Vector Ambiguity and Freeness Problems in SL (2, ℤ)

We study the vector ambiguity problem and the vector freeness problem in SL\((2,\mathbb {Z})\). Given a finitely generated \(n \times n\) matrix semigroup S and an n-dimensional vector \(\mathbf {x}\), the vector ambiguity problem is to decide whether for every target vector \(\mathbf {y} = M\mathbf {x}\), where \(M \in S\), M is unique. We also consider the vector freeness problem which is to show that every matrix M which is transforming \(\mathbf {x}\) to \(M \mathbf {x}\) has a unique factorization with respect to the generator of S. We show that both problems are NP-complete in SL\((2,\mathbb {Z})\), which is the set of \(2 \times 2\) integer matrices with determinant 1. Moreover, we generalize the vector ambiguity problem and extend to the finite and k-vector ambiguity problems where we consider the degree of vector ambiguity of matrix semigroups.