Achilles and the Tortoise Climbing up the Hyper-Arithmetical Hierarchy

In this paper, we characterize the computational power of dynamical systems with piecewise constant derivatives (PCD) considered as computational machines working on a continuous real space with a continuous real time: we prove that piecewise constant derivative systems recognize precisely the languages of the ω th (respectively: ω + 1 th ) level of the hyper-arithmetical hierarchy in dimension d = 2k + 3 (respectively: d = 2k + 4), k ≥ 0. Hence we prove that the reachability problem for PCD systems of dimension d = 2k + 3 (resp. d = 2k + 4), k ≥ 1, is hyper-arithmetical and is Σωk -complete (resp. Σωk+1-complete).