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.
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Asarin, E., Maler, O. (1995). Achilles and the tortoise climbing up the arithmetical hierarchy. In: Thiagarajan, P.S. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1995. Lecture Notes in Computer Science, vol 1026. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60692-0_68
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DOI: https://doi.org/10.1007/3-540-60692-0_68
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