Matrix Semigroup Freeness Problems in $\mathrm{SL}(2,\mathbb{Z})$

TL;DR

This paper investigates the decidability and computational complexity of freeness and factorization problems in matrix semigroups within the special linear group SL(2,Z), revealing NP-hardness and PSPACE-hardness results.

Contribution

It establishes the NP-hardness of the non-freeness problem and PSPACE-hardness of factorization bounds in SL(2,Z), advancing understanding of matrix semigroup decision problems.

Findings

Deciding non-freeness is NP-hard.

Determining if matrices have at most k factorizations is PSPACE-hard.

Some factorization problems are computationally very hard.

Abstract

In this paper we study decidability and complexity of decision problems on matrices from the special linear group $\mathrm{SL}(2,\mathbb{Z})$. In particular, we study the freeness problem: given a finite set of matrices $G$ generating a multiplicative semigroup $S$, decide whether each element of $S$ has at most one factorization over $G$. In other words, is $G$ a code? We show that the problem of deciding whether a matrix semigroup in $\mathrm{SL}(2,\mathbb{Z})$ is non-free is NP-hard. Then, we study questions about the number of factorizations of matrices in the matrix semigroup such as the finite freeness problem, the recurrent matrix problem, the unique factorizability problem, etc. Finally, we show that some factorization problems could be even harder in $\mathrm{SL}(2,\mathbb{Z})$, for example we show that to decide whether every prime matrix has at most $k$ factorizations is PSPACE-hard.