Abstract geometrical computation 9: Exact discretization of 3-speed rational signal machines
Résumé
In the context of abstract geometrical computation, signal
machines have been developed as a continuous counterpart of
cellular automata capturing the notions of particles, signals and
collisions. An important issue is the automatic generation of a
CA "mimicking" the dynamics of a given signal machine. On the one
hand, ad hoc/manual conversions exist. On the other hand, it is
not always possible since some signal machines exhibit
behaviors (like fractal constructions or relying on an unbounded
density of information) that are just impossible with (discrete)
cellular automata.
This article provides a solution to automatically generate a
cellular automata for a "maximal" class of signal machines that
cannot exhibit such behaviors. This scheme encompasses the
discretization of initial configuration so that the original
continuous dynamics and the discrete one generated are identical
up to a linear deformation. The considered signal machines are
the ones that use only three speeds and allow only rational
speeds and initial positions. Signals in these machines are
always trapped inside a regular mesh. The construction relies on
generating the corresponding discrete mesh. The soundness of the
construction is proved and experimental results are provided.