Complexity Column

Abstract

Algorithmic problems in groups and semigroups are a classic research area that has attracted quite a bit of attention since the seminal works by Markov and others. These results demonstrated that very natural questions such as the Group Membership problem can be undecidable even in groups of matrices, and even in decidable cases can be hard. Moreover, the exact complexity of such problems can be difficult to determine. The collection of computational problems being studied is rich and include, apart from the Group and Semigroup Membership problems, the Word problem, the Intersection problem for subgroups and subsemigroups, and many more.