Number Conservation via Particle Flow in One-dimensional Cellular Automata

Article type: Research Article

Authors: Redeker, Markus^{; *}

Affiliations: Hamburg, Germany. markus2.redeker@mail.de

Correspondence: [*] Address for correspondence: markus2.redeker@mail.de

Abstract: A number-conserving cellular automaton is a simplified model for a system of interacting particles. This paper contains two related constructions by which one can find all one-dimensional number-conserving cellular automata with one kind of particle. The output of both methods is a “flow function”, which describes the movement of the particles. In the first method, one puts increasingly stronger restrictions on the particle flow until a single flow function is specified. There are no dead ends, every choice of restriction steps ends with a flow. The second method uses the fact that the flow functions can be ordered and then form a lattice. This method consists of a recipe for the slowest flow that enforces a given minimal particle speed in one given neighbourhood. All other flow functions are then maxima of sets of these flows. Other questions, like that about the nature of non-deterministic number-conserving rules, are treated briefly at the end.

DOI: 10.3233/FI-222129

Journal: Fundamenta Informaticae, vol. 187, no. 1, pp. 31-59, 2022