Instability,complexity, and evolution |
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Authors: | S Vakulenko D Grigoriev |
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Institution: | (1) Institute of Mechanical Engineering Problems; North Western Institute of Printing, St. Petersburg State University of Technology and Design, St. Petersburg, Russia;(2) Université de Lille, Villeneuve d’Ascq, France |
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Abstract: | In this paper, we consider a new class of random dynamical systems that contains, in particular, neural networks and complicated
circuits. For these systems, we consider the viability problem: we suppose that the system survives only the system state
is in a prescribed domain Π of the phase space. The approach developed here is based on some fundamental ideas proposed by
A. Kolmogorov, R. Thom, M. Gromov, L. Valiant, L. Van Valen, and others. Under some conditions it is shown that almost all
systems from this class with fixed parameters are unstable in the following sense: the probability P
t
to leave Π within the time interval 0, t] tends to 1 as t → ∞. However, it is allowed to change these parameters sometimes (“evolutionary” case), then it may happen that P
t
< 1 − δ < 1 for all t (“stable evolution”). Furthermore, we study the properties of such a stable evolution assuming that the system parameters
are encoded by a dicsrete code. This allows us to apply complexity theory, coding, algorithms, etc. Evolution is a Markov
process of modification of this code. Under some conditions we show that the stable evolution of unstable systems possesses
the following general fundamental property: the relative Kolmogorov complexity of the code cannot be bounded by a constant
as t → ∞. For circuit models, we define complexity characteristics of these circuits. We find that these complexities also have
a tendency to increase during stable evolution. We give concrete examples of stable evolution. Bibliography: 80 titles.
To the memory of A. N. Livshitz
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 31–69. |
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