also called the estimate of mathematical expectation in the literature) in the case of infinitely increasing random variables. The Kolmogorov concept…
The concept of effective complexity of an object as the minimal description length of its regularities has been initiated by Gell-Mann and Lloyd. The regularities are modeled by means of ensembles, that is probability distributions on…
We prove a strong Symmetry of Information relation for random strings (in the sense of Kolmogorov complexity) and establish tight bounds on the amount on nonuniformity that is necessary for extracting a string with randomness rate 1 from a…
We survey concepts at the frontier of research connecting artificial, animal and human cognition to computation and information processing---from the Turing test to Searle's Chinese Room argument, from Integrated Information Theory to…
We present a fully automatic method for music classification, based only on compression of strings that represent the music pieces. The method uses no background knowledge about music whatsoever: it is completely general and can, without…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
We propose a numerical test of fundamental physics based on the complexity measure of a general set of functions, which is directly related to the Kolmogorov (or algorithmic) complexity studied in mathematics and computer science. The…
Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard average-case analysis is ammenable to this method. Here we…
We propose an information-theoretic framework for matrix completion. The theory goes beyond the low-rank structure and applies to general matrices of "low description complexity". Specifically, we consider $m\times n$ random matrices…