The Reachability Problem for Affine Functions on the Integers

We consider the problem of determining, given x, y ∈ Z and a finite set F of affine functions on Z, whether y is reachable from x by applying the functions F . We also consider the analogous problem over N. These problems are known to be undecidable for k ≥ 2. We give 2-EXPTIME algorithms for both problems in the remaining case k = 1. The exact complexities remain open, although we show a simple NP lower bound.