Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy

Abstract

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 k th (respectively ( ω k + 1)th) level of the hyper-arithmetical hierarchy in dimension d = 2 k + 3 (respectively d = 2 k + 4), k ⩾ 0. Hence we prove that the reachability problem for PCD systems of dimension d = 2 k + 3 (resp. d = 2 k + 4), k ⩾ 1, is hyper-arithmetical and is ∑ ω k -complete (resp. ∑ ω k -complete).