Chapter 4 CELLULAR AUTOMATA WITH ERRORS: PROBLEMS for STUDENTS of PROBABILITY

ABSTRACT. This is a survey of some problems and methods in the theory of probabilistic cellular automata. It is addressed to those students who love to learn a theory by solving problems. The only prerequisite is a standard course in probability. General methods are illustrated by examples, some of which have played important roles in the development of these methods. More than a hundred exercises and problems and more than a dozen of unsolved problems are discussed. Special attention is paid to the computational aspect; several pseudo-codes are given to show how to program processes in question. We consider probabilistic systems with local interactions, which may be thought of as special kinds of Markov processes. Informally speaking, they describe the functioning of a finite or infinite system of automata, all of which update their states at every moment of discrete time, according to some deterministic or stochastic rule depending on their neighbors’ states. From statistical physics came the question of uniqueness of the limit as t→∞ behavior of the infinite system. Existence of at least one limit follows from the famous fixed-point theorem and is proved here as Theorem 1. Systems which have a unique limit behavior and converge to it from any initial condition may be said to forget everything when time tends to infinity, while the others remember something forever. The former systems are called ergodic, the latter non-ergodic. In statistical physics the non-ergodic systems help us understand conservation of non-symmetry in macro-objects. Computer scientists prefer to consider finite systems. Although all nondegenerate finite systems converge, some of them converge enormously slower than others. Some converge fast, i.e. lose practically all the information about the initial state in a time which does not depend on their size (or grows as the logarithm of their size, according to another definition), whereas some others converge slowly, i.e.can retain information during time which grows exponentially in their size. The latter may help to solve the problem of designing reliable systems consisting of unreliable elements. There is much in common between the discrete-time and continuous-time approaches to interacting processes. American readers are better acquainted with the continuous-time approach, summarized, for example, in [5]. Methods