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Jean-Paul Delahaye

Publications and source records attributed to Jean-Paul Delahaye.

18 recordsLinked to original sources
arXiv2018

Organized Complexity: is Big History a Big Computation?

The concept of "logical depth" introduced by Charles H. Bennett (1988) seems to capture, at least partially, the notion of organized complexity, so central in big history. More precisely, the increase in organized complexity refers here to the wealth, variety and intricacy of structures, and should not be confused with the increase of random complexity, formalized by Kolmogorov (1965). If Bennett is right in proposing to assimilate organized complexity with "computational content", then the fundamental cause of the increase of complexity in the universe is the existence of computing mechanisms with memory, and able to cumulatively create and preserve computational contents. In this view, the universe computes, remembers its calculations, and reuses them to conduct further computations. Evolutionary mechanisms are such forms of cumulative computation with memory and we owe them the organized complexity of life. Language, writing, culture, science and technology can also be analyzed as computation mechanisms generating, preserving and accelerating the increase in organized complexity. The main unifying theme for big history is the energy rate density, a metric based on thermodynamics. However useful, this metric does not provide much insight into the role that information and computation play in our universe. The concept of "logical depth" provides a new lens to examine the increase of organized complexity. We argue in this paper that organized complexity is a valid and useful way to make sense of big history. Additionally, logical depth has a rigorous formal definition in theoretical computer science that hints at a broader research program to quantify complexity in the universe. Keywords: organized complexity, Kolmogorov complexity, logical depth, big history, cosmic evolution, evolution, complexity, complexification, computation, artificial life, philosophy of information

cs.OH↗
arXiv2015

Two-Dimensional Kolmogorov Complexity and Validation of the Coding Theorem Method by Compressibility

We propose a measure based upon the fundamental theoretical concept in algorithmic information theory that provides a natural approach to the problem of evaluating $n$-dimensional complexity by using an $n$-dimensional deterministic Turing machine. The technique is interesting because it provides a natural algorithmic process for symmetry breaking generating complex $n$-dimensional structures from perfectly symmetric and fully deterministic computational rules producing a distribution of patterns as described by algorithmic probability. Algorithmic probability also elegantly connects the frequency of occurrence of a pattern with its algorithmic complexity, hence effectively providing estimations to the complexity of the generated patterns. Experiments to validate estimations of algorithmic complexity based on these concepts are presented, showing that the measure is stable in the face of some changes in computational formalism and that results are in agreement with the results obtained using lossless compression algorithms when both methods overlap in their range of applicability. We then use the output frequency of the set of 2-dimensional Turing machines to classify the algorithmic complexity of the space-time evolutions of Elementary Cellular Automata.

cs.CC↗
arXiv2015

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.

q-fin.CP↗
arXiv2014

Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines

Drawing on various notions from theoretical computer science, we present a novel numerical approach, motivated by the notion of algorithmic probability, to the problem of approximating the Kolmogorov-Chaitin complexity of short strings. The method is an alternative to the traditional lossless compression algorithms, which it may complement, the two being serviceable for different string lengths. We provide a thorough analysis for all $\sum_{n=1}^{11} 2^n$ binary strings of length $n<12$ and for most strings of length $12\leq n \leq16$ by running all $\sim 2.5 \times 10^{13}$ Turing machines with 5 states and 2 symbols ($8\times 22^9$ with reduction techniques) using the most standard formalism of Turing machines, used in for example the Busy Beaver problem. We address the question of stability and error estimation, the sensitivity of the continued application of the method for wider coverage and better accuracy, and provide statistical evidence suggesting robustness. As with compression algorithms, this work promises to deliver a range of applications, and to provide insight into the question of complexity calculation of finite (and short) strings.

cs.IT↗
arXiv2013

Algorithmic Complexity for Short Binary Strings Applied to Psychology: A Primer

Since human randomness production has been studied and widely used to assess executive functions (especially inhibition), many measures have been suggested to assess the degree to which a sequence is random-like. However, each of them focuses on one feature of randomness, leading authors to have to use multiple measures. Here we describe and advocate for the use of the accepted universal measure for randomness based on algorithmic complexity, by means of a novel previously presented technique using the the definition of algorithmic probability. A re-analysis of the classical Radio Zenith data in the light of the proposed measure and methodology is provided as a study case of an application.

cs.CC↗
arXiv2013

Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures

We show that real-value approximations of Kolmogorov-Chaitin (K_m) using the algorithmic Coding theorem as calculated from the output frequency of a large set of small deterministic Turing machines with up to 5 states (and 2 symbols), is in agreement with the number of instructions used by the Turing machines producing s, which is consistent with strict integer-value program-size complexity. Nevertheless, K_m proves to be a finer-grained measure and a potential alternative approach to lossless compression algorithms for small entities, where compression fails. We also show that neither K_m nor the number of instructions used shows any correlation with Bennett's Logical Depth LD(s) other than what's predicted by the theory. The agreement between theory and numerical calculations shows that despite the undecidability of these theoretical measures, approximations are stable and meaningful, even for small programs and for short strings. We also announce a first Beta version of an Online Algorithmic Complexity Calculator (OACC), based on a combination of theoretical concepts, as a numerical implementation of the Coding Theorem Method.

cs.IT↗
arXiv2012

Unpredictability and Computational Irreducibility

We explore several concepts for analyzing the intuitive notion of computational irreducibility and we propose a robust formal definition, first in the field of cellular automata and then in the general field of any computable function f from N to N. We prove that, through a robust definition of what means "to be unable to compute the nth step without having to follow the same path than simulating the automaton or the function", this implies genuinely, as intuitively expected, that if the behavior of an object is computationally irreducible, no computation of its nth state can be faster than the simulation itself.

cs.CC↗
arXiv2011

Numerical Evaluation of Algorithmic Complexity for Short Strings: A Glance into the Innermost Structure of Randomness

We describe an alternative method (to compression) that combines several theoretical and experimental results to numerically approximate the algorithmic (Kolmogorov-Chaitin) complexity of all $\sum_{n=1}^82^n$ bit strings up to 8 bits long, and for some between 9 and 16 bits long. This is done by an exhaustive execution of all deterministic 2-symbol Turing machines with up to 4 states for which the halting times are known thanks to the Busy Beaver problem, that is 11019960576 machines. An output frequency distribution is then computed, from which the algorithmic probability is calculated and the algorithmic complexity evaluated by way of the (Levin-Zvonkin-Chaitin) coding theorem.

cs.IT↗
arXiv2011

Un metodo estable para la evaluacion de la complejidad algoritmica de cadenas cortas

It is discussed and surveyed a numerical method proposed before, that alternative to the usual compression method, provides an approximation to the algorithmic (Kolmogorov) complexity, particularly useful for short strings for which compression methods simply fail. The method shows to be stable enough and useful to conceive and compare patterns in an algorithmic models. (article in Spanish)

cs.CC↗
arXiv2011

Image Characterization and Classification by Physical Complexity

We present a method for estimating the complexity of an image based on Bennett's concept of logical depth. Bennett identified logical depth as the appropriate measure of organized complexity, and hence as being better suited to the evaluation of the complexity of objects in the physical world. Its use results in a different, and in some sense a finer characterization than is obtained through the application of the concept of Kolmogorov complexity alone. We use this measure to classify images by their information content. The method provides a means for classifying and evaluating the complexity of objects by way of their visual representations. To the authors' knowledge, the method and application inspired by the concept of logical depth presented herein are being proposed and implemented for the first time.

cs.CC↗
arXiv2011

Sloane's Gap. Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS

The Online Encyclopedia of Integer Sequences (OEIS) is made up of thousands of numerical sequences considered particularly interesting by some mathematicians. The graphic representation of the frequency with which a number n as a function of n appears in that database shows that the underlying function decreases fast, and that the points are distributed in a cloud, seemingly split into two by a clear zone that will be referred to here as "Sloane's Gap". The decrease and general form are explained by mathematics, but an explanation of the gap requires further considerations.

math.PR↗
arXiv2010

On the Kolmogorov-Chaitin Complexity for short sequences

A drawback of Kolmogorov-Chaitin complexity (K) as a function from s to the shortest program producing s is its noncomputability which limits its range of applicability. Moreover, when strings are short, the dependence of K on a particular universal Turing machine U can be arbitrary. In practice one can approximate it by computable compression methods. However, such compression methods do not always provide meaningful approximations--for strings shorter, for example, than typical compiler lengths. In this paper we suggest an empirical approach to overcome this difficulty and to obtain a stable definition of the Kolmogorov-Chaitin complexity for short sequences. Additionally, a correlation in terms of distribution frequencies was found across the output of two models of abstract machines, namely unidimensional cellular automata and deterministic Turing machine.

cs.CC↗
arXiv2010

An algorithmic information-theoretic approach to the behaviour of financial markets

Using frequency distributions of daily closing price time series of several financial market indexes, we investigate whether the bias away from an equiprobable sequence distribution found in the data, predicted by algorithmic information theory, may account for some of the deviation of financial markets from log-normal, and if so for how much of said deviation and over what sequence lengths. We do so by comparing the distributions of binary sequences from actual time series of financial markets and series built up from purely algorithmic means. Our discussion is a starting point for a further investigation of the market as a rule-based system with an 'algorithmic' component, despite its apparent randomness, and the use of the theory of algorithmic probability with new tools that can be applied to the study of the market price phenomenon. The main discussion is cast in terms of assumptions common to areas of economics in agreement with an algorithmic view of the market.

q-fin.TR↗
arXiv2010

On the Algorithmic Nature of the World

We propose a test based on the theory of algorithmic complexity and an experimental evaluation of Levin's universal distribution to identify evidence in support of or in contravention of the claim that the world is algorithmic in nature. To this end we have undertaken a statistical comparison of the frequency distributions of data from physical sources on the one hand--repositories of information such as images, data stored in a hard drive, computer programs and DNA sequences--and the frequency distributions generated by purely algorithmic means on the other--by running abstract computing devices such as Turing machines, cellular automata and Post Tag systems. Statistical correlations were found and their significance measured.

cs.CC↗
arXiv2010

Towards a stable definition of Kolmogorov-Chaitin complexity

Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the Kolmogorov-Chaitin complexity of a string s. Some attempts have been made to arrive at a framework stable enough for a concrete definition of K, independent of any constant under a programming language, by appealing to the "naturalness" of the language in question. The aim of this paper is to present an approach to overcome the problem by looking at a set of models of computation converging in output probability distribution such that that "naturalness" can be inferred, thereby providing a framework for a stable definition of K under the set of convergent models of computation.

cs.IT↗
arXiv2009

Scatter and regularity imply Benford's law... and more

A random variable (r.v.) X is said to follow Benford's law if log(X) is uniform mod 1. Many experimental data sets prove to follow an approximate version of it, and so do many mathematical series and continuous random variables. This phenomenon received some interest, and several explanations have been put forward. Most of them focus on specific data, depending on strong assumptions, often linked with the log function. Some authors hinted - implicitly - that the two most important characteristics of a random variable when it comes to Benford are regularity and scatter. In a first part, we prove two theorems, making up a formal version of this intuition: scattered and regular r.v.'s do approximately follow Benford's law. The proofs only need simple mathematical tools, making the analysis easy. Previous explanations thus become corollaries of a more general and simpler one. These results suggest that Benford's law does not depend on properties linked with the log function. We thus propose and test a general version of the Benford's law. The success of these tests may be viewed as an a posteriori validation of the analysis formulated in the first part.

math.PR↗
arXiv2009

French Roadmap for complex Systems 2008-2009

This second issue of the French Complex Systems Roadmap is the outcome of the Entretiens de Cargese 2008, an interdisciplinary brainstorming session organized over one week in 2008, jointly by RNSC, ISC-PIF and IXXI. It capitalizes on the first roadmap and gathers contributions of more than 70 scientists from major French institutions. The aim of this roadmap is to foster the coordination of the complex systems community on focused topics and questions, as well as to present contributions and challenges in the complex systems sciences and complexity science to the public, political and industrial spheres.

nlin.AO↗