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.

翻译：本文综述了矩阵半群算法理论的最新进展。该研究领域的主要目标是为无限群$G$（通常表示为矩阵群）的有限生成子半群构造判定其各种性质的算法。此类问题在一般情况下可能不可判定。事实上，它们曾催生了算法理论中最早的一批不可判定性结果。然而，当群$G$满足额外约束条件时，情况会发生变化。本综述概述了当$G$为低维矩阵群或具有交换性、幂零性、可解性等附加结构的群时，若干算法问题的可判定性与计算复杂性。