# On Affine Reachability Problems

## TL;DR

This paper investigates the computational complexity of affine reachability problems in low dimensions, establishing PSPACE-completeness and NP-completeness results for specific cases, and explores related matrix semigroup problems.

## Contribution

It proves PSPACE-completeness for 1-register affine reachability and NP-completeness for the mortality problem in 2D matrices with specific determinants, advancing understanding of these problems.

## Key findings

- Reachability for 1-register machines is PSPACE-complete.
- Mortality problem for 2D matrices with determinants +1 and 0 is NP-complete.
- Complexity results for reachability in semigroups of 2D upper-triangular matrices.

## Abstract

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.