The Reachability Problem for Affine Functions on the Integers

TL;DR

This paper investigates the reachability problem for affine functions on integers and natural numbers, providing 2-EXPTIME algorithms for the one-dimensional case and discussing the complexity for higher dimensions.

Contribution

It introduces 2-EXPTIME algorithms for the reachability problem in one dimension and explores the complexity landscape for higher dimensions.

Findings

Decidability for k=1 with 2-EXPTIME algorithms

Undecidability for k >= 2

NP lower bound for the problem complexity

Abstract

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