Asymptotic Properties of the Behavior of Elementary Automata in a Game

The game for automata outlined in [M. L. Tsetlin, Uspekhi Mat. Nauk, 1963, vol. 18, no. 4, pp. 3–28] can be described by a Markov chain whose states are the sets of states of all of the automata in the game. The behavior of simple automata with deep memories can be described approximately by a Markov chain whose states are the sets of outputs (and not states) of the players. This makes it possible to investigate analytically the asymptotic properties of specific games for automata, which we will demonstrate with reference to a game of two automata and a Goore game [V. A. Borovikov and V. I. Bryzgalov, Avtomat. Telemekh., 1965, vol. 26, no. 4, pp. 683–687].