While Kolmogorov complexity is the accepted absolute measure of information content in an individual finite object, a similarly absolute notion is needed for the information distance between two individual objects, for example, two pictures. We give several natural definitions of a universal information metric, based on length of shortest programs for either ordinary computations or reversible (dissipationless) computations. It turns out that these definitions are equivalent up to an additive logarithmic term. We show that the information distance is a universal cognitive similarity distance. We investigate the maximal correlation of the shortest programs involved, the maximal uncorrelation of programs (a generalization of the Slepian-Wolf theorem of classical information theory), and the density properties of the discrete metric spaces induced by the information distances. A related distance measures the amount of nonreversibility of a computation. Using the physical theo...

The average complexity of any algorithm whatsoever (provided it always terminates) under the universal distribution is of the same order of magnitude as the worst-case complexity. This holds both for time complexity and for space complexity. To focus our discussion, we use as illustrations the particular case of sorting algorithms, and the general case of the average case complexity of NPcomplete problems. 1

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Any notion of effective randomness that is defined with respect to arbitrary computable probability measures canonically induces an equivalence relation on such measures for which two measures are considered equivalent if their respective classes of random elements coincide. Elaborating on work of Bienvenu [1], we determine all the implications that hold between the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness, and weak randomness, and the equivalence and consistency relations of probability measures, except that we do not know whether two computable probability measures need to be equivalent in case their respective concepts of weakly randomness coincide. Keywords: computable probability measures, Martin-Löf randomness, computable randomness, Schnorr randomness, weak randomness, equivalence of probability measures, consistency of probability measures.

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Our main aim is to propose a new characterization for the software development process. We suggest that software development methodology has some limits. These limits are a clue that software development process is more subjective and empirical than objective and formal. We use Kolmogorov complexity to develop the formal argument and to outline the informal conclusions. Kolmogorov complexity is based on the size in bits of the smallest e ective description of an object and is a suitable quantitative measure of the object's information content.

Abstract. We study the equivalence relations on probability measures corresponding respectively to having the same Martin-Löf random reals, having the same Kolmogorov-Loveland random reals, and having the same computably random reals. In particular, we show that, when restricted to the class of strongly positive generalized Bernoulli measures, they all coincide with the classical equivalence, which requires that two measures have the same nullsets. 1

Abstract VLSI f-models allow the switching time to decrease to f(D) when the length of all wires is restricted by D.

Abstract This paper offers some new results on randomness with respect to classes of measures, along with a didactical exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli

This paper presents a proposal for the application of Kolmogorov complexity to the characterization of systems and processes, and the evaluation of computational models. The methodology developed represents a theoretical tool to solve problems from systems science. Two applications of the methodology are presented in order to illustrate the proposal, both of which were developed by the authors. The first one is related to the software development process, the second to computer animation models. In the end a third application of the method is briefly introduced, with the intention of characterizing dynamic systems of chaotic behavior, which clearly demonstrates the potentials of the methodology.