The Polyhedron-Hitting Problem

Ventsislav Chonev; Joël Ouaknine; James Worrell

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arXiv · 1407.1889

The Polyhedron-Hitting Problem

Ventsislav Chonev·Joël Ouaknine·James Worrell

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Abstract

We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC '80 and JACM '86)---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 Q^m. In the context of program verification, very similar reachability questions were also considered and left open by Lee and Yannakakis in (STOC '92). 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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Ventsislav Chonev, Joël Ouaknine, James Worrell. 2014-07-07. The Polyhedron-Hitting Problem. https://arxiv.org/abs/1407.1889

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