Time Flies When Looking out of the Window: Timed Games with Window Parity Objectives

The window mechanism was introduced by Chatterjee et al. to reinforce mean-payoff and total-payoff objectives with time bounds in two-player turn-based games on graphs [17]. It has since proved useful in a variety of settings, including parity objectives in games [14] and both mean-payoff and parity objectives in Markov decision processes [12]. We study window parity objectives in timed automata and timed games: given a bound on the window size, a path satisfies such an objective if, in all states along the path, we see a sufficiently small window in which the smallest priority is even. We show that checking that all time-divergent paths of a timed automaton satisfy such a window parity objective can be done in polynomial space, and that the corresponding timed games can be solved in exponential time. This matches the complexity class of timed parity games, while adding the ability to reason about time bounds. We also consider multi-dimensional objectives and show that the complexity class does not increase. To the best of our knowledge, this is the first study of the window mechanism in a real-time setting. 2012 ACM Subject Classification Theory of computation → Formal languages and automata theory

[1]  Joseph Sifakis,et al.  On the Synthesis of Discrete Controllers for Timed Systems (An Extended Abstract) , 1995, STACS.

[2]  Véronique Bruyère,et al.  On the Complexity of Heterogeneous Multidimensional Games , 2016, CONCUR.

[3]  Petr Novotný,et al.  Stability in Graphs and Games , 2016, CONCUR.

[4]  Krishnendu Chatterjee,et al.  Looking at mean-payoff and total-payoff through windows , 2015, Inf. Comput..

[5]  Mickael Randour,et al.  Automated synthesis of reliable and efficient systems through game theory: a case study , 2012, ArXiv.

[6]  Mickael Randour,et al.  Life is Random, Time is Not: Markov Decision Processes with Window Objectives , 2019, CONCUR.

[7]  Patricia Bouyer,et al.  Decisiveness of Stochastic Systems and its Application to Hybrid Models , 2020, GandALF.

[8]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[9]  Rajeev Alur,et al.  A Theory of Timed Automata , 1994, Theor. Comput. Sci..

[10]  Thomas A. Henzinger,et al.  Discrete-Time Control for Rectangular Hybrid Automata , 1997, ICALP.

[11]  Cristian S. Calude,et al.  Deciding parity games in quasipolynomial time , 2017, STOC.

[12]  Arno Pauly,et al.  Extending finite-memory determinacy by Boolean combination of winning conditions , 2018, FSTTCS.

[13]  Krishnendu Chatterjee,et al.  Timed Parity Games: Complexity and Robustness , 2008, FORMATS.

[14]  Krishnendu Chatterjee,et al.  Finitary Winning in omega-Regular Games , 2006, TACAS.

[15]  Véronique Bruyère,et al.  Meet Your Expectations With Guarantees: Beyond Worst-Case Synthesis in Quantitative Games , 2013, STACS.

[16]  Guillermo A. Pérez,et al.  Looking at mean payoff through foggy windows , 2017, Acta Informatica.

[17]  Thomas A. Henzinger,et al.  Discrete-Time Control for Rectangular Hybrid Automata , 1997, Theor. Comput. Sci..

[18]  Christel Baier.  Reasoning About Cost-Utility Constraints in Probabilistic Models , 2015, RP.

[19]  Mickael Randour,et al.  Threshold Constraints with Guarantees for Parity Objectives in Markov Decision Processes , 2017, ICALP.

[20]  Rajeev Alur,et al.  Model-Checking in Dense Real-time , 1993, Inf. Comput..

[21]  Krishnendu Chatterjee,et al.  Trading Performance for Stability in Markov Decision Processes , 2013, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science.

[22]  Thomas A. Henzinger,et al.  The Element of Surprise in Timed Games , 2003, CONCUR.

[23]  Christel Baier,et al.  Principles of model checking , 2008 .

[24]  Véronique Bruyère,et al.  Window Parity Games: An Alternative Approach Toward Parity Games with Time Bounds (Full Version) , 2016, GandALF.

[25]  Christel Baier,et al.  Weight monitoring with linear temporal logic: complexity and decidability , 2014, CSL-LICS.