Finite and Infinite Computations and a Classification of Two-Dimensional Cellular Automata Using Infinite Computations

This paper proposes an application of the Infinite Unit Axiom and grossone, introduced by Yaroslav Sergeyev (see [19-23]), to the development and classification of two-dimensional cellular automata. This application establishes, by the application of grossone, a new and more precise nonarchimedean metric on the space of definition for two-dimensional cellular automata, whereby the accuracy of computations is increased. Using this new metric, open disks are defined and the number of points in each disk computed. The forward dynamics of a cellular automaton map are also studied by defined sets. It is also shown that using the Infinite Unit Axiom, the number of configurations that follow a given configuration, under the forward iterations of the cellular automaton map, can now be computed and hence a classification scheme developed based on this computation.