E.Asarin, O. Maler, **Achilles and the Tortoise Climbing Up
the Arithmetical Hierarchy.** In this paper we show how to
construct for every set *R* of integers in the arithmetical
hierarchy a dynamical system *H* with piecewise-constant
derivatives (PCD) such that deciding membership in *R* can
be reduced to solving the reachability problem between two
rational points for *H*. The ability of such simple
dynamical systems to "simulate" highly undecidable
problems is closely related to *Zeno'*s paradox dealing with
the ability to pack infinitely many discrete steps in a bounded
interval of time. [Pdf]