Knapsack problem for automaton groups

TL;DR

This paper investigates the computational complexity of the knapsack problem within automaton groups, proving undecidability for certain classes and decidability for others, thus advancing understanding of algorithmic problems in group theory.

Contribution

It establishes the undecidability of the knapsack problem for groups generated by Mealy automata and shows decidability for bounded automaton groups.

Findings

Knapsack problem is undecidable for groups generated by Mealy automata.

Decidability of the knapsack problem is proven for bounded automaton groups.

A graph construction helps determine solutions in specific automaton groups.

Abstract

The knapsack problem is a classic optimisation problem that has been recently extended in the setting of groups. Its study reveals to be interesting since it provides many different behaviours, depending on the considered class of groups. In this paper we deal with groups generated by Mealy automata-a class that is often used to study group-theoretical conjectures-and prove that the knapsack problem is undecidable for this class. In a second time, we construct a graph that, if finite, provides a solution to the knapsack problem. We deduce that the knapsack problem is decidable for the so-called bounded automaton groups, a class where the order and conjugacy problems are already known to be decidable.