Abstract geometrical computation 9: Exact discretization of 3-speed rational signal machines

International audienceIn the context of abstract geometrical computation, signalmachines have been developed as a continuous counterpart ofcellular automata capturing the notions of particles, signals andcollisions. An important issue is the automatic generation of aCA "mimicking" the dynamics of a given signal machine. On the onehand, ad hoc/manual conversions exist. On the other hand, it isnot always possible since some signal machines exhibitbehaviors (like fractal constructions or relying on an unboundeddensity of information) that are just impossible with (discrete)cellular automata.This article provides a solution to automatically generate acellular automata for a "maximal" class of signal machines thatcannot exhibit such behaviors. This scheme encompasses thediscretization of initial configuration so that the originalcontinuous dynamics and the discrete one generated are identicalup to a linear deformation. The considered signal machines arethe ones that use only three speeds and allow only rationalspeeds and initial positions. Signals in these machines arealways trapped inside a regular mesh. The construction relies ongenerating the corresponding discrete mesh. The soundness of theconstruction is proved and experimental results are provided