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

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

Complexité de Kolmogorov, une mise en perspective. Partie II : Classification, Traitement de l' Information et Dualité.

50 pagesInternational audienceNous exposons différentes approches du concept d'information : depuis la notion d'entropie de Shannon jusqu'à la théorie de la complexité de Kolmogorov. Nous présentons deux des principales applications de la complexité de Kolmogorov : l'aléatoirité et la classification. Cet exposé est divisé en deux parties publiées dans un même volume. La partie II est consacrée à la relation entre la logique et les systèmes d'information, dans le cadre de la théorie algorithmique de l'information de Kolmogorov. Nous exposons une application récente de la complexité de Kolmogorov : la classification par compression, qui met en oeuvre une implémentation audacieuse de la complexité de Kolmogorov, par des auteurs comme Bennett, Vitányi et Cilibrasi. Cela permet, en outre, de dégager en quoi la complexité de Kolmogorov est liée à la classification, tout comme elle fonde l'aléatoirité. Nous présentons également une autre approche de la classification, la “Google classification”. Celle-ci utilise une autre idée originale et particulièrement intéressante, et qui est connectée à la classification par compression et à la complexité de Kolmogorov d'un point de vue conceptuel. Nous présentons et unifions ces différentes approches de la classification en termes de modes opératoires Bottom-Up versus Top-Down dont nous indiquons les principes fondamentaux et la dualité sous-jacente. Nous étudions comment ces modes duals sont utilisés, dans l'appréhension des systèmes d'information, et tout particulièrement dans le modèle relationnel des bases de données introduit par Codd dans les années 70. Nous réinterprétons en outre ces modes opératoires dans le contexte du schéma de compréhension de la théorie axiomatique des ensembles ZF. Ceci nous amène à développer en quoi la complexité de Kolmogorov est liée à l'intentionnalité, à l'abstraction, à la classification et aux systèmes d'information

Refinment of the "up to a constant" ordering using contructive co-immunity and alike. Application to the Min/Max hierarchy of Kolmogorov complexities

We introduce orderings between total functions f,g: N -> N which refine the pointwise "up to a constant" ordering <=cte and also insure that f(x) is often much less thang(x). With such orderings, we prove a strong hierarchy theorem for Kolmogorov complexities obtained with jump oracles and/or Max or Min of partial recursive functions. We introduce a notion of second order conditional Kolmogorov complexity which yields a uniform bound for the "up to a constant" comparisons involved in the hierarchy theorem.Comment: 41 page

Set theoretical Representations of Integers, I

We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations), in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple representation of integers that we suitably effectivize in order to develop an associated Kolmogorov theory. Such effectivizations are particular instances of a general notion of "self-enumerated system" that we introduce in this paper. Our main result asserts that, with such effectivizations, Kolmogorov theory allows to quantitatively distinguish the underlying semantics. We characterize the families obtained by such effectivizations and prove that the associated Kolmogorov complexities constitute a hierarchy which coincides with that of Kolmogorov complexities defined via jump oracles and/or infinite computations. This contrasts with the well-known fact that usual Kolmogorov complexity does not depend (up to a constant) on the chosen arithmetic representation of integers, let it be in any base unary, binary et so on. Also, in a conceptual point of view, our result can be seen as a mean to measure the degree of abstraction of these diverse semantics.Comment: 56 page

Kolmogorov complexity in perspective

We survey the diverse approaches to the notion of information content: from Shannon entropy to Kolmogorov complexity. The main applications of Kolmogorov complexity are presented namely, the mathematical notion of randomness (which goes back to the 60's with the work of Martin-Lof, Schnorr, Chaitin, Levin), and classification, which is a recent idea with provocative implementation by Vitanyi and Cilibrasi.Comment: 37 page