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 Q. In the context of program veri cation, 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. m = 1 m = 2 m = 3 m = 4 m = k m ≥ k + 1 k = 0 P P P P P P k = 1 PSPACE PSPACE PSPACE PSPACE PSPACE PSPACE k = 2 PSPACE PSPACE PSPACE PSPACE PSPACE k = 3 PSPACE S5 PSPACE S5 k = 4 D D D & S5 k ≥ 5 D D & Sk+1 Figure 1: Decidability and hardness for instances of the Polyhedron-Hitting Problem in ambient dimensionm with a k-dimensional target. The row k = 0 corresponds to Kannan and Lipton's Orbit Problem [14, 13]. Hardness for certain Diophantine-approximation problems (detailed precisely in Sec. 2.2) is denoted by D, whereas hardness for Skolem's Problem of order d (de ned in Sec. 2.2) is indicated by Sd.