The Polyhedron-Hitting Problem

We consider polyhedral versions of Kannan and Lipton's Orbit Problem [14, 13]---determining whether a target polyhedron V may be reached from a starting point x under repeated applications of a linear transformation A in an ambient vector space Qm. In the context of program verification, very similar reachability questions were also considered and left open by Lee and Yannakakis in [15], and by Braverman in [4]. We present what amounts to a complete characterisation of the decidability landscape for the Polyhedron-Hitting Problem, expressed as a function of the dimension m of the ambient space, together with the dimension of the polyhedral target V: more precisely, for each pair of dimensions, we either establish decidability, or show hardness for longstanding number-theoretic open problems.

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