Achilles and the tortoise climbing up the hyper-arithmetical hiearchy
Résumé
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 ${\omega ^{k}} ^{th}$ (respectively: ${\omega ^{k}+1} ^{th}$) level of the hyper-arithmetical hierarchy in dimension $d=2k+3$ (respectively: $d=2k+4$), $k \ge 0$. Hence we prove that the reachability problem for PCD systems of dimension $d=2k+3$ (resp. $d=2k+4$), $k \ge 1$, is hyper-arithmetical and is $\Sigma_{\omega ^{k}} $-complete (resp. $\Sigma_{\omega ^{k}+1}$-complete).