Reachability Problems in Nondeterministic Polynomial Maps on the Integers

We study the reachability problems in various nondeterministic polynomial maps in \(\mathbb {Z}^n\). We prove that the reachability problem for very simple three-dimensional affine maps (with independent variables) is undecidable and is \(\texttt {PSPACE}\)-hard for two-dimensional quadratic maps. Then we show that the complexity of the reachability problem for maps without functions of the form \(\pm x+b\) is lower. In this case the reachability problem is \(\texttt {PSPACE}\)-complete in general, and \(\mathtt{NP}\)-hard for any fixed dimension. Finally we extend the model by considering maps as language acceptors and prove that the universality problem is undecidable for two-dimensional affine maps.

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