The Reachability Problem for Affine Functions on the Integers

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We consider the problem of determining, given x, y in Zᵏ and a finite set F of affine functions on Zᵏ, whether y is reachable from x by applying the functions F. We also consider the analogous problem over Nᵏ. 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.