Integer Gradient for Cellular Automata: Principle and Examples

Abstract

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. Due to the locality of communication, only an estimation of the distance can be made at each time step, and each location. When no assumption is made on the size of the medium, that distance takes its values in N, 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 N-fields into a finite set of Z/nZfields. 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 two rules using the modulo representation of distance field: Voronoi Diagram and Convex Hull. The two rules applies for moving sources, although at the moment, the convex hull has been implemented only for static sources.

Moore and Von Neumann Neighborhood N-Dimensional Generalized Firing Squad Solutions Using Fields

“…As we can easily encode the derivative with very few values (-1, 0 or +1), then the finiteness of the distance fields is guaranteed. This has been much more intensively described in [18].…”

Distributed Execution of Automata Networks on a Computing Medium: Introducing IfAny Machines

“…The computation of the hop count can be done in bounded state using a separate distance layer on the medium, the method is presented in details in [7], it also work in a dynamic environment, the resulting spanning DAG will then be automatically updated if the support and/or the leader move during execution. Note that however, the method assumes a synchronous update, it can nevertheless always be made to work on the quasi synchronous medium using standard synchronisation algorithms.…”

Gabriel Graphs in Arbitrary Metric Space and their Cellular Automaton for Many Grids

“…The purpose of the distance field is to give access to d(c, P) in a way that allows two neighbor points to compare their values. In Maignan and Gruau [2008], we describe a systematic way to transform such a field so that it has only a finite number of states. This transformation allows our field to use only three states, instead of whole infinite set of integers.…”

“…This transformation allows our field to use only three states, instead of whole infinite set of integers. For the sake of brevity, we only give the original distance field; interested readers can consult Maignan and Gruau [2008] for more information:…”