A theorem about the algorithm of minimization of differences for multicomponent cellular automata

Abstract

Multicomponent Cellular Automata, also known as Macroscopic Cellular Automata, characterize a methodological approach for modeling complex systems, that need many components both for the states (substates) to account for different properties of the cell and for the transition function (elementary processes) in order to account for various different dynamics. Many applications were developed for modeling complex natural phenomena, particularly macroscopic ones, e.g., large scale surface flows. Minimizing the differences of a certain quantity in the cell neighborhood, by distribution from the cell to the other neighboring cells, is a basic component of many transition functions in this context. The Algorithm for the Minimization of Differences (AMD) was applied in different ways to many models. A fundamental theorem about AMD is proved in this paper; it shows that AMD properties are more extended than the previous demonstrated theorem. © 2012 Springer-Verlag Berlin Heidelberg.