In the monograph we present certain results of the work we have done in mathematical theory of the classical homogeneous structures (HS; synonym "Cellular Automata") during 1968 - 2008. Much of our research has been stimulated by scientific programs of Tallinn Research Group and International Academy of Noosphere. At present, Cellular Automata (CA) problems is the well enough developed independent field of the modern mathematical cybernetics, which has considerable sphere of applications. Above all, it is necessary to emphasize, that in Russian terminology (basics of which in our monograph were presented for the first time) for the concept "Cellular Automata (CA)" the most widespread synonym "Homogeneous Structures (HS)" is used. Therefore this term is used most widely for the space of the given monograph.

The (HS) is 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 should be borne in mind that cellular automata, 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 structural and dynamical theory of the infinite automata. (HS) can serve as the 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 modeling. At present, the (HS) theory forms an original part of the modern mathematical cybernetics.

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