  We consider the Membership and the Half-Space Reachability problems for
matrices in dimensions two and three. Our first main result is that the
Membership Problem is decidable for finitely generated sub-semigroups of the
Heisenberg group over rational numbers. Furthermore, we prove two decidability
results for the Half-Space Reachability Problem. Namely, we show that this
problem is decidable for sub-semigroups of $\mathrm{GL}(2,\mathbb{Z})$ and of
the Heisenberg group over rational numbers.
