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Algorithmic Complexity and Stochastic Properties of Finite Binary Sequences
, 1999
"... This paper is a survey of concepts and results related to simple Kolmogorov complexity, prefix complexity and resourcebounded complexity. We also consider a new type of complexity statistical complexity closely related to mathematical statistics. Unlike other discoverers of algorithmic complexit ..."
Abstract

Cited by 17 (0 self)
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This paper is a survey of concepts and results related to simple Kolmogorov complexity, prefix complexity and resourcebounded complexity. We also consider a new type of complexity statistical complexity closely related to mathematical statistics. Unlike other discoverers of algorithmic complexity, A. N. Kolmogorov's leading motive was developing on its basis a mathematical theory more adequately substantiating applications of probability theory, mathematical statistics and information theory. Kolmogorov wanted to deduce properties of a random object from its complexity characteristics without use of the notion of probability. In the first part of this paper we present several results in this direction. Though the subsequent development of algorithmic complexity and randomness was different, algorithmic complexity has successful applications in a traditional probabilistic framework. In the second part of the paper we consider applications to the estimation of parameters and the definition of Bernoulli sequences. All considerations have finite combinatorial character. 1.
What is a Random Sequence
 The Mathematical Association of America, Monthly
, 2002
"... there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a ..."
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Cited by 4 (1 self)
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there laws of randomness? These old and deep philosophical questions still stir controversy today. Some scholars have suggested that our difficulty in dealing with notions of randomness could be gauged by the comparatively late development of probability theory, which had a
A random walk on water*
"... According to the traditional notion of randomness and uncertainty, natural phenomena are separated into two mutually exclusive components, random (or stochastic) and deterministic. Within this dichotomous logic, the deterministic part supposedly represents causeeffect relationships and, thus, is ph ..."
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Cited by 2 (0 self)
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According to the traditional notion of randomness and uncertainty, natural phenomena are separated into two mutually exclusive components, random (or stochastic) and deterministic. Within this dichotomous logic, the deterministic part supposedly represents causeeffect relationships and, thus, is physics and science (the “good”), whereas randomness has little relationship with science and no relationship with understanding (the “evil”). We argue that such views should be reconsidered by admitting that uncertainty is an intrinsic property of nature, that causality implies dependence of natural processes in time, thus suggesting predictability, but even the tiniest uncertainty (e.g., in initial conditions) may result in unpredictability after a certain time horizon. On these premises it is possible to shape a consistent stochastic representation of natural processes, in which predictability (suggested by deterministic laws) and unpredictability (randomness) coexist and are not separable or additive components. Deciding which of the two dominates is simply a matter of specifying the time horizon of the prediction. Long horizons of prediction are inevitably associated with high uncertainty, whose quantification relies on understanding the longterm stochastic properties of the processes.
Kolmogorov complexity and symmetric relational structures
, 2008
"... We study partitions of Fraïssé limits of classes of finite relational structures where the partitions are encoded by infinite binary strings which are random in the sense of KolmogorovChaitin. 1 ..."
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We study partitions of Fraïssé limits of classes of finite relational structures where the partitions are encoded by infinite binary strings which are random in the sense of KolmogorovChaitin. 1
The Algorithmic Theory of Randomness
, 2001
"... this paper we won't discuss this very important topic. We will focus instead on the admittedly less ambitious but more manageable question of whether it is possible at least to obtain a mathematically rigourous (and reasonable) denition of randomness. That is, in the hope of clarifying the conc ..."
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this paper we won't discuss this very important topic. We will focus instead on the admittedly less ambitious but more manageable question of whether it is possible at least to obtain a mathematically rigourous (and reasonable) denition of randomness. That is, in the hope of clarifying the concept of chance, one tries to examine a mathematical model or idealization, that might (or might not) capture some of the intuitive properties associated with randomness. In the process of rening our intuition and circunscribing our concepts, we might be able to arrive at some fundamental notions. With luck (no pun intended) , these might in turn bring some insight into the deeper problems mentioned before. At least it could help one to discard some of our previous intuitions or to decide for the need of yet another mathematical model.