Abstract
Concavely-priced timed automata, a generalization of linearly-priced timed automata, are introduced. Computing the minimum value of a number of cost functions—including reachability price, discounted price, average time, average price, price-per-time average, and price-per-reward average—is considered in a uniform fashion for concavely-priced timed automata. All the corresponding decision problems are shown to be PSPACE-complete. This paper generalises the recent work of Bouyer et al. on deciding the minimum reachability price and the minimum ratio-price for linearly-priced timed automata.
A new type of a region graph—the boundary region graph—is defined, which generalizes the corner-point abstraction of Bouyer et al. A broad class of cost functions—concave-regular cost functions—is introduced, and the boundary region graph is shown to be a correct abstraction for deciding the minimum value of concave-regular cost functions for concavely-priced timed automata.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abdeddaïm, Y., Maler, O.: Job-shop scheduling using timed automata. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 478–492. Springer, Heidelberg (2001)
Alur, R., Courcoubetis, C., Dill, D.: Model-checking in dense real-time. Information and Computation 104(1), 2–34 (1993)
Alur, R., Courcoubetis, C., Henzinger, T.A.: Computing accumulated delays in real-time systems. Formal Methods in System Design 11(2), 137–155 (1997)
Alur, R., Dill, D.: A theory of timed automata. In: Theoretical Computer Science, vol. 126, pp. 183–235 (1994)
Alur, R., La Torre, S., Pappas, G.J.: Optimal paths in weighted timed automata. Theoretical Computer Science 318(3), 297–322 (2004)
Asarin, E., Maler, O.: As soon as possible: Time optimal control for timed automata. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 19–30. Springer, Heidelberg (1999)
Behrmann, G., Fehnker, A., Hune, T., Larsen, K.G., Pettersson, P., Romijn, J., 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)
Bertsekas, D.P., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific (2003)
Bouyer, P.: Weighted timed automata: Model-checking and games. Electr. Notes Theor. Comput. Sci. 158, 3–17 (2006)
Bouyer, P., Brihaye, T., Bruyère, V., Raskin, J.: On the optimal reachability problem on weighted timed automata. Formal Methods in System Design 31(2), 135–175 (2007)
Bouyer, P., Brinksma, E., Larsen, K.G.: Optimal infinite scheduling for multi-priced timed automata. Formal Methods in System Design 32(1), 3–23 (2008)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Brihaye, T., Henzinger, T.A., Prabhu, V.S., Raskin, J.: Minimum-time reachability in timed games. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 825–837. Springer, Heidelberg (2007)
Courcoubetis, C., Yannakakis, M.: Minimum and maximum delay problems in real-time systems. In: FMSD 1992, vol. 1, pp. 385–415. Kluwer, Dordrecht (1992)
Jurdziński, M., Trivedi, A.: Reachability-time games on timed automata. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 838–849. Springer, Heidelberg (2007)
Kesten, Y., Pnueli, A., Sifakis, J., Yovine, S.: Decidable integration graphs. Information and Computation 150(2), 209–243 (1999)
Larsen, K.G., Behrmann, G., Brinksma, E., Fehnker, A., Hune, T., Pettersson, P., Romijn, J.: As cheap as possible: Efficient cost-optimal reachability for priced timed automata. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 493–505. Springer, Heidelberg (2001)
Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill Series in Systems Science. McGraw-Hill, New York (1969)
Niebert, P., Tripakis, S., Yovine, S.: Minimum-time reachability for timed automata. In: MED 2000. IEEE Comp. Soc. Press, Los Alamitos (2000)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jurdziński, M., Trivedi, A. (2008). Concavely-Priced Timed Automata. In: Cassez, F., Jard, C. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2008. Lecture Notes in Computer Science, vol 5215. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85778-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-85778-5_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85777-8
Online ISBN: 978-3-540-85778-5
eBook Packages: Computer ScienceComputer Science (R0)