In the monograph we present certain results of the work we have done in mathematical theory of the classical homogeneous structures (HS; English synonym "Cellular Automata (CA)") during 1968-2008. These results form the essential enough part of the HS problems. In particular, we have considered such problems as problem of nonconstructibility, decomposition of global transition functions, extremal constructive opportunities, complexity of finite configurations, parallel formal grammars and languages defined by HS, problem of modeling in classical HS, certain applied aspects of the HS problems, etc. At present, the HS problems is the well enough developed independent sphere of the modern mathematical cybernetics, which has considerable field of numerous applications.

The HS is a parallel information processing system consisting of intercommunicating identical finite automata. Although homogeneous structures (HS) are used throughout this monograph as the usual term, it should be borne in mind that cellular automata, iterative networks, systolic structures, 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 be considered 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.