Costs and Rewards in Priced Timed Automata

We consider Pareto analysis of reachable states of multi-priced timed automata (MPTA): timed automata equipped with multiple observers that keep track of costs (to be minimised) and rewards (to be maximised) along a computation. Each observer has a constant non-negative derivative which may depend on the location of the MPTA. We study the Pareto Domination Problem, which asks whether it is possible to reach a target location via a run in which the accumulated costs and rewards Pareto dominate a given objective vector. We show that this problem is undecidable in general, but decidable for MPTA with at most three observers. For MPTA whose observers are all costs or all rewards, we show that the Pareto Domination Problem is PSPACE-complete. We also consider an epsilon-approximate Pareto Domination Problem that is decidable without restricting the number and types of observers. We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert's 10th Problem, while decidability for three observers is shown by a translation to a fragment of arithmetic involving quadratic forms.

[1]  Kim G. Larsen,et al.  Optimal infinite scheduling for multi-priced timed automata , 2008, Formal Methods Syst. Des..

[2]  Kim G. Larsen,et al.  Infinite Runs in Weighted Timed Automata with Energy Constraints , 2008, FORMATS.

[3]  Véronique Bruyère,et al.  On model-checking timed automata with stopwatch observers , 2006, Inf. Comput..

[4]  F. Grunewald,et al.  On the integer solutions of quadratic equations , 2004 .

[5]  Anthony Widjaja Lin,et al.  Parikh Images of Grammars: Complexity and Applications , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

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

[7]  Pravin Varaiya,et al.  What's decidable about hybrid automata? , 1995, STOC '95.

[8]  Véronique Bruyère,et al.  On the optimal reachability problem of weighted timed automata , 2007, Formal Methods Syst. Des..

[9]  Kim G. Larsen,et al.  Pareto Optimal Reachability Analysis for Simple Priced Timed Automata , 2017, ICFEM.

[10]  Mihalis Yannakakis,et al.  Small Approximate Pareto Sets for Biobjective Shortest Paths and Other Problems , 2009, SIAM J. Comput..

[11]  James P. Jones.  Undecidable diophantine equations , 1980 .

[12]  Kim G. Larsen,et al.  As Cheap as Possible: Efficient Cost-Optimal Reachability for Priced Timed Automata , 2001, CAV.

[13]  Karin Quaas.  Kleene-Schützenberger and Büchi theorems for weighted timed automata , 2010 .

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

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

[16]  Kim G. Larsen,et al.  Model Checking One-clock Priced Timed Automata , 2007, Log. Methods Comput. Sci..

[17]  Vitaly Perevoshchikov.  Multi-weighted automata models and quantitative logics , 2014, Joint Workshop of the German Research Training Groups in Computer Science.

[18]  Martin Fränzle,et al.  Revisiting Decidability and Optimum Reachability for Multi-Priced Timed Automata , 2009, FORMATS.

[19]  Anthony Widjaja To,et al.  Parikh Images of Regular Languages: Complexity and Applications , 2010, 1002.1464.

[20]  Christoph Haase,et al.  Integer Vector Addition Systems with States , 2014, RP.

[21]  Kim G. Larsen,et al.  Optimal reachability for multi-priced timed automata , 2008, Theor. Comput. Sci..