DROPS

Document

DOI: 10.4230/LIPIcs.STACS.2021.36

Reachability in Two-Parametric Timed Automata with One Parameter Is EXPSPACE-Complete

Authors: Stefan Göller and Mathieu Hilaire

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract

Parametric timed automata (PTA) have been introduced by Alur, Henzinger, and Vardi as an extension of timed automata in which clocks can be compared against parameters. The reachability problem asks for the existence of an assignment of the parameters to the non-negative integers such that reachability holds in the underlying timed automaton. The reachability problem for PTA is long known to be undecidable, already over three parametric clocks. A few years ago, Bundala and Ouaknine proved that for PTA over two parametric clocks and one parameter the reachability problem is decidable and also showed a lower bound for the complexity class PSPACE^NEXP. Our main result is that the reachability problem for parametric timed automata over two parametric clocks and one parameter is EXPSPACE-complete. For the EXPSPACE lower bound we make use of deep results from complexity theory, namely a serializability characterization of EXPSPACE (in turn based on Barrington’s Theorem) and a logspace translation of numbers in Chinese Remainder Representation to binary representation due to Chiu, Davida, and Litow. It is shown that with small PTA over two parametric clocks and one parameter one can simulate serializability computations. For the EXPSPACE upper bound, we first give a careful exponential time reduction from PTA over two parametric clocks and one parameter to a (slight subclass of) parametric one-counter automata over one parameter based on a minor adjustment of a construction due to Bundala and Ouaknine. For solving the reachability problem for parametric one-counter automata with one parameter, we provide a series of techniques to partition a fictitious run into several carefully chosen subruns that allow us to prove that it is sufficient to consider a parameter value of exponential magnitude only. This allows us to show a doubly-exponential upper bound on the value of the only parameter of a PTA over two parametric clocks and one parameter. We hope that extensions of our techniques lead to finally establishing decidability of the long-standing open problem of reachability in parametric timed automata with two parametric clocks (and arbitrarily many parameters) and, if decidability holds, determining its precise computational complexity.

Cite as

Stefan Göller and Mathieu Hilaire. Reachability in Two-Parametric Timed Automata with One Parameter Is EXPSPACE-Complete. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{goller_et_al:LIPIcs.STACS.2021.36,
  author =	{G\"{o}ller, Stefan and Hilaire, Mathieu},
  title =	{{Reachability in Two-Parametric Timed Automata with One Parameter Is EXPSPACE-Complete}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{36:1--36:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.36},
  URN =		{urn:nbn:de:0030-drops-136817},
  doi =		{10.4230/LIPIcs.STACS.2021.36},
  annote =	{Keywords: Parametric Timed Automata, Computational Complexity, Reachability}
}

Document

DOI: 10.4230/LIPIcs.FUN.2016.10

Spy-Game on Graphs

Authors: Nathann Cohen, Mathieu Hilaire, Nícolas A. Martins, Nicolas Nisse, and Stéphane Pérennes

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Abstract

We define and study the following two-player game on a graph G. Let k in N^*. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s in N^* is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d in N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy? This game generalizes two well-studied games: Cops and robber games (when s=1) and Eternal Dominating Set (when s is unbounded). We consider the computational complexity of the problem, showing that it is NP-hard and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards and the required distance d when G is a path or a cycle. Our main result is that there exists beta>0 such that Omega(n^{1+beta}) guards are required to win in any n*n grid.

Cite as

Nathann Cohen, Mathieu Hilaire, Nícolas A. Martins, Nicolas Nisse, and Stéphane Pérennes. Spy-Game on Graphs. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

Copy BibTex To Clipboard

@InProceedings{cohen_et_al:LIPIcs.FUN.2016.10,
  author =	{Cohen, Nathann and Hilaire, Mathieu and Martins, N{\'\i}colas A. and Nisse, Nicolas and P\'{e}rennes, St\'{e}phane},
  title =	{{Spy-Game on Graphs}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.10},
  URN =		{urn:nbn:de:0030-drops-58782},
  doi =		{10.4230/LIPIcs.FUN.2016.10},
  annote =	{Keywords: graph, two-player games, cops and robber games, complexity}
}

Search Results

Documents authored by Hilaire, Mathieu

Reachability in Two-Parametric Timed Automata with One Parameter Is EXPSPACE-Complete

Abstract

Cite as

Spy-Game on Graphs

Abstract

Cite as

Thanks for your feedback!

Could not send message