Formal Methods in System Design

, Volume 31, Issue 2, pp 135–175 | Cite as

On the optimal reachability problem of weighted timed automata

  • Patricia Bouyer
  • Thomas Brihaye
  • Véronique Bruyère
  • Jean-François RaskinEmail author


We study the cost-optimal reachability problem for weighted timed automata such that positive and negative costs are allowed on edges and locations. By optimality, we mean an infimum cost as well as a supremum cost. We show that this problem is PSpace-Complete. Our proof uses techniques of linear programming, and thus exploits an important property of optimal runs: their time-transitions use a time τ which is arbitrarily close to an integer. We then propose an extension of the region graph, the weighted discrete graph, whose structure gives light on the way to solve the  cost-optimal reachability problem. We also give an application of the  cost-optimal reachability problem in the context of timed games.


Weighted timed automaton Cost-optimal reachability problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur R, Dill DL (1994) A theory of timed automata. Theor Comput Sci 126(2):183–235 zbMATHCrossRefGoogle Scholar
  2. 2.
    Alur R, Courcoubetis C, Dill DL (1993) Model-checking in dense real-time. Inf Comput 104(1):2–34 zbMATHCrossRefGoogle Scholar
  3. 3.
    Alur R, Courcoubetis C, Henzinger TA (1993) Computing accumulated delays in real-time systems. In: CAV’93: computer aided verification. Lecture notes in computer science, vol 697. Springer, Berlin, pp 181–193 Google Scholar
  4. 4.
    Alur R, La Torre S, Pappas GJ (2001) Optimal paths in weighted timed automata. In: HSCC’01: hybrid systems: computation and control. Lecture notes in computer science, vol 2034. Springer, Berlin, pp 49–62 Google Scholar
  5. 5.
    Alur R, Bernadsky M, Madhusudan P (2004) Optimal reachability for weighted timed games. In: ICALP’04: automata, languages, and programming. Lecture notes in computer science, vol 3142. Springer, Berlin, pp 122–133 Google Scholar
  6. 6.
    Asarin E, Maler O (1999) As soon as possible: time optimal control for timed automata. In: HSCC’99: hybrid systems: computation and control. Lecture notes in computer science, vol 1569. Springer, Berlin, pp 19–30 CrossRefGoogle Scholar
  7. 7.
    Behrmann G, Fehnker A, Hune T, Larsen KG, Pettersson P, Romijn J, Vaandrager FW (2001) Minimum-cost reachability for priced timed automata. In: HSCC’01: hybrid systems: computation and control. Lecture notes in computer science, vol 2034. Springer, Berlin, pp 147–161 Google Scholar
  8. 8.
    Bérard B, Diekert V, Gastin P, Petit A (1998) Characterization of the expressive power of silent transitions in timed automata. Fundam Inf 36(2–3):145–182 zbMATHGoogle Scholar
  9. 9.
    Bouyer P, Brinksma E, Larsen KG (2004) Staying alive as cheaply as possible. In: HSSC’04: hybrid systems: computation and control. Lecture notes in computer science, vol 2993. Springer, Berlin, pp 203–218 Google Scholar
  10. 10.
    Bouyer P, Cassez F, Fleury E, Larsen KG (2004) Optimal strategies in priced timed game automata. In: FST&TCS’04: foundations of software technology and theoretical computer science. Lecture notes in computer science, vol 3328. Springer, Berlin, pp 148–160 Google Scholar
  11. 11.
    Brihaye T, Bruyère V, Raskin J-F (2005) On optimal timed strategies. In: FORMATS’05: formal modelling and analysis of timed systems. Lecture notes in computer science, vol 3829. Springer, Berlin, pp 49–64 CrossRefGoogle Scholar
  12. 12.
    Henzinger TA (1996) The theory of hybrid automata. In: LICS’96: logic in computer science. IEEE Computer Society Press, pp 278–292 Google Scholar
  13. 13.
    Henzinger TA, Ho P-H, Wong-Toi H (1995) A user guide to HyTech. In: TACAS’95: tools and algorithms for the construction and analysis of systems. Lecture notes in computer science, vol 1019. Springer, Berlin, pp 41–71 Google Scholar
  14. 14.
    Henzinger TA, Kopke PW, Puri A, Varaiya P (1995) What’s decidable about hybrid automata? In: Proceedings of the 27th annual symposium on theory of computing. ACM Press, pp 373–382 Google Scholar
  15. 15.
    Kesten Y, Pnueli A, Sifakis J, Yovine S (1999) Decidable integration graphs. Inf Comput 150(2):209–243 zbMATHCrossRefGoogle Scholar
  16. 16.
    La Torre S, Mukhopadhyay S, Murano A (2002) Optimal-reachability and control for acyclic weighted timed automata. In: IFIP TCS’02: foundations of information technology in the era of networking and mobile computing. IFIP conference proceedings, vol 223. Kluwer, Dordrecht, pp 485–497 Google Scholar
  17. 17.
    Larsen KG, Pettersson P, Yi W (1997) Uppaal in a nutshell. Int J Softw Tools Technol Transf 1(1–2):134–152 zbMATHCrossRefGoogle Scholar
  18. 18.
    Larsen KG, Rasmussen JI (2005) Optimal conditional reachability for multi-priced timed automata. In: FoSSaCS’05: foundations of software science and computational structures. Lecture notes in computer science, vol 3441. Springer, Berlin, pp 234–249 Google Scholar
  19. 19.
    Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley-interscience series in discrete mathematics and optimization. Wiley, New York zbMATHGoogle Scholar
  20. 20.
    Raskin J-F (2005) An introduction to hybrid automata. In: Handbook of networked and embedded control systems. Birkhäuser, Basel, pp 491–518 Google Scholar
  21. 21.
    Rockafellar RT (1970) Convex analysis. Princeton Univ. Press, New Jersey zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Patricia Bouyer
    • 1
  • Thomas Brihaye
    • 2
  • Véronique Bruyère
    • 2
  • Jean-François Raskin
    • 3
    Email author
  1. 1.LSV–CNRS & ENS de CachanCachanFrance
  2. 2.Faculté des SciencesUniversité de Mons-HainautMonsBelgium
  3. 3.Département d’InformatiqueUniversité Libre de BruxellesBruxellesBelgium

Personalised recommendations