Complexity and stable evolution of circuits

We consider the viability problem for random dynamical systems, in particular, for circuits. A system is viable only if the system state stays in a prescribed domain Π of a phase space. We assume that the circuit structure is coded by a code evolving in time. We introduce the notion of stable evolution of the code and the system: evolution is stable if there is a δ > 0 such that the probability PT to be in Π within time interval [0, T ] satisfies PT > δ as T →∞. We show that for certain large classes of systems, the stable evolution has the following fundamental property: the Kolmogorov complexity of the code cannot be bounded by a constant as time t → ∞. For circuit models, we describe examples of stable evolution of complicated boolean networks for a difficult case when the domain Π is unknown. We dedicate this paper to Professor Grisha Mints. We admire the breadth of his interests.

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