In the monograph, we present our main results of the work we have done in mathematical theory of the classical homogeneous structures (HS; synonym Cellular Automata) during 1968 - 2008. Homogeneous Structure (HS) - a parallel information processing system consisting of intercommunicating identical finite automata. Although homogeneous structures will be used throughout this monograph as the usual term, it is necessary to keep in mind, that cellular automata (CA), iterative networks etc. are essentially synonyms. We can interpret HS as a theoretical framework of artificial parallel information processing systems. From the logical point of view the HS is an infinite automaton with characteristic internal structure. The HS theory can be considered as a structural and dynamical theory of the infinite automata. HS can serve as the excellent basis for modeling of many discrete processes and they present interesting enough independent objects for investigations as well. In recent years has arisen undoubted interest to the HS theory, and in this direction many remarkable results have been obtained. Much of this work has been activated by the growing interest in computer science and mathematical modelling. At present, the HS theory forms an original part of the modern mathematical cybernetics.

HS (Cellular Automata) are suitable models for study and simulation of complex dynamical behaviors of systems with a great number of interacting components. The existence of many fields of applications (specifically, computer science, biology, physics, mathematics, social sciences, and large quantity of other fields) proves this relevance and interest. The possibility of simulation on modern parallel computers allows us to make easy numerical experiments, and the usage of classical tools in combinatorial theory, graph theory, and theory of discrete random processes lead to a great number of theoretical results.

The monograph is the second revised and supplemented edition of our previous monograph, founded on a special lecture course "Classical Homogeneous Structures (Classical Cellular Automata)" that has been given for students and graduate students of faculty of mathematics and informatics at the Grodno State University during April - May, 2008.

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