The book represents mathematical theory and applications of classical cellular automata.
First of all, a few words about the terminology in the book. Today, the problematics of Cellular automata (CA) well enough is advanced, being quite independent field of modern mathematical cybernetics, having own terminology and axiomatics at existence of a rather broad sphere of various appendices. At the same time, it is necessary to emphasize, that at the assimilation of the given problematics in the Soviet Union in Russian-lingual terminology, whose the basis for the first time have been laid by us at 1970, for the concept «Cellular automata» the term «Homogeneous structures» (HS; HS-models) has been determined which nowadays is the generally accepted term together with a whole series of other notions, denotations and definitions [1,16, 119]. So, during the present monograph along with the given term its well-known Anglo-lingual equivalent the «Cellular Automata - CA» is used too.
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), iterativenetworks etc. are essentially synonyms. We can interpret HS as a theoretical basis of artificial parallel information processing systems. From the logical standpoint a HS is an infinite automaton with specific internal structure . The HS-theory can be considered as a structural and dynamical theory of the infinite automata. HS-models can serve as an excellent basis for modeling of many discrete processes , representing interesting enough independent objects for research too. Recently, the undoubted interest to the HS-problematics has arisen anew and in the given direction many remarkable results have been obtained.
So, the HS-axiomatics provides such three fundamental properties as homogeneity, localness and parallelism of functioning. If in a similar computing model we shall with each elementary automaton associate a separate microprocessor then it is possible to unrestrictedly increase the sizes of such computing system without any essential increase of temporal and constructive expenses, required for each new expansion of the computing space, and also without an y overheads connected to coordination of functioning of an arbitrary supplementary quantity of elementary microprocessors. Similar high-parallel computing models admit9practical realizations consisting of rather large number of rather elementary microprocessors which are limited not so much by certain architectural reasons as by a lot of especially economic and technologic reasons defined by a modern level of development of microelectronic technology, however with the great potentialities in the future, first of all, in light of rather intensive works in field of nanotechnology [536].
The above three such features as high homogeneity, high parallelism and locality of interactions are provided by the HS-axiomatics itself, while such property important from the physical standpoint as reversibility of dynamics is given by program way. In light of the listed properties even classical HS are high-abstract models of the real physical world, which function in a space and time. Therefore, they in many respects better than many others formal architectures can be mapped onto a lot of physical realities in their modern understanding. Moreover the HS-concept itself is enough well adapted to solution of various problems of modelling in such areas as mathematics, cybernetics, development biology, theoretical physics, computing sciences, discrete synergetics, dynamic systems theory, robotics, etc. Told and numerous examples available for today lead us to the conclusion that the HS can represent a rather serious interest as a new perspective environment of modelling and research of many discrete processes and phenomena, determined by the above properties; in addition, raising the HS-problematics onto a new interdisciplinary level and, on the other hand, as an interesting enough independent formal mathematical object of researches.
