Continuous-Valued Cellular Automata in Two Dimensions

We explore a variety of two-dimensional continuous-valued cellular automata (CAs). We discuss how to derive CA schemes from differential equations and look at CAs based on several kinds of nonlinear wave equations. In addition we cast some of Hans Meinhardt’s activator-inhibitor reaction-diffusion rules into two dimensions. Some illustrative runs of CAPOW, a. CA simulator, are presented. A cellular automaton, or CA, is a computation made up of finite elements called cells. Each cell contains the same type of state. The cells are updated in parallel, using a rule which is homogeneous, and local. In slightly different words, a CA is a computation based upon a grid of cells, with each cell containing an object called a state. The states are updated in discrete steps, with all the cells being effectively updated at the same time. Each cell uses the same algorithm for its update rule. The update algorithm computes a cell’s new state by using information about the states of the cell’s nearby space-time neighbors, that is, using the state of the cell itself, using the states of the cell’s nearby neighbors, and using the recent prior states of the cell and its neighbors. The states do not necessarily need to be single numbers, they can also be data structures built up from numbers. A CA is said to be discrete valued if its states are built from integers, and a CA is continuous valued if its states are built from real numbers. As Norman Margolus and Tommaso Toffoli have pointed out, CAs are well suited for modeling nature [7]. The parallelism of the CA update process mirrors the uniform flow of time. The homogeneity of the CA update rule across all the cells corresponds to the universality of natural law. And the locality of CAs reflect the fact that nature seems to forbid action at a distance. The use of finite space-time elements for CAs are a necessary evil so that we can compute at all. But one might argue that the use of discrete states is an unnecessary evil.