Related papers: Kolmogorov Last Discovery? (Kolmogorov and Algorit…

In classical information theory, entropy rate and Kolmogorov complexity per symbol are related by a theorem of Brudno. In this paper, we prove a quantum version of this theorem, connecting the von Neumann entropy rate and two notions of…

We present an overview of some results about characterization of compactness in which the concept of approximation scheme has had a role. In particular, we present several results that were proved by the second author, jointly with Luther,…

There is a parallelism between Shannon information theory and algorithmic information theory. In particular, the same linear inequalities are true for Shannon entropies of tuples of random variables and Kolmogorov complexities of tuples of…

April 25, 2003, marked the 100th anniversary of the birth of Andrei Nikolaevich Kolmogorov, the twentieth century's foremost contributor to the mathematical and philosophical foundations of probability. The year 2003 was also the 70th…

We survey and introduce concepts and tools located at the intersection of information theory and network biology. We show that Shannon's information entropy, compressibility and algorithmic complexity quantify different local and global…

Randomness extraction is the process of constructing a source of randomness of high quality from one or several sources of randomness of lower quality. The problem can be modeled using probability distributions and min-entropy to measure…

In a 1985 commentary to his collected works, Kolmogorov informed the reader that his 1932 paper 'On the interpretation of intuitionistic logic' "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic…

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…

Since Kolmogorov's theory, turbulence has been studied using various methods, many of which could be now be understood in a probabilistic framework. Herein, a comprehensive review of the advances made on stochastic theory of turbulence…

The idea to find the "maximal number that can be named" can be traced back to Archimedes (see his Psammit). From the viewpoint of computation theory the natural question is "which number can be described by at most n bits"? This question…

Research into active networking has provided the incentive to re-visit what has traditionally been classified as distinct properties and characteristics of information transfer such as protocol versus service; at a more fundamental level…

We study in which way Kolmogorov complexity and instance complexity affect properties of r.e. sets. We show that the well-known 2log n upper bound on the Kolmogorov complexity of initial segments of r.e.\ sets is optimal and characterize…

So-called separation automata are in the core of several recently invented quasi-polynomial time algorithms for parity games. An explicit $q$-state separation automaton implies an algorithm for parity games with running time polynomial in…

The main result is that: function descriptions are not made equal, and they can be categorised in at least two categories using various computational methods for function evaluation. The result affects Kolmogorov complexity and Random…

We study the relations between the notions of highness, lowness and logical depth in the setting of complexity theory. We introduce a new notion of polylog depth based on time bounded Kolmogorov complexity. We show polylog depth satisfies…

We provide a refined characterization of the super-Turing computational power of analog, evolving, and stochastic neural networks based on the Kolmogorov complexity of their real weights, evolving weights, and real probabilities,…

Andrei Kolmogorov's Grundbegriffe der Wahrscheinlichkeits-rechnung put probability's modern mathematical formalism in place. It also provided a philosophy of probability--an explanation of how the formalism can be connected to the world of…

Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program…

Mutual information I in infinite sequences (and in their finite prefixes) is essential in theoretical analysis of many situations. Yet its right definition has been elusive for a long time. I address it by generalizing Kolmogorov Complexity…

We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing…