Exact Discretization of 3-Speed Rational Signal Machines into Cellular Automata

Cellular Automata (CA) operate in discrete time and space whereas Signal Machines (SM) have been developed as a continuous idealization of CA capturing the key concept of signals/particles and collisions. Inside a Euclidean space, dimensionless signals move freely; collisions are instantaneous. Today's issue is the automatic generation of a CA mimicking a given SM. On the one hand, many ad hoc manual conversions exist. On the other hand, some irrational or 4+-speed SM exhibit Zeno-like behaviors/space-time compression or rely on information being locally unbounded, both being incompatible with CA. This article provides a solution to automatically generate an exactly mimicking CA for a restricted class of SM: the ones that uses only three rational speeds, and rational initial positions. In these SM, signals are always contained inside a regular mesh. The discretization brings forth the corresponding discrete mesh. The simulation is valid on any infinite run and preserves the relative position of collisions.