Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume. Part I is dedicated to information theory and the mathematical formalization of randomness based on Kolmogorov complexity. This last application goes back to the 60's and 70's with the work of Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last years.Comment: 40 page

Topological arguments for Kolmogorov complexity

We present several application of simple topological arguments in problems of Kolmogorov complexity. Basically we use the standard fact from topology that the disk is simply connected. It proves to be enough to construct strings with some nontrivial algorithmic properties.Comment: Extended versio

Algorithmic statistics revisited

The mission of statistics is to provide adequate statistical hypotheses (models) for observed data. But what is an "adequate" model? To answer this question, one needs to use the notions of algorithmic information theory. It turns out that for every data string $x$ one can naturally define "stochasticity profile", a curve that represents a trade-off between complexity of a model and its adequacy. This curve has four different equivalent definitions in terms of (1)~randomness deficiency, (2)~minimal description length, (3)~position in the lists of simple strings and (4)~Kolmogorov complexity with decompression time bounded by busy beaver function. We present a survey of the corresponding definitions and results relating them to each other

Estimating the Algorithmic Complexity of Stock Markets

Randomness and regularities in Finance are usually treated in probabilistic terms. In this paper, we develop a completely different approach in using a non-probabilistic framework based on the algorithmic information theory initially developed by Kolmogorov (1965). We present some elements of this theory and show why it is particularly relevant to Finance, and potentially to other sub-fields of Economics as well. We develop a generic method to estimate the Kolmogorov complexity of numeric series. This approach is based on an iterative "regularity erasing procedure" implemented to use lossless compression algorithms on financial data. Examples are provided with both simulated and real-world financial time series. The contributions of this article are twofold. The first one is methodological : we show that some structural regularities, invisible with classical statistical tests, can be detected by this algorithmic method. The second one consists in illustrations on the daily Dow-Jones Index suggesting that beyond several well-known regularities, hidden structure may in this index remain to be identified

Limit complexities revisited [once more]

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 same argument to prove similar results for prefix complexity, a priori probability on binary tree, to prove Conidis' theorem about limits of effectively open sets, and also to improve the results of Muchnik about limit frequencies. As a by-product, we get a criterion of 2-randomness proved by Miller: a sequence $X$ is 2-random if and only if there exists $c$ such that any prefix $x$ of $X$ is a prefix of some string $y$ such that $C(y)\ge |y|-c$. (In the 1960ies this property was suggested in Kolmogorov as one of possible randomness definitions.) We also get another 2-randomness criterion by Miller and Nies: $X$ is 2-random if and only if $C(x)\ge |x|-c$ for some $c$ and infinitely many prefixes $x$ of $X$. This is a modified version of our old paper that contained a weaker (and cumbersome) version of Conidis' result, and the proof used low basis theorem (in quite a strange way). The full version was formulated there as a conjecture. This conjecture was later proved by Conidis. Bruno Bauwens (personal communication) noted that the proof can be obtained also by a simple modification of our original argument, and we reproduce Bauwens' argument with his permission.Comment: See http://arxiv.org/abs/0802.2833 for the old pape

Applying MDL to Learning Best Model Granularity

The Minimum Description Length (MDL) principle is solidly based on a provably ideal method of inference using Kolmogorov complexity. We test how the theory behaves in practice on a general problem in model selection: that of learning the best model granularity. The performance of a model depends critically on the granularity, for example the choice of precision of the parameters. Too high precision generally involves modeling of accidental noise and too low precision may lead to confusion of models that should be distinguished. This precision is often determined ad hoc. In MDL the best model is the one that most compresses a two-part code of the data set: this embodies ``Occam's Razor.'' In two quite different experimental settings the theoretical value determined using MDL coincides with the best value found experimentally. In the first experiment the task is to recognize isolated handwritten characters in one subject's handwriting, irrespective of size and orientation. Based on a new modification of elastic matching, using multiple prototypes per character, the optimal prediction rate is predicted for the learned parameter (length of sampling interval) considered most likely by MDL, which is shown to coincide with the best value found experimentally. In the second experiment the task is to model a robot arm with two degrees of freedom using a three layer feed-forward neural network where we need to determine the number of nodes in the hidden layer giving best modeling performance. The optimal model (the one that extrapolizes best on unseen examples) is predicted for the number of nodes in the hidden layer considered most likely by MDL, which again is found to coincide with the best value found experimentally.Comment: LaTeX, 32 pages, 5 figures. Artificial Intelligence journal, To appea

Kolmogorov Complexity in perspective. Part II: Classification, Information Processing and Duality

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts published in a same volume. Part II is dedicated to the relation between logic and information system, within the scope of Kolmogorov algorithmic information theory. We present a recent application of Kolmogorov complexity: classification using compression, an idea with provocative implementation by authors such as Bennett, Vitanyi and Cilibrasi. This stresses how Kolmogorov complexity, besides being a foundation to randomness, is also related to classification. Another approach to classification is also considered: the so-called "Google classification". It uses another original and attractive idea which is connected to the classification using compression and to Kolmogorov complexity from a conceptual point of view. We present and unify these different approaches to classification in terms of Bottom-Up versus Top-Down operational modes, of which we point the fundamental principles and the underlying duality. We look at the way these two dual modes are used in different approaches to information system, particularly the relational model for database introduced by Codd in the 70's. This allows to point out diverse forms of a fundamental duality. These operational modes are also reinterpreted in the context of the comprehension schema of axiomatic set theory ZF. This leads us to develop how Kolmogorov's complexity is linked to intensionality, abstraction, classification and information system.Comment: 43 page