  We analyze affine reachability problems in dimensions 1 and 2. We show that
the reachability problem for 1-register machines over the integers with affine
updates is PSPACE-hard, hence PSPACE-complete, strengthening a result by Finkel
et al. that required polynomial updates. Building on recent results on
two-dimensional integer matrices, we prove NP-completeness of the mortality
problem for 2-dimensional integer matrices with determinants +1 and 0.
Motivated by tight connections with 1-dimensional affine reachability problems
without control states, we also study the complexity of a number of
reachability problems in finitely generated semigroups of 2-dimensional
upper-triangular integer matrices.
