A Language for Particle Interactions in One-dimensional Cellular Automata

TL;DR

This paper develops a formal language to describe and analyze particle interactions in one-dimensional cellular automata, exemplified by Rule 54, enabling precise computation of collision outcomes and particle stability.

Contribution

It introduces a universal formalism for particles in 1D cellular automata, allowing detailed analysis of collisions, stability, and growth of structures, with explicit formulas for Rule 54.

Findings

Formulas for four main particles in Rule 54

All two-particle collision outcomes identified

No additional particles arise beyond the four studied

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.