Achilles and the tortoise climbing up the arithmetical hierarchy

Abstract

In this paper we show how to construct for every set R of integers in the arithmetical hierarchy a dynamical system $\mathcal{H}$with piecewise-constant derivatives (PCD) such that deciding membership in R can be reduced to solving the reachability problem between two rational points for $\mathcal{H}$. 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.

This research was supported in part by the European Community projects BRA-REACT(6021), HYBRID EC-US-043 and INTAS-94-697 as well as by Research Grant #93-012-884 of Russian Foundation of Fundamental Research. Verimag is a joint laboratory of cnrs, inpg, ujf and verilog sa. spectre is a project of inria.