Symmetric Large Systems of Finite Automata

A definition is introduced for convergence as $n\to\infty$ of symmetric systems consisting of a large number $n$ of stochastic finite automata, depending on which time intervals this convergence is being examined. Sufficient conditions are derived for the existence in a specified sense of a complete hierarchy of intervals, tending to infinity, on which the behavior of the systems has a limit as $n\to\infty$. A class of systems is put into consideration, in which the interaction of the automata is based on their forming random groups conducting themselves independently of each other; for such systems sufficient conditions are given for their convergence and formulas are given for the limit systems.