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On some relations between dynamical systems and transition systems

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Automata, Languages and Programming (ICALP 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 820))

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Abstract

In this paper we define a precise notion of abstraction relation between continuous dynamical systems and discrete state-transition systems. Our main result states that every Turing Machine can be realized by a dynamical system with piecewise-constant derivatives in a 3-dimensional space and thus the reachability problem for such systems is undecidable for 3 dimensions. A decision procedure for 2-dimensional systems has been recently reported by Maler and Pnueli. On the other hand we show that some non-deterministic finite automata cannot be realized by any continuous dynamical system with less than 3 dimensions.

This research was supported in part by the France-Israel project for cooperation in Computer Science, by the European Community ESPRIT Basic Research Action Projects REACT (6021) and by Research Grant #93-012-884 of the Russian Foundation of Fundamental Research. Verimag is a joint laboratory of cnrs, inpg, ujf and Verilog sa. Spectre is a project of inria. The final version was prepared while the first author was visiting the Dept. of Mathematics of the University Paris 12 — Val de Marne.

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Authors and Affiliations

  1. Institute for Information Transmission Problems, 19 Ermolovoy st., Moscow, Russia

    Eugene Asarin

  2. Spectre - Verimag Miniparc-zirst, 38330, Montbonnot, France

    Oded Maler

Authors
  1. Eugene Asarin
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  2. Oded Maler
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Serge Abiteboul Eli Shamir

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© 1994 Springer-Verlag Berlin Heidelberg

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Asarin, E., Maler, O. (1994). On some relations between dynamical systems and transition systems. In: Abiteboul, S., Shamir, E. (eds) Automata, Languages and Programming. ICALP 1994. Lecture Notes in Computer Science, vol 820. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58201-0_58

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  • DOI: https://doi.org/10.1007/3-540-58201-0_58

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  • Print ISBN: 978-3-540-58201-4

  • Online ISBN: 978-3-540-48566-7

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