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 …

In this paper, we show that the problem of determining if the identity matrix belongs to a
finitely generated semigroup of 2× 2 matrices from the modular group PSL2 (ℤ) and thus the …

In this paper we study several closely related fundamental problems for words and matrices.
First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of …

The main result of this paper is the decidability of the membership problem for 2× 2
nonsingular integer matrices. Namely, we will construct the first algorithm that for any …

We study the identity problem for matrices, ie, whether the identity matrix is in a semigroup
generated by a given set of generators. In particular we consider the identity problem for the …

This paper deals with the decidability of semigroup freeness. More precisely, the freeness
problem over a semigroup S is defined as: given a finite subset X⊆ S, decide whether each …

We show that the problem of determining if the identity matrix belongs to a finitely generated
semigroup of 2× 2 matrices from the modular group PSL 2 (Z), the Special Linear group SL 2 …

Most computational problems for matrix semigroups and groups are inherently difficult to
solve and even undecidable starting from dimension three. The questions about the …

We study the freeness problem for matrix semigroups. We show that the freeness problem is
decidable for upper-triangular 2× 2 matrices with rational entries when the products are …

The decision problems on matrices were intensively studied for many decades as matrix
products play an essential role in the representation of various computational processes …