A Language for Particle Interactions in Rule 54 and Other Cellular Automata

Abstract

This is a study of localised structures in one-dimensional cellular automata, with the elementary cellular automaton Rule 54 as a guiding example. A formalism for particles on a periodic background is derived, applicable to all one-dimensional cellular automata. One can compute which particles collide and in how many ways. One can also compute the fate of a particle after an unlimited number of collisions - whether they only produce other particles, or the result is a growing structure that destroys the background pattern. For Rule 54, formulas for the four most common particles are given and all two-particle collisions are found. We show that no other particles arise, which particles are stable and which can be created, provided that only two particles interact at a time. More complex behaviour of Rule 54 requires therefore multi-particle collisions.