Directional movement of a collective of compassless automata on square lattice of width 2

TL;DR

This paper investigates the ability of a collective of simple, compassless automata and pebbles to sustain directed movement on a narrow two-dimensional lattice, establishing the minimal conditions needed for such movement.

Contribution

It demonstrates that a collective of one automaton and four pebbles can maintain directed movement, whereas fewer pebbles are insufficient, clarifying the minimal requirements for directionality.

Findings

A single automaton with up to three pebbles cannot sustain directed movement.

A single automaton with four pebbles can maintain directed movement.

The study defines minimal conditions for collective movement without a compass.

Abstract

We study the following problem: Can a collective of finite automata maintain directed movement on a two-dimensional integer lattice of width 2, where the elements (vertices) are anonymous? The automata do not distinguish between vertices based on their coordinates of direction (that means each automaton has no compass). We considered collectives consisting of an automaton and some pebbles, which are automata of the simplest form, whose positions are entirely determined by the automaton. We demonstrate that a collective of one automaton and a maximum of three pebbles cannot maintain a direction of movement on the lattice. However, a collective of one automaton and four pebbles can do so.