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DOI: 10.4230/LIPIcs.CONCUR.2017.21

Infinite-Duration Bidding Games

Authors: Guy Avni, Thomas A. Henzinger, and Ventsislav Chonev

Published in: LIPIcs, Volume 85, 28th International Conference on Concurrency Theory (CONCUR 2017)

Abstract

Two-player games on graphs are widely studied in formal methods as they model the interaction between a system and its environment. The game is played by moving a token throughout a graph to produce an infinite path. There are several common modes to determine how the players move the token through the graph; e.g., in turn-based games the players alternate turns in moving the token. We study the bidding mode of moving the token, which, to the best of our knowledge, has never been studied in infinite-duration games. Both players have separate budgets, which sum up to $1$. In each turn, a bidding takes place. Both players submit bids simultaneously, and a bid is legal if it does not exceed the available budget. The winner of the bidding pays his bid to the other player and moves the token. For reachability objectives, repeated bidding games have been studied and are called Richman games [Lazarus1999,Lazarus2012]. There, a central question is the existence and computation of threshold budgets; namely, a value t \in [0,1] such that if \PO's budget exceeds t, he can win the game, and if \PT's budget exceeds 1-t, he can win the game. We focus on parity games and mean-payoff games. We show the existence of threshold budgets in these games, and reduce the problem of finding them to Richman games. We also determine the strategy-complexity of an optimal strategy. Our most interesting result shows that memoryless strategies suffice for mean-payoff bidding games.

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Guy Avni, Thomas A. Henzinger, and Ventsislav Chonev. Infinite-Duration Bidding Games. In 28th International Conference on Concurrency Theory (CONCUR 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 85, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{avni_et_al:LIPIcs.CONCUR.2017.21,
  author =	{Avni, Guy and Henzinger, Thomas A. and Chonev, Ventsislav},
  title =	{{Infinite-Duration Bidding Games}},
  booktitle =	{28th International Conference on Concurrency Theory (CONCUR 2017)},
  pages =	{21:1--21:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-048-4},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{85},
  editor =	{Meyer, Roland and Nestmann, Uwe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2017.21},
  URN =		{urn:nbn:de:0030-drops-77741},
  doi =		{10.4230/LIPIcs.CONCUR.2017.21},
  annote =	{Keywords: Bidding Games, Parity Games, Mean-Payoff Games, Richman Games}
}

Document

DOI: 10.4230/LIPIcs.ICALP.2016.100

On the Skolem Problem for Continuous Linear Dynamical Systems

Authors: Ventsislav Chonev, Joël Ouaknine, and James Worrell

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

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 — 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.

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Ventsislav Chonev, Joël Ouaknine, and James Worrell. On the Skolem Problem for Continuous Linear Dynamical Systems. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 100:1-100:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chonev_et_al:LIPIcs.ICALP.2016.100,
  author =	{Chonev, Ventsislav and Ouaknine, Jo\"{e}l and Worrell, James},
  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 =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.100},
  URN =		{urn:nbn:de:0030-drops-62357},
  doi =		{10.4230/LIPIcs.ICALP.2016.100},
  annote =	{Keywords: differential equations, reachability, Baker’s Theorem, Schanuel’s Conjecture, semi-algebraic sets}
}

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Infinite-Duration Bidding Games

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On the Skolem Problem for Continuous Linear Dynamical Systems

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