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DOI: 10.4230/LIPIcs.ICALP.2016.100
URN: urn:nbn:de:0030-drops-62357
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6235/
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Chonev, Ventsislav ; Ouaknine, JoŽl ; Worrell, James

On the Skolem Problem for Continuous Linear Dynamical Systems

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Abstract

The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems, such as linear hybrid automata and continuoustime Markov chains. Decidability of the problem is currently open ó indeed decidability is open even for the sub-problem in which a zero is sought in a bounded interval. In this paper we show decidability of the bounded problem subject to Schanuel's Conjecture, a unifying conjecture in transcendental number theory. We furthermore analyse the unbounded problem in terms of the frequencies of the differential equation, that is, the imaginary parts of the characteristic roots. We show that the unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. We complete the picture by showing that decidability of the unbounded problem in the case of two (or more) rationally linearly independent frequencies would entail a major new effectiveness result in Diophantine approximation, namely computability of the Diophantine-approximation types of all real algebraic numbers.

BibTeX - Entry

@InProceedings{chonev_et_al:LIPIcs:2016:6235,
  author =	{Ventsislav Chonev and Jo{\"e}l Ouaknine and James Worrell},
  title =	{{On the Skolem Problem for Continuous Linear Dynamical Systems}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{100:1--100:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6235},
  URN =		{urn:nbn:de:0030-drops-62357},
  doi =		{10.4230/LIPIcs.ICALP.2016.100},
  annote =	{Keywords: differential equations, reachability, Bakerís Theorem, Schanuelís Conjecture, semi-algebraic sets}
}

Keywords: differential equations, reachability, Bakerís Theorem, Schanuelís Conjecture, semi-algebraic sets
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 17.08.2016


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