The Polyhedron-Hitting Problem
  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. 
Ventsislav Chonev, Jo\"el Ouaknine, James Worrell
cs.CC