Cellular automata composition techniques for spatial dynamics simulation

Abstract

Till now, no mathematical method and no promising approach is known to CA synthesis from a given description of its evolution. Nevertheless, some way out should be found. A simple one is to follow a well known approach used in PDE theory which implies composing PDE systems out of a set of differential operators and functions. Such an approach seems to be expedient when considering the following similarities between CA composition and PDE system construction. For example, first order and second order differential operators in PDEs over the space have their CA counterparts in the form of shift and diffusion local operators, respectively. And in both cases for obtaining a mathematical model of reaction? diffusion process those operators are composed with nonlinear reaction functions. Unfortunately, the above similarities are only particular cases. In general, there is no formal procedure to obtain a CA simulating space-time nonlinear behavior. It is just the fact that has provoked the development of compatible algebraic operations on cellular arrays, allowing to integrate continuous functions into a CA composition techniques. But the most important destination of CA composition is not in presenting another way of simulating processes which may be described in terms of PDE, but in obtaining capability of constructing mathematical models for those phenomena, for whom no other mathematical description is known. Such a processes are mostly associated with the fields of science which are in the initial stage of development. For example, plant growth mechanisms, embryo fetation, cellular division, morphogenesis? from biology; surface oxidation, chemical reaction on catalyst, dissociation, adsorption ? from chemistry; epitaxial growth, crack formation, rubber deformation, robotics ? from engineering; tumor growth ? from medicine, etc. Of course, the available experience in science and engineering is not sufficient to forecast the future of CA simulation methodology. Anyway, now it is clear that only a small part of the huge amount of CA-models have evolutions which resemble natural phenomena, and, hence, may be used for simulation. Moreover, those, which occur to be helpful, ought to be enriched by some additional properties, such as probability in transition functions, complicated modes of operations, composite alphabet, nonhomogeneous cellular space, etc. All these, being oriented to obtain CA-models of complex phenomena, require a unique formalism for composing complex CAmodels from a number of more simple ones. The above considerations allow to hope that the presented attempt to construct a systematic approach to CA composition is not futile. © 2010 Springer-Verlag Berlin Heidelberg.