On Optimal Timed Strategies

In this paper, we study timed games played on weighted timed automata. In this context, the reachability problem asks if, given a set T of locations and a cost C, Player 1 has a strategy to force the game into T with a cost less than C no matter how Player 2 behaves. Recently, this problem has been studied independently by Alur et al and by Bouyer et al. In those two works, a semi-algorithm is proposed to solve the reachability problem, which is proved to terminate under a condition imposing the non-zenoness of cost. In this paper, we show that in the general case the existence of a strategy for Player 1 to win the game with a bounded cost is undecidable. Our undecidability result holds for weighted timed game automata with five clocks. On the positive side, we show that if we restrict the number of clocks to one and we limit the form of the cost on locations, then the semi-algorithm proposed by Bouyer et al always terminates.

[1]  George J. Pappas,et al.  Optimal Paths in Weighted Timed Automata , 2001, HSCC.

[2]  Kim G. Larsen,et al.  Minimum-Cost Reachability for Priced Timed Automata , 2001, HSCC.

[3]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[4]  Supratik Mukhopadhyay,et al.  Optimal-Reachability and Control for Acyclic Weighted Timed Automata , 2002, IFIP TCS.

[5]  Cnrs Fre,et al.  Model Checking a Path (Preliminary Report) , 2003 .

[6]  Kousha Etessami,et al.  Verifying Probabilistic Procedural Programs , 2004, FSTTCS.

[7]  Eugene Asarin,et al.  As Soon as Possible: Time Optimal Control for Timed Automata , 1999, HSCC.

[8]  Rajeev Alur,et al.  Optimal Reachability for Weighted Timed Games , 2004, ICALP.

[9]  Rajeev Alur,et al.  Decision Problems for Timed Automata: A Survey , 2004, SFM.

[10]  Yassine Lakhnech,et al.  Formal Techniques, Modelling and Analysis of Timed and Fault-Tolerant Systems , 2004, Lecture Notes in Computer Science.

[11]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

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

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

[14]  Véronique Bruyère,et al.  Model-Checking for Weighted Timed Automata , 2004, FORMATS/FTRTFT.

[15]  R. Milner,et al.  Bigraphical Reactive Systems , 2001, CONCUR.

[16]  Joseph Sifakis,et al.  Tools and Applications II: The IF Toolset , 2004 .

[17]  Kim G. Larsen,et al.  Optimal Strategies in Priced Timed Game Automata , 2004, FSTTCS.

[18]  Thomas A. Henzinger,et al.  Symbolic Algorithms for Infinite-State Games , 2001, CONCUR.

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