License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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DOI: 10.4230/LIPIcs.STACS.2019.6
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10245/
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Almagor, Shaull ; Ouaknine, Joël ; Worrell, James

The Semialgebraic Orbit Problem

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LIPIcs-STACS-2019-6.pdf (0.5 MB)


Abstract

The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d in N, a square matrix A in Q^{d x d}, and semialgebraic source and target sets S,T subseteq R^d. The question is whether there exists x in S and n in N such that A^nx in T. The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d <= 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory - Baker's theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

BibTeX - Entry

@InProceedings{almagor_et_al:LIPIcs:2019:10245,
  author =	{Shaull Almagor and Jo{\"e}l Ouaknine and James Worrell},
  title =	{{The Semialgebraic Orbit Problem}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{6:1--6:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Rolf Niedermeier and Christophe Paul},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10245},
  doi =		{10.4230/LIPIcs.STACS.2019.6},
  annote =	{Keywords: linear dynamical systems, Orbit Problem, first order theory of the reals}
}

Keywords: linear dynamical systems, Orbit Problem, first order theory of the reals
Collection: 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)
Issue Date: 2019
Date of publication: 12.03.2019


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