Adding Negative Prices to Priced Timed Games
Priced timed games (PTGs) are two-player zero-sum games played on the infinite graph of configurations of priced timed automata where two players take turns to choose transitions in order to optimize cost to reach target states. Bouyer et al. and Alur, Bernadsky, and Madhusudan independently proposed algorithms to solve PTGs with nonnegative prices under certain divergence restriction over prices. Brihaye, Bruyère, and Raskin later provided a justification for such a restriction by showing the undecidability of the optimal strategy synthesis problem in the absence of this divergence restriction. This problem for PTGs with one clock has long been conjectured to be in polynomial time, however the current best known algorithm, by Hansen, Ibsen-Jensen, and Miltersen, is exponential. We extend this picture by studying PTGs with both negative and positive prices. We refine the undecidability results for optimal strategy synthesis problem, and show undecidability for several variants of optimal reachability cost objectives including reachability cost, time-bounded reachability cost, and repeated reachability cost objectives. We also identify a subclass with bi-valued price-rates and give a pseudo-polynomial (polynomial when prices are nonnegative) algorithm to partially answer the conjecture on the complexity of one-clock PTGs.
KeywordsPolynomial Time Hybrid Automaton Game Graph Counter Machine Clock Constraint
Unable to display preview. Download preview PDF.
- 4.Behrmann, G., Fehnker, A., Hune, T., Larsen, K.G., Pettersson, P., Romijn, J.M.T., Vaandrager, F.W.: Minimum-cost reachability for priced timed automata. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 147–161. Springer, Heidelberg (2001)Google Scholar
- 9.Bouyer, P., Cassez, F., Fleury, E., Larsen, K.G.: Optimal strategies in priced timed game automata. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 148–160. Springer, Heidelberg (2004)Google Scholar
- 10.Bouyer, P., Larsen, K.G., Markey, N., Rasmussen, J.I.: Almost optimal strategies in one-clock priced timed games. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 345–356. Springer, Heidelberg (2006)Google Scholar
- 13.Brihaye, T., Geeraerts, G., Krishna, S.N., Manasa, L., Monmege, B., Trivedi, A.: Reachability-cost games with negative weights. Technical report (2014), http://arxiv.org/abs/1404.5894
- 14.Brihaye, T., Geeraerts, G., Monmege, B.: Reachability-cost games with negative weights. Technical report (2014), http://www.ulb.ac.be/di/verif/monmege/download/priced-games.pdf
- 17.Minsky, M.L.: Computation: Finite and infinite machines. Prentice-Hall, Inc. (1967)Google Scholar
- 20.Rutkowski, M.: Two-player reachability-price games on single-clock timed automata. In: QAPL 2011. EPTCS, vol. 57, pp. 31–46 (2011)Google Scholar