On the Undecidability of Freeness of Matrix Semigroups

We slightly improve the result of Klarner, Birget and Satterfield, showing that the freeness of finitely presented multiplicative semigroups of 3×3 matrices over ℕ is undecidable even for triangular matrices. This is achieved by proving a new variant of Post correspondence problem. We also consider the freeness problem for 2×2 matrices. On the one hand, we show that it cannot be proved undecidable using the above methods which work in higher dimensions, and, on the other hand, we give some evidence that its decidability, if so, is also a challenging problem.