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Journal Articles Theoretical Computer Science Year : 1999

Achilles and the tortoise climbing up the hyper-arithmetical hiearchy

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).

Dates and versions

inria-00100818 , version 1 (26-09-2006)

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Cite

Olivier Bournez. Achilles and the tortoise climbing up the hyper-arithmetical hiearchy. Theoretical Computer Science, 1999, 210 (1), pp.21-71. ⟨10.1016/S0304-3975(98)00096-6⟩. ⟨inria-00100818⟩
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