Classical Cellular Automata: Mathematical Theory and Applications book cover

Classical Cellular Automata: Mathematical Theory and Applications

In the book we present some of results of the work we have done in theory of classical Cellular Automata (CA) and their appendices during 1969–2013 in truth with considerable pauses. These results at present form essential constituent of the CA problems. In particular, we have studied such problems as the nonconstructability problem in CA, decomposition of global transition functions in CA, extremal constructive possibilities, complexity of finite configurations and global transition functions, parallel formal grammars along with languages defined by CA, the modelling problem in the classical CA, computer simulation of CA, certain applied aspects of CA, etc. At present, the CA problems is a rather well developed independent sphere of the mathematical cybernetics which has considerable field of numerous appendices. At that, with the equal right the CA problems can be considered as a component of such fields as discrete mathematics, the discrete parallel dynamic systems, complex systems

Genre: Math, Science
Year:
2014
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Introduction Chapter 1. The basic concepts of classical cellular automata Chapter 2. Nonconstructability problem in classical cellular automata (classical CA models) 2.1. Preliminary information on the CA problematics 2.2. The nonconstructability types for classical CA models 2.3. Existence criteria of the basic nonconstructability types in classical CA models 2.4. The nonconstructability problem for finite CA models and CA models on splitting 2.5. The reversibility problem of dynamics of classical CA models 2.6. Algorithmical aspects of the nonconstructability problem and some connected questions of dynamics of classical CA models Chapter 3. Extremal constructive opportunities of classical cellular automata 3.1. Universal finite configurations in classical CA models 3.2. Self–reproduction of finite configurations in classical CA models Chapter 4. The complexity problem of finite configurations in classical CA models Chapter 5. Parallel formal grammars and languages determined by classical cellular automata (CA models) 5.1. The basic properties of the parallel languages, determined by classical cellular automata 5.2. Parallel grammars determined by classical CA models in comparison with formal grammars of some other classes and types 5.3. Parallel grammars defined by nondeterministic CA models 5.4. Algorithmical problems of the theory of parallel grammars, determined by classical CA models Chapter 6. The modelling problem in classical cellular automata (CA) along with the related questions 6.1. Concepts of modelling in classical CA 6.2. Modelling of the well–known formal processing algorithms of words in finite alphabets by means of CA models 6.3. Simulating of classical CA models by means of CA models of the same class 6.4. The formal parallel algorithms determined by classical one–dimensional CA models 6.5. Special questions of simulating in classical CA models concerning their dynamics 6.6. Sketch on sofrware oriented on CA simulating Chapter 7. The decomposition problem of global transition functions in classical CA models 7.1. Decomposition of special global transition functions in classical CA models 7.2. Some approaches to solution of the general decomposition problem of global transition functions 7.3. Questions of solvability of the decomposition problem for global transition functions of CA models 7.4. The complexity problem for global transition functions in classical CA models Chapter 8. Certain applied aspects of the CA problematics 8.1. Solution of the Steinhaus combinatory problem 8.2. Solution of the Ulam problem from number theory 8.3. Certain applied aspects of CA models in biological sciences Conclusion References About the author
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Victor Aladjev

Aladjev Victor Zakharovich – President of the Baltic Branch of the International Academy of Noosphere, Prof., DSc in Mathematics. He was born on June 14, 1942 in Grodno and was baptized on July 14, 1942 in the Farny catholic Church in Grodno (West Belarus). In 1959 he entered the first year of the Physics and Mathematics Faculty of the University of Grodno, and in 1962 he was transferred to the Mathematics Department of the University of Tartu (Estonia), which he successfully graduated in 1966 with a degree in Mathematics. In 1969 Aladjev entered the graduate school of the Academy of Sciences of the ESSR with a degree in Theory of Probability and Mathematical Statistics, which successfully graduated in 1972 in two specialties at once, Theoretical Cybernetics and Technical Cybernetics. In 1972 he was awarded a doctorate in mathematics (DSc) from Prof. R. Bellman (USA) for his work “Mathematical Theory of the Homogenious Structures and Their Applications”; the given work was recognized as the best in the Academy Sciences of the ESSR at 1972. Since 1970, Aladjev V. – President of the Tallinn Research Group (TRG) organized by him, whose scientific results subsequently have received certain international recognition, primarily in the field of researches on the mathematical theory of homogeneous structures (Cellular Automata). Cellular automata (CAs), initially being the root cause of the formation of the TRG, over time became a less significant field of scientifical and applied activity of the Group, inferior to more priority fields, in particular computer mathematics systems, although quite often interest in the CAs problematics arose again for particular periods of duration; however, the intensity of researches in this direction also decreased over time. In 1972, V. Aladyev published the first monograph on the homogeneous structures theory in the USSR, which was recognized as one of the best monographical publication of the Estonian Academy of Sciences in the same year, and in 1977 was noted in Soviet Mathematical Encyclopedia and in Encyclopaedia of Physical Science and Technology. The monograph not only presented a number of original results on this problematics, but introduced the basic Russian–language terminology on cellular automata too, which is now generally accepted. From 1972 up to 1990, he held senior positions (chief engineer, deputy director for science) in a number of design, technological and research organizations in Tallinn (Estonia). Aladjev’s activities at these posts were repeatedly awarded and prizes by the Council of Ministers of the USSR, the Central statistics Committee of the USSR, the All–Union Project and Tecnological Institute of the Central Statistics Committee of the USSR, etc. Prof. V.Z. Aladjev is the basic author more than 500 scientific and scientific and technical works (including 90 monographs, textbooks and collections of articles) published in the USSR, Russia, Germany, Belarus, Estonia, Lithuania, Ukraine, the GDR, Czechoslovakia, Hungary, Japan, the USA, Holland, Bulgaria and Great Britain. Since 1972, he is referent and member of editorial board of the international mathematical journal “Zentralblatt für Mathematik” and since 1980, he is a member of IAMM (International Accociation on Mathematical Modelling). Prof. V. Aladjev is a member of the editorial boards of a number of scientific journals. He created the Estonian School for the mathematical theory of homogeneous structures, whose fundamental results received international recognition and have made certain contributions in the basis of a new division of the modern mathematical cybernetics. A lot of applied works of Aladjev V.Z. refers to computer science among which it is worth noting widely known textbooks on computer mathematics systems. Along with these original editions, he developed a large UserLib6789 library of new software tools (more than 850) for which he was won the Smart Award network award, and a large unified MathToolBox package (more 1420 tools) for Maple and Mathematica systems, respectively, which rather essentially expand the functionality of these systems. During the preparation of these books and creation of software tools for the Maple and Mathematica systems, a sufficiently wide range of the proposals for organization, functioning and set of standard tools that improve both systems was registered, certain of which were subsequently included in subsequent versions of the systems. In a number of fields (mathematics, computer science, cellular automata, mathematical packages, etc.) Aladjev V.Z. collaborates with a number of universities in the CIS under the program “Visiting Professor”. As a part of the given program and in the process of preparing a series of books on Maple and Mathematica systems, Aladjev V. for a lot of years gave cycles of lectures on these systems for students, graduate students and teachers of the universities of the Baltic States and Belarus, which became rather famous. more…

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