Achilles and the Tortoise Climbing up the Hyper-arithmetical Hierarchy Ecole Normale Supérieure De Lyon Achilles and the Tortoise Climbing up the Hyper-arithmetical Hierarchy

We pursue the study of the computational power of Piecewise Constant Derivative PCD systems started in PCD systems are dynamical systems de ned by a piecewise constant di erential equation and can be considered as computational machines working on a continuous space with a continuous time We prove that the languages recognized by rational PCD systems in dimension d k respectively d k k in nite continuous time are precisely the languages of the k th resp k th level of the hyper arithmetical hierarchy Hence the reachability problem for rational PCD systems of dimension d k resp d k k is hyper arithmetical and is k complete resp k complete