Recent advances in algorithmic problems for semigroups

TL;DR

This survey reviews recent progress on the decidability and complexity of algorithmic problems in matrix semigroups, especially within groups with specific structures like low-dimensional, commutative, nilpotent, or solvable groups.

Contribution

It provides a comprehensive overview of the current state of decidability results and complexity classifications for algorithmic problems in structured matrix semigroups.

Findings

Decidability varies with group constraints and structure.

Certain problems become decidable in low-dimensional or structured groups.

The survey highlights open problems and recent breakthroughs in the field.

Abstract

In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite group $G$, often represented as a matrix group. Such problems might not be decidable in general. In fact, they gave rise to some of the earliest undecidability results in algorithmic theory. However, the situation changes when the group $G$ satisfies additional constraints. In this survey, we give an overview of the decidability and the complexity of several algorithmic problems in the cases where $G$ is a low-dimensional matrix group, or a group with additional structures such as commutativity, nilpotency and solvability.