Integer Gradient for Cellular Automata: Principle and Examples

When programming a spatial computing medium such as a cellular automaton, the hop count distance to some set of sources (particles) is an often used information. In particular, we consider the case where the sources themselves are moving. When no assumption is made on the size of the medium, that distance takes its values in the set of integers, which is not desirable, because it does not lead to finite state. This paper shows how to use the modulo operation to project that set of integer fields into a set of finite state fields. Using the modulo stored at each site, we show that we are still able to compute the local differential of the original field, allowing to manipulate the former as a directional gradient. It allows us to evaluate the direction of the nearest source, provided the sources move at bounded speed, less than one site per time unit. This information can be used to solve several problems of spatial nature. In the particular case of cellular automata, we present rules for two such problems: Voronoi diagram of moving points and convex hull.