# Achilles and the tortoise climbing up the hyper-arithmetical hiearchy

1 PROTHEO - Constraints, automatic deduction and software properties proofs
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : 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).
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Type de document :
Article dans une revue
Theoretical Computer Science, Elsevier, 1999, 210 (1), pp.21-71
Domaine :

https://hal.inria.fr/inria-00100818
Contributeur : Publications Loria <>
Soumis le : mardi 26 septembre 2006 - 14:51:33
Dernière modification le : jeudi 17 mai 2018 - 12:52:03

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• HAL Id : inria-00100818, version 1

### Citation

Olivier Bournez. Achilles and the tortoise climbing up the hyper-arithmetical hiearchy. Theoretical Computer Science, Elsevier, 1999, 210 (1), pp.21-71. 〈inria-00100818〉

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